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The influence of binaries on galactic chemical evolution
New Astronomy Reviews 48 (2004) 861–975 www.elsevier.com/locate/newastrev
The influence of binaries on galactic chemical evolution Erwin De Donder *, Dany Vanbeveren Astrophysical Institute, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium Available online 25 August 2004
Abstract Understanding the galaxy in which we live is one of the great intellectual challenges facing modern science. With the advent of high quality observational data, the chemical evolution modeling of our galaxy has been the subject of numerous studies in the last years. However, all these studies have one missing element which is the evolution of close binaries. Reason: their evolution is very complex and single stars only perhaps can do the job. (Un)Fortunately at present we know that the majority of the observed stars are members of a binary or multiple system and that certain objects can only be formed through binary evolution. Therefore galactic studies that do not account for close binary evolution may be far from realistic. Because of the large expertise developed through the years in stellar evolution in general and binary evolution in particular at the Brussels Astrophysical Institute, we found ourselves in a privileged position to be the first to do chemical evolutionary simulations with the inclusion of detailed binary evolution. The complexity of close binary evolution has kept many astronomers from including binary stars into their studies. However, it is not always the easiest way of living that gives you the most excitement and satisfaction. 2004 Elsevier B.V. All rights reserved.
Abbreviations: AGB, asymptotic giant branch; BH, black hole; BHG, blue Hertzsprung gap; BSG, blue supergiant; CE, common envelope; CEM, chemical evolutionary model; CHB, core hydrogen burning; CHeB, core helium burning; CO WD, carbon–oxygen white dwarf; E-AGB, early asymptotic giant branch; GWR, gravitational wave radiation; HBB, hot bottom burning; HeSB, helium shell burning; HN, hypernova; HRD, Hertzsprung–Russell diagram; HSB, hydrogen shell burning; IMA, instantaneous mixing approximation; IMCB, intermediate mass close binary; ISM, interstellar medium; LBV, luminous blue variable; LMC, Large Magellanic Cloud; MCB, massive close binary; MS, main sequence; NLTE, non-local thermodynamic equilibrium; NS, neutron star; PN, planetary nebula; PNS, population number synthesis; RLOF, Roche lobe overflow; RSG, red supergiant; SMC, Small Magellanic _ _ Cloud; SN, supernova; SW, stellar wind; TP-AGB, thermal pulsating asymptotic giant branch; TZO, Thorne–Zytkov object; WD, white dwarf; WR, Wolf–Rayet star; YSG, yellow supergiant; ZAMS, zero age main sequence. * Corresponding author. E-mail address: [email protected] (E. De Donder). 1387-6473/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.newar.2004.07.001
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Contents 1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Galactic chemical evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Binary effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Goal and outline of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The population synthesis model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. General concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Single star evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Low and intermediate mass stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Massive stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Summary of the single star evolutionary model . . . . . . . . . . . . . . . . . . . . . . . 2.3. Binary star evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Mass transfer and accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Evolution of the binary orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. Detailed binary evolutionary model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. PNS predictions of massive star populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The chemical evolutionary model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The two-infall scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The star formation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The initial mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. The binary fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. The isotopic evolution model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Computation of the supernova rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. The stellar yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1. Single star yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2. Binary star yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3. Yields from one stellar generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.
The chemical evolution of the solar neighbourhood. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Star formation rate and surface densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Intermezzo: the WR/O number ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The age–metallicity relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. The G-dwarf disk metallicity distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. The solar abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. The abundance ratios evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1. The CNO elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2. The a-elements Mg, Si, S and Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3. Comparison to other studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.4. Overall conclusion on the time evolution of the abundance ratios . . . . . . . . . . 4.6.5. The influence of rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. The effect of BH formation above 40 Mx and metallicity dependent SW mass loss . . . 4.7.1. Comparison to other studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8. The supernova rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1. Observed rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2. Overview of the progenitor populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3. The evolutionary clock of supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4. The galactic evolution of the SN rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.5. Comparison to other studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.9.
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The evolution of the r-process elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1. The formation of NS + NS and NS + BH binaries . . . . . . . . . . . . . . . . . . . . 4.9.2. The theoretically predicted population of double compact star binaries . . . . . 4.9.3. The predicted evolution of [Eu/Fe] vs. [Fe/H] . . . . . . . . . . . . . . . . . . . . . . . 4.9.4. Comparison to other studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Summary and conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965 Appendix A.
Numerical solution of the isotope evolution equation . . . . . . . . . . . . . . . . . . . . . . . . . . 968
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970
1. Introduction 1.1. Galactic chemical evolution To understand the behaviour of the Universe we need to understand the formation and evolution of galaxies, which are the main constituents of the Universe. Galaxies are made of stars, interstellar gas and dark matter. Stars evolve and therefore galaxies evolve. A galaxy evolves simultaneously dynamically, thermally, (spectro) photometrically and chemically, which makes its evolution a very complex problem. In this work, we focus on the chemical evolution of our galaxy, the Milky Way, which entails with reconstructing the history of the chemical composition of the interstellar gas. To study the chemical evolution of a galaxy, the cosmological picture is adopted in which it is assumed that all deuterium, a major part of helium and some lithium has been produced during the Big Bang whereas all the heavier elements 1 are made in stars. A galaxy initially forms from a huge lump of gas that as a result of self-gravity decouples from the general expansion of the Universe. The enrichment and spatial distribution of the chemical species within the galaxy occur via various galactic and stellar processes that take place in the course of time. These processes were first
1 As is common in astrophysics, all elements heavier than helium are referred to as metals.
identified by Tinsley (1980). Following the outline, a first generation of metal free stars (= population III stars) forms from the collapsing protogalaxy. They process the matter nuclearly and return it partially or completely to the interstellar medium (ISM) via stellar winds (SWs) and/or supernova (SN) explosions. The ISM becomes enriched in new elements heavier than were initially present. Meanwhile the galaxy may exchange gas with the intergalactic medium or with nearby galaxies whereby its mass, chemical composition, structure and star formation rate change. Successive generations of stars are born, each with a different initial chemical composition and therefore a different evolution. The cyclic process of stellar birth and death continues until no more gas is left to create new stars. Interactions with neighbour galaxies may reactivate and reshape the galaxy. So, in order to simulate the chemical evolution of a galaxy we have to consider the processes of galaxy formation, star formation and stellar evolution. To decipher the formation and evolutionary history of an observed galaxy, the chemical and dynamical properties of its stars are analyzed. The atmospheric composition of unevolved stars reflects in most cases the chemical composition of the interstellar gas from which the stars are formed, whereas stellar kinematics says something about the physical condition of the gas at the moment that the stars were formed. Many details are known about the stellar population in the solar neighbourhood, which is defined as a cylindric region of about 1 kpc around the Sun. Evidence for the chemical evolution in the solar neighbourhood
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Fig. 1. Cross section of the vertical structure of the Milky Way (from Carraro, 2000). Stars with abundances identical to the Sun have [Fe/H] = 0.0; less metal rich stars have negative values and more metal rich stars have positive values.
and elsewhere in the Milky Way is given by the presence of stars with different chemical compositions and ages. The stellar populations that are observed in the various parts of the Milky Way (i.e. the halo, the thick and thin disk and the bulge) all differ in chemical composition, kinematics and dynamics, suggesting that they were formed during different evolutionary phases of the Milky Way (e.g. Wyse and Gilmore, 1992; Beers and Sommer-Larsen, 1995; Gratton et al., 1996). An edgeon cross section of the spatial structure with the average metallicity (relative to the solar value) Æ[Fe/H]æ 2 is given in Fig. 1. The Sun is a typical youngish (4.5 Gyr) thin disk star positioned at 8 kpc from the galactic center and is composed (by mass) of 70% hydrogen (=Xx), 28% helium (=Yx) and 2% metals (=Zx) mainly in the form of carbon, nitrogen, oxygen and iron. The older thin disk stars which represent most of the mass, have on average lower metallicities, higher random motions and lower rotation velocities. About 98% of the local stars are a member of the thin disk. The thick disk stars (2%) dominate at 1–2 kpc above the galactic plane and have a metallicity of about one quarter of the solar value. Only
2
[Fe/H] = log(Fe/H)% log(Fe/H)x.
0.1% of the local stars belong to the stellar halo. They are metal poor subdwarfs (<0.1Zx) with a high velocity, formed in globular clusters. The stars in the bulge have sub- to supersolar metallicities, with the majority having solar metallicity. The first qualitative model for the evolution of the Milky Way was proposed by Eggen et al. (1962). From observationally derived eccentricities and angular momenta of stellar orbits, they deduced that the collapse of the protogalaxy must have started approximately 10 Gyr ago and that both halo and bulge formed within only a few times 108 yr. Much later, Sandage and Fouts (1987) studied the kinematical and chemical properties of a large sample of subdwarfs and concluded contrary to Eggen et al. (1962) that the halo formed during a slow collapse and that spin-up occurred with an increasing collapse factor. In correspondence with the results of Gilmore and Reid (1983) they found a halo, a thick and thin disk component from which they concluded that the dissipation rate changed during the galactic evolution. From a study of globular clusters, Searle and Zinn (1978) did not find any significant radial abundance gradient in the outer halo and proposed a model in which the central region forms first, followed by the infall of gas fragments. The infalling gas accumulates in high density regions which become the sites of cluster and halo
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star formation. The remaining gas undergoes dissipation and is swept into the galactic disk. For the formation of the galactic bulge several scenarios have been invoked: in the model of Eggen et al. (1962) the bulge is formed during the initial contraction of the protogalactic cloud; Wyse and Gilmore (1992) considered its formation from the infall of remaining halo gas; Sofue and Habe (1992) proposed that a significant fraction of bulge stars could have their origin in gas clouds that were ejected by a central starburst; Pfenninger and Norman (1990) showed that a barred potential in the central region can account for the heating of a flat stellar disk. At present it is still not very clear how the different galactic components relate to each other and how they may have mutually affected their formation and evolution. In addition to these qualitative pictures that have been presented in the literature, several numerical simulations of galactic evolution have been performed in order to explain the observations like the local stellar metallicity distribution, abundance gradients, etc. Since different parts of a galaxy influence each other by gas flows and large-scale heating (McCray and Snow, 1979; Norman and Ikeuchi, 1989; Irwin and Seaquist, 1990; Li and Ikeuchi, 1992), the whole Milky Way has to be treated at once in order to obtain a self-consistent description of galactic evolution. Thereto models are needed that treat in detail star formation/ evolution and all important dynamical (e.g. gas flows, rotation) and energetical (e.g. dissipation, heating) processes as well as their dependence on the metallicity (e.g. cooling, stellar evolution). Because of the numerical complexity and the many (unknown) parameters involved in modeling all these processes, the chemical and dynamical evolution of the Milky Way have usually been treated separately. Purely chemical evolutionary models (CEMs) (e.g. Matteucci and Franc¸ois, 1989; Carigi, 1994; Giovanoli and Tosi, 1995; Prantzos and Aubert, 1995; Chiappini et al., 1997, 1999; Goswami and Prantzos, 2000) provide to study in detail the chemical properties of the Milky Way but lack self-consistent gas and star dynamics, while purely dynamical models (e.g. Larson, 1975, 1976; Burkert and Hensler, 1987, 1988) follow the dynamical evolution of the gas and the
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stars but neglect the interactions between them. Dynamical models have mainly been adopted to simulate the structural formation of the Milky Way, while CEMs have been and are still widely used to constrain star formation histories, supernova rates and abundances in the ISM, in the stars and in the intracluster medium. The basic philosophy behind CEMs is that reproducing observed element abundances may already put significant constraints on galaxy formation without considering complicated dynamical processes. Among all the CEMs that have been constructed so far, the most simple ones are closed box models which were first invented and elaborated by Tinsely (1974, 1981). They simulate the Galaxy as an isolated system with a total constant mass, initially composed of pure primordial gas (i.e. metal free), in which the stars and gas are perfectly mixed at all times. Although these models span a wide range of complexity and help to understand how the stars and the ISM affect each other locally, they cannot account for the observed metallicity distribution of long-lived stars in the solar neighbourhood (known as the G-dwarf problem, van den Bergh, 1962; Schmidt, 1963) which is a major observational constraint that needs to be fulfilled by any CEM. Open box models which consider the inflow of primordial or metal poor gas, offer a solution to the G-dwarf problem and are based on the results of dynamical models (Larson, 1972). The main assumption in these models is that the galactic disk is formed over a long timescale (several Gyr, much longer than the halo and bulge formation timescale) by the gradual accretion of unprocessed or partially processed infalling material which only starts to form stars after it has fallen into the disk. The infalling gas is believed to come from the gaseous halo as exemplified by the high-velocity HI clouds, and/or from outside the galaxy. Today, these open infall models are classified as standard chemical evolutionary models since they can satisfactorily (within a factor of 2) reproduce all major observational constraints in the solar neighbourhood and beyound (see Tosi, 2000 for an overview). Beside the lack of dynamics, a main shortcoming of most CEMs is that they follow only large scale and long-term phenomena and not cloud-to-cloud and star-to-star fluctuations
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and therefore can only produce average evolutionary trends. To explain the abundance variations observed among stars of the same age, only a few CEMs have been presented that account for local chemical inhomogeneities by considering inhomogeneous evolution and non-instantaneous mixing of the stellar ejecta with the interstellar gas (e.g. van den Hoek and de Jong, 1997; Argast et al., 2000, 2002, 2004). To meet the shortcomings of purely chemical and dynamical models, chemo-dynamical models have been constructed which try to couple the chemical evolution with the dynamical evolution (e.g. Theis et al., 1992; Hensler et al., 1993; Steinmetz and Mu¨ller, 1994; Samland, 1994; Samland et al., 1997; Berczik, 1999). Due to limited computer capacities, most chemo-dynamical models evolve non-rotating galaxies in one dimension and include a less sophisticated treatment of chemical evolution than classic CEMs. By consequence, only some of the most obvious observational constraints, such as the age–metallicity relation have so far been confirmed. Current chemo-dynamical models make use of the smooth particle dynamics (SPH) (e.g. Lia et al., 2002) which allows to treat the gas and the stars as individual particles and to follow them each in space and time. These sophisticated chemo-dynamical models are at present the most realistic models to study the effects of dynamics and show the way that galactic evolution studies should go in the future. In this work we investigate the influence of binary evolution on galactic chemical evolution, in the framework of a two-infall galactic model. The latter was first proposed by Chiappini et al. (1997) and is at present one of the most sophisticated models among the class of CEMs. The major motivation behind the concept of the formation of the Milky Way in two separated infall periods and its success, is that it explains the observed different metallicity and angular momentum distributions of the different galactic components (mainly halo vs. disk) (Wyse and Gilmore, 1992; Beers and Sommer-Larsen, 1995; Gratton et al., 1996). During the first period, the primordial gas collapses very quickly (1 Gyr) leading to the formation of the halo whereby the gas lost from the halo rapidly accumulates in the center and creates the
bulge. During the second episode a much slower infall of primordial gas (and some traces of halo gas) forms the disk whereby the gas accumulates faster in the inner than in the outer regions. In this scenario the halo and disk are almost complete independently formed and thus allows for the reproduction of the completely different metallicity distributions in the different components. The inside–out mechanism to build up the disk, is based on dynamical simulations of disk formation and quite successfully reproduces the main observed features of the Milky Way as well as of external galaxies especially concerning abundance gradients (see Chiappini et al., 1997; Prantzos and Boissier, 2000). It is important to note that the exponential declining (with time) infall rate and the considered long timescale (8 Gyr) for the formation of the solar neighbourhood (10 kpc from the galactic center) adopted in the two-infall model is supported by recent cosmological chemodynamical simulations of galaxy formation (Sommer-Larsen et al., 2003). Another major feature of the two-infall galactic model is that it includes a star formation threshold in the gas density below which the star formation rate becomes zero. Main consequences of this threshold are that a star formation stop occurs at the end of the first infall period which is in agreement with the hiatus in the star formation suggested by the dynamical model of Larson (1976) and that it prevents gas consumption on a timescale shorter than the age of the Milky Way, which was also a problem encountered in closed box models. A weakness of the two-infall model is the treatment of the galactic disk as an ensemble of independently evolving cylindric rings which form by infall of gas at a different rate without accounting for possible radial flows of gas or stars along the disk (= the one-zone formulation, Talbot and Arnett, 1971). This approximation is often applied in CEMs that focus on the evolution of the galactic disk and the solar neighbourhood. Radial flows may occur as a result of angular momentum transfer due to gas viscosity and the infall of gas with a specific angular momentum different from the one of the underlying disk and as a result from gravitational interaction between the disk gas and spiral density waves that leads to large-scale shocks and dissipation.
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Therefore to assure the conservation of angular momentum during the disk formation, radial gas flows should be included. The effects that radial flows may have on galactic evolution has been explored by several authors (Mayor and Vigroux, 1981; Lacey and Fall, 1985; Clarke, 1989; Chamcham and Tayler, 1994; Portinari and Chiosi, 2000) by adopting parameterizations of the viscosity and flow velocity profiles that are quite arbitrary and ad hoc. From these simulations it turns out that radial flows are expected to be inflows over most of the disk and mainly affect gas and metallicity gradients along the disk. Although they are generally found to be not the main cause of the existence of gradients along the disk, they help to build up gradients but seemingly only under specific combinations of the parameters of infall, radial inflow and star formation rate. However, in a recent paper Chiappini et al. (2001) presented simulations with the two-infall model but with a more detailed treatment of the halo, and they illustrated that the radial profiles of stars and gas observed along the disk can be well reproduced without invoking radial gas flows. Finally, we notice that the two-infall model does not account for possible gas outflows from the Milky Way which could be driven by supernova explosions. Gas outflows or galactic winds are observed in dwarf irregular starburst galaxies and explain the solar iron abundance in the chemical composition of the intracluster medium which indicates that galaxies in clusters should loose a substantial amount of their interstellar gas. For the evolution of the Milky Way it is not clear whether or not galactic winds played a significant role. From the few simulations that have been made so far with models that include gas outflows, it seems that they do not reproduce the abundances and abundance ratios observed in field stars (Tosi et al., 1998). However the physical conditions to create powerful winds are not well known and these results should be considered as preliminary. 1.2. Binary effects It is evident that to correctly interpret the observed abundances and to compute the chemical
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enrichment of the ISM by the stars, reliable models for stellar evolution are needed. The evolution of a star and its contribution to the chemical enrichment of the ISM depend on its initial mass, initial chemical composition, mass loss history and last but not least on whether or not it evolves in isolation or in the presence of a close companion. The effects of binaries is generally ignored in present CEMs and most of them use detailed evolutionary computations of single stars only. Still these simplified models can predict the observations in the solar neighbourhood to within a factor of two (e.g. Timmes et al., 1995; Yoshii et al., 1996; Chiappini et al., 1997, 1999; Portinari et al., 1998; Boissier and Prantzos, 2000; Prantzos and Boissier, 2000). Though, despite this putative success, the importance of binaries in stellar astrophysics has been proven by both observations and theory: (i) A large fraction of all stars observed in the Milky Way have a close companion (e.g. Mermilliod, 2001; Mason et al., 2001) and will interact with consequences for the evolution and appearance of the stars as well as the nature of the orbit (e.g. Iben, 1991; Vanbeveren et al., 1998b). (ii) The existence of many observed stellar objects like type Ia SNe, X-ray binaries, cataclysmic variables, binary pulsars, etc. can only be explained via binary evolution. When a star has a close companion its evolution may become affected in a critical way. Due to strong tidal interaction it may lose or gain large amounts of matter. By consequence it will form a carbon–oxygen (CO) core that is different in mass than expected from single star evolution and therefore may produce different chemical yields. Also the nature of the compact object that is finally left is a function of the mass of the CO core. In this way a binary component of initially intermediate mass may transform into a massive star, produce iron and finally explode as a SN instead of ending up as a white dwarf. During the accretion process part of the star may become mixed with new material that may alter the chemical profile of the outer layers and the SW ejecta.
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Type Ia SN explosions, which are responsible for a large fraction of the present iron abundance in the ISM, are currently believed to be made exclusively via binary evolution. They are exploding carbon–oxygen white dwarfs (CO WDs) which are formed via mass exchange in close binary systems. It is common in chemical evolution modeling to insert the SNIa rate as a free parameter (see Greggio and Renzini, 1983) in order to reproduce the observed iron abundance and the evolution of the a-element abundance ratios [a/Fe] in the solar neighbourhood. Any details on the pre-SN evolution of the progenitor binary system is omitted and the evolution of the system is simulated by the evolution of two non-interacting single stars. The SNIa rate is finally fixed by matching the observed local SNIa rate and iron abundance. Although this artificial procedure seems to give satisfying results, it is certainly not correct. Moreover, it has been shown by De Donder and Vanbeveren (1998) that the SN rates depend critically on the properties of the underlying binary and single star population, which implies that the evolution of the progenitor population must be followed in detail when computing the chemical evolution of the Milky Way as a function of time. Rapid neutron capture reactions 3 on iron-peak nuclei (= r-process) are (partly) responsible for the existence of elements heavier than iron (Suess and Urey, 1956; Burbridge et al., 1957; Cameron, 1957). Because of the required high neutron density and temperature region to initiate the r-process, the current main candidate sources are merging neutron star (NS) binaries (Lattimer and Schramm, 1974, 1976; Symbalisty and Schramm, 1982; Freiburghaus et al., 1999a) and neutrinoheated ejecta from core-collapse SNe (Truran, 1981; Woosley and Hoffman, 1992; Meyer et al., 1992; Takahashi et al., 1994; Woosley et al., 1994). At present, SN simulations do not produce the observed r-process abundance pattern for nuclei with atomic number A < 120 and have a too low entropy to produce the heavier nuclei (Frei-
3 The neutron capture timescale is much smaller than the bdecay timescale.
burghaus et al., 1997, 1999a), while these problems are seemingly not met in current models of decompressed ejecta of merging NS binaries (Freiburghaus et al., 1999b). However, due to the uncertainties in the theoretical models it is at present too early to make any definitive conclusion on which model is most favourable. Observations of r-process abundances in very metal-poor halo stars indicate that the enrichment of the ISM in r-process elements already started early in the evolution of the Milky Way. The presence of a large scatter at low metallicity (up to [Fe/ H] = 1) suggests that incomplete mixing of the ISM with the ejecta of a very rare event, like the merging of a NS binary and much rarer than the actual core-collapse SN rate, took place. The average evolutionary trend is a positive [r/Fe] ratio that increases with decreasing metallicity for [Fe/ H] > 2(2.5) and declines to subsolar values below [Fe/H] 2(2.5). In trying to reproduce the observed behaviour, several chemical evolutionary computations (with simple CEMs) for the r-process enrichment have been carried out (e.g. Andreani et al., 1988; Mathews et al., 1992; Pagel and Tautvasˇien_e, 1997a; Travaglio et al., 1999, 2001; Ishimaru and Wanajo, 1999). They all promote (under the assumption that core-collapse SNe do synthesize all r-process elements) low mass SNe (7–8 Mx) as the major enrichers in r-elements especially in the early phases of the Milky Way, on the basis of the observed delayed increase of the [r/Fe] ratio as a function of [Fe/H]. Ishimaru and Wanajo (1999) also added the explosion of P 30 Mx stars to explain the large dispersion in [r/Fe] for halo stars. Only in the work of Mathews et al. (1992) were merging NS binaries included, though in a too qualitative way without using detailed binary evolutionary computations. By arbitrarily varying the timescale (within crudely estimated limits) for merging and thus for the ejection of the r-elements, they can roughly fit the average observed trend. Very recently the galactic enrichment in r-process elements has been computed by Argast et al. (2004) with a CEM that includes a stochastic star formation model in order to reproduce the observed scatter at low metallicity. We will discuss their results in comparison with our computational results in Section 4.9.
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The orbital decay of a freshly born NS binary is driven by angular momentum and orbital energy loss via gravitational wave radiation (GWR) and it is well known from general relativity that the timescale for decay sensitively depends on the mass, orbital period and eccentricity of the system. Therefore to make reliable predictions on the merger rate of NS systems, the initial properties of the NS binary population must be well known. This requires a detailed modeling of the progenitor massive binary population, which is not included in present CEMs. 1.3. Goal and outline of this work In order to make reliable predictions of the SN rates and the number of stars of various classes, one needs a population number synthesis (PNS) model that combines the initial properties of stars with a stellar evolutionary model. With a PNS code the evolution of a stellar population is simulated in detail as a function of time. By comparing the predicted population with the observations, the adequacy of the adopted stellar evolutionary models is checked and constraints are derived on the associated model parameters. The first numerical PNS codes were developed in the 1980s and mainly used to study binary populations (e.g. Kornilov and Lipunov, 1983; Lipunov and Postnov, 1987; Dewey and Cordes, 1987; Eggleton et al., 1989; Politano, 1988; Tutukov and Yungelson, 1987). Meanwhile PNS has become very popular and many articles have been published on various stellar objects produced by binary star evolution. The present work describes how to implement the detailed evolution of all type of binaries into a galactic CEM. We do this by combining a PNS model with a model for galaxy formation and star formation. The star formation model regulates the birthrate of stars, the PNS model follows the evolution of the stars and their chemical output and the galactic formation model dictates the structural formation of the Milky Way. In combination they allow the computation of the time evolution of the chemical composition of the ISM, the supernova rates and the merger rates of double neutron star and black hole (BH) systems.
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Section 2 describes the PNS model. We discuss single star and binary star evolution with special attention to the SW mass loss rate prescriptions during the different evolutionary phases of massive stars, to the formation of BHs with or without a prior SN outburst, to the formation and merger timescale of double compact star binaries and to the progenitor models of SNIas, SNIIs, SNIb/cs. In Section 3 we present our CEM and we discuss in detail the nucleosynthesis yields from single and binary stars. Using these yields we derive the R-values which are the fractions of the stars that are returned to the ISM in the form of the various chemical species and which are key inputs in our CEM. Finally, Section 4 deals with the time evolution, from the moment of the birth of the Milky Way to the present, of the SN rates of the different types, of the merger rate of double neutron star and black hole systems, of the elements H, He, C, N, O, Ne, Mg, S, Si, Ca, Fe and the r-process elements. As a major constraint on galactic CEMs we also simulate the distribution of G-type dwarfs, which represent a record of the evolutionary history of the metal content of the ISM. All our model results are compared to the observations in the solar neighbourhood.
2. The population synthesis model 2.1. General concept The purpose of PNS is to compute birthrates and expected numbers of various classes of stellar objects as a function of time. By comparing the predictions with the observations, the input theoretical PNS model is checked on its reliability and constraints are derived on the range of values that the parameters in the model can have. In this way PNS provides a tool to improve current stellar evolutionary models and to retrieve information on the evolutionary history and properties of different groups of stars. The general approach in PNS is to evolve in time an entire population of stars under a set of
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assumptions. First an initial population of unevolved single and binary stars is generated with physical properties according to predetermined distribution functions that are derived from observations. With the PNS code we follow the evolution of each single and binary star from its initial state until it becomes a compact object: a WD, an NS or a BH. The product of the birthrate of stars with a certain set of initial parameters and of the lifetime in any particular evolutionary stage gives the number of stars in this evolutionary stage. All the physical properties of the single or binary star are at each moment computed and stored. Exploring numerically the whole phase space of stellar parameters, the total number of stars of particular types in the considered region of star formation is obtained and compared to the observed number. In the following sections we discuss the basic inputs in our PNS model, e.g. the stellar evolutionary models to evolve single and binary stars, detailed evolutionary computations of single and binary stars, the evolutionary definition of the different classes of stars that are studied, the distribution functions of the stellar parameters to initialise the stellar population. 2.2. Single star evolution On the basis of their fate and the physical processes that dominate during their evolution, stars can be classified as low mass stars, intermediate mass stars and massive stars. The single star models that we use cover the evolution of stars in the mass range 0.8 6 M0/ Mx 6 120 with an initial metallicity Z in the interval [0.002, 0.02], from the Zero Age Main Sequence (ZAMS) until the formation of a WD, an NS or a BH. Throughout this work we indicate the value of a stellar parameter at birth (= ZAMS) with the index 0 (e.g. M0). 2.2.1. Low and intermediate mass stars To follow the evolution of low and intermediate mass stars we use the evolutionary computations of the Geneva group, which are presented and discussed in Schaller et al. (1992), in combination
with the evolutionary computations of van den Hoek and Groenewegen (1997). Starting from the tracks of Schaller et al. (1992), van den Hoek and Groenewegen (1997) computed the evolution during the whole AGB phase with the synthetic AGB model of Groenewegen and de Jong (1993) to complete the tracks of Schaller et al. (1992). 2.2.2. Massive stars To model the evolution of massive stars we use our own evolutionary computations which we will refer to further in this work as the Brussels tracks. We discuss successively the SW mass loss rates, the treatment of convection and rotation and the computation of the final masses after core collapse, which are all main issues in massive star evolution. 2.2.2.1. Stellar wind mass loss rates. Mass loss by _ expressed in M =yrÞ stellar winds ðthe rate ¼ M substantially influence the evolution of stars by modifying their evolutionary timescales, luminosities, masses and their surface abundances. They are also potential input sources of energy, momentum and nuclearly processed material to the ISM. The Brussels tracks are computed with the following mass loss rate prescriptions which fit well most of the observational constraints (Vanbeveren, 2001). We consider successively SW mass loss during the Main Sequence phase (MS), the Luminous Blue Variable phase (LBV), the Red Supergiant phase (RSG) and the Wolf–Rayet phase (WR). The MS phase prior to the LBV stage. From the compilation of observed mass loss rates for Population I stars (= metal rich disk stars like the Sun) by de Jager et al. (1988) and Vanbeveren et al. (1998b) selected the OB-type stars and updated _ values of the 24 stars that have been studied the M by Puls et al. (1996). The following best fit relation _ L (in the unit Lx) and Teff can be debetween M; rived: _ ¼ 1:67 log L 1:55 log T eff 8:29: logðMÞ
ð2:1Þ
In our evolutionary computations, Eq. (2.1) is applied throughout the CHB phase prior to the LBV phase. _ values computed by Puls We remark that the M et al. (1996) are not corrected for rotation and pos-
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sible clumping and therefore they may be overestimated up to a factor of two (Petrenz and Puls, 1996). Since metallicity affects the opacity (bound-free and bound-bound) in the very external layers of massive stars, it is expected that the wind strength is metallicity dependent. Observations of massive stars in the Magellanic Clouds confirm this (Garmany and Conti, 1985; Prinja, 1987). To scale the mass loss p rate ffiffiffi with the metallicity we use the _ / Z which is predicted by radiation relation M driven wind models for O stars (Kudritzki et al., 1989; Lamers and Cassinelli, 1996). Anticipating, overall PNS results and chemical evolutionary simulations only marginally depend on the massive star SW mass loss during CHB before the LBV phase. The RSG phase. After CHB, hydrogen is ignited in a shell outside the helium core. The star expands on a thermal timescale and if it has not yet lost most of its hydrogen rich envelope on the MS, it moves to the yellow/red part of the Hertzsprung– Russell (HR) diagram where it starts to burn helium in its core. The star is now observed either as a yellow supergiant (YSG) 4 (spectral type FG) or as an RSG (spectral type KM). The most luminous ones are called hypergiants. During this evolutionary phase the SW blows at a low velocity (10–40 km/s, Jura and Kleinmann, 1990). The observed mass loss rates are very uncertain (due to the poor understanding of circumstellar envelopes) and the driving wind mechanism is not well known. From infrared data on RSGs, Jura proposed a mass loss rate formula (Jura, 1987) that relates the mass loss to the wind velocity, the distance, the luminosity, the flux and the average wavelength of the flux distribution. Using this relation, Reid et al. (1990) estimated the mass loss rates of 16 RSGs in the Large Magellanic Cloud (LMC) with Mbol 6 7, which have been observed by IRAS. 5 With these mass loss rates and adopting an average distance modulus to the
4 The yellow supergiants form an intermediate stage between the blue and the red supergiant phase. 5 IRAS: Infrared Astronomical Satellite.
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LMC of 18.55 mag, Vanbeveren et al. (1998c) derived for the sample of 16 RSGs the following linear relation (with a correlation coe¨fficient of 0.96) between the logarithmic value of the mass loss rate and the logarithmic value of the luminosity: _ ¼ 0:8 log L C; logðMÞ
ð2:2Þ
with C = 8.7. Because of the possible large uncertainties in the derived RSG mass loss rates, the va_ lue of C could be as well 9 which implies M-values that are a factor of two smaller than predicted with C = 8.7. When relation (2.2) is implemented in a stellar evolutionary code with C = 8.7 (resp. 9), it follows that massive stars with an initial mass as low as 15 Mx (resp. 20 Mx) may lose most of their hydrogen rich envelope during the RSG phase (see Section 2.2.3). There are presently no models or observations that may give any reliable information on a rela_ and Z for YSGs/RSGs. If the tion between M _ same M–Z relation holds as for O stars, the RSG SW mass loss rate of YSGs/RSGs for Z = 0.02 may be 1.6 times larger than predicted by relation (2.2) with C = 8.7, implying that even a 12 Mx may lose most of its hydrogen rich layers in the RSG phase. To investigate the effect of a metallicity dependent RSG SW on our computations, we made stellar evolutionary calculations for two cases: a Z independent RSG SW which means that we apply relation (2.2) for all Z values, pffiffiffiand an RSG SW that is scaled proportional to Z . We notice that there are no RSGs and YSGs observed with Mbol 6 9.5 (Humphreys and McElroy, 1984) which may indicate that the most massive stars do not become yellow and/or red supergiants due to previous strong mass loss. The LBV phase. Luminous blue variables are observed as supergiants with a spectral type between O9 and A (Humphreys and Davidson, 1994) and occupy the upper most part of the HR diagram. The most luminous ones (Mbol 6 9.5) are particularly interesting from evolutionary point of view. They are characterized by extreme instability and violent eruptions causing mass losses of the order of (103–102) Mx/yr. In between the eruptions, the LBVs still lose mass at rates of typically (107–104) Mx/yr.
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Because the mass loss rate is poorly known and LBV mass loss may explain the lack of RSGs with Mbol 6 9.5 (corresponding with stars with an initial mass P 40 Mx) we use as a working hypothesis for evolution: _ during the LBV + RSG phase of a star with The M Mbol 6 9.5 must be sufficiently large to assure an RSG phase that is short enough to explain the lack of observed RSGs with Mbol 6 9.5. This formalism was first introduced by Vanbeveren (1991) and has been widely accepted and applied in most PNS computations. The absence of YSGs and RSGs with Mbol 6 9.5 is also observed in the Magellanic Clouds and our working hypothesis may apply at metallicities lower than Z = 0.02 as well. The WR phase. It is generally accepted that WR stars are massive CHeB stars that have lost all or most of their hydrogen rich envelope in the previous RSG or LBV phase (or in case of a binary system by mass transfer to the companion), consequently showing products of H burning (WN types) or He burning (WC or WO types) at their surface. The emitted spectrum of WR stars is characterized by strong emission lines due to a _ 105 M =yrÞ. Because of very intense SW ðM the dense SW, the hydrostatic surface of WR stars is not directly observable, which makes it very hard to determine and define their effective temperatures. It is therefore not an obvious task to link theoretical stellar evolutionary models to observed WR stars. To decide upon an evolutionary definition of a WR star we account for the following points: Observed WR stars have an atmospheric hydrogen abundance Xatm 6 0.30.4, which indicates that most of the WR stars are CHeB stars that have lost most of their hydrogen rich layers either by LBV SW, RSG SW or mass transfer in a close binary. The majority of the observed WR stars seem to have luminosities log(L/Lx) P 4.5 and temperatures log Teff P 4.4 (Hamann, 1994; Hamann and Koesterke, 1998a). From the theoretically predicted mass luminosity relation of massive
hydrogen poor CHeB stars (Vanbeveren and Packet, 1979; Vanbeveren et al., 1998b) it follows that the majority of the observed single WR stars must have masses above 5 Mx. Most of the WR stars in observed WR + OB binaries in the Milky Way have an estimated mass larger than 8–9 Mx (Vanbeveren et al., 1998b). In our evolutionary computations we assume that a hydrogen deficient CHeB star (Xatm 6 0.3) is observed as a WR star when its mass is larger than M WR min , for which we consider two possible values: 5 Mx and 8 Mx. We will perform PNS computations for both values separately. Once a WR star is formed, its further evolution depends critically on the SW mass loss during CHeB. Using a hydrodynamic atmosphere code where the SW is assumed to be homogeneous, _ values Hamann et al. (1995) determined the M for a large number of WR stars. Since then evidence has grown that these winds are clumpy and that a homogeneous wind model overesti_ typically by a factor of 2–4 (Hillier, mates M 1996; Moffat, 1996; Schmutz, 1996; Hamann and Koesterke, 1998b). Adopting these lower WR mass loss rates, Vanbeveren et al. (1998b) _ and proposed a linear relation between logðMÞ log L, _ ¼ log L 10; logðMÞ
ð2:3Þ
whereby the following observational data (known at that time) are taken as constraints: The WN5 star HD 50896 (WR 6) has a luminos_ ¼ 4:4 ity log L ¼ 5:6–5:7 and a logðMÞ 0:15 (Schmutz, 1997). _ of the WNE component of the The logðMÞ binary V444 Cyg (WR 139) deduced from the observed orbital period variation is 5 (Khaliullin et al., 1984; Underhill et al., 1990). Given its orbital mass of 9 Mx and using a mass-luminosity relation holding for WNE-binary components (Vanbeveren and Packet, 1979; Langer, 1989) it follows that log L = 5. The observed masses of BH components in Xray binaries indicate that stars with an initial mass >40 Mx should end their life with a mass
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Table 1 The luminosity and mass loss rates of observed WR stars in the Milky Way, computed with NLTE atmosphere models including clumping WR number
log L
_ logðMÞ
Eq. (2.3)
References
WR 6 WR 147 WR 111 WR 90 WR 135 WR 146 WR 11 WR 123 WR 139 OB10-WR1
5.45 5.65 5.3 5.5 5.2 5.7 5 5.7 5 5.7
4.4 4.6 4.8 4.6 4.9 4.5 5.1 4.14 5 4.3
4.5 4.3 4.7 4.5 4.8 4.3 5.0 4.3 5 4.3
Schmutz (1997) Morris et al. (2000) Hillier and Miller (1999) Dessart et al. (2000) Dessart et al. (2000) Dessart et al. (2000) De Marco et al. (2000) Nugis et al. (1998) Underhill et al. (1990) Smartt et al. (2001)
The values are compared with the ones predicted by Eq. (2.3).
larger than 10 Mx, which is the mass of the star at the end of CHeB. In the solar neighbourhood the observed WN/ WC number ratio is 1. The theoretically predicted ratio largely depends on the adopted _ and an M _ that is too large (too small) preM; dicts a too small (too large) WN/WC ratio. Since then more WR stars have been observed and investigated with detailed atmosphere codes that include clumping effects. They also fit relation (2.3) well. The observed WR stars with their luminosity and mass loss rates computed with NLTE atmosphere codes that account for clumping are _ values predicted listed in Table 1. In the table the M by relation (2.3) are also listed for comparison. It is not clear whether or not the WR mass loss rate is metallicity dependent. If the WR wind is radiatively driven, the iron group elements are expected to be the main wind drivers (Crowther, 2002). Because the iron elements are neither created nor destroyed during the pre-SN evolution, the iron abundance in the envelope of the WR star is roughly proportional to the total metallicity Z of the gas out of which the star was initially formed. Therefore the WR mass loss rate may scale with some power of Z similar to that of OB-type stars. Hamann and Koesterke (2000) investigated 18 WN stars in the LMC. Taking Z = 0.006 for the LMC and assuming a linear relation similar to _ Eq. (2.3), Vanbeveren (2001) found that the M–L data of the observed WN stars can be fitted with the relation,
sffiffiffiffiffiffi _ ¼ log L 10 þ log Z : logðMÞ Z
ð2:4Þ
To investigate to what extent the PNS and chemical evolutionary computations depend on the variation of the WR SW with the total metallicity Z of the interstellar gas out of which the WR progenitor population is formed, we explore two cases: the WR SW mass loss rate is independent of Z and Eq. (2.3) is applied for all Z values, the WR SW mass loss rate is Z dependent and is computed with relation (2.4). 2.2.2.2. Convection and rotation. We use the local pressure scale height Hp to constrain empirically the amount of convective core overshoot with the assumption that the radial size of the overshoot region is proportional to Hp at the edge of the convective core, i.e. dover = aHp. We take a = 0.2, a value that has been suggested by studies of open clusters (e.g. Kozhurina-Platais et al., 1997) and observed detached eclipsing binaries (Ribas et al., 2000). Moderate convective overshooting can also explain the simultaneous presence of Blue Supergiants (BSGs) and RSGs in starburst regions 6 (Meynet et al., 1993).
6 A starburst region is a region that in a recent past has known a short period of very high star formation activity.
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Detailed calculations of rotating massive stars show that rapid rotation can have important effects on the evolution (e.g. Maeder and Meynet, 2000a). Two major consequences of rotation as far as the internal structure and evolution is concerned are the outward diffusion of nuclearly processed material to the outer layers and an enlargement of the convective core on the MS. Naturally the effects are more pronounced at higher rotation rates. The first effect can explain the observed He and N excesses in the atmospheres of fast rotating massive O-type and B-type stars and in giants and supergiants (Herrero et al., 1998; Lyubimkov, 1996). The rotational effects are less significant in the post-CHB phase because nuclear evolution occurs too fast for rotation to become important. From test computations made by the Geneva group (e.g. Maeder and Meynet, 2000a) it follows that the evolution of a massive star with an equatorial rotational velocity of 150 km/s corresponding with the observed average equatorial velocity 100–150 km/s of OB-type stars (Penny, 1996; Vanbeveren et al., 1998c,b) resembles the evolution of a non-rotating star with a convective core overshooting distance during CHB of dover 0.2Hp. Therefore it is fair to state that as far as PNS simulations is concerned the main effect of rotation on single star evolution is similar to the effect of moderate convective core overshooting during CHB in non-rotating models. Our evolutionary computations are made without including rotation but with moderate convective core overshooting. Maeder and Meynet (2000a) suggest a third effect of rotation on stellar evolution, e.g. the effect of rotation on the stellar wind mass loss rate formalisms used in stellar evolutionary codes. A close inspection of the calculations presented by the Geneva group reveals that the way how they treat this effect critically affects their results. However as argued by Vanbeveren (2001) a study of the effect _ and thus on evolution is meanof rotation on M _ is used which is based on a theingful only if a M oretical model that accounts for all physical processes including rotation. Since semi-empirical SW mass loss rates are used in evolutionary codes, the effect of rotation on these empirical rates is hard to determine. Even more, empirical rates that
were determined without accounting for rotation overestimate the real values (Petrenz and Puls, 1996). Notice however that rotation (and magnetic fields) may be very important in order to understand the physics of the core collapse and supernova explosions, but this is beyound the scope of the present work. 2.2.2.3. Supernova explosion and black hole formation. When hydrogen lines are present in the optical spectrum the SN is classified as an SNII and as an SNIbc when no hydrogen is detectable (Minkowski, 1941). Depending on the mass of the collapsing material the remnant star is either an NS or a BH. If the final core mass exceeds the maximum mass it can sustain, it will collapse to a BH rather than evolve to a stable NS. The maximum mass for a stable NS configuration ð¼ M NS max Þ depends on the equation of state of matter at nuclear densities. Different theoretical models have been explored giving gravitational masses that range from 1.5 up to 2.9 Mx (Lattimer and Swesty, 1991; Thorsson et al., 1994; Kalogera and Baym, 1996; Akmal, 1998). The average mass of NSs observed in binary systems is 1.35 (±0.04 Mx) (Thorsett and Chakrabarty, 1999) while measured BH masses in binaries range from 3 to 20 Mx (Orosz, 2001; McClintock, 2001; Froning and Robinson, 2001; Wagner et al., 2001). In our PNS model we treat M NS max as a parameter with a value in the mass interval [1.5, 3] Mx. It is unclear in which initial mass range BHs form and whether or not they are preceded by an SN explosion. We summarise the information presently available. From core collapse simulations, Fryer (1999) concluded that stars with an initial mass between 20 and 40 Mx form BHs with an SN outburst while for stars initially more massive than 40 Mx the pre-collapse star directly implodes to form a BH and no SN event takes place. However Fryer did not include SW mass loss in his computations, which increases the limit on the initial mass for BH formation. Indirect evidence that the BH candidate (4.1– 7.9 Mx) in the X-ray binary system GRO J1655-
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40 (Nova Scorpii) was formed in an SN explosion has been reported by Israelian et al. (1999). They base their inference on the relatively high abundance of O, Mg, Si and S (6–10 times the solar value) on the surface of the companion star, which has a mass (1.6–3.1 Mx) that is too low to have produced such abundances by thermonuclear burning. Assuming that those elements were deposited on the companion by the SN explosion of the primary star provides indirect evidence of a BH formed in an SN. The lightcurve of hypernovae (= very energetic SNe like SN 1999ef) and faint SNe (e.g. SN 1997D), which deviates from a standard observed SN, is thought to be associated with BH formation. On the basis of the estimated ejected mass of 56Ni, Nomoto et al. (2003) suggest that stars with an initial mass P 20–25 Mx form a BH and produce either a hypernova or a faint SN while the massive stars of lower mass produce a classical SN with the formation of an NS. In our PNS model we define the parameter M BH 0;s as the initial mass limit above which a single star forms a BH without an SN explosion. We will make computations for M BH 0;s ¼ 25 or 40 M . Below this mass limit low mass BHs may form in an SN explosion. Notice however that in Section 4 we will critically discuss the effects on chemical evolution of galaxies when all BH are preceded by an SN explosion. Remnant masses. To compute the mass of the formed NS or BH we proceed as follows. Above the limit M BH 0;s we take the remnant mass (= mass of the BH) equal to the mass of the star at the moment of core collapse. Below the limit we use the stellar model computations of Woosley and Weaver (1995). To link the latter with our evolutionary computations (which end at the beginning of carbon ignition in the core), we proceed as follows. Woosley and Weaver (1995) computed the evolution and explosion of massive single stars with masses in the range (11–40) Mx and for various initial metallicities. However they do not account for SW mass loss and do not include convective core overshooting, which means that stars of initially the same mass form different CO core masses in their computations than they do in
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our models. Therefore to compute the remnant masses for our models, we use the relation between the CO core mass and the compact remnant mass that follows from the model computations of Woosley and Weaver (1995). The iron core mass and remnant mass as a function of the CO core mass as given by the computations from Woosley and Weaver (1995) are shown in Fig. 2. The letters A, B, C correspond with the different explosion models which are considered by Woosley and Weaver (1995) in order to account for the uncertainty in the explosion energy. With respect to their model A, in which the final ejecta gain the canonical kinetic energy at infinity of 1.2 · 1051 erg, the explosion energy in model B is enhanced by a factor 1.5 for M0 P 30 Mx and by a factor 2 in model C for M0 P 35 Mx. We use model B as it represents a reasonable compromise between lower and higher energy models but we will explore the effects on chemical evolution when different models apply. The pre-collapse mass (Mf), the CO core mass (MCO) and the remnant mass (Mr) as a function of the initial mass (M0) for our computed models are given in Fig. 3 for the case that M BH 0;s ¼ 40 M . Supernova kicks. There are several observations that indicate that SN explosions are asymmetric (e.g. van den Heuvel and van Paradijs, 1997; Lai, 2001). A direct effect of the asymmetry in the SN explosion is the contribution of a kick to the formed remnant star, which is reflected in the high space velocities of observed single pulsars. A few percent asymmetry in the ejection of the SN shell can already lead to kick velocities of the order of 400 km/s (e.g. Shklovskii, 1969). The origin of the asymmetry is not clear from present SN simulations. Possible physical mechanisms for generating an initial asymmetry are non-isotropic neutrino emission induced by strong magnetic fields (Lai and Qian, 1998) and magnetorotational core collapses (Khokhlov et al., 1999). As will be discussed later in Section 2.3.2, SN kicks can be catastrophic in binary systems, with a large impact on the formation of double NS systems (e.g. Dewey and Cordes, 1987; Fryer and Kalogera, 1997; De Donder and Vanbeveren, 1998) and should be included in PNS studies. Since
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M(Fe)
13
A
12
B
11
C
Z=0.02
10 9
Mr
8 7 6 5 4 3 2 1 0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
13
14
MCO 10
M(Fe) 9
Z=0.002
A
8
B
7
C
6
Mr
5 4 3 2 1 0 0
1
2
3
4
5
6
7
8
9
10
11
12
MCO Fig. 2. The iron core mass (MFe), remnant mass (Mr) as a function of the CO core mass (MCO) as computed by Woosley and Weaver (1995) for Z = 0.02 and Z = 0.002. The letters A, B and C correspond with the different explosion models that have been explored by Woosley and Weaver (1995).
there are no clear predictions from theory on the magnitude and direction of kicks one has to rely on observations. Therefore the results of recent studies on statistical analyses of observed pulsar velocities are used to parameterize the kick velocity. Pulsars radiate in the radio region and form a well-defined observed population. Although observational biases limit the observed population to relatively nearby objects, nearly all pulsars have been discovered in large surveys giving a sufficiently large sample for a meaningful statistical
analyses. Detailed studies of measured proper motions and distances of pulsars give an average space velocity at birth of the order of 250–450 km/s, with a tail extending to velocities larger than 800 km/s (Lyne and Lorimer, 1994; Hansen and Phinney, 1997; Cordes and Chernoff, 1997; Lorimer et al., 1997). We take as input kick velocity distribution a v2-like distribution, which describes well the 3D pulsar space velocity distribution that has been derived by Lyne and Lorimer (1994) and has an average kick velocity vk of 450 km/s, i.e.
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877
100
Z=0.02 Mf Mco Mr 10
1
0.1 1
10
100
M0 100
Mf Z=0.002 Mco Mr 10
1
0.1
1
10
100
M0 Fig. 3. Our computed pre-collapse mass (Mf = final mass at the end of CHeB), the CO core mass (MCO) and the remnant mass (Mr) vs. the initial mass for pffiffiffiZ = 0.02 and Z = 0.002. For Z = 0.002 we scaled the SW mass loss rate during the OB, RSG and CHeB phase proportionally to Z . In the low and intermediate mass range the computations of van den Hoek and Groenewegen (1997) are plotted.
3
3vk
Cðvk Þ ¼ 1:96 106 v2k e 514 :
ð2:5Þ
The direction of the kick is taken arbitrarily in space. As small proper motions are hard to measure and errors may not be negligible it cannot be excluded that the average velocity could be lower than suggested by present observations. Therefore we will also make computations for smaller average values (150 km/s) whereby keeping the shape of a v2-distribution, i.e.
3
vk
Cðvk Þ ¼ 2:70 105 v2k e 60 :
ð2:6Þ
If BHs form with an SN explosion the kick is taken from the input kick velocity distribution and weighted with the amount of fallback material which equals the BH mass minus the mass of the pre-SN iron core. The kick velocity is then given by vk ¼ ð1 dm Þvk , with vk the kick velocity that is selected from the input kick velocity distribution and dm the mass fraction of the initially ejected stellar envelope that falls back. This implies that BHs may receive small but non-zero kicks, which
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0.5
probability
0.4
0.3
0.2
0.1
0 0
200
400
600
800
1000
1200
1400
1600
vk(km/s)
Fig. 4. The kick velocity distribution described by Eq. (2.5) (full line) and by Eq. (2.6) (dashed line).
Fig. 5. Overview of the single star evolutionary model.
is consistent with the velocity dispersion of 40 km/s of X-ray transients with BH candidates (White and van Paradijs, 1996). More recently the velocity distribution of isolated radio pulsars has been re-analysed by Arzoumanian et al. (2002). They promote a twocomponent distribution with characteristic velocities of 90 and 500 km/s. Test calculations illustrate that the PNS results with the latter distribution are somewhere in between the results holding for the
two distributions given by Eqs. (2.5) and (2.6). Fig. 4 shows the two distributions. 2.2.3. Summary of the single star evolutionary model Fig. 5 summarizes our adopted single star evolutionary scenario. The main evolutionary data are given in Table 2. Obviously the details of the evolutionary calculations are available upon request.
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Table 2 Single star evolutionary data for Z = 0.02 and for Z = 0.002 (italic) TCHB
TO
TRSG
TWR
TWN
TWC
TCHeB
9961.7 6263.6 1827.3 855.6 1115.9 855.6 352.5 290.8 94.5 88.3 43.2 45.1 26.4 28.8 16.0 18.1 14.8 16.1 10.1
– – – – – – – – – – – – – – – – – – 3.88
– – – – – – – – – – – – 0.94 1.1 0.85 1.0 0.46 1.16 0.254
11.5 8.2
6.8 5.33
0.77 0.194
25 30
9.2 6.9
6.71 5.16
0.66 0.165
30 40 40
7.9 5.5 6.2
6.23 3.98 5.27
0.54 0.109 0.42
60 60
3.4 3.9
2.96 3.44
100 100
2.41 2.94
2.96 2.65
– 0 0 – 0
– – – – – – – – – – – – – – – – – – 0.413 0 – 0.298 0.180 – 0.204 0.204 – 0.123 0.172 0 0.077 0.075 0.276 0.069 0.051 0.198
– – – – – – – – – – – – – – – – – – –
20 25
– – – – – – – – – – – – – – – – – – 0.413 0 – 0.483 0.180 – 0.428 0.311 – 0.373 0.500 0.437 0.421 0.427 0.360 0.338 0.352 0.318
– – – – 240.9 140.6 86.2 45.2 12.4 10.9 4.7 4.7 2.6 2.7 1.6 1.6 1.12 1.16 0.841 0.840 20 0.677 0.677 – 0.593 0.593 0.54 0.482 0.500 39 0.421 0.427 0.360 0.338 0.352 0.318
M0 1 1 1.7 1.7 2 2 3 3 5 5 7 7 9 9 12 12 15 15 20
– 0.185 0 – 0.224 0.107 – 0.250 0.328 0.437 0.344 0.352 0.084 0.269 0.301 0.120
MCHBe
MWRb
1 1 1.7 1.7 2
– – – – –
3
–
5 5 7 7 9
– – – – –
– –
– 12 – 15 – 19.5
– – 7.1
– 24
9.9
28
12.9
29.5 36 39 26.1 48 53 53 74 81 81
– 19.2 26.1 32 36 36 43 56 56
TCHB = CHB lifetime, TO = O lifetime, TRSG = RSG lifetime, TWR = WR lifetime, TWR = WN lifetime, TWC = WC lifetime, TCHeB = CHeB lifetime, MCHB,e = mass at the end of CHB and MWR,b = mass at the beginning of the WR phase. For Z = 0.02 and 6 WR M0 = 20, 25 and 30 Mx the first(second) row corresponds with pffiffiffi M 0;s ¼ 5 ð8Þ M . Masses are expressed in solar mass and times in 10 yr. For Z = 0.002 the RSG SW is scaled pffiffiffiproportionally to Z , the LBV SW is taken Z independent and the WR SW is taken once Z independent (= first row) and once / Z (= second row) for M0 = 40, 60, 100 Mx.
2.3. Binary star evolution By definition, close binaries are pairs of stars that experience during their evolution at least one phase of mass transfer. At birth when the binary is still unevolved or detached (both stars are on the ZAMS), the system is completely defined by the mass M1,0 of the most massive component called the primary star, the
mass ratio q0 = M2,0/M1,0 with M2,0 the mass of the secondary star which is the less massive component, the orbital period P0 and the eccentricity e0 of the orbit. The values of M1,0, q0, P0 and (to a lesser degree) e0 determine the evolutionary path of the binary system. If the binary orbit is wide enough for both individual stars not to be affected by each other, the stars behave approximately as single stars.
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However when they become close enough to interact, gas may flow from one star to the other. Mass exchange or transfer in binary systems is commonly treated in the Roche model, which describes, in a frame rotating with a circular and synchronized binary system, the equipotential or Roche surfaces of the binary under the assumption that the gravitational field generated by the two stars is that of point-masses. In this Roche geometry, each star can expand to a maximum radius, called the Roche radius RR. When the radius of a star becomes larger than RR it starts to lose mass through the first Lagrangian point L1 in which the effective force of gravity corrected for centrifugal forces is zero. A detailed description of the Roche model can be found in the work of Kopal (1972). The Roche radius of the primary star relative to the orbital separation A of the system has been fitted by Eggleton (1983). The Roche radius of the secondary is obtained by changing q into 1/q into the latter fit. The outer Lagrangian surface is the equipotential surface that surrounds the entire binary and has an intersection point at L2, called the second Lagrangian point. Through L2 matter can escape most easily from the gravitational field of the binary. The Roche model is used in all binary evolutionary computations. Although this may only be a rough approximation of reality, binary evolutionary computations show that the existence of the Roche radius is of primary importance rather than its exact value. 2.3.1. Mass transfer and accretion Mass transfer occurs in a binary when a component exceeds its Roche radius and performs Roche lobe overflow (RLOF). The characteristics of the RLOF process are determined by the behaviour of the Roche radius as a function of the mass ratio and by the reaction of a star to mass loss or accretion. Following the classification of Kippenhahn and Weigert (1967) and Lauterborn (1970) we distinguish three main cases of mass transfer, which correspond to the three major expansion phases during stellar evolution: case A mass transfer which takes place during CHB, case B mass transfer which occurs during the hydrogen shell burning
phase prior to central helium burning and case C mass transfer which begins after helium has been depleted in the core. Case B RLOF is further divided into early case B or case Br where at the onset of RLOF the envelope of the primary is mostly radiative and late case B or case Bc where the primary has a deep convective envelope at the beginning of the RLOF phase. In case B binaries the RLOF remnant star may fill its Roche lobe for a second time during helium shell burning and perform case BB RLOF (Delgado and Thomas, 1981). When the RLOF takes place according to case A/case Br, mass transfer proceeds on the nuclear/ thermal timescale and occurs dynamically stable. The overflowing material passes through L1 and is accreted (all or partly) by the secondary. Although the accretion process in a binary is by no means spherically symmetric, it is commonly assumed that once the matter falls onto the secondary it is redistributed homogeneously over its surface on a timescale that is short compared to the evolutionary timescale of the star. Binary evolutionary computations have been made by De Loore and Vanbeveren (1992, 1994, 1995) and Vanbeveren and De Loore (1995), using two accretion models in order to follow the temporal behaviour of the mass gainer: the standard accretion model (Neo et al., 1977) and the accretion induced full mixing model (Vanbeveren et al., 1994; Vanbeveren and De Loore, 1994). If the primary has deep surface convection layers (i.e. in case Bc and case C) mass transfer is dynamically unstable because the radius of the primary increases as a result of mass loss while the Roche radius normally decreases when the primary is more massive than the secondary. In this case the RLOF process is accompanied by a common-envelope (CE) phase (Paczynski, 1976) and a spiral-in of the secondary through the envelope of the primary. The outcome of the spiral-in phase depends on how much energy is released in the orbital decay of the binary and is deposited in the envelope. 2.3.2. Evolution of the binary orbit The orbital binary period changes continuously in time and is decisive for the final fate of the bin-
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ary. We compute the evolution of the orbital period in detail, whereby accounting for the effect of SW mass loss, conservative and non-conservative RLOF, CE with spiral-in evolution, the impact of asymmetric SN explosions and the effect of gravitational wave radiation. We use the following prescriptions to compute the period variation caused by the different processes. In what follows the initial resp. final values are indicated with the subscript index i resp. f. 2.3.2.1. Stellar wind mass loss. As already discussed for single stars, stars continuously lose mass by an SW throughout their lives. The SW from a binary component affects the binary parameters via the loss of mass and angular momentum. Some of the wind material may be accreted by the companion star. To study the effect of SW mass loss on the orbital period we assume that the mass outflow is isotropic. The specific angular momentum of the wind lwind in units of Avorb (with A the orbital separation between the two stars in the binary system) is given by the relation, J_ ¼ lwind Avorb : ð2:7Þ _ wind M If the wind velocity is larger than the orbital velocity such that the wind cannot extract angular momentum from the orbital motion by torque, the specific angular momentum of the wind can be approximated by the specific angular momentum of the matter at the moment it leaves the mass losing star, i.e. 2 M2 lwind ¼ : ð2:8Þ M1 þ M2 For the general case that both binary components lose mass by a spherical symmetric SW, combination of Eqs. (2.7) and (2.8) and using Keplers third law results into the following initial– final period relation: 2 Pf M 1;i þ M 2;i ¼ : ð2:9Þ Pi M 1;f þ M 2;f It follows immediately that in this case the orbital period increases.
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However, if the wind blows at velocities of magnitude comparable to the orbital velocity, the wind interacts with the orbit and the ejected mass carries away angular momentum. In this case lwind is completely unknown and currently no prescription is available. The net result may be a strong reduction of the orbital period. 2.3.2.2. Roche lobe overflow. The variation of the orbital period during RLOF depends on the fraction b of mass that is lost by the primary star and is accreted by the secondary star. It is commonly assumed that during RLOF the influence of a possible SW on the gasstream and period is negligible, which is a good approximation for the majority of close binaries. The following 4 cases are distinguished. Dynamically stable RLOF: the conservative case b = 1 In case of a conservative mass transfer the total mass and angular momentum of the system is conserved: dJ = 0 and dM1 = dM2. The relative orbital period change yields then dP 3ð1 q2 Þ dðM 1 þ M 2 Þ ¼ ; P q M1 þ M2
ð2:10Þ
with q = M2/M1. At the onset of the RLOF, the primary star is more massive than the secondary star and the orbital period decreases until the mass ratio equals unity. Subsequently the mass ratio becomes reversed and the orbital period increases again. Integrating Eq. (2.10) over the whole RLOF phase gives the period relation, 3 Pf M 1;i M 2;i ¼ : ð2:11Þ Pi M 1;f M 2;f The final period is longer or shorter than the initial period depending on the final mass ratio. Dynamically stable RLOF: the non-conservative case 0 6 b\1 If during RLOF the accreting secondary star gets out of thermal equilibrium and swells up to its own Roche lobe, a contact binary is formed. The inner contact Roche surface becomes completely filled and gas may reach the outer Lagrangian point L2, where it can leave the system.
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When a particle leaves the binary through L2 its energy is too small to escape to infinity. However, it is no longer forced to corotate with the binary and, for most mass ratios, acquires enough energy by gravitational interaction with the binary to spiral to infinity. In our model we assume that the escaping matter forms a ring around the system and we approximate the loss of angular momentum by the angular momentum of the ring (see van den Heuvel, 1993; Soberman et al., 1997). Because of the frequent collisions among the particles themselves it is expected that the ring will be circular. In a first approximation the particles in the ring may be regarded as moving under the gravitational attraction of a source of mass M1 + M2 located at the center of mass of the system. Assuming that the ring revolves around the center of mass with a radius Aring, and in the same sense as the binary motion, the angular momentum per unit mass of the ring is equal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2:12Þ jring ¼ GðM 1 þ M 2 ÞAring : Since jring must come from the binary itself, an amount of matter = (1 b) dM1 that escapes via L2 causes a change in the total orbital angular momentum of the binary that is given by dJ ¼ jring ð1 bÞdM 1 :
ð2:13Þ
The distance Aring of the L2 point from the center of mass is a slowly varying function of the mass ratio q. For the ring to be stable within the gravitational potential it must be sufficiently wide, i.e. Aring > 2A. We take Aring = 2.3A. The relative change of the period is obtained by combining Keplers third law, the expression for J and Eq. (2.12), dP ð1 bÞdM 1 dM 1 dM 2 ¼ ð1 þ 3cÞ 3 3 ; P M1 þ M2 M1 M2 ðM 1 þM 2 Þ2 M1M2
qffiffiffiffiffiffiffi
Aring . A
ð2:14Þ
with c ¼ Integrating this equation gives the period relation:for b > 0: Þ 3½pffiffigð1bÞ1 3½pffiffigð1b þ1 b Pf M 1;f þ M 2;f M 1;f M 2;f ¼ Pi M 1;i þ M 2;i M 1;i M 2;i ð2:15Þ for b = 0:
Pf ¼ Pi
3½pffiffig1 pffiffiM 1;f M 1;i 3 g M 1;f þ M 2;f M 1;f M 2;i ; e M 1;i þ M 2;i M 1;i ð2:16Þ
with g = Aring/A. Unlike spherical symmetric mass outflow, mass loss from the L2 point causes the orbit to shrink by a factor that sensitively depends on the value of b. Dynamically unstable RLOF In case of dynamically unstable mass transfer, the lower mass star is dragged into the envelope of the primary star and the ensuing CE evolution is dominated by frictional forces due to differential rotation of the embedded binary with respect to the envelope. The global outcome is expected to be a strong reduction of the binary separation with a possible ejection of the whole envelope if the binding energy that is released by the shrinking binary is large enough to do this. The spiral-in process is highly complicated and a correct treatment requires detailed 3D hydrodynamic computations consistently from start to finish. Currently, neither the initial phases (i.e. the formation of the CE) nor the final ejection of the envelope has been computed consistently. However, an estimate of the size of the final orbit can be obtained by equating the binding energy of the stellar envelope to the difference in orbital energy of the binary at the beginning and end of the spiral-in process. Assuming that the envelope binding energy and structure is that of the unperturbed giant star at the onset of spiral-in, we have that GM 1;i ðM 1;i M 1;f Þ GM 1;f M 2;i GM 1;i M 2;i ¼a þ ; kðRR =Ai ÞAi 2Af 2Ai ð2:17Þ with RR the Roche radius of the primary star, k a factor describing the density structure of the primarys envelope and a the efficiency of converting orbital energy into kinetic energy for the expulsion of the envelope (Webbink, 1984; de Kool, 1990). The mass of the secondary star is assumed to remain constant. The value of k is derived from stellar evolutionary calculations and lies within 0.7 and 1.5 for convective envelopes and between 0.3 and 0.5 for radiative envelopes. The value of a is
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necessarily 6 1 and probably ranges from 0.1 to 1 as suggested by hydrodynamic calculations (e.g. Yorke et al., 1995) and observational studies of binary planetary nuclei (= close double white dwarf systems with a planetary nebula) (Bragaglia et al., 1990). Elaboration of Eq. (2.17) and using Keplers third law leads to the following equation for the evolution of the orbit: 0 13=2 3=2 1=2 Pf M 1;f M M 1;i þ M 2;i 2;i @ A ¼ : 2ðM M Þ Pi M 1;i M 1;f þ M 2;i M 2;i þ 1;i 1;f
In this case matter leaves the system with the specific orbital angular momentum of the accretion star (e.g. Bhattacharya and van den Heuvel, 1991; van den Heuvel, 1994; Soberman et al., 1997). When a fraction f of the transferred mass is ejected isotropically from the vicinity of the companion star, the induced orbital period change is given by the relation Pf ¼ Pi
3f 3 7:5þ1f qi 1 þ qi 1 þ ð1 fÞqf ; qf 1 þ qf 1 þ ð1 fÞqi
ð2:19Þ
kðRR =Ai Þa
ð2:18Þ
Whether or not the binary finally survives the spiral-in process depends on the amount of orbital energy that is deposited into the envelope. If the efficiency is too small or if the binding energy of the envelope is too large, spiral-in may continue until both stars merge. Otherwise, the outcome is a small period binary whereby the core of the donor star revolves around its companion. Since the spiral-in phase is expected to proceed very fast (102–103 yr, Iben and Livio, 1993), it is highly probable that no accretion of mass will take place onto the companion star. Therefore we assume that b = 0. When the primary star has turned into a compact object (WD, NS or BH) and the secondary expands, the compact star may spiral-in into the envelope of the secondary. The efficiency of the conversion of orbital into potential energy may here be different because of the different nature of the spiral-in object. Therefore we consider in our PNS model different efficiencies for binaries in different evolutionary stages: a1 for non-evolved binaries with q0 6 0.2 (see later on in Section 2.3.3), a2 for binaries with q0 > 0.2 that perform dynamically unstable RLOF, a3 for systems with a compact component (WD, NS or BH). Isotropic re-emission An alternative way by which transferred mass may leave the system is via an isotropic fast wind that develops in the vicinity of the accretion star.
883
with q the ratio of the mass of the Roche lobe overflowing star to the mass of the companion star. For f = 1 the period variation is 3 Pf qi 1 þ qi ¼ ð2:20Þ expð3ðqf qi ÞÞ: Pi qf 1 þ qf This mode of mass transfer is often applied to model the orbital evolution of X-ray binaries in which matter is transferred to the NS in a conservative way before it is ejected isotropically. 2.3.2.3. Asymmetric supernova explosion. In a binary, an SN event is treated as an instantaneous mass loss of the exploding star. It is straightforward to show that if the explosion is symmetric, a circularised binary will be disrupted if more than half of the systems mass is ejected (e.g. Boersma, 1960). From conservation of momentum in the frame moving with the binarys original center of mass it follows that the sudden mass loss gives the binary system a kick velocity that is equal to vk ¼
M 2;i M 1;i þ M 2;i
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M 1;i M 1;f GðM 1;i þ M 2;i Þ ; Ai M 1;f þ M 2;i ð2:21Þ
with M1,f the remnant mass of the exploding progenitor star of mass M1,i. The post-SN period Pf is related to the pre-SN period Pi (in case of no disruption) by Pf ¼ Pi
1=2 M 1;i þ M 2;i ð1 eÞ3=2 ; M 1 þ M 2;i
ð2:22Þ
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with e the induced eccentricity equal to (M1,i M1,f)/(M1,f + M2,i). In general the impact of the ejected SN shell on the companion star can be neglected (Fryxell and Arnett, 1981). It is interesting to note that when mass exchange has taken place before the SN event, the ejected mass shell will be a smaller fraction of the binarys total mass and therefore a lower probability for disruption. When the explosion is asymmetric, the resulting orbit depends on the degree and direction of the non-isotropy. The kick that is imparted to the remnant mass of the exploding star can favour or disfavour the disruption of the binary system. The dynamical effects of mass loss and kicks on the binary system have been studied in several works (e.g. Sutantyo, 1978; Brandt and Podsiadlowski, 1995; Tauris and Takens, 1998). We use the formalism of Tauris and Takens (1998) to compute the post-SN orbital parameters. The geometry of an asymmetric SN outburst in a binary system is given in Fig. 6. The exploding star 1 is positioned in the origin of a cartesian reference frame (x, y, z). The direction of the z-axis equals that of the pre-SN orbital angular momentum vector ~ J , while the y-axis points towards the secondary star 2. The positive x-axis is aligned with the pre-SN relative velocity vector ~ v of the primary star with respect to the secondary star. The velocities and positions of both stars relative to the initial center of mass (c.m.) frame are ~ v11 ;~ v21 ; ~ r11 and ~ r21 . To simulate the explosion the mass of the exploding star is changed and an arbitrary oriented kick velocity ~ vk ðh; /Þ is added to its orbi-
tal velocity in 3D. We neglect the impact of the SN shell on the companion star. If the system stays bound after the SN explosion, the post-SN orbital period is given by P f pffiffiffiffi ~ ½2 m ~ 1 þ 2~v cos h cos / þ ~v2 3=2 ; ¼ m Pi ð2:23Þ with ~v the ratio of the kick velocity vk to the initial ~ the total mass relative orbital velocity vorb and m of the system before the explosion relative to the total mass after the explosion. The induced eccentricity of the orbit is ~ m ~ 1 þ 2~v cos h cos / þ ~v2 e2 ¼ 1 m½2 ½ð1 þ ~v cos h cos /Þ þ ~v2 sin2 h :
ð2:24Þ
2.3.2.4. Gravitational wave radiation. According to general relativity binary stars are radiators of gravitational waves. The radiation occurs because the binary has a time dependent mass quadrupole moment. The system loses energy and orbital angular momentum. As will be discussed in Section 3, the evolution of double compact star binaries is mainly governed by GWR. For two masses m1 and m2 moving in a circular orbit with a separation A and angular velocity x, the rate of orbital energy (E) and angular momentum (J) loss is given by dE 32G m1 m2 ¼ A4 x 6 ; 5 dt 5c ðm1 þ m2 Þ2
ð2:25Þ
dJ ðm1 m2 Þ2 ¼ 3:47 1067 P 7=3 ; 2=3 dt ðm1 þ m2 Þ
ð2:26Þ
with c the velocity of light in vacuum and with all the quantities expressed in cgs units (Landau and Lifshitz, 1951). When taking the masses as invariable and converting the cgs units into solar units (Rx,Mx), the change of orbital period P and the decay time Td (= timescale on which the orbital distance A ! 0) is given by Fig. 6. Geometry of an asymmetric supernova explosion in a circular binary.
dP M 1M 2 ¼ 3:68 106 P 5=3 ; 1=3 dt ðM 1 þ M 2 Þ
ð2:27Þ
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T d ¼ 3:22 103
ðM 1 þ M 2 Þ M 1M 2
1=3
P 8=3 ;
is computed by solving Eqs. (2.31) and (2.32) for A ! 0 and e ! 0. To illustrate the strong influence of the eccentricity on the decay time, Fig. 7 gives the ratio of the decay time of the eccentric system to the decay time of the circular system. Notice that this ratio is independent of the initial semimajor axis.
ð2:28Þ
with P and t in seconds and Td in years. If the binary has an eccentricity e, the loss of orbital energy and angular momentum averaged over one orbit is given by e2 þ 37 e4 dE 32G4 m21 m22 ðm1 þ m2 Þ 1 þ 73 24 96 ; ¼ 5 7=2 5 dt 5c A ð1 e2 Þ ð2:29Þ
dJ dt
¼
32G7=2 m21 m22 ðm1 þ m2 Þ 5c5 A7=2
1=2
2.3.3. Detailed binary evolutionary model Our PNS model follows the evolution of binaries with M1,0 2 ]1, 120] Mx, q0 2 ]0, 1] and P0 6 10 yr. Both components are assumed to be formed at the same time with the same initial chemical composition and in a circular orbit. To model the evolution of both components we use an extended set of detailed binary evolutionary computations that are made with the Brussels binary code, in which the evolution of both components is computed until the end of central helium burning. An extensive discussion of the binary tracks has been given in the papers by De Loore and Vanbeveren (1992, 1994, 1995), Vanbeveren and De Loore (1995) and Vanbeveren et al. (1998a). We start from the ZAMS and treat both stars as single stars until the primary star fills its Roche lobe. When meanwhile the system suffers from mass loss by a SW from one or both components, the orbital period variation is computed with Eq.
ð1 þ 78 e2 Þ 2
ð1 e2 Þ
ð2:30Þ (Peters, 1964). The corresponding change of orbital distance A and eccentricity e are e2 þ 37 e4 dA 64G3 m1 m2 ðm1 þ m2 Þ 1 þ 73 24 96 ; ¼ 3 7=2 5 dt 5c A ð1 e2 Þ ð2:31Þ
d ln e dt
¼
885
e2 Þ 304G3 m1 m2 ðm1 þ m2 Þ ð1 þ 121 304 : 5=2 15c5 A4 ð1 e2 Þ ð2:32Þ
Since the orbital distance evolution is coupled to the eccentricity evolution, the orbital decay time 1
Td,e /Td,c
0.1
0.01
0.001 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
e Fig. 7. The ratio of the decay time of the eccentric system to the decay time of the circular system (=Td,e/Td,c) as a function of the initial eccentricity.
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(2.9). If no binary interaction takes place, both stars evolve as single stars until they are both transformed into compact objects. For the interacting binaries we follow the next scheme. 2.3.3.1. The evolution of close binaries with q0 > 0.2 until the primary forms a compact star (WD, NS or BH). Primaries with M1,0 > 40 Mx. We assume that a primary initially more massive than 40 Mx loses all its hydrogen rich layers due to a spherically symmetric SW (OB wind and LBV wind) and that RLOF does not occur: the LBV scenario (Vanbeveren, 1991). The period variation is computed with Eq. (2.9). At the end of the SW mass loss phase the star is at the beginning of CHeB and becomes a WR star. The evolution of the primary is identical to that of a single star with the same mass. Similar as for single stars, we assume that the LBV scenario is independent from the metallicity. Primaries with M1,0 6 40 Mx. Case A and Br systems Primary star. We treat case A and case Br RLOF in the same way. Mass transfer occurs dynamically stably and stops when helium is burning in the core and the atmospheric hydrogen abundance has dropped to 0.2–0.3. The mass of the primary after RLOF is given by the relations 8 2 > < 0:0387M i 0:041M i þ 0:202 M i < 3 3 6 M i 6 7; M f ¼ 0:1ð0:124ÞM 1:32 i > : 1:44ð1:7Þ 0:09ð0:048ÞM i M i > 7: The numbers between the round brackets hold when Z = 0.002. The masses Mi resp. Mf are the masses before resp. after RLOF. If the system does not merge, the post-RLOF evolution of a massive primary star depends critically on the adopted SW mass loss formalism during CHeB. To decide whether or not a hydrogen deficient CHeB star is a WR star we use the same criterium adopted for single stars. The computed WR lifetimes are given in Table 3. For Z = 0.02 Eq. (2.3) is applied while for Z = 0.002 we consider the two cases, i.e. a Z-independent WR SW (Eq. (2.3) is applied) and a Z-dependent WR SW (Eq. (2.4) is applied).
Secondary star. The evolution of the secondary star that has gained matter lost by the primary during RLOF is followed by detailed binary evolutionary computations. We use an extensive set of evolutionary tracks that are computed with the two earlier mentioned accretion models: the accretion induced full mixing model and the standard accretion model where the effect of convection and semiconvection on top of the convective core of the gainer is treated as a very fast diffusion process. All the evolutionary data (relevant for PNS computations) have been published in the paper of Vanbeveren et al. (1998a). We apply the accretion induced full mixing model when a Keplerian accretion disk forms, and the standard accretion model when the gasstream directly hits the secondary. Orbital period variation. Since the mass loss rate due to RLOF is much larger than the mass loss rate due to a possible SW, we neglect the effect of SW mass loss on the variation of the binary period during RLOF. The evolution of the period during RLOF depends on the fraction b of matter lost by the primary and accreted by the secondary. To compute b one has to solve the magnetohydrodynamic equations (in 3D) that describe the mass transfer process, which is a very complex problem that is presently unsolved. In order to investigate the effect of b on the PNS results, we use the following formalism, which is frequently used in PNS studies:
b¼
bmax
q P 0:4;
5bmax ðq 0:2Þ 0:2 6 q < 0:4;
with bmax a constant that we either take = 1 (= conservative RLOF) or =0.5 (= non-conservative RLOF). We apply Eq. (2.11) for the case b = 1, Eq. (2.15) for 0 < b < 1 and Eq. (2.16) for b = 0. Case Bc and C systems Primary star. The initial orbital period is large enough for the primary to become a R(S)G prior to the onset of the RLOF. This means that we have to account for the effect of SW during the R(S)G phase. If a massive star loses all its hydrogen rich layers by SW during the YSG/RSG phase the total
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Table 3 The core He burning lifetime (TCHeB) and the WR lifetime (TWR) together with the duration of the subphases (TWN, TWC) as a function of the primary masses M1,0 for case B systems M1,0
TCHeB
TWR
TWN
TWC
3
50.3 62 10.81 15.31 5.82 4.92 2.95 2.89 1.59 1.58 1.35 1.25 1.04 1.02 0.82 0.697 0.635 0.541 0.549 0.500 0.437
– – – – – – – – – – – – 0.372/0 0.429/0 0.802/0 0.697/0.288 0.635/0.374 0.541/0.541 0.549/0.455 0.500/0.500 0.437/0.437
– – – – – – – – – – – – 0.372/0 0.429/0 0.802/0 0.276/0.276 0.289/0.289 0.541/0.541 0.148/0.148 0.172/0.172 0.437/0.437
– – – – – – – – – – – – – – – 0.421/0.012 0.346/0.085 – 0.401/0.307 0.328/0.328 –
5 7 9 12 15 20
30
40
WR For the WR lifetimes, the first number is computed with M WR min ¼ 5 M and the second with M min ¼ 8 M . For the primary masses 20, 30 p and 40 M the second row is computed with a Z independent WR SW and the third row with a WR SW that is scaled proportional x ffiffiffi to Z . The lifetimes are given in units of 106 yr.
mass loss due to RLOF is largely reduced, this is called the RSG scenario (Vanbeveren, 1996). In our PNS model we assume that massive primary stars with an initial mass between 15–20 and 40 Mx for Z = 0.02 and larger than 40 Mx for Z = 0.002 avoid the HeSB expansion phase due to SW mass loss and therefore do not perform case C mass transfer. Similar to a case Br binary, the mass loss process in a case Bc/C binary stops when most of the hydrogen rich layers of the primary star are removed. If the system does not merge, the further evolution of the primary is similar to the evolution of a primary in a case Br system. Secondary star. Because of the convective envelope of the primary star, mass transfer occurs dynamically unstably and soon after the onset of the RLOF a common envelope is formed. We assume that during CE evolution no significant matter accretion occurs and that the mass of the secondary remains constant (i.e. b = 0). Orbital period variation. To follow the period evolution during CE evolution and spiral-in we ap-
ply Eq. (2.18) with k = 1. For the parameter a2 we consider the values 0.5 and 1. 2.3.3.2. The evolution of close binaries with q0 6 0.2. When the primary evolves it first expands on the nuclear timescale during CHB and then on the Kelvin–Helmholtz timescale during HSB. The expansion causes the star to spin down and rotation becomes asynchronised with the orbital motion. Via tidal interaction, the star tries to absorb orbital angular momentum of the component in order to spin up and regain synchronism. If the available orbital angular momentum is not sufficient to meet the need of the primary, the low mass companion is swallowed by the primary (Darwin, 1908; Kopal, 1972; Counselman, 1973; Sparks and Stecher, 1974). The outcome is the merging of both components. From evolutionary computations of binaries it has been found that although a massive star shrinks in response to the loss of matter, it may not shrink fast enough to compete with the rapid shrinkage of the Roche radius when initially the
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binary has a mass ratio q smaller than 0.2. It is therefore conceivable that in most of the binaries with q 6 0.2 the secondary is engulfed by the primary during RLOF and both components merge. We evolve systems with q 6 0.2 by means of the common-envelope formalism whereby applying Eq. (2.18) with k = 0.5. For the parameter a1 we consider the values 0.5 and 1.
Re ¼ 0:62 104 M 3 0:49 102 M 2 þ 0:18M þ 0:17;
ð2:33Þ
which follows from our detailed binary evolutionary computations and hardly depends on the metallicity.
q0 6 0.2 mergers: we assume that the further evolution of the merger star, which has a mass equal to the sum of the masses of both components, is similar to the evolution of a single star with the same helium core mass. Since at the moment of the merging process the lower mass component has amply left the ZAMS and thus is almost homogeneous, it is reasonable to simulate the evolution of the merger star by means of the evolution of a star that has accreted an amount of mass equal to the mass of the smaller component. q0 > 0.2 mergers: we assume that the inspiraling secondary star (which is in most cases a CHB star) mixes with the hydrogen deficient CHeB primary star in an homogeneous way. The total mass of the merger star is taken equal to the mass of the primary and secondary at the end of RLOF.
2.3.3.4. The evolution of close binaries after the primary has turned into a WD, NS or BH. Massive primaries in close binaries have lost most of their hydrogen rich layers at the end of their life and likely explode as SNIbcs. To compute the remnant mass of a massive primary star we proceed in the same way as for single stars, i.e. we link our computed CO cores to the SN model computations of Woosley and Weaver (1995) using their explosion model B. Similar as for single stars we introduce an initial mass limit M BH 0;b above which a massive star finally forms a massive BH without an SN explosion. Because interacting massive primaries form smaller CO cores than isolated single stars of the same initial mass, BH formation is likely to happen at higher initial masses. We consider 40 Mx as a minimum value for M BH 0;b . The remnant masses together with the CO core masses are plotted in Fig. 9. The primary forms a NS or a BH. If the SN disrupts the binary, the secondary star further evolves as a single star. When the system is not disrupted, the post-SN orbital period is given by Eq. (2.23). In case of direct BH formation whereby no SN explosion takes place, the orbital period stays unaffected during the formation of the BH. The further evolution of the system depends on the orbital period, the mass of the secondary and the mass ratio:
To decide whether or not the binary merged during non-conservative RLOF, we check on the radius of both components after RLOF. When one of the Roche radii is smaller than the equilibrium radius of the corresponding component, we judge that both stars merged before the end of the RLOF. The equilibrium radius Re of a hydrogen deficient CHeB star with a mass M is computed with the relation
If the secondary is more massive than 40 Mx we assume that all its hydrogen layers are removed by the LBV SW and no spiral-in phase takes place. The orbital period variation is computed with Eq. (2.9). If the mass of the secondary is smaller than 40 Mx and the period is small enough that the secondary fills its Roche lobe before it becomes an R(S)G, then similar as for binaries with q0 6 0.2 this will result in a CE where due to
2.3.3.3. The formation and evolution of mergers. Merger stars form from binaries with q0 6 0.2 that do not survive the spiral-in phase and from binaries with q0 > 0.2 in which both components coalescence after non-conservative case A, case B or case C RLOF. How mergers look like and how they will further evolve is very uncertain.
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viscosity, the NS or BH will start to spiral in, into the envelope of the secondary. We apply Eq. (2.18) to compute the final period. When the mass of the massive secondary star is smaller than 40 Mx and the period is sufficiently large to allow the star to evolve into an RSG before it starts filling its Roche lobe, we first compute the period variation due to RSG SW mass loss with Eq. (2.9). When the spiral-in starts the further period evolution is computed with Eq. (2.18). We assume that during the CE and spiral-in phase the NS or BH does not accrete matter from the envelope of the secondary. To determine the outcome of the spiral-in phase we check the equilibrium radius Re of the secondary star. When the Roche radius is smaller than Re we assume that both stars merged before the removal of the whole hydrogen rich envelope. The system consists of an HSB star with a compact star _ in its He core, which is called a Thorne–Zytkow _ _ object (TZO) (Thorne and Zytkow, 1975, 1977). From detailed structure models it follows that _ TZOs have the structure of an RSG (Biehle, 1991; Cannon et al., 1992). In our PNS computa_ tions we use a TZO model which is based on the latter two studies. However, anticipating, we _ notice that the number of forming TZOs is small and that our assumption on their evolution is not critical for our chemical evolutionary computations. If the NS or BH does not merge with the secondary during spiral-in, a CHeB + NS or BH system is formed. Obviously, the further evolution of the CHeB component is taken similar to that of the post RLOF CHeB evolution of a primary with the same mass. At the moment that the secondary star explodes it is reasonable to assume that the binary has been circularised. The effects of an asymmetric SN explosion are again included in detail, similar as for the SN explosion of the primary. In the case that a double compact binary forms, i.e. an NS + NS, NS + BH, BH + NS or BH + BH system, the further evolution of the system is driven by orbital energy and angular momentum loss due to gravitational wave radiation. We follow the period and eccentricity
889
evolution with Eqs. (2.31) and (2.32) and compute the timescale on which the orbit decays. The primary forms a WD. The evolution of WD + MS systems is critical for the production of type Ia SNe. During evolution the secondary star may fill its Roche lobe and transfer mass to the WD. When the WD accretes matter mainly three things can happen depending on the accretion rate. For a very restricted range of the accretion rates stable burning on the WD surface can happen and the mass of the WD will increase. Depending on its mass prior to accretion the WD may reach the Chandrasekhar mass and explode, possibly as a SNIa in which large amounts of Fe are ejected. The evolution of these systems together with the accretion rates is discussed in more detail in the next paragraph. If the accretion rate is below the required minimum rate to have stable burning on the WD, the WD undergoes nova explosions whereby some of the inner core is ejected leading to a reduction of the WD mass. We do not follow them since we do not account for the enrichment by novae in our chemical evolutionary model. At accretion rates above the allowed maximum rate to have stable burning on the WD, a red-giant-like envelope forms around the WD with stable burning of hydrogen in a shell around the WD. As both stars are now larger than their Roche lobes, the binary enters a stage of CE evolution. The CE will be removed at the expense of orbital energy and spiral-in starts. The orbital evolution of the system is computed with Eq. (2.18). The outcome of the spiral-in process is either the coalescence of the WD with the CHeB companion star or a short period WD + hydrogen deficient CHeB star system where, depending on its mass, the CHeB star evolves into an SN or a WD. In case that both components merge, the result of the merging process is unknown. It was suggested by Sparks and Stecher (1974) on the basis of energy computations that an NS may form with a possible SN explosion. Another possibility may be that the intruding CO WD ignites carbon with
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a flash and completely desintegrates. Whatever the outcome is, if an SN explosion takes place with or without leaving a remnant, it will show hydrogen in its spectrum if during spiral-in the envelope of the companion giant star was not completely ejected. In our PNS code we assume that an SN takes place if the combined mass of the WD and the core of the companion is larger than 1.4 Mx. We classify the SN as a type II (we use the notation SNIIWD). 2.3.3.5. Type Ia SN progenitor scenarios. Type Ia SNe (no H and He lines but strong O, Si and Fe lines) are the brightest of all SN types. Theoretically there is the consensus that an SNIa is the explosion and complete disintegration of a CO WD that has a mass of the order of the Chandrasekhar limiting mass MCh (1.4 Mx). In this model the rapid conversion of about 0.8 Mx of C/O into 56 Ni that decays finally to 56Fe provides the right amount of energy to power the observed explosion and releases it on the right timescale to explain the observed light curve. The deflagration of the CO WD also produces a substantial amount of intermediate mass heavy elements (Ca,Si,. . .) and leaves a fraction of carbon and oxygen in the outer layers unburned, which is in good correspondence with the observed SNIa spectrum (e.g. Nomoto et al., 1984). Helium WDs with masses <0.5 Mx explode if they can grow to 0.6–1.0 Mx. However they eject mainly helium and iron peak elements (Nomoto and Sugimoto, 1977; Woosley and Weaver, 1986) and do not produce the observed SNIa spectrum and light curve. Oxygen–neon–magnesium (ONeMg) WDs which originate from primaries around 10 Mx, are expected to collapse into a NS by induced electron capture on Ne and Mg (Gutierrez et al., 1996). Isolated CO WDs can never become massive enough to explode since they are born with masses between 0.5 and 1.2 Mx. They slowly cool down and end up as dark matter. Only in close binary systems may CO WDs grow to the Chandrasekhar mass by accreting matter from a companion star. The kind of binary systems that can bring the WD mass to the Chandrasekhar
Fig. 8. SNIa progenitor evolutionary paths.
mass are presently not completely identified. Two evolutionary scenarios have been worked out. We summarize them together with the arguments in favour and against them. The corresponding main evolutionary paths are sketched in Fig. 8. Double degenerate scenario: DD scenario. In this scenario the SNIa arises from the merging of two close CO WDs that have a combined mass larger than or equal to MCh. Both dwarfs are brought together by GWR on a timescale Td until the less massive WD fills its Roche lobe and starts to transfer mass. It was first proposed by Iben and Tutukov (1984) and by Webbink (1984) that such binaries exist as a consequence of binary stellar evolution and would explain the absence of hydrogen and provide an easy way to approach the critical mass for explosion. The observed sample of close double WD systems is small and the systems are either not massive enough to explode or have an orbital period that is too long for merging to occur within Hubble time (for a recent review see Livio, 2000; Marsh, 2001). However it has been estimated by Yungelson et al. (2001) (within an uncertainty of a factor of 2–3) that about 1000 WDs with a visual magnitude V 6 15 should be surveyed in order
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891
100
Mf Mco Mr
Z=0.02
10
1
0.1 1
10
100
M1,0 100
Mf Mco Mr
Z=0.002
10
1
0.1 1
10
100
M1,0 Fig. 9. The final mass after case B RLOF (Mf), the CO core mass (MCO) and the mass of the remnant compact star (Mr) vs. initial primary mass (M1,0) for Z = 0.02 (upper figure) and Z = 0.002 (lower figure). For Z = 0.002 the computations are made with the SW pffiffiffi mass loss rate during the OB, RSG and CHeB phase proportional to Z .
to detect a WD pair that fulfills the conditions to produce an SNIa. The theoretical uncertainty in the DD model is the outcome of the merger process and it is at present unclear whether a SNIa (without a NS) or a NS is formed (e.g. Nomoto and Iben, 1985; Segretain et al., 1997; Saio and Nomoto, 1985, 1998; Fryer et al., 1999; Benz et al., 1990; Mochkovitch and Livio, 1990). The time elapse between the moment of birth (on the ZAMS) of the binary progenitor and the moment that the SNIa goes off is equal to the sum of the timescale on which the secondary star
becomes a WD and the time Td for orbital decay by GWR. For the SNIa to occur within Hubble time the orbital period of the initial WD + WD system must be sufficiently short. For example a (1 Mx + 1 Mx) DD binary will merge within Hubble time if the orbital period at the moment that the secondary has become a WD is smaller than 6.5 h. Single degenerate scenario: SD scenario. The SD scenario was first introduced by Whelan and Iben (1973) and later on promoted and further elaborated by the Tokyo group (see Nomoto et al., 1999 for an overview). The progenitor binary
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system is made up of a CO WD that accretes hydrogen or helium from a Roche lobe filling non-degenerate companion star. For the CO WD to reach MCh, hydrogen and helium burning on top of the carbon–oxygen core must occur stable or with weak flashes. Computations show that this can happen when _ l6 the accretion rate is in the interval M _ 6M _ u with M _ u 8:5 107 ðM WD 0:52Þ M = M _ l ¼ 0:4 M _ u (Nomoto et al., 1979; Fujiyr and M _ l the WD moto, 1982). At rates lower than M undergoes nova eruptions caused by hydrogen flashes. In the outburst more mass is lost than the WD had prior to it (Kovetz and Prialnik, _ u the 1994) and MCh is never reached. Above M mass transfer occurs too fast for accretion and an extended H rich red giant envelope forms around the WD, which is eventually lost via interaction (spiral-in) with the companion star (Nomoto et al., 1979). However, common envelope formation can be prohibited by a strong wind that the WD blows when it accretes at a rate >4 · 106 Mx/yr. This so-called strong wind model was first proposed by Hachisu et al. (1996) and can stabilize the accretion rate. The systems with the appropriate initial parameters, i.e. mass of the WD, mass of the companion and orbital period at the moment that mass transfer starts, for producing an SNIa according to the SD scenario have been theoretically identified by Hachisu et al. (1999). The latter will be used in the present work. To compute with our PNS code the SNIa rate, our binary evolutionary model follows the evolution of all intermediate mass binaries and it computes the properties of the population of WD + MS/evolved companion binaries. At the moment that the secondary star starts to fill its Roche lobe we check on the mass of the WD, the mass of the secondary and on the orbital period to determine whether the system will produce an SNIa or not according to the progenitor criteria that have been derived by Hachisu et al. (1999). This allows us to compute the expected SNIa rate produced by the SD scenario. With the DD scenario we compute the number of CO WD + CO WD systems that forms with a total mass P 1.4 Mx.
2.3.3.6. Case BB evolution. After case B mass transfer, the remnant helium star may grow to giant dimensions during HeSB and fill its Roche lobe for a second time, initiating a phase of case BB mass transfer (Habets, 1986a,b; Avila Reese, 1993). The star loses its remaining hydrogen layers and most of its helium layers on top of the helium burning shell. The mass loss rates of the Roche lobe filling component during case BB RLOF are considerably smaller than during case B RLOF (Dewi et al., 2002). This implies that if during case BB RLOF the accretion star is a normal star, mass transfer will proceed more or less conservatively. However if the accretion star is a compact star (WD, NS or BH) the situation is more complex. The evolution of helium star + NS binaries during case BB RLOF has been studied in detail by Dewi et al. (2002), Dewi and Pols (2003) and by Ivanova et al. (2002). Similar as in Habets (1986a,b), the authors illustrate the importance of the convective envelope of the donor on the mass transfer process. It is expected that in many cases a CE will be formed and the further evolution will be governed by the spiral-in process. However, the papers listed above agree upon the fact that when certain conditions are fulfilled, a CE may be avoided during the case BB RLOF. In this latter case, when the mass transfer rate becomes larger than the critical Eddington accretion rate, the excess mass leaves the binary as a NS stellar wind with the specific orbital angular momentum of the NS (= isotropic re-emission mode, section Roche lobe overflow). Notice however that the conditions mentioned above rely on postCHeB stellar evolutionary calculations and uncertainties in the latter imply uncertainties in the former. The differences in the papers of Dewi et al. (2002), Dewi and Pols (2003) and of Ivanova et al. (2002) illustrate possible consequences of these uncertainties. Dewi et al. (2002) and Dewi and Pols (2003) account for stellar wind mass loss during CHeB but they use a formalism which implies that SW mass loss during CHeB is very small for a post-RLOF He star with a mass 6 6 Mx and hardly affects its CHeB and post CHeB evolution. However, this formalism is very uncertain and, to illustrate, if
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our Eq. (2.4) is extrapolated downwards, it cannot be excluded that at Z = 0.02 this mass loss is sufficiently large in order to suppress case BB RLOF in massive binaries, i.e. the mass that would leave the star due to case BB RLOF is lost by SW during CHeB prior to the onset of case BB. As has been outlined in Vanbeveren et al. (1998b), one of the most important differences between case BB RLOF (and the applied physics of mass transfer/ mass loss from the system) vs. stellar wind mass loss in relation to population synthesis is the orbital period evolution of the binary. To illustrate, when the companion is a compact star, case BB RLOF may be governed by the spiral-in process which may result in a significant hardening of the binary. The latter has a significant effect on the probability for the binary to remain bound after the second SN explosion, and thus on the birth rate of double compact star systems. Notice that when the WR mass loss rate scales with the iron abundance according to Eq. (2.4), the SW at Z = 0.002 is too small to suppress case BB RLOF. Accounting for the discussion above, to demonstrate the importance of the assumption of case BB or stellar wind mass loss and to illustrate the effects of the physics used to describe the mass transfer during case BB, we consider the following five scenarios: scenario 1: case BB RLOF is suppressed by the SW mass loss of the helium star. The SW mass loss is independent from the metallicity and case BB is suppressed at low Z as well. The period variation is given by Eq. (2.9). scenario 2: case BB RLOF is suppressed by the SW mass loss of the helium star only for Z = 0.02. The SW mass loss is metallicity dependent and satisfies Eq. (2.4), which implies that case BB RLOF is not suppressed at Z = 0.002. Because no detailed case BB evolutionary computations exist for low metallicities, we use the computations made for Z = 0.02 for low Z as well. The period evolution is followed by assuming that matter leaves the binary in the form of an NS SW with the specific orbital angular momentum of the NS (see Dewi et al., 2002). The period change is computed with Eq. (2.20).
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scenario 3: similar as scenario 2 but we assume that the mass transfer to the NS results in a CE/ spiral-in evolution and the period variation is given by Eq. (2.18). scenario 4: the SW mass loss for hydrogen deficient CHeB stars with a mass smaller than 5 Mx can be neglected and case BB RLOF does take place independently of the metallicity. The period evolution during case BB mass transfer to the NS is treated in the same way it is in scenario 2. scenario 5: similar as scenario 4 but case BB mass transfer to the NS is assumed always to result into CE/spiral-in evolution. We remark that when the accretion star is a normal star, the case BB RLOF is always treated as a conservative process in our PNS model. A complete overview of the various binary evolutionary channels is given in Figs. 10–12.
2.4. Distribution functions To set up the initial stellar population for PNS computations, distributions of the properties of the stars at birth are needed. For single stars we need a distribution only for the initial mass M0 while for binaries we need distributions for the initial primary mass M1,0, the initial mass ratio q0, the initial orbital period P0 and the initial eccentricity (e0). The orbits of semidetached binaries are generally circularised by tidal forces on a timescale which is much smaller than the nuclear timescale on which the binaries evolve. Also during stable RLOF the binary is expected to become circularised. Therefore we take e0 = 0. Anticipating, our PNS computations marginally depend on the initial eccentricity. For the initial metallicity we always take a single chemical composition. Assuming that the initial properties of the stars are time-independent we have that the number of single stars formed in the time interval [t, t + dt] and mass interval [M0, M0 + dM0] is
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Fig. 10. Evolution of close binaries with q0 6 0.2 (left part) and >0.2 (right part) starting from the ZAMS until the beginning of CHeB.
Fig. 11. The evolution from CHeB until the formation of a compact object.
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dN s ðM 0 ; tÞ ¼ fs WðtÞuðM 0 ÞdM 0 dt;
ð2:34Þ
with fs the single star fraction among the zeroage population, W(t) the star formation rate (SFR) that gives the rate at which gas is converted into stars and u(M0) the initial mass function (IMF) that describes the distribution of stellar masses at birth the number of binaries formed in the time interval [t, t + dt] with a primary mass in the interval [M1,0, M1,0 + dM1,0], a mass-ratio in the interval [q0, q0 + dq0] and orbital period in the interval [P0, P0 + dP0] is dN b ðM 1;0 ; q0 ; P 0 ; tÞ ¼ fb WðtÞuðM 1;0 Þ/ðq0 ÞpðP 0 ÞdM 1;0 dq0 dP 0 dt; ð2:35Þ with fb the binary fraction among the zero-age population, u(M1,0) the initial mass function for the initial primary mass, /(q0) the initial mass ratio distribution and p(P0) the initial orbital period distribution. We assume that the initial distribution functions of the parameters are independent (an assumption that is generally made in PNS) since the observations give no clear indications for a possible correlation between them. a. The single and binary star fraction: fs and fb. At each moment the fraction of single resp. binary stars on the ZAMS is given by fs resp. fb with fs + fb = 1. From observational studies on spectroscopic binaries in the solar neighbourhood it is known that about 33% (±13%) of the O-type stars are the primaries of massive close binaries with a mass ratio q > 0.2 and an orbital period P 6 100 days (Garmany et al., 1980). A similar conclusion has been drawn for the intermediate mass B stars (Vanbeveren et al., 1998b). More recently the binary frequency among O stars in some young open clusters and OB associations has been estimated to be larger than 50% (up to 80%) (Mermilliod, 2001). The observed binarity among low mass G stars is about 57% (±9%) (Duquennoy and Mayor, 1991) and 42% (±9%) among M stars (Fischer and Marcy, 1992). It should be clear that due to
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selection effects these estimated percentages are underlimits of the true binary frequency. The fraction of initially formed binaries can differ significantly from the observed binary fraction. For example an observed single star may have had its origin in a binary system that was disrupted by an SN explosion or may be the merger product of two stars in a close binary system. In our PNS model we treat fb as a free parameter and look for the initial binary fraction that is required to reproduce the observed fractions among the different stellar objects. b. The initial mass function: u(M0). In the solar neighbourhood the IMF can be well described by a power law of the form uðM 0 Þ / M x 0 with x ¼ 2:35 for M 0 6 2 M (Salpeter, 1955) and x = 2.7 for M0 > 2 Mx (Scalo, 1986). The IMF is normally not corrected for unresolved binaries and may not represent truly the mass distribution of real single stars (Vanbeveren, 1982). However, in PNS studies for reasons of simplicity, it is generally assumed that the IMF of real single stars and of primary stars in binaries is the same. It was shown by Vanbeveren (1982) that this only holds if the binary fraction and the mass ratio distribution are independent of the total mass of the binary. It is obvious that the range of allowed primary masses depends on the finite mass of the parent gascloud out of which the binary forms. In our PNS model we take the same IMF for the primary masses as for the single star masses. c. The initial mass ratio distribution: /(q0). The mass ratio distribution of unevolved binaries is quite uncertain since the observed q-distribution is strongly affected by selection effects. From observed double-lined spectroscopic binaries (SB2s) for which the orbital motion of both stars can be measured, the mass ratio q0 can be determined directly. However, SB2s are heavily biased towards systems with mass ratios near unity because in lower q0 binaries the signal of the fainter secondary star (M2,0) is swamped by that of the brighter and more massive primary star (M1,0). If the observed binary spectrum is single-lined (=SB1s), q0 cannot be determined directly but instead the mass function f(M2,0) is measured, which is given by f ðM 2;0 Þ ¼
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M 1;0 sin3 iq30 =ð1 þ q0 Þ with i the orbital inclination equal to the angle between the line of sight and the axis perpendicular to the orbital plane. The primary mass (M1,0) can be derived from its spectral type and luminosity class, while the inclination is unknown and has to be accounted for statistically (see e.g. Abt and Levy, 1985; Vanbeveren et al., 1998b). Extensive studies on the shape of the mass ratio distribution for spectroscopic binaries (SB1s and SB2s) of different spectral type, whereby accounting for the observational selection effects, have been carried out by several people (e.g. Garmany et al., 1980; Hogeveen, 1991, 1992; Mazeh and Goldberg, 1992; Van Rensbergen, 2001). Different distributions (due to different interpretations of the data) are found: a flat one where every possible mass ratio occurs with a constant probability, one that decreases towards q0 = 1 or one that peaks near q0 = 1. Presently, the observational selection effects are not known sufficiently well to select the distribution that is closest to reality, and therefore it is advisable to parameterize the mass ratio distribu-
tion as /ðq0 Þ / qc0 whereby the value of c is varied to account for the possible different shapes. In our PNS computations we use the distributions with the following prescriptions: /F(q0) = 1 (flat). /H ðq0 Þ / q0:2 ðq0 > 0:3Þ ¼ constant ðq0 6 0:3Þ 0 (Hogeveen, 1991, 1992). /G ðq0 Þ / q0:5 0 (Garmany et al., 1980). We notice that a Hogeveen type distribution peaks at small q values, which is also predicted by a model where the distribution of the sum of the binary component masses corresponds to the overall IMF. This distribution is also preferred for reproducing the number of low mass X-ray binaries (LMXBs) which are interacting systems in which an NS or BH is accreting mass from a lowR mass companion. For the normalization we 1 set 0 /ðqÞdq ¼ 1. d. The initial orbital period distribution: p(P0). Although observational selections are very important here as well, the observations appear to be consistent with a flat distribution in log P0 for or-
Fig. 12. Evolution of close binaries with a WD companion (left part) and NS or BH companion (right part).
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bital periods of between a few days and several thousand days (Popova et al., 1982; Abt, 1983). More recently a new period distribution was estimated by Mason et al. (2001) using the existing observational data in literature in combination with the results of the speckle interferometric survey of galactic O-type binaries. Their derived period distribution appears to be bimodal in log P0, with peaks at 81 days and at 150 yr, which correspond respectively to the most common spectroscopic and visual binaries. However, the absent systems with periods of years to decades are the most difficult ones to observe and a flat distribution in log P0 is not excluded. In this work the initial orbital periods are distributed according to P(P0) dP0/P0 between P0,min and P0,max = 10 yrs, corresponding with the minimum period required to avoid contact of both stars on the ZAMS (on average 1 day) and a considered maximum period for binary interaction. The R 3650period distribution is normalized according to P min PðP ÞdP ¼ 1. e. The kick velocity distribution: C(vk). For exploding binary components we use the same kick velocity distribution as for single stars (Section 2.2.2). 2.5. PNS predictions of massive star populations Many studies have been published on simulations of massive star populations. Curiously, a non-negligible number denies the existence of binaries and therefore we do not consider them furtheron. The PNS code described in the previous sections has been used to study massive star populations that are formed in regions of continuous star formation and in starburst regions. The results have been published in the series of papers (Vanbeveren et al., 1997, 1998a; De Donder et al., 1997; De Donder and Vanbeveren, 1998, 1999, 2002, 2003a,b,c; Van Bever and Vanbeveren, 1997, 1998, 2000, 2003; Van Bever et al., 1999). They contain a long list of references of population synthesis work done by other groups. Although it is difficult to promote one unique PNS model, when we consider all studies it follows that to explain the observations of the massive star population in the solar neighbourhood and in the
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Magellanic Clouds we find a best agreement with PNS simulations where it is assumed that more than half of the massive stars form in interacting binaries, during the supernova explosion the average kick velocity 450 km/s, mass loss by SW during CHeB is metallicity dependent, the binary fraction is a function of metallicity. Any severe restriction on the fraction b of material transferred during RLOF and accreted by the companion star cannot be deduced from our present PNS computations. Nevertheless it can be concluded that if b = 0, no mass gainers are produced but that a large number of mergers may be formed due to orbital decay caused by angular momentum loss when matter leaves the system. Although mergers are observed as single stars, their behaviour may deviate significantly from a normal single star. In the opposite case when b = 1, a significant number of mass gainers are produced, which rejuvenates the population. More observations and sophisticated evolutionary computations are needed to identify the real b law. It has been concluded by Vanbeveren (2001) that although rotation may play a key role in understanding the properties of individual stars (e.g. the CNO surface abundances), for the overall evolution of massive single stars and primaries the effect of rotation can be simulated by moderate convective core overshooting. Therefore it is expected that the inclusion of rotation will hardly change conclusions that result from massive star PNS for the Milky Way where evolutionary computations are used with a moderate amount of overshooting.
3. The chemical evolutionary model Supernova rates in galaxies depend on the physics of galaxy formation, on the overall star formation rate and on stellar evolution. At least the latter depends on the metallicity. Therefore, to calculate the temporal evolution of the SN rates in galaxies, it is essential to combine a star formation
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model, a galaxy formation model and a PNS model which all together form a galactic chemical evolutionary model. We combine our PNS model, which follows in detail the evolution of a stellar population, consisting of single stars and close binaries, with the two-infall Galaxy formation model of Chiappini et al. (1997) to study the time evolution in the solar neighbourhood of the star formation rate, the iron relative to hydrogen abundance ratio [Fe/H], the G-dwarf disk metallicity distribution, the supernova rates of the different spectral types, the merger rates of binaries consisting of only NSs and BHs, the abundance ratios [X/ Fe] 7 with X = He, C, N, O, Ne, Mg, Si, S, Ca and r-process elements. 3.1. Introduction By means of a CEM the evolution of the chemical composition of the ISM and of the abundance distributions in the stars is followed in space and time. The galaxy is formed and evolved according to specific prescriptions for the initial conditions, the star formation rate, the behaviour of the stars and their chemical output and for possible gas inand/or outflow processes. The prescriptions are derived from theory and/or from galactic and extragalactic observations. A CEM is considered as successful when it reproduces the observational characteristics of the modeled galaxy within a factor of two, whereby accounting for the uncertainties in the input theory and in the handling and interpretation of the observational data. At present, the availability of a large amount of high quality data on the major features of the Milky Way and of the chemical yields for stars of all masses at different initial metallicities, allow the construction of successful models for the chemical evolution of the solar neighbourhood (e.g. Boissier and Prantzos, 1999; Chang et al., 1999; Chiappini et al., 1999; Portinari and Chiosi, 1999). A unique scenario for the evolution of the entire Milky Way does not yet exist as there are still some issues that are not fully understood (for an overview see Tosi, 2000).
7
[X/Fe] = log(X/Fe)% log(X/Fe)x.
In trying to fit the observational constraints in the solar neighbourhood, some basic assumptions have been invoked in the course of time which are now standard in most CEMs. A major assumption is that the Milky Way has been formed by infall of gas with primordial abundances (or gas that is only slightly enriched in heavy elements Tosi, 1988). Infall provides a solution to the G-dwarf problem that was originally discovered by van den Bergh (1962) and Schmidt (1963) and occurs when the Milky Way is modeled as a closed box. In this case theory predicts too many long-lived disk dwarfs at low metallicity when compared to the observed G-type dwarf metallicity distribution in the solar neighbourhood. The open infall model (i.e. gas can flow in and out of the galaxy) was first proposed by Larson (1972) and Sciama (1972) and still forms the basis of most CEMs. A second basic assumption that is generally made is the homogeneous evolution of the ISM, which is referred to as the instantaneous mixing approximation (IMA). It implies that at any moment the ISM has a uniform composition, which strongly simplifies the basic equations for chemical evolution. Homogeneous evolution does not allow to treat scatter in abundances due to processes like orbital diffusion, non-instantaneous mixing of enriched material, self propagation of star formation etc., but only gives average trends. Although this can be considered as a limitation or drawback, it is often preferred to the non-negligible number of extra free parameters that are introduced when including these dynamical processes. In the present work we adopt the two-infall model of Chiappini et al. (1997) in which the Milky Way is assembled by infalling galactic and extragalactic gas. For a more extended discussion on the latter model in terms of the dynamical evolution of galaxies, see Section 1. An overview of the general approach in chemical evolutionary studies is given in Fig. 13. 3.2. The two-infall scenario In our CEM the Milky Way is assumed to form by infall of primordial gas (i.e. X = 0.76, Y = 0.24 and Z = 0) with no outflows. In a first period the
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Fig. 13. Workscheme for chemical evolution modeling.
halo and thick disk form during a fast collapse. At the same time gas that escapes from the halo accumulates in the center of the spheroid and creates the bulge. Once the halo, thick disk is almost formed (after 1 Gyr), a second much longer period of major infall starts during which the thin disk assembles mainly from primordial gas and some halo gas. The infall rate at a distance r from the galactic center is given by the relation drg;inf ðr; tÞ ¼ AðrÞet=tI þ BðrÞeðttmax Þ=tII ðrÞ ; dt
ð3:1Þ
with rg,inf the surface gas density of the infalling gas, which is assumed to have a primordial chemical composition; tI and tII are the timescales for mass accretion onto the halo, thick disk and thin disk components respectively; tmax the time of maximum mass accretion onto the thin disk which coincides with the end of the halo, thick disk phase. Other types of infall rates have also been explored in the literature, but not all of them can solve the G-dwarf problem. For example, a constant infall rate that just balances star formation produces too many stars at high metallicity. An exponentially decreasing one seems to give the best
overall results and agrees with recent results of hydrodynamical simulation (Sommer-Larsen et al., 2003). In the thin disk we assume that the inner parts are built much more rapidly than the outer ones according to the inside–out scenario of Larson (1976), which ensures the formation of abundance gradients along the disk. This implies that the timescale tII is a function of the galactocentric distance r. We adopt the relation tII ¼ 0:875r 0:75 ðGyrÞ:
ð3:2Þ
(Matteucci and Franc¸ois, 1989; Burkert et al., 1992) which gives a timescale for the bulge formation of 1 Gyr and agrees with the results of Matteucci and Brocato (1990), and a timescale of 8 Gyr for the solar neighbourhood (rx = 10 kpc) which reproduces well the G-type dwarf metallicity distribution. Notice that these timescales are also supported by the recent hydrodynamical modeling of the Milky Way by Sommer-Larsen et al. (2003). The coe¨fficients A and B are fixed by reproducing the current total surface mass density distribution in the solar neighbourhood. For the disk we use the distribution given by Rana
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(1991) while we take rI(tnow) 10 Mx/pc2 which is an average value that follows from observational studies on the density ratio of the thick and thin disk (Kuijken and Gilmore, 1989; Reid and Majewski, 1993; Robin et al., 1996; Buser et al., 1998). In our CEM we take tI = 1 Gyr, tII = 8 Gyr for r = rx = 10 kpc, tmax = 2 Gyr and tnow = 15 Gyr as fixed values. We remind that this set of values correspond to the best model A in the study of Chiappini et al. (1997) and are obtained without the inclusion of binaries. When including binaries these values may not give the best fit with the observations, however it is beyound the scope of this work to investigate the influence of binaries on the parameters involved in galaxy formation. The occurrence of two distinct infall periods has been suggested by observational studies on the kinematics and chemistry of a large sample of stars in the solar neighbourhood. Major observational indications are a different angular momentum distribution of the spheroidal (bulge and halo) and disk (thick and thin) components and the metalweak tail ([Fe/H] 6 1.5) of the thick disk population (Beers and Sommer-Larsen, 1995; Ibata and Gilmore, 1995; Gratton et al., 1996). Also the observed [a/Fe] distribution seems to indicate a short timescale for the evolution of the halo and thick disk phases and a sudden decrease in the star formation in the period preceding the formation of the thin disk (Gratton et al., 1996; Bernkopf and Fuhrmann, 1998). These observational results are difficult to reconcile with the picture in which the Milky Way forms through a continuously dissipative collapse (during which halo, thick and thin disk formed simultaneously at different rates) or by one episode of infall. We remark that the rather long timescale for the formation of the solar neighbourhood implies that the disk must have been formed mainly out of extragalactic gas. The long timescale is required to reproduce the metallicity distribution of disk stars by Wyse and Gilmore (1995) and RochaPinto and Maciel (1996). 3.3. The star formation rate In order to study the evolution of the Milky Way we obviously need to know the temporal evo-
lution of the star formation rate. However, the history of star formation is not well known and different analytical prescriptions have been explored in terms of intrinsic parameters of spiral galaxies. Most often the Schmidt law (Schmidt, 1959) is used, which assumes that the star formation rate is proportional to some power of the surface gas density rg i.e. WðtÞ / rkg ðtÞ with k 1–2. The Schmidt law is compatible with the observed disk averaged star formation rate in spirals and starbursts but does not describe well the local star formation process (Kennicutt, 1998). Locally, star formation is regulated by the balance between cooling due to gas accretion onto the disk, which enhances star formation, and gas heating due to the feedback of massive stars via SWs and SN explosions which reduces the star formation efficiency. This means that the star formation rate is a function of the local gravitational potential and thus also depends on the total surface mass density r i.e. WðtÞ / rk1 rkg ðtÞ (Talbot and Arnett, 1975). As shown by Dopita and Ryder (1994) this kind of star formation law is in better agreement with the observations compared to the simple Schmidt law. Moreover, self-regulated star formation is also obtained in chemodynamical models (Burkert et al., 1992) and can accommodate the observed correlation between the surface brightness and the oxygen abundance in late spiral disks (Edmunds and Pagel, 1984; Ryder, 1995). We use the following prescription for the star formation rate, which is based on the Talbott and Arnett formulation but has been adapted to the infall model by Chiosi (1980): ðk1Þ 2ðk1Þ k rðr;tÞ rðr;tnow Þ rg ðr;tÞ Wðr;tÞ ¼~m rðr ;tÞ rðr;tÞ rðr;tnow Þ ¼ 0 when rg ðr;tÞ67 M =pc2 ; with ~m the efficiency of star formation expressed in Gyr1. Both parameters ~m and k have different values for the halo, thick disk and thin disk phase, i.e. ð~mI ; k I Þ and ð~mII ; k II Þ, for which we adopt respectively the values (2, 1.5) and (1, 1.5) corresponding with model A in the study of Chi-
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appini et al. (1997). The adopted value for the surface gas density exponent (k) also agrees with the observational results of Kennicutt (1998) and the N-body simulations of Gerritsen and Icke (1997). A star formation stop at surface gas densities lower than 7 Mx/pc2 has been suggested by observational studies on massive star formation in external galaxies (Kennicutt, 1989; Chamcham and Tayler, 1994; Gratton et al., 1996; de Blok and McGaugh, 1996). At these low densities the gas is theoretically expected to be stable against density condensations which are the seeds for star formation. It has been demonstrated and emphasized by Chiappini et al. (1997) that the threshold causes a gap in the SFR between the end of the halo, thick disk phase and the beginning of the thin disk phase, which naturally explains the observed steep increase of [Fe/O] and [Fe/Mg] at a certain value of [O/H] and [Mg/H] (Gratton et al., 2000). A temporal sink in the O and Mg production due to a stop in the formation of new massive stars easily explains the observed behaviour of [Fe/O] and [Fe/Mg]. It is interesting to note that the threshold may depend on the metallicity because the opacity, which depends on the metallicity, acts as a cooling agent and may play an important role in the cooling and collapse of a protostellar cloud. Therefore a lower threshold density can be expected at lower metallicities. However, there are currently no prescriptions for a possible threshold density-metallicity relation. In our model we take the threshold density independently of the metallicity. 3.4. The initial mass function We use the same IMF as the one adopted in our PNS model but extend the mass range downwards to 0.1 Mx. Although these very low mass stars do not participate in the chemical enrichment of the ISM, they represent a non-negligible amount of matter. We normalise the IMF by Rmass on the stellar 2 mass range [0.1, 120] Mx i.e. 0:1 c1 M 1:35 dM 0 þ 0 R 120 1.35 1.7 1:7 c2 M 0 dM 0 ¼ 1 with c1 Æ 2 = c2 Æ 2 . 2 The time behaviour of the IMF since the birth of the Milky Way is not known. It has been shown by Chiappini et al. (2000) that the combination of in-
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fall with an invariable IMF gives the best fit with all the observations. Therefore and for simplicity we assume that the IMF is constant in space and time. 3.5. The binary fraction Whether or not binary star formation depends on metallicity is totally unknown. Several binary star formation mechanisms like fragmentation and fission (for a review see Clarke, 1996) have been proposed but the role of metallicity and gas density in these formation processes is not clear. In our model we use two prescriptions: fb,1: a binary fraction (on the ZAMS) that is constant in space and time. We consider two values: 40% and 70%. fb,2: a binary fraction (on the ZAMS) that is a linear function of the metallicity Z of the interstellar gas out of which the ZAMS-population is formed at that moment. We assume that at Z = 0 the binary fraction is zero and grows linearly to a value of 70% at Z = 0.02. The choice of a current binary fraction (on the ZAMS) of 70% follows from our PNS computations and the comparison with observations (see also Section 2.5), which show that such a high massive binary fraction is needed to reproduce the observed overall massive binary star population in the solar neighbourhood. From our detailed PNS study of the WR star population we concluded that the initial massive binary fraction in the SMC could be much smaller than in the Milky Way and therefore being a function of the metallicity. For simplicity we consider a linear relation between the binary frequency and metallicity and take the same binary fraction among low, intermediate and massive stars. 3.6. The isotopic evolution model To compute the chemical composition of the ISM, the galactic disk is divided into concentric, independently evolving rings of 2 kpc wide. The solar neighbourhood corresponds with the ring at a galactocentric distance r = 10 kpc. Within each ring it is assumed that the stellar ejecta are
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completely and instantaneously mixed with the ISM, implying that each ring is always composed of a homogeneous mixture of gas, stars and stellar remnants. Thus at any time the local ISM is characterized by a unique composition. Under these assumptions the physical quantities that describe the state of each ring like the surface mass density, the gas and stellar masses, the chemical abundances, etc. are only a function of the galactocentric radius and time, which is called the one-zone formulation (Talbot and Arnett, 1971). We treat in detail the delay in the chemical enrichment due to the finite stellar lifetimes (=delayed production approximation). The evolution of the galactic disk is most conveniently described in terms of the total surface mass density r(r,t), which equals the sum of the surface gas density rg(r,t) and the surface star density rs(r,t). In CEMs that model the Milky Way as a closed system, the total mass surface density is constant and all relevant quantities are normalized to it. However this is not true for open models, for which it is more suitable to normalize with respect to the present total surface mass density r(r,tnow). Therefore the normalized surface gas density is defined: G(r,t) = rg(r, t)/r(r,tnow). For each isotope i the normalized gas density is then given as Gi(r,t) = Xi Æ G(r,t) with Xi the abundance by mass of the isotope i. The time evolution of the isotope i is obtained by solving the following set of nonlinear-integro differential equations (see Appendix A): dGi ðr; tÞ dt ¼ X i Wðr; tÞ Z Mu þ ð1 fb ðt tM ÞÞWðr; t tM ÞuðMÞ Ml
RM;i ðt tM ÞdM Z M 1;u Z qu Z P u þ fb ðt tM 1 ÞuðM 1 Þ/ðqÞ M 1;l
ql
Pl
PðP Þ½Wðr; t tM 1 ÞRM 1;i ðt tM 1 Þ þ Wðr; t tM 2 ÞRM 2;i ðt tM 2 Þ dP dq dM 1 þ
dGi;inf ðr; tÞ ; dt
ð3:3Þ
with on the right the first term representing the depletion of isotope i from the ISM by star formation, the second and respectively third term giving the enrichment of the ISM in isotope i by single stars respectively binary stars and the last term gives the contribution from the infalling gas. In the equations, t is the galactic lifetime, tM the evolutionary lifetime of a star with mass M (0 < tM 6 total lifetime of the star), RM,i(t tM) the mass fraction of a star of mass M formed at the moment (t tM) and which is ejected at galactic lifetime t into the ISM in the form of the isotope i, and fb(t) equals the fraction of binaries formed on the ZAMS at time t. We notice that the integrations in term 3 account for all the primaries and secondaries that are formed in the past but eject their material at moment t. This implies that all these primaries and secondaries may not have been born in the same binary system and therefore can have a different star formation rate. For our galactic evolutionary computations however we rewrite terms 2 and 3 with respect to time rather than with respect to mass in order to account for the dependency on the metallicity of all the quantities present in the equation. This has been worked out in the appendix (see Eq. (A.14)). The returned mass fraction RM,i is the ejected amount of processed and non-processed matter divided by the total initial mass of the star and is computed with the formula, N X M ij;exp RM;i ¼ ð3:4Þ X j; X jM j¼1 with Xj the initial abundance (by mass) of isotope j; Mij,exp the amount of isotope i synthesized from M isotope j and eventually expelled and Xii;exp is the iM unprocessed fraction of isotope i that is eventually re-ejected. The ejection takes place during the evolution of a star as a consequence of SW mass loss or as a consequence of the RLOF process when it is a binary component, and at the end of stellar evolution when an SN outburst occurs or a PN is ejected. We explicitly account for these different evolutionary moments. For single stars (second term in the equation) RM,i is only function of the initial mass and metallicity. In the case of a binary system (third term in the equation) we separate the
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975
mass fraction of the primary ðRM 1 ;i Þ and of the secondary ðRM 2 ;i Þ star. The returned mass fractions are function of the initial binary parameters (M1,0, q0, P0) and initial metallicity. They also depend on the adopted values for the binary evolutionary parameters ði:e: bmax ; a; vk Þ. We have constructed complete tables with the R-values for single stars and binary stars and they are available upon request. 3.7. Computation of the supernova rates For single stars the computation of the SN rates is given by a simple continuous integration over the progenitor initial mass range. However, in case of binaries, the SN rates depend critically (as we will show later on in Section 4.8) on the characteristics of the progenitor binary population and must be computed with a PNS model. In most CEMs only the SNIas (for which the SD scenario is commonly adopted) are considered to be formed by binaries and their rate is generally computed by solving the integral Z
M max
A M min
uðMÞ M
Z
0:5
f ðlÞWðt sM 2 Þdl dM
ð3:5Þ
lmin
(Greggio and Renzini, 1983) with M = the total mass of the progenitor binary system, l = M2/ M = the mass ratio of the secondarys mass to the total mass of the binary, f(l) = the mass ratio distribution and A a free parameter that represents the fraction of the mass range [Mmin, Mmax] that forms binaries which produce SNIas. In the model computations the parameter A is then finally fixed to match the observed SNIa rate and iron abundance. The evolution of these binary systems is in most CEMs approximated by that of two single stars whereby the total ejecta is taken equal to the sum of the ejecta of a single star with the same initial mass as the primary star and the ejecta of a typical SNIa. It follows from binary evolutionary theory that whether or not a binary component produces an SN, of which spectral type (II, Ib/c, Ia) the SN will be,
903
on which timescale the SN forms, the realisation frequencies depend on the binary mass, mass ratio, orbital period, the adopted distribution functions and evolutionary parameters and cannot be evaluated by a simple integral as given by Eq. (3.5). In particular for the SNIas, the conditions on the binary parameters are so confined that a detailed modeling of the binary population is needed. Therefore the treatment of the SNIas in current CEMs is oversimplified and the models should be linked with an PNS model that can evaluate in detail the evolution of the properties of the single and binary star population as a function of time. Our PNS model in combination with the adopted star formation rate allows the computation of the time evolution of the SNII, SNIbc, SNIa rates over the whole galactic lifetime. 3.8. The stellar yields It is obvious that the amounts of matter in the form of isotope i ejected by a star are key ingredients in a CEM. Rather than discussing the ejecta which are used in the CEM, we prefer to consider here the stellar yields in order to have an idea of the nucleosynthetic production due to the star itself. The stellar yield Mpi is the mass of a newly synthesized isotope i that is produced and ejected by a star of initial mass M. It is computed according to: Z T ðMÞ _ MðM; tÞ½X i;s X i;0 dt Mpi ¼ 0
þ
Z
Mf
½X i ðM 0 Þ X i;0 dM 0 ;
ð3:6Þ
Mr
with T(M) the total lifetime of the star, Xi,0, Xi,s and Xi(M 0 ) the mass fraction of isotope i at resp. the moment of birth, the surface and at mass level M 0 in the star (Maeder, 1992). The first term gives the SW yield and the second one the SN yield with Mf the final mass of the star at core collapse and Mr the mass of the remnant compact object. If the isotope is converted into other species rather than being produced, its yield is negative. The total amount of isotope i with the inclusion of its initial abundance, ejected by the star (=ejecta) is given by the relation:
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Ei;M ¼ ðM M r ÞX i;0 þ Mpi :
ð3:7Þ
3.8.1. Single star yields Stars initially less massive than 0.8 Mx do not produce yields and thus do not contribute to the chemical enrichment of the ISM. They only function as depots of matter. Low and intermediate mass stars enrich the ISM via their SWs during the RGB, AGB and the PN phase in 4He, 12C, 13 C, 14N, 16O, 17O and heavy s-process elements like Ba and Sr, which are essentially made during HeSB. The yields in this mass range are most sensitive to the adopted mixing length parameter, which determines the depth of the convective stellar envelope and the extent of hot bottom burning (HBB). When the convective envelope moves inward, convective motion dredges up material from the inner region to the outer layers. It takes place on the RGB (= first dredge-up), on the E-AGB (= second dredge-up) and finally during the TPAGB (= third dredge-up). 8 The efficiency of dredge-up determines the amount of material that is mixed into the outer layers. Hot bottom burning occurs at the base of the convective envelope during the interpulse phase on the AGB if the temperature is high enough, which implies a minimum initial mass (Iben and Renzini, 1983). During HBB carbon and oxygen are converted into nitrogen. Observations indicate that HBB occurs in AGB stars heavier than 3.5 Mx. For low and intermediate mass stars the yields from Renzini and Voli (1981) have been extensively used in CEMs. However, in recent years new evolutionary calculations have been presented with updated physical models (Marigo et al., 1996, 1998; Marigo, 2001; van den Hoek and Groenewegen, 1997). Here we use the stellar yields from the work of van den Hoek and Groenewegen (1997), which are computed for stars with an initial mass between 0.9 and 8 Mx and initial metallicities Z = 0.001, 0.004, 0.008, 0.02 and 0.04. In their evolutionary computations they account for the three dredge-
8 In intermediate mass stars the first dredge-up occurs only if Z > 0.001 (Dominguez et al., 1999) while low mass stars do not undergo the second dredge-up.
up phases, HBB and treat in detail mass loss as a function of the initial metallicity. The free parameters in their stellar models, which are related to mass loss, the minimum core mass for dredge-up and HBB, and the third dredge-up efficiency, are fixed in order to get the best fit with the observations of AGB stars in the Milky Way and LMC. We use their standard model calculations. The yields are shown in Fig. 14 for Z = 0.001 and Z = 0.02. From these figures it can be concluded that All stars in this mass range enrich the ISM in 4 He in more or less equal amounts. The enrichment in 12C mainly comes from the low mass AGB stars (M 6 4 Mx) while the higher mass stars predominantly contribute to the 14N enrichment. This is mainly due to the presence of HBB in stars above 4 Mx. The 12C and 16O yields generally increase with decreasing initial Z. At lower Z the SW mass loss rates are smaller, which results in larger core masses at the first thermal pulse and longer AGB lifetimes, leading to more pulses. All together the amount of material dredged-up from the core to the envelope is substantially larger in AGB stars of initial lower Z. The 14N yields slightly increase with rising Z since 14N is mainly formed as a secondary element (i.e. its production is proportional to the initial CNO abundance) during CNO burning by consuming 12C and 16O. The oxygen enrichment by low and intermediate mass stars is very small. Massive stars produce the bulk of the heavy elements with 16O as the most abundant one. The dominant elements in their SN ejecta are the a-elements 12C, 16O, 20Ne, 24Mg, 28Si, 32S, 40Ca and 56 Fe that derives from 56Ni which is synthesized during explosive Si-burning. Explosive nucleosynthesis mainly affects the abundances of Si, S and Ca. Supernova nucleosynthesis computations with the inclusion of detailed explosive nucleosynthesis have been performed by Woosley and Weaver (1995) for various initial metallicities (Z = 0, 2e6, 2e4, 0.002 and 0.02) and masses (8–40 Mx), by Thielemann et al. (1996) and Nomoto et al. (1997) for helium cores that correspond to
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975
905
Mpi
Mpi
0.025
0.35
C
12
He
4
0.3
0.02 0.015
0.25 0.2
0.01
0.15
0.005
0.1
0
0.05
-0.005 -0.01
0 0
1
2
3
4
5
6
7
0
8
1
2
3
Mi
4
5
6
7
8
5
6
7
8
Mi Mpi
Mpi
0.004
0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -0.01
14
16
0.002 0
N
O
-0.002 -0.004 -0.006 -0.008 -0.01 -0.012 -0.014 -0.016 0
1
2
3
4
5
6
7
8
0
1
2
3
Mi
4
Mi
Fig. 14. Stellar yields (expressed in solar masses) from low and intermediate mass stars from the work of HG97 for Z = 0.02 (full lines) and Z = 0.001 (dashed lines).
single stars of initial solar metallicity with masses up to 70 Mx and by Limongi et al. (2000) for Z = 0, 0.001 and 0.02 in the mass range (13–25) Mx. The yields from Woosley and Weaver (1995) and Thielemann et al. have been widely used in CEMs. At solar metallicity both works give similar yields for the CNO isotopes but significant differences for other important isotopes (especially iron and magnesium) due to different input physics like the treatment of convection. Because of the wider considered interval of masses and metallicities, we use the Woosley and Weaver (1995) computations to derive the SN yields of our exploding massive stars according to the CO core mass vs. remnant mass relation that was discussed earlier in Section 2.2.2. From the tabulated ejecta in Woosley and Weaver (1995), the CO core ejecta are calculated according to the method outlined by Portinari et al. (1998). It assumes that the H and He ejecta
given by Woosley and Weaver (1995) solely come from the layers outside the CO-core and that the global metallicity in the layers outside the CO core is equal to the initial (ZAMS) one. Under these assumptions the mass of the CO core is given by M CO ¼ M
EH þ EHe 1Z
and the ejecta of the CO core can be computed with simple relations as given in Portinari et al. (1998). Despite the incoherences of this method (discussed by Portinari et al., 1998) it is sufficient for the scope of the present work. For the remaining layers above the CO core we assume that they are not affected by the outgoing SN shock wave and take the chemical composition from our computed stellar models. For a given pre-core collapse mass, the uncertainties in the SN yields mainly come from the adopted nuclear reaction rates and the mass-cut
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which defines the border between the remnant mass and the ejected envelope. The latter determines the amount of iron that is ejected. The uncertainty is about a factor of 2 for 16O and higher for the heavier isotopes. In the phases prior to the SN, a massive star ejects elements via SW mass loss. Our SW yields are computed with the SW mass loss rate prescriptions discussed in Section 2. They are given in Fig. 15 as a function of the initial mass and for Z = 0.02 and pffiffiffi 0.001. The computations are made _ / Z during the MS, RSG and CHeB with M phase. We also present computations that are made with a Z-independent SW mass loss rate during the CHeB phase. To compare, the SW yields derived from the Geneva tracks (Schaller et al., 1992) are also given.
Mpi
The figures illustrate the following conclusions: Stellar winds contribute significantly to the enrichment in 4He. Due to the transformation of 12C and 16O into 14 N during the CNO cycle, the 12C and 16O stellar wind yields are negative in the mass range below 25 Mx. The 12C (16O) yields are positive when SW mass loss is large enough so that He burning products appear on the surface, i.e. when the star becomes a WC star. The later the WC stage starts, the more He is processed into 12C and 16O and the more 12C and 16O is ejected by the wind. Because in our stellar models 14N is created as a secondary element, the 14N stellar wind yields are larger at higher Z.
Mp i 2.2
2.2
He/10 Z=0.02 N
2 1.8 1.6
1.6
1.4
1.4 1.2
1.2
1
1 0.8 0.6
0.8 0.6
0.4
0.4
0.2 0
0.2 0
-0.2 0
C/10 Z=0.02 O/10 Ne
2 1.8
-0.2 10
20
30
40
50
60
70
80
90
100
110
120
0
10
20
30
40
50
Mi
60
70
80
90
100
110
120
70
80
90
100
110
120
Mi
Mpi
Mpi 4
2.2
He/10 Z=0.001 N*10
2 1.8 1.6
C O Ne
3.5 3
Z=0.001
2.5
1.4 1.2
2
1 1.5
0.8 0.6
1
0.4
0.5
0.2 0
0 -0.2 0
-0.5 10
20
30
40
50
60
Mi
70
80
90
100
110
120
0
10
20
30
40
50
60
Mi
Fig. 15. The SW yields (expressed in solar masses) vs. the initial mass for Z = 0.02 and Z = 0.001. The SW yields from the computations of Schaller et al. (1992) are given by the dashed lines. The computations performed with a Z-independent SW during the CHeB phase for Z = 0.001 are given by the thick lines.
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975
The 20Ne stellar wind yields are negligible because 20Ne is made by a-capture on 16O in a region inside the star that is never exposed at the surface even when extreme mass loss occurs. Non-negligible amounts of 22Ne are ejected by WR stars. However, for relatively low masses and/or metallicities the 22Ne yields are negligible. At Z = 0.02 the stellar wind yields from the Geneva tracks (Schaller et al., 1992) in C and O are on average larger due to the adopted WR SW mass loss formalism of Langer (1989), which overestimates the mass loss by a factor of 2–4. For Z = 0.001, our yields are lar_ is ger especially when a Z-independent M adopted during the CHeB phase. We remark that in the computations of Schaller et al. (1992) for Z = 0.001, WR stars only form above an initial mass of 80 Mx, which is much higher than the 40 Mx mass-limit in our modelspfor ffiffiffi the case that the SW mass loss scales with Z . 3.8.2. Binary star yields Non-interacting binary components behave as single stars and therefore produce the same yields. In particular this is also the case for the most massive binaries evolving according to the LBV scenario. Interacting binary stars experience the same various nuclear burning phases as single stars do, but their chemical profile may become significantly altered after interaction with the companion star. We discuss the different aspects of interacting binary evolution and their effects on the yields. 3.8.2.1. Roche lobe overflow. Additional to SW mass loss, RLOF enhances the mass loss rate of an interacting binary component. The process of mass transfer affects the core development of both the mass losing and mass gaining star and by consequence their yields. During RLOF a large part of the H rich envelope is lost on nuclear, thermal or dynamical timescale. Table 4 gives the amount of mass lost by RLOF in the form of the CNO elements that was originally in the convective hydrogen burning core. After CHB the helium core grows due to H shell burning. However, because of the rapid mass loss
907
Table 4 The amount of mass lost in the form of the CNO elements, during case B RLOF that has been part of the convective core during H burning as a function of the initial mass Mi
MCNO
3 5 7 9 12 15 20 25 40
0.31 0.70 0.92 1.18 1.63 2.08 3.04 3.98 7.43
during RLOF the growth of the He core is less progressive and a smaller He core is formed when compared to a single star of the same initial mass. Since RLOF removes very rapidly most of the hydrogen rich layers of a massive primary, the effects of a WR-like SW during the post-RLOF CHeB phase are quite different compared to the WR-SW-effects in a single star with the same initial mass, so are the SW yields. It can be concluded that in general interacting primaries form smaller He and CO cores and by consequence produce a lower amount of metals compared to a single star of the same initial mass. Due to case A/B (BB) RLOF, low and intermediate mass primaries may avoid the third dredge-up phase and therefore produce much lower 12C yields compared to single stars of the same initial mass. Fig. 16 gives the SW yields after case B RLOF as a function of the initial primary mass (on the ZAMS). After mass accretion, the secondarys atmosphere may become enriched in 14N and 4He. When the accreted material has already been nuclearly processed by the primary star, thermohaline mixing is induced and the accreted material is mixed with the unaffected outer layers of the accretion star. Depending on the amount of accreted mass and on the amount of mixed material, the accretion star may show moderately or extremely altered CNO surface abundances. As an illustration, Table 5 gives the surface He and N abundances of the secondary after partly-conservative (b = 0.5) and conservative (b = 1) RLOF as predicted by our binary evolutionary computations.
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Mp,i 0.6 0.5 0.4 0.3
He/10 C N O Ne
0.2 0.1 0 -0.1 10
15
20
25
30
35
40
30
35
40
M1,0 Mp,i 0.12 0.1 0.08
He/10 C N O Ne
0.06 0.04 0.02 0 10
15
20
25
M1,0 Fig. 16. The SW yields of massive primaries vs. the initial primary pffiffiffi mass for Z = 0.02 (upper figure) and Z = 0.002 (lower figure). For _ during all phases are scaled according to Z . Z = 0.002 the SW M
The results are given for both accretion models, i.e. the standard accretion model and the full mixing model. Further mass loss by stellar wind may remove these layers and they will enrich the ISM in nitrogen and helium. Another important effect of accretion is the enlargement of the convective core. In particular, due to accretion the core of an intermediate mass secondary may sufficiently grow in mass to undergo the complete nuclear burning cycle and to end up as an SN. For example, an 8 + 7 Mx close binary evolves into a 1 + 14 Mx system whereby the 14 Mx accretion star will explode as an SNII. This
is an aspect of binary evolution that is not met in single star evolution. Furthermore, massive secondaries may acquire after accretion a core mass that surpasses the limit for BH formation and will by consequence eject less matter. In general, accretion stars produce heavier remnants at the end of their evolution than they would have done had no accretion taken place. Table 6 gives the convective core masses of the mass gainers before and after RLOF. 3.8.2.2. Common envelope evolution. We assume that no material is accreted by the inspiraling star
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975 Table 5 The 4He and
909
14
N abundances of the mass gainer after case B RLOF expressed relative to the solar values (=Xi/Xi,x)
Initial system
N/Nx
He/Hex
40 + 36 40 + 24 30 + 27 30 + 18 20 + 18 20 + 12 15 + 14 15 + 9 12 + 11 12 + 7 9 + 8.1 9 + 5.4 6 + 5.4 6 + 3.6 3 + 2.7 3 + 1.8
b = 0.5
b=1
b = 0.5
b=1
1.29(1.94) 1.24(1.61) 1.14(1.72) 1.10(1.39) 1.14(1.60) 1.14(1.32) 1.09(1.46) 1.09(1.22) 1.07(1.39) 1.10(1.20) 1.07(1.33) 1.09(1.18) 1.13(1.33) 1.17(1.22) 1.04(1.20) 1.06(1.10)
1.28(1.80) 1.25(1.53) 1.01(1.46) 1.01(1.23) 1.21(1.57) 1.21(1.34) 1.12(1.41) 1.14((1.23) 1.10(1.34) 1.13(1.21) 1.10(1.30) 1.13(1.18) 1.01(1.34) 1.01(1.26) 1.09(1.20) 1.08(1.11)
2.50(2.03) 2.79(2.24) 1.95(1.69) 2.12(1.83) 1.98(1.76) 2.22(1.96) 1.79(1.65) 2.02(1.82) 1.73(1.62) 1.99(1.81) 1.69(1.60) 1.93(1.78) 1.831.73) 2.11(1.95) 1.47(1.43) 1.62(1.55)
3.04(2.54) 3.38(2.82) 2.08(1.87) 2.33(2.08) 2.49(2.23) 2.74(2.46) 2.17(2.01) 2.43(2.18) 2.09(1.95) 2.36(2.18) 2.03(1.92) 2.27(2.12) 2.23(2.11) 2.51(2.35) 1.73(1.68) 1.84(1.78)
The results given between round brackets are computed for full mixing after accretion. The initial metallicity Z is taken equal to 0.02.
Table 6 The convective core mass of the accreting secondary star before (=RLOFi) and after RLOF (=RLOFe) Initial system
MCC(RLOFi)
MCC(RLOFe)
40 + 36 40 + 24 30 + 27 30 + 18 20 + 18 20 + 12 15 + 14 15 + 9 12 + 11 12 + 7 9 + 8.1 9 + 5.4 6 + 5.4 6 + 3.6 3 + 2.7 3 + 1.8
12.7 9.7 8.9 6.6 5.1 3.8 3.2 2.7 2.4 2.1 1.6 1.4 0.95 0.88 0.35 0.32
16.4 14.3 13.4 13.6 9.7/14.3 8.4/12.1 10.3/10.5 6.3/10.1 7.4/8.1 4.4/7.5 3.2/5.7 3.3/4.9 2.1/3.4 1.9/3.2 0.7/0.9 0.8/1.2
In the third column the first number is computed for b = 0.5 and the second one for b = 1.
during CE evolution. If the system survives, the CE is ejected from the system with the chemical composition it had at the start of the CE phase. 3.8.2.3. Supernovae. To compute the SN yields of massive interacting binary stars we proceed in
Table 7 The SNIa yields produced by a thermonuclearly exploding CO WD (Iwamoto, 1999) compared with the yields of a typical SNII Element
SNIa
SNII
12
4.83e2 1.16e6 1.43e1 2.02e3 8.50e3 1.54e1 8.46e2 1.19e2 6.26e1
7.93e2 1.56e3 1.80 2.12e1 8.83e2 1.05e1 3.84e2 5.77e3 8.44e2
C 14 N 16 O 20 Ne 24 Mg 28 Si 32 S 40 Ca 56 Fe
the same way as for single stars, i.e. we link our computed CO cores with the nucleosynthesis computations (model B) of Woosley and Weaver (1995). For the type Ia SNe we use the yields from the work of Iwamoto (1999) for their model W7, which is an updated version of the classical carbon deflagration W7 model (Nomoto et al., 1984; Thielemann et al., 1986). We use the same yields for different metallicities and different binary progenitors.
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Table 7 gives the SNIa yields of the major elements together with the yields of a typical SNII. It follows that core collapse SNe dominate the production of oxygen while the SNIas dominate the production of iron.
3.8.3. Yields from one stellar generation To compare the single star yields with the binary star yields we compute the relative chemical contribution from one generation of 100% single stars and from one of 100% binary stars. The quantity that expresses the relative contribution is R 120 Mp ðMÞuðMÞdM s ð3:8Þ y i ¼ 0:8R 120 i MuðMÞdM 0:8
3.8.2.4. Mergers. Our treatment of mergers was discussed earlier in Section 2.3.3 in Section 2. No detailed nucleosynthesis computations of binary mergers exist. We compute their yields as follows.
for the population of single stars and
q0 6 0.2 mergers: we take the yields of a single star with the same mass. q0 > 0.2 mergers: both stars are mixed in a homogeneous way. merging WD + non-degenerate secondary: if the resulting CO core mass is larger than 1.4 Mx we assume that the star explodes and take the SN yields for the corresponding CO core in the same way we do for massive stars. If the CO core is not massive enough to ignite carbon non-degenerately we assume that the layers outside the final CO core are ejected in a PN like SW and a WD is left.
y bi ¼
R 120 R 1 R 3650 0:8
0
Mpi ðMÞuðM 1 Þ/ðqÞPðP ÞdP dqdM 1 R 120 R 1 MuðM 1 Þ/ðqÞdqdM 1 0:8 0
P min
ð3:9Þ for the population of binary stars. In the latter formula, M is the total mass of the binary and Pmin is the minimum initial period that a binary must have to avoid contact of both stars on the ZAMS and Mpi(M) is the sum of the stellar yield of the primary and of the secondary. We take 0.8 Mx as the lower mass limit since less massive stars do not contribute (or in negligible amounts) to the chemical enrichment of the ISM. Though they must be included when doing the complete chemical evolution as a function of time since they lock up a non-negligible amount of matter.
3.8.2.5. Novae. We do not include the yields from nova explosions since their contribution to the enrichment in the elements that we investigate in this work is very small. They mainly produce isotopes like 7Li, 13C, 15N, 17O, etc. which are made during explosive H burning.
Table 8 The different parameter sets for which the total integrated yields are computed (ni = not included, i = included) Set
fb
/(q)
bmax
a1,
1 2 3 4 5 6 7 8 9 10 11 12
0 0 1 1 1 1 1 1 1 1 1 1
– – /F /H /G /F /F /F /F /F /F /F
– – 1 1 1 0.5 1 1 0.5 1 1 1
– – 1 1 1 1 0.5 1 0.5 1 1 1
2
a3
vk
M BH 0;s
M BH 0;b
SNIIWD
– – 1 1 1 1 1 0.5 0.5 1 1 1
– – 450 450 450 450 450 450 450 150 450 450
25 40 25 25 25 25 25 25 25 25 25 40
– – 40 40 40 40 40 40 40 40 40 40
ni ni ni ni ni ni ni ni ni ni i ni
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975
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Table 9 Integrated yields (relative to the total yields from the whole mass range) from the massa range [0.8, 9] Mx computed for the different elements and for different PNS sets Set
1 3 4 5 6 7 8 9 11
Element He
C
N
O
Ne
Mg
Si
S
Ca
Fe
8.03e3 7.67e3 2.73e3 3.17e3 1.46e3 1.86e3 3.33e3 3.78e3 4.64e3 5.14e3 3.26e3 3.81e3 2.73e3 3.17e3 5.16e3 5.80e3 2.85e3 3.77e3
1.08e3 5.42e4 2.90e4 2.72e4 1.33e4 1.28e4 3.54e4 3.32e4 4.05e4 4.07e4 3.32e4 3.23e4 2.90e4 2.72e4 4.54e4 4.58e4 3.05e4 3.43e4
3.88e4 8.68e4 2.32e4 3.56e4 1.68e4 2.28e4 2.62e4 4.14e4 3.18e4 5.11e4 2.56e4 3.95e4 2.32e4 3.55e4 3.43e4 5.53e4 2.34e4 3.86e4
1.05e4 3.98e5 1.29e4 1.81e4 3.69e5 6.50e5 1.70e4 2.38e4 2.26e4 4.10e4 1.51e4 2.59e4 1.29e4 1.81e4 2.49e4 4.89e4 1.54e4 3.80e4
– – 1.70e5 4.21e5 5.13e6 2.07e5 2.26e5 5.33e5 3.22e5 9.29e5 2.00e5 6.03e5 1.70e5 4.21e5 3.52e5 1.11e5 1.99e5 7.12e5
– – 6.13e6 1.15e5 1.98e6 6.11e6 8.12e6 1.43e5 1.21e5 2.71e5 7.44e6 1.75e5 6.13e6 1.15e5 1.34e5 3.31e5 7.40e6 1.97e5
– – 2.60e5 3.51e5 1.12e5 1.83e5 3.35e5 4.40e5 5.65e5 7.34e5 3.62e5 4.81e5 2.60e5 3.51e5 6.67e5 8.63e5 6.31e5 1.00e4
– – 1.89e5 2.04e5 1.11e5 9.93e6 2.33e5 2.59e5 4.22e5 4.76e5 2.76e5 3.05e5 1.89e5 2.04e5 5.10e5 5.76e5 3.93e5 7.16e5
– – 3.84e6 2.76e6 2.43e6 1.30e6 4.69e6 3.51e6 8.68e6 6.48e6 5.71e6 4.14e6 3.84e6 2.76e6 1.05e5 7.86e6 7.71e6 1.20e5
– – 2.10e4 1.31e4 1.49e4 9.07e5 2.51e4 1.56e4 4.89e4 3.21e4 3.23e4 2.10e4 2.10e4 1.31e4 6.03e4 4.00e4 3.56e4 2.07e4
In each row the first line gives the values computed for Z = 0.002 and the second line for Z = 0.02.
Tables 9 and 10 list y si and y bi as a function of metallicity for the different elements. The y bi values are computed for different sets of binary parameters to investigate their influence. The parameter sets are defined in Table 8. Set 11 includes the yields from exploding mergers of a WD with a MS or RG companion. To have a clearer view on the relative enrichment we have separated the contribution from stars in the mass range [0.8, 9] Mx from those in the mass range ]9, 120] Mx. We have not included the contribution from SNIas in order to measure their weight in the iron production. The integrated SNIa yields relative to the total initial mass of the stellar generation are given in Tables 11 and 12. Remark that the integrated yield is given by the SNIa rate multiplied with the ejecta which are taken the same for each SNIa, whether it is formed via the SD or DD scenario. However the SNIa rate depends on the adopted scenario and therefore the integrated yields are computed for both progenitor scenarios. For the SD scenario we took the metallicity dependent case. The model number refers to the corresponding binary parameter set.
3.8.3.1. Conclusions. 1.
2.
3.
Low and intermediate mass binaries enrich less in carbon than single stars do. This is due to the fact that the dredge-up phase on the TP-AGB (which is typical in single stars) is suppressed by the RLOF in most of the components of interacting binaries. Single stars of low and intermediate mass eject more helium and nitrogen than binaries in the same mass range do. Single stars of intermediate mass hardly produce a-elements, while a non-negligible amount can be synthesized in binaries of intermediate mass, i.e. accretion stars of intermediate mass that grow sufficiently in mass to burn non-degenerately carbon and oxygen and finally explode, exploding merger stars, type Ia SNe. Their relative contribution to the enrichment in a-elements depends on the adopted binary parameters.
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Table 10 Integrated yields (relative to the total yields from the whole mass range) from the massa range ]9, 120] Mx computed for the different elements and for different PNS sets Set
1 2 3 4 5 6 7 8 9 10 11 12
Element He
C
N
O
Ne
Mg
Si
S
Ca
Fe
1.46e2 1.32e2 1.91e2 1.45e2 9.81e3 2.04e2 1.16e2 2.06e2 9.32e3 2.05e2 1.05e2 2.31e2 9.88e3 2.02e2 9.77e3 2.03e2 1.05e2 2.29e2 9.57e3 2.06e2 9.81e3 2.02e2 1.45e2 2.02e2
1.41e3 5.30e3 1.70e3 5.39e3 1.16e3 5.93e3 1.33e3 6.24e3 1.12e3 5.86e3 1.26e3 6.52e3 1.17e3 6.00e3 1.16e3 5.92e3 1.26e3 6.57e3 1.28e3 5.97e3 1.17e3 5.94e3 1.48e3 6.02e3
1.04e4 7.16e4 1.15e4 7.44e4 7.38e5 6.36e4 8.76e5 6.92e4 7.00e5 6.25e4 8.89e5 7.51e4 7.34e5 6.43e4 7.32e5 6.33e4 8.78e5 7.54e4 7.44e5 6.36e4 7.39e5 6.35e4 8.50e5 6.61e4
4.19e3 6.07e3 8.53e3 8.12e3 3.44e3 5.59e3 4.28e3 6.08e3 3.21e3 5.47e3 4.16e3 6.03e3 3.46e3 5.71e3 3.44e3 5.58e3 4.17e3 6.13e3 3.97e3 5.70e3 3.45e3 5.66e3 7.98e3 6.99e3
5.28e4 7.30e4 1.16e3 2.15e3 3.19e4 7.59e4 4.37e4 7.56e4 2.84e4 7.69e4 3.99e4 7.45e4 3.22e4 7.60e4 3.19e4 7.56e4 4.01e4 7.43e4 3.19e4 7.41e4 3.20e4 7.68e4 9.57e4 1.76e3
1.47e4 1.12e4 3.83e4 1.52e4 7.75e5 9.64e5 1.18e4 1.01e4 6.56e5 9.56e5 1.06e4 9.10e5 7.83e5 9.42e5 7.74e5 9.59e5 1.06e4 8.82e5 7.67e5 9.30e5 7.79e5 9.84e5 3.27e4 1.24e4
6.37e4 6.79e4 8.50e4 9.55e4 4.44e4 4.90e4 5.91e4 5.60e4 4.02e4 4.73e4 5.67e4 5.41e4 4.47e4 5.09e4 4.44e4 4.89e4 5.70e4 5.57e4 4.41e4 4.75e4 4.46e4 5.02e4 6.82e4 6.74e4
3.43e4 3.61e4 4.27e4 4.94e4 2.29e4 2.09e4 3.03e4 2.56e4 2.08e4 1.96e4 2.93e4 2.50e4 2.31e4 2.23e4 2.29e4 2.08e4 2.94e4 2.63e4 2.28e4 2.02e4 2.30e4 2.15e4 3.26e4 2.97e4
5.20e5 4.23e5 6.21e5 5.53e5 3.34e5 1.99e5 4.35e5 2.66e5 3.05e5 1.80e5 4.06e5 2.65e5 3.36e5 2.19e5 3.33e5 1.98e5 4.08e5 2.83e5 3.31e5 1.91e5 3.35e5 2.05e5 4.50e5 2.85e5
1.39e3 8.14e4 1.51e3 8.91e4 8.99e4 6.35e4 9.88e4 6.44e4 8.80e4 6.37e4 8.12e4 5.28e4 9.10e4 6.18e4 8.96e4 6.33e4 8.21e4 5.09e4 8.87e4 6.25e4 9.06e4 6.48e4 1.04e3 6.85e4
In each row the first line gives the values computed for Z = 0.002 and the second line for Z = 0.02.
Table 11 Integrated yields (relative to the total yields from the whole mass range) from SNIas computed for the different elements, for different PNS sets and for the SD (metallicity dependent) scenario Set
3 4 5 6 7 8 9
Element C
N
O
Ne
Mg
Si
S
Ca
Fe
3.06e6 2.91e5 6.84e6 4.62e5 2.00e6 2.43e5 3.06e6 2.91e5 7.93e7 1.48e5 3.06e6 2.91e5 5.29e7 9.85e6
7.34e11 6.98e10 1.64e10 1.11e9 4.79e11 5.83e10 7.34e11 6.98e10 1.90e11 3.55e10 7.34e11 6.98e10 1.27e11 2.37e10
9.05e6 8.61e5 2.02e5 1.37e4 5.91e6 7.18e5 9.05e6 8.61e5 2.35e6 4.37e5 9.05e6 8.61e5 1.57e6 2.92e5
1.28e7 1.22e6 2.86e7 1.93e6 8.35e8 1.01e6 1.28e7 1.22e6 3.32e8 6.18e7 1.28e7 1.22e6 2.21e8 4.12e7
5.38e7 5.12e6 1.20e6 8.12e6 3.51e7 4.27e6 5.38e7 5.12e6 1.40e7 2.60e6 5.38e7 5.12e6 9.30e8 1.73e6
9.74e6 9.27e5 2.18e5 1.47e4 6.36e6 7.74e5 9.75e6 9.27e5 2.53e6 4.71e5 9.75e6 9.27e5 1.69e6 3.14e5
5.35e6 5.09e5 1.20e5 8.09e5 3.50e6 4.25e5 5.35e6 5.09e5 1.39e6 2.59e5 5.35e6 5.09e5 9.26e7 1.73e5
7.53e7 7.16e6 1.68e6 1.14e5 4.92e7 5.98e6 7.53e7 7.16e6 1.95e7 3.64e6 7.53e7 7.16e6 1.30e7 2.43e6
3.96e5 3.77e4 8.86e5 5.98e4 2.59e5 3.14e4 3.96e5 3.77e4 1.03e5 1.91e4 3.96e5 3.77e4 6.85e6 1.28e4
In each row the first line gives the values computed for Z = 0.002 and the second line for Z = 0.02.
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Table 12 Integrated yields (relative to the total yields from the whole mass range) from SNIas computed for the different elements, for different PNS sets and for the DD scenario Set
3 4 5 6 7 8 9
Element C
N
O
Ne
Mg
Si
S
Ca
Fe
6.26e5 4.07e5 1.86e5 1.30e5 8.19e5 5.28e5 2.76e5 2.09e5 5.87e5 3.80e5 8.56e5 6.81e6 8.69e6 6.17e6
1.50e9 9.77e10 4.48e10 3.11e10 1.97e9 1.27e9 6.63e10 5.01e10 1.41e9 9.13e10 2.05e9 1.64e9 2.09e10 1.48e10
1.85e4 1.20e4 5.52e5 3.84e5 2.43e4 1.56e4 8.17e5 6.17e5 1.74e4 1.13e4 2.53e4 2.02e4 2.57e5 1.83e5
2.62e6 1.70e6 7.79e7 5.42e7 3.43e6 2.21e6 1.15e6 8.72e7 2.45e6 1.59e6 3.58e6 2.85e6 3.63e7 2.58e7
1.10e5 7.16e6 3.28e6 2.28e6 1.44e5 9.29e6 4.86e6 3.67e6 1.03e5 6.69e6 1.51e5 1.20e5 1.53e6 1.09e6
2.00e4 1.30e4 5.94e5 4.13e5 2.61e4 1.68e4 8.80e5 6.65e5 1.87e4 1.21e4 2.73e4 2.17e4 2.77e5 1.97e5
1.10e4 7.12e5 3.26e5 2.27e5 1.43e4 9.25e5 4.83e5 3.65e5 1.03e4 6.66e5 1.50e4 1.19e4 1.52e5 1.08e5
1.54e5 1.00e5 4.59e6 3.19e6 2.02e5 1.30e5 6.80e6 5.14e6 1.45e5 9.37e6 2.11e5 1.68e5 2.14e6 1.52e6
8.11e4 5.27e4 2.42e4 1.68e4 1.06e3 6.85e4 3.58e4 2.70e4 7.61e4 4.93e4 1.11e3 8.83e4 1.13e4 7.99e5
In each row the first line gives the values computed for Z = 0.002 and the second line for Z = 0.02.
4.
5.
6.
7.
8.
Due to RLOF the overall He enrichment by massive binaries is larger than it is by massive single stars. Massive stars that perform RLOF develop smaller helium and CO cores than single stars of the same initial mass, which results in a lower production of a-elements. However due to mass accretion, secondary stars may form larger cores and thus synthesize larger amounts of metals than expected from their initial mass before accretion. The variations caused in the enrichment when different binary parameter values are adopted are generally within a factor of 2 except in the a-element enrichment by close binaries of low and intermediate mass, where they may increase up to a factor of 5. The limit on the initial mass of a massive single star to form a BH without a prior SN outburst significantly affects the a-element production, which is illustrated by set 1 and 2. In summary, except for the a-element enrichment in the intermediate mass range, the overall differences between the integrated single and binary star yields are within a factor of 2 and 3.
4. The chemical evolution of the solar neighbourhood In this last section we use our CEM to study the time evolution of the SN rates, the merger rate of double NS/BH systems and of the abundance ratios [X/Fe] in the solar neighbourhood, the region for which most observational data is available. We first repeat the main inputs of our CEM: Assumptions
infall of primordial gas, instantaneous mixing, delayed recycling approximation, continuous SFR with threshold, SFR efficiency: ~mI ¼ 2:0 and ~mII ¼ 1:0, SFR exponent: kI = 1.5 and kII = 1.5, two-component disk: tI = 1.0 Gyr and tII = 8.0 Gyr, time of max. accretion onto disk: tmax = 2.0 Gyr, position of the Sun: rx = 10 kpc, birth of the Sun: tx = 10.5 Gyr, age of the Milky Way: tnow = 15 Gyr. Stellar evolution:
single stars: M0 = [0.1, 120] Mx,
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binary stars: M1,0 = ]1, 120] Mx; q0 > 0; 0 < P0 6 10 yr, IMF: / Mx, x = 2.35 (M0 < 2) and = 2.7 (M0 P 2), SNIa progenitor model: SD or DD, binary frequency: constant (fb,1) or linear with Z (fb,2). Since our main aim is to study the importance of binaries in chemical evolution modeling and because of the large number of parameters, we make our simulations for fixed values of the parameters that directly involve the model for star and Galaxy formation. This means that all our simulations are made with: ~mI ¼ 2:0; ~mII ¼ 1:0; k I ¼ 1:5; k II ¼ 1:5; tI ¼ 1:0 Gyr; tII ¼ 8:0 Gyr; tmax ¼ 2:0 Gyr; uðMÞ / M x with x ¼ 2:35 ðM 0 < 2Þ and ¼ 2:7 ðM 0 P 2Þ. We remind again that this value set has been retained by Chiappini et al. (1997) as the model that gives the best fit with the observations in the solar neighbourhood, despite their computations being made without detailed binary evolution. Notice that the adopted age of the Galaxy may be too large accounting for more recent age determinations of the Universe (about 13.7 Gyr) which implies that the Galaxy is probably not older than 12–13 Gyr. However, it is easy to understand that
for the main conclusions of this section, the uncertainty in the real age of the Milky Way is not very important. We therefore preferred to retain the parameter set of Chiappini et al. (1997). For the stellar evolutionary parameters we define the following standard PNS model which has been proven to reproduce well the observed OBtype and WR star population: the SW mass loss pffiffiffi rates of RSGs are metallicity dependent ð/ Z Þ; although the observations of the present population of massive stars in the Galaxy and in the Magellanic Clouds is better reproduced with a PNS model where the WR SW is Z-dependent (Section 2.5), we consider both a Z-independent WR SW and a Z-dependent WR SW during CHeB to illustrate its effect on the abundances evolution (if not explicitly mentioned the WR SW is taken Z-independent),
a Hogeveen type mass ratio distribution, bmax = 1, maximum CE ejection efficiency: a = 1, average kick velocity: vk ¼ 450 km=s, maximum NS mass: M NS max ¼ 2 M , mass limit for direct BH formation: BH M BH 0;s ¼ 25 M and M 0;b ¼ 40 M .
Fig. 17. The star formation rate in the solar neighbourhood as a function of time, predicted by the standard PNS model with a constant binary fraction of 70%. The SFRs predicted with other PNS models and binary fractions hardly differ (within a few percent) from the one presented here.
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If the value of a parameter in a simulation is not explicitly mentioned, then its value in the standard model is assumed. For the SD scenario we consider two different cases i.e. the WD wind is either metallicity independent (=SD1) or metallicity dependent (=SD2) (see Section 2.3.3 and Kobayashi et al., 1998). 4.1. Star formation rate and surface densities Fig. 17 shows the time evolution of the SFR in the solar neighbourhood predicted by the standard PNS model combined with a constant binary frequency of 70% throughout the galactic evolution. The shape is typical for the two-infall model: a first short and very active period of star formation (= the halo-thick disk phase) followed by a long per-
915
iod of moderate and after some time constant level of star formation (= the thin disk phase). The required minimum density for star formation to start is reached after 0.6 Gyr. Due to the included star formation threshold we find that star formation especially during the last 3 Gyr of the thin disk evolution alternately goes on (for a period of 6 Myr) and off (for a period of 4 Myr). During the period of non-activity the dying stars replenish the ISM with gas and again increase the surface gas density. We remark that not all points of zero star formation during galactic evolution are shown (due to a limited capacity of plotting points in the adopted software). The predicted values for the present-day gas fraction, the SFR and surface densities are within the observed value ranges and hardly depend on
Table 13 Predicted and observed present values in the solar neighbourhood Quantity
Predicted
Observed
References
W(Mx pc2 Gyr1) rg(Mx pc2) rs(Mx pc2) rg/r r_ inf ðM pc2 Gyr1 Þ
2.54 7.01 45.48 0.14 1.08
2–5 7–13 30–40 0.05–0.20 0.3–1.5
Gu¨sten and Mezger (1982) Dickey (1993), Flynn et al. (1999) Gilmore et al. (1989) Portinari et al. (1998)
Fig. 18. The time evolution of the surface densities of the total mass (r), visible stars (rs) and of the gas (rg) predicted by the standard PNS model and a constant binary fraction of 70%. The predicted values with other PNS models and binary fractions are within a few percent.
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Fig. 19. Evolution of the SFR during the last Gyr of galactic evolution.
the stellar evolutionary parameters. Their values are given in Table 13 and their time evolution is shown in Fig. 18. Main conclusion: The influence of binaries on the star formation rate is insignificant even when adopting a constant high binary frequency. 4.2. Intermezzo: the WR/O number ratio It is obvious that a massive star formation rate that fluctuates on a timescale shorter than the evolutionary timescale of a massive star, may have important consequences on the temporal variation of the O-type and WR star populations in the solar neighbourhood. Fig. 19 shows a blow up of the predicted SFR during the last Gyr of galactic evolution. The corresponding predicted evolution of the WR/O number ratio (computed with the standard PNS model and a constant binary fraction of 70%) is given in Fig. 20. We observe an overall fluctuating behaviour of the ratio with an amplitude of a factor of 10 in a time interval of at most a few million years. This result implies that although it is easy to recover the observed WR/O ratio, the strongly fluctuation makes that a comparison between observations and theoretical predictions in order to test stellar evolutionary models, is ambiguous. The evolution of the number ratios WC/WN and (WR + OB)/WR are much less affected.
Remark Because we use a uniform density model, abrupt star formation stops occur when the average surface gas density is smaller than the adopted threshold value. However, in reality the surface gas density may be locally larger than the threshold value and it can be expected that the transition from high to low star formation occurs much smoother. Furthermore, the way how the Kennicut condition has been implemented in the model of star formation used in the present work can be questioned. Maybe the path to follow is to incorporate this Kennicut condition into the star formation rate itself, for instance modulating its efficiency in such a way that it should get lower and lower as the gas density approaches the Kennicut regime. Both effects discussed above should smooth the temporal behaviour of the WR/O number ratio but not extinguish the fluctuations on a timescale which is of the order of the evolutionary timescale of a massive star. 4.3. The age–metallicity relation Stellar abundance determinations are usually discussed in terms of the element abundance relative to iron [X/Fe] as a function of the ironto-hydrogen ratio [Fe/H]. This is because [Fe/H] is easy to measure in stars and can be considered as a cosmic clock since the accumulation of iron
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Fig. 20. The predicted time evolution of the WR/O number ratio computed for the standard PNS model and with a constant binary fraction of 70%. The computations shown in the upper (lower) figure are made with M WR min ¼ 5 M ð8 M Þ.
in the ISM increases monotonically with time (Wheeler et al., 1989). The evolution of [Fe/H] as a function of time forms the basis of the age–metallicity relation 9 (AMR) which is an important relation to be repro-
9 As earlier defined in Section 1, the metallicity is the mass fraction of all elements heavier than helium denoted by Z. However this is not always practical for observers because information usually does not exist for all the elements. Therefore in observational astronomy, the word metallicity often refers to the iron abundance.
duced by CEMs. It links the variety of the physical timescales involved like the infall rate, star formation rate and stellar lifetimes. Because all the nucleosynthesized iron is ejected during SN explosions, the evolutionary behaviour of [Fe/H] is directly related to the time evolution of the SN rates. Various methods have been used to obtain the AMR from the observations. A comparative discussion on these methods is given by Carraro et al. (1998). Edvardsson et al. (1993) derived the AMR of 189 nearby field F- and G-type disk dwarfs with very accurately determined abundances. The trend is a decreasing metallicity with
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increasing stellar ages and with a considerable scatter about the mean value that is substantially larger than the observational error. Subsequent studies of the AMR in the disk (e.g. Rocha-Pinto et al., 2000; Feltzing et al., 1989; Ibukiyama and Arimoto, 2002) have confirmed the presence of the large scatter which is believed to be physically real and may originate from processes like star formation by sporadic gas infall, local inhomogeneous mixing, orbital diffusion, etc. (e.g. Edvardsson et al., 1993; Matteucci, 1996; Pagel and Tautvasˇien_e, 1995). Very recently the AMR in the disk has been studied by Nordstro¨m et al. (2004) with new data from the large Geneva– Copenhagen survey of the solar neighbourhood for a sample of 16682 local F and G dwarfs. In their sample a reasonable number of old metal-rich stars is present which were not included in the study of Edvardsson et al. (1993). The resulting mean metallicity is more or less constant at all ages oppositely to the average AMR of Edvardsson et al. (1993) that declines at stellar ages >10 Gyr. However it should be realised that the derived ages for these old metal-rich stars are the most uncertain (Feltzing et al., 1989). Since we assume homogeneous chemical evolution in our model, we can only compute a mean relation without any scatter. This means that although the AMR does not constitute a very tight constraint on our model predictions due to the large observational dispersion, our model predictions should fit the average run of the observed data. We computed the AMR for a number of models defined in Table 14. They illustrate the effects of binaries. Model 1 corresponds to the situation in
Table 14 The different models for which the AMR has been computed Model
fb
PNS model
SNIa
Fe yields
1 2(3) 4(5) 6(7) 8(9) 10(11) 12(13) 14(15) 16(17)
0 fb,1(fb,2) fb,1(fb,2) fb,1(fb,2) fb,1(fb,2) fb,1(fb,2) fb,1(fb,2) fb,1(fb,2) fb,1(fb,2)
stand. stand. stand. stand. stand. but UF stand. stand. stand. stand.
– SD1 SD2 DD DD SD1 SD2 DD SD1
WW95 WW95 WW95 WW95 WW95 WW95:2 WW95:2 WW95:2 WW95
which only single stars are formed and thus no SNIas. In models 16 and 17 only the binary progenitors of the SNIas are included together with single stars. These latter models correspond with the models commonly adopted in CEMs but differ by the treatment of the SNIas, i.e. we compute the SNIa rate by a detailed modeling of the evolution of the SNIa binary progenitors instead of including the SNIa rate as a free parameter that is finally fixed to obtain the observed solar iron abundance. To account for the uncertainty in the iron corecollapse yields (which is at least a factor of two) computed by Woosley and Weaver (1995), Timmes et al. (1995) made some chemical evolutionary simulations with the iron core-collapse yields reduced by a factor of two and found a general better agreement with the observed evolution of the abundance ratios. Lower iron core-collapse yields are also supported by observational estimates of the iron synthesized in SN 1987A (= explosion of a 20 Mx with Z 0.1Zx in the LMC) and in SN 1993J (=explosion of a 14 Mx in the galaxy M81 where Z Zx). Therefore we also consider models with the iron yields from Woosley and Weaver (1995) reduced by a factor of two (indicated by WW95:2). The predicted AMR is shown in Figs. 21 and 22 together with the observational data points from the work of Edvardsson et al. (1993). The error bars (shown on the figures in the lower right corner) give the typical mean measurement errors on the observed data. All the predictions show an overall trend of a fast increase during the first 2– 3 Gyr and much more slowly afterwards. Compared to the observations of Edvardsson et al. (1993) we are inclined to conclude that the predicted mean relations fit better the observed average trend when the reduced iron core-collapse yields are implemented, confirming the conclusions of Timmes et al. (1995). However, in order to reproduce an average [Fe/ H] ratio above 0.3 at the beginning of disk formation in correspondence with the observations of Nordstro¨m et al. (2004), the models with the high iron-core collapse yields are more satisfactory. It is obvious that the contribution of SNIas to the iron enrichment is non-negligible. Their relative weight in the total iron production depends
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Fig. 21. The theoretically predicted AMR for the models 1, 3, 5, 7, 9, 11, 13, 15, 17. The error bars give the typical measurement errors on the observed data.
on the adopted iron core-collapse yields and the SNIa rate which on its turn depends on the adopted progenitor scenario and on the adopted values for the binary PNS parameters like the mass ratio distribution, common envelope ejection efficiency ai and the RLOF accretion parameter bmax. If the reduced iron core-collapse yields apply it follows that SN Ias are responsible for at least 50% of all the iron of the Galaxy. It is clear that SNIas play an important role in the iron evolution and therefore their formation rates must be computed in a correct way which requires the use of a PNS
model. We remark that the AMR predicted with a constant binary frequency in combination with the Z-dependent SD model shows a similar behaviour to the one computed with the Z-independent SD model and a binary frequency that increases linearly with Z which is illustrated by models 3 and 4. When we compare our model predictions with binaries to those without binaries but with SNIas (i.e. models 2 and 3 vs. models 16 and 17) we conclude that binaries significantly affect the iron evolution mainly through SNIas.
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Fig. 22. The theoretically predicted AMR for the models 2, 4, 6, 8, 10, 12, 14, 16. The error bars give the typical measurement errors on the observed data.
The low [a/Fe] ratios (solar values) observed in damped Lyman a (DLA) systems 10 at high redshift with [Fe/H] < 1, suggest that SNIas already form at low metallicity since they eject sufficient amounts of iron and therefore prevent large over-
10 Damped Lyman a systems are high column density (N(HI) P 2 · 1020 cm2) absorbers detected in the optical spectra of quasars up to relatively high redshift (up to z 5). They have metal abundances that range from 102Zx to 1/ 3Zx.
abundances of a-elements relative to iron (compared to the solar value). The moment during galactic evolution at which SNIas start to form depends on the progenitor scenario. In case of the metallicity dependent SD scenario, the first SNIas only appear around [Fe/H] > 1.1 and it would be difficult to explain the observed low [a/ Fe] ratios in DLAs at high redshift. However, in the DD scenario Type Ia SNe already appear 40 Myr after the beginning of star formation (the characteristic timescales on which SNIas are formed in the different scenarios are discussed in
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Fig. 23. The predicted AMR computed with model 12 (=green line), with model 14 (=blue line) and both models combined (=red line). The error bars give the typical measurement errors on the observed data.
Section 4.8.3) and affect the [a/Fe] ratios at metallicities lower than 1.0 dex. When the SD scenario is taken independent of the metallicity SNIas also
form at lower metallicities though their maximum rate occurs 1 Gyr after the beginning of star formation and therefore largely affect the [a/Fe] ratio
Fig. 24. The theoretically predicted G-type dwarf metallicity distribution in the solar neighbourhood for the models 2–9. The observationally derived histograms are from Wyse and Gilmore (1995) (bold line) and from Rocha-Pinto and Maciel (1996) (thin line).
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around [Fe/H] > 1. From these considerations it follows that if SNIas are indeed responsible for the observed low [a/Fe] ratios, they are mainly produced by merging CO WDs, especially at low metallicity. However, this does not imply that no SNIas or only a small fraction forms via the SD scenario. As we point out later on in Section 4.8.4 where we compute the SNII, SNIbc and SNIa rates, the high Ia/Ibc supernova number ratio (1.6) observed in early and late type spirals is difficult to explain with one of the two progenitor scenarios but can be reproduced when SNIas are formed via both the DD and SD scenario. Therefore we also computed the AMR with a model in which SNIas are formed by the DD and the SD scenario. The results are given in Fig. 23 for the combination of model 12 with model 14. The main effect is obviously that the predicted SNIa rate increases and we conclude: if SNIas are formed by both the single degenerate and the double degenerate binary scenario, they are responsible for at least 70% of all the iron in the Milky Way.
4.4. The G-dwarf disk metallicity distribution Stars of spectral type G have main-sequence lifetimes of the order of the age of the Milky Way (10–15 Gyr) and many of them that have ever been formed still exist, even those formed when the Galaxy was very young. Therefore these stars provide information on the chemical evolutionary history from the early phases until now. Latest compilations of the metallicity of G-dwarfs in the thin disk have been performed by Wyse and Gilmore (1995) and Rocha-Pinto and Maciel (1996). The results of both groups are in good agreement and show a narrow distribution with a prominent peak around [Fe/H] 0.2. Figs. 24 and 25 show the G-type dwarf distribution predicted by the different models defined in the previous section in Table 14, together with the observed distribution from Wyse and Gilmore (1995) and Rocha-Pinto and Maciel (1996). The model prediction with both SNIa scenarios together is given in Fig. 26.
Fig. 25. The same as Fig. 24 but for the models 10–17.
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Fig. 26. The predicted G-dwarf disk metallicity distribution computed with model 12 (from Table 17) (= bold black line), with model 14 but (= dashed black line) and both models 11 and 15 combined (= red line).
When comparing the predicted distributions with the observed ones, we find that the G-dwarf metallicity distribution depends sensitively on the evolution of the iron enrichment and thus on the adopted SNIa binary model and core-collapse SN iron yields. With the iron core-collapse yields of Woosley and Weaver (1995) we find an overall underproduction of G-dwarfs with [Fe/H] < 0.5 with in some cases a compensating overproduction at [Fe/H]-values >0 which is due to a high SNIa rate already in the beginning of galactic evolution causing a very fast iron evolution. A better agreement (based on a statistical v2 test) with the observed distributions is obtained when the reduced iron core-collapse yields are used and a constant binary frequency (resp. linearly increasing binary frequency) is combined with the metallicity dependent SD or DD model (resp. metallicity independent SD model). The best fit results from the combination of model 12 and 14 i.e. when both the DD and SD scenario produce SNIas. The model predictions that only include single stars and SNIas (= models 16 and 17) show that except for the SNIas, binaries do not much affect the Gdwarf disk metallicity distribution. 4.5. The solar abundances The solar abundances are determined from photospheric and meteoretic analysis. For most
of the elements they agree within 0.2 dex. Table 15 lists the observed data from Anders and Grevesse (1989) updated by Grevesse and Sauval (1998) for the isotopes that are considered in this work. The uncertainties on the derived abundances (by mass) are within 10%. Hydrogen (1H) and helium (4He) represent 98% of all the mass with the remnant 2% contained in metals, of which half is in the form of 16O. In Fig. 27 we show the calculated mass fractions (= computed/observed value) for the considered isotopes at the moment of the birth of the Sun in our model. Since we assume that the Milky Way is 15 Gyr old (remind however that this number is not essential and using 13 Gyr would lead to very similar conclusions), the Sun was born 10.5 Gyr after the Big Bang. Given the uncertainties in the observed values (while isotopic ratios for most elements are known precisely, elemental abundances are less certain) and uncertainties in the theoretical values (as a consequence of uncertain physics in the models) we can consider that our model predictions are in agreement with the observed value within a factor of two difference. 11
11 It is a daunting task to make a formal analysis of the total uncertainty and therefore the convention that isotopes falling within a factor of 2 of their solar value are successes, is commonly adopted.
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Table 15 The observed data (Grevesse and Sauval, 1998) together with the predicted values at t = tx (second column) and at t = tnow (third BH column) for model 2 with M BH 0;s ¼ M 0;b ¼ 40 M Element
Nucleon fraction (Xi) Observation
Theory (t = tx)
Theory (t = tnow)
H He 12 C 14 N 16 O 20 Ne 24 Mg 28 Si 32 S 40 Ca 56 Fe
7.06e1 2.75e1 3.03e3 1.11e3 9.59e3 1.55e3 5.13e4 6.53e4 3.96e4 5.99e5 1.17e3
7.11e1 2.72e1 4.57e3 9.24e4 7.48e3 1.21e3 2.90e4 7.88e4 4.22e4 6.51e5 1.35e3
7.07e1 2.75e1 4.75e3 1.06e3 8.10e3 1.40e3 3.27e4 8.94e4 4.86e4 7.45e5 1.62e3
Z
1.89e2
1.71e2
1.88e2
1 4
Fig. 27. The theoretically predicted solar abundances relative to the observed solar values (=Xi/Xi,x) as predicted by model 2 with BH M BH 0;s ¼ 25 M (filled circles), with M 0;s ¼ 40 M (open circles) and by model 17 (crosses). The red dot gives the iron mass fraction when the iron core-collapse yields are reduced by a factor of two (= model 10).
This factor of two difference is denoted in Fig. 27 by the horizontal dotted lines. The mass fractions are nearly equal to unity for H and He, independent of the adopted model. For carbon we find oversolar values which is due to the high C production by low and intermediate mass stars between 1 and 4 Mx. The overabundance is smaller when a large binary fraction is adopted, because the majority of interacting binaries do
not undergo (due to previous mass transfer) the dredge-up phase at the AGB during which large amounts of carbon are brought up to the surface and ejected. The predicted solar abundances of the elements O, Ne and Mg depend sensitively on the adopted mass limit for BH formation without a SN outburst. If direct BH formation for single stars already starts at 25 Mx less O, Ne and Mg is ejected because the final CO core is swal-
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lowed by the BH, which results in computed solar values that are lower than observed (given by the filled dots in Fig. 27). The effect of BH formation is much smaller on the predicted abundances of the elements Si, S, Ca and Fe since the bulk production of these elements comes from stars <30 Mx. For the iron mass fraction the best result is obtained when the lower iron core-collapse yields (i.e. Fe(WW95):2) are used in combination with model 2 (= model 10) which is given in Fig. 27 by the red dot. Higher Ne solar abundances are predicted when large binary fractions are adopted. While among the single stars Ne is almost exclusively produced by the massive ones, intermediate mass stars in binaries may also contribute to the Ne enrichment when they become massive stars after accreting matter from or merging with the companion star. Magnesium is underproduced compared to its observed value, which is a well known result in chemical evolutionary models that implement the Mg core-collapse yields from Woosley and Weaver (1995). The underestimation of Mg has been discussed many times in literature (e.g. Timmes et al., 1995; Thomas et al., 1998). One of the arguments is that uncertainties in SN nucleosynthesis computations still allow for rather different yields among different authors, especially in the case of 24Mg production. For example the 24 Mg yields from Thielemann et al. (1996) are sensitively higher than those from Woosley and Weaver (1995) and are found to better reproduce the observed solar value (e.g. Thomas et al., 1998; Chiappini et al., 1999). Our computations with the other models reveal similar results, all predicting isotopic solar abundances that fall within a factor of two of their solar value. Table 15 lists the predicted abundances at the moment of the birth of the Sun in our model and for the adopted present age of the Milky Way. We conclude that since the formation of the Sun, the chemical evolution of the solar neighbourhood has been very slow, which is in good agreement with the observed constancy of the oxygen abundance in the last 4.5 Gyr (e.g. Moos et al., 2002; Sofia and Meyer, 2001; Peimbert, 1999). The slow chemical enrichment is mainly due to the included threshold in the star formation rate, which shows an oscillatory behaviour during the latest
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phases of galactic evolution. We notice that for other values of bmax ; ai ; vk , than adopted in the standard PNS model, the predicted solar values do not change much, as could be expected from our PNS parameter study on the binary yields in the previous section (Section 3.8.3). When the special class of type II SNe (=SNIIWD) which are produced by merging WD + MS/RG binaries are included, the a-element abundances are increased, though the results stay within a factor of two of their observed solar value. However, the nature of these merger objects is still speculative and it is too early to draw any confirmative conclusion on their effects on galactic chemical evolution. As a main conclusion on the effect of binaries, we find that except for the differences in carbon and neon, a model with single stars only but with a correct treatment of the SNIa population predicts similar solar abundances compared to a model in which the effects of all binaries are included in detail. We also keep in mind that the enrichment in O, Ne and Mg is affected by the mass limit for direct BH formation. 4.6. The abundance ratios evolution In this section we consider abundance ratios rather than absolute abundances, because they depend less on the galactic model parameters but mainly on the nucleosynthesis yields. The evolution of the abundance ratios [X/Fe] as a function of metallicity is often considered as a diagnostic of the IMF and SFR parameters and the timescale for the chemical evolution of stellar systems. Extended discussions and reviews on observed abundance ratios have been given by e.g. Wheeler et al. (1989), Timmes et al. (1995) and McWilliam (1997). The most abundant elements provide important clues for understanding stellar and galactic evolution. For example, observations of the CNO abundances in main-sequence stars are valuable for finding what types of stars have been responsible for the CNO nucleosynthesis during the different galactic evolutionary phases. We compute the time evolution of the abundance ratios [C/ Fe], [N/Fe], [O/Fe], [Mg/Fe], [Si/Fe], [S/Fe] and [Ca/Fe] as a function of [Fe/H] and compare them with observational data of spectroscopic abundances in nearby stars. There are no observational
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data for neon. Elemental neon only has optical transitions from very high excitation levels, and apparently there are no stellar abundance determinations in either dwarf or field giants (except for the Sun). We note that in the computation of the [X/Fe] ratio we use the observed solar abundance ratio (X/Fe)x and not the one predicted by our model. In view of the effect of direct BH formation on the predicted solar abundances as discussed in the previous section, we take M BH 0;s ¼ 40 M for the computation of the abundance ratios. Before discussing our results, the following general comment on the observations must be kept in mind. Examinations of elemental abundance ratios as a function of [Fe/H] show that star-to-star differences exist and rise with decreasing metallicity (e.g. Ryan et al., 1996). Especially at low metallicity the observational spread of abundance ratios is large and unlikely to be only the result of the increased uncertainty in the estimates of elemental abundances in metal-poor stars. Therefore the spread at low metallicity is strongly believed to be physically real. At present the following scenario is generally considered to explain the observed spread. During the beginning of galactic evolution (i.e. halo phase) the enrichment is determined by the number of exploding massive stars and the efficiency by which the ejecta are mixed with the surrounding ISM. If the ejected metals are distributed over a large volume, a spatially homogeneous enrichment takes place. If on the contrary only a small volume is contaminated, the ISM in the vicinity of the SN explosion becomes highly enriched while the ISM further out remains metal-poor. In this case the ISM evolves inhomogeneously whereby new stars at a given galactic age may form with a different chemical composition, depending on where they form. This scenario has recently been explored in detail by Argast et al. (2000, 2002, 2004). By applying a stochastic formation model for the halo they computed the early chemical enrichment of the ISM of the halo. They find that the halo ISM at [Fe/H] > 3.0 is unmixed and dominated by local inhomogeneities caused by individual SN explosions. For [Fe/ H] > 3.0 the ISM becomes much better mixed resulting into a scatter in the abundance ratios
that decreases with increasing [Fe/H] values, in agreement with the behaviour of the observed scatter. Concerning the magnitude of the scatter, their model cannot account for the observed scatter of some elements like O and Mg which however may be due to uncertainties in the adopted nucleosynthesis yields. In order to quantify the observed scatter in the measured elemental abundances and to allow a meaningful comparison between observations and model predictions that do not account for inhomogeneous mixing, Chiappini et al. (1999) applied a statistical method (see Ryan et al., 1996; Cleveland and Kleiner, 1975 for further details) to a large consistent set of observational data. They computed the midmean line which gives the average of all data between the quartiles of the distribution of relative abundances over a given [Fe/ H] interval; the lower semi-midmean which is the midmean of all observations below the median and finally the upper semi-midmean corresponding to the midmean of all observations above the median. If the scatter around the midmean is Gausssian, the semi-midmeans are an estimate of the true quartiles. As has been checked by Chiappini et al. (1999) this last assumption seems to be globally valid for each of the abundance ratios that are considered in their dataset. The computed summary lines from the work of Chiappini et al. (1999) are shown in Figs. 28 and 29 (right panels) together with the observational data they used for their statistical analysis (left panels). Because we consider in this work the same abundance ratios and almost the same data sources (except for the addition of recent data on oxygen abundances) we use the presented summary lines as a reference for comparing our model predictions with the best fit to the observed data. 4.6.1. The CNO elements Carbon is produced as a primary element by the triple-a process in stars over the whole mass range. Observations of carbon abundances in halo and disk dwarfs show that [C/Fe] is almost constant in time, with a value approximately solar (Laird, 1985; Carbon et al., 1987; Tomkin et al., 1992), although there is a significant dispersion in the data at all metallicities. More recently, Gustafsson et al.
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Fig. 28. Abundance ratios of the elements O, Mg and Ca to Fe vs. [Fe/H] from the work of Chiappini et al. (1999). The left panels show the observed data from different sources but all normalized to the same solar abundances (Anders and Grevesse, 1989). The right panels give the midmean line (solid line) and the lower and upper semi-midmean lines (dotted lines) resulting from the statistical analyses of the observed data set.
(1999) found a slow decrease of [C/Fe] with increasing [Fe/H], with an overall slope of 0.17 ± 0.03 dex. The observed and predicted evolution of [C/Fe] as a function of [Fe/H] is given in Fig. 30. The model predictions show a decline of the [C/ Fe] ratio till the end of the halo, thick disk phase ([Fe/H] 1) followed by an increment that produces a bump in the interval 1 < [Fe/H] < 0. The slow decline during the halo, thick disk epoch indicates that intermediate mass stars that evolve during the late halo phase do not contribute significantly to the carbon enrichment. The carbon bump that appears during the thin disk epoch is created by intermediate and in particular low mass stars (<3 Mx) that start to eject large amounts of carbon and no iron. Because type Ia SNe which produce large amounts of iron and few carbon also start to appear around the same epoch, the strong increase in carbon enrichment is finally compen-
sated and the [C/Fe] ratio decreases to the solar value. This general behaviour corresponds with the trend line that gives the best fit to the observations (as given in Fig. 29), though the carbon bump is less pronounced than predicted by the models with a low binary fraction during the halo, thick disk phase (= models 3, 7, 9, 11 and 17 in Fig. 30). The bump is smaller for the model predictions with a high constant binary fraction (= models 2, 4, 6, 8 and 10 in Fig. 30). This effect is explained by the fact that the majority of low and intermediate mass interacting binaries do not evolve up to the TP-AGB during which, in single star evolution, by dredge-up processes large amounts of carbon are mixed into the outer stellar layers and finally ejected. However the C yields of low and intermediate mass stars may depend sensitively on the treatment of physical processes like hot bottom burning which are not yet well known. When the lower iron core-collapse yields are adopted the
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Fig. 29. The same as Fig. 28 but for the elements C, N, Si and S.
carbon bump is shifted to lower [Fe/H] values which is in better agreement with the observed mean trend. Reducing the iron core-collapse results in a lower average [Fe/H] evolution with time which is clearly seen in the predictions of the AMR with models that use the WW95 iron yields reduced with a factor of two (e.g. model 11 in Fig. 22). As a major effect of binary evolution on the [C/ Fe] relation, we find that models with a constant high binary fraction predict a flatter [C/Fe] evolution during the disk phase compared to model predictions that except for SNIas do not include binaries. The trend of the best fit to the observed data indicates an almost constant [C/Fe] evolution during the disk phase and therefore models with binaries better fit the average data run. Nitrogen is made from 12C and 16O during CNO-burning, which takes place in all stars with mass >1 Mx. Except for some primary production in AGB stars, 14N is mostly produced as a secondary element. Observational surveys show that despite the large scatter, [N/Fe] remains more or
less constant (comparable with the solar value) down to [Fe/H] 2 (Laird, 1985; Carbon et al., 1987; Israelian et al., 2000) indicating the production of primary nitrogen over this metallicity range. Primary nitrogen production by massive stars at low metallicity is not predicted by current evolutionary models of massive stars, which is a well-known problem in CEMs. Timmes et al. (1995) obtained primary nitrogen from metal poor massive stars with an initial mass P30 Mx by considering enhanced convective overshooting into the hydrogen shell from the helium convective burning shell. Synthesis of primary nitrogen by the injection of protons into helium burning zones could also happen via rotation (Meynet and Maeder, 2002). We propose here a third mechanism via binary evolution. In a zero-metallicity close binary with a massive primary, the secondary star may become enriched in carbon by accreting material that has been ejected during the SN explosion of the primary star. Test calculations show that through thermohaline mixing, the envelope of the accretion
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Fig. 30. The predicted and observed [C/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Laird (1985), Gratton and Ortolani (1986), Tamkin et al. (1986), Carbon et al. (1987) and Carretta et al. (2000).
star may become strongly enriched in carbon that will be converted partly into nitrogen during hydrogen shell burning. This scenario was also suggested by Vanbeveren (1994) to explain OBC stars. Detailed stellar evolutionary computations where this effect is included have not yet been made, but we would like to suggest that the enrichment of the ISM in primary nitrogen during early galactic evolution may be caused by massive close binary evolution. The predicted evolution of [N/Fe] as a function of [Fe/H] is displayed in Fig. 31. Our results give
a rapid increase of [N/Fe] at low metallicity, which clearly shows that metal poor stars do not produce primary nitrogen. The ratio steadily increases up to [Fe/H] 0.5 due to the progressive contribution of mostly secondary nitrogen ejected by intermediate and low mass stars, until the iron production of SNIas compensates the ejecta of intermediate mass stars and flattens the evolution of [N/Fe]. Except for the possible production of primary nitrogen and the compensating effect by the large iron injection form SNIas, the differences in the [N/Fe] evolution as a result of binary evolution are very small.
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Fig. 31. The predicted and observed [N/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Laird (1985), Gratton and Ortolani (1986), Carbon et al. (1987) and Carretta et al. (2000).
Oxygen is primarily produced by massive stars through a-processing. Iron instead is produced by stars with a variety of masses, with a large production by evolving close binaries of intermediate mass that form SNIas. The general observed trend is [O/Fe] > 0 for [Fe/ H] > 1 and a gradual decline in the disk to the solar value. The occurrence of the decline could be caused by the time delayed injection of iron by type Ia SNe. However, this interpretation has been questioned recently on the basis of new observations of oxygen abundances in the low metallicity range. For the behaviour of [O/Fe] in metal poor stars,
two different results have been presented in the literature: one shows a nearly flat plateau at [O/ Fe] 0.5 (e.g. Gratton et al., 2000; Carretta et al., 2000; Nissen et al., 2001, 2002), while the other is a steady increase up to [O/Fe] 1, as [Fe/H] decreases from 1 down to 3 (Israelian et al., 1998; Boesgaard et al., 1999; Mishenina et al., 2000). If the latter results are real, there is an overall continuous decline of [O/Fe] as a function of [Fe/H] without a pronounced knick at [Fe/H] (10.5). We remark that these latter results were not included in the set of observational data on which Chiappini et al. (1999) applied their statistical anal-
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Fig. 32. The predicted and observed [O/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Gratton et al. (1996), Boesgaard et al. (1999), Mishekina et al. (2000), Carretta et al. (2000) and Nissen et al. (2002).
yses. Therefore the trend lines shown in Fig. 28 do not hold for the observed data plotted in Fig. 32. Our model predictions, given in Fig. 32, show an overall underproduction of oxygen relative to iron during the early galactic epoch. The downturn to the solar value appears at values of [Fe/H] higher than 1 (on average at 0.3) corresponding with the maximum formation rate of the SNIas as predicted by the progenitor models. When the DD scenario is applied the decrease is stronger because SNIas appear earlier (already at 40 Myr) than in the SD scenario. Higher
[O/Fe] ratios are obtained with the reduced Fe yields (= models 10 and 11). Although the results are closer to the observed average, we still find an underproduction of oxygen during the halo, thick disk evolution. Anticipating our results presented in the next Section 4.7, a higher production of oxygen is achieved if stars with an initial mass above 40 Mx undergo an SN like outburst instead of a direct collapse into a BH which we have assumed here. Also here we may conclude that except for the iron contribution of SNIas, the [O/ Fe] is not much affected by binary evolution.
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4.6.2. The a-elements Mg, Si, S and Ca The observed evolution of the ratios [a/Fe] for the Mg, Si, S and Ca are similar to the one of [O/ Fe] with an almost flat evolution in the halo and a gradual decline in the disk (McWilliam et al., 1995; Ryan et al., 1996). The a-elements are only produced in massive stars and type Ia SNe. Our results for the evolution of the [a/Fe] ratios are given in Figs. 33–36. We find an overall evolution similar to oxygen. The evolutionary behaviour is in general agreement with the average observed trend, though the ratios are rather low, especially for magnesium and sulphur at low met-
allicity, whether binaries are included or not. The underproduction of Mg is a generally known result. It has been pointed out by Goswami and Prantzos (2000) that the use of the metallicity dependent yields of Limongi et al. (2000) may solve the problem or that a supplementary source for Mg production is needed (Timmes et al., 1995). 4.6.3. Comparison to other studies The oxygen discrepancy during the early galactic evolution was also found by Timmes et al. (1995) who also uses the yields from Woosley and Weaver (1995) for their case B explosion
Fig. 33. The predicted and observed [Mg/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Gratton and Sneden (1987), Magain (1987, 1989) and Edvardsson et al. (1993).
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Fig. 34. The predicted and observed [Si/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Tomkin et al. (1985), Franc¸ois (1986), Gratton and Sneden and Edvardson et al. (1993).
model. When the figures of Chiappini et al. (1997) are rescaled relative to the observed solar ratio, the discrepancy is also visible in their results. At first glance it was surprising that this discrepancy was not found by some other groups (e.g. Chiappini et al., 1999; Boissier and Prantzos, 2000; Prantzos and Boissier, 2000; Portinari et al., 1998). Although most groups use the yields from Woosley and Weaver (1995), it is not always clear whether the case A or case B SN explosion model is implemented in their CEM. Portinari et al. (1998) use the yields obtained with the explosion
model A which corresponds to our model with the reduced iron yields. The library of Woosley and Weaver (1995) provides yields for stars only with an initial mass lower than 40 Mx. To obtain the yields for stars above 40 Mx some of the groups mentioned above extrapolate the yields from Woosley and Weaver (1995). In the overall IMF this mass range is insignificant, but during the early phases of galactic evolution mainly the massive stars enrich, and as we will show in the next section stars with a mass P 40 Mx play a very significant role in the oxygen enrichment.
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Fig. 35. The predicted and observed [S/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Clegg et al. (1981) and Franc¸ois (1987, 1988).
Kobayashi et al. (1998) applied the different SNIa scenarios in a standard chemical evolutionary model and computed the run of [O/Fe] with [Fe/H]. They conclude that a best fit to the observations is obtained with the metallicity dependent SD model since it nicely reproduces the observed downturn in [O/Fe] at [Fe/H] 1. Their result is difficult to understand since the change in the slope corresponds with the maximum SNIa rate, while in the metallicity dependent SD model the first SNIas form around [Fe/H] 1.1 and therefore produce a maximum at [Fe/H] > 1. This issue has also been discussed by Matteucci and
Recchi (2001) who made chemical evolutionary computations with the metallicity dependent SD scenario and for various CEMs. With the twoinfall model they obtain a predicted behaviour of the [O/Fe] vs. [Fe/H] relation similar to ours (= model 4), which clearly shows that a break in the [O/Fe] ratio corresponding with the moment of a maximum SNIa rate, occurs at [Fe/H] > 1. 4.6.4. Overall conclusion on the time evolution of the abundance ratios In the same way as we have done for the AMR, G-type dwarf metallicity distribution and the solar
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Fig. 36. The predicted and observed [Ca/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Tomkin et al. (1985), Gratton and Sneden and Edvardsson et al. (1993).
abundances, we also made test computations with a CEM where, except for the details of the SNIa progenitors, we consider all other stars as single (i.e. models 16 and 17). We conclude that the time evolution of the elements He, O, Ne, Mg, Si, S, and Ca predicted by galactic evolution, where the effect of binaries is considered in detail, differs by no more than a factor of two or three from the results of models in which most of the stars are treated as single stars and in which the effect of binaries is simulated only to account for the SNIa population; however, intermediate mass close binaries may be
essential in order to understand the time evolution of Fe and C. 4.6.5. The influence of rotation Due to rotation larger He and CO cores form and consequently a higher production of helium and a-elements (Heger, 1998). However, as mentioned earlier in Section 2.2.2 the same effect is obtained by mild convective overshooting. When the He core is larger, the 12C(a,c)16O reaction is more active at the end of He burning. As a consequence the fraction of carbon left in
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the CO core is lower than in the non-rotating case. This effect increases the oxygen yield. Since the carbon burning phase is shortened the stellar core has less time to remove its entropy through heavy neutrino losses, which favours the formation of a BH above an NS (Woosley, 1986). Recently, Fryer and Heger (2000) concluded that the net effect of rotation is a delay of the SN explosion and a weaker explosion, which finally leads to the formation of larger compact remnants and thus a lower initial mass limit for BH formation. Rotational diffusion can extract newly synthesized elements from the core, mix them with the outer layers and save them from further processing in the next burning phases in the core. Although this increases the yields, quantitatively the overall yields that are expelled during the whole evolution of the star are generally not much affected. Rotational diffusion may be an important way to produce primary nitrogen in zero metallicity stars, which seems to be required to reproduce the observed N/O ratio in the galactic halo stars (e.g. Timmes et al., 1995). The primary 14N can be produced when 12C that has been synthesized by the 3a-reaction in the He burning core is transported outwards by rotational diffusion into the CNO burning H shell. Because the H shell must be rather thick and long living, primary 14N is mainly produced in massive stars that do not suffer too strong mass loss by an SW. Primary 14N may also be produced in He burning zones through 12 C(p, c)13N(b+)13C(p, c)14N when protons are injected by rotational mixing. However, this primary 14 N is quickly destroyed to produce 18O via 14 N(a,c)18F(b+)18O. We remind that primary 14N can be produced by the secondary star in a binary, after it has accreted 12C rich material during the SN explosion of the primary star. 4.7. The effect of BH formation above 40 Mx and metallicity dependent SW mass loss In many CEMs the evolution of stars with an initial mass >40 Mx has been claimed to be unimportant because of their low formation rate as dictated by the adopted law for the stellar mass distribution at birth. As far as number statistics is
concerned these stars are indeed of less importance but this may not be true when we consider their contribution to the chemical enrichment of the ISM during the early phases of the Milky Way. The evolution of stars above 40 Mx is in most CEMs treated in a rather sloppy way, i.e. either they are not included or an extrapolation is made from stars below 40 Mx. Moreover, SW mass loss in this high mass range is often not accounted for or is rather qualitatively included. An exception is the work of Portinari et al. (1998). The carbon and oxygen yields of massive stars strongly depend on the SW mass loss rate and the final fate of the star after the core collapse. If the SW is radiatively driven, the mass loss rate mainly depends on the abundance of the iron group elements in the star, which are the main drivers. Therefore the dependency of the SW mass loss rate on the initial metallicity of the gas out of which the star is formed can be crucial. The WR and/or RSG SW mass loss has a major effect on the evolution of the CO core of a massive star. When the WR mass loss is small or when the RSG mass loss is too small for a massive single star to become a WR star, the star retains most of its mass until the end of its evolution. This means that the CO core increases in mass during the whole core helium burning (CHeB) phase and at the end mainly consists of oxygen that is ejected if an SN outburst takes place. On the contrary if the WR mass loss is high, a large part of the mass is ejected with the wind, essentially in the form of helium and carbon. The CO core decreases in mass and a smaller amount of matter is turned into oxygen. In this case a large amount of carbon and little oxygen will be ejected when an SN outburst occurs. The situation of low mass loss rates prevails at low metallicity when the WR mass loss depends on the initially metallicity of the star, while the second situation holds if the WR mass loss is independent from the metallicity. In summary, the enrichment of carbon and oxygen in the early phases of galactic evolution may depend on whether or not one uses massive star evolutionary computations with WR mass loss rates that are metallicity dependent or not.
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Whether stars above 40 Mx first explode and then form a BH or directly collapse into a massive BH without the ejection of material is presently not clear. The runaway status of some low and high mass X-ray binaries with a BH component (e.g. Nelemans et al., 1999) and the presence of heavy elements in the atmosphere of the companion star of the BH X-ray binary GRO J1655-40 (Israelian et al., 1999) suggest that massive BHs form with a prior SN like outburst during which some material is ejected. Also the hypernovae observed in external galaxies have been associated with the formation of a BH. The issue of massive BH formation with or without matter ejection could be important for the chemical evolution in the early Galaxy when lower mass stars have not yet evolved. In our previous computations we have taken the WR SW mass loss independent of the metallicity and assumed that stars (single and binary) more massive than 40 Mx do not undergo an SN explosion but directly collapse into a BH. To illustrate the importance of the evolution of stars with an initial mass P 40 Mx and the effect of metallicity dependent WR SW mass loss on the chemical evolution of the Galaxy, we make the following simulations: Simulation 1: All stars above 40 Mx form massive BHs without a preceding SN outburst and the WR SW mass loss rates do not depend on the initial metallicity of the gas out of which the stars are formed. Simulation 2: All stars above 40 Mx form massive BHs without a preceding SN outburst p but ffiffiffi the WR SW mass loss rates are scaled with Z . Simulation 3: All stars above 40 Mx form massive BHs with the ejection of 0.7 Mx of 56Ni p and ffiffiffi the WR SW mass loss rates are scaled with Z . Simulation 4: All stars above 40 Mx form massive BHs with the ejection of oxygen and carbon but no nickel. The amount of ejected oxygen is taken once 7 Mx (a) and once 4 Mx (b). To determine the ejected amount of carbon we adapt the C/O ratio that follows from the hypernova models computed by Nakamura for the corresponding He core. The WR p SW ffiffiffi mass loss rates are in both cases scaled with Z .
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In simulation 3 and 4 we consider two extremes for the ejected amount of 56Ni. An amount of 0.7 Mx has been predicted by the super-energetic explosion of a 16 Mx He star with E 3– 6 · 1052 ergs in order to explain the lightcurve of the observed hypernova SN 1998bw (Nakamura et al., 2001). The corresponding ejected oxygen mass is 7 Mx. Low amounts of 56Ni are predicted by hypernova models in which the masscut is placed at a large distance from the center of the star (>4 Mx) or where the explosion occurs asymmetrically (Maeda et al., 2002). The latter models with few 56Ni ejection can explain the low Fe abundance and high Ti, S, Si, Mg and O abundances in the companion star of the X-ray BH binary Nova Sco (GRO J1655-40). In all our simulations we take a constant binary frequency of 70% and use the iron yields from Woosley and Weaver (1995) reduced by a factor of two. The results are shown in Figs. 37 and 38. We conclude:
1. The effect of stars with an initial mass P 40 Mx on the chemical evolution at [Fe/H] < 2 is very significant. Therefore it is important to implement the most up to date SW mass loss rate formalisms (for all masses) during the various evolutionary phases. 2. The observations of [O/Fe] and [C/Fe] at low metallicity are not reproduced when large amounts of 56Ni are ejected when the BH forms. 3. When the WR mass loss rates are taken to be independent of the initial metallicity a significant overenrichment in carbon shows up during the early evolution of the solar neighbourhood. We interpret this as indirect evidence that the strength of these winds is strongly sensitive to the initial metallicity of the gas, which corresponds with the prediction from stellar wind theory. 4. The evolution of stars with an initial mass above 40 Mx in general, the detailed physics of BH formation and the ejected amount of material in the preceding SN like outburst determine the real carbon, oxygen and iron enrichment of the Galaxy during the early phases.
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Fig. 37. The predicted evolutionary behaviour of [C/Fe] as function of time for the different simulations. The upper figure shows Sim1 (thin line) and Sim2 (bold line), and the lower figure shows Sim3 (bold line) and Sim4 with the ejection of 4 Mx resp. 7 Mx of oxygen (dashed thin line resp. full thin line).
4.7.1. Comparison to other studies Metallicity dependent stellar yields from massive stars with the inclusion of SW mass loss have been computed by Maeder (1992) and Portinari et al. (1998). These yields were respectively implemented into a CEM by Prantzos et al. (1994), Portinari et al. (1998) and by Gustafsson et al. (1999) in order to study their influence on the evolutionary behaviour of carbon and oxygen in the Galaxy. They all concluded that the [C/O] vs. [Fe/H] relation for disk stars can be well explained by the metallicity-dependent behaviour of radiatively driven SWs of massive stars and that there is no real need for a moderate carbon contribution
from low and intermediate mass stars. However, Maeder (1992) and Portinari et al. (1998) used mass loss rate prescriptions in their stellar evolutionary computations that are out of date. For the WR SW mass loss they used the Langer formalism (Langer, 1989), which has been proven to overestimate largely (up to a factor of 4) the mass loss rate. Furthermore they assume a metallicity dependent SW only during the pre-WR phase, which implies that at Z 6 0.002 only the most massive stars (>80 Mx) may enter the WR phase and that He, C and O enrichment by SW mass loss only becomes important at high metallicity. In our preferred evolutionary model all stars initially
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Fig. 38. The predicted evolutionary behaviour of [O/Fe] as function of time for the different simulations. The upper figure shows Sim1 (thin line) and Sim2 (bold line), and the lower figure shows Sim3 (bold line) and Sim4 with the ejection of 4 Mx resp. 7 Mx of oxygen (dashed thin line resp. full thin line).
more massive than 40 Mx become WR stars independently of the initial metallicity due to severe mass loss during the LBV evolutionary phase (= LBV scenario). Additionally, Maeder (1992) also concluded that direct BH formation should already start at (25–30) Mx to avoid an overproduction of oxygen and to obtain a large helium to metallicity enrichment ratio DY/DZ. Extrapolating this result to low metallicities would imply that stars above 40 Mx are totally irrelevant for the chemical evolution during the early phases of galactic evolution. However, Maeder (1992) did not do the chemical evolution over the whole life-
time of the Galaxy and did not include type Ia SNe. Prantzos (1994) reanalysed Maeders study and concluded that the observations are better fitted when all massive stars up to 100 Mx contribute, via SWs and/or SNe, to the chemical enrichment of the Milky Way, which is in agreement with our results. 4.8. The supernova rates 4.8.1. Observed rates The SN rates in different types of galaxies have been estimated by Cappellaro et al. (1999) and are
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Table 16 The SN rates (expressed in SNu with the Hubble constant H0 = 75 km/s/Mpc) in different types of galaxies. Others includes Sm, Irregular and Peculiar type galaxies (ref. Cappellaro et al. (1999)) Galaxy type
E-S0 S0a-Sb Sbc-Sd Others All
Nr. of SNe
SN rate
Ia
Ib/c
II
Ia
Ibc
II
All
22.0 18.5 22.4 6.8 69.6
5.5 7.1 2.2 14.9
16.0 31.5 5.0 52.5
0.18 ± 0.06 0.18 ± 0.07 0.21 ± 0.08 0.40 ± 0.16 0.20 ± 0.06
<0.01 0.11 ± 0.06 0.14 ± 0.07 0.22 ± 0.16 0.08 ± 0.04
<0.02 0.42 ± 0.19 0.86 ± 0.35 0.65 ± 0.39 0.40 ± 0.19
0.18 ± 0.06 0.72 ± 0.21 1.21 ± 0.37 1.26 ± 0.45 0.68 ± 0.20
summarised in Table 16. The rates are expressed in the SN unit Snu = 1 SN Æ (100 yr)1 Æ (1010 LB,x)1 with LB,x the luminosity of the Sun in the photometric B band. Considering the Milky Way as a type Sb-Sbc with a total blue luminosity LB = 2 · 1010 Lx (van der Kruit, 1987), the present galactic SN rate is estimated at 2 SNe per century, of which 20% type Ias, 13% type Ibcs and 67% type IIs. Taking a disk radius of 18 kpc, this corresponds roughly to a local rate of 1.96 · 102 SNe/pc2/Gyr. 4.8.2. Overview of the progenitor populations In our PNS model, the different SN types are produced by the following stars. SNII progenitors single born massive stars and massive non-interacting binary components with 8 < M 0 =M X ¼0 < M X0;s¼0 ; where M 0;s is defined as the minimum initial ZAMS mass of a single star that ends its life without hydrogen; our single star evolutionary calculations of massive single stars with stellar wind mass loss give the value of X ¼0 M 0;s from first principles, binary components with 8 < M 0 =M < M X0;s¼0 that became single after the first SN explosion that disrupted the binary system, binary star mergers with masses after merging X ¼0 between 8 M and M 0;s , merging WD + MS/RG binaries that form a single star with an MCO P 1.4 Mx (=SNIIWD) The initial properties (on the ZAMS) of the binary progenitors of the merging WD + MS/RG binaries are shown in Fig. 40. They are computed for the standard PNS model. The majority of the
binary progenitors are initial short period systems (case A/Br systems) with an intermediate mass ratio that undergo a quasi-conservative first RLOF. SNIbc progenitors single born massive stars, massive single stars with a binary history and non-interacting massive binX ¼0 ary components with M 0;s < M 0 = M < M BH 0;s , interacting binary components with 10 < M 0 = M < M BH 0;b , massive secondaries that have lost their hydrogen envelope via SW during the RSG phase and/or LBV phase or in a spiral-in phase with the compact primary star during which the CE has been ejected, binary star mergers that became more massive X ¼0 than M 0;s after merging and further evolved like a normal single star. SNIa progenitors SD scenario: CO WD + MS/RG binaries with the right properties (see Hachisu et al., 1999) to produce an SNIa, DD scenario: CO WD + CO WD binaries with (M1 + M2) P 1.4 Mx and a lifetime 6 tnow. Fig. 39 shows the initial properties (on the ZAMS) of the binary progenitors for both scenarios. They are computed with our standard PNS model. In the SD scenario, the binary progenitors are mainly case Bc/C systems with an initial intermediate mass ratio that will evolve through a CE phase. The SNIa progenitors in the DD scenario have initial periods between (5–100) days that either evolve through a CE phase or perform quasi-conservative RLOF.
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Fig. 39. The initial (on the ZAMS) primary mass (upper figure), mass ratio (middle figure) and orbital period (lower figure) of the SNIa progenitors for the SD scenario (bold lines) and the DD scenario (thin lines). The computations are made with /H, bmax = 1, ai = 1 and Z = 0.02.
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Fig. 40. The initial (on the ZAMS) primary mass (upper figure), mass ratio (middle figure) and orbital period (lower figure) of the progenitors of WD + MS/RG mergers. The computations are made with /H, bmax = 1, ai = 1 and Z = 0.02.
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Hypernova (HN) progenitors Some of the observed SNIcs (SN1997dq, SN1997ef, SN1998bw, SN1998ey, SN99as, SN2002ap) have extremely high peak luminosities (1.6 · 1043 erg/s) and expansion velocities (>6 · 104 km/s) (e.g. Galama et al., 1998, 1999; Bloom, 1999; Reichart, 1999; van Paradijs, 1999). They have been classified as hypernovae. With current SN models these peculiar characteristics can be reproduced if explosion energies P 1052 ergs are used in stars with an initial mass P 25 Mx. Depending on the location of the adopted mass-cut, the ejected amount of 56Ni for these high energies can increase up to 0.4–0.7 Mx (matching with SN1998bw), which is 10 times larger than the 56 Ni-ejectum of a standard core-collapse SN (Nakamura et al., 2001). Hypernovae are currently believed to be associated with the formation of massive BHs since the large explosion energy can be explained by a rapidly rotating BH with a strong magnetic field (e.g. MacFadyen and Woosley, 1999). Although their occurrence rate in the nearby universe is quite low, they could have been more frequent in earlier epochs when star formation might have been higher. Since they produce large amounts of iron they might have a strong im-
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pact on the evolution of the [a/Fe] abundance ratios. Therefore we compute their ratio to the core-collapse SNe. To compute the number of HNe we assume that they arise from massive single (binary) stars with BH an initial mass P M BH 0;s ðM 0;b Þ. Notice that in our standard PNS model these massive single (binary) stars are assumed to form massive BHs without an SN explosion.
4.8.3. The evolutionary clock of supernovae In order to illustrate the typical timescales on which SNe occur, we computed the time evolution of the SN rates after an instantaneous starburst. To allow an easy scaling of our results to any starburst of arbitrary total mass, we took the total mass of the starburst equal to 1 Mx. We separately consider a burst in which only single stars are formed and one in which only binary stars are formed. The time evolution of the core collapse SN rates is given in Figs. 41 and 42 and of the SNIas in the Figs. 43 and 44 for different metallicities. For the SD scenario we assumed a metallicity dependent WD wind (i.e. SD2). Since the single and binary
Fig. 41. The time evolution of the core collapse SN rates after an instantaneous starburst for Z = 0.02. The curves produced by the starburst with 100% single stars are indicated with the letter s between round brackets while the curves labeled with the letter b correspond with a burst of 100% binaries. The computations are made for the standard PNS model. The meaning of the colour lines is given in the legend.
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Fig. 42. The same as Fig. 41 but for Z = 0.002.
Fig. 43. The time evolution of the SNIa rates after an instantaneous starburst (with 100% binaries) for Z = 0.02. We separately give the timescales of the SNIas produced by the SD2 scenario (black lines) and those by the DD scenario (grey lines). The computations made with bmax = 0.5 and a2,3 = 0.5 are given by the bold lines.
SN rates scale linearly with the single star fraction and binary star fraction at birth (i.e. on the ZAMS), it is straightforward to make predictions for any single/binary star fraction. We performed several simulations by changing the PNS parameter values. For the core collapse SNe it can be concluded that the time evolution
hardly depends on the adopted PNS parameter values. Therefore we show in Figs. 41 and 42 only the results computed for the standard PNS model. However, the timescales on which SNIas form, depend sensitively on the orbital period evolution of the progenitor binaries which is mostly affected by the accretion parameter during RLOF (i.e. bmax)
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Fig. 44. The same as Fig. 43 but for Z = 0.002.
and the CE ejection efficiency (i.e. a). Therefore we show in Figs. 43 and 44 also the results obtained with bmax = 0.5 and a2,3 = 0.5. From our simulations we conclude that: 1. SNIas form in the SD model on a timescale of 0.3–4.5 Gyr after the initial burst, while in the DD scenario the first SNIas already appear when the starburst is 40 Myr old with a maximum rate at 0.3 Gyr and 5 Gyr. The two peaks correspond with the two different evolutionary channels through which SNIas are formed in the DD scenario. The first maximum contains mainly the SNIas of which the progenitor binary experienced two CE phases (MS + MS ! CE ! WD + MS ! CE ! WD + WD). This maximum corresponds with the results of Iben and Tutukov (1984). The second maximum contains the SNIas of which the progenitor binary first evolved through a quasi-conservative RLOF (MS+MS ! RLOF ! WD + MS ! CE ! WD + WD). The WD + WD systems formed via the second channel have longer final periods and therefore longer lifetimes. This evolutionary channel was not considered by Iben and Tutukov (1984) although, according to our simulations, it may be an important
one. However, when bmax = 0.5 is adopted this second group of long period systems is strongly reduced because a significant amount of orbital angular momentum is taken away from the system when matter leaves the binary during RLOF. Due to this mass and angular momentum loss the periods are on average smaller and some fraction may merge. 2. When binaries are formed in the starburst, the phase during which the core collapse SNe appear lasts much longer than when only single stars are formed, i.e. 250 Myr vs. 40 Myr. This is mainly due to the mass transfer in interacting binaries transforming secondaries of initially intermediate mass into massive stars, and to the explosion of merging WDs with their non-degenerate companion star. 3. The situation for the core collapse SNe is at Z = 0.002 much the same as it is at Z = 0.02. In our adopted standard PNS model no SNIbcs are formed by single stars _ at Z = 0.002 because of the lower pffiffiffi SW M _ (we used here a WR SW M / Z ) and the adopted mass limit for direct BH formation (25 or 40 Mx). The metallicity effect is more pronounced for the SNIa timescales. In the SD scenario the first SNIa appears only after
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Table 17 The different models for which the time evolution of the SN rates is computed PNS model
Model
fb
1 (7) 2 (8) 3 (9) 4 (10) 5 (11) 6 (12) 13
40% 40% 40% 40% 40% 40% fb,2
(70%) (70%) (70%) (70%) (70%) (70%)
Standard standard but /F standard but /G Standard but bmax = 0.5, a1,2,3 = 0.5 Standard but a1,2,3 = 0.5 Standard but vk ¼ 150 Standard
The model numbers between round brackets correspond to a constant binary frequency of 70%, while model 13 assumes a binary frequency that increases linearly with the metallicity.
1 Gyr instead of 0.3 Gyr, which is due to the lower upper mass limit on the WD companion star for Z = 0.002.
4.8.4. The galactic evolution of the SN rates Because the SN rates depend on the metallicity and the star formation rate we computed the evolution of the SN rates as a function of time during the galactic evolution. Our simulations are made for the models listed in Table 17. We take the iron yields from Woosley and Weaver (1995) decreased by a factor ofptwo ffiffiffi and the WR SW mass loss proportional to Z . Except for model 13, when the SD scenario is applied we assume that the WD wind is metallicity dependent (i.e. SD2).
Obviously, the absolute SN rates predicted by our simulations depend on the parameters of the star formation and the galaxy formation model. However, relative rates hardly depend on them. Also in the relative observed rates, the uncertainty in the Hubble constant is excluded. Table 18 lists the present (i.e. predicted at t = tnow) relative SN rates in the solar neighbourhood for the different models. It also gives the fraction A among the total intermediate mass binary population (on the ZAMS) that produces an SNIa. As mentioned earlier in Section 3.7, this fraction A is inserted in most current CEMs as a free parameter that is regulated to finally meet the observations. Figs. 46 and 47 show the time evolution of the SNIa rates throughout the galactic evolution for all the
Table 18 The present relative SN rates predicted by different models for the solar neighbourhood Model
SNIIWD/II
SNII/SNIbc
HN/(SNII + SNIbc)
SNIa/SNIbc
SNIa/(SNII + SNIbc)
A (%)
1 2 3 4 5 6 7 8 9 10 11 12 13
0.07 0.12 0.14 0.05 0.06 0.07 0.10 0.25 0.27 0.10 0.10 0.10 0.09
5.57 3.81 3.31 5.98 6.13 5.07 4.20 2.44 2.03 4.72 4.92 3.82 4.31
0.147 0.147 0.148 0.149 0.149 0.137 0.140 0.141 0.142 0.140 0.140 0.130 0.150
0.53j0.38 0.24j0.69 0.17j0.69 0.21j0.05 0.21j0.36 0.49j0.35 0.78j0.60 0.30j1.05 0.24j0.90 0.34j0.07 0.35j0.65 0.73j0.56 0.81j0.50
0.08j0.06 0.05j0.17 0.04j0.16 0.03j0.01 0.03j0.05 0.08j0.06 0.15j0.12 0.09j0.31 0.08j0.30 0.06j0.01 0.06j0.01 0.15j0.12 0.16j0.10
3.06j1.11 2.05j2.36 1.58j3.44 1.72j0.19 1.04j4.61 3.06j1.11 3.06j1.11 2.05j2.36 1.58j3.44 1.72j0.19 1.04j4.61 3.06j1.11 3.06j1.11
In the last three columns the first number is computed when the SD scenario is applied and the second number when the DD scenario is used. The last column contains the fraction A among the initial intermediate mass binary population (on the ZAMS) that produces an SNIa.
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Fig. 45. The time evolution of the SNII (upper figure) and SNIbc (lower figure) rate predicted by the models 1, 7 and 13.
models except for models 6 and 12 since the average kick magnitude is not involved in the evolution of the SNIa binary progenitor population. Because the evolutionary behaviour of the SNII and SNIbc rates depends less sensitively on the PNS model parameters, we show their galactic evolution in Fig. 45 only for the models 1, 7 and 13. We remark that in our computed SNII rate the merging WD + MS/RG binaries (=SNIIWD) are not included. The evolution of the ratio SNIa/(SNII + SNIbc) is given in Figs. 48 and 49. From our simu-
lations and comparison with the observations we draw the following conclusions: 1. Independent from the adopted parameters in the PNS model, the predicted present total SN rate is 1–2 SNe per century, which agrees well with the observations. However, the relative contribution of the different SN types differs for different PNS models, especially the fraction of SNIas, which may vary by a factor 2–3.
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Fig. 46. The time evolution of the SNIa rate predicted by the models 1, 2, 3, 4 and 5 with the SD scenario (upper figure) and the DD scenario (lower figure).
2. In order to obtain the present SNIa rate (within the observational error) an intermediate mass binary frequency (on the ZAMS) of at least 70% is required when the SD scenario is used and (30–50)% if the DD scenario is applied. With the SD scenario a maximum SNIa rate is obtained when a Hogeveen type mass ratio distribution is used in combination with a maximum CE ejection efficiency (i.e. ai = 1). In the case of the DD scenario we find
the highest number of SNIas for a Garmany type mass ratio distribution combined with bmax = 1 and ai = 1. 3. Only a small percentage of the total zero-age population of intermediate mass binaries finally produces an SNIa. In most CEMs this fraction is included as a free parameter (and not explicitly computed) that is fixed in order to meet the observations. The commonly adopted value for A in these CEMs varies
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Fig. 47. The time evolution of the SNIa rate predicted by the models 7, 8, 9, 10, 11 and 13 with the SD scenario (upper figure) and the DD scenario (lower figure).
between 0.7% and 8% (e.g. Timmes et al., 1995; Portinari et al., 1998), which covers our predicted values. 4. Depending on the binary properties, the number of SNIIs produced by merging WD + MS/RG binaries is between 5% and 30% of the number of SNIIs originating from exploding massive stars. 5. The expected relative rate of hypernovae to core-collapse SNe is about 0.15, which corresponds with an absolute rate of 2–4 HNe per millennium.
6. The SNIa/(SNII + SNIbc) number ratio increases (after 3 Gyr) almost linearly as a function of time, irrespective of the adopted PNS model. Because SNIas form on much longer timescales than core collapse SNe do, the SNIas and SNIIs and SNIbcs that appear at the same moment during galactic evolution have progenitors that were formed in a different earlier epoch of star formation. This explains why the ratio is not constant in time. The effect is pronounced in the SD scenario where a significant fraction of the
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Fig. 48. The time evolution of the ratio SNIa/(SNII + SNIbc) predicted by the models 1, 2, 3, 4 and 5 with the SD scenario (upper figure) and the DD scenario (lower figure).
SNIas progenitors, formed during the first Gyr of galactic evolution, only appears after 2 Gyr, when the star formation rate is much lower. 7. The number of SNIIs relative to SNIbcs depends significantly on the properties of the massive binary population (the massive binary frequency and mass ratio distribution). However, the ratio hardly depends on the metallicity and therefore the ratio varies very slowly as a function of time.
8. Since most of the SNIbc progenitors are binary components, the observed ratio SNIa/ SNIbc may be an indication that the massive binary population relative to the intermediate mass binary population is similar in early and late spirals. However, due to the fact that the SNII/SNIbc ratio is significantly different, this could be an indication that the overall binary frequency (or the overall binary population) in both types of spiral galaxies is significantly different.
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9. The observed SNIa/SNIbc ratio in our Galaxy is best approached if the efficiency of CE ejection is very large (i.e. close to 1). 10. When we account for all observations of the OB-type stars in the solar neighbourhood (including the OB-type binary population), accounting for statistical biases when interpreting the binary population data which implies that the interacting binary frequency is at least 50% whereas the mass ratio distribution is flat or peaks at small values of q, then a best correspondence between the simulated and the observed SN population in the solar neighbourhood is obtained when the CE ejection is very efficient (a 1), the RLOF in intermediate mass binaries with a mass ratio q P 0.4 is almost purely conservative (bmax = 1), the merging of a WD with an MS/RG companion star produces an SNII, both the SD and the DD scenario produce SNIas. Remarks 1. Hachisu et al. (1999) proposed that binaries with an initial period larger than 10 years may become short period WD + RG binaries via wind-orbit interaction and produce SNIas. They estimate that about one third of all SNIas could have been produced via this evolutionary channel. It is straightforward to estimate that if we include these systems, the required binary frequency to obtain the observed present SNIa rate with the SD scenario is reduced to 40–50%. 2. The properties of the binary population and the metallicity may be different in different galaxies of the same morphological type. Because the observed SN rate in our Galaxy is computed as an average over a sample of external galaxies that are of the same morphological type as our Galaxy but possibly may harbor a stellar population with different properties, the average value may not be representative of our Galaxy. 3. For all galaxies together in the observed sample of Cappellaro et al. (1999) the ratio II/Ibc is 5. Since in the sample all the main morphological
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types of galaxies are included, this value could be considered as some cosmological average. It follows from our simulations that to recover this average, we need a binary fraction (on the ZAMS) between 40% and 70% which could imply that the cosmological massive binary formation frequency may be of the order of 50%.
4.8.5. Comparison to other studies Tutukov et al. (1992) computed the SN rates of different spectral types assuming a constant star formation rate over the lifetime of our Galaxy and a 100% initial binary frequency. For the SNIas they only consider the DD scenario. Scaling their results to a binary fraction of 70% we find that, within the differences in the adopted PNS prescriptions, their predicted relative rates are close to our predicted values except for the ratio SNIa/SNIbc, for which we find a higher value. The explanation for this difference is twofold: (1) Tutukov et al. (1992) predict an SNIbc rate of (0.6–1) per century which is higher than our estimated rate of (0.3–0.7) per century. Since they assume a symmetric SN explosion, the majority of their binary systems remains bound after the explosion of the primary massive star. This implies that the secondary star (if massive) in many cases loses its hydrogen envelope via RLOF and explodes as an SNIbc. It is obvious that when the SN explosion occurs asymmetrically (which is assumed in our PNS model), most binary systems (about 80% when using an average kick magnitude of 450 km/s) are disrupted after the explosion of the primary star, and the secondary star further evolves as a single star. According to our single star evolutionary scenario the secondary star explodes as an SNIbc only if its mass is larger than (17–20) Mx for Z = 0.02, which results in a lower SNIbc rate. (2) In our DD model we predict that a significant fraction of the SNIas form with lifetimes >1 Gyr coming from binary progenitors that evolved through a first quasi-conservative RLOF phase and a CE phase during the second RLOF. In the evolutionary scenario
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Fig. 49. The time evolution of the ratio SNIa/(SNII + SNIbc) predicted by the models 7, 8, 9, 10, 11 and 13 with the SD scenario (upper figure) and the DD scenario (lower figure).
of Iben and Tutukov (1984) (which is used in the work of Tutukov et al., 1992) SNIas are only produced by binary progenitors that underwent CE evolution during the first and second RLOF. Therefore the fraction of SNIas formed in their evolutionary scenario with a lifetime >1 Gyr is small. By consequence our predicted SNIa rate is generally larger by up to a factor of 1.5–2, depending on the adopted PNS parameter values. Only when model 4 is used do we find a good cor-
respondence with the results of Tutukov et al. (1992). In this model bmax = 0.5, which strongly reduces the fraction of SNIas with lifetimes >1 Gyr, as we explained earlier. We notice that due to our higher predicted rate of SNIas and lower rate of SNIbcs we better approach the observed SNIa/SNIbc ratio. In their PNS study of binary compact objects Belczynski et al. (2002) report SNII/SNIbc ratios between 1.8 and 3.7 for Z = 0.02 and an initial bin-
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ary fraction of 50%, which agrees with our predicted ratios. The time evolution of the SNII and SNIa rate has been computed by Kobayashi et al. (2000) for elliptical and spiral galaxies. They propose an explanation for the higher SNII rate (a factor of two) observed in late type spirals compared to early types based on a different star formation rate. However in their galactic evolutionary model they adopt a very simple population synthesis model and do not separate the SN IIs from the SN Ibcs. Because of the latter assumption, their explanation may not be true. A different star formation rate would also imply a different SNIbc rate in both spiral types, which is not observed. On this point we think that our proposition of a different overall binary formation rate in both types of spiral galaxies offers a better explanation. The evolutionary behaviour of the SNIa/SNII ratio (to be compared with our SNIa/(SNII + SNIbc) ratio) as a function of time predicted by Kobayashi et al. (2000) is similar to our results for both the DD and SD model, though their predicted ratio-values are generally higher especially during the early evolutionary phases of the Galaxy. For the present (t = 15 Gyr) ratio we find a best correspondence with their predictions when we consider our models that assume a constant binary frequency of 70%. We remark that Kobayashi et al. (2000) uses the evolutionary model of Iben and Tutukov (1984) for the DD scenario. The lack of a detailed discussion on the adopted stellar evolutionary models in the work of Kobayashi et al. (2000) does not allow for a thorough discussion on the differences between their work and ours.
4.9. The evolution of the r-process elements In this final section we study the galactic enrichment in r-process elements by merging NS + NS and NS + BH pairs in which the BH is formed either by the primary or by the secondary star. We first compute the time evolution of the formation and merger rate of these double compact star binaries and then compute the resulting chemical enrichment of r-process elements as a function of time.
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4.9.1. The formation of NS + NS and NS + BH binaries The standard scenario for the formation of these pairs of compact objects starts with an initial massive binary system in which the secondary is a massive star or an intermediate mass star that becomes massive after accretion during RLOF by the primary. The primary first evolves off the main sequence, overfills its Roche lobe and transfers mass to the secondary. The primary which has become a hydrogen deficient CHeB star further evolves (in some cases performing a second RLOF during helium shell burning = case BB RLOF) and finally collapses into an NS or BH, with or without a SN explosion. If the binary remains bound after the SN explosion, an NS/BH + OB binary is formed. This system passes through an X-ray binary phase and evolves through a CE phase as the secondary expands. During this CE phase the NS/BH spirals into the secondarys envelope forming a NS/BH + He star binary. If the helium star expands sufficiently during helium shell burning or beyound, it may overflow its Roche lobe for a second time. Depending on the actual properties of the system (orbital period, mass ratio, . . .) this mass transfer phase increases or decreases the orbital distance between both compact objects. After the secondary helium star has turned into an NS or BH and the system is not disrupted by an SN explosion, an NS/BH + NS/BH remains, of which the orbit decays via gravitational wave radiation, which finally leads to the merger of both compact objects. The occurrence of a case BB RLOF phase from the helium star to the compact companion is crucial for the orbital parameters of the resulting double compact star binary. If this mass transfer phase proceeds dynamically unstably, ultra compact NS/ BH + NS pairs can be formed which decay on a very short timescale (Myr). It is obvious that the timescale on which the orbital decay takes place determines the moment at which merging NS + NS and NS + BH pairs start to enrich the Galaxy with r-process elements. Because of the uncertainty in the treatment of case BB RLOF and its importance for the enrichment in r-process elements we make simulations for the different case BB scenarios that were defined and discussed
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earlier in Section 2.3.3. To refresh we repeat them here: scenario 1: case BB RLOF is suppressed by the SW mass loss of the helium star. The SW mass loss is independent from the metallicity and case BB is suppressed at low Z as well. The period variation is given by Eq. (2.9). scenario 2: case BB RLOF is suppressed by the SW mass loss of the helium star only for Z = 0.02. The SW mass loss is metallicity dependent and satisfies Eq. (2.4), which implies that case BB RLOF is not suppressed at Z = 0.002. Because no detailed case BB evolutionary computations exist for low metallicities, we use the computations made for Z = 0.02 for low Z as well. The period evolution is followed by assuming that matter leaves the binary in the form of an NS SW with the specific orbital angular momentum of the NS (see Dewi et al., 2002). scenario 3: the same as scenario 2 but we assume that the mass transfer to the NS results in a CE/spiral-in evolution and the period variation is given by Eq. (2.18). scenario 4: the SW mass loss for hydrogen deficient CHeB stars with a mass smaller than 5 Mx can be neglected and case BB RLOF does take place independent from the metallicity. The period evolution during case BB mass transfer to the NS is treated in the same way as in scenario 2. scenario 5: the same as scenario 4 but case BB mass transfer to the NS is assumed always to result into CE/spiral-in evolution. It has been argued that an NS would be able to increase its mass by accreting matter at a superEddington rate during CE and spiral-in evolution and to become a BH (e.g. Blondin, 1986; Chevalier, 1989, 1993; Brown, 1995). If this is true, it becomes very difficult to form NS + NS pairs via the CE evolution of HMXBs. However, this scenario is very uncertain and strong accretion onto the NS can be avoided, for example if the donor star is very extended or if rotational effects are taken into account (e.g. Chevalier, 1996). In our evolutionary model we do not account for hyper-critical accretion.
In order to determine the properties of a binary after the SN explosion of one of the components, we assume that prior to the SN explosion the system has been circularised. Since we treat the effects of the SN explosion on the binary system in full 3D our PNS code is able to compute the post-SN period, the space velocity and eccentricity of the binary. The knowledge of the post-SN eccentricity is essential in computing the timescale on which a double compact binary decays and finally merges due to GWR (see Section 2.3.2). 4.9.2. The theoretically predicted population of double compact star binaries 4.9.2.1. The predicted distribution of the orbital period, eccentricity, orbital decay timescale and space velocity. In order to illustrate the typical properties of the NS/BH + NS/BH population, we compute for an instantaneous starburst of 100% binaries the distribution of the orbital period (P), eccentricity (e) and space velocity (v) at the moment that the secondary has turned into a compact object and the corresponding timescale for orbital decay by GWR (Td). Our computations are made for the different PNS models defined in Table 19 and for two initial metallicities: Z = 0.02 and 0.002. We always take M NS max ¼ 2 M . We give the results made for the case BB scenarios 1 and 5. The other case BB scenarios give intermediate results. Figs. 50–53 and 56 give the normalized distribution of the orbital period, eccentricity and space velocity of the double compact star binaries at the time of their formation. The timescales for orbital decay are given in Figs. 54 and 55. We conclude that 1. The period distribution of double compact star binaries at birth in the period range <100 days is dominated by the NS + NS and BH + NS systems, while the majority of NS + BH and BH + BH systems form with periods above 100 days. The shape of the period distributions does not depend much on the choice of the PNS parameters. 2. At low Z double compact star binaries are expected to have smaller eccentricities. Compared to BH + NS systems, the NS + NS binaries
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takes place and leads to CE evolution, the timescales are strongly reduced to 104 and 108 yr, illustrating the important effect of case BB RLOF. 4. The space velocity distribution of the double compact star binaries is mainly determined by the orbital speed at the moment that the secondary explodes. The NS + NS systems acquire the highest speeds, with a broad peak around 300 km/s. The tightest systems (especially formed at Z = 0.002) are accelerated up to velocities of (500–1000) km/s. The velocity distribution of BH + NS systems is double peaked, at 30 km/s and 150 kms. The second peak is produced by the group of BH + NS binaries with the lowest periods (<25 days for Z = 0.002 and <10 days for Z = 0.02) which are clearly distinguishable in Fig. 51. The
Table 19 The different PNS sets for which the population of double compact star binaries is computed Set
bmax
a1,2
a3
vk
/(q)
1 2 3 4 5 6
1 1 1 0.5 1 1
1 1 1 1 1 1
1 1 1 1 0.5 1
450 450 450 450 450 150
/F /H /G /F /F /F
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have much higher eccentric orbits. The shape of the eccentricity distribution is rather insensitive to the PNS parameters except for the average kick velocity. 3. The orbit of the majority of NS + NS systems decays within 108 and 1011 yr if no case BB RLOF takes place. However, if case BB RLOF
N/Ntot
N/Ntot 0.18
0.12 0.1
set 1
set 2
set 3
set 4
set 5
set 6
Z=0.002 NS+NS
0.08
0.16
Z=0.002
0.14
NS+BH
0.12 0.1
0.06
0.08 0.06
0.04
0.04
0.02
0.02
0 -3
-2
-1
0
1
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-2
-1
0
1
log P(days)
2
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7
log P(days)
N/Ntot
N/Ntot 0.12
0.07
Z=0.002
Z=0.002
0.06
0.1
BH+NS
0.05
BH+BH
0.08
0.04 0.06
7
0.03
0.04
0.02
0.02
0.01
0
0 -3
-2
-1
0
1
2 log P(days)
3
4
5
6
7
-3
-2
-1
0
1
2 log P(days)
Fig. 50. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) period distribution of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.002. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 5. The red line corresponds with case BB scenario 1 in combination with PNS set 1.
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N/Ntot
N/Ntot
0.07
0.14
Z=0.02 NS+NS
0.06
Z=0.02 0.12
0.05
0.1
0.04
0.08
0.03
0.06
0.02
0.04
0.01
0.02
NS+BH
0
0 -3
-2
-1
0
1
2
3
4
5
6
-3
7
-2
-1
0
log P(days)
1
2
3
4
5
6
7
3
4
5
6
7
log P(days)
N/Ntot
N/Ntot
0.09
0.08
Z=0.02 BH+BH
Z=0.02 BH+NS
0.08 0.07
0.06
0.06 0.05
0.04
0.04 0.03
0.02 0.02 0.01
0
0 -3
-2
-1
0
1
2
3
4
5
6
log P(days)
7
-3
-2
-1
0
1
2
log P(days)
Fig. 51. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) period distribution of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.02. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 1. The blue line corresponds with case BB scenario 5 in combination with PNS set 1.
BH + NS systems have larger space velocities than NS + BH systems because the NS is formed during the second SN explosion. Also, the space velocity acquired during the first SN explosion is much lower (10–50 km/s) due to the larger periods of the CheB + OB systems. Double BH binaries have the lowest space velocities of all (<10 km/s). As a major result we find that the adopted case BB evolutionary scenario in He + NS/BH binaries significantly influences the predicted distribution of the orbital decay timescales of double compact star binaries. 4.9.2.2. The predicted formation and merger rate. Since the galactic enrichment history of the r-process elements is expressed as a function of the metallicity, we display the evolution of the galactic merger rate of double compact star binaries vs.
[Fe/H]. The predicted evolution of the merger rate of double NS and mixed (=BH + NS and NS + BH together) pairs is given in Fig. 57 for the different case BB scenarios. Fig. 58 shows the effect of the PNS parameters on the merger rate. In Table 20 we list the predicted annual formation rate for the metallicities Z = 0.002 and 0.02. From our simulations we draw the following main conclusions. 1. The predicted present formation rate of NS + NS systems and their merger rates range within (106–104) yr1. Their formation is most influenced by the CE efficiency during the spiral-in phase after the first SN explosion (a factor of 2), the kick velocity (a factor of 20 variation), the adopted case BB RLOF scenario (a factor of 1.2–4 variation) and the amount of mass and angular momentum
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957
N/Ntot
N/Ntot
1
0.35
Z=0.002 NS+NS
0.3
Z=0.002 NS+BH
0.9 0.8
0.25
0.7 0.6
0.2
0.5 0.15
0.4 0.3
0.1
0.2 0.05
0.1 0
0 0
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0.9
0
1
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0.5
0.6
0.7
0.8
0.9
1
e
e N/Ntot
N/Ntot 0.25
1
Z=0.002 BH+NS
Z=0.002 BH+BH
0.9
0.2
0.8 0.7
0.15
0.6 0.5
0.1
0.4 0.3
0.05
0.2 0.1
0
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
e
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
e
Fig. 52. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) eccentricity distribution of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.002. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 5. The red line corresponds with case BB scenario 1 in combination with PNS set 1.
lost from the system during non-conservative RLOF (a factor of 2–3 variation). These predicted rates agree with the observational estimated values (Belczynski et al., 2002). Notice that the merging of BH + NS binaries happens earlier than the one of NS + NS binaries. 2. NS + BH systems form at higher rates (up to a factor of 10–20) than NS + NS systems do because BHs form with smaller or no kicks, which means a lower chance for the binary to become disrupted. Double BH systems form the dominant class with birthrates of 104 yr1. They originate from binaries that avoid RLOF because of mass loss via SW (RSG and LBV scenario). They have large final periods and are not expected to merge within Hubble time.
4.9.3. The predicted evolution of [Eu/Fe] vs. [Fe/H] Detailed r-process calculations for merging neutron stars have been performed recently by Freiburghaus et al. (1999b). Merging neutron stars potentially can provide in a natural way the large neutron fluxes that are required for the build-up of heavy elements through rapid neutron capture. Newtonian calculations of merging NSs show that a few times 103–102 Mx of r-process matter might be ejected in one merger event. This amount is significantly larger than the typical values of 105–106 Mx of r-process material that is thought to be ejected in a SN event (e.g. Wanajo et al., 2001). However since the rate of NS mergers in the Milky Way is significantly lower than that of type II SNe, either of these two sources may account for the total amount of r-process matter in the Milky Way. We remark that recent NS merger
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N/Ntot
1
0.45
Z=0.02
0.4
Z=0.02 NS+BH
0.9
NS+NS
0.8
0.35
0.7
0.3
0.6
0.25 0.5
0.2
0.4
0.15
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0.1 0
0 0
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0
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e
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0.9
1
e N/Ntot
N/Ntot 0.25
1
Z=0.02 BH+NS
0.2
Z=0.02 BH+BH
0.9 0.8 0.7 0.6
0.15
0.5 0.4
0.1
0.3 0.2
0.05
0.1 0
0 0
0.1
0.2
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0.4
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0.6
0.7
0.8
0.9
e
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
e
Fig. 53. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) eccentricity distribution of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.02. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 1. The blue line corresponds with case BB scenario 5 in combination with PNS set 1.
computations that include general relativistic effects suggest lower amounts of ejected r-process material, i.e. 5 · 105–2 · 104 Mx (Oechslin et al., 2002). We compute the galactic evolution of the abundance of r-process elements assuming that merging NS + NS and NS + BH systems are the major rprocess source and that SNe do not contribute to the enrichment in r-process elements. Additionally we assume that the amount of matter ejected during the merger event is independent from the total mass of the system and the kind (NS or BH) of merging compact objects, all the ejected material is r-process material. Our computations are made with a constant binary fraction of 70% during galactic evolution,
the metallicity dependent SD scenario for the SNIas and with the iron yields from Woosley and Weaver (1995) reduced by a factor of two. For the solar r-abundance we take Xr,x 107 (Ka¨ppeler et al., 1989). Remark that the moment at which the merger releases its r-process yields is given by the timescale after which the original binary is transformed into a NS + NS or BH pair plus the timescale of orbital decay by GWR. In order to compare our predictions with the observations, we consider the element europium, which is almost purely (95%) produced by the r-process and therefore has a behaviour that is typical for an r-process element. Due to its numerous clean atomic lines in the visible part of the spectrum, it is one of the few elements (among the existing r-process elements) that has been observed.
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N/Ntot 0.1
0.09
Z=0.002 NS+NS
0.08
Z=0.002 NS+BH
0.09 0.08
0.07
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log Td(yr)
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8
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log Td(yr) N/Ntot
N/Ntot 0.035
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Z=0.002 BH+NS
0.03
Z=0.002 BH+BH 0.04
0.025 0.03
0.02 0.015
0.02
0.01 0.01
0.005 0
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11
12
13
14
0
15
1
2
3
4
5
log Td(yr)
6
7
8
9
log Td(yr)
Fig. 54. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) distribution of the orbital decay timescale of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.002. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 5. The red line corresponds with case BB scenario 1 in combination with PNS set 1. N/Ntot
N/Ntot
0.08
0.09
Z=0.02 NS+NS
0.07
Z=0.02 BH+NS
0.08 0.07
0.06
0.06
0.05 0.05
0.04 0.04
0.03 0.03
0.02
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0
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14
15
log Td(yr)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
log Td(yr)
Fig. 55. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) distribution of the orbital decay timescale of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.02. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 1. The blue line corresponds with case BB scenario 5 in combination with PNS set 1.
Fig. 59 compares the predicted [Eu/Fe] vs. [Fe/ H] relation (for the different case BB scenarios and PNS set 1) with the observational data from
Woolf et al. (1995) for the disk stars and from Burris et al. (2000) for the halo stars. The error bars (shown below in the left corner of the figure)
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N/Ntot
N/Ntot
0.25
0.18
Z=0.002 NS+NS
Z=0.002 BH+NS
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0.2
0.14 0.12
0.15 0.1 0.08
0.1
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300
v(km/s)
Fig. 56. The normalized (respectively to the total number of NS + NS and BH + NS systems) space velocity distribution of NS + NS and BH + NS at the time of their formation for Z = 0.002 and Z = 0.02. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 3. The red (blue) line corresponds with case BB scenario 1 (5) in combination with PNS set 1.
give the typical measurement error on the data for the disk stars. For the halo stars no errors are reported by Burris et al. (2000), though they argue that the observed spread at low Z is certainly not the result of measurements errors but due to inhomogeneous evolution. The high [Eu/ Fe] ratio (and also of other r-process elements) observed in a few halo giants (= triangles on the figure) indicates that local enrichments took place in an unmixed halo. The effect of the PNS parameters on the predicted evolution of the [Eu/Fe] ratio is given in Fig. 60. When assuming that mass transfer from the helium star to the NS always results in CE evolution and that at least 0.004 Mx of r-process material is ejected during merging, the observed [Eu/Fe] trend is well reproduced for [Fe/ H] P 2.5. Below [Fe/H] = 2.5 the predicted [Eu/Fe] ratio decreases steeply with [Fe/H] and is underproduced compared to the average ob-
served values. Larger ratios but with an overproduction at metallicities [Fe/H] > 2.5, are obtained for Mej = 0.04 Mx which is according to present hydrodynamical computations the maximum amount of r-process material that can be ejected during a merger event. If instead of CE evolution, case BB mass transfer mainly leads to the formation of a NS SW and no CE phase is encountered, the enrichment in r-process elements starts too late during galactic evolution, due to the larger orbital periods with which double NS/BH pairs are formed and consequently larger merging timescales. The [Eu/Fe] vs. [Fe/ H] relation is affected by the PNS parameters in a similar way as the merger rate of NS + NS or BH systems. A large average kick velocity ði:e: vk ¼ 450 km=sÞ and/or a small CE ejection efficiency (i.e. a = 0.5) reduces the merger rate of NS + NS or BH systems and therefore also the enrichment in r-process elements. From these
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Fig. 57. The predicted time evolution of the galactic merger rate of NS + NS (upper figure) and mixed (=BH + NS and NS + BH together) (lower figure) pairs for the different case BB scenarios in combination with PNS set 1.
main results we basically conclude that the contribution from merging NS + NS or BH pairs to the galactic enrichment in r-process elements depends critically on the effects on the orbital binary parameters of an asymmetric SN explosion, the physics of CE evolution, the adopted scenario and physics of case BB evolution,
the amount of ejected r-process material during the merger event. According to current theoretical evolutionary computations for Z = 0.02 (Dewi and Pols, 2003; Ivanova et al., 2002) the majority of the systems with a NS component and a Roche lobe filling He star, evolves through a stable mass transfer rather than through a CE phase due to a SW that is assumed to develop in the vicinity of the NS and
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Fig. 58. The predicted time evolution of the galactic merger rate of NS + NS (upper figure) and mixed (=BH + NS and NS + BH together) (lower figure) pairs for the different PNS sets in combination with case BB scenario 5.
to expel isotropically all transferred matter once the Eddington accretion limit is exceeded. This implies that the number of very short period NS + NS and NS + BH systems significantly decreases and according to our simulations, we would be inclined to conclude that merging NSs are very likely not the main contributors to the galactic enrichment in r-process elements.
However the NS SW is mainly driven by the iron lines and it is not unreasonable to think that at low metallicity the wind may not be strong enough to expel the transferred matter and a common envelope may form instead. In this case NS + NS and NS + BH systems form with merging timescales that are sufficiently short to produce significant amounts of r-elements at very low met-
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Table 20 The predicted actual (t = 15 Gyr) formation rate (in 106 yr1) of the different kinds of double compact star binaries for different PNS parameter sets combined with case BB scenario 1 (first number) and 5 (second number) Set
NS + NS
NS + BH
BH + NS
BH + BH
1 2 3 4 5 6
3.68j14.6 2.31j8.80 4.18j16.9 1.45j5.97 1.86j3.98 69.4j110
65.9j92.4 23.2j30.4 86.4j122 6.89j9.24 65.9j92.4 217j268
0.81j5.02 0.56j1.83 0.92j4.93 0.70j3.62 1.11j0 6.05j7.14
408j408 142j174 412j519 261j330 293j409 326j409
Fig. 59. The predicted [Eu/Fe] vs. [Fe/H] relation for the different case BB scenarios, PNS set 1 and Mej = 0.04 (upper figure) and 0.004 Mx (lower figure). The error bars give the typical measurement errors on the observed data for the disk stars.
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Fig. 60. The predicted [Eu/Fe] vs. [Fe/H] relation computed for the different PNS models in combination with case BB scenario 5 and Mej = 0.04 (upper figure) and 0.004 Mx (lower figure). The error bars give the typical measurement errors on the observed data for the disk stars.
allicities in the early Galaxy that correspond with the observations. 4.9.4. Comparison to other studies Accounting for the different PNS assumptions, our predicted ranges of the formation/merger rate of double compact star binaries and the distributions of their physical properties do not much differ from those of other studies (e.g. Portegies Zwart
and Yungelson, 1998; Fryer et al., 1999; Belczynski et al., 2002). The formation rate of NS + BH and BH + BH systems are often not reported in PNS studies because of their negligible small merger rate. Non-zero BH + BH merger rates of the order of the NS + NS merger rate have been predicted by (e.g. Tutukov and Yungelson, 1993; Lipunov et al., 1996; Belczynski et al., 2002). They form when mass loss by SW is strongly reduced, which results in
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smaller periods and therefore shorter merger times. This situation corresponds with our computations at low metallicity. In their model G2, which correspond with higher mass losses similar to our adopted rates at Z = 0.02, Belczynski et al. (2002) find a zero merger rate, which is in agreement with our results. Also remember that in many PNS studies the Langer SW formalism for helium stars is adopted (e.g. Portegies Zwart and Yungelson, 1998), which reduces strongly the formation rate of BHs. It is currently accepted that these rates overestimate the mass loss by a factor of 2–4. Belczynski et al. (2002) investigated the effect of hyper-critical accretion during CE phases on the NS + NS merger rate, which we have not included in our computations. In general, hyper-critical accretion leads to heavier inspiraling compact objects and larger post-CE orbital periods, because less envelope mass has to be ejected and therefore less orbital energy is needed. The merger rate of NS + NS systems is increased or decreased depending on the adopted maximum NS mass. From their results Belczynski et al. (2002) conclude that when M NS max 6 2 M , (40–80)% of the NSs in the CE phase turn into a BH and thus strongly reduces (by up to a factor of 6) the NS + NS merger rate. On the contrary for M NS max ¼ 3 M the rate increases with 20%. Accounting for inhomogeneous chemical evolution, Argast et al. (2004) computed recently the galactic enrichment in r-process elements whereby considering (separately) supernovae and double NS mergers as the production sources of r-elements. Despite they do not perform PNS modeling in order to compute in a correct way the formation and merger rate of double NSs, they come to the general conclusion that double NS mergers are probably not the dominant r-process. They argue that due to their low occurrence rate and high ejected r-process yields (compared to the SN case) merging double NSs start to enrich too late during galactic evolution and cause considerable chemical inhomogeneities in the ISM during the thin disk phase, which are not observed. In their paper Argast et al. (2004) only show the [Ba/Fe] vs. [Fe/ H] relation (whereby only accounting for the rcontribution to the Ba abundance) computed with their double NS merger model and it is therefore
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difficult to make a direct comparison with our simulations. Moreover, they adopt different iron core-collapse yields and a very simplified method to compute the SNIa rate, which results in a different [Fe/H] evolution with time. An important result from their study to mention, is that with their inhomogeneous chemical evolutionary model the observed scatter in the abundance ratios at low metallicity can be reproduced.
5. Summary and conclusions In this work, detailed binary evolution is included in chemical evolutionary computations for the first time, with the aim to study the effects that binaries may have on galactic chemical evolution. The formation of the Milky Way is simulated with the two-infall model of Chiappini et al. (1997). Although this model does not include star and gas dynamics, its main assumptions on the formation of the halo and disk (i.e. timescale and rate of infall, star formation) are supported by recent hydrodynamical simulations of the Milky Way. However, since comparisons are made mutually between model predictions with and without binaries but always with the inclusion of SNIas, any weaknesses or shortcomings in the galactic model that are linked to dynamics, affect evenly all our model predictions. Therefore our conclusions on the pure chemical effects of binaries hardly depend on them. Since we assume chemical homogeneous evolution, which is only expected during the evolution of the galactic disk and the late halo phase, our model results are averages which are compared to the observed average trends. As is usually done in chemical evolutionary computations, we computed abundance ratios rather than absolute abundances because they only depend on nucleosynthesis, stellar lifetimes and the initial mass function. To include binaries in our chemical evolutionary calculations we combined a PNS model with the galactic chemical evolutionary model. The PNS code models in detail the evolution of single and binary stars whereby accounting for all binary processes. Since several aspects of binary evolution are not yet completely understood, the modelation of the different binary interaction processes
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contains a number of free parameters which introduces some extra uncertainties. A major free parameter is the birth frequency of binaries. The physics of binary formation is not well known and we have to rely on the observations. In order to constrain the parameters and to set up a standard PNS model for our chemical evolutionary computations, we first computed with our PNS model the massive star population in the solar neighbourhood and the Magellanic Clouds. The observed average metallicity in the Magellanic Clouds is lower than the solar one and therefore the Clouds are excellent objects to measure the influence of metallicity on PNS modeling. We focused on the population of OB-type and Wolf– Rayet stars, the formation rate of SNe and the space velocity distribution of single radio pulsars. The best agreement with the observations in the solar neighbourhood is obtained when the initial fraction of interacting massive binaries is larger than 50% and when the majority of massive binaries is disrupted after the first SN explosion implying a preferred average kick velocity of 450 km/s. The required initial fraction of interacting massive binaries in order to reproduce the observed WR star population in the SMC, is much lower (20%) and could indicate that the binary formation rate is a function of metallicity. Furtheron, we find good results for the SMC when the WR p SW ffiffiffi mass loss rate relates to the metallicity Z as Z . Any modifications in our adopted RSG and WR wind mass loss formalisms leads to problems in reproducing the WC/WN number ratio. Although these comparisons with the observations can not really tell us if our adopted models for single and binary star evolution are correct, they give at least an indication that our models are reliable. Combining our PNS model with the two-infall galactic model, we simulated the chemical evolution of the solar neighbourhood for different assumptions on the binary fraction, i.e. a constant high binary fraction of 70% and one that grows linearly from zero at Z = 0 to 70% at Z = 0.02. In both cases the same binary fraction has been taken over the whole mass range. Both assumptions are inspired by our PNS predictions on the massive star population in the solar neighbourhood and Magellanic Clouds.
Binary evolution mainly affects the galactic evolution of carbon and iron. Interacting binaries of low and intermediate mass produce less carbon than their single counterparts because the majority of them avoid the dredge-up phases during late AGB evolution. In this particular evolutionary phase large amounts of carbon are transported to the surface and ejected by a superwind. This binary effect translates into a flatter evolution of the [C/Fe] abundance ratio during the thin disk phase compared to model predictions with only single stars and SNIas. A small bump in the [C/ Fe] abundance ratio during the disk evolution is also present in the averaged observations though much less pronounced than predicted by the model with exclusively single stars and SNIas. Except for the assumed binary fraction, the predicted evolution of carbon does not much depend on the adopted values for the binary parameters. The production and evolution of iron mainly depends on the production rate and formation timescale of SNIas and also on the adopted iron yields of massive stars. In this work we adopted the core-collapse iron yields from Woosley and Weaver (1995) and the single and/or double degenerate SNIa model. When combining the WW95 iron yields with the standard SNIa iron ejecta (i.e. 0.7 Mx) and an SNIa model that reproduces the observed local SNIa rate, we find [Fe/H] ratios as a function of time that are in general (for T > 1 Gyr) larger than the observed averages and [X/Fe] ratios that underproduce the observations especially during the halo, thick disk phase. The observed averaged run of the [Fe/H] and [X/Fe] ratios is better reproduced if the core-collapse iron yields are reduced with a factor of two. The use of lower iron yields was also suggested in earlier studies on chemical evolution and is supported by recent detailed spectral analysis of some well observed core-collapse SNe. The contribution of SNIas, which are formed by interacting intermediate mass binaries, to the total iron production is about 30% and increases to 50% when the WW95 iron yields are reduced with a factor of two. The precise value of these fractions depend on the used SNIa model (SD or DD model) and the related binary parameters, i.e. the common envelope ejection efficiency a and the accretion parameter bmax during RLOF.
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In order to explain the observed Ia/Ibc SN number ratio in early and late type spirals (which is too low when only the SD model or DD model is assumed) and the observed low [a/Fe] ratios in DLAs at high redshift (which requires the production of SNIas at [Fe/H] < 1), we suggest in this work that SNIas are formed by both the SD and DD model. In this case the SNIas dominate the galactic iron production by 70%. The moment at which the SNIas start to influence the iron evolution and by consequence the [X/ Fe] ratios is determined by the timescale on which SNIas are formed. We computed with our PNS model the timescales for both SNIa scenarios. The output is very important: the moment during galactic evolution at which the first SNIas start to form and the moment at which they reach a maximum sensitively depends on the adopted SNIa model and the values of the related binary parameters. Therefore we emphasize in this work that in order to compute in a correct way the time evolution of the SN rates, detailed PNS modeling must be included into a galactic chemical evolutionary model. This conclusion is enforced by computing the G-dwarf disk metallicity distribution which is one of the major constraints on CEMs and critically depends on the iron evolution. We find the best agreement with the observed distribution for the scenario in which SNIas are produced by both the SD and DD model. Binaries may provide the production of primary nitrogen which is needed to explain the observed [N/Fe] ratio during the halo, thick disk phase. We propose the following scenario: when the massive primary star explodes, carbon enriched material may be accreted by the secondary from the SN shell and consequently be CNO processed into nitrogen. Next to some single star scenarios like rotation that may offer a solution to the nitrogen discrepancy as well, the binary mechanism deserves to be further investigated. With our adopted SW mass loss rates, massive BHs (>10 Mx) are formed by stars with an initial mass above 40 Mx which explains well the observed masses of BH components in X-ray binaries. Whether these BHs are formed with or without a SN outburst is presently a matter of debate. If an explosion occurs they are thought to
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produce hypernovae which are observed as very energetic SN explosions. In order to reproduce the observed lightcurve of hypernovae with current supernova model computations large amounts of iron (10 times the amount ejected by a standard SN) are assumed to be ejected. We made some simulations with these hypernova models and find a strong underproduction of the [X/Fe] ratios during the halo evolution due to the large iron ejecta. Because of their short lifetimes stars above 40 Mx mainly affect the galactic enrichment during the halo phase. If these stars indeed explode we illustrate that mainly oxygen and only a small amount (<0.07 Mx) of iron should be ejected to recover the observations. Since we expect (from our PNS computations of the WC/ WN number ratio in the SMC) that the RSG and WR SW mass loss rates decrease with metallicity, the mass layers of the CO-core that are ejected are mainly composed of oxygen. If the SW mass loss would be independent from the metallicity we expect an overproduction of carbon during the major part of galactic evolution. The ejection of large amounts of a-elements and few iron during BH formation is in agreement with the observed large (oversolar) abundances of a-elements and a normal iron abundance (solar) in the atmosphere of the BH companion in the X-ray binary Nova Scorpii. Since the companion star is a low mass star, these extra amounts of a-elements are very likely accreted from the material ejected during the formation of the BH. Finally, we computed the galactic enrichment in r-process elements assuming that only merging NS + NS and NS + BH binaries produce r-process elements. Their contribution depends critically on the treatment of mass transfer in He star + NS binaries, the common envelope ejection efficiency, the supernova kick and the amount of ejected rprocess material during merging. Our simulations reveal that even accounting for the uncertainties in the r-process yields and the PNS parameters, binary mergers cannot reproduce the overall observed galactic evolution of r-process elements if the majority of He star + NS or BH binaries avoids CE evolution and this independent of the metallicity. However present uncertainties in the evolution of He + NS or BH systems do not allow
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to firmly conclude that NS + NS/BH mergers are not a dominant source of r-process elements. As a general conclusion, we can state that CEM predictions with binaries differ by no more than a factor of 2–3 from the predictions by models that only include single stars and SNIas. Since we computed abundance ratios, our results hardly depend on the adopted galactic model parameters and therefore should also hold for simulations with other CEMs that are different from the two-infall model.
Appendix A. Numerical solution of the isotope evolution equation
the general solution of Eq. (A.3) is Z t Gi ðr; tÞ ¼ exp gðr; t Þdt "Z Z 0 t
This general solution can be further transformed into a first order linear recurrence relation an + 1 = ban + c. This is done as follows: Z t1 Gi ðr; t1 Þ ¼ exp gðr; t Þdt
Z Wðr; tÞ dt I:F ¼ exp Gðr; tÞ Z ¼ exp gðr; tÞdt
Z
t1
gðr; t Þdt
0
t2
ðA:5Þ
gðr;t Þdt
0
exp
Z
Z ¼ exp
Z
t1 0 t1
gðr;t Þdt
t2
gðr;t Þdt
t1
ðA:1Þ þ
Z
t1
0
W i ðr;t Þexp
Z
0
þ
Z
!
t0
gðr;t Þdt
t2
0
W i ðr;t Þexp
Z
!
t0
gðr;t Þdt
0
t1
Z exp
t2
gðr;t Þdt
t1
Z þ exp
t1
0 t2
gðr;t Þdt
gðr;t Þdt
t1
t2
0
W i ðr;t Þexp
Z
Gi ðr;t1 Þ
t1
Z
dt0
0
Z þ const ¼ exp
ðA:3Þ
W i ðr;t0 Þ
gðr;t Þdt dt0 þ const
0
ðA:2Þ
t2 0
Z exp
or
Taking the integrating factor
W i ðr; t Þ exp
Z Gi ðr;t2 Þ ¼ exp
can be written as a linear differential equation:
dGi ðr; tÞ Wðr; tÞ ¼ Gi ðr; tÞ þ W i ðr; tÞ: dt Gðr; tÞ
0
and
þ Wðr; t tM 2 ÞRM 2;i ðt tM 2 Þ dP dq dM 1
dGi ðr; tÞ ¼ X i ðr; tÞWðr; tÞ þ W i ðr; tÞ dt
t1
dt þ const
Pl
dGi;inf ðr; tÞ dt
0
Z
0
PðP Þ½Wðr; t tM 1 ÞRM 1;i ðt tM 1 Þ
þ
dt0 ðA:4Þ
0
RM;i ðt tM ÞdM Z M 1;u Z qu Z P u þ fb ðt tM 1 ÞuðM 1 Þ/ðqÞ ql
gðr; t Þdt
þconst :
Ml
M 1;l
W i ðr; t Þ exp
#
For each isotope i the Eq. (3.3) dGi ðr; tÞ dt ¼ X i Wðr; tÞ Z Mu þ ð1 fb ðt tM ÞÞWðr; t tM ÞuðMÞ
!
t
0
!
t0
gðr;t Þdt
dt0 :
0
ðA:6Þ
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975
Finally, the analytical solution can be written as
Z Gi ðr; tnþ1 Þ ¼ Gi ðr; tn Þ exp Z
tnþ1
gðr; t Þdt
tn
W i ðr; t0 Þ tn Z tnþ1 exp gðr; t Þdt dt0 ; þ
with Gi(r)(k) the best estimate for Gi(r)n + 1 at the kth iteration and diðkþ1Þ ¼
tnþ1
969
Gi ðrÞðkÞ Ai bi Gi ðrÞ
ðkÞ
ðA:11Þ
;
W i ðrÞ W i ðrÞ ; Ai ðrÞ ¼ expðgðrÞDtÞ Gi ðr; tn Þ þ gðrÞ gðrÞ
t0
ðA:12Þ
ðA:7Þ
with gðr; tÞ ¼
bi ðrÞ ¼
Wðr; tÞ Gðr; tÞ
and W i ðr; tÞ ¼
Z
ðA:13Þ
Mu
ð1 fb ðt tM ÞÞWðr; t tM ÞuðMÞ
Ml
RM;i ðt tM ÞdM Z M 1;u Z qu Z P u fb ðt tM 1 Þ þ M 1;l
ql
Pl
uðM 1 Þ/ðqÞPðP Þ Wðr; t tM 1 ÞRM 1;i ðt tM 1 Þ þWðr; t tM 2 ÞRM 2;i ðt tM 2 Þ dP dq dM 1 þ
dGi;inf ðr; tÞ : dt
Gi ðr; tnþ1 Þ ¼ Gi ðr; tn Þ expðgðrÞDtÞ þ ½1 expðgðrÞDtÞ :
W i ðrÞ gðrÞ ðA:9Þ
Because g and Wi are not constant over Dt we need to iterate to obtain convergency. We perform our iteration on g. This mainly because Wi is less sensitive to the value of Gi(r,tn+1) (it is an integral over all the past values of Gi) than g. As shown by numerical computations, the improvement that is obtained by iterating on both terms g and Wi is small. The solution is now obtained by executing the following successive corrections: ðkþ1Þ
ðkÞ
¼ Gi ðrÞ ½1 þ dðkþ1Þ ; i
The iteration is stopped when diðkþ1Þ < dmax . The length of the timestep is chosen such that DGi(r)/ Gi(r) and Dr(r)/r(r) is 6 0.05. For the evaluation of the Wis we need to know which stars, previously formed, will contribute to the chemical enrichment at the considered age t of the model. Therefore we rewrite the equation for Wi(r,t) as follows:
ðA:8Þ
When integrating over a timestep Dt = tn + 1 tn we take g and Wi constant and equal to their value estimated at tn + 1/2 = tn + Dt/2 which gives
Gi ðrÞ
1 o ln gðrÞ W i ðrÞ ðexpðgðrÞDtÞ 1Þ 2 o ln GðrÞ gðrÞ W i ðrÞ gðrÞDt expðgðrÞDtÞ Gi ðtn Þ : gðrÞ
ðA:10Þ
W i ðr; tÞ Z ttðM u Þ ð1 fb ðt0 ÞÞWðr; t0 Þ ¼ 0 dM uðMÞRM;i ðt0 Þ dt0 dtM 0 Mðtt Þ Z ttmin Z qu Z P u þ /ðqÞPðP Þfb ðt0 Þ 0
ql
Pl
dM 1 Wðr; t ÞuðM 1 Þ RMi;1 ðt t Þ dtM 1 M 1 ðtt0 Þ ! dM 2 þ RMi;2 ðt t0 Þ dtM 2 M 2 ðtt0 Þ 0
dP dqdt0
dGi;inf ðtÞ : dt
0
ðA:14Þ
The upperlimit for the time integration in case of binaries depend on the fate of the considered binary system. If the primary evolves the fastest we have tmin ¼ tM 1;u , if the secondary accretes that
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much matter lost from the primary that it becomes more massive than the primary initially was it may end its evolution before the primary and is tmin ¼ tqM 1;u , and in case of an early merger the merger object may have a shorter lifetime than the original primary star, so we have then tmin ¼ tM merger . It is clear that since all the quantities which are involved in Eq. (A.14), depend on the metallicity, all the values of Gi(r,tn), Xi(r,tn) and Z(r,tn) (in a specific ring) on every past time tn of the model must be kept in memory throughout the calculation. For the integration over time we approximate the integral by a sum over series of time intervals [t(k 1), t(k)] with t0 = 0 and tðkmax Þ ¼ tnow . Within each time interval, the integrand is taken constant and estimated at the midpoint of the interval. For single stars (first integral) the integral is calculated on the path of the (M, t) plane which is fixed by the relation between mass and lifetime. For binary stars however the situation is more complicated. There we need to calculate the integral for each fixed pair of q and P, on the (M1, t) plane and (M2, t) plane. References Abt, H.A., 1983. ARA&A 21, 343. Abt, H.A., Levy, S.G., 1985. ApJSS 59, 229. Akmal, A., 1998. PhD. Thesis, University of Illinois at UrbanaChampaign. Anders, E., Grevesse, M., 1989. Geochim. Cosmochin. Acta 53, 197. Andreani, P., Vangioni-Flam, E., Audouze, J., 1988. ApJ 334, 698–706. Argast, D., Samland, M., Gerhard, O.E., Thielemann, F.-K., 2000. A&A 356, 873. Argast, D., Samland, M., Thielemann, F.-K., Gerhard, O.E., 2002. A&A 388, 842. Argast, D., Samland, D., Thielemann, F.-K., Qian, Y.-Z., 2004. A&A 416, 997. Arzoumanian, Z., Chernoff, D.F., Cordes, J.M., 2002. ApJ 568 (1), 289–301. Avila Reese, V.A., 1993. Rev. Mexicana Astron. Astrof. 25, 79– 89. Beers, T.C., Sommer-Larsen, J., 1995. ApJS 96, 175. Belczynski, K., Kalogera, V., Bulik, T., 2002. ApJ 572, 407. Benz, W., Bowers, R.L., Cameron, A.G.W., Press, W.H., 1990. ApJ 348, 647–667. Berczik, P., 1999. A&A 348, 371. Bernkopf, J., Fuhrmann, K., 1998. In: Spite, M. (Ed.), Galaxy Evolution. Editions Frontie`res, Meudon, Paris.
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The influence of binaries on galactic chemical evolution Erwin De Donder *, Dany Vanbeveren Astrophysical Institute, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium Available online 25 August 2004
Abstract Understanding the galaxy in which we live is one of the great intellectual challenges facing modern science. With the advent of high quality observational data, the chemical evolution modeling of our galaxy has been the subject of numerous studies in the last years. However, all these studies have one missing element which is the evolution of close binaries. Reason: their evolution is very complex and single stars only perhaps can do the job. (Un)Fortunately at present we know that the majority of the observed stars are members of a binary or multiple system and that certain objects can only be formed through binary evolution. Therefore galactic studies that do not account for close binary evolution may be far from realistic. Because of the large expertise developed through the years in stellar evolution in general and binary evolution in particular at the Brussels Astrophysical Institute, we found ourselves in a privileged position to be the first to do chemical evolutionary simulations with the inclusion of detailed binary evolution. The complexity of close binary evolution has kept many astronomers from including binary stars into their studies. However, it is not always the easiest way of living that gives you the most excitement and satisfaction. 2004 Elsevier B.V. All rights reserved.
Abbreviations: AGB, asymptotic giant branch; BH, black hole; BHG, blue Hertzsprung gap; BSG, blue supergiant; CE, common envelope; CEM, chemical evolutionary model; CHB, core hydrogen burning; CHeB, core helium burning; CO WD, carbon–oxygen white dwarf; E-AGB, early asymptotic giant branch; GWR, gravitational wave radiation; HBB, hot bottom burning; HeSB, helium shell burning; HN, hypernova; HRD, Hertzsprung–Russell diagram; HSB, hydrogen shell burning; IMA, instantaneous mixing approximation; IMCB, intermediate mass close binary; ISM, interstellar medium; LBV, luminous blue variable; LMC, Large Magellanic Cloud; MCB, massive close binary; MS, main sequence; NLTE, non-local thermodynamic equilibrium; NS, neutron star; PN, planetary nebula; PNS, population number synthesis; RLOF, Roche lobe overflow; RSG, red supergiant; SMC, Small Magellanic _ _ Cloud; SN, supernova; SW, stellar wind; TP-AGB, thermal pulsating asymptotic giant branch; TZO, Thorne–Zytkov object; WD, white dwarf; WR, Wolf–Rayet star; YSG, yellow supergiant; ZAMS, zero age main sequence. * Corresponding author. E-mail address: [email protected] (E. De Donder). 1387-6473/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.newar.2004.07.001
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Contents 1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Galactic chemical evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Binary effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Goal and outline of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The population synthesis model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. General concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Single star evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Low and intermediate mass stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Massive stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Summary of the single star evolutionary model . . . . . . . . . . . . . . . . . . . . . . . 2.3. Binary star evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Mass transfer and accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Evolution of the binary orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. Detailed binary evolutionary model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. PNS predictions of massive star populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The chemical evolutionary model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The two-infall scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The star formation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The initial mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. The binary fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. The isotopic evolution model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Computation of the supernova rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. The stellar yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1. Single star yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2. Binary star yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3. Yields from one stellar generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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The chemical evolution of the solar neighbourhood. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Star formation rate and surface densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Intermezzo: the WR/O number ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The age–metallicity relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. The G-dwarf disk metallicity distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. The solar abundances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. The abundance ratios evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1. The CNO elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2. The a-elements Mg, Si, S and Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3. Comparison to other studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.4. Overall conclusion on the time evolution of the abundance ratios . . . . . . . . . . 4.6.5. The influence of rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. The effect of BH formation above 40 Mx and metallicity dependent SW mass loss . . . 4.7.1. Comparison to other studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8. The supernova rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1. Observed rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2. Overview of the progenitor populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3. The evolutionary clock of supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4. The galactic evolution of the SN rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.5. Comparison to other studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.9.
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The evolution of the r-process elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1. The formation of NS + NS and NS + BH binaries . . . . . . . . . . . . . . . . . . . . 4.9.2. The theoretically predicted population of double compact star binaries . . . . . 4.9.3. The predicted evolution of [Eu/Fe] vs. [Fe/H] . . . . . . . . . . . . . . . . . . . . . . . 4.9.4. Comparison to other studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Numerical solution of the isotope evolution equation . . . . . . . . . . . . . . . . . . . . . . . . . . 968
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 970
1. Introduction 1.1. Galactic chemical evolution To understand the behaviour of the Universe we need to understand the formation and evolution of galaxies, which are the main constituents of the Universe. Galaxies are made of stars, interstellar gas and dark matter. Stars evolve and therefore galaxies evolve. A galaxy evolves simultaneously dynamically, thermally, (spectro) photometrically and chemically, which makes its evolution a very complex problem. In this work, we focus on the chemical evolution of our galaxy, the Milky Way, which entails with reconstructing the history of the chemical composition of the interstellar gas. To study the chemical evolution of a galaxy, the cosmological picture is adopted in which it is assumed that all deuterium, a major part of helium and some lithium has been produced during the Big Bang whereas all the heavier elements 1 are made in stars. A galaxy initially forms from a huge lump of gas that as a result of self-gravity decouples from the general expansion of the Universe. The enrichment and spatial distribution of the chemical species within the galaxy occur via various galactic and stellar processes that take place in the course of time. These processes were first
1 As is common in astrophysics, all elements heavier than helium are referred to as metals.
identified by Tinsley (1980). Following the outline, a first generation of metal free stars (= population III stars) forms from the collapsing protogalaxy. They process the matter nuclearly and return it partially or completely to the interstellar medium (ISM) via stellar winds (SWs) and/or supernova (SN) explosions. The ISM becomes enriched in new elements heavier than were initially present. Meanwhile the galaxy may exchange gas with the intergalactic medium or with nearby galaxies whereby its mass, chemical composition, structure and star formation rate change. Successive generations of stars are born, each with a different initial chemical composition and therefore a different evolution. The cyclic process of stellar birth and death continues until no more gas is left to create new stars. Interactions with neighbour galaxies may reactivate and reshape the galaxy. So, in order to simulate the chemical evolution of a galaxy we have to consider the processes of galaxy formation, star formation and stellar evolution. To decipher the formation and evolutionary history of an observed galaxy, the chemical and dynamical properties of its stars are analyzed. The atmospheric composition of unevolved stars reflects in most cases the chemical composition of the interstellar gas from which the stars are formed, whereas stellar kinematics says something about the physical condition of the gas at the moment that the stars were formed. Many details are known about the stellar population in the solar neighbourhood, which is defined as a cylindric region of about 1 kpc around the Sun. Evidence for the chemical evolution in the solar neighbourhood
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Fig. 1. Cross section of the vertical structure of the Milky Way (from Carraro, 2000). Stars with abundances identical to the Sun have [Fe/H] = 0.0; less metal rich stars have negative values and more metal rich stars have positive values.
and elsewhere in the Milky Way is given by the presence of stars with different chemical compositions and ages. The stellar populations that are observed in the various parts of the Milky Way (i.e. the halo, the thick and thin disk and the bulge) all differ in chemical composition, kinematics and dynamics, suggesting that they were formed during different evolutionary phases of the Milky Way (e.g. Wyse and Gilmore, 1992; Beers and Sommer-Larsen, 1995; Gratton et al., 1996). An edgeon cross section of the spatial structure with the average metallicity (relative to the solar value) Æ[Fe/H]æ 2 is given in Fig. 1. The Sun is a typical youngish (4.5 Gyr) thin disk star positioned at 8 kpc from the galactic center and is composed (by mass) of 70% hydrogen (=Xx), 28% helium (=Yx) and 2% metals (=Zx) mainly in the form of carbon, nitrogen, oxygen and iron. The older thin disk stars which represent most of the mass, have on average lower metallicities, higher random motions and lower rotation velocities. About 98% of the local stars are a member of the thin disk. The thick disk stars (2%) dominate at 1–2 kpc above the galactic plane and have a metallicity of about one quarter of the solar value. Only
2
[Fe/H] = log(Fe/H)% log(Fe/H)x.
0.1% of the local stars belong to the stellar halo. They are metal poor subdwarfs (<0.1Zx) with a high velocity, formed in globular clusters. The stars in the bulge have sub- to supersolar metallicities, with the majority having solar metallicity. The first qualitative model for the evolution of the Milky Way was proposed by Eggen et al. (1962). From observationally derived eccentricities and angular momenta of stellar orbits, they deduced that the collapse of the protogalaxy must have started approximately 10 Gyr ago and that both halo and bulge formed within only a few times 108 yr. Much later, Sandage and Fouts (1987) studied the kinematical and chemical properties of a large sample of subdwarfs and concluded contrary to Eggen et al. (1962) that the halo formed during a slow collapse and that spin-up occurred with an increasing collapse factor. In correspondence with the results of Gilmore and Reid (1983) they found a halo, a thick and thin disk component from which they concluded that the dissipation rate changed during the galactic evolution. From a study of globular clusters, Searle and Zinn (1978) did not find any significant radial abundance gradient in the outer halo and proposed a model in which the central region forms first, followed by the infall of gas fragments. The infalling gas accumulates in high density regions which become the sites of cluster and halo
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star formation. The remaining gas undergoes dissipation and is swept into the galactic disk. For the formation of the galactic bulge several scenarios have been invoked: in the model of Eggen et al. (1962) the bulge is formed during the initial contraction of the protogalactic cloud; Wyse and Gilmore (1992) considered its formation from the infall of remaining halo gas; Sofue and Habe (1992) proposed that a significant fraction of bulge stars could have their origin in gas clouds that were ejected by a central starburst; Pfenninger and Norman (1990) showed that a barred potential in the central region can account for the heating of a flat stellar disk. At present it is still not very clear how the different galactic components relate to each other and how they may have mutually affected their formation and evolution. In addition to these qualitative pictures that have been presented in the literature, several numerical simulations of galactic evolution have been performed in order to explain the observations like the local stellar metallicity distribution, abundance gradients, etc. Since different parts of a galaxy influence each other by gas flows and large-scale heating (McCray and Snow, 1979; Norman and Ikeuchi, 1989; Irwin and Seaquist, 1990; Li and Ikeuchi, 1992), the whole Milky Way has to be treated at once in order to obtain a self-consistent description of galactic evolution. Thereto models are needed that treat in detail star formation/ evolution and all important dynamical (e.g. gas flows, rotation) and energetical (e.g. dissipation, heating) processes as well as their dependence on the metallicity (e.g. cooling, stellar evolution). Because of the numerical complexity and the many (unknown) parameters involved in modeling all these processes, the chemical and dynamical evolution of the Milky Way have usually been treated separately. Purely chemical evolutionary models (CEMs) (e.g. Matteucci and Franc¸ois, 1989; Carigi, 1994; Giovanoli and Tosi, 1995; Prantzos and Aubert, 1995; Chiappini et al., 1997, 1999; Goswami and Prantzos, 2000) provide to study in detail the chemical properties of the Milky Way but lack self-consistent gas and star dynamics, while purely dynamical models (e.g. Larson, 1975, 1976; Burkert and Hensler, 1987, 1988) follow the dynamical evolution of the gas and the
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stars but neglect the interactions between them. Dynamical models have mainly been adopted to simulate the structural formation of the Milky Way, while CEMs have been and are still widely used to constrain star formation histories, supernova rates and abundances in the ISM, in the stars and in the intracluster medium. The basic philosophy behind CEMs is that reproducing observed element abundances may already put significant constraints on galaxy formation without considering complicated dynamical processes. Among all the CEMs that have been constructed so far, the most simple ones are closed box models which were first invented and elaborated by Tinsely (1974, 1981). They simulate the Galaxy as an isolated system with a total constant mass, initially composed of pure primordial gas (i.e. metal free), in which the stars and gas are perfectly mixed at all times. Although these models span a wide range of complexity and help to understand how the stars and the ISM affect each other locally, they cannot account for the observed metallicity distribution of long-lived stars in the solar neighbourhood (known as the G-dwarf problem, van den Bergh, 1962; Schmidt, 1963) which is a major observational constraint that needs to be fulfilled by any CEM. Open box models which consider the inflow of primordial or metal poor gas, offer a solution to the G-dwarf problem and are based on the results of dynamical models (Larson, 1972). The main assumption in these models is that the galactic disk is formed over a long timescale (several Gyr, much longer than the halo and bulge formation timescale) by the gradual accretion of unprocessed or partially processed infalling material which only starts to form stars after it has fallen into the disk. The infalling gas is believed to come from the gaseous halo as exemplified by the high-velocity HI clouds, and/or from outside the galaxy. Today, these open infall models are classified as standard chemical evolutionary models since they can satisfactorily (within a factor of 2) reproduce all major observational constraints in the solar neighbourhood and beyound (see Tosi, 2000 for an overview). Beside the lack of dynamics, a main shortcoming of most CEMs is that they follow only large scale and long-term phenomena and not cloud-to-cloud and star-to-star fluctuations
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and therefore can only produce average evolutionary trends. To explain the abundance variations observed among stars of the same age, only a few CEMs have been presented that account for local chemical inhomogeneities by considering inhomogeneous evolution and non-instantaneous mixing of the stellar ejecta with the interstellar gas (e.g. van den Hoek and de Jong, 1997; Argast et al., 2000, 2002, 2004). To meet the shortcomings of purely chemical and dynamical models, chemo-dynamical models have been constructed which try to couple the chemical evolution with the dynamical evolution (e.g. Theis et al., 1992; Hensler et al., 1993; Steinmetz and Mu¨ller, 1994; Samland, 1994; Samland et al., 1997; Berczik, 1999). Due to limited computer capacities, most chemo-dynamical models evolve non-rotating galaxies in one dimension and include a less sophisticated treatment of chemical evolution than classic CEMs. By consequence, only some of the most obvious observational constraints, such as the age–metallicity relation have so far been confirmed. Current chemo-dynamical models make use of the smooth particle dynamics (SPH) (e.g. Lia et al., 2002) which allows to treat the gas and the stars as individual particles and to follow them each in space and time. These sophisticated chemo-dynamical models are at present the most realistic models to study the effects of dynamics and show the way that galactic evolution studies should go in the future. In this work we investigate the influence of binary evolution on galactic chemical evolution, in the framework of a two-infall galactic model. The latter was first proposed by Chiappini et al. (1997) and is at present one of the most sophisticated models among the class of CEMs. The major motivation behind the concept of the formation of the Milky Way in two separated infall periods and its success, is that it explains the observed different metallicity and angular momentum distributions of the different galactic components (mainly halo vs. disk) (Wyse and Gilmore, 1992; Beers and Sommer-Larsen, 1995; Gratton et al., 1996). During the first period, the primordial gas collapses very quickly (1 Gyr) leading to the formation of the halo whereby the gas lost from the halo rapidly accumulates in the center and creates the
bulge. During the second episode a much slower infall of primordial gas (and some traces of halo gas) forms the disk whereby the gas accumulates faster in the inner than in the outer regions. In this scenario the halo and disk are almost complete independently formed and thus allows for the reproduction of the completely different metallicity distributions in the different components. The inside–out mechanism to build up the disk, is based on dynamical simulations of disk formation and quite successfully reproduces the main observed features of the Milky Way as well as of external galaxies especially concerning abundance gradients (see Chiappini et al., 1997; Prantzos and Boissier, 2000). It is important to note that the exponential declining (with time) infall rate and the considered long timescale (8 Gyr) for the formation of the solar neighbourhood (10 kpc from the galactic center) adopted in the two-infall model is supported by recent cosmological chemodynamical simulations of galaxy formation (Sommer-Larsen et al., 2003). Another major feature of the two-infall galactic model is that it includes a star formation threshold in the gas density below which the star formation rate becomes zero. Main consequences of this threshold are that a star formation stop occurs at the end of the first infall period which is in agreement with the hiatus in the star formation suggested by the dynamical model of Larson (1976) and that it prevents gas consumption on a timescale shorter than the age of the Milky Way, which was also a problem encountered in closed box models. A weakness of the two-infall model is the treatment of the galactic disk as an ensemble of independently evolving cylindric rings which form by infall of gas at a different rate without accounting for possible radial flows of gas or stars along the disk (= the one-zone formulation, Talbot and Arnett, 1971). This approximation is often applied in CEMs that focus on the evolution of the galactic disk and the solar neighbourhood. Radial flows may occur as a result of angular momentum transfer due to gas viscosity and the infall of gas with a specific angular momentum different from the one of the underlying disk and as a result from gravitational interaction between the disk gas and spiral density waves that leads to large-scale shocks and dissipation.
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Therefore to assure the conservation of angular momentum during the disk formation, radial gas flows should be included. The effects that radial flows may have on galactic evolution has been explored by several authors (Mayor and Vigroux, 1981; Lacey and Fall, 1985; Clarke, 1989; Chamcham and Tayler, 1994; Portinari and Chiosi, 2000) by adopting parameterizations of the viscosity and flow velocity profiles that are quite arbitrary and ad hoc. From these simulations it turns out that radial flows are expected to be inflows over most of the disk and mainly affect gas and metallicity gradients along the disk. Although they are generally found to be not the main cause of the existence of gradients along the disk, they help to build up gradients but seemingly only under specific combinations of the parameters of infall, radial inflow and star formation rate. However, in a recent paper Chiappini et al. (2001) presented simulations with the two-infall model but with a more detailed treatment of the halo, and they illustrated that the radial profiles of stars and gas observed along the disk can be well reproduced without invoking radial gas flows. Finally, we notice that the two-infall model does not account for possible gas outflows from the Milky Way which could be driven by supernova explosions. Gas outflows or galactic winds are observed in dwarf irregular starburst galaxies and explain the solar iron abundance in the chemical composition of the intracluster medium which indicates that galaxies in clusters should loose a substantial amount of their interstellar gas. For the evolution of the Milky Way it is not clear whether or not galactic winds played a significant role. From the few simulations that have been made so far with models that include gas outflows, it seems that they do not reproduce the abundances and abundance ratios observed in field stars (Tosi et al., 1998). However the physical conditions to create powerful winds are not well known and these results should be considered as preliminary. 1.2. Binary effects It is evident that to correctly interpret the observed abundances and to compute the chemical
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enrichment of the ISM by the stars, reliable models for stellar evolution are needed. The evolution of a star and its contribution to the chemical enrichment of the ISM depend on its initial mass, initial chemical composition, mass loss history and last but not least on whether or not it evolves in isolation or in the presence of a close companion. The effects of binaries is generally ignored in present CEMs and most of them use detailed evolutionary computations of single stars only. Still these simplified models can predict the observations in the solar neighbourhood to within a factor of two (e.g. Timmes et al., 1995; Yoshii et al., 1996; Chiappini et al., 1997, 1999; Portinari et al., 1998; Boissier and Prantzos, 2000; Prantzos and Boissier, 2000). Though, despite this putative success, the importance of binaries in stellar astrophysics has been proven by both observations and theory: (i) A large fraction of all stars observed in the Milky Way have a close companion (e.g. Mermilliod, 2001; Mason et al., 2001) and will interact with consequences for the evolution and appearance of the stars as well as the nature of the orbit (e.g. Iben, 1991; Vanbeveren et al., 1998b). (ii) The existence of many observed stellar objects like type Ia SNe, X-ray binaries, cataclysmic variables, binary pulsars, etc. can only be explained via binary evolution. When a star has a close companion its evolution may become affected in a critical way. Due to strong tidal interaction it may lose or gain large amounts of matter. By consequence it will form a carbon–oxygen (CO) core that is different in mass than expected from single star evolution and therefore may produce different chemical yields. Also the nature of the compact object that is finally left is a function of the mass of the CO core. In this way a binary component of initially intermediate mass may transform into a massive star, produce iron and finally explode as a SN instead of ending up as a white dwarf. During the accretion process part of the star may become mixed with new material that may alter the chemical profile of the outer layers and the SW ejecta.
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Type Ia SN explosions, which are responsible for a large fraction of the present iron abundance in the ISM, are currently believed to be made exclusively via binary evolution. They are exploding carbon–oxygen white dwarfs (CO WDs) which are formed via mass exchange in close binary systems. It is common in chemical evolution modeling to insert the SNIa rate as a free parameter (see Greggio and Renzini, 1983) in order to reproduce the observed iron abundance and the evolution of the a-element abundance ratios [a/Fe] in the solar neighbourhood. Any details on the pre-SN evolution of the progenitor binary system is omitted and the evolution of the system is simulated by the evolution of two non-interacting single stars. The SNIa rate is finally fixed by matching the observed local SNIa rate and iron abundance. Although this artificial procedure seems to give satisfying results, it is certainly not correct. Moreover, it has been shown by De Donder and Vanbeveren (1998) that the SN rates depend critically on the properties of the underlying binary and single star population, which implies that the evolution of the progenitor population must be followed in detail when computing the chemical evolution of the Milky Way as a function of time. Rapid neutron capture reactions 3 on iron-peak nuclei (= r-process) are (partly) responsible for the existence of elements heavier than iron (Suess and Urey, 1956; Burbridge et al., 1957; Cameron, 1957). Because of the required high neutron density and temperature region to initiate the r-process, the current main candidate sources are merging neutron star (NS) binaries (Lattimer and Schramm, 1974, 1976; Symbalisty and Schramm, 1982; Freiburghaus et al., 1999a) and neutrinoheated ejecta from core-collapse SNe (Truran, 1981; Woosley and Hoffman, 1992; Meyer et al., 1992; Takahashi et al., 1994; Woosley et al., 1994). At present, SN simulations do not produce the observed r-process abundance pattern for nuclei with atomic number A < 120 and have a too low entropy to produce the heavier nuclei (Frei-
3 The neutron capture timescale is much smaller than the bdecay timescale.
burghaus et al., 1997, 1999a), while these problems are seemingly not met in current models of decompressed ejecta of merging NS binaries (Freiburghaus et al., 1999b). However, due to the uncertainties in the theoretical models it is at present too early to make any definitive conclusion on which model is most favourable. Observations of r-process abundances in very metal-poor halo stars indicate that the enrichment of the ISM in r-process elements already started early in the evolution of the Milky Way. The presence of a large scatter at low metallicity (up to [Fe/ H] = 1) suggests that incomplete mixing of the ISM with the ejecta of a very rare event, like the merging of a NS binary and much rarer than the actual core-collapse SN rate, took place. The average evolutionary trend is a positive [r/Fe] ratio that increases with decreasing metallicity for [Fe/ H] > 2(2.5) and declines to subsolar values below [Fe/H] 2(2.5). In trying to reproduce the observed behaviour, several chemical evolutionary computations (with simple CEMs) for the r-process enrichment have been carried out (e.g. Andreani et al., 1988; Mathews et al., 1992; Pagel and Tautvasˇien_e, 1997a; Travaglio et al., 1999, 2001; Ishimaru and Wanajo, 1999). They all promote (under the assumption that core-collapse SNe do synthesize all r-process elements) low mass SNe (7–8 Mx) as the major enrichers in r-elements especially in the early phases of the Milky Way, on the basis of the observed delayed increase of the [r/Fe] ratio as a function of [Fe/H]. Ishimaru and Wanajo (1999) also added the explosion of P 30 Mx stars to explain the large dispersion in [r/Fe] for halo stars. Only in the work of Mathews et al. (1992) were merging NS binaries included, though in a too qualitative way without using detailed binary evolutionary computations. By arbitrarily varying the timescale (within crudely estimated limits) for merging and thus for the ejection of the r-elements, they can roughly fit the average observed trend. Very recently the galactic enrichment in r-process elements has been computed by Argast et al. (2004) with a CEM that includes a stochastic star formation model in order to reproduce the observed scatter at low metallicity. We will discuss their results in comparison with our computational results in Section 4.9.
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The orbital decay of a freshly born NS binary is driven by angular momentum and orbital energy loss via gravitational wave radiation (GWR) and it is well known from general relativity that the timescale for decay sensitively depends on the mass, orbital period and eccentricity of the system. Therefore to make reliable predictions on the merger rate of NS systems, the initial properties of the NS binary population must be well known. This requires a detailed modeling of the progenitor massive binary population, which is not included in present CEMs. 1.3. Goal and outline of this work In order to make reliable predictions of the SN rates and the number of stars of various classes, one needs a population number synthesis (PNS) model that combines the initial properties of stars with a stellar evolutionary model. With a PNS code the evolution of a stellar population is simulated in detail as a function of time. By comparing the predicted population with the observations, the adequacy of the adopted stellar evolutionary models is checked and constraints are derived on the associated model parameters. The first numerical PNS codes were developed in the 1980s and mainly used to study binary populations (e.g. Kornilov and Lipunov, 1983; Lipunov and Postnov, 1987; Dewey and Cordes, 1987; Eggleton et al., 1989; Politano, 1988; Tutukov and Yungelson, 1987). Meanwhile PNS has become very popular and many articles have been published on various stellar objects produced by binary star evolution. The present work describes how to implement the detailed evolution of all type of binaries into a galactic CEM. We do this by combining a PNS model with a model for galaxy formation and star formation. The star formation model regulates the birthrate of stars, the PNS model follows the evolution of the stars and their chemical output and the galactic formation model dictates the structural formation of the Milky Way. In combination they allow the computation of the time evolution of the chemical composition of the ISM, the supernova rates and the merger rates of double neutron star and black hole (BH) systems.
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Section 2 describes the PNS model. We discuss single star and binary star evolution with special attention to the SW mass loss rate prescriptions during the different evolutionary phases of massive stars, to the formation of BHs with or without a prior SN outburst, to the formation and merger timescale of double compact star binaries and to the progenitor models of SNIas, SNIIs, SNIb/cs. In Section 3 we present our CEM and we discuss in detail the nucleosynthesis yields from single and binary stars. Using these yields we derive the R-values which are the fractions of the stars that are returned to the ISM in the form of the various chemical species and which are key inputs in our CEM. Finally, Section 4 deals with the time evolution, from the moment of the birth of the Milky Way to the present, of the SN rates of the different types, of the merger rate of double neutron star and black hole systems, of the elements H, He, C, N, O, Ne, Mg, S, Si, Ca, Fe and the r-process elements. As a major constraint on galactic CEMs we also simulate the distribution of G-type dwarfs, which represent a record of the evolutionary history of the metal content of the ISM. All our model results are compared to the observations in the solar neighbourhood.
2. The population synthesis model 2.1. General concept The purpose of PNS is to compute birthrates and expected numbers of various classes of stellar objects as a function of time. By comparing the predictions with the observations, the input theoretical PNS model is checked on its reliability and constraints are derived on the range of values that the parameters in the model can have. In this way PNS provides a tool to improve current stellar evolutionary models and to retrieve information on the evolutionary history and properties of different groups of stars. The general approach in PNS is to evolve in time an entire population of stars under a set of
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assumptions. First an initial population of unevolved single and binary stars is generated with physical properties according to predetermined distribution functions that are derived from observations. With the PNS code we follow the evolution of each single and binary star from its initial state until it becomes a compact object: a WD, an NS or a BH. The product of the birthrate of stars with a certain set of initial parameters and of the lifetime in any particular evolutionary stage gives the number of stars in this evolutionary stage. All the physical properties of the single or binary star are at each moment computed and stored. Exploring numerically the whole phase space of stellar parameters, the total number of stars of particular types in the considered region of star formation is obtained and compared to the observed number. In the following sections we discuss the basic inputs in our PNS model, e.g. the stellar evolutionary models to evolve single and binary stars, detailed evolutionary computations of single and binary stars, the evolutionary definition of the different classes of stars that are studied, the distribution functions of the stellar parameters to initialise the stellar population. 2.2. Single star evolution On the basis of their fate and the physical processes that dominate during their evolution, stars can be classified as low mass stars, intermediate mass stars and massive stars. The single star models that we use cover the evolution of stars in the mass range 0.8 6 M0/ Mx 6 120 with an initial metallicity Z in the interval [0.002, 0.02], from the Zero Age Main Sequence (ZAMS) until the formation of a WD, an NS or a BH. Throughout this work we indicate the value of a stellar parameter at birth (= ZAMS) with the index 0 (e.g. M0). 2.2.1. Low and intermediate mass stars To follow the evolution of low and intermediate mass stars we use the evolutionary computations of the Geneva group, which are presented and discussed in Schaller et al. (1992), in combination
with the evolutionary computations of van den Hoek and Groenewegen (1997). Starting from the tracks of Schaller et al. (1992), van den Hoek and Groenewegen (1997) computed the evolution during the whole AGB phase with the synthetic AGB model of Groenewegen and de Jong (1993) to complete the tracks of Schaller et al. (1992). 2.2.2. Massive stars To model the evolution of massive stars we use our own evolutionary computations which we will refer to further in this work as the Brussels tracks. We discuss successively the SW mass loss rates, the treatment of convection and rotation and the computation of the final masses after core collapse, which are all main issues in massive star evolution. 2.2.2.1. Stellar wind mass loss rates. Mass loss by _ expressed in M =yrÞ stellar winds ðthe rate ¼ M substantially influence the evolution of stars by modifying their evolutionary timescales, luminosities, masses and their surface abundances. They are also potential input sources of energy, momentum and nuclearly processed material to the ISM. The Brussels tracks are computed with the following mass loss rate prescriptions which fit well most of the observational constraints (Vanbeveren, 2001). We consider successively SW mass loss during the Main Sequence phase (MS), the Luminous Blue Variable phase (LBV), the Red Supergiant phase (RSG) and the Wolf–Rayet phase (WR). The MS phase prior to the LBV stage. From the compilation of observed mass loss rates for Population I stars (= metal rich disk stars like the Sun) by de Jager et al. (1988) and Vanbeveren et al. (1998b) selected the OB-type stars and updated _ values of the 24 stars that have been studied the M by Puls et al. (1996). The following best fit relation _ L (in the unit Lx) and Teff can be debetween M; rived: _ ¼ 1:67 log L 1:55 log T eff 8:29: logðMÞ
ð2:1Þ
In our evolutionary computations, Eq. (2.1) is applied throughout the CHB phase prior to the LBV phase. _ values computed by Puls We remark that the M et al. (1996) are not corrected for rotation and pos-
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sible clumping and therefore they may be overestimated up to a factor of two (Petrenz and Puls, 1996). Since metallicity affects the opacity (bound-free and bound-bound) in the very external layers of massive stars, it is expected that the wind strength is metallicity dependent. Observations of massive stars in the Magellanic Clouds confirm this (Garmany and Conti, 1985; Prinja, 1987). To scale the mass loss p rate ffiffiffi with the metallicity we use the _ / Z which is predicted by radiation relation M driven wind models for O stars (Kudritzki et al., 1989; Lamers and Cassinelli, 1996). Anticipating, overall PNS results and chemical evolutionary simulations only marginally depend on the massive star SW mass loss during CHB before the LBV phase. The RSG phase. After CHB, hydrogen is ignited in a shell outside the helium core. The star expands on a thermal timescale and if it has not yet lost most of its hydrogen rich envelope on the MS, it moves to the yellow/red part of the Hertzsprung– Russell (HR) diagram where it starts to burn helium in its core. The star is now observed either as a yellow supergiant (YSG) 4 (spectral type FG) or as an RSG (spectral type KM). The most luminous ones are called hypergiants. During this evolutionary phase the SW blows at a low velocity (10–40 km/s, Jura and Kleinmann, 1990). The observed mass loss rates are very uncertain (due to the poor understanding of circumstellar envelopes) and the driving wind mechanism is not well known. From infrared data on RSGs, Jura proposed a mass loss rate formula (Jura, 1987) that relates the mass loss to the wind velocity, the distance, the luminosity, the flux and the average wavelength of the flux distribution. Using this relation, Reid et al. (1990) estimated the mass loss rates of 16 RSGs in the Large Magellanic Cloud (LMC) with Mbol 6 7, which have been observed by IRAS. 5 With these mass loss rates and adopting an average distance modulus to the
4 The yellow supergiants form an intermediate stage between the blue and the red supergiant phase. 5 IRAS: Infrared Astronomical Satellite.
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LMC of 18.55 mag, Vanbeveren et al. (1998c) derived for the sample of 16 RSGs the following linear relation (with a correlation coe¨fficient of 0.96) between the logarithmic value of the mass loss rate and the logarithmic value of the luminosity: _ ¼ 0:8 log L C; logðMÞ
ð2:2Þ
with C = 8.7. Because of the possible large uncertainties in the derived RSG mass loss rates, the va_ lue of C could be as well 9 which implies M-values that are a factor of two smaller than predicted with C = 8.7. When relation (2.2) is implemented in a stellar evolutionary code with C = 8.7 (resp. 9), it follows that massive stars with an initial mass as low as 15 Mx (resp. 20 Mx) may lose most of their hydrogen rich envelope during the RSG phase (see Section 2.2.3). There are presently no models or observations that may give any reliable information on a rela_ and Z for YSGs/RSGs. If the tion between M _ same M–Z relation holds as for O stars, the RSG SW mass loss rate of YSGs/RSGs for Z = 0.02 may be 1.6 times larger than predicted by relation (2.2) with C = 8.7, implying that even a 12 Mx may lose most of its hydrogen rich layers in the RSG phase. To investigate the effect of a metallicity dependent RSG SW on our computations, we made stellar evolutionary calculations for two cases: a Z independent RSG SW which means that we apply relation (2.2) for all Z values, pffiffiffiand an RSG SW that is scaled proportional to Z . We notice that there are no RSGs and YSGs observed with Mbol 6 9.5 (Humphreys and McElroy, 1984) which may indicate that the most massive stars do not become yellow and/or red supergiants due to previous strong mass loss. The LBV phase. Luminous blue variables are observed as supergiants with a spectral type between O9 and A (Humphreys and Davidson, 1994) and occupy the upper most part of the HR diagram. The most luminous ones (Mbol 6 9.5) are particularly interesting from evolutionary point of view. They are characterized by extreme instability and violent eruptions causing mass losses of the order of (103–102) Mx/yr. In between the eruptions, the LBVs still lose mass at rates of typically (107–104) Mx/yr.
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Because the mass loss rate is poorly known and LBV mass loss may explain the lack of RSGs with Mbol 6 9.5 (corresponding with stars with an initial mass P 40 Mx) we use as a working hypothesis for evolution: _ during the LBV + RSG phase of a star with The M Mbol 6 9.5 must be sufficiently large to assure an RSG phase that is short enough to explain the lack of observed RSGs with Mbol 6 9.5. This formalism was first introduced by Vanbeveren (1991) and has been widely accepted and applied in most PNS computations. The absence of YSGs and RSGs with Mbol 6 9.5 is also observed in the Magellanic Clouds and our working hypothesis may apply at metallicities lower than Z = 0.02 as well. The WR phase. It is generally accepted that WR stars are massive CHeB stars that have lost all or most of their hydrogen rich envelope in the previous RSG or LBV phase (or in case of a binary system by mass transfer to the companion), consequently showing products of H burning (WN types) or He burning (WC or WO types) at their surface. The emitted spectrum of WR stars is characterized by strong emission lines due to a _ 105 M =yrÞ. Because of very intense SW ðM the dense SW, the hydrostatic surface of WR stars is not directly observable, which makes it very hard to determine and define their effective temperatures. It is therefore not an obvious task to link theoretical stellar evolutionary models to observed WR stars. To decide upon an evolutionary definition of a WR star we account for the following points: Observed WR stars have an atmospheric hydrogen abundance Xatm 6 0.30.4, which indicates that most of the WR stars are CHeB stars that have lost most of their hydrogen rich layers either by LBV SW, RSG SW or mass transfer in a close binary. The majority of the observed WR stars seem to have luminosities log(L/Lx) P 4.5 and temperatures log Teff P 4.4 (Hamann, 1994; Hamann and Koesterke, 1998a). From the theoretically predicted mass luminosity relation of massive
hydrogen poor CHeB stars (Vanbeveren and Packet, 1979; Vanbeveren et al., 1998b) it follows that the majority of the observed single WR stars must have masses above 5 Mx. Most of the WR stars in observed WR + OB binaries in the Milky Way have an estimated mass larger than 8–9 Mx (Vanbeveren et al., 1998b). In our evolutionary computations we assume that a hydrogen deficient CHeB star (Xatm 6 0.3) is observed as a WR star when its mass is larger than M WR min , for which we consider two possible values: 5 Mx and 8 Mx. We will perform PNS computations for both values separately. Once a WR star is formed, its further evolution depends critically on the SW mass loss during CHeB. Using a hydrodynamic atmosphere code where the SW is assumed to be homogeneous, _ values Hamann et al. (1995) determined the M for a large number of WR stars. Since then evidence has grown that these winds are clumpy and that a homogeneous wind model overesti_ typically by a factor of 2–4 (Hillier, mates M 1996; Moffat, 1996; Schmutz, 1996; Hamann and Koesterke, 1998b). Adopting these lower WR mass loss rates, Vanbeveren et al. (1998b) _ and proposed a linear relation between logðMÞ log L, _ ¼ log L 10; logðMÞ
ð2:3Þ
whereby the following observational data (known at that time) are taken as constraints: The WN5 star HD 50896 (WR 6) has a luminos_ ¼ 4:4 ity log L ¼ 5:6–5:7 and a logðMÞ 0:15 (Schmutz, 1997). _ of the WNE component of the The logðMÞ binary V444 Cyg (WR 139) deduced from the observed orbital period variation is 5 (Khaliullin et al., 1984; Underhill et al., 1990). Given its orbital mass of 9 Mx and using a mass-luminosity relation holding for WNE-binary components (Vanbeveren and Packet, 1979; Langer, 1989) it follows that log L = 5. The observed masses of BH components in Xray binaries indicate that stars with an initial mass >40 Mx should end their life with a mass
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Table 1 The luminosity and mass loss rates of observed WR stars in the Milky Way, computed with NLTE atmosphere models including clumping WR number
log L
_ logðMÞ
Eq. (2.3)
References
WR 6 WR 147 WR 111 WR 90 WR 135 WR 146 WR 11 WR 123 WR 139 OB10-WR1
5.45 5.65 5.3 5.5 5.2 5.7 5 5.7 5 5.7
4.4 4.6 4.8 4.6 4.9 4.5 5.1 4.14 5 4.3
4.5 4.3 4.7 4.5 4.8 4.3 5.0 4.3 5 4.3
Schmutz (1997) Morris et al. (2000) Hillier and Miller (1999) Dessart et al. (2000) Dessart et al. (2000) Dessart et al. (2000) De Marco et al. (2000) Nugis et al. (1998) Underhill et al. (1990) Smartt et al. (2001)
The values are compared with the ones predicted by Eq. (2.3).
larger than 10 Mx, which is the mass of the star at the end of CHeB. In the solar neighbourhood the observed WN/ WC number ratio is 1. The theoretically predicted ratio largely depends on the adopted _ and an M _ that is too large (too small) preM; dicts a too small (too large) WN/WC ratio. Since then more WR stars have been observed and investigated with detailed atmosphere codes that include clumping effects. They also fit relation (2.3) well. The observed WR stars with their luminosity and mass loss rates computed with NLTE atmosphere codes that account for clumping are _ values predicted listed in Table 1. In the table the M by relation (2.3) are also listed for comparison. It is not clear whether or not the WR mass loss rate is metallicity dependent. If the WR wind is radiatively driven, the iron group elements are expected to be the main wind drivers (Crowther, 2002). Because the iron elements are neither created nor destroyed during the pre-SN evolution, the iron abundance in the envelope of the WR star is roughly proportional to the total metallicity Z of the gas out of which the star was initially formed. Therefore the WR mass loss rate may scale with some power of Z similar to that of OB-type stars. Hamann and Koesterke (2000) investigated 18 WN stars in the LMC. Taking Z = 0.006 for the LMC and assuming a linear relation similar to _ Eq. (2.3), Vanbeveren (2001) found that the M–L data of the observed WN stars can be fitted with the relation,
sffiffiffiffiffiffi _ ¼ log L 10 þ log Z : logðMÞ Z
ð2:4Þ
To investigate to what extent the PNS and chemical evolutionary computations depend on the variation of the WR SW with the total metallicity Z of the interstellar gas out of which the WR progenitor population is formed, we explore two cases: the WR SW mass loss rate is independent of Z and Eq. (2.3) is applied for all Z values, the WR SW mass loss rate is Z dependent and is computed with relation (2.4). 2.2.2.2. Convection and rotation. We use the local pressure scale height Hp to constrain empirically the amount of convective core overshoot with the assumption that the radial size of the overshoot region is proportional to Hp at the edge of the convective core, i.e. dover = aHp. We take a = 0.2, a value that has been suggested by studies of open clusters (e.g. Kozhurina-Platais et al., 1997) and observed detached eclipsing binaries (Ribas et al., 2000). Moderate convective overshooting can also explain the simultaneous presence of Blue Supergiants (BSGs) and RSGs in starburst regions 6 (Meynet et al., 1993).
6 A starburst region is a region that in a recent past has known a short period of very high star formation activity.
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Detailed calculations of rotating massive stars show that rapid rotation can have important effects on the evolution (e.g. Maeder and Meynet, 2000a). Two major consequences of rotation as far as the internal structure and evolution is concerned are the outward diffusion of nuclearly processed material to the outer layers and an enlargement of the convective core on the MS. Naturally the effects are more pronounced at higher rotation rates. The first effect can explain the observed He and N excesses in the atmospheres of fast rotating massive O-type and B-type stars and in giants and supergiants (Herrero et al., 1998; Lyubimkov, 1996). The rotational effects are less significant in the post-CHB phase because nuclear evolution occurs too fast for rotation to become important. From test computations made by the Geneva group (e.g. Maeder and Meynet, 2000a) it follows that the evolution of a massive star with an equatorial rotational velocity of 150 km/s corresponding with the observed average equatorial velocity 100–150 km/s of OB-type stars (Penny, 1996; Vanbeveren et al., 1998c,b) resembles the evolution of a non-rotating star with a convective core overshooting distance during CHB of dover 0.2Hp. Therefore it is fair to state that as far as PNS simulations is concerned the main effect of rotation on single star evolution is similar to the effect of moderate convective core overshooting during CHB in non-rotating models. Our evolutionary computations are made without including rotation but with moderate convective core overshooting. Maeder and Meynet (2000a) suggest a third effect of rotation on stellar evolution, e.g. the effect of rotation on the stellar wind mass loss rate formalisms used in stellar evolutionary codes. A close inspection of the calculations presented by the Geneva group reveals that the way how they treat this effect critically affects their results. However as argued by Vanbeveren (2001) a study of the effect _ and thus on evolution is meanof rotation on M _ is used which is based on a theingful only if a M oretical model that accounts for all physical processes including rotation. Since semi-empirical SW mass loss rates are used in evolutionary codes, the effect of rotation on these empirical rates is hard to determine. Even more, empirical rates that
were determined without accounting for rotation overestimate the real values (Petrenz and Puls, 1996). Notice however that rotation (and magnetic fields) may be very important in order to understand the physics of the core collapse and supernova explosions, but this is beyound the scope of the present work. 2.2.2.3. Supernova explosion and black hole formation. When hydrogen lines are present in the optical spectrum the SN is classified as an SNII and as an SNIbc when no hydrogen is detectable (Minkowski, 1941). Depending on the mass of the collapsing material the remnant star is either an NS or a BH. If the final core mass exceeds the maximum mass it can sustain, it will collapse to a BH rather than evolve to a stable NS. The maximum mass for a stable NS configuration ð¼ M NS max Þ depends on the equation of state of matter at nuclear densities. Different theoretical models have been explored giving gravitational masses that range from 1.5 up to 2.9 Mx (Lattimer and Swesty, 1991; Thorsson et al., 1994; Kalogera and Baym, 1996; Akmal, 1998). The average mass of NSs observed in binary systems is 1.35 (±0.04 Mx) (Thorsett and Chakrabarty, 1999) while measured BH masses in binaries range from 3 to 20 Mx (Orosz, 2001; McClintock, 2001; Froning and Robinson, 2001; Wagner et al., 2001). In our PNS model we treat M NS max as a parameter with a value in the mass interval [1.5, 3] Mx. It is unclear in which initial mass range BHs form and whether or not they are preceded by an SN explosion. We summarise the information presently available. From core collapse simulations, Fryer (1999) concluded that stars with an initial mass between 20 and 40 Mx form BHs with an SN outburst while for stars initially more massive than 40 Mx the pre-collapse star directly implodes to form a BH and no SN event takes place. However Fryer did not include SW mass loss in his computations, which increases the limit on the initial mass for BH formation. Indirect evidence that the BH candidate (4.1– 7.9 Mx) in the X-ray binary system GRO J1655-
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40 (Nova Scorpii) was formed in an SN explosion has been reported by Israelian et al. (1999). They base their inference on the relatively high abundance of O, Mg, Si and S (6–10 times the solar value) on the surface of the companion star, which has a mass (1.6–3.1 Mx) that is too low to have produced such abundances by thermonuclear burning. Assuming that those elements were deposited on the companion by the SN explosion of the primary star provides indirect evidence of a BH formed in an SN. The lightcurve of hypernovae (= very energetic SNe like SN 1999ef) and faint SNe (e.g. SN 1997D), which deviates from a standard observed SN, is thought to be associated with BH formation. On the basis of the estimated ejected mass of 56Ni, Nomoto et al. (2003) suggest that stars with an initial mass P 20–25 Mx form a BH and produce either a hypernova or a faint SN while the massive stars of lower mass produce a classical SN with the formation of an NS. In our PNS model we define the parameter M BH 0;s as the initial mass limit above which a single star forms a BH without an SN explosion. We will make computations for M BH 0;s ¼ 25 or 40 M . Below this mass limit low mass BHs may form in an SN explosion. Notice however that in Section 4 we will critically discuss the effects on chemical evolution of galaxies when all BH are preceded by an SN explosion. Remnant masses. To compute the mass of the formed NS or BH we proceed as follows. Above the limit M BH 0;s we take the remnant mass (= mass of the BH) equal to the mass of the star at the moment of core collapse. Below the limit we use the stellar model computations of Woosley and Weaver (1995). To link the latter with our evolutionary computations (which end at the beginning of carbon ignition in the core), we proceed as follows. Woosley and Weaver (1995) computed the evolution and explosion of massive single stars with masses in the range (11–40) Mx and for various initial metallicities. However they do not account for SW mass loss and do not include convective core overshooting, which means that stars of initially the same mass form different CO core masses in their computations than they do in
875
our models. Therefore to compute the remnant masses for our models, we use the relation between the CO core mass and the compact remnant mass that follows from the model computations of Woosley and Weaver (1995). The iron core mass and remnant mass as a function of the CO core mass as given by the computations from Woosley and Weaver (1995) are shown in Fig. 2. The letters A, B, C correspond with the different explosion models which are considered by Woosley and Weaver (1995) in order to account for the uncertainty in the explosion energy. With respect to their model A, in which the final ejecta gain the canonical kinetic energy at infinity of 1.2 · 1051 erg, the explosion energy in model B is enhanced by a factor 1.5 for M0 P 30 Mx and by a factor 2 in model C for M0 P 35 Mx. We use model B as it represents a reasonable compromise between lower and higher energy models but we will explore the effects on chemical evolution when different models apply. The pre-collapse mass (Mf), the CO core mass (MCO) and the remnant mass (Mr) as a function of the initial mass (M0) for our computed models are given in Fig. 3 for the case that M BH 0;s ¼ 40 M . Supernova kicks. There are several observations that indicate that SN explosions are asymmetric (e.g. van den Heuvel and van Paradijs, 1997; Lai, 2001). A direct effect of the asymmetry in the SN explosion is the contribution of a kick to the formed remnant star, which is reflected in the high space velocities of observed single pulsars. A few percent asymmetry in the ejection of the SN shell can already lead to kick velocities of the order of 400 km/s (e.g. Shklovskii, 1969). The origin of the asymmetry is not clear from present SN simulations. Possible physical mechanisms for generating an initial asymmetry are non-isotropic neutrino emission induced by strong magnetic fields (Lai and Qian, 1998) and magnetorotational core collapses (Khokhlov et al., 1999). As will be discussed later in Section 2.3.2, SN kicks can be catastrophic in binary systems, with a large impact on the formation of double NS systems (e.g. Dewey and Cordes, 1987; Fryer and Kalogera, 1997; De Donder and Vanbeveren, 1998) and should be included in PNS studies. Since
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M(Fe)
13
A
12
B
11
C
Z=0.02
10 9
Mr
8 7 6 5 4 3 2 1 0 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
13
14
MCO 10
M(Fe) 9
Z=0.002
A
8
B
7
C
6
Mr
5 4 3 2 1 0 0
1
2
3
4
5
6
7
8
9
10
11
12
MCO Fig. 2. The iron core mass (MFe), remnant mass (Mr) as a function of the CO core mass (MCO) as computed by Woosley and Weaver (1995) for Z = 0.02 and Z = 0.002. The letters A, B and C correspond with the different explosion models that have been explored by Woosley and Weaver (1995).
there are no clear predictions from theory on the magnitude and direction of kicks one has to rely on observations. Therefore the results of recent studies on statistical analyses of observed pulsar velocities are used to parameterize the kick velocity. Pulsars radiate in the radio region and form a well-defined observed population. Although observational biases limit the observed population to relatively nearby objects, nearly all pulsars have been discovered in large surveys giving a sufficiently large sample for a meaningful statistical
analyses. Detailed studies of measured proper motions and distances of pulsars give an average space velocity at birth of the order of 250–450 km/s, with a tail extending to velocities larger than 800 km/s (Lyne and Lorimer, 1994; Hansen and Phinney, 1997; Cordes and Chernoff, 1997; Lorimer et al., 1997). We take as input kick velocity distribution a v2-like distribution, which describes well the 3D pulsar space velocity distribution that has been derived by Lyne and Lorimer (1994) and has an average kick velocity vk of 450 km/s, i.e.
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877
100
Z=0.02 Mf Mco Mr 10
1
0.1 1
10
100
M0 100
Mf Z=0.002 Mco Mr 10
1
0.1
1
10
100
M0 Fig. 3. Our computed pre-collapse mass (Mf = final mass at the end of CHeB), the CO core mass (MCO) and the remnant mass (Mr) vs. the initial mass for pffiffiffiZ = 0.02 and Z = 0.002. For Z = 0.002 we scaled the SW mass loss rate during the OB, RSG and CHeB phase proportionally to Z . In the low and intermediate mass range the computations of van den Hoek and Groenewegen (1997) are plotted.
3
3vk
Cðvk Þ ¼ 1:96 106 v2k e 514 :
ð2:5Þ
The direction of the kick is taken arbitrarily in space. As small proper motions are hard to measure and errors may not be negligible it cannot be excluded that the average velocity could be lower than suggested by present observations. Therefore we will also make computations for smaller average values (150 km/s) whereby keeping the shape of a v2-distribution, i.e.
3
vk
Cðvk Þ ¼ 2:70 105 v2k e 60 :
ð2:6Þ
If BHs form with an SN explosion the kick is taken from the input kick velocity distribution and weighted with the amount of fallback material which equals the BH mass minus the mass of the pre-SN iron core. The kick velocity is then given by vk ¼ ð1 dm Þvk , with vk the kick velocity that is selected from the input kick velocity distribution and dm the mass fraction of the initially ejected stellar envelope that falls back. This implies that BHs may receive small but non-zero kicks, which
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0.5
probability
0.4
0.3
0.2
0.1
0 0
200
400
600
800
1000
1200
1400
1600
vk(km/s)
Fig. 4. The kick velocity distribution described by Eq. (2.5) (full line) and by Eq. (2.6) (dashed line).
Fig. 5. Overview of the single star evolutionary model.
is consistent with the velocity dispersion of 40 km/s of X-ray transients with BH candidates (White and van Paradijs, 1996). More recently the velocity distribution of isolated radio pulsars has been re-analysed by Arzoumanian et al. (2002). They promote a twocomponent distribution with characteristic velocities of 90 and 500 km/s. Test calculations illustrate that the PNS results with the latter distribution are somewhere in between the results holding for the
two distributions given by Eqs. (2.5) and (2.6). Fig. 4 shows the two distributions. 2.2.3. Summary of the single star evolutionary model Fig. 5 summarizes our adopted single star evolutionary scenario. The main evolutionary data are given in Table 2. Obviously the details of the evolutionary calculations are available upon request.
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Table 2 Single star evolutionary data for Z = 0.02 and for Z = 0.002 (italic) TCHB
TO
TRSG
TWR
TWN
TWC
TCHeB
9961.7 6263.6 1827.3 855.6 1115.9 855.6 352.5 290.8 94.5 88.3 43.2 45.1 26.4 28.8 16.0 18.1 14.8 16.1 10.1
– – – – – – – – – – – – – – – – – – 3.88
– – – – – – – – – – – – 0.94 1.1 0.85 1.0 0.46 1.16 0.254
11.5 8.2
6.8 5.33
0.77 0.194
25 30
9.2 6.9
6.71 5.16
0.66 0.165
30 40 40
7.9 5.5 6.2
6.23 3.98 5.27
0.54 0.109 0.42
60 60
3.4 3.9
2.96 3.44
100 100
2.41 2.94
2.96 2.65
– 0 0 – 0
– – – – – – – – – – – – – – – – – – 0.413 0 – 0.298 0.180 – 0.204 0.204 – 0.123 0.172 0 0.077 0.075 0.276 0.069 0.051 0.198
– – – – – – – – – – – – – – – – – – –
20 25
– – – – – – – – – – – – – – – – – – 0.413 0 – 0.483 0.180 – 0.428 0.311 – 0.373 0.500 0.437 0.421 0.427 0.360 0.338 0.352 0.318
– – – – 240.9 140.6 86.2 45.2 12.4 10.9 4.7 4.7 2.6 2.7 1.6 1.6 1.12 1.16 0.841 0.840 20 0.677 0.677 – 0.593 0.593 0.54 0.482 0.500 39 0.421 0.427 0.360 0.338 0.352 0.318
M0 1 1 1.7 1.7 2 2 3 3 5 5 7 7 9 9 12 12 15 15 20
– 0.185 0 – 0.224 0.107 – 0.250 0.328 0.437 0.344 0.352 0.084 0.269 0.301 0.120
MCHBe
MWRb
1 1 1.7 1.7 2
– – – – –
3
–
5 5 7 7 9
– – – – –
– –
– 12 – 15 – 19.5
– – 7.1
– 24
9.9
28
12.9
29.5 36 39 26.1 48 53 53 74 81 81
– 19.2 26.1 32 36 36 43 56 56
TCHB = CHB lifetime, TO = O lifetime, TRSG = RSG lifetime, TWR = WR lifetime, TWR = WN lifetime, TWC = WC lifetime, TCHeB = CHeB lifetime, MCHB,e = mass at the end of CHB and MWR,b = mass at the beginning of the WR phase. For Z = 0.02 and 6 WR M0 = 20, 25 and 30 Mx the first(second) row corresponds with pffiffiffi M 0;s ¼ 5 ð8Þ M . Masses are expressed in solar mass and times in 10 yr. For Z = 0.002 the RSG SW is scaled pffiffiffiproportionally to Z , the LBV SW is taken Z independent and the WR SW is taken once Z independent (= first row) and once / Z (= second row) for M0 = 40, 60, 100 Mx.
2.3. Binary star evolution By definition, close binaries are pairs of stars that experience during their evolution at least one phase of mass transfer. At birth when the binary is still unevolved or detached (both stars are on the ZAMS), the system is completely defined by the mass M1,0 of the most massive component called the primary star, the
mass ratio q0 = M2,0/M1,0 with M2,0 the mass of the secondary star which is the less massive component, the orbital period P0 and the eccentricity e0 of the orbit. The values of M1,0, q0, P0 and (to a lesser degree) e0 determine the evolutionary path of the binary system. If the binary orbit is wide enough for both individual stars not to be affected by each other, the stars behave approximately as single stars.
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However when they become close enough to interact, gas may flow from one star to the other. Mass exchange or transfer in binary systems is commonly treated in the Roche model, which describes, in a frame rotating with a circular and synchronized binary system, the equipotential or Roche surfaces of the binary under the assumption that the gravitational field generated by the two stars is that of point-masses. In this Roche geometry, each star can expand to a maximum radius, called the Roche radius RR. When the radius of a star becomes larger than RR it starts to lose mass through the first Lagrangian point L1 in which the effective force of gravity corrected for centrifugal forces is zero. A detailed description of the Roche model can be found in the work of Kopal (1972). The Roche radius of the primary star relative to the orbital separation A of the system has been fitted by Eggleton (1983). The Roche radius of the secondary is obtained by changing q into 1/q into the latter fit. The outer Lagrangian surface is the equipotential surface that surrounds the entire binary and has an intersection point at L2, called the second Lagrangian point. Through L2 matter can escape most easily from the gravitational field of the binary. The Roche model is used in all binary evolutionary computations. Although this may only be a rough approximation of reality, binary evolutionary computations show that the existence of the Roche radius is of primary importance rather than its exact value. 2.3.1. Mass transfer and accretion Mass transfer occurs in a binary when a component exceeds its Roche radius and performs Roche lobe overflow (RLOF). The characteristics of the RLOF process are determined by the behaviour of the Roche radius as a function of the mass ratio and by the reaction of a star to mass loss or accretion. Following the classification of Kippenhahn and Weigert (1967) and Lauterborn (1970) we distinguish three main cases of mass transfer, which correspond to the three major expansion phases during stellar evolution: case A mass transfer which takes place during CHB, case B mass transfer which occurs during the hydrogen shell burning
phase prior to central helium burning and case C mass transfer which begins after helium has been depleted in the core. Case B RLOF is further divided into early case B or case Br where at the onset of RLOF the envelope of the primary is mostly radiative and late case B or case Bc where the primary has a deep convective envelope at the beginning of the RLOF phase. In case B binaries the RLOF remnant star may fill its Roche lobe for a second time during helium shell burning and perform case BB RLOF (Delgado and Thomas, 1981). When the RLOF takes place according to case A/case Br, mass transfer proceeds on the nuclear/ thermal timescale and occurs dynamically stable. The overflowing material passes through L1 and is accreted (all or partly) by the secondary. Although the accretion process in a binary is by no means spherically symmetric, it is commonly assumed that once the matter falls onto the secondary it is redistributed homogeneously over its surface on a timescale that is short compared to the evolutionary timescale of the star. Binary evolutionary computations have been made by De Loore and Vanbeveren (1992, 1994, 1995) and Vanbeveren and De Loore (1995), using two accretion models in order to follow the temporal behaviour of the mass gainer: the standard accretion model (Neo et al., 1977) and the accretion induced full mixing model (Vanbeveren et al., 1994; Vanbeveren and De Loore, 1994). If the primary has deep surface convection layers (i.e. in case Bc and case C) mass transfer is dynamically unstable because the radius of the primary increases as a result of mass loss while the Roche radius normally decreases when the primary is more massive than the secondary. In this case the RLOF process is accompanied by a common-envelope (CE) phase (Paczynski, 1976) and a spiral-in of the secondary through the envelope of the primary. The outcome of the spiral-in phase depends on how much energy is released in the orbital decay of the binary and is deposited in the envelope. 2.3.2. Evolution of the binary orbit The orbital binary period changes continuously in time and is decisive for the final fate of the bin-
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ary. We compute the evolution of the orbital period in detail, whereby accounting for the effect of SW mass loss, conservative and non-conservative RLOF, CE with spiral-in evolution, the impact of asymmetric SN explosions and the effect of gravitational wave radiation. We use the following prescriptions to compute the period variation caused by the different processes. In what follows the initial resp. final values are indicated with the subscript index i resp. f. 2.3.2.1. Stellar wind mass loss. As already discussed for single stars, stars continuously lose mass by an SW throughout their lives. The SW from a binary component affects the binary parameters via the loss of mass and angular momentum. Some of the wind material may be accreted by the companion star. To study the effect of SW mass loss on the orbital period we assume that the mass outflow is isotropic. The specific angular momentum of the wind lwind in units of Avorb (with A the orbital separation between the two stars in the binary system) is given by the relation, J_ ¼ lwind Avorb : ð2:7Þ _ wind M If the wind velocity is larger than the orbital velocity such that the wind cannot extract angular momentum from the orbital motion by torque, the specific angular momentum of the wind can be approximated by the specific angular momentum of the matter at the moment it leaves the mass losing star, i.e. 2 M2 lwind ¼ : ð2:8Þ M1 þ M2 For the general case that both binary components lose mass by a spherical symmetric SW, combination of Eqs. (2.7) and (2.8) and using Keplers third law results into the following initial– final period relation: 2 Pf M 1;i þ M 2;i ¼ : ð2:9Þ Pi M 1;f þ M 2;f It follows immediately that in this case the orbital period increases.
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However, if the wind blows at velocities of magnitude comparable to the orbital velocity, the wind interacts with the orbit and the ejected mass carries away angular momentum. In this case lwind is completely unknown and currently no prescription is available. The net result may be a strong reduction of the orbital period. 2.3.2.2. Roche lobe overflow. The variation of the orbital period during RLOF depends on the fraction b of mass that is lost by the primary star and is accreted by the secondary star. It is commonly assumed that during RLOF the influence of a possible SW on the gasstream and period is negligible, which is a good approximation for the majority of close binaries. The following 4 cases are distinguished. Dynamically stable RLOF: the conservative case b = 1 In case of a conservative mass transfer the total mass and angular momentum of the system is conserved: dJ = 0 and dM1 = dM2. The relative orbital period change yields then dP 3ð1 q2 Þ dðM 1 þ M 2 Þ ¼ ; P q M1 þ M2
ð2:10Þ
with q = M2/M1. At the onset of the RLOF, the primary star is more massive than the secondary star and the orbital period decreases until the mass ratio equals unity. Subsequently the mass ratio becomes reversed and the orbital period increases again. Integrating Eq. (2.10) over the whole RLOF phase gives the period relation, 3 Pf M 1;i M 2;i ¼ : ð2:11Þ Pi M 1;f M 2;f The final period is longer or shorter than the initial period depending on the final mass ratio. Dynamically stable RLOF: the non-conservative case 0 6 b\1 If during RLOF the accreting secondary star gets out of thermal equilibrium and swells up to its own Roche lobe, a contact binary is formed. The inner contact Roche surface becomes completely filled and gas may reach the outer Lagrangian point L2, where it can leave the system.
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When a particle leaves the binary through L2 its energy is too small to escape to infinity. However, it is no longer forced to corotate with the binary and, for most mass ratios, acquires enough energy by gravitational interaction with the binary to spiral to infinity. In our model we assume that the escaping matter forms a ring around the system and we approximate the loss of angular momentum by the angular momentum of the ring (see van den Heuvel, 1993; Soberman et al., 1997). Because of the frequent collisions among the particles themselves it is expected that the ring will be circular. In a first approximation the particles in the ring may be regarded as moving under the gravitational attraction of a source of mass M1 + M2 located at the center of mass of the system. Assuming that the ring revolves around the center of mass with a radius Aring, and in the same sense as the binary motion, the angular momentum per unit mass of the ring is equal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2:12Þ jring ¼ GðM 1 þ M 2 ÞAring : Since jring must come from the binary itself, an amount of matter = (1 b) dM1 that escapes via L2 causes a change in the total orbital angular momentum of the binary that is given by dJ ¼ jring ð1 bÞdM 1 :
ð2:13Þ
The distance Aring of the L2 point from the center of mass is a slowly varying function of the mass ratio q. For the ring to be stable within the gravitational potential it must be sufficiently wide, i.e. Aring > 2A. We take Aring = 2.3A. The relative change of the period is obtained by combining Keplers third law, the expression for J and Eq. (2.12), dP ð1 bÞdM 1 dM 1 dM 2 ¼ ð1 þ 3cÞ 3 3 ; P M1 þ M2 M1 M2 ðM 1 þM 2 Þ2 M1M2
qffiffiffiffiffiffiffi
Aring . A
ð2:14Þ
with c ¼ Integrating this equation gives the period relation:for b > 0: Þ 3½pffiffigð1bÞ1 3½pffiffigð1b þ1 b Pf M 1;f þ M 2;f M 1;f M 2;f ¼ Pi M 1;i þ M 2;i M 1;i M 2;i ð2:15Þ for b = 0:
Pf ¼ Pi
3½pffiffig1 pffiffiM 1;f M 1;i 3 g M 1;f þ M 2;f M 1;f M 2;i ; e M 1;i þ M 2;i M 1;i ð2:16Þ
with g = Aring/A. Unlike spherical symmetric mass outflow, mass loss from the L2 point causes the orbit to shrink by a factor that sensitively depends on the value of b. Dynamically unstable RLOF In case of dynamically unstable mass transfer, the lower mass star is dragged into the envelope of the primary star and the ensuing CE evolution is dominated by frictional forces due to differential rotation of the embedded binary with respect to the envelope. The global outcome is expected to be a strong reduction of the binary separation with a possible ejection of the whole envelope if the binding energy that is released by the shrinking binary is large enough to do this. The spiral-in process is highly complicated and a correct treatment requires detailed 3D hydrodynamic computations consistently from start to finish. Currently, neither the initial phases (i.e. the formation of the CE) nor the final ejection of the envelope has been computed consistently. However, an estimate of the size of the final orbit can be obtained by equating the binding energy of the stellar envelope to the difference in orbital energy of the binary at the beginning and end of the spiral-in process. Assuming that the envelope binding energy and structure is that of the unperturbed giant star at the onset of spiral-in, we have that GM 1;i ðM 1;i M 1;f Þ GM 1;f M 2;i GM 1;i M 2;i ¼a þ ; kðRR =Ai ÞAi 2Af 2Ai ð2:17Þ with RR the Roche radius of the primary star, k a factor describing the density structure of the primarys envelope and a the efficiency of converting orbital energy into kinetic energy for the expulsion of the envelope (Webbink, 1984; de Kool, 1990). The mass of the secondary star is assumed to remain constant. The value of k is derived from stellar evolutionary calculations and lies within 0.7 and 1.5 for convective envelopes and between 0.3 and 0.5 for radiative envelopes. The value of a is
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necessarily 6 1 and probably ranges from 0.1 to 1 as suggested by hydrodynamic calculations (e.g. Yorke et al., 1995) and observational studies of binary planetary nuclei (= close double white dwarf systems with a planetary nebula) (Bragaglia et al., 1990). Elaboration of Eq. (2.17) and using Keplers third law leads to the following equation for the evolution of the orbit: 0 13=2 3=2 1=2 Pf M 1;f M M 1;i þ M 2;i 2;i @ A ¼ : 2ðM M Þ Pi M 1;i M 1;f þ M 2;i M 2;i þ 1;i 1;f
In this case matter leaves the system with the specific orbital angular momentum of the accretion star (e.g. Bhattacharya and van den Heuvel, 1991; van den Heuvel, 1994; Soberman et al., 1997). When a fraction f of the transferred mass is ejected isotropically from the vicinity of the companion star, the induced orbital period change is given by the relation Pf ¼ Pi
3f 3 7:5þ1f qi 1 þ qi 1 þ ð1 fÞqf ; qf 1 þ qf 1 þ ð1 fÞqi
ð2:19Þ
kðRR =Ai Þa
ð2:18Þ
Whether or not the binary finally survives the spiral-in process depends on the amount of orbital energy that is deposited into the envelope. If the efficiency is too small or if the binding energy of the envelope is too large, spiral-in may continue until both stars merge. Otherwise, the outcome is a small period binary whereby the core of the donor star revolves around its companion. Since the spiral-in phase is expected to proceed very fast (102–103 yr, Iben and Livio, 1993), it is highly probable that no accretion of mass will take place onto the companion star. Therefore we assume that b = 0. When the primary star has turned into a compact object (WD, NS or BH) and the secondary expands, the compact star may spiral-in into the envelope of the secondary. The efficiency of the conversion of orbital into potential energy may here be different because of the different nature of the spiral-in object. Therefore we consider in our PNS model different efficiencies for binaries in different evolutionary stages: a1 for non-evolved binaries with q0 6 0.2 (see later on in Section 2.3.3), a2 for binaries with q0 > 0.2 that perform dynamically unstable RLOF, a3 for systems with a compact component (WD, NS or BH). Isotropic re-emission An alternative way by which transferred mass may leave the system is via an isotropic fast wind that develops in the vicinity of the accretion star.
883
with q the ratio of the mass of the Roche lobe overflowing star to the mass of the companion star. For f = 1 the period variation is 3 Pf qi 1 þ qi ¼ ð2:20Þ expð3ðqf qi ÞÞ: Pi qf 1 þ qf This mode of mass transfer is often applied to model the orbital evolution of X-ray binaries in which matter is transferred to the NS in a conservative way before it is ejected isotropically. 2.3.2.3. Asymmetric supernova explosion. In a binary, an SN event is treated as an instantaneous mass loss of the exploding star. It is straightforward to show that if the explosion is symmetric, a circularised binary will be disrupted if more than half of the systems mass is ejected (e.g. Boersma, 1960). From conservation of momentum in the frame moving with the binarys original center of mass it follows that the sudden mass loss gives the binary system a kick velocity that is equal to vk ¼
M 2;i M 1;i þ M 2;i
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M 1;i M 1;f GðM 1;i þ M 2;i Þ ; Ai M 1;f þ M 2;i ð2:21Þ
with M1,f the remnant mass of the exploding progenitor star of mass M1,i. The post-SN period Pf is related to the pre-SN period Pi (in case of no disruption) by Pf ¼ Pi
1=2 M 1;i þ M 2;i ð1 eÞ3=2 ; M 1 þ M 2;i
ð2:22Þ
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with e the induced eccentricity equal to (M1,i M1,f)/(M1,f + M2,i). In general the impact of the ejected SN shell on the companion star can be neglected (Fryxell and Arnett, 1981). It is interesting to note that when mass exchange has taken place before the SN event, the ejected mass shell will be a smaller fraction of the binarys total mass and therefore a lower probability for disruption. When the explosion is asymmetric, the resulting orbit depends on the degree and direction of the non-isotropy. The kick that is imparted to the remnant mass of the exploding star can favour or disfavour the disruption of the binary system. The dynamical effects of mass loss and kicks on the binary system have been studied in several works (e.g. Sutantyo, 1978; Brandt and Podsiadlowski, 1995; Tauris and Takens, 1998). We use the formalism of Tauris and Takens (1998) to compute the post-SN orbital parameters. The geometry of an asymmetric SN outburst in a binary system is given in Fig. 6. The exploding star 1 is positioned in the origin of a cartesian reference frame (x, y, z). The direction of the z-axis equals that of the pre-SN orbital angular momentum vector ~ J , while the y-axis points towards the secondary star 2. The positive x-axis is aligned with the pre-SN relative velocity vector ~ v of the primary star with respect to the secondary star. The velocities and positions of both stars relative to the initial center of mass (c.m.) frame are ~ v11 ;~ v21 ; ~ r11 and ~ r21 . To simulate the explosion the mass of the exploding star is changed and an arbitrary oriented kick velocity ~ vk ðh; /Þ is added to its orbi-
tal velocity in 3D. We neglect the impact of the SN shell on the companion star. If the system stays bound after the SN explosion, the post-SN orbital period is given by P f pffiffiffiffi ~ ½2 m ~ 1 þ 2~v cos h cos / þ ~v2 3=2 ; ¼ m Pi ð2:23Þ with ~v the ratio of the kick velocity vk to the initial ~ the total mass relative orbital velocity vorb and m of the system before the explosion relative to the total mass after the explosion. The induced eccentricity of the orbit is ~ m ~ 1 þ 2~v cos h cos / þ ~v2 e2 ¼ 1 m½2 ½ð1 þ ~v cos h cos /Þ þ ~v2 sin2 h :
ð2:24Þ
2.3.2.4. Gravitational wave radiation. According to general relativity binary stars are radiators of gravitational waves. The radiation occurs because the binary has a time dependent mass quadrupole moment. The system loses energy and orbital angular momentum. As will be discussed in Section 3, the evolution of double compact star binaries is mainly governed by GWR. For two masses m1 and m2 moving in a circular orbit with a separation A and angular velocity x, the rate of orbital energy (E) and angular momentum (J) loss is given by dE 32G m1 m2 ¼ A4 x 6 ; 5 dt 5c ðm1 þ m2 Þ2
ð2:25Þ
dJ ðm1 m2 Þ2 ¼ 3:47 1067 P 7=3 ; 2=3 dt ðm1 þ m2 Þ
ð2:26Þ
with c the velocity of light in vacuum and with all the quantities expressed in cgs units (Landau and Lifshitz, 1951). When taking the masses as invariable and converting the cgs units into solar units (Rx,Mx), the change of orbital period P and the decay time Td (= timescale on which the orbital distance A ! 0) is given by Fig. 6. Geometry of an asymmetric supernova explosion in a circular binary.
dP M 1M 2 ¼ 3:68 106 P 5=3 ; 1=3 dt ðM 1 þ M 2 Þ
ð2:27Þ
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T d ¼ 3:22 103
ðM 1 þ M 2 Þ M 1M 2
1=3
P 8=3 ;
is computed by solving Eqs. (2.31) and (2.32) for A ! 0 and e ! 0. To illustrate the strong influence of the eccentricity on the decay time, Fig. 7 gives the ratio of the decay time of the eccentric system to the decay time of the circular system. Notice that this ratio is independent of the initial semimajor axis.
ð2:28Þ
with P and t in seconds and Td in years. If the binary has an eccentricity e, the loss of orbital energy and angular momentum averaged over one orbit is given by e2 þ 37 e4 dE 32G4 m21 m22 ðm1 þ m2 Þ 1 þ 73 24 96 ; ¼ 5 7=2 5 dt 5c A ð1 e2 Þ ð2:29Þ
dJ dt
¼
32G7=2 m21 m22 ðm1 þ m2 Þ 5c5 A7=2
1=2
2.3.3. Detailed binary evolutionary model Our PNS model follows the evolution of binaries with M1,0 2 ]1, 120] Mx, q0 2 ]0, 1] and P0 6 10 yr. Both components are assumed to be formed at the same time with the same initial chemical composition and in a circular orbit. To model the evolution of both components we use an extended set of detailed binary evolutionary computations that are made with the Brussels binary code, in which the evolution of both components is computed until the end of central helium burning. An extensive discussion of the binary tracks has been given in the papers by De Loore and Vanbeveren (1992, 1994, 1995), Vanbeveren and De Loore (1995) and Vanbeveren et al. (1998a). We start from the ZAMS and treat both stars as single stars until the primary star fills its Roche lobe. When meanwhile the system suffers from mass loss by a SW from one or both components, the orbital period variation is computed with Eq.
ð1 þ 78 e2 Þ 2
ð1 e2 Þ
ð2:30Þ (Peters, 1964). The corresponding change of orbital distance A and eccentricity e are e2 þ 37 e4 dA 64G3 m1 m2 ðm1 þ m2 Þ 1 þ 73 24 96 ; ¼ 3 7=2 5 dt 5c A ð1 e2 Þ ð2:31Þ
d ln e dt
¼
885
e2 Þ 304G3 m1 m2 ðm1 þ m2 Þ ð1 þ 121 304 : 5=2 15c5 A4 ð1 e2 Þ ð2:32Þ
Since the orbital distance evolution is coupled to the eccentricity evolution, the orbital decay time 1
Td,e /Td,c
0.1
0.01
0.001 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
e Fig. 7. The ratio of the decay time of the eccentric system to the decay time of the circular system (=Td,e/Td,c) as a function of the initial eccentricity.
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(2.9). If no binary interaction takes place, both stars evolve as single stars until they are both transformed into compact objects. For the interacting binaries we follow the next scheme. 2.3.3.1. The evolution of close binaries with q0 > 0.2 until the primary forms a compact star (WD, NS or BH). Primaries with M1,0 > 40 Mx. We assume that a primary initially more massive than 40 Mx loses all its hydrogen rich layers due to a spherically symmetric SW (OB wind and LBV wind) and that RLOF does not occur: the LBV scenario (Vanbeveren, 1991). The period variation is computed with Eq. (2.9). At the end of the SW mass loss phase the star is at the beginning of CHeB and becomes a WR star. The evolution of the primary is identical to that of a single star with the same mass. Similar as for single stars, we assume that the LBV scenario is independent from the metallicity. Primaries with M1,0 6 40 Mx. Case A and Br systems Primary star. We treat case A and case Br RLOF in the same way. Mass transfer occurs dynamically stably and stops when helium is burning in the core and the atmospheric hydrogen abundance has dropped to 0.2–0.3. The mass of the primary after RLOF is given by the relations 8 2 > < 0:0387M i 0:041M i þ 0:202 M i < 3 3 6 M i 6 7; M f ¼ 0:1ð0:124ÞM 1:32 i > : 1:44ð1:7Þ 0:09ð0:048ÞM i M i > 7: The numbers between the round brackets hold when Z = 0.002. The masses Mi resp. Mf are the masses before resp. after RLOF. If the system does not merge, the post-RLOF evolution of a massive primary star depends critically on the adopted SW mass loss formalism during CHeB. To decide whether or not a hydrogen deficient CHeB star is a WR star we use the same criterium adopted for single stars. The computed WR lifetimes are given in Table 3. For Z = 0.02 Eq. (2.3) is applied while for Z = 0.002 we consider the two cases, i.e. a Z-independent WR SW (Eq. (2.3) is applied) and a Z-dependent WR SW (Eq. (2.4) is applied).
Secondary star. The evolution of the secondary star that has gained matter lost by the primary during RLOF is followed by detailed binary evolutionary computations. We use an extensive set of evolutionary tracks that are computed with the two earlier mentioned accretion models: the accretion induced full mixing model and the standard accretion model where the effect of convection and semiconvection on top of the convective core of the gainer is treated as a very fast diffusion process. All the evolutionary data (relevant for PNS computations) have been published in the paper of Vanbeveren et al. (1998a). We apply the accretion induced full mixing model when a Keplerian accretion disk forms, and the standard accretion model when the gasstream directly hits the secondary. Orbital period variation. Since the mass loss rate due to RLOF is much larger than the mass loss rate due to a possible SW, we neglect the effect of SW mass loss on the variation of the binary period during RLOF. The evolution of the period during RLOF depends on the fraction b of matter lost by the primary and accreted by the secondary. To compute b one has to solve the magnetohydrodynamic equations (in 3D) that describe the mass transfer process, which is a very complex problem that is presently unsolved. In order to investigate the effect of b on the PNS results, we use the following formalism, which is frequently used in PNS studies:
b¼
bmax
q P 0:4;
5bmax ðq 0:2Þ 0:2 6 q < 0:4;
with bmax a constant that we either take = 1 (= conservative RLOF) or =0.5 (= non-conservative RLOF). We apply Eq. (2.11) for the case b = 1, Eq. (2.15) for 0 < b < 1 and Eq. (2.16) for b = 0. Case Bc and C systems Primary star. The initial orbital period is large enough for the primary to become a R(S)G prior to the onset of the RLOF. This means that we have to account for the effect of SW during the R(S)G phase. If a massive star loses all its hydrogen rich layers by SW during the YSG/RSG phase the total
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Table 3 The core He burning lifetime (TCHeB) and the WR lifetime (TWR) together with the duration of the subphases (TWN, TWC) as a function of the primary masses M1,0 for case B systems M1,0
TCHeB
TWR
TWN
TWC
3
50.3 62 10.81 15.31 5.82 4.92 2.95 2.89 1.59 1.58 1.35 1.25 1.04 1.02 0.82 0.697 0.635 0.541 0.549 0.500 0.437
– – – – – – – – – – – – 0.372/0 0.429/0 0.802/0 0.697/0.288 0.635/0.374 0.541/0.541 0.549/0.455 0.500/0.500 0.437/0.437
– – – – – – – – – – – – 0.372/0 0.429/0 0.802/0 0.276/0.276 0.289/0.289 0.541/0.541 0.148/0.148 0.172/0.172 0.437/0.437
– – – – – – – – – – – – – – – 0.421/0.012 0.346/0.085 – 0.401/0.307 0.328/0.328 –
5 7 9 12 15 20
30
40
WR For the WR lifetimes, the first number is computed with M WR min ¼ 5 M and the second with M min ¼ 8 M . For the primary masses 20, 30 p and 40 M the second row is computed with a Z independent WR SW and the third row with a WR SW that is scaled proportional x ffiffiffi to Z . The lifetimes are given in units of 106 yr.
mass loss due to RLOF is largely reduced, this is called the RSG scenario (Vanbeveren, 1996). In our PNS model we assume that massive primary stars with an initial mass between 15–20 and 40 Mx for Z = 0.02 and larger than 40 Mx for Z = 0.002 avoid the HeSB expansion phase due to SW mass loss and therefore do not perform case C mass transfer. Similar to a case Br binary, the mass loss process in a case Bc/C binary stops when most of the hydrogen rich layers of the primary star are removed. If the system does not merge, the further evolution of the primary is similar to the evolution of a primary in a case Br system. Secondary star. Because of the convective envelope of the primary star, mass transfer occurs dynamically unstably and soon after the onset of the RLOF a common envelope is formed. We assume that during CE evolution no significant matter accretion occurs and that the mass of the secondary remains constant (i.e. b = 0). Orbital period variation. To follow the period evolution during CE evolution and spiral-in we ap-
ply Eq. (2.18) with k = 1. For the parameter a2 we consider the values 0.5 and 1. 2.3.3.2. The evolution of close binaries with q0 6 0.2. When the primary evolves it first expands on the nuclear timescale during CHB and then on the Kelvin–Helmholtz timescale during HSB. The expansion causes the star to spin down and rotation becomes asynchronised with the orbital motion. Via tidal interaction, the star tries to absorb orbital angular momentum of the component in order to spin up and regain synchronism. If the available orbital angular momentum is not sufficient to meet the need of the primary, the low mass companion is swallowed by the primary (Darwin, 1908; Kopal, 1972; Counselman, 1973; Sparks and Stecher, 1974). The outcome is the merging of both components. From evolutionary computations of binaries it has been found that although a massive star shrinks in response to the loss of matter, it may not shrink fast enough to compete with the rapid shrinkage of the Roche radius when initially the
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binary has a mass ratio q smaller than 0.2. It is therefore conceivable that in most of the binaries with q 6 0.2 the secondary is engulfed by the primary during RLOF and both components merge. We evolve systems with q 6 0.2 by means of the common-envelope formalism whereby applying Eq. (2.18) with k = 0.5. For the parameter a1 we consider the values 0.5 and 1.
Re ¼ 0:62 104 M 3 0:49 102 M 2 þ 0:18M þ 0:17;
ð2:33Þ
which follows from our detailed binary evolutionary computations and hardly depends on the metallicity.
q0 6 0.2 mergers: we assume that the further evolution of the merger star, which has a mass equal to the sum of the masses of both components, is similar to the evolution of a single star with the same helium core mass. Since at the moment of the merging process the lower mass component has amply left the ZAMS and thus is almost homogeneous, it is reasonable to simulate the evolution of the merger star by means of the evolution of a star that has accreted an amount of mass equal to the mass of the smaller component. q0 > 0.2 mergers: we assume that the inspiraling secondary star (which is in most cases a CHB star) mixes with the hydrogen deficient CHeB primary star in an homogeneous way. The total mass of the merger star is taken equal to the mass of the primary and secondary at the end of RLOF.
2.3.3.4. The evolution of close binaries after the primary has turned into a WD, NS or BH. Massive primaries in close binaries have lost most of their hydrogen rich layers at the end of their life and likely explode as SNIbcs. To compute the remnant mass of a massive primary star we proceed in the same way as for single stars, i.e. we link our computed CO cores to the SN model computations of Woosley and Weaver (1995) using their explosion model B. Similar as for single stars we introduce an initial mass limit M BH 0;b above which a massive star finally forms a massive BH without an SN explosion. Because interacting massive primaries form smaller CO cores than isolated single stars of the same initial mass, BH formation is likely to happen at higher initial masses. We consider 40 Mx as a minimum value for M BH 0;b . The remnant masses together with the CO core masses are plotted in Fig. 9. The primary forms a NS or a BH. If the SN disrupts the binary, the secondary star further evolves as a single star. When the system is not disrupted, the post-SN orbital period is given by Eq. (2.23). In case of direct BH formation whereby no SN explosion takes place, the orbital period stays unaffected during the formation of the BH. The further evolution of the system depends on the orbital period, the mass of the secondary and the mass ratio:
To decide whether or not the binary merged during non-conservative RLOF, we check on the radius of both components after RLOF. When one of the Roche radii is smaller than the equilibrium radius of the corresponding component, we judge that both stars merged before the end of the RLOF. The equilibrium radius Re of a hydrogen deficient CHeB star with a mass M is computed with the relation
If the secondary is more massive than 40 Mx we assume that all its hydrogen layers are removed by the LBV SW and no spiral-in phase takes place. The orbital period variation is computed with Eq. (2.9). If the mass of the secondary is smaller than 40 Mx and the period is small enough that the secondary fills its Roche lobe before it becomes an R(S)G, then similar as for binaries with q0 6 0.2 this will result in a CE where due to
2.3.3.3. The formation and evolution of mergers. Merger stars form from binaries with q0 6 0.2 that do not survive the spiral-in phase and from binaries with q0 > 0.2 in which both components coalescence after non-conservative case A, case B or case C RLOF. How mergers look like and how they will further evolve is very uncertain.
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viscosity, the NS or BH will start to spiral in, into the envelope of the secondary. We apply Eq. (2.18) to compute the final period. When the mass of the massive secondary star is smaller than 40 Mx and the period is sufficiently large to allow the star to evolve into an RSG before it starts filling its Roche lobe, we first compute the period variation due to RSG SW mass loss with Eq. (2.9). When the spiral-in starts the further period evolution is computed with Eq. (2.18). We assume that during the CE and spiral-in phase the NS or BH does not accrete matter from the envelope of the secondary. To determine the outcome of the spiral-in phase we check the equilibrium radius Re of the secondary star. When the Roche radius is smaller than Re we assume that both stars merged before the removal of the whole hydrogen rich envelope. The system consists of an HSB star with a compact star _ in its He core, which is called a Thorne–Zytkow _ _ object (TZO) (Thorne and Zytkow, 1975, 1977). From detailed structure models it follows that _ TZOs have the structure of an RSG (Biehle, 1991; Cannon et al., 1992). In our PNS computa_ tions we use a TZO model which is based on the latter two studies. However, anticipating, we _ notice that the number of forming TZOs is small and that our assumption on their evolution is not critical for our chemical evolutionary computations. If the NS or BH does not merge with the secondary during spiral-in, a CHeB + NS or BH system is formed. Obviously, the further evolution of the CHeB component is taken similar to that of the post RLOF CHeB evolution of a primary with the same mass. At the moment that the secondary star explodes it is reasonable to assume that the binary has been circularised. The effects of an asymmetric SN explosion are again included in detail, similar as for the SN explosion of the primary. In the case that a double compact binary forms, i.e. an NS + NS, NS + BH, BH + NS or BH + BH system, the further evolution of the system is driven by orbital energy and angular momentum loss due to gravitational wave radiation. We follow the period and eccentricity
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evolution with Eqs. (2.31) and (2.32) and compute the timescale on which the orbit decays. The primary forms a WD. The evolution of WD + MS systems is critical for the production of type Ia SNe. During evolution the secondary star may fill its Roche lobe and transfer mass to the WD. When the WD accretes matter mainly three things can happen depending on the accretion rate. For a very restricted range of the accretion rates stable burning on the WD surface can happen and the mass of the WD will increase. Depending on its mass prior to accretion the WD may reach the Chandrasekhar mass and explode, possibly as a SNIa in which large amounts of Fe are ejected. The evolution of these systems together with the accretion rates is discussed in more detail in the next paragraph. If the accretion rate is below the required minimum rate to have stable burning on the WD, the WD undergoes nova explosions whereby some of the inner core is ejected leading to a reduction of the WD mass. We do not follow them since we do not account for the enrichment by novae in our chemical evolutionary model. At accretion rates above the allowed maximum rate to have stable burning on the WD, a red-giant-like envelope forms around the WD with stable burning of hydrogen in a shell around the WD. As both stars are now larger than their Roche lobes, the binary enters a stage of CE evolution. The CE will be removed at the expense of orbital energy and spiral-in starts. The orbital evolution of the system is computed with Eq. (2.18). The outcome of the spiral-in process is either the coalescence of the WD with the CHeB companion star or a short period WD + hydrogen deficient CHeB star system where, depending on its mass, the CHeB star evolves into an SN or a WD. In case that both components merge, the result of the merging process is unknown. It was suggested by Sparks and Stecher (1974) on the basis of energy computations that an NS may form with a possible SN explosion. Another possibility may be that the intruding CO WD ignites carbon with
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a flash and completely desintegrates. Whatever the outcome is, if an SN explosion takes place with or without leaving a remnant, it will show hydrogen in its spectrum if during spiral-in the envelope of the companion giant star was not completely ejected. In our PNS code we assume that an SN takes place if the combined mass of the WD and the core of the companion is larger than 1.4 Mx. We classify the SN as a type II (we use the notation SNIIWD). 2.3.3.5. Type Ia SN progenitor scenarios. Type Ia SNe (no H and He lines but strong O, Si and Fe lines) are the brightest of all SN types. Theoretically there is the consensus that an SNIa is the explosion and complete disintegration of a CO WD that has a mass of the order of the Chandrasekhar limiting mass MCh (1.4 Mx). In this model the rapid conversion of about 0.8 Mx of C/O into 56 Ni that decays finally to 56Fe provides the right amount of energy to power the observed explosion and releases it on the right timescale to explain the observed light curve. The deflagration of the CO WD also produces a substantial amount of intermediate mass heavy elements (Ca,Si,. . .) and leaves a fraction of carbon and oxygen in the outer layers unburned, which is in good correspondence with the observed SNIa spectrum (e.g. Nomoto et al., 1984). Helium WDs with masses <0.5 Mx explode if they can grow to 0.6–1.0 Mx. However they eject mainly helium and iron peak elements (Nomoto and Sugimoto, 1977; Woosley and Weaver, 1986) and do not produce the observed SNIa spectrum and light curve. Oxygen–neon–magnesium (ONeMg) WDs which originate from primaries around 10 Mx, are expected to collapse into a NS by induced electron capture on Ne and Mg (Gutierrez et al., 1996). Isolated CO WDs can never become massive enough to explode since they are born with masses between 0.5 and 1.2 Mx. They slowly cool down and end up as dark matter. Only in close binary systems may CO WDs grow to the Chandrasekhar mass by accreting matter from a companion star. The kind of binary systems that can bring the WD mass to the Chandrasekhar
Fig. 8. SNIa progenitor evolutionary paths.
mass are presently not completely identified. Two evolutionary scenarios have been worked out. We summarize them together with the arguments in favour and against them. The corresponding main evolutionary paths are sketched in Fig. 8. Double degenerate scenario: DD scenario. In this scenario the SNIa arises from the merging of two close CO WDs that have a combined mass larger than or equal to MCh. Both dwarfs are brought together by GWR on a timescale Td until the less massive WD fills its Roche lobe and starts to transfer mass. It was first proposed by Iben and Tutukov (1984) and by Webbink (1984) that such binaries exist as a consequence of binary stellar evolution and would explain the absence of hydrogen and provide an easy way to approach the critical mass for explosion. The observed sample of close double WD systems is small and the systems are either not massive enough to explode or have an orbital period that is too long for merging to occur within Hubble time (for a recent review see Livio, 2000; Marsh, 2001). However it has been estimated by Yungelson et al. (2001) (within an uncertainty of a factor of 2–3) that about 1000 WDs with a visual magnitude V 6 15 should be surveyed in order
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100
Mf Mco Mr
Z=0.02
10
1
0.1 1
10
100
M1,0 100
Mf Mco Mr
Z=0.002
10
1
0.1 1
10
100
M1,0 Fig. 9. The final mass after case B RLOF (Mf), the CO core mass (MCO) and the mass of the remnant compact star (Mr) vs. initial primary mass (M1,0) for Z = 0.02 (upper figure) and Z = 0.002 (lower figure). For Z = 0.002 the computations are made with the SW pffiffiffi mass loss rate during the OB, RSG and CHeB phase proportional to Z .
to detect a WD pair that fulfills the conditions to produce an SNIa. The theoretical uncertainty in the DD model is the outcome of the merger process and it is at present unclear whether a SNIa (without a NS) or a NS is formed (e.g. Nomoto and Iben, 1985; Segretain et al., 1997; Saio and Nomoto, 1985, 1998; Fryer et al., 1999; Benz et al., 1990; Mochkovitch and Livio, 1990). The time elapse between the moment of birth (on the ZAMS) of the binary progenitor and the moment that the SNIa goes off is equal to the sum of the timescale on which the secondary star
becomes a WD and the time Td for orbital decay by GWR. For the SNIa to occur within Hubble time the orbital period of the initial WD + WD system must be sufficiently short. For example a (1 Mx + 1 Mx) DD binary will merge within Hubble time if the orbital period at the moment that the secondary has become a WD is smaller than 6.5 h. Single degenerate scenario: SD scenario. The SD scenario was first introduced by Whelan and Iben (1973) and later on promoted and further elaborated by the Tokyo group (see Nomoto et al., 1999 for an overview). The progenitor binary
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system is made up of a CO WD that accretes hydrogen or helium from a Roche lobe filling non-degenerate companion star. For the CO WD to reach MCh, hydrogen and helium burning on top of the carbon–oxygen core must occur stable or with weak flashes. Computations show that this can happen when _ l6 the accretion rate is in the interval M _ 6M _ u with M _ u 8:5 107 ðM WD 0:52Þ M = M _ l ¼ 0:4 M _ u (Nomoto et al., 1979; Fujiyr and M _ l the WD moto, 1982). At rates lower than M undergoes nova eruptions caused by hydrogen flashes. In the outburst more mass is lost than the WD had prior to it (Kovetz and Prialnik, _ u the 1994) and MCh is never reached. Above M mass transfer occurs too fast for accretion and an extended H rich red giant envelope forms around the WD, which is eventually lost via interaction (spiral-in) with the companion star (Nomoto et al., 1979). However, common envelope formation can be prohibited by a strong wind that the WD blows when it accretes at a rate >4 · 106 Mx/yr. This so-called strong wind model was first proposed by Hachisu et al. (1996) and can stabilize the accretion rate. The systems with the appropriate initial parameters, i.e. mass of the WD, mass of the companion and orbital period at the moment that mass transfer starts, for producing an SNIa according to the SD scenario have been theoretically identified by Hachisu et al. (1999). The latter will be used in the present work. To compute with our PNS code the SNIa rate, our binary evolutionary model follows the evolution of all intermediate mass binaries and it computes the properties of the population of WD + MS/evolved companion binaries. At the moment that the secondary star starts to fill its Roche lobe we check on the mass of the WD, the mass of the secondary and on the orbital period to determine whether the system will produce an SNIa or not according to the progenitor criteria that have been derived by Hachisu et al. (1999). This allows us to compute the expected SNIa rate produced by the SD scenario. With the DD scenario we compute the number of CO WD + CO WD systems that forms with a total mass P 1.4 Mx.
2.3.3.6. Case BB evolution. After case B mass transfer, the remnant helium star may grow to giant dimensions during HeSB and fill its Roche lobe for a second time, initiating a phase of case BB mass transfer (Habets, 1986a,b; Avila Reese, 1993). The star loses its remaining hydrogen layers and most of its helium layers on top of the helium burning shell. The mass loss rates of the Roche lobe filling component during case BB RLOF are considerably smaller than during case B RLOF (Dewi et al., 2002). This implies that if during case BB RLOF the accretion star is a normal star, mass transfer will proceed more or less conservatively. However if the accretion star is a compact star (WD, NS or BH) the situation is more complex. The evolution of helium star + NS binaries during case BB RLOF has been studied in detail by Dewi et al. (2002), Dewi and Pols (2003) and by Ivanova et al. (2002). Similar as in Habets (1986a,b), the authors illustrate the importance of the convective envelope of the donor on the mass transfer process. It is expected that in many cases a CE will be formed and the further evolution will be governed by the spiral-in process. However, the papers listed above agree upon the fact that when certain conditions are fulfilled, a CE may be avoided during the case BB RLOF. In this latter case, when the mass transfer rate becomes larger than the critical Eddington accretion rate, the excess mass leaves the binary as a NS stellar wind with the specific orbital angular momentum of the NS (= isotropic re-emission mode, section Roche lobe overflow). Notice however that the conditions mentioned above rely on postCHeB stellar evolutionary calculations and uncertainties in the latter imply uncertainties in the former. The differences in the papers of Dewi et al. (2002), Dewi and Pols (2003) and of Ivanova et al. (2002) illustrate possible consequences of these uncertainties. Dewi et al. (2002) and Dewi and Pols (2003) account for stellar wind mass loss during CHeB but they use a formalism which implies that SW mass loss during CHeB is very small for a post-RLOF He star with a mass 6 6 Mx and hardly affects its CHeB and post CHeB evolution. However, this formalism is very uncertain and, to illustrate, if
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our Eq. (2.4) is extrapolated downwards, it cannot be excluded that at Z = 0.02 this mass loss is sufficiently large in order to suppress case BB RLOF in massive binaries, i.e. the mass that would leave the star due to case BB RLOF is lost by SW during CHeB prior to the onset of case BB. As has been outlined in Vanbeveren et al. (1998b), one of the most important differences between case BB RLOF (and the applied physics of mass transfer/ mass loss from the system) vs. stellar wind mass loss in relation to population synthesis is the orbital period evolution of the binary. To illustrate, when the companion is a compact star, case BB RLOF may be governed by the spiral-in process which may result in a significant hardening of the binary. The latter has a significant effect on the probability for the binary to remain bound after the second SN explosion, and thus on the birth rate of double compact star systems. Notice that when the WR mass loss rate scales with the iron abundance according to Eq. (2.4), the SW at Z = 0.002 is too small to suppress case BB RLOF. Accounting for the discussion above, to demonstrate the importance of the assumption of case BB or stellar wind mass loss and to illustrate the effects of the physics used to describe the mass transfer during case BB, we consider the following five scenarios: scenario 1: case BB RLOF is suppressed by the SW mass loss of the helium star. The SW mass loss is independent from the metallicity and case BB is suppressed at low Z as well. The period variation is given by Eq. (2.9). scenario 2: case BB RLOF is suppressed by the SW mass loss of the helium star only for Z = 0.02. The SW mass loss is metallicity dependent and satisfies Eq. (2.4), which implies that case BB RLOF is not suppressed at Z = 0.002. Because no detailed case BB evolutionary computations exist for low metallicities, we use the computations made for Z = 0.02 for low Z as well. The period evolution is followed by assuming that matter leaves the binary in the form of an NS SW with the specific orbital angular momentum of the NS (see Dewi et al., 2002). The period change is computed with Eq. (2.20).
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scenario 3: similar as scenario 2 but we assume that the mass transfer to the NS results in a CE/ spiral-in evolution and the period variation is given by Eq. (2.18). scenario 4: the SW mass loss for hydrogen deficient CHeB stars with a mass smaller than 5 Mx can be neglected and case BB RLOF does take place independently of the metallicity. The period evolution during case BB mass transfer to the NS is treated in the same way it is in scenario 2. scenario 5: similar as scenario 4 but case BB mass transfer to the NS is assumed always to result into CE/spiral-in evolution. We remark that when the accretion star is a normal star, the case BB RLOF is always treated as a conservative process in our PNS model. A complete overview of the various binary evolutionary channels is given in Figs. 10–12.
2.4. Distribution functions To set up the initial stellar population for PNS computations, distributions of the properties of the stars at birth are needed. For single stars we need a distribution only for the initial mass M0 while for binaries we need distributions for the initial primary mass M1,0, the initial mass ratio q0, the initial orbital period P0 and the initial eccentricity (e0). The orbits of semidetached binaries are generally circularised by tidal forces on a timescale which is much smaller than the nuclear timescale on which the binaries evolve. Also during stable RLOF the binary is expected to become circularised. Therefore we take e0 = 0. Anticipating, our PNS computations marginally depend on the initial eccentricity. For the initial metallicity we always take a single chemical composition. Assuming that the initial properties of the stars are time-independent we have that the number of single stars formed in the time interval [t, t + dt] and mass interval [M0, M0 + dM0] is
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Fig. 10. Evolution of close binaries with q0 6 0.2 (left part) and >0.2 (right part) starting from the ZAMS until the beginning of CHeB.
Fig. 11. The evolution from CHeB until the formation of a compact object.
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dN s ðM 0 ; tÞ ¼ fs WðtÞuðM 0 ÞdM 0 dt;
ð2:34Þ
with fs the single star fraction among the zeroage population, W(t) the star formation rate (SFR) that gives the rate at which gas is converted into stars and u(M0) the initial mass function (IMF) that describes the distribution of stellar masses at birth the number of binaries formed in the time interval [t, t + dt] with a primary mass in the interval [M1,0, M1,0 + dM1,0], a mass-ratio in the interval [q0, q0 + dq0] and orbital period in the interval [P0, P0 + dP0] is dN b ðM 1;0 ; q0 ; P 0 ; tÞ ¼ fb WðtÞuðM 1;0 Þ/ðq0 ÞpðP 0 ÞdM 1;0 dq0 dP 0 dt; ð2:35Þ with fb the binary fraction among the zero-age population, u(M1,0) the initial mass function for the initial primary mass, /(q0) the initial mass ratio distribution and p(P0) the initial orbital period distribution. We assume that the initial distribution functions of the parameters are independent (an assumption that is generally made in PNS) since the observations give no clear indications for a possible correlation between them. a. The single and binary star fraction: fs and fb. At each moment the fraction of single resp. binary stars on the ZAMS is given by fs resp. fb with fs + fb = 1. From observational studies on spectroscopic binaries in the solar neighbourhood it is known that about 33% (±13%) of the O-type stars are the primaries of massive close binaries with a mass ratio q > 0.2 and an orbital period P 6 100 days (Garmany et al., 1980). A similar conclusion has been drawn for the intermediate mass B stars (Vanbeveren et al., 1998b). More recently the binary frequency among O stars in some young open clusters and OB associations has been estimated to be larger than 50% (up to 80%) (Mermilliod, 2001). The observed binarity among low mass G stars is about 57% (±9%) (Duquennoy and Mayor, 1991) and 42% (±9%) among M stars (Fischer and Marcy, 1992). It should be clear that due to
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selection effects these estimated percentages are underlimits of the true binary frequency. The fraction of initially formed binaries can differ significantly from the observed binary fraction. For example an observed single star may have had its origin in a binary system that was disrupted by an SN explosion or may be the merger product of two stars in a close binary system. In our PNS model we treat fb as a free parameter and look for the initial binary fraction that is required to reproduce the observed fractions among the different stellar objects. b. The initial mass function: u(M0). In the solar neighbourhood the IMF can be well described by a power law of the form uðM 0 Þ / M x 0 with x ¼ 2:35 for M 0 6 2 M (Salpeter, 1955) and x = 2.7 for M0 > 2 Mx (Scalo, 1986). The IMF is normally not corrected for unresolved binaries and may not represent truly the mass distribution of real single stars (Vanbeveren, 1982). However, in PNS studies for reasons of simplicity, it is generally assumed that the IMF of real single stars and of primary stars in binaries is the same. It was shown by Vanbeveren (1982) that this only holds if the binary fraction and the mass ratio distribution are independent of the total mass of the binary. It is obvious that the range of allowed primary masses depends on the finite mass of the parent gascloud out of which the binary forms. In our PNS model we take the same IMF for the primary masses as for the single star masses. c. The initial mass ratio distribution: /(q0). The mass ratio distribution of unevolved binaries is quite uncertain since the observed q-distribution is strongly affected by selection effects. From observed double-lined spectroscopic binaries (SB2s) for which the orbital motion of both stars can be measured, the mass ratio q0 can be determined directly. However, SB2s are heavily biased towards systems with mass ratios near unity because in lower q0 binaries the signal of the fainter secondary star (M2,0) is swamped by that of the brighter and more massive primary star (M1,0). If the observed binary spectrum is single-lined (=SB1s), q0 cannot be determined directly but instead the mass function f(M2,0) is measured, which is given by f ðM 2;0 Þ ¼
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M 1;0 sin3 iq30 =ð1 þ q0 Þ with i the orbital inclination equal to the angle between the line of sight and the axis perpendicular to the orbital plane. The primary mass (M1,0) can be derived from its spectral type and luminosity class, while the inclination is unknown and has to be accounted for statistically (see e.g. Abt and Levy, 1985; Vanbeveren et al., 1998b). Extensive studies on the shape of the mass ratio distribution for spectroscopic binaries (SB1s and SB2s) of different spectral type, whereby accounting for the observational selection effects, have been carried out by several people (e.g. Garmany et al., 1980; Hogeveen, 1991, 1992; Mazeh and Goldberg, 1992; Van Rensbergen, 2001). Different distributions (due to different interpretations of the data) are found: a flat one where every possible mass ratio occurs with a constant probability, one that decreases towards q0 = 1 or one that peaks near q0 = 1. Presently, the observational selection effects are not known sufficiently well to select the distribution that is closest to reality, and therefore it is advisable to parameterize the mass ratio distribu-
tion as /ðq0 Þ / qc0 whereby the value of c is varied to account for the possible different shapes. In our PNS computations we use the distributions with the following prescriptions: /F(q0) = 1 (flat). /H ðq0 Þ / q0:2 ðq0 > 0:3Þ ¼ constant ðq0 6 0:3Þ 0 (Hogeveen, 1991, 1992). /G ðq0 Þ / q0:5 0 (Garmany et al., 1980). We notice that a Hogeveen type distribution peaks at small q values, which is also predicted by a model where the distribution of the sum of the binary component masses corresponds to the overall IMF. This distribution is also preferred for reproducing the number of low mass X-ray binaries (LMXBs) which are interacting systems in which an NS or BH is accreting mass from a lowR mass companion. For the normalization we 1 set 0 /ðqÞdq ¼ 1. d. The initial orbital period distribution: p(P0). Although observational selections are very important here as well, the observations appear to be consistent with a flat distribution in log P0 for or-
Fig. 12. Evolution of close binaries with a WD companion (left part) and NS or BH companion (right part).
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bital periods of between a few days and several thousand days (Popova et al., 1982; Abt, 1983). More recently a new period distribution was estimated by Mason et al. (2001) using the existing observational data in literature in combination with the results of the speckle interferometric survey of galactic O-type binaries. Their derived period distribution appears to be bimodal in log P0, with peaks at 81 days and at 150 yr, which correspond respectively to the most common spectroscopic and visual binaries. However, the absent systems with periods of years to decades are the most difficult ones to observe and a flat distribution in log P0 is not excluded. In this work the initial orbital periods are distributed according to P(P0) dP0/P0 between P0,min and P0,max = 10 yrs, corresponding with the minimum period required to avoid contact of both stars on the ZAMS (on average 1 day) and a considered maximum period for binary interaction. The R 3650period distribution is normalized according to P min PðP ÞdP ¼ 1. e. The kick velocity distribution: C(vk). For exploding binary components we use the same kick velocity distribution as for single stars (Section 2.2.2). 2.5. PNS predictions of massive star populations Many studies have been published on simulations of massive star populations. Curiously, a non-negligible number denies the existence of binaries and therefore we do not consider them furtheron. The PNS code described in the previous sections has been used to study massive star populations that are formed in regions of continuous star formation and in starburst regions. The results have been published in the series of papers (Vanbeveren et al., 1997, 1998a; De Donder et al., 1997; De Donder and Vanbeveren, 1998, 1999, 2002, 2003a,b,c; Van Bever and Vanbeveren, 1997, 1998, 2000, 2003; Van Bever et al., 1999). They contain a long list of references of population synthesis work done by other groups. Although it is difficult to promote one unique PNS model, when we consider all studies it follows that to explain the observations of the massive star population in the solar neighbourhood and in the
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Magellanic Clouds we find a best agreement with PNS simulations where it is assumed that more than half of the massive stars form in interacting binaries, during the supernova explosion the average kick velocity 450 km/s, mass loss by SW during CHeB is metallicity dependent, the binary fraction is a function of metallicity. Any severe restriction on the fraction b of material transferred during RLOF and accreted by the companion star cannot be deduced from our present PNS computations. Nevertheless it can be concluded that if b = 0, no mass gainers are produced but that a large number of mergers may be formed due to orbital decay caused by angular momentum loss when matter leaves the system. Although mergers are observed as single stars, their behaviour may deviate significantly from a normal single star. In the opposite case when b = 1, a significant number of mass gainers are produced, which rejuvenates the population. More observations and sophisticated evolutionary computations are needed to identify the real b law. It has been concluded by Vanbeveren (2001) that although rotation may play a key role in understanding the properties of individual stars (e.g. the CNO surface abundances), for the overall evolution of massive single stars and primaries the effect of rotation can be simulated by moderate convective core overshooting. Therefore it is expected that the inclusion of rotation will hardly change conclusions that result from massive star PNS for the Milky Way where evolutionary computations are used with a moderate amount of overshooting.
3. The chemical evolutionary model Supernova rates in galaxies depend on the physics of galaxy formation, on the overall star formation rate and on stellar evolution. At least the latter depends on the metallicity. Therefore, to calculate the temporal evolution of the SN rates in galaxies, it is essential to combine a star formation
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model, a galaxy formation model and a PNS model which all together form a galactic chemical evolutionary model. We combine our PNS model, which follows in detail the evolution of a stellar population, consisting of single stars and close binaries, with the two-infall Galaxy formation model of Chiappini et al. (1997) to study the time evolution in the solar neighbourhood of the star formation rate, the iron relative to hydrogen abundance ratio [Fe/H], the G-dwarf disk metallicity distribution, the supernova rates of the different spectral types, the merger rates of binaries consisting of only NSs and BHs, the abundance ratios [X/ Fe] 7 with X = He, C, N, O, Ne, Mg, Si, S, Ca and r-process elements. 3.1. Introduction By means of a CEM the evolution of the chemical composition of the ISM and of the abundance distributions in the stars is followed in space and time. The galaxy is formed and evolved according to specific prescriptions for the initial conditions, the star formation rate, the behaviour of the stars and their chemical output and for possible gas inand/or outflow processes. The prescriptions are derived from theory and/or from galactic and extragalactic observations. A CEM is considered as successful when it reproduces the observational characteristics of the modeled galaxy within a factor of two, whereby accounting for the uncertainties in the input theory and in the handling and interpretation of the observational data. At present, the availability of a large amount of high quality data on the major features of the Milky Way and of the chemical yields for stars of all masses at different initial metallicities, allow the construction of successful models for the chemical evolution of the solar neighbourhood (e.g. Boissier and Prantzos, 1999; Chang et al., 1999; Chiappini et al., 1999; Portinari and Chiosi, 1999). A unique scenario for the evolution of the entire Milky Way does not yet exist as there are still some issues that are not fully understood (for an overview see Tosi, 2000).
7
[X/Fe] = log(X/Fe)% log(X/Fe)x.
In trying to fit the observational constraints in the solar neighbourhood, some basic assumptions have been invoked in the course of time which are now standard in most CEMs. A major assumption is that the Milky Way has been formed by infall of gas with primordial abundances (or gas that is only slightly enriched in heavy elements Tosi, 1988). Infall provides a solution to the G-dwarf problem that was originally discovered by van den Bergh (1962) and Schmidt (1963) and occurs when the Milky Way is modeled as a closed box. In this case theory predicts too many long-lived disk dwarfs at low metallicity when compared to the observed G-type dwarf metallicity distribution in the solar neighbourhood. The open infall model (i.e. gas can flow in and out of the galaxy) was first proposed by Larson (1972) and Sciama (1972) and still forms the basis of most CEMs. A second basic assumption that is generally made is the homogeneous evolution of the ISM, which is referred to as the instantaneous mixing approximation (IMA). It implies that at any moment the ISM has a uniform composition, which strongly simplifies the basic equations for chemical evolution. Homogeneous evolution does not allow to treat scatter in abundances due to processes like orbital diffusion, non-instantaneous mixing of enriched material, self propagation of star formation etc., but only gives average trends. Although this can be considered as a limitation or drawback, it is often preferred to the non-negligible number of extra free parameters that are introduced when including these dynamical processes. In the present work we adopt the two-infall model of Chiappini et al. (1997) in which the Milky Way is assembled by infalling galactic and extragalactic gas. For a more extended discussion on the latter model in terms of the dynamical evolution of galaxies, see Section 1. An overview of the general approach in chemical evolutionary studies is given in Fig. 13. 3.2. The two-infall scenario In our CEM the Milky Way is assumed to form by infall of primordial gas (i.e. X = 0.76, Y = 0.24 and Z = 0) with no outflows. In a first period the
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Fig. 13. Workscheme for chemical evolution modeling.
halo and thick disk form during a fast collapse. At the same time gas that escapes from the halo accumulates in the center of the spheroid and creates the bulge. Once the halo, thick disk is almost formed (after 1 Gyr), a second much longer period of major infall starts during which the thin disk assembles mainly from primordial gas and some halo gas. The infall rate at a distance r from the galactic center is given by the relation drg;inf ðr; tÞ ¼ AðrÞet=tI þ BðrÞeðttmax Þ=tII ðrÞ ; dt
ð3:1Þ
with rg,inf the surface gas density of the infalling gas, which is assumed to have a primordial chemical composition; tI and tII are the timescales for mass accretion onto the halo, thick disk and thin disk components respectively; tmax the time of maximum mass accretion onto the thin disk which coincides with the end of the halo, thick disk phase. Other types of infall rates have also been explored in the literature, but not all of them can solve the G-dwarf problem. For example, a constant infall rate that just balances star formation produces too many stars at high metallicity. An exponentially decreasing one seems to give the best
overall results and agrees with recent results of hydrodynamical simulation (Sommer-Larsen et al., 2003). In the thin disk we assume that the inner parts are built much more rapidly than the outer ones according to the inside–out scenario of Larson (1976), which ensures the formation of abundance gradients along the disk. This implies that the timescale tII is a function of the galactocentric distance r. We adopt the relation tII ¼ 0:875r 0:75 ðGyrÞ:
ð3:2Þ
(Matteucci and Franc¸ois, 1989; Burkert et al., 1992) which gives a timescale for the bulge formation of 1 Gyr and agrees with the results of Matteucci and Brocato (1990), and a timescale of 8 Gyr for the solar neighbourhood (rx = 10 kpc) which reproduces well the G-type dwarf metallicity distribution. Notice that these timescales are also supported by the recent hydrodynamical modeling of the Milky Way by Sommer-Larsen et al. (2003). The coe¨fficients A and B are fixed by reproducing the current total surface mass density distribution in the solar neighbourhood. For the disk we use the distribution given by Rana
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(1991) while we take rI(tnow) 10 Mx/pc2 which is an average value that follows from observational studies on the density ratio of the thick and thin disk (Kuijken and Gilmore, 1989; Reid and Majewski, 1993; Robin et al., 1996; Buser et al., 1998). In our CEM we take tI = 1 Gyr, tII = 8 Gyr for r = rx = 10 kpc, tmax = 2 Gyr and tnow = 15 Gyr as fixed values. We remind that this set of values correspond to the best model A in the study of Chiappini et al. (1997) and are obtained without the inclusion of binaries. When including binaries these values may not give the best fit with the observations, however it is beyound the scope of this work to investigate the influence of binaries on the parameters involved in galaxy formation. The occurrence of two distinct infall periods has been suggested by observational studies on the kinematics and chemistry of a large sample of stars in the solar neighbourhood. Major observational indications are a different angular momentum distribution of the spheroidal (bulge and halo) and disk (thick and thin) components and the metalweak tail ([Fe/H] 6 1.5) of the thick disk population (Beers and Sommer-Larsen, 1995; Ibata and Gilmore, 1995; Gratton et al., 1996). Also the observed [a/Fe] distribution seems to indicate a short timescale for the evolution of the halo and thick disk phases and a sudden decrease in the star formation in the period preceding the formation of the thin disk (Gratton et al., 1996; Bernkopf and Fuhrmann, 1998). These observational results are difficult to reconcile with the picture in which the Milky Way forms through a continuously dissipative collapse (during which halo, thick and thin disk formed simultaneously at different rates) or by one episode of infall. We remark that the rather long timescale for the formation of the solar neighbourhood implies that the disk must have been formed mainly out of extragalactic gas. The long timescale is required to reproduce the metallicity distribution of disk stars by Wyse and Gilmore (1995) and RochaPinto and Maciel (1996). 3.3. The star formation rate In order to study the evolution of the Milky Way we obviously need to know the temporal evo-
lution of the star formation rate. However, the history of star formation is not well known and different analytical prescriptions have been explored in terms of intrinsic parameters of spiral galaxies. Most often the Schmidt law (Schmidt, 1959) is used, which assumes that the star formation rate is proportional to some power of the surface gas density rg i.e. WðtÞ / rkg ðtÞ with k 1–2. The Schmidt law is compatible with the observed disk averaged star formation rate in spirals and starbursts but does not describe well the local star formation process (Kennicutt, 1998). Locally, star formation is regulated by the balance between cooling due to gas accretion onto the disk, which enhances star formation, and gas heating due to the feedback of massive stars via SWs and SN explosions which reduces the star formation efficiency. This means that the star formation rate is a function of the local gravitational potential and thus also depends on the total surface mass density r i.e. WðtÞ / rk1 rkg ðtÞ (Talbot and Arnett, 1975). As shown by Dopita and Ryder (1994) this kind of star formation law is in better agreement with the observations compared to the simple Schmidt law. Moreover, self-regulated star formation is also obtained in chemodynamical models (Burkert et al., 1992) and can accommodate the observed correlation between the surface brightness and the oxygen abundance in late spiral disks (Edmunds and Pagel, 1984; Ryder, 1995). We use the following prescription for the star formation rate, which is based on the Talbott and Arnett formulation but has been adapted to the infall model by Chiosi (1980): ðk1Þ 2ðk1Þ k rðr;tÞ rðr;tnow Þ rg ðr;tÞ Wðr;tÞ ¼~m rðr ;tÞ rðr;tÞ rðr;tnow Þ ¼ 0 when rg ðr;tÞ67 M =pc2 ; with ~m the efficiency of star formation expressed in Gyr1. Both parameters ~m and k have different values for the halo, thick disk and thin disk phase, i.e. ð~mI ; k I Þ and ð~mII ; k II Þ, for which we adopt respectively the values (2, 1.5) and (1, 1.5) corresponding with model A in the study of Chi-
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appini et al. (1997). The adopted value for the surface gas density exponent (k) also agrees with the observational results of Kennicutt (1998) and the N-body simulations of Gerritsen and Icke (1997). A star formation stop at surface gas densities lower than 7 Mx/pc2 has been suggested by observational studies on massive star formation in external galaxies (Kennicutt, 1989; Chamcham and Tayler, 1994; Gratton et al., 1996; de Blok and McGaugh, 1996). At these low densities the gas is theoretically expected to be stable against density condensations which are the seeds for star formation. It has been demonstrated and emphasized by Chiappini et al. (1997) that the threshold causes a gap in the SFR between the end of the halo, thick disk phase and the beginning of the thin disk phase, which naturally explains the observed steep increase of [Fe/O] and [Fe/Mg] at a certain value of [O/H] and [Mg/H] (Gratton et al., 2000). A temporal sink in the O and Mg production due to a stop in the formation of new massive stars easily explains the observed behaviour of [Fe/O] and [Fe/Mg]. It is interesting to note that the threshold may depend on the metallicity because the opacity, which depends on the metallicity, acts as a cooling agent and may play an important role in the cooling and collapse of a protostellar cloud. Therefore a lower threshold density can be expected at lower metallicities. However, there are currently no prescriptions for a possible threshold density-metallicity relation. In our model we take the threshold density independently of the metallicity. 3.4. The initial mass function We use the same IMF as the one adopted in our PNS model but extend the mass range downwards to 0.1 Mx. Although these very low mass stars do not participate in the chemical enrichment of the ISM, they represent a non-negligible amount of matter. We normalise the IMF by Rmass on the stellar 2 mass range [0.1, 120] Mx i.e. 0:1 c1 M 1:35 dM 0 þ 0 R 120 1.35 1.7 1:7 c2 M 0 dM 0 ¼ 1 with c1 Æ 2 = c2 Æ 2 . 2 The time behaviour of the IMF since the birth of the Milky Way is not known. It has been shown by Chiappini et al. (2000) that the combination of in-
901
fall with an invariable IMF gives the best fit with all the observations. Therefore and for simplicity we assume that the IMF is constant in space and time. 3.5. The binary fraction Whether or not binary star formation depends on metallicity is totally unknown. Several binary star formation mechanisms like fragmentation and fission (for a review see Clarke, 1996) have been proposed but the role of metallicity and gas density in these formation processes is not clear. In our model we use two prescriptions: fb,1: a binary fraction (on the ZAMS) that is constant in space and time. We consider two values: 40% and 70%. fb,2: a binary fraction (on the ZAMS) that is a linear function of the metallicity Z of the interstellar gas out of which the ZAMS-population is formed at that moment. We assume that at Z = 0 the binary fraction is zero and grows linearly to a value of 70% at Z = 0.02. The choice of a current binary fraction (on the ZAMS) of 70% follows from our PNS computations and the comparison with observations (see also Section 2.5), which show that such a high massive binary fraction is needed to reproduce the observed overall massive binary star population in the solar neighbourhood. From our detailed PNS study of the WR star population we concluded that the initial massive binary fraction in the SMC could be much smaller than in the Milky Way and therefore being a function of the metallicity. For simplicity we consider a linear relation between the binary frequency and metallicity and take the same binary fraction among low, intermediate and massive stars. 3.6. The isotopic evolution model To compute the chemical composition of the ISM, the galactic disk is divided into concentric, independently evolving rings of 2 kpc wide. The solar neighbourhood corresponds with the ring at a galactocentric distance r = 10 kpc. Within each ring it is assumed that the stellar ejecta are
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completely and instantaneously mixed with the ISM, implying that each ring is always composed of a homogeneous mixture of gas, stars and stellar remnants. Thus at any time the local ISM is characterized by a unique composition. Under these assumptions the physical quantities that describe the state of each ring like the surface mass density, the gas and stellar masses, the chemical abundances, etc. are only a function of the galactocentric radius and time, which is called the one-zone formulation (Talbot and Arnett, 1971). We treat in detail the delay in the chemical enrichment due to the finite stellar lifetimes (=delayed production approximation). The evolution of the galactic disk is most conveniently described in terms of the total surface mass density r(r,t), which equals the sum of the surface gas density rg(r,t) and the surface star density rs(r,t). In CEMs that model the Milky Way as a closed system, the total mass surface density is constant and all relevant quantities are normalized to it. However this is not true for open models, for which it is more suitable to normalize with respect to the present total surface mass density r(r,tnow). Therefore the normalized surface gas density is defined: G(r,t) = rg(r, t)/r(r,tnow). For each isotope i the normalized gas density is then given as Gi(r,t) = Xi Æ G(r,t) with Xi the abundance by mass of the isotope i. The time evolution of the isotope i is obtained by solving the following set of nonlinear-integro differential equations (see Appendix A): dGi ðr; tÞ dt ¼ X i Wðr; tÞ Z Mu þ ð1 fb ðt tM ÞÞWðr; t tM ÞuðMÞ Ml
RM;i ðt tM ÞdM Z M 1;u Z qu Z P u þ fb ðt tM 1 ÞuðM 1 Þ/ðqÞ M 1;l
ql
Pl
PðP Þ½Wðr; t tM 1 ÞRM 1;i ðt tM 1 Þ þ Wðr; t tM 2 ÞRM 2;i ðt tM 2 Þ dP dq dM 1 þ
dGi;inf ðr; tÞ ; dt
ð3:3Þ
with on the right the first term representing the depletion of isotope i from the ISM by star formation, the second and respectively third term giving the enrichment of the ISM in isotope i by single stars respectively binary stars and the last term gives the contribution from the infalling gas. In the equations, t is the galactic lifetime, tM the evolutionary lifetime of a star with mass M (0 < tM 6 total lifetime of the star), RM,i(t tM) the mass fraction of a star of mass M formed at the moment (t tM) and which is ejected at galactic lifetime t into the ISM in the form of the isotope i, and fb(t) equals the fraction of binaries formed on the ZAMS at time t. We notice that the integrations in term 3 account for all the primaries and secondaries that are formed in the past but eject their material at moment t. This implies that all these primaries and secondaries may not have been born in the same binary system and therefore can have a different star formation rate. For our galactic evolutionary computations however we rewrite terms 2 and 3 with respect to time rather than with respect to mass in order to account for the dependency on the metallicity of all the quantities present in the equation. This has been worked out in the appendix (see Eq. (A.14)). The returned mass fraction RM,i is the ejected amount of processed and non-processed matter divided by the total initial mass of the star and is computed with the formula, N X M ij;exp RM;i ¼ ð3:4Þ X j; X jM j¼1 with Xj the initial abundance (by mass) of isotope j; Mij,exp the amount of isotope i synthesized from M isotope j and eventually expelled and Xii;exp is the iM unprocessed fraction of isotope i that is eventually re-ejected. The ejection takes place during the evolution of a star as a consequence of SW mass loss or as a consequence of the RLOF process when it is a binary component, and at the end of stellar evolution when an SN outburst occurs or a PN is ejected. We explicitly account for these different evolutionary moments. For single stars (second term in the equation) RM,i is only function of the initial mass and metallicity. In the case of a binary system (third term in the equation) we separate the
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975
mass fraction of the primary ðRM 1 ;i Þ and of the secondary ðRM 2 ;i Þ star. The returned mass fractions are function of the initial binary parameters (M1,0, q0, P0) and initial metallicity. They also depend on the adopted values for the binary evolutionary parameters ði:e: bmax ; a; vk Þ. We have constructed complete tables with the R-values for single stars and binary stars and they are available upon request. 3.7. Computation of the supernova rates For single stars the computation of the SN rates is given by a simple continuous integration over the progenitor initial mass range. However, in case of binaries, the SN rates depend critically (as we will show later on in Section 4.8) on the characteristics of the progenitor binary population and must be computed with a PNS model. In most CEMs only the SNIas (for which the SD scenario is commonly adopted) are considered to be formed by binaries and their rate is generally computed by solving the integral Z
M max
A M min
uðMÞ M
Z
0:5
f ðlÞWðt sM 2 Þdl dM
ð3:5Þ
lmin
(Greggio and Renzini, 1983) with M = the total mass of the progenitor binary system, l = M2/ M = the mass ratio of the secondarys mass to the total mass of the binary, f(l) = the mass ratio distribution and A a free parameter that represents the fraction of the mass range [Mmin, Mmax] that forms binaries which produce SNIas. In the model computations the parameter A is then finally fixed to match the observed SNIa rate and iron abundance. The evolution of these binary systems is in most CEMs approximated by that of two single stars whereby the total ejecta is taken equal to the sum of the ejecta of a single star with the same initial mass as the primary star and the ejecta of a typical SNIa. It follows from binary evolutionary theory that whether or not a binary component produces an SN, of which spectral type (II, Ib/c, Ia) the SN will be,
903
on which timescale the SN forms, the realisation frequencies depend on the binary mass, mass ratio, orbital period, the adopted distribution functions and evolutionary parameters and cannot be evaluated by a simple integral as given by Eq. (3.5). In particular for the SNIas, the conditions on the binary parameters are so confined that a detailed modeling of the binary population is needed. Therefore the treatment of the SNIas in current CEMs is oversimplified and the models should be linked with an PNS model that can evaluate in detail the evolution of the properties of the single and binary star population as a function of time. Our PNS model in combination with the adopted star formation rate allows the computation of the time evolution of the SNII, SNIbc, SNIa rates over the whole galactic lifetime. 3.8. The stellar yields It is obvious that the amounts of matter in the form of isotope i ejected by a star are key ingredients in a CEM. Rather than discussing the ejecta which are used in the CEM, we prefer to consider here the stellar yields in order to have an idea of the nucleosynthetic production due to the star itself. The stellar yield Mpi is the mass of a newly synthesized isotope i that is produced and ejected by a star of initial mass M. It is computed according to: Z T ðMÞ _ MðM; tÞ½X i;s X i;0 dt Mpi ¼ 0
þ
Z
Mf
½X i ðM 0 Þ X i;0 dM 0 ;
ð3:6Þ
Mr
with T(M) the total lifetime of the star, Xi,0, Xi,s and Xi(M 0 ) the mass fraction of isotope i at resp. the moment of birth, the surface and at mass level M 0 in the star (Maeder, 1992). The first term gives the SW yield and the second one the SN yield with Mf the final mass of the star at core collapse and Mr the mass of the remnant compact object. If the isotope is converted into other species rather than being produced, its yield is negative. The total amount of isotope i with the inclusion of its initial abundance, ejected by the star (=ejecta) is given by the relation:
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Ei;M ¼ ðM M r ÞX i;0 þ Mpi :
ð3:7Þ
3.8.1. Single star yields Stars initially less massive than 0.8 Mx do not produce yields and thus do not contribute to the chemical enrichment of the ISM. They only function as depots of matter. Low and intermediate mass stars enrich the ISM via their SWs during the RGB, AGB and the PN phase in 4He, 12C, 13 C, 14N, 16O, 17O and heavy s-process elements like Ba and Sr, which are essentially made during HeSB. The yields in this mass range are most sensitive to the adopted mixing length parameter, which determines the depth of the convective stellar envelope and the extent of hot bottom burning (HBB). When the convective envelope moves inward, convective motion dredges up material from the inner region to the outer layers. It takes place on the RGB (= first dredge-up), on the E-AGB (= second dredge-up) and finally during the TPAGB (= third dredge-up). 8 The efficiency of dredge-up determines the amount of material that is mixed into the outer layers. Hot bottom burning occurs at the base of the convective envelope during the interpulse phase on the AGB if the temperature is high enough, which implies a minimum initial mass (Iben and Renzini, 1983). During HBB carbon and oxygen are converted into nitrogen. Observations indicate that HBB occurs in AGB stars heavier than 3.5 Mx. For low and intermediate mass stars the yields from Renzini and Voli (1981) have been extensively used in CEMs. However, in recent years new evolutionary calculations have been presented with updated physical models (Marigo et al., 1996, 1998; Marigo, 2001; van den Hoek and Groenewegen, 1997). Here we use the stellar yields from the work of van den Hoek and Groenewegen (1997), which are computed for stars with an initial mass between 0.9 and 8 Mx and initial metallicities Z = 0.001, 0.004, 0.008, 0.02 and 0.04. In their evolutionary computations they account for the three dredge-
8 In intermediate mass stars the first dredge-up occurs only if Z > 0.001 (Dominguez et al., 1999) while low mass stars do not undergo the second dredge-up.
up phases, HBB and treat in detail mass loss as a function of the initial metallicity. The free parameters in their stellar models, which are related to mass loss, the minimum core mass for dredge-up and HBB, and the third dredge-up efficiency, are fixed in order to get the best fit with the observations of AGB stars in the Milky Way and LMC. We use their standard model calculations. The yields are shown in Fig. 14 for Z = 0.001 and Z = 0.02. From these figures it can be concluded that All stars in this mass range enrich the ISM in 4 He in more or less equal amounts. The enrichment in 12C mainly comes from the low mass AGB stars (M 6 4 Mx) while the higher mass stars predominantly contribute to the 14N enrichment. This is mainly due to the presence of HBB in stars above 4 Mx. The 12C and 16O yields generally increase with decreasing initial Z. At lower Z the SW mass loss rates are smaller, which results in larger core masses at the first thermal pulse and longer AGB lifetimes, leading to more pulses. All together the amount of material dredged-up from the core to the envelope is substantially larger in AGB stars of initial lower Z. The 14N yields slightly increase with rising Z since 14N is mainly formed as a secondary element (i.e. its production is proportional to the initial CNO abundance) during CNO burning by consuming 12C and 16O. The oxygen enrichment by low and intermediate mass stars is very small. Massive stars produce the bulk of the heavy elements with 16O as the most abundant one. The dominant elements in their SN ejecta are the a-elements 12C, 16O, 20Ne, 24Mg, 28Si, 32S, 40Ca and 56 Fe that derives from 56Ni which is synthesized during explosive Si-burning. Explosive nucleosynthesis mainly affects the abundances of Si, S and Ca. Supernova nucleosynthesis computations with the inclusion of detailed explosive nucleosynthesis have been performed by Woosley and Weaver (1995) for various initial metallicities (Z = 0, 2e6, 2e4, 0.002 and 0.02) and masses (8–40 Mx), by Thielemann et al. (1996) and Nomoto et al. (1997) for helium cores that correspond to
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975
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Mpi
Mpi
0.025
0.35
C
12
He
4
0.3
0.02 0.015
0.25 0.2
0.01
0.15
0.005
0.1
0
0.05
-0.005 -0.01
0 0
1
2
3
4
5
6
7
0
8
1
2
3
Mi
4
5
6
7
8
5
6
7
8
Mi Mpi
Mpi
0.004
0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -0.01
14
16
0.002 0
N
O
-0.002 -0.004 -0.006 -0.008 -0.01 -0.012 -0.014 -0.016 0
1
2
3
4
5
6
7
8
0
1
2
3
Mi
4
Mi
Fig. 14. Stellar yields (expressed in solar masses) from low and intermediate mass stars from the work of HG97 for Z = 0.02 (full lines) and Z = 0.001 (dashed lines).
single stars of initial solar metallicity with masses up to 70 Mx and by Limongi et al. (2000) for Z = 0, 0.001 and 0.02 in the mass range (13–25) Mx. The yields from Woosley and Weaver (1995) and Thielemann et al. have been widely used in CEMs. At solar metallicity both works give similar yields for the CNO isotopes but significant differences for other important isotopes (especially iron and magnesium) due to different input physics like the treatment of convection. Because of the wider considered interval of masses and metallicities, we use the Woosley and Weaver (1995) computations to derive the SN yields of our exploding massive stars according to the CO core mass vs. remnant mass relation that was discussed earlier in Section 2.2.2. From the tabulated ejecta in Woosley and Weaver (1995), the CO core ejecta are calculated according to the method outlined by Portinari et al. (1998). It assumes that the H and He ejecta
given by Woosley and Weaver (1995) solely come from the layers outside the CO-core and that the global metallicity in the layers outside the CO core is equal to the initial (ZAMS) one. Under these assumptions the mass of the CO core is given by M CO ¼ M
EH þ EHe 1Z
and the ejecta of the CO core can be computed with simple relations as given in Portinari et al. (1998). Despite the incoherences of this method (discussed by Portinari et al., 1998) it is sufficient for the scope of the present work. For the remaining layers above the CO core we assume that they are not affected by the outgoing SN shock wave and take the chemical composition from our computed stellar models. For a given pre-core collapse mass, the uncertainties in the SN yields mainly come from the adopted nuclear reaction rates and the mass-cut
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which defines the border between the remnant mass and the ejected envelope. The latter determines the amount of iron that is ejected. The uncertainty is about a factor of 2 for 16O and higher for the heavier isotopes. In the phases prior to the SN, a massive star ejects elements via SW mass loss. Our SW yields are computed with the SW mass loss rate prescriptions discussed in Section 2. They are given in Fig. 15 as a function of the initial mass and for Z = 0.02 and pffiffiffi 0.001. The computations are made _ / Z during the MS, RSG and CHeB with M phase. We also present computations that are made with a Z-independent SW mass loss rate during the CHeB phase. To compare, the SW yields derived from the Geneva tracks (Schaller et al., 1992) are also given.
Mpi
The figures illustrate the following conclusions: Stellar winds contribute significantly to the enrichment in 4He. Due to the transformation of 12C and 16O into 14 N during the CNO cycle, the 12C and 16O stellar wind yields are negative in the mass range below 25 Mx. The 12C (16O) yields are positive when SW mass loss is large enough so that He burning products appear on the surface, i.e. when the star becomes a WC star. The later the WC stage starts, the more He is processed into 12C and 16O and the more 12C and 16O is ejected by the wind. Because in our stellar models 14N is created as a secondary element, the 14N stellar wind yields are larger at higher Z.
Mp i 2.2
2.2
He/10 Z=0.02 N
2 1.8 1.6
1.6
1.4
1.4 1.2
1.2
1
1 0.8 0.6
0.8 0.6
0.4
0.4
0.2 0
0.2 0
-0.2 0
C/10 Z=0.02 O/10 Ne
2 1.8
-0.2 10
20
30
40
50
60
70
80
90
100
110
120
0
10
20
30
40
50
Mi
60
70
80
90
100
110
120
70
80
90
100
110
120
Mi
Mpi
Mpi 4
2.2
He/10 Z=0.001 N*10
2 1.8 1.6
C O Ne
3.5 3
Z=0.001
2.5
1.4 1.2
2
1 1.5
0.8 0.6
1
0.4
0.5
0.2 0
0 -0.2 0
-0.5 10
20
30
40
50
60
Mi
70
80
90
100
110
120
0
10
20
30
40
50
60
Mi
Fig. 15. The SW yields (expressed in solar masses) vs. the initial mass for Z = 0.02 and Z = 0.001. The SW yields from the computations of Schaller et al. (1992) are given by the dashed lines. The computations performed with a Z-independent SW during the CHeB phase for Z = 0.001 are given by the thick lines.
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975
The 20Ne stellar wind yields are negligible because 20Ne is made by a-capture on 16O in a region inside the star that is never exposed at the surface even when extreme mass loss occurs. Non-negligible amounts of 22Ne are ejected by WR stars. However, for relatively low masses and/or metallicities the 22Ne yields are negligible. At Z = 0.02 the stellar wind yields from the Geneva tracks (Schaller et al., 1992) in C and O are on average larger due to the adopted WR SW mass loss formalism of Langer (1989), which overestimates the mass loss by a factor of 2–4. For Z = 0.001, our yields are lar_ is ger especially when a Z-independent M adopted during the CHeB phase. We remark that in the computations of Schaller et al. (1992) for Z = 0.001, WR stars only form above an initial mass of 80 Mx, which is much higher than the 40 Mx mass-limit in our modelspfor ffiffiffi the case that the SW mass loss scales with Z . 3.8.2. Binary star yields Non-interacting binary components behave as single stars and therefore produce the same yields. In particular this is also the case for the most massive binaries evolving according to the LBV scenario. Interacting binary stars experience the same various nuclear burning phases as single stars do, but their chemical profile may become significantly altered after interaction with the companion star. We discuss the different aspects of interacting binary evolution and their effects on the yields. 3.8.2.1. Roche lobe overflow. Additional to SW mass loss, RLOF enhances the mass loss rate of an interacting binary component. The process of mass transfer affects the core development of both the mass losing and mass gaining star and by consequence their yields. During RLOF a large part of the H rich envelope is lost on nuclear, thermal or dynamical timescale. Table 4 gives the amount of mass lost by RLOF in the form of the CNO elements that was originally in the convective hydrogen burning core. After CHB the helium core grows due to H shell burning. However, because of the rapid mass loss
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Table 4 The amount of mass lost in the form of the CNO elements, during case B RLOF that has been part of the convective core during H burning as a function of the initial mass Mi
MCNO
3 5 7 9 12 15 20 25 40
0.31 0.70 0.92 1.18 1.63 2.08 3.04 3.98 7.43
during RLOF the growth of the He core is less progressive and a smaller He core is formed when compared to a single star of the same initial mass. Since RLOF removes very rapidly most of the hydrogen rich layers of a massive primary, the effects of a WR-like SW during the post-RLOF CHeB phase are quite different compared to the WR-SW-effects in a single star with the same initial mass, so are the SW yields. It can be concluded that in general interacting primaries form smaller He and CO cores and by consequence produce a lower amount of metals compared to a single star of the same initial mass. Due to case A/B (BB) RLOF, low and intermediate mass primaries may avoid the third dredge-up phase and therefore produce much lower 12C yields compared to single stars of the same initial mass. Fig. 16 gives the SW yields after case B RLOF as a function of the initial primary mass (on the ZAMS). After mass accretion, the secondarys atmosphere may become enriched in 14N and 4He. When the accreted material has already been nuclearly processed by the primary star, thermohaline mixing is induced and the accreted material is mixed with the unaffected outer layers of the accretion star. Depending on the amount of accreted mass and on the amount of mixed material, the accretion star may show moderately or extremely altered CNO surface abundances. As an illustration, Table 5 gives the surface He and N abundances of the secondary after partly-conservative (b = 0.5) and conservative (b = 1) RLOF as predicted by our binary evolutionary computations.
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Mp,i 0.6 0.5 0.4 0.3
He/10 C N O Ne
0.2 0.1 0 -0.1 10
15
20
25
30
35
40
30
35
40
M1,0 Mp,i 0.12 0.1 0.08
He/10 C N O Ne
0.06 0.04 0.02 0 10
15
20
25
M1,0 Fig. 16. The SW yields of massive primaries vs. the initial primary pffiffiffi mass for Z = 0.02 (upper figure) and Z = 0.002 (lower figure). For _ during all phases are scaled according to Z . Z = 0.002 the SW M
The results are given for both accretion models, i.e. the standard accretion model and the full mixing model. Further mass loss by stellar wind may remove these layers and they will enrich the ISM in nitrogen and helium. Another important effect of accretion is the enlargement of the convective core. In particular, due to accretion the core of an intermediate mass secondary may sufficiently grow in mass to undergo the complete nuclear burning cycle and to end up as an SN. For example, an 8 + 7 Mx close binary evolves into a 1 + 14 Mx system whereby the 14 Mx accretion star will explode as an SNII. This
is an aspect of binary evolution that is not met in single star evolution. Furthermore, massive secondaries may acquire after accretion a core mass that surpasses the limit for BH formation and will by consequence eject less matter. In general, accretion stars produce heavier remnants at the end of their evolution than they would have done had no accretion taken place. Table 6 gives the convective core masses of the mass gainers before and after RLOF. 3.8.2.2. Common envelope evolution. We assume that no material is accreted by the inspiraling star
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975 Table 5 The 4He and
909
14
N abundances of the mass gainer after case B RLOF expressed relative to the solar values (=Xi/Xi,x)
Initial system
N/Nx
He/Hex
40 + 36 40 + 24 30 + 27 30 + 18 20 + 18 20 + 12 15 + 14 15 + 9 12 + 11 12 + 7 9 + 8.1 9 + 5.4 6 + 5.4 6 + 3.6 3 + 2.7 3 + 1.8
b = 0.5
b=1
b = 0.5
b=1
1.29(1.94) 1.24(1.61) 1.14(1.72) 1.10(1.39) 1.14(1.60) 1.14(1.32) 1.09(1.46) 1.09(1.22) 1.07(1.39) 1.10(1.20) 1.07(1.33) 1.09(1.18) 1.13(1.33) 1.17(1.22) 1.04(1.20) 1.06(1.10)
1.28(1.80) 1.25(1.53) 1.01(1.46) 1.01(1.23) 1.21(1.57) 1.21(1.34) 1.12(1.41) 1.14((1.23) 1.10(1.34) 1.13(1.21) 1.10(1.30) 1.13(1.18) 1.01(1.34) 1.01(1.26) 1.09(1.20) 1.08(1.11)
2.50(2.03) 2.79(2.24) 1.95(1.69) 2.12(1.83) 1.98(1.76) 2.22(1.96) 1.79(1.65) 2.02(1.82) 1.73(1.62) 1.99(1.81) 1.69(1.60) 1.93(1.78) 1.831.73) 2.11(1.95) 1.47(1.43) 1.62(1.55)
3.04(2.54) 3.38(2.82) 2.08(1.87) 2.33(2.08) 2.49(2.23) 2.74(2.46) 2.17(2.01) 2.43(2.18) 2.09(1.95) 2.36(2.18) 2.03(1.92) 2.27(2.12) 2.23(2.11) 2.51(2.35) 1.73(1.68) 1.84(1.78)
The results given between round brackets are computed for full mixing after accretion. The initial metallicity Z is taken equal to 0.02.
Table 6 The convective core mass of the accreting secondary star before (=RLOFi) and after RLOF (=RLOFe) Initial system
MCC(RLOFi)
MCC(RLOFe)
40 + 36 40 + 24 30 + 27 30 + 18 20 + 18 20 + 12 15 + 14 15 + 9 12 + 11 12 + 7 9 + 8.1 9 + 5.4 6 + 5.4 6 + 3.6 3 + 2.7 3 + 1.8
12.7 9.7 8.9 6.6 5.1 3.8 3.2 2.7 2.4 2.1 1.6 1.4 0.95 0.88 0.35 0.32
16.4 14.3 13.4 13.6 9.7/14.3 8.4/12.1 10.3/10.5 6.3/10.1 7.4/8.1 4.4/7.5 3.2/5.7 3.3/4.9 2.1/3.4 1.9/3.2 0.7/0.9 0.8/1.2
In the third column the first number is computed for b = 0.5 and the second one for b = 1.
during CE evolution. If the system survives, the CE is ejected from the system with the chemical composition it had at the start of the CE phase. 3.8.2.3. Supernovae. To compute the SN yields of massive interacting binary stars we proceed in
Table 7 The SNIa yields produced by a thermonuclearly exploding CO WD (Iwamoto, 1999) compared with the yields of a typical SNII Element
SNIa
SNII
12
4.83e2 1.16e6 1.43e1 2.02e3 8.50e3 1.54e1 8.46e2 1.19e2 6.26e1
7.93e2 1.56e3 1.80 2.12e1 8.83e2 1.05e1 3.84e2 5.77e3 8.44e2
C 14 N 16 O 20 Ne 24 Mg 28 Si 32 S 40 Ca 56 Fe
the same way as for single stars, i.e. we link our computed CO cores with the nucleosynthesis computations (model B) of Woosley and Weaver (1995). For the type Ia SNe we use the yields from the work of Iwamoto (1999) for their model W7, which is an updated version of the classical carbon deflagration W7 model (Nomoto et al., 1984; Thielemann et al., 1986). We use the same yields for different metallicities and different binary progenitors.
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Table 7 gives the SNIa yields of the major elements together with the yields of a typical SNII. It follows that core collapse SNe dominate the production of oxygen while the SNIas dominate the production of iron.
3.8.3. Yields from one stellar generation To compare the single star yields with the binary star yields we compute the relative chemical contribution from one generation of 100% single stars and from one of 100% binary stars. The quantity that expresses the relative contribution is R 120 Mp ðMÞuðMÞdM s ð3:8Þ y i ¼ 0:8R 120 i MuðMÞdM 0:8
3.8.2.4. Mergers. Our treatment of mergers was discussed earlier in Section 2.3.3 in Section 2. No detailed nucleosynthesis computations of binary mergers exist. We compute their yields as follows.
for the population of single stars and
q0 6 0.2 mergers: we take the yields of a single star with the same mass. q0 > 0.2 mergers: both stars are mixed in a homogeneous way. merging WD + non-degenerate secondary: if the resulting CO core mass is larger than 1.4 Mx we assume that the star explodes and take the SN yields for the corresponding CO core in the same way we do for massive stars. If the CO core is not massive enough to ignite carbon non-degenerately we assume that the layers outside the final CO core are ejected in a PN like SW and a WD is left.
y bi ¼
R 120 R 1 R 3650 0:8
0
Mpi ðMÞuðM 1 Þ/ðqÞPðP ÞdP dqdM 1 R 120 R 1 MuðM 1 Þ/ðqÞdqdM 1 0:8 0
P min
ð3:9Þ for the population of binary stars. In the latter formula, M is the total mass of the binary and Pmin is the minimum initial period that a binary must have to avoid contact of both stars on the ZAMS and Mpi(M) is the sum of the stellar yield of the primary and of the secondary. We take 0.8 Mx as the lower mass limit since less massive stars do not contribute (or in negligible amounts) to the chemical enrichment of the ISM. Though they must be included when doing the complete chemical evolution as a function of time since they lock up a non-negligible amount of matter.
3.8.2.5. Novae. We do not include the yields from nova explosions since their contribution to the enrichment in the elements that we investigate in this work is very small. They mainly produce isotopes like 7Li, 13C, 15N, 17O, etc. which are made during explosive H burning.
Table 8 The different parameter sets for which the total integrated yields are computed (ni = not included, i = included) Set
fb
/(q)
bmax
a1,
1 2 3 4 5 6 7 8 9 10 11 12
0 0 1 1 1 1 1 1 1 1 1 1
– – /F /H /G /F /F /F /F /F /F /F
– – 1 1 1 0.5 1 1 0.5 1 1 1
– – 1 1 1 1 0.5 1 0.5 1 1 1
2
a3
vk
M BH 0;s
M BH 0;b
SNIIWD
– – 1 1 1 1 1 0.5 0.5 1 1 1
– – 450 450 450 450 450 450 450 150 450 450
25 40 25 25 25 25 25 25 25 25 25 40
– – 40 40 40 40 40 40 40 40 40 40
ni ni ni ni ni ni ni ni ni ni i ni
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975
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Table 9 Integrated yields (relative to the total yields from the whole mass range) from the massa range [0.8, 9] Mx computed for the different elements and for different PNS sets Set
1 3 4 5 6 7 8 9 11
Element He
C
N
O
Ne
Mg
Si
S
Ca
Fe
8.03e3 7.67e3 2.73e3 3.17e3 1.46e3 1.86e3 3.33e3 3.78e3 4.64e3 5.14e3 3.26e3 3.81e3 2.73e3 3.17e3 5.16e3 5.80e3 2.85e3 3.77e3
1.08e3 5.42e4 2.90e4 2.72e4 1.33e4 1.28e4 3.54e4 3.32e4 4.05e4 4.07e4 3.32e4 3.23e4 2.90e4 2.72e4 4.54e4 4.58e4 3.05e4 3.43e4
3.88e4 8.68e4 2.32e4 3.56e4 1.68e4 2.28e4 2.62e4 4.14e4 3.18e4 5.11e4 2.56e4 3.95e4 2.32e4 3.55e4 3.43e4 5.53e4 2.34e4 3.86e4
1.05e4 3.98e5 1.29e4 1.81e4 3.69e5 6.50e5 1.70e4 2.38e4 2.26e4 4.10e4 1.51e4 2.59e4 1.29e4 1.81e4 2.49e4 4.89e4 1.54e4 3.80e4
– – 1.70e5 4.21e5 5.13e6 2.07e5 2.26e5 5.33e5 3.22e5 9.29e5 2.00e5 6.03e5 1.70e5 4.21e5 3.52e5 1.11e5 1.99e5 7.12e5
– – 6.13e6 1.15e5 1.98e6 6.11e6 8.12e6 1.43e5 1.21e5 2.71e5 7.44e6 1.75e5 6.13e6 1.15e5 1.34e5 3.31e5 7.40e6 1.97e5
– – 2.60e5 3.51e5 1.12e5 1.83e5 3.35e5 4.40e5 5.65e5 7.34e5 3.62e5 4.81e5 2.60e5 3.51e5 6.67e5 8.63e5 6.31e5 1.00e4
– – 1.89e5 2.04e5 1.11e5 9.93e6 2.33e5 2.59e5 4.22e5 4.76e5 2.76e5 3.05e5 1.89e5 2.04e5 5.10e5 5.76e5 3.93e5 7.16e5
– – 3.84e6 2.76e6 2.43e6 1.30e6 4.69e6 3.51e6 8.68e6 6.48e6 5.71e6 4.14e6 3.84e6 2.76e6 1.05e5 7.86e6 7.71e6 1.20e5
– – 2.10e4 1.31e4 1.49e4 9.07e5 2.51e4 1.56e4 4.89e4 3.21e4 3.23e4 2.10e4 2.10e4 1.31e4 6.03e4 4.00e4 3.56e4 2.07e4
In each row the first line gives the values computed for Z = 0.002 and the second line for Z = 0.02.
Tables 9 and 10 list y si and y bi as a function of metallicity for the different elements. The y bi values are computed for different sets of binary parameters to investigate their influence. The parameter sets are defined in Table 8. Set 11 includes the yields from exploding mergers of a WD with a MS or RG companion. To have a clearer view on the relative enrichment we have separated the contribution from stars in the mass range [0.8, 9] Mx from those in the mass range ]9, 120] Mx. We have not included the contribution from SNIas in order to measure their weight in the iron production. The integrated SNIa yields relative to the total initial mass of the stellar generation are given in Tables 11 and 12. Remark that the integrated yield is given by the SNIa rate multiplied with the ejecta which are taken the same for each SNIa, whether it is formed via the SD or DD scenario. However the SNIa rate depends on the adopted scenario and therefore the integrated yields are computed for both progenitor scenarios. For the SD scenario we took the metallicity dependent case. The model number refers to the corresponding binary parameter set.
3.8.3.1. Conclusions. 1.
2.
3.
Low and intermediate mass binaries enrich less in carbon than single stars do. This is due to the fact that the dredge-up phase on the TP-AGB (which is typical in single stars) is suppressed by the RLOF in most of the components of interacting binaries. Single stars of low and intermediate mass eject more helium and nitrogen than binaries in the same mass range do. Single stars of intermediate mass hardly produce a-elements, while a non-negligible amount can be synthesized in binaries of intermediate mass, i.e. accretion stars of intermediate mass that grow sufficiently in mass to burn non-degenerately carbon and oxygen and finally explode, exploding merger stars, type Ia SNe. Their relative contribution to the enrichment in a-elements depends on the adopted binary parameters.
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Table 10 Integrated yields (relative to the total yields from the whole mass range) from the massa range ]9, 120] Mx computed for the different elements and for different PNS sets Set
1 2 3 4 5 6 7 8 9 10 11 12
Element He
C
N
O
Ne
Mg
Si
S
Ca
Fe
1.46e2 1.32e2 1.91e2 1.45e2 9.81e3 2.04e2 1.16e2 2.06e2 9.32e3 2.05e2 1.05e2 2.31e2 9.88e3 2.02e2 9.77e3 2.03e2 1.05e2 2.29e2 9.57e3 2.06e2 9.81e3 2.02e2 1.45e2 2.02e2
1.41e3 5.30e3 1.70e3 5.39e3 1.16e3 5.93e3 1.33e3 6.24e3 1.12e3 5.86e3 1.26e3 6.52e3 1.17e3 6.00e3 1.16e3 5.92e3 1.26e3 6.57e3 1.28e3 5.97e3 1.17e3 5.94e3 1.48e3 6.02e3
1.04e4 7.16e4 1.15e4 7.44e4 7.38e5 6.36e4 8.76e5 6.92e4 7.00e5 6.25e4 8.89e5 7.51e4 7.34e5 6.43e4 7.32e5 6.33e4 8.78e5 7.54e4 7.44e5 6.36e4 7.39e5 6.35e4 8.50e5 6.61e4
4.19e3 6.07e3 8.53e3 8.12e3 3.44e3 5.59e3 4.28e3 6.08e3 3.21e3 5.47e3 4.16e3 6.03e3 3.46e3 5.71e3 3.44e3 5.58e3 4.17e3 6.13e3 3.97e3 5.70e3 3.45e3 5.66e3 7.98e3 6.99e3
5.28e4 7.30e4 1.16e3 2.15e3 3.19e4 7.59e4 4.37e4 7.56e4 2.84e4 7.69e4 3.99e4 7.45e4 3.22e4 7.60e4 3.19e4 7.56e4 4.01e4 7.43e4 3.19e4 7.41e4 3.20e4 7.68e4 9.57e4 1.76e3
1.47e4 1.12e4 3.83e4 1.52e4 7.75e5 9.64e5 1.18e4 1.01e4 6.56e5 9.56e5 1.06e4 9.10e5 7.83e5 9.42e5 7.74e5 9.59e5 1.06e4 8.82e5 7.67e5 9.30e5 7.79e5 9.84e5 3.27e4 1.24e4
6.37e4 6.79e4 8.50e4 9.55e4 4.44e4 4.90e4 5.91e4 5.60e4 4.02e4 4.73e4 5.67e4 5.41e4 4.47e4 5.09e4 4.44e4 4.89e4 5.70e4 5.57e4 4.41e4 4.75e4 4.46e4 5.02e4 6.82e4 6.74e4
3.43e4 3.61e4 4.27e4 4.94e4 2.29e4 2.09e4 3.03e4 2.56e4 2.08e4 1.96e4 2.93e4 2.50e4 2.31e4 2.23e4 2.29e4 2.08e4 2.94e4 2.63e4 2.28e4 2.02e4 2.30e4 2.15e4 3.26e4 2.97e4
5.20e5 4.23e5 6.21e5 5.53e5 3.34e5 1.99e5 4.35e5 2.66e5 3.05e5 1.80e5 4.06e5 2.65e5 3.36e5 2.19e5 3.33e5 1.98e5 4.08e5 2.83e5 3.31e5 1.91e5 3.35e5 2.05e5 4.50e5 2.85e5
1.39e3 8.14e4 1.51e3 8.91e4 8.99e4 6.35e4 9.88e4 6.44e4 8.80e4 6.37e4 8.12e4 5.28e4 9.10e4 6.18e4 8.96e4 6.33e4 8.21e4 5.09e4 8.87e4 6.25e4 9.06e4 6.48e4 1.04e3 6.85e4
In each row the first line gives the values computed for Z = 0.002 and the second line for Z = 0.02.
Table 11 Integrated yields (relative to the total yields from the whole mass range) from SNIas computed for the different elements, for different PNS sets and for the SD (metallicity dependent) scenario Set
3 4 5 6 7 8 9
Element C
N
O
Ne
Mg
Si
S
Ca
Fe
3.06e6 2.91e5 6.84e6 4.62e5 2.00e6 2.43e5 3.06e6 2.91e5 7.93e7 1.48e5 3.06e6 2.91e5 5.29e7 9.85e6
7.34e11 6.98e10 1.64e10 1.11e9 4.79e11 5.83e10 7.34e11 6.98e10 1.90e11 3.55e10 7.34e11 6.98e10 1.27e11 2.37e10
9.05e6 8.61e5 2.02e5 1.37e4 5.91e6 7.18e5 9.05e6 8.61e5 2.35e6 4.37e5 9.05e6 8.61e5 1.57e6 2.92e5
1.28e7 1.22e6 2.86e7 1.93e6 8.35e8 1.01e6 1.28e7 1.22e6 3.32e8 6.18e7 1.28e7 1.22e6 2.21e8 4.12e7
5.38e7 5.12e6 1.20e6 8.12e6 3.51e7 4.27e6 5.38e7 5.12e6 1.40e7 2.60e6 5.38e7 5.12e6 9.30e8 1.73e6
9.74e6 9.27e5 2.18e5 1.47e4 6.36e6 7.74e5 9.75e6 9.27e5 2.53e6 4.71e5 9.75e6 9.27e5 1.69e6 3.14e5
5.35e6 5.09e5 1.20e5 8.09e5 3.50e6 4.25e5 5.35e6 5.09e5 1.39e6 2.59e5 5.35e6 5.09e5 9.26e7 1.73e5
7.53e7 7.16e6 1.68e6 1.14e5 4.92e7 5.98e6 7.53e7 7.16e6 1.95e7 3.64e6 7.53e7 7.16e6 1.30e7 2.43e6
3.96e5 3.77e4 8.86e5 5.98e4 2.59e5 3.14e4 3.96e5 3.77e4 1.03e5 1.91e4 3.96e5 3.77e4 6.85e6 1.28e4
In each row the first line gives the values computed for Z = 0.002 and the second line for Z = 0.02.
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975
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Table 12 Integrated yields (relative to the total yields from the whole mass range) from SNIas computed for the different elements, for different PNS sets and for the DD scenario Set
3 4 5 6 7 8 9
Element C
N
O
Ne
Mg
Si
S
Ca
Fe
6.26e5 4.07e5 1.86e5 1.30e5 8.19e5 5.28e5 2.76e5 2.09e5 5.87e5 3.80e5 8.56e5 6.81e6 8.69e6 6.17e6
1.50e9 9.77e10 4.48e10 3.11e10 1.97e9 1.27e9 6.63e10 5.01e10 1.41e9 9.13e10 2.05e9 1.64e9 2.09e10 1.48e10
1.85e4 1.20e4 5.52e5 3.84e5 2.43e4 1.56e4 8.17e5 6.17e5 1.74e4 1.13e4 2.53e4 2.02e4 2.57e5 1.83e5
2.62e6 1.70e6 7.79e7 5.42e7 3.43e6 2.21e6 1.15e6 8.72e7 2.45e6 1.59e6 3.58e6 2.85e6 3.63e7 2.58e7
1.10e5 7.16e6 3.28e6 2.28e6 1.44e5 9.29e6 4.86e6 3.67e6 1.03e5 6.69e6 1.51e5 1.20e5 1.53e6 1.09e6
2.00e4 1.30e4 5.94e5 4.13e5 2.61e4 1.68e4 8.80e5 6.65e5 1.87e4 1.21e4 2.73e4 2.17e4 2.77e5 1.97e5
1.10e4 7.12e5 3.26e5 2.27e5 1.43e4 9.25e5 4.83e5 3.65e5 1.03e4 6.66e5 1.50e4 1.19e4 1.52e5 1.08e5
1.54e5 1.00e5 4.59e6 3.19e6 2.02e5 1.30e5 6.80e6 5.14e6 1.45e5 9.37e6 2.11e5 1.68e5 2.14e6 1.52e6
8.11e4 5.27e4 2.42e4 1.68e4 1.06e3 6.85e4 3.58e4 2.70e4 7.61e4 4.93e4 1.11e3 8.83e4 1.13e4 7.99e5
In each row the first line gives the values computed for Z = 0.002 and the second line for Z = 0.02.
4.
5.
6.
7.
8.
Due to RLOF the overall He enrichment by massive binaries is larger than it is by massive single stars. Massive stars that perform RLOF develop smaller helium and CO cores than single stars of the same initial mass, which results in a lower production of a-elements. However due to mass accretion, secondary stars may form larger cores and thus synthesize larger amounts of metals than expected from their initial mass before accretion. The variations caused in the enrichment when different binary parameter values are adopted are generally within a factor of 2 except in the a-element enrichment by close binaries of low and intermediate mass, where they may increase up to a factor of 5. The limit on the initial mass of a massive single star to form a BH without a prior SN outburst significantly affects the a-element production, which is illustrated by set 1 and 2. In summary, except for the a-element enrichment in the intermediate mass range, the overall differences between the integrated single and binary star yields are within a factor of 2 and 3.
4. The chemical evolution of the solar neighbourhood In this last section we use our CEM to study the time evolution of the SN rates, the merger rate of double NS/BH systems and of the abundance ratios [X/Fe] in the solar neighbourhood, the region for which most observational data is available. We first repeat the main inputs of our CEM: Assumptions
infall of primordial gas, instantaneous mixing, delayed recycling approximation, continuous SFR with threshold, SFR efficiency: ~mI ¼ 2:0 and ~mII ¼ 1:0, SFR exponent: kI = 1.5 and kII = 1.5, two-component disk: tI = 1.0 Gyr and tII = 8.0 Gyr, time of max. accretion onto disk: tmax = 2.0 Gyr, position of the Sun: rx = 10 kpc, birth of the Sun: tx = 10.5 Gyr, age of the Milky Way: tnow = 15 Gyr. Stellar evolution:
single stars: M0 = [0.1, 120] Mx,
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binary stars: M1,0 = ]1, 120] Mx; q0 > 0; 0 < P0 6 10 yr, IMF: / Mx, x = 2.35 (M0 < 2) and = 2.7 (M0 P 2), SNIa progenitor model: SD or DD, binary frequency: constant (fb,1) or linear with Z (fb,2). Since our main aim is to study the importance of binaries in chemical evolution modeling and because of the large number of parameters, we make our simulations for fixed values of the parameters that directly involve the model for star and Galaxy formation. This means that all our simulations are made with: ~mI ¼ 2:0; ~mII ¼ 1:0; k I ¼ 1:5; k II ¼ 1:5; tI ¼ 1:0 Gyr; tII ¼ 8:0 Gyr; tmax ¼ 2:0 Gyr; uðMÞ / M x with x ¼ 2:35 ðM 0 < 2Þ and ¼ 2:7 ðM 0 P 2Þ. We remind again that this value set has been retained by Chiappini et al. (1997) as the model that gives the best fit with the observations in the solar neighbourhood, despite their computations being made without detailed binary evolution. Notice that the adopted age of the Galaxy may be too large accounting for more recent age determinations of the Universe (about 13.7 Gyr) which implies that the Galaxy is probably not older than 12–13 Gyr. However, it is easy to understand that
for the main conclusions of this section, the uncertainty in the real age of the Milky Way is not very important. We therefore preferred to retain the parameter set of Chiappini et al. (1997). For the stellar evolutionary parameters we define the following standard PNS model which has been proven to reproduce well the observed OBtype and WR star population: the SW mass loss pffiffiffi rates of RSGs are metallicity dependent ð/ Z Þ; although the observations of the present population of massive stars in the Galaxy and in the Magellanic Clouds is better reproduced with a PNS model where the WR SW is Z-dependent (Section 2.5), we consider both a Z-independent WR SW and a Z-dependent WR SW during CHeB to illustrate its effect on the abundances evolution (if not explicitly mentioned the WR SW is taken Z-independent),
a Hogeveen type mass ratio distribution, bmax = 1, maximum CE ejection efficiency: a = 1, average kick velocity: vk ¼ 450 km=s, maximum NS mass: M NS max ¼ 2 M , mass limit for direct BH formation: BH M BH 0;s ¼ 25 M and M 0;b ¼ 40 M .
Fig. 17. The star formation rate in the solar neighbourhood as a function of time, predicted by the standard PNS model with a constant binary fraction of 70%. The SFRs predicted with other PNS models and binary fractions hardly differ (within a few percent) from the one presented here.
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If the value of a parameter in a simulation is not explicitly mentioned, then its value in the standard model is assumed. For the SD scenario we consider two different cases i.e. the WD wind is either metallicity independent (=SD1) or metallicity dependent (=SD2) (see Section 2.3.3 and Kobayashi et al., 1998). 4.1. Star formation rate and surface densities Fig. 17 shows the time evolution of the SFR in the solar neighbourhood predicted by the standard PNS model combined with a constant binary frequency of 70% throughout the galactic evolution. The shape is typical for the two-infall model: a first short and very active period of star formation (= the halo-thick disk phase) followed by a long per-
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iod of moderate and after some time constant level of star formation (= the thin disk phase). The required minimum density for star formation to start is reached after 0.6 Gyr. Due to the included star formation threshold we find that star formation especially during the last 3 Gyr of the thin disk evolution alternately goes on (for a period of 6 Myr) and off (for a period of 4 Myr). During the period of non-activity the dying stars replenish the ISM with gas and again increase the surface gas density. We remark that not all points of zero star formation during galactic evolution are shown (due to a limited capacity of plotting points in the adopted software). The predicted values for the present-day gas fraction, the SFR and surface densities are within the observed value ranges and hardly depend on
Table 13 Predicted and observed present values in the solar neighbourhood Quantity
Predicted
Observed
References
W(Mx pc2 Gyr1) rg(Mx pc2) rs(Mx pc2) rg/r r_ inf ðM pc2 Gyr1 Þ
2.54 7.01 45.48 0.14 1.08
2–5 7–13 30–40 0.05–0.20 0.3–1.5
Gu¨sten and Mezger (1982) Dickey (1993), Flynn et al. (1999) Gilmore et al. (1989) Portinari et al. (1998)
Fig. 18. The time evolution of the surface densities of the total mass (r), visible stars (rs) and of the gas (rg) predicted by the standard PNS model and a constant binary fraction of 70%. The predicted values with other PNS models and binary fractions are within a few percent.
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Fig. 19. Evolution of the SFR during the last Gyr of galactic evolution.
the stellar evolutionary parameters. Their values are given in Table 13 and their time evolution is shown in Fig. 18. Main conclusion: The influence of binaries on the star formation rate is insignificant even when adopting a constant high binary frequency. 4.2. Intermezzo: the WR/O number ratio It is obvious that a massive star formation rate that fluctuates on a timescale shorter than the evolutionary timescale of a massive star, may have important consequences on the temporal variation of the O-type and WR star populations in the solar neighbourhood. Fig. 19 shows a blow up of the predicted SFR during the last Gyr of galactic evolution. The corresponding predicted evolution of the WR/O number ratio (computed with the standard PNS model and a constant binary fraction of 70%) is given in Fig. 20. We observe an overall fluctuating behaviour of the ratio with an amplitude of a factor of 10 in a time interval of at most a few million years. This result implies that although it is easy to recover the observed WR/O ratio, the strongly fluctuation makes that a comparison between observations and theoretical predictions in order to test stellar evolutionary models, is ambiguous. The evolution of the number ratios WC/WN and (WR + OB)/WR are much less affected.
Remark Because we use a uniform density model, abrupt star formation stops occur when the average surface gas density is smaller than the adopted threshold value. However, in reality the surface gas density may be locally larger than the threshold value and it can be expected that the transition from high to low star formation occurs much smoother. Furthermore, the way how the Kennicut condition has been implemented in the model of star formation used in the present work can be questioned. Maybe the path to follow is to incorporate this Kennicut condition into the star formation rate itself, for instance modulating its efficiency in such a way that it should get lower and lower as the gas density approaches the Kennicut regime. Both effects discussed above should smooth the temporal behaviour of the WR/O number ratio but not extinguish the fluctuations on a timescale which is of the order of the evolutionary timescale of a massive star. 4.3. The age–metallicity relation Stellar abundance determinations are usually discussed in terms of the element abundance relative to iron [X/Fe] as a function of the ironto-hydrogen ratio [Fe/H]. This is because [Fe/H] is easy to measure in stars and can be considered as a cosmic clock since the accumulation of iron
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Fig. 20. The predicted time evolution of the WR/O number ratio computed for the standard PNS model and with a constant binary fraction of 70%. The computations shown in the upper (lower) figure are made with M WR min ¼ 5 M ð8 M Þ.
in the ISM increases monotonically with time (Wheeler et al., 1989). The evolution of [Fe/H] as a function of time forms the basis of the age–metallicity relation 9 (AMR) which is an important relation to be repro-
9 As earlier defined in Section 1, the metallicity is the mass fraction of all elements heavier than helium denoted by Z. However this is not always practical for observers because information usually does not exist for all the elements. Therefore in observational astronomy, the word metallicity often refers to the iron abundance.
duced by CEMs. It links the variety of the physical timescales involved like the infall rate, star formation rate and stellar lifetimes. Because all the nucleosynthesized iron is ejected during SN explosions, the evolutionary behaviour of [Fe/H] is directly related to the time evolution of the SN rates. Various methods have been used to obtain the AMR from the observations. A comparative discussion on these methods is given by Carraro et al. (1998). Edvardsson et al. (1993) derived the AMR of 189 nearby field F- and G-type disk dwarfs with very accurately determined abundances. The trend is a decreasing metallicity with
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increasing stellar ages and with a considerable scatter about the mean value that is substantially larger than the observational error. Subsequent studies of the AMR in the disk (e.g. Rocha-Pinto et al., 2000; Feltzing et al., 1989; Ibukiyama and Arimoto, 2002) have confirmed the presence of the large scatter which is believed to be physically real and may originate from processes like star formation by sporadic gas infall, local inhomogeneous mixing, orbital diffusion, etc. (e.g. Edvardsson et al., 1993; Matteucci, 1996; Pagel and Tautvasˇien_e, 1995). Very recently the AMR in the disk has been studied by Nordstro¨m et al. (2004) with new data from the large Geneva– Copenhagen survey of the solar neighbourhood for a sample of 16682 local F and G dwarfs. In their sample a reasonable number of old metal-rich stars is present which were not included in the study of Edvardsson et al. (1993). The resulting mean metallicity is more or less constant at all ages oppositely to the average AMR of Edvardsson et al. (1993) that declines at stellar ages >10 Gyr. However it should be realised that the derived ages for these old metal-rich stars are the most uncertain (Feltzing et al., 1989). Since we assume homogeneous chemical evolution in our model, we can only compute a mean relation without any scatter. This means that although the AMR does not constitute a very tight constraint on our model predictions due to the large observational dispersion, our model predictions should fit the average run of the observed data. We computed the AMR for a number of models defined in Table 14. They illustrate the effects of binaries. Model 1 corresponds to the situation in
Table 14 The different models for which the AMR has been computed Model
fb
PNS model
SNIa
Fe yields
1 2(3) 4(5) 6(7) 8(9) 10(11) 12(13) 14(15) 16(17)
0 fb,1(fb,2) fb,1(fb,2) fb,1(fb,2) fb,1(fb,2) fb,1(fb,2) fb,1(fb,2) fb,1(fb,2) fb,1(fb,2)
stand. stand. stand. stand. stand. but UF stand. stand. stand. stand.
– SD1 SD2 DD DD SD1 SD2 DD SD1
WW95 WW95 WW95 WW95 WW95 WW95:2 WW95:2 WW95:2 WW95
which only single stars are formed and thus no SNIas. In models 16 and 17 only the binary progenitors of the SNIas are included together with single stars. These latter models correspond with the models commonly adopted in CEMs but differ by the treatment of the SNIas, i.e. we compute the SNIa rate by a detailed modeling of the evolution of the SNIa binary progenitors instead of including the SNIa rate as a free parameter that is finally fixed to obtain the observed solar iron abundance. To account for the uncertainty in the iron corecollapse yields (which is at least a factor of two) computed by Woosley and Weaver (1995), Timmes et al. (1995) made some chemical evolutionary simulations with the iron core-collapse yields reduced by a factor of two and found a general better agreement with the observed evolution of the abundance ratios. Lower iron core-collapse yields are also supported by observational estimates of the iron synthesized in SN 1987A (= explosion of a 20 Mx with Z 0.1Zx in the LMC) and in SN 1993J (=explosion of a 14 Mx in the galaxy M81 where Z Zx). Therefore we also consider models with the iron yields from Woosley and Weaver (1995) reduced by a factor of two (indicated by WW95:2). The predicted AMR is shown in Figs. 21 and 22 together with the observational data points from the work of Edvardsson et al. (1993). The error bars (shown on the figures in the lower right corner) give the typical mean measurement errors on the observed data. All the predictions show an overall trend of a fast increase during the first 2– 3 Gyr and much more slowly afterwards. Compared to the observations of Edvardsson et al. (1993) we are inclined to conclude that the predicted mean relations fit better the observed average trend when the reduced iron core-collapse yields are implemented, confirming the conclusions of Timmes et al. (1995). However, in order to reproduce an average [Fe/ H] ratio above 0.3 at the beginning of disk formation in correspondence with the observations of Nordstro¨m et al. (2004), the models with the high iron-core collapse yields are more satisfactory. It is obvious that the contribution of SNIas to the iron enrichment is non-negligible. Their relative weight in the total iron production depends
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Fig. 21. The theoretically predicted AMR for the models 1, 3, 5, 7, 9, 11, 13, 15, 17. The error bars give the typical measurement errors on the observed data.
on the adopted iron core-collapse yields and the SNIa rate which on its turn depends on the adopted progenitor scenario and on the adopted values for the binary PNS parameters like the mass ratio distribution, common envelope ejection efficiency ai and the RLOF accretion parameter bmax. If the reduced iron core-collapse yields apply it follows that SN Ias are responsible for at least 50% of all the iron of the Galaxy. It is clear that SNIas play an important role in the iron evolution and therefore their formation rates must be computed in a correct way which requires the use of a PNS
model. We remark that the AMR predicted with a constant binary frequency in combination with the Z-dependent SD model shows a similar behaviour to the one computed with the Z-independent SD model and a binary frequency that increases linearly with Z which is illustrated by models 3 and 4. When we compare our model predictions with binaries to those without binaries but with SNIas (i.e. models 2 and 3 vs. models 16 and 17) we conclude that binaries significantly affect the iron evolution mainly through SNIas.
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Fig. 22. The theoretically predicted AMR for the models 2, 4, 6, 8, 10, 12, 14, 16. The error bars give the typical measurement errors on the observed data.
The low [a/Fe] ratios (solar values) observed in damped Lyman a (DLA) systems 10 at high redshift with [Fe/H] < 1, suggest that SNIas already form at low metallicity since they eject sufficient amounts of iron and therefore prevent large over-
10 Damped Lyman a systems are high column density (N(HI) P 2 · 1020 cm2) absorbers detected in the optical spectra of quasars up to relatively high redshift (up to z 5). They have metal abundances that range from 102Zx to 1/ 3Zx.
abundances of a-elements relative to iron (compared to the solar value). The moment during galactic evolution at which SNIas start to form depends on the progenitor scenario. In case of the metallicity dependent SD scenario, the first SNIas only appear around [Fe/H] > 1.1 and it would be difficult to explain the observed low [a/ Fe] ratios in DLAs at high redshift. However, in the DD scenario Type Ia SNe already appear 40 Myr after the beginning of star formation (the characteristic timescales on which SNIas are formed in the different scenarios are discussed in
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Fig. 23. The predicted AMR computed with model 12 (=green line), with model 14 (=blue line) and both models combined (=red line). The error bars give the typical measurement errors on the observed data.
Section 4.8.3) and affect the [a/Fe] ratios at metallicities lower than 1.0 dex. When the SD scenario is taken independent of the metallicity SNIas also
form at lower metallicities though their maximum rate occurs 1 Gyr after the beginning of star formation and therefore largely affect the [a/Fe] ratio
Fig. 24. The theoretically predicted G-type dwarf metallicity distribution in the solar neighbourhood for the models 2–9. The observationally derived histograms are from Wyse and Gilmore (1995) (bold line) and from Rocha-Pinto and Maciel (1996) (thin line).
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around [Fe/H] > 1. From these considerations it follows that if SNIas are indeed responsible for the observed low [a/Fe] ratios, they are mainly produced by merging CO WDs, especially at low metallicity. However, this does not imply that no SNIas or only a small fraction forms via the SD scenario. As we point out later on in Section 4.8.4 where we compute the SNII, SNIbc and SNIa rates, the high Ia/Ibc supernova number ratio (1.6) observed in early and late type spirals is difficult to explain with one of the two progenitor scenarios but can be reproduced when SNIas are formed via both the DD and SD scenario. Therefore we also computed the AMR with a model in which SNIas are formed by the DD and the SD scenario. The results are given in Fig. 23 for the combination of model 12 with model 14. The main effect is obviously that the predicted SNIa rate increases and we conclude: if SNIas are formed by both the single degenerate and the double degenerate binary scenario, they are responsible for at least 70% of all the iron in the Milky Way.
4.4. The G-dwarf disk metallicity distribution Stars of spectral type G have main-sequence lifetimes of the order of the age of the Milky Way (10–15 Gyr) and many of them that have ever been formed still exist, even those formed when the Galaxy was very young. Therefore these stars provide information on the chemical evolutionary history from the early phases until now. Latest compilations of the metallicity of G-dwarfs in the thin disk have been performed by Wyse and Gilmore (1995) and Rocha-Pinto and Maciel (1996). The results of both groups are in good agreement and show a narrow distribution with a prominent peak around [Fe/H] 0.2. Figs. 24 and 25 show the G-type dwarf distribution predicted by the different models defined in the previous section in Table 14, together with the observed distribution from Wyse and Gilmore (1995) and Rocha-Pinto and Maciel (1996). The model prediction with both SNIa scenarios together is given in Fig. 26.
Fig. 25. The same as Fig. 24 but for the models 10–17.
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Fig. 26. The predicted G-dwarf disk metallicity distribution computed with model 12 (from Table 17) (= bold black line), with model 14 but (= dashed black line) and both models 11 and 15 combined (= red line).
When comparing the predicted distributions with the observed ones, we find that the G-dwarf metallicity distribution depends sensitively on the evolution of the iron enrichment and thus on the adopted SNIa binary model and core-collapse SN iron yields. With the iron core-collapse yields of Woosley and Weaver (1995) we find an overall underproduction of G-dwarfs with [Fe/H] < 0.5 with in some cases a compensating overproduction at [Fe/H]-values >0 which is due to a high SNIa rate already in the beginning of galactic evolution causing a very fast iron evolution. A better agreement (based on a statistical v2 test) with the observed distributions is obtained when the reduced iron core-collapse yields are used and a constant binary frequency (resp. linearly increasing binary frequency) is combined with the metallicity dependent SD or DD model (resp. metallicity independent SD model). The best fit results from the combination of model 12 and 14 i.e. when both the DD and SD scenario produce SNIas. The model predictions that only include single stars and SNIas (= models 16 and 17) show that except for the SNIas, binaries do not much affect the Gdwarf disk metallicity distribution. 4.5. The solar abundances The solar abundances are determined from photospheric and meteoretic analysis. For most
of the elements they agree within 0.2 dex. Table 15 lists the observed data from Anders and Grevesse (1989) updated by Grevesse and Sauval (1998) for the isotopes that are considered in this work. The uncertainties on the derived abundances (by mass) are within 10%. Hydrogen (1H) and helium (4He) represent 98% of all the mass with the remnant 2% contained in metals, of which half is in the form of 16O. In Fig. 27 we show the calculated mass fractions (= computed/observed value) for the considered isotopes at the moment of the birth of the Sun in our model. Since we assume that the Milky Way is 15 Gyr old (remind however that this number is not essential and using 13 Gyr would lead to very similar conclusions), the Sun was born 10.5 Gyr after the Big Bang. Given the uncertainties in the observed values (while isotopic ratios for most elements are known precisely, elemental abundances are less certain) and uncertainties in the theoretical values (as a consequence of uncertain physics in the models) we can consider that our model predictions are in agreement with the observed value within a factor of two difference. 11
11 It is a daunting task to make a formal analysis of the total uncertainty and therefore the convention that isotopes falling within a factor of 2 of their solar value are successes, is commonly adopted.
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Table 15 The observed data (Grevesse and Sauval, 1998) together with the predicted values at t = tx (second column) and at t = tnow (third BH column) for model 2 with M BH 0;s ¼ M 0;b ¼ 40 M Element
Nucleon fraction (Xi) Observation
Theory (t = tx)
Theory (t = tnow)
H He 12 C 14 N 16 O 20 Ne 24 Mg 28 Si 32 S 40 Ca 56 Fe
7.06e1 2.75e1 3.03e3 1.11e3 9.59e3 1.55e3 5.13e4 6.53e4 3.96e4 5.99e5 1.17e3
7.11e1 2.72e1 4.57e3 9.24e4 7.48e3 1.21e3 2.90e4 7.88e4 4.22e4 6.51e5 1.35e3
7.07e1 2.75e1 4.75e3 1.06e3 8.10e3 1.40e3 3.27e4 8.94e4 4.86e4 7.45e5 1.62e3
Z
1.89e2
1.71e2
1.88e2
1 4
Fig. 27. The theoretically predicted solar abundances relative to the observed solar values (=Xi/Xi,x) as predicted by model 2 with BH M BH 0;s ¼ 25 M (filled circles), with M 0;s ¼ 40 M (open circles) and by model 17 (crosses). The red dot gives the iron mass fraction when the iron core-collapse yields are reduced by a factor of two (= model 10).
This factor of two difference is denoted in Fig. 27 by the horizontal dotted lines. The mass fractions are nearly equal to unity for H and He, independent of the adopted model. For carbon we find oversolar values which is due to the high C production by low and intermediate mass stars between 1 and 4 Mx. The overabundance is smaller when a large binary fraction is adopted, because the majority of interacting binaries do
not undergo (due to previous mass transfer) the dredge-up phase at the AGB during which large amounts of carbon are brought up to the surface and ejected. The predicted solar abundances of the elements O, Ne and Mg depend sensitively on the adopted mass limit for BH formation without a SN outburst. If direct BH formation for single stars already starts at 25 Mx less O, Ne and Mg is ejected because the final CO core is swal-
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lowed by the BH, which results in computed solar values that are lower than observed (given by the filled dots in Fig. 27). The effect of BH formation is much smaller on the predicted abundances of the elements Si, S, Ca and Fe since the bulk production of these elements comes from stars <30 Mx. For the iron mass fraction the best result is obtained when the lower iron core-collapse yields (i.e. Fe(WW95):2) are used in combination with model 2 (= model 10) which is given in Fig. 27 by the red dot. Higher Ne solar abundances are predicted when large binary fractions are adopted. While among the single stars Ne is almost exclusively produced by the massive ones, intermediate mass stars in binaries may also contribute to the Ne enrichment when they become massive stars after accreting matter from or merging with the companion star. Magnesium is underproduced compared to its observed value, which is a well known result in chemical evolutionary models that implement the Mg core-collapse yields from Woosley and Weaver (1995). The underestimation of Mg has been discussed many times in literature (e.g. Timmes et al., 1995; Thomas et al., 1998). One of the arguments is that uncertainties in SN nucleosynthesis computations still allow for rather different yields among different authors, especially in the case of 24Mg production. For example the 24 Mg yields from Thielemann et al. (1996) are sensitively higher than those from Woosley and Weaver (1995) and are found to better reproduce the observed solar value (e.g. Thomas et al., 1998; Chiappini et al., 1999). Our computations with the other models reveal similar results, all predicting isotopic solar abundances that fall within a factor of two of their solar value. Table 15 lists the predicted abundances at the moment of the birth of the Sun in our model and for the adopted present age of the Milky Way. We conclude that since the formation of the Sun, the chemical evolution of the solar neighbourhood has been very slow, which is in good agreement with the observed constancy of the oxygen abundance in the last 4.5 Gyr (e.g. Moos et al., 2002; Sofia and Meyer, 2001; Peimbert, 1999). The slow chemical enrichment is mainly due to the included threshold in the star formation rate, which shows an oscillatory behaviour during the latest
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phases of galactic evolution. We notice that for other values of bmax ; ai ; vk , than adopted in the standard PNS model, the predicted solar values do not change much, as could be expected from our PNS parameter study on the binary yields in the previous section (Section 3.8.3). When the special class of type II SNe (=SNIIWD) which are produced by merging WD + MS/RG binaries are included, the a-element abundances are increased, though the results stay within a factor of two of their observed solar value. However, the nature of these merger objects is still speculative and it is too early to draw any confirmative conclusion on their effects on galactic chemical evolution. As a main conclusion on the effect of binaries, we find that except for the differences in carbon and neon, a model with single stars only but with a correct treatment of the SNIa population predicts similar solar abundances compared to a model in which the effects of all binaries are included in detail. We also keep in mind that the enrichment in O, Ne and Mg is affected by the mass limit for direct BH formation. 4.6. The abundance ratios evolution In this section we consider abundance ratios rather than absolute abundances, because they depend less on the galactic model parameters but mainly on the nucleosynthesis yields. The evolution of the abundance ratios [X/Fe] as a function of metallicity is often considered as a diagnostic of the IMF and SFR parameters and the timescale for the chemical evolution of stellar systems. Extended discussions and reviews on observed abundance ratios have been given by e.g. Wheeler et al. (1989), Timmes et al. (1995) and McWilliam (1997). The most abundant elements provide important clues for understanding stellar and galactic evolution. For example, observations of the CNO abundances in main-sequence stars are valuable for finding what types of stars have been responsible for the CNO nucleosynthesis during the different galactic evolutionary phases. We compute the time evolution of the abundance ratios [C/ Fe], [N/Fe], [O/Fe], [Mg/Fe], [Si/Fe], [S/Fe] and [Ca/Fe] as a function of [Fe/H] and compare them with observational data of spectroscopic abundances in nearby stars. There are no observational
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data for neon. Elemental neon only has optical transitions from very high excitation levels, and apparently there are no stellar abundance determinations in either dwarf or field giants (except for the Sun). We note that in the computation of the [X/Fe] ratio we use the observed solar abundance ratio (X/Fe)x and not the one predicted by our model. In view of the effect of direct BH formation on the predicted solar abundances as discussed in the previous section, we take M BH 0;s ¼ 40 M for the computation of the abundance ratios. Before discussing our results, the following general comment on the observations must be kept in mind. Examinations of elemental abundance ratios as a function of [Fe/H] show that star-to-star differences exist and rise with decreasing metallicity (e.g. Ryan et al., 1996). Especially at low metallicity the observational spread of abundance ratios is large and unlikely to be only the result of the increased uncertainty in the estimates of elemental abundances in metal-poor stars. Therefore the spread at low metallicity is strongly believed to be physically real. At present the following scenario is generally considered to explain the observed spread. During the beginning of galactic evolution (i.e. halo phase) the enrichment is determined by the number of exploding massive stars and the efficiency by which the ejecta are mixed with the surrounding ISM. If the ejected metals are distributed over a large volume, a spatially homogeneous enrichment takes place. If on the contrary only a small volume is contaminated, the ISM in the vicinity of the SN explosion becomes highly enriched while the ISM further out remains metal-poor. In this case the ISM evolves inhomogeneously whereby new stars at a given galactic age may form with a different chemical composition, depending on where they form. This scenario has recently been explored in detail by Argast et al. (2000, 2002, 2004). By applying a stochastic formation model for the halo they computed the early chemical enrichment of the ISM of the halo. They find that the halo ISM at [Fe/H] > 3.0 is unmixed and dominated by local inhomogeneities caused by individual SN explosions. For [Fe/ H] > 3.0 the ISM becomes much better mixed resulting into a scatter in the abundance ratios
that decreases with increasing [Fe/H] values, in agreement with the behaviour of the observed scatter. Concerning the magnitude of the scatter, their model cannot account for the observed scatter of some elements like O and Mg which however may be due to uncertainties in the adopted nucleosynthesis yields. In order to quantify the observed scatter in the measured elemental abundances and to allow a meaningful comparison between observations and model predictions that do not account for inhomogeneous mixing, Chiappini et al. (1999) applied a statistical method (see Ryan et al., 1996; Cleveland and Kleiner, 1975 for further details) to a large consistent set of observational data. They computed the midmean line which gives the average of all data between the quartiles of the distribution of relative abundances over a given [Fe/ H] interval; the lower semi-midmean which is the midmean of all observations below the median and finally the upper semi-midmean corresponding to the midmean of all observations above the median. If the scatter around the midmean is Gausssian, the semi-midmeans are an estimate of the true quartiles. As has been checked by Chiappini et al. (1999) this last assumption seems to be globally valid for each of the abundance ratios that are considered in their dataset. The computed summary lines from the work of Chiappini et al. (1999) are shown in Figs. 28 and 29 (right panels) together with the observational data they used for their statistical analysis (left panels). Because we consider in this work the same abundance ratios and almost the same data sources (except for the addition of recent data on oxygen abundances) we use the presented summary lines as a reference for comparing our model predictions with the best fit to the observed data. 4.6.1. The CNO elements Carbon is produced as a primary element by the triple-a process in stars over the whole mass range. Observations of carbon abundances in halo and disk dwarfs show that [C/Fe] is almost constant in time, with a value approximately solar (Laird, 1985; Carbon et al., 1987; Tomkin et al., 1992), although there is a significant dispersion in the data at all metallicities. More recently, Gustafsson et al.
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Fig. 28. Abundance ratios of the elements O, Mg and Ca to Fe vs. [Fe/H] from the work of Chiappini et al. (1999). The left panels show the observed data from different sources but all normalized to the same solar abundances (Anders and Grevesse, 1989). The right panels give the midmean line (solid line) and the lower and upper semi-midmean lines (dotted lines) resulting from the statistical analyses of the observed data set.
(1999) found a slow decrease of [C/Fe] with increasing [Fe/H], with an overall slope of 0.17 ± 0.03 dex. The observed and predicted evolution of [C/Fe] as a function of [Fe/H] is given in Fig. 30. The model predictions show a decline of the [C/ Fe] ratio till the end of the halo, thick disk phase ([Fe/H] 1) followed by an increment that produces a bump in the interval 1 < [Fe/H] < 0. The slow decline during the halo, thick disk epoch indicates that intermediate mass stars that evolve during the late halo phase do not contribute significantly to the carbon enrichment. The carbon bump that appears during the thin disk epoch is created by intermediate and in particular low mass stars (<3 Mx) that start to eject large amounts of carbon and no iron. Because type Ia SNe which produce large amounts of iron and few carbon also start to appear around the same epoch, the strong increase in carbon enrichment is finally compen-
sated and the [C/Fe] ratio decreases to the solar value. This general behaviour corresponds with the trend line that gives the best fit to the observations (as given in Fig. 29), though the carbon bump is less pronounced than predicted by the models with a low binary fraction during the halo, thick disk phase (= models 3, 7, 9, 11 and 17 in Fig. 30). The bump is smaller for the model predictions with a high constant binary fraction (= models 2, 4, 6, 8 and 10 in Fig. 30). This effect is explained by the fact that the majority of low and intermediate mass interacting binaries do not evolve up to the TP-AGB during which, in single star evolution, by dredge-up processes large amounts of carbon are mixed into the outer stellar layers and finally ejected. However the C yields of low and intermediate mass stars may depend sensitively on the treatment of physical processes like hot bottom burning which are not yet well known. When the lower iron core-collapse yields are adopted the
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Fig. 29. The same as Fig. 28 but for the elements C, N, Si and S.
carbon bump is shifted to lower [Fe/H] values which is in better agreement with the observed mean trend. Reducing the iron core-collapse results in a lower average [Fe/H] evolution with time which is clearly seen in the predictions of the AMR with models that use the WW95 iron yields reduced with a factor of two (e.g. model 11 in Fig. 22). As a major effect of binary evolution on the [C/ Fe] relation, we find that models with a constant high binary fraction predict a flatter [C/Fe] evolution during the disk phase compared to model predictions that except for SNIas do not include binaries. The trend of the best fit to the observed data indicates an almost constant [C/Fe] evolution during the disk phase and therefore models with binaries better fit the average data run. Nitrogen is made from 12C and 16O during CNO-burning, which takes place in all stars with mass >1 Mx. Except for some primary production in AGB stars, 14N is mostly produced as a secondary element. Observational surveys show that despite the large scatter, [N/Fe] remains more or
less constant (comparable with the solar value) down to [Fe/H] 2 (Laird, 1985; Carbon et al., 1987; Israelian et al., 2000) indicating the production of primary nitrogen over this metallicity range. Primary nitrogen production by massive stars at low metallicity is not predicted by current evolutionary models of massive stars, which is a well-known problem in CEMs. Timmes et al. (1995) obtained primary nitrogen from metal poor massive stars with an initial mass P30 Mx by considering enhanced convective overshooting into the hydrogen shell from the helium convective burning shell. Synthesis of primary nitrogen by the injection of protons into helium burning zones could also happen via rotation (Meynet and Maeder, 2002). We propose here a third mechanism via binary evolution. In a zero-metallicity close binary with a massive primary, the secondary star may become enriched in carbon by accreting material that has been ejected during the SN explosion of the primary star. Test calculations show that through thermohaline mixing, the envelope of the accretion
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Fig. 30. The predicted and observed [C/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Laird (1985), Gratton and Ortolani (1986), Tamkin et al. (1986), Carbon et al. (1987) and Carretta et al. (2000).
star may become strongly enriched in carbon that will be converted partly into nitrogen during hydrogen shell burning. This scenario was also suggested by Vanbeveren (1994) to explain OBC stars. Detailed stellar evolutionary computations where this effect is included have not yet been made, but we would like to suggest that the enrichment of the ISM in primary nitrogen during early galactic evolution may be caused by massive close binary evolution. The predicted evolution of [N/Fe] as a function of [Fe/H] is displayed in Fig. 31. Our results give
a rapid increase of [N/Fe] at low metallicity, which clearly shows that metal poor stars do not produce primary nitrogen. The ratio steadily increases up to [Fe/H] 0.5 due to the progressive contribution of mostly secondary nitrogen ejected by intermediate and low mass stars, until the iron production of SNIas compensates the ejecta of intermediate mass stars and flattens the evolution of [N/Fe]. Except for the possible production of primary nitrogen and the compensating effect by the large iron injection form SNIas, the differences in the [N/Fe] evolution as a result of binary evolution are very small.
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Fig. 31. The predicted and observed [N/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Laird (1985), Gratton and Ortolani (1986), Carbon et al. (1987) and Carretta et al. (2000).
Oxygen is primarily produced by massive stars through a-processing. Iron instead is produced by stars with a variety of masses, with a large production by evolving close binaries of intermediate mass that form SNIas. The general observed trend is [O/Fe] > 0 for [Fe/ H] > 1 and a gradual decline in the disk to the solar value. The occurrence of the decline could be caused by the time delayed injection of iron by type Ia SNe. However, this interpretation has been questioned recently on the basis of new observations of oxygen abundances in the low metallicity range. For the behaviour of [O/Fe] in metal poor stars,
two different results have been presented in the literature: one shows a nearly flat plateau at [O/ Fe] 0.5 (e.g. Gratton et al., 2000; Carretta et al., 2000; Nissen et al., 2001, 2002), while the other is a steady increase up to [O/Fe] 1, as [Fe/H] decreases from 1 down to 3 (Israelian et al., 1998; Boesgaard et al., 1999; Mishenina et al., 2000). If the latter results are real, there is an overall continuous decline of [O/Fe] as a function of [Fe/H] without a pronounced knick at [Fe/H] (10.5). We remark that these latter results were not included in the set of observational data on which Chiappini et al. (1999) applied their statistical anal-
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Fig. 32. The predicted and observed [O/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Gratton et al. (1996), Boesgaard et al. (1999), Mishekina et al. (2000), Carretta et al. (2000) and Nissen et al. (2002).
yses. Therefore the trend lines shown in Fig. 28 do not hold for the observed data plotted in Fig. 32. Our model predictions, given in Fig. 32, show an overall underproduction of oxygen relative to iron during the early galactic epoch. The downturn to the solar value appears at values of [Fe/H] higher than 1 (on average at 0.3) corresponding with the maximum formation rate of the SNIas as predicted by the progenitor models. When the DD scenario is applied the decrease is stronger because SNIas appear earlier (already at 40 Myr) than in the SD scenario. Higher
[O/Fe] ratios are obtained with the reduced Fe yields (= models 10 and 11). Although the results are closer to the observed average, we still find an underproduction of oxygen during the halo, thick disk evolution. Anticipating our results presented in the next Section 4.7, a higher production of oxygen is achieved if stars with an initial mass above 40 Mx undergo an SN like outburst instead of a direct collapse into a BH which we have assumed here. Also here we may conclude that except for the iron contribution of SNIas, the [O/ Fe] is not much affected by binary evolution.
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4.6.2. The a-elements Mg, Si, S and Ca The observed evolution of the ratios [a/Fe] for the Mg, Si, S and Ca are similar to the one of [O/ Fe] with an almost flat evolution in the halo and a gradual decline in the disk (McWilliam et al., 1995; Ryan et al., 1996). The a-elements are only produced in massive stars and type Ia SNe. Our results for the evolution of the [a/Fe] ratios are given in Figs. 33–36. We find an overall evolution similar to oxygen. The evolutionary behaviour is in general agreement with the average observed trend, though the ratios are rather low, especially for magnesium and sulphur at low met-
allicity, whether binaries are included or not. The underproduction of Mg is a generally known result. It has been pointed out by Goswami and Prantzos (2000) that the use of the metallicity dependent yields of Limongi et al. (2000) may solve the problem or that a supplementary source for Mg production is needed (Timmes et al., 1995). 4.6.3. Comparison to other studies The oxygen discrepancy during the early galactic evolution was also found by Timmes et al. (1995) who also uses the yields from Woosley and Weaver (1995) for their case B explosion
Fig. 33. The predicted and observed [Mg/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Gratton and Sneden (1987), Magain (1987, 1989) and Edvardsson et al. (1993).
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Fig. 34. The predicted and observed [Si/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Tomkin et al. (1985), Franc¸ois (1986), Gratton and Sneden and Edvardson et al. (1993).
model. When the figures of Chiappini et al. (1997) are rescaled relative to the observed solar ratio, the discrepancy is also visible in their results. At first glance it was surprising that this discrepancy was not found by some other groups (e.g. Chiappini et al., 1999; Boissier and Prantzos, 2000; Prantzos and Boissier, 2000; Portinari et al., 1998). Although most groups use the yields from Woosley and Weaver (1995), it is not always clear whether the case A or case B SN explosion model is implemented in their CEM. Portinari et al. (1998) use the yields obtained with the explosion
model A which corresponds to our model with the reduced iron yields. The library of Woosley and Weaver (1995) provides yields for stars only with an initial mass lower than 40 Mx. To obtain the yields for stars above 40 Mx some of the groups mentioned above extrapolate the yields from Woosley and Weaver (1995). In the overall IMF this mass range is insignificant, but during the early phases of galactic evolution mainly the massive stars enrich, and as we will show in the next section stars with a mass P 40 Mx play a very significant role in the oxygen enrichment.
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Fig. 35. The predicted and observed [S/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Clegg et al. (1981) and Franc¸ois (1987, 1988).
Kobayashi et al. (1998) applied the different SNIa scenarios in a standard chemical evolutionary model and computed the run of [O/Fe] with [Fe/H]. They conclude that a best fit to the observations is obtained with the metallicity dependent SD model since it nicely reproduces the observed downturn in [O/Fe] at [Fe/H] 1. Their result is difficult to understand since the change in the slope corresponds with the maximum SNIa rate, while in the metallicity dependent SD model the first SNIas form around [Fe/H] 1.1 and therefore produce a maximum at [Fe/H] > 1. This issue has also been discussed by Matteucci and
Recchi (2001) who made chemical evolutionary computations with the metallicity dependent SD scenario and for various CEMs. With the twoinfall model they obtain a predicted behaviour of the [O/Fe] vs. [Fe/H] relation similar to ours (= model 4), which clearly shows that a break in the [O/Fe] ratio corresponding with the moment of a maximum SNIa rate, occurs at [Fe/H] > 1. 4.6.4. Overall conclusion on the time evolution of the abundance ratios In the same way as we have done for the AMR, G-type dwarf metallicity distribution and the solar
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Fig. 36. The predicted and observed [Ca/Fe] evolution. The error bars give the typical measurement errors on the observed data. The observed data are taken from Tomkin et al. (1985), Gratton and Sneden and Edvardsson et al. (1993).
abundances, we also made test computations with a CEM where, except for the details of the SNIa progenitors, we consider all other stars as single (i.e. models 16 and 17). We conclude that the time evolution of the elements He, O, Ne, Mg, Si, S, and Ca predicted by galactic evolution, where the effect of binaries is considered in detail, differs by no more than a factor of two or three from the results of models in which most of the stars are treated as single stars and in which the effect of binaries is simulated only to account for the SNIa population; however, intermediate mass close binaries may be
essential in order to understand the time evolution of Fe and C. 4.6.5. The influence of rotation Due to rotation larger He and CO cores form and consequently a higher production of helium and a-elements (Heger, 1998). However, as mentioned earlier in Section 2.2.2 the same effect is obtained by mild convective overshooting. When the He core is larger, the 12C(a,c)16O reaction is more active at the end of He burning. As a consequence the fraction of carbon left in
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the CO core is lower than in the non-rotating case. This effect increases the oxygen yield. Since the carbon burning phase is shortened the stellar core has less time to remove its entropy through heavy neutrino losses, which favours the formation of a BH above an NS (Woosley, 1986). Recently, Fryer and Heger (2000) concluded that the net effect of rotation is a delay of the SN explosion and a weaker explosion, which finally leads to the formation of larger compact remnants and thus a lower initial mass limit for BH formation. Rotational diffusion can extract newly synthesized elements from the core, mix them with the outer layers and save them from further processing in the next burning phases in the core. Although this increases the yields, quantitatively the overall yields that are expelled during the whole evolution of the star are generally not much affected. Rotational diffusion may be an important way to produce primary nitrogen in zero metallicity stars, which seems to be required to reproduce the observed N/O ratio in the galactic halo stars (e.g. Timmes et al., 1995). The primary 14N can be produced when 12C that has been synthesized by the 3a-reaction in the He burning core is transported outwards by rotational diffusion into the CNO burning H shell. Because the H shell must be rather thick and long living, primary 14N is mainly produced in massive stars that do not suffer too strong mass loss by an SW. Primary 14N may also be produced in He burning zones through 12 C(p, c)13N(b+)13C(p, c)14N when protons are injected by rotational mixing. However, this primary 14 N is quickly destroyed to produce 18O via 14 N(a,c)18F(b+)18O. We remind that primary 14N can be produced by the secondary star in a binary, after it has accreted 12C rich material during the SN explosion of the primary star. 4.7. The effect of BH formation above 40 Mx and metallicity dependent SW mass loss In many CEMs the evolution of stars with an initial mass >40 Mx has been claimed to be unimportant because of their low formation rate as dictated by the adopted law for the stellar mass distribution at birth. As far as number statistics is
concerned these stars are indeed of less importance but this may not be true when we consider their contribution to the chemical enrichment of the ISM during the early phases of the Milky Way. The evolution of stars above 40 Mx is in most CEMs treated in a rather sloppy way, i.e. either they are not included or an extrapolation is made from stars below 40 Mx. Moreover, SW mass loss in this high mass range is often not accounted for or is rather qualitatively included. An exception is the work of Portinari et al. (1998). The carbon and oxygen yields of massive stars strongly depend on the SW mass loss rate and the final fate of the star after the core collapse. If the SW is radiatively driven, the mass loss rate mainly depends on the abundance of the iron group elements in the star, which are the main drivers. Therefore the dependency of the SW mass loss rate on the initial metallicity of the gas out of which the star is formed can be crucial. The WR and/or RSG SW mass loss has a major effect on the evolution of the CO core of a massive star. When the WR mass loss is small or when the RSG mass loss is too small for a massive single star to become a WR star, the star retains most of its mass until the end of its evolution. This means that the CO core increases in mass during the whole core helium burning (CHeB) phase and at the end mainly consists of oxygen that is ejected if an SN outburst takes place. On the contrary if the WR mass loss is high, a large part of the mass is ejected with the wind, essentially in the form of helium and carbon. The CO core decreases in mass and a smaller amount of matter is turned into oxygen. In this case a large amount of carbon and little oxygen will be ejected when an SN outburst occurs. The situation of low mass loss rates prevails at low metallicity when the WR mass loss depends on the initially metallicity of the star, while the second situation holds if the WR mass loss is independent from the metallicity. In summary, the enrichment of carbon and oxygen in the early phases of galactic evolution may depend on whether or not one uses massive star evolutionary computations with WR mass loss rates that are metallicity dependent or not.
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Whether stars above 40 Mx first explode and then form a BH or directly collapse into a massive BH without the ejection of material is presently not clear. The runaway status of some low and high mass X-ray binaries with a BH component (e.g. Nelemans et al., 1999) and the presence of heavy elements in the atmosphere of the companion star of the BH X-ray binary GRO J1655-40 (Israelian et al., 1999) suggest that massive BHs form with a prior SN like outburst during which some material is ejected. Also the hypernovae observed in external galaxies have been associated with the formation of a BH. The issue of massive BH formation with or without matter ejection could be important for the chemical evolution in the early Galaxy when lower mass stars have not yet evolved. In our previous computations we have taken the WR SW mass loss independent of the metallicity and assumed that stars (single and binary) more massive than 40 Mx do not undergo an SN explosion but directly collapse into a BH. To illustrate the importance of the evolution of stars with an initial mass P 40 Mx and the effect of metallicity dependent WR SW mass loss on the chemical evolution of the Galaxy, we make the following simulations: Simulation 1: All stars above 40 Mx form massive BHs without a preceding SN outburst and the WR SW mass loss rates do not depend on the initial metallicity of the gas out of which the stars are formed. Simulation 2: All stars above 40 Mx form massive BHs without a preceding SN outburst p but ffiffiffi the WR SW mass loss rates are scaled with Z . Simulation 3: All stars above 40 Mx form massive BHs with the ejection of 0.7 Mx of 56Ni p and ffiffiffi the WR SW mass loss rates are scaled with Z . Simulation 4: All stars above 40 Mx form massive BHs with the ejection of oxygen and carbon but no nickel. The amount of ejected oxygen is taken once 7 Mx (a) and once 4 Mx (b). To determine the ejected amount of carbon we adapt the C/O ratio that follows from the hypernova models computed by Nakamura for the corresponding He core. The WR p SW ffiffiffi mass loss rates are in both cases scaled with Z .
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In simulation 3 and 4 we consider two extremes for the ejected amount of 56Ni. An amount of 0.7 Mx has been predicted by the super-energetic explosion of a 16 Mx He star with E 3– 6 · 1052 ergs in order to explain the lightcurve of the observed hypernova SN 1998bw (Nakamura et al., 2001). The corresponding ejected oxygen mass is 7 Mx. Low amounts of 56Ni are predicted by hypernova models in which the masscut is placed at a large distance from the center of the star (>4 Mx) or where the explosion occurs asymmetrically (Maeda et al., 2002). The latter models with few 56Ni ejection can explain the low Fe abundance and high Ti, S, Si, Mg and O abundances in the companion star of the X-ray BH binary Nova Sco (GRO J1655-40). In all our simulations we take a constant binary frequency of 70% and use the iron yields from Woosley and Weaver (1995) reduced by a factor of two. The results are shown in Figs. 37 and 38. We conclude:
1. The effect of stars with an initial mass P 40 Mx on the chemical evolution at [Fe/H] < 2 is very significant. Therefore it is important to implement the most up to date SW mass loss rate formalisms (for all masses) during the various evolutionary phases. 2. The observations of [O/Fe] and [C/Fe] at low metallicity are not reproduced when large amounts of 56Ni are ejected when the BH forms. 3. When the WR mass loss rates are taken to be independent of the initial metallicity a significant overenrichment in carbon shows up during the early evolution of the solar neighbourhood. We interpret this as indirect evidence that the strength of these winds is strongly sensitive to the initial metallicity of the gas, which corresponds with the prediction from stellar wind theory. 4. The evolution of stars with an initial mass above 40 Mx in general, the detailed physics of BH formation and the ejected amount of material in the preceding SN like outburst determine the real carbon, oxygen and iron enrichment of the Galaxy during the early phases.
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Fig. 37. The predicted evolutionary behaviour of [C/Fe] as function of time for the different simulations. The upper figure shows Sim1 (thin line) and Sim2 (bold line), and the lower figure shows Sim3 (bold line) and Sim4 with the ejection of 4 Mx resp. 7 Mx of oxygen (dashed thin line resp. full thin line).
4.7.1. Comparison to other studies Metallicity dependent stellar yields from massive stars with the inclusion of SW mass loss have been computed by Maeder (1992) and Portinari et al. (1998). These yields were respectively implemented into a CEM by Prantzos et al. (1994), Portinari et al. (1998) and by Gustafsson et al. (1999) in order to study their influence on the evolutionary behaviour of carbon and oxygen in the Galaxy. They all concluded that the [C/O] vs. [Fe/H] relation for disk stars can be well explained by the metallicity-dependent behaviour of radiatively driven SWs of massive stars and that there is no real need for a moderate carbon contribution
from low and intermediate mass stars. However, Maeder (1992) and Portinari et al. (1998) used mass loss rate prescriptions in their stellar evolutionary computations that are out of date. For the WR SW mass loss they used the Langer formalism (Langer, 1989), which has been proven to overestimate largely (up to a factor of 4) the mass loss rate. Furthermore they assume a metallicity dependent SW only during the pre-WR phase, which implies that at Z 6 0.002 only the most massive stars (>80 Mx) may enter the WR phase and that He, C and O enrichment by SW mass loss only becomes important at high metallicity. In our preferred evolutionary model all stars initially
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Fig. 38. The predicted evolutionary behaviour of [O/Fe] as function of time for the different simulations. The upper figure shows Sim1 (thin line) and Sim2 (bold line), and the lower figure shows Sim3 (bold line) and Sim4 with the ejection of 4 Mx resp. 7 Mx of oxygen (dashed thin line resp. full thin line).
more massive than 40 Mx become WR stars independently of the initial metallicity due to severe mass loss during the LBV evolutionary phase (= LBV scenario). Additionally, Maeder (1992) also concluded that direct BH formation should already start at (25–30) Mx to avoid an overproduction of oxygen and to obtain a large helium to metallicity enrichment ratio DY/DZ. Extrapolating this result to low metallicities would imply that stars above 40 Mx are totally irrelevant for the chemical evolution during the early phases of galactic evolution. However, Maeder (1992) did not do the chemical evolution over the whole life-
time of the Galaxy and did not include type Ia SNe. Prantzos (1994) reanalysed Maeders study and concluded that the observations are better fitted when all massive stars up to 100 Mx contribute, via SWs and/or SNe, to the chemical enrichment of the Milky Way, which is in agreement with our results. 4.8. The supernova rates 4.8.1. Observed rates The SN rates in different types of galaxies have been estimated by Cappellaro et al. (1999) and are
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Table 16 The SN rates (expressed in SNu with the Hubble constant H0 = 75 km/s/Mpc) in different types of galaxies. Others includes Sm, Irregular and Peculiar type galaxies (ref. Cappellaro et al. (1999)) Galaxy type
E-S0 S0a-Sb Sbc-Sd Others All
Nr. of SNe
SN rate
Ia
Ib/c
II
Ia
Ibc
II
All
22.0 18.5 22.4 6.8 69.6
5.5 7.1 2.2 14.9
16.0 31.5 5.0 52.5
0.18 ± 0.06 0.18 ± 0.07 0.21 ± 0.08 0.40 ± 0.16 0.20 ± 0.06
<0.01 0.11 ± 0.06 0.14 ± 0.07 0.22 ± 0.16 0.08 ± 0.04
<0.02 0.42 ± 0.19 0.86 ± 0.35 0.65 ± 0.39 0.40 ± 0.19
0.18 ± 0.06 0.72 ± 0.21 1.21 ± 0.37 1.26 ± 0.45 0.68 ± 0.20
summarised in Table 16. The rates are expressed in the SN unit Snu = 1 SN Æ (100 yr)1 Æ (1010 LB,x)1 with LB,x the luminosity of the Sun in the photometric B band. Considering the Milky Way as a type Sb-Sbc with a total blue luminosity LB = 2 · 1010 Lx (van der Kruit, 1987), the present galactic SN rate is estimated at 2 SNe per century, of which 20% type Ias, 13% type Ibcs and 67% type IIs. Taking a disk radius of 18 kpc, this corresponds roughly to a local rate of 1.96 · 102 SNe/pc2/Gyr. 4.8.2. Overview of the progenitor populations In our PNS model, the different SN types are produced by the following stars. SNII progenitors single born massive stars and massive non-interacting binary components with 8 < M 0 =M X ¼0 < M X0;s¼0 ; where M 0;s is defined as the minimum initial ZAMS mass of a single star that ends its life without hydrogen; our single star evolutionary calculations of massive single stars with stellar wind mass loss give the value of X ¼0 M 0;s from first principles, binary components with 8 < M 0 =M < M X0;s¼0 that became single after the first SN explosion that disrupted the binary system, binary star mergers with masses after merging X ¼0 between 8 M and M 0;s , merging WD + MS/RG binaries that form a single star with an MCO P 1.4 Mx (=SNIIWD) The initial properties (on the ZAMS) of the binary progenitors of the merging WD + MS/RG binaries are shown in Fig. 40. They are computed for the standard PNS model. The majority of the
binary progenitors are initial short period systems (case A/Br systems) with an intermediate mass ratio that undergo a quasi-conservative first RLOF. SNIbc progenitors single born massive stars, massive single stars with a binary history and non-interacting massive binX ¼0 ary components with M 0;s < M 0 = M < M BH 0;s , interacting binary components with 10 < M 0 = M < M BH 0;b , massive secondaries that have lost their hydrogen envelope via SW during the RSG phase and/or LBV phase or in a spiral-in phase with the compact primary star during which the CE has been ejected, binary star mergers that became more massive X ¼0 than M 0;s after merging and further evolved like a normal single star. SNIa progenitors SD scenario: CO WD + MS/RG binaries with the right properties (see Hachisu et al., 1999) to produce an SNIa, DD scenario: CO WD + CO WD binaries with (M1 + M2) P 1.4 Mx and a lifetime 6 tnow. Fig. 39 shows the initial properties (on the ZAMS) of the binary progenitors for both scenarios. They are computed with our standard PNS model. In the SD scenario, the binary progenitors are mainly case Bc/C systems with an initial intermediate mass ratio that will evolve through a CE phase. The SNIa progenitors in the DD scenario have initial periods between (5–100) days that either evolve through a CE phase or perform quasi-conservative RLOF.
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Fig. 39. The initial (on the ZAMS) primary mass (upper figure), mass ratio (middle figure) and orbital period (lower figure) of the SNIa progenitors for the SD scenario (bold lines) and the DD scenario (thin lines). The computations are made with /H, bmax = 1, ai = 1 and Z = 0.02.
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Fig. 40. The initial (on the ZAMS) primary mass (upper figure), mass ratio (middle figure) and orbital period (lower figure) of the progenitors of WD + MS/RG mergers. The computations are made with /H, bmax = 1, ai = 1 and Z = 0.02.
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Hypernova (HN) progenitors Some of the observed SNIcs (SN1997dq, SN1997ef, SN1998bw, SN1998ey, SN99as, SN2002ap) have extremely high peak luminosities (1.6 · 1043 erg/s) and expansion velocities (>6 · 104 km/s) (e.g. Galama et al., 1998, 1999; Bloom, 1999; Reichart, 1999; van Paradijs, 1999). They have been classified as hypernovae. With current SN models these peculiar characteristics can be reproduced if explosion energies P 1052 ergs are used in stars with an initial mass P 25 Mx. Depending on the location of the adopted mass-cut, the ejected amount of 56Ni for these high energies can increase up to 0.4–0.7 Mx (matching with SN1998bw), which is 10 times larger than the 56 Ni-ejectum of a standard core-collapse SN (Nakamura et al., 2001). Hypernovae are currently believed to be associated with the formation of massive BHs since the large explosion energy can be explained by a rapidly rotating BH with a strong magnetic field (e.g. MacFadyen and Woosley, 1999). Although their occurrence rate in the nearby universe is quite low, they could have been more frequent in earlier epochs when star formation might have been higher. Since they produce large amounts of iron they might have a strong im-
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pact on the evolution of the [a/Fe] abundance ratios. Therefore we compute their ratio to the core-collapse SNe. To compute the number of HNe we assume that they arise from massive single (binary) stars with BH an initial mass P M BH 0;s ðM 0;b Þ. Notice that in our standard PNS model these massive single (binary) stars are assumed to form massive BHs without an SN explosion.
4.8.3. The evolutionary clock of supernovae In order to illustrate the typical timescales on which SNe occur, we computed the time evolution of the SN rates after an instantaneous starburst. To allow an easy scaling of our results to any starburst of arbitrary total mass, we took the total mass of the starburst equal to 1 Mx. We separately consider a burst in which only single stars are formed and one in which only binary stars are formed. The time evolution of the core collapse SN rates is given in Figs. 41 and 42 and of the SNIas in the Figs. 43 and 44 for different metallicities. For the SD scenario we assumed a metallicity dependent WD wind (i.e. SD2). Since the single and binary
Fig. 41. The time evolution of the core collapse SN rates after an instantaneous starburst for Z = 0.02. The curves produced by the starburst with 100% single stars are indicated with the letter s between round brackets while the curves labeled with the letter b correspond with a burst of 100% binaries. The computations are made for the standard PNS model. The meaning of the colour lines is given in the legend.
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Fig. 42. The same as Fig. 41 but for Z = 0.002.
Fig. 43. The time evolution of the SNIa rates after an instantaneous starburst (with 100% binaries) for Z = 0.02. We separately give the timescales of the SNIas produced by the SD2 scenario (black lines) and those by the DD scenario (grey lines). The computations made with bmax = 0.5 and a2,3 = 0.5 are given by the bold lines.
SN rates scale linearly with the single star fraction and binary star fraction at birth (i.e. on the ZAMS), it is straightforward to make predictions for any single/binary star fraction. We performed several simulations by changing the PNS parameter values. For the core collapse SNe it can be concluded that the time evolution
hardly depends on the adopted PNS parameter values. Therefore we show in Figs. 41 and 42 only the results computed for the standard PNS model. However, the timescales on which SNIas form, depend sensitively on the orbital period evolution of the progenitor binaries which is mostly affected by the accretion parameter during RLOF (i.e. bmax)
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Fig. 44. The same as Fig. 43 but for Z = 0.002.
and the CE ejection efficiency (i.e. a). Therefore we show in Figs. 43 and 44 also the results obtained with bmax = 0.5 and a2,3 = 0.5. From our simulations we conclude that: 1. SNIas form in the SD model on a timescale of 0.3–4.5 Gyr after the initial burst, while in the DD scenario the first SNIas already appear when the starburst is 40 Myr old with a maximum rate at 0.3 Gyr and 5 Gyr. The two peaks correspond with the two different evolutionary channels through which SNIas are formed in the DD scenario. The first maximum contains mainly the SNIas of which the progenitor binary experienced two CE phases (MS + MS ! CE ! WD + MS ! CE ! WD + WD). This maximum corresponds with the results of Iben and Tutukov (1984). The second maximum contains the SNIas of which the progenitor binary first evolved through a quasi-conservative RLOF (MS+MS ! RLOF ! WD + MS ! CE ! WD + WD). The WD + WD systems formed via the second channel have longer final periods and therefore longer lifetimes. This evolutionary channel was not considered by Iben and Tutukov (1984) although, according to our simulations, it may be an important
one. However, when bmax = 0.5 is adopted this second group of long period systems is strongly reduced because a significant amount of orbital angular momentum is taken away from the system when matter leaves the binary during RLOF. Due to this mass and angular momentum loss the periods are on average smaller and some fraction may merge. 2. When binaries are formed in the starburst, the phase during which the core collapse SNe appear lasts much longer than when only single stars are formed, i.e. 250 Myr vs. 40 Myr. This is mainly due to the mass transfer in interacting binaries transforming secondaries of initially intermediate mass into massive stars, and to the explosion of merging WDs with their non-degenerate companion star. 3. The situation for the core collapse SNe is at Z = 0.002 much the same as it is at Z = 0.02. In our adopted standard PNS model no SNIbcs are formed by single stars _ at Z = 0.002 because of the lower pffiffiffi SW M _ (we used here a WR SW M / Z ) and the adopted mass limit for direct BH formation (25 or 40 Mx). The metallicity effect is more pronounced for the SNIa timescales. In the SD scenario the first SNIa appears only after
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Table 17 The different models for which the time evolution of the SN rates is computed PNS model
Model
fb
1 (7) 2 (8) 3 (9) 4 (10) 5 (11) 6 (12) 13
40% 40% 40% 40% 40% 40% fb,2
(70%) (70%) (70%) (70%) (70%) (70%)
Standard standard but /F standard but /G Standard but bmax = 0.5, a1,2,3 = 0.5 Standard but a1,2,3 = 0.5 Standard but vk ¼ 150 Standard
The model numbers between round brackets correspond to a constant binary frequency of 70%, while model 13 assumes a binary frequency that increases linearly with the metallicity.
1 Gyr instead of 0.3 Gyr, which is due to the lower upper mass limit on the WD companion star for Z = 0.002.
4.8.4. The galactic evolution of the SN rates Because the SN rates depend on the metallicity and the star formation rate we computed the evolution of the SN rates as a function of time during the galactic evolution. Our simulations are made for the models listed in Table 17. We take the iron yields from Woosley and Weaver (1995) decreased by a factor ofptwo ffiffiffi and the WR SW mass loss proportional to Z . Except for model 13, when the SD scenario is applied we assume that the WD wind is metallicity dependent (i.e. SD2).
Obviously, the absolute SN rates predicted by our simulations depend on the parameters of the star formation and the galaxy formation model. However, relative rates hardly depend on them. Also in the relative observed rates, the uncertainty in the Hubble constant is excluded. Table 18 lists the present (i.e. predicted at t = tnow) relative SN rates in the solar neighbourhood for the different models. It also gives the fraction A among the total intermediate mass binary population (on the ZAMS) that produces an SNIa. As mentioned earlier in Section 3.7, this fraction A is inserted in most current CEMs as a free parameter that is regulated to finally meet the observations. Figs. 46 and 47 show the time evolution of the SNIa rates throughout the galactic evolution for all the
Table 18 The present relative SN rates predicted by different models for the solar neighbourhood Model
SNIIWD/II
SNII/SNIbc
HN/(SNII + SNIbc)
SNIa/SNIbc
SNIa/(SNII + SNIbc)
A (%)
1 2 3 4 5 6 7 8 9 10 11 12 13
0.07 0.12 0.14 0.05 0.06 0.07 0.10 0.25 0.27 0.10 0.10 0.10 0.09
5.57 3.81 3.31 5.98 6.13 5.07 4.20 2.44 2.03 4.72 4.92 3.82 4.31
0.147 0.147 0.148 0.149 0.149 0.137 0.140 0.141 0.142 0.140 0.140 0.130 0.150
0.53j0.38 0.24j0.69 0.17j0.69 0.21j0.05 0.21j0.36 0.49j0.35 0.78j0.60 0.30j1.05 0.24j0.90 0.34j0.07 0.35j0.65 0.73j0.56 0.81j0.50
0.08j0.06 0.05j0.17 0.04j0.16 0.03j0.01 0.03j0.05 0.08j0.06 0.15j0.12 0.09j0.31 0.08j0.30 0.06j0.01 0.06j0.01 0.15j0.12 0.16j0.10
3.06j1.11 2.05j2.36 1.58j3.44 1.72j0.19 1.04j4.61 3.06j1.11 3.06j1.11 2.05j2.36 1.58j3.44 1.72j0.19 1.04j4.61 3.06j1.11 3.06j1.11
In the last three columns the first number is computed when the SD scenario is applied and the second number when the DD scenario is used. The last column contains the fraction A among the initial intermediate mass binary population (on the ZAMS) that produces an SNIa.
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Fig. 45. The time evolution of the SNII (upper figure) and SNIbc (lower figure) rate predicted by the models 1, 7 and 13.
models except for models 6 and 12 since the average kick magnitude is not involved in the evolution of the SNIa binary progenitor population. Because the evolutionary behaviour of the SNII and SNIbc rates depends less sensitively on the PNS model parameters, we show their galactic evolution in Fig. 45 only for the models 1, 7 and 13. We remark that in our computed SNII rate the merging WD + MS/RG binaries (=SNIIWD) are not included. The evolution of the ratio SNIa/(SNII + SNIbc) is given in Figs. 48 and 49. From our simu-
lations and comparison with the observations we draw the following conclusions: 1. Independent from the adopted parameters in the PNS model, the predicted present total SN rate is 1–2 SNe per century, which agrees well with the observations. However, the relative contribution of the different SN types differs for different PNS models, especially the fraction of SNIas, which may vary by a factor 2–3.
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Fig. 46. The time evolution of the SNIa rate predicted by the models 1, 2, 3, 4 and 5 with the SD scenario (upper figure) and the DD scenario (lower figure).
2. In order to obtain the present SNIa rate (within the observational error) an intermediate mass binary frequency (on the ZAMS) of at least 70% is required when the SD scenario is used and (30–50)% if the DD scenario is applied. With the SD scenario a maximum SNIa rate is obtained when a Hogeveen type mass ratio distribution is used in combination with a maximum CE ejection efficiency (i.e. ai = 1). In the case of the DD scenario we find
the highest number of SNIas for a Garmany type mass ratio distribution combined with bmax = 1 and ai = 1. 3. Only a small percentage of the total zero-age population of intermediate mass binaries finally produces an SNIa. In most CEMs this fraction is included as a free parameter (and not explicitly computed) that is fixed in order to meet the observations. The commonly adopted value for A in these CEMs varies
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Fig. 47. The time evolution of the SNIa rate predicted by the models 7, 8, 9, 10, 11 and 13 with the SD scenario (upper figure) and the DD scenario (lower figure).
between 0.7% and 8% (e.g. Timmes et al., 1995; Portinari et al., 1998), which covers our predicted values. 4. Depending on the binary properties, the number of SNIIs produced by merging WD + MS/RG binaries is between 5% and 30% of the number of SNIIs originating from exploding massive stars. 5. The expected relative rate of hypernovae to core-collapse SNe is about 0.15, which corresponds with an absolute rate of 2–4 HNe per millennium.
6. The SNIa/(SNII + SNIbc) number ratio increases (after 3 Gyr) almost linearly as a function of time, irrespective of the adopted PNS model. Because SNIas form on much longer timescales than core collapse SNe do, the SNIas and SNIIs and SNIbcs that appear at the same moment during galactic evolution have progenitors that were formed in a different earlier epoch of star formation. This explains why the ratio is not constant in time. The effect is pronounced in the SD scenario where a significant fraction of the
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Fig. 48. The time evolution of the ratio SNIa/(SNII + SNIbc) predicted by the models 1, 2, 3, 4 and 5 with the SD scenario (upper figure) and the DD scenario (lower figure).
SNIas progenitors, formed during the first Gyr of galactic evolution, only appears after 2 Gyr, when the star formation rate is much lower. 7. The number of SNIIs relative to SNIbcs depends significantly on the properties of the massive binary population (the massive binary frequency and mass ratio distribution). However, the ratio hardly depends on the metallicity and therefore the ratio varies very slowly as a function of time.
8. Since most of the SNIbc progenitors are binary components, the observed ratio SNIa/ SNIbc may be an indication that the massive binary population relative to the intermediate mass binary population is similar in early and late spirals. However, due to the fact that the SNII/SNIbc ratio is significantly different, this could be an indication that the overall binary frequency (or the overall binary population) in both types of spiral galaxies is significantly different.
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9. The observed SNIa/SNIbc ratio in our Galaxy is best approached if the efficiency of CE ejection is very large (i.e. close to 1). 10. When we account for all observations of the OB-type stars in the solar neighbourhood (including the OB-type binary population), accounting for statistical biases when interpreting the binary population data which implies that the interacting binary frequency is at least 50% whereas the mass ratio distribution is flat or peaks at small values of q, then a best correspondence between the simulated and the observed SN population in the solar neighbourhood is obtained when the CE ejection is very efficient (a 1), the RLOF in intermediate mass binaries with a mass ratio q P 0.4 is almost purely conservative (bmax = 1), the merging of a WD with an MS/RG companion star produces an SNII, both the SD and the DD scenario produce SNIas. Remarks 1. Hachisu et al. (1999) proposed that binaries with an initial period larger than 10 years may become short period WD + RG binaries via wind-orbit interaction and produce SNIas. They estimate that about one third of all SNIas could have been produced via this evolutionary channel. It is straightforward to estimate that if we include these systems, the required binary frequency to obtain the observed present SNIa rate with the SD scenario is reduced to 40–50%. 2. The properties of the binary population and the metallicity may be different in different galaxies of the same morphological type. Because the observed SN rate in our Galaxy is computed as an average over a sample of external galaxies that are of the same morphological type as our Galaxy but possibly may harbor a stellar population with different properties, the average value may not be representative of our Galaxy. 3. For all galaxies together in the observed sample of Cappellaro et al. (1999) the ratio II/Ibc is 5. Since in the sample all the main morphological
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types of galaxies are included, this value could be considered as some cosmological average. It follows from our simulations that to recover this average, we need a binary fraction (on the ZAMS) between 40% and 70% which could imply that the cosmological massive binary formation frequency may be of the order of 50%.
4.8.5. Comparison to other studies Tutukov et al. (1992) computed the SN rates of different spectral types assuming a constant star formation rate over the lifetime of our Galaxy and a 100% initial binary frequency. For the SNIas they only consider the DD scenario. Scaling their results to a binary fraction of 70% we find that, within the differences in the adopted PNS prescriptions, their predicted relative rates are close to our predicted values except for the ratio SNIa/SNIbc, for which we find a higher value. The explanation for this difference is twofold: (1) Tutukov et al. (1992) predict an SNIbc rate of (0.6–1) per century which is higher than our estimated rate of (0.3–0.7) per century. Since they assume a symmetric SN explosion, the majority of their binary systems remains bound after the explosion of the primary massive star. This implies that the secondary star (if massive) in many cases loses its hydrogen envelope via RLOF and explodes as an SNIbc. It is obvious that when the SN explosion occurs asymmetrically (which is assumed in our PNS model), most binary systems (about 80% when using an average kick magnitude of 450 km/s) are disrupted after the explosion of the primary star, and the secondary star further evolves as a single star. According to our single star evolutionary scenario the secondary star explodes as an SNIbc only if its mass is larger than (17–20) Mx for Z = 0.02, which results in a lower SNIbc rate. (2) In our DD model we predict that a significant fraction of the SNIas form with lifetimes >1 Gyr coming from binary progenitors that evolved through a first quasi-conservative RLOF phase and a CE phase during the second RLOF. In the evolutionary scenario
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Fig. 49. The time evolution of the ratio SNIa/(SNII + SNIbc) predicted by the models 7, 8, 9, 10, 11 and 13 with the SD scenario (upper figure) and the DD scenario (lower figure).
of Iben and Tutukov (1984) (which is used in the work of Tutukov et al., 1992) SNIas are only produced by binary progenitors that underwent CE evolution during the first and second RLOF. Therefore the fraction of SNIas formed in their evolutionary scenario with a lifetime >1 Gyr is small. By consequence our predicted SNIa rate is generally larger by up to a factor of 1.5–2, depending on the adopted PNS parameter values. Only when model 4 is used do we find a good cor-
respondence with the results of Tutukov et al. (1992). In this model bmax = 0.5, which strongly reduces the fraction of SNIas with lifetimes >1 Gyr, as we explained earlier. We notice that due to our higher predicted rate of SNIas and lower rate of SNIbcs we better approach the observed SNIa/SNIbc ratio. In their PNS study of binary compact objects Belczynski et al. (2002) report SNII/SNIbc ratios between 1.8 and 3.7 for Z = 0.02 and an initial bin-
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ary fraction of 50%, which agrees with our predicted ratios. The time evolution of the SNII and SNIa rate has been computed by Kobayashi et al. (2000) for elliptical and spiral galaxies. They propose an explanation for the higher SNII rate (a factor of two) observed in late type spirals compared to early types based on a different star formation rate. However in their galactic evolutionary model they adopt a very simple population synthesis model and do not separate the SN IIs from the SN Ibcs. Because of the latter assumption, their explanation may not be true. A different star formation rate would also imply a different SNIbc rate in both spiral types, which is not observed. On this point we think that our proposition of a different overall binary formation rate in both types of spiral galaxies offers a better explanation. The evolutionary behaviour of the SNIa/SNII ratio (to be compared with our SNIa/(SNII + SNIbc) ratio) as a function of time predicted by Kobayashi et al. (2000) is similar to our results for both the DD and SD model, though their predicted ratio-values are generally higher especially during the early evolutionary phases of the Galaxy. For the present (t = 15 Gyr) ratio we find a best correspondence with their predictions when we consider our models that assume a constant binary frequency of 70%. We remark that Kobayashi et al. (2000) uses the evolutionary model of Iben and Tutukov (1984) for the DD scenario. The lack of a detailed discussion on the adopted stellar evolutionary models in the work of Kobayashi et al. (2000) does not allow for a thorough discussion on the differences between their work and ours.
4.9. The evolution of the r-process elements In this final section we study the galactic enrichment in r-process elements by merging NS + NS and NS + BH pairs in which the BH is formed either by the primary or by the secondary star. We first compute the time evolution of the formation and merger rate of these double compact star binaries and then compute the resulting chemical enrichment of r-process elements as a function of time.
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4.9.1. The formation of NS + NS and NS + BH binaries The standard scenario for the formation of these pairs of compact objects starts with an initial massive binary system in which the secondary is a massive star or an intermediate mass star that becomes massive after accretion during RLOF by the primary. The primary first evolves off the main sequence, overfills its Roche lobe and transfers mass to the secondary. The primary which has become a hydrogen deficient CHeB star further evolves (in some cases performing a second RLOF during helium shell burning = case BB RLOF) and finally collapses into an NS or BH, with or without a SN explosion. If the binary remains bound after the SN explosion, an NS/BH + OB binary is formed. This system passes through an X-ray binary phase and evolves through a CE phase as the secondary expands. During this CE phase the NS/BH spirals into the secondarys envelope forming a NS/BH + He star binary. If the helium star expands sufficiently during helium shell burning or beyound, it may overflow its Roche lobe for a second time. Depending on the actual properties of the system (orbital period, mass ratio, . . .) this mass transfer phase increases or decreases the orbital distance between both compact objects. After the secondary helium star has turned into an NS or BH and the system is not disrupted by an SN explosion, an NS/BH + NS/BH remains, of which the orbit decays via gravitational wave radiation, which finally leads to the merger of both compact objects. The occurrence of a case BB RLOF phase from the helium star to the compact companion is crucial for the orbital parameters of the resulting double compact star binary. If this mass transfer phase proceeds dynamically unstably, ultra compact NS/ BH + NS pairs can be formed which decay on a very short timescale (Myr). It is obvious that the timescale on which the orbital decay takes place determines the moment at which merging NS + NS and NS + BH pairs start to enrich the Galaxy with r-process elements. Because of the uncertainty in the treatment of case BB RLOF and its importance for the enrichment in r-process elements we make simulations for the different case BB scenarios that were defined and discussed
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earlier in Section 2.3.3. To refresh we repeat them here: scenario 1: case BB RLOF is suppressed by the SW mass loss of the helium star. The SW mass loss is independent from the metallicity and case BB is suppressed at low Z as well. The period variation is given by Eq. (2.9). scenario 2: case BB RLOF is suppressed by the SW mass loss of the helium star only for Z = 0.02. The SW mass loss is metallicity dependent and satisfies Eq. (2.4), which implies that case BB RLOF is not suppressed at Z = 0.002. Because no detailed case BB evolutionary computations exist for low metallicities, we use the computations made for Z = 0.02 for low Z as well. The period evolution is followed by assuming that matter leaves the binary in the form of an NS SW with the specific orbital angular momentum of the NS (see Dewi et al., 2002). scenario 3: the same as scenario 2 but we assume that the mass transfer to the NS results in a CE/spiral-in evolution and the period variation is given by Eq. (2.18). scenario 4: the SW mass loss for hydrogen deficient CHeB stars with a mass smaller than 5 Mx can be neglected and case BB RLOF does take place independent from the metallicity. The period evolution during case BB mass transfer to the NS is treated in the same way as in scenario 2. scenario 5: the same as scenario 4 but case BB mass transfer to the NS is assumed always to result into CE/spiral-in evolution. It has been argued that an NS would be able to increase its mass by accreting matter at a superEddington rate during CE and spiral-in evolution and to become a BH (e.g. Blondin, 1986; Chevalier, 1989, 1993; Brown, 1995). If this is true, it becomes very difficult to form NS + NS pairs via the CE evolution of HMXBs. However, this scenario is very uncertain and strong accretion onto the NS can be avoided, for example if the donor star is very extended or if rotational effects are taken into account (e.g. Chevalier, 1996). In our evolutionary model we do not account for hyper-critical accretion.
In order to determine the properties of a binary after the SN explosion of one of the components, we assume that prior to the SN explosion the system has been circularised. Since we treat the effects of the SN explosion on the binary system in full 3D our PNS code is able to compute the post-SN period, the space velocity and eccentricity of the binary. The knowledge of the post-SN eccentricity is essential in computing the timescale on which a double compact binary decays and finally merges due to GWR (see Section 2.3.2). 4.9.2. The theoretically predicted population of double compact star binaries 4.9.2.1. The predicted distribution of the orbital period, eccentricity, orbital decay timescale and space velocity. In order to illustrate the typical properties of the NS/BH + NS/BH population, we compute for an instantaneous starburst of 100% binaries the distribution of the orbital period (P), eccentricity (e) and space velocity (v) at the moment that the secondary has turned into a compact object and the corresponding timescale for orbital decay by GWR (Td). Our computations are made for the different PNS models defined in Table 19 and for two initial metallicities: Z = 0.02 and 0.002. We always take M NS max ¼ 2 M . We give the results made for the case BB scenarios 1 and 5. The other case BB scenarios give intermediate results. Figs. 50–53 and 56 give the normalized distribution of the orbital period, eccentricity and space velocity of the double compact star binaries at the time of their formation. The timescales for orbital decay are given in Figs. 54 and 55. We conclude that 1. The period distribution of double compact star binaries at birth in the period range <100 days is dominated by the NS + NS and BH + NS systems, while the majority of NS + BH and BH + BH systems form with periods above 100 days. The shape of the period distributions does not depend much on the choice of the PNS parameters. 2. At low Z double compact star binaries are expected to have smaller eccentricities. Compared to BH + NS systems, the NS + NS binaries
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takes place and leads to CE evolution, the timescales are strongly reduced to 104 and 108 yr, illustrating the important effect of case BB RLOF. 4. The space velocity distribution of the double compact star binaries is mainly determined by the orbital speed at the moment that the secondary explodes. The NS + NS systems acquire the highest speeds, with a broad peak around 300 km/s. The tightest systems (especially formed at Z = 0.002) are accelerated up to velocities of (500–1000) km/s. The velocity distribution of BH + NS systems is double peaked, at 30 km/s and 150 kms. The second peak is produced by the group of BH + NS binaries with the lowest periods (<25 days for Z = 0.002 and <10 days for Z = 0.02) which are clearly distinguishable in Fig. 51. The
Table 19 The different PNS sets for which the population of double compact star binaries is computed Set
bmax
a1,2
a3
vk
/(q)
1 2 3 4 5 6
1 1 1 0.5 1 1
1 1 1 1 1 1
1 1 1 1 0.5 1
450 450 450 450 450 150
/F /H /G /F /F /F
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have much higher eccentric orbits. The shape of the eccentricity distribution is rather insensitive to the PNS parameters except for the average kick velocity. 3. The orbit of the majority of NS + NS systems decays within 108 and 1011 yr if no case BB RLOF takes place. However, if case BB RLOF
N/Ntot
N/Ntot 0.18
0.12 0.1
set 1
set 2
set 3
set 4
set 5
set 6
Z=0.002 NS+NS
0.08
0.16
Z=0.002
0.14
NS+BH
0.12 0.1
0.06
0.08 0.06
0.04
0.04
0.02
0.02
0 -3
-2
-1
0
1
2
3
4
5
6
7
0
-3
-2
-1
0
1
log P(days)
2
3
4
5
6
7
3
4
5
6
7
log P(days)
N/Ntot
N/Ntot 0.12
0.07
Z=0.002
Z=0.002
0.06
0.1
BH+NS
0.05
BH+BH
0.08
0.04 0.06
7
0.03
0.04
0.02
0.02
0.01
0
0 -3
-2
-1
0
1
2 log P(days)
3
4
5
6
7
-3
-2
-1
0
1
2 log P(days)
Fig. 50. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) period distribution of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.002. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 5. The red line corresponds with case BB scenario 1 in combination with PNS set 1.
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N/Ntot
N/Ntot
0.07
0.14
Z=0.02 NS+NS
0.06
Z=0.02 0.12
0.05
0.1
0.04
0.08
0.03
0.06
0.02
0.04
0.01
0.02
NS+BH
0
0 -3
-2
-1
0
1
2
3
4
5
6
-3
7
-2
-1
0
log P(days)
1
2
3
4
5
6
7
3
4
5
6
7
log P(days)
N/Ntot
N/Ntot
0.09
0.08
Z=0.02 BH+BH
Z=0.02 BH+NS
0.08 0.07
0.06
0.06 0.05
0.04
0.04 0.03
0.02 0.02 0.01
0
0 -3
-2
-1
0
1
2
3
4
5
6
log P(days)
7
-3
-2
-1
0
1
2
log P(days)
Fig. 51. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) period distribution of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.02. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 1. The blue line corresponds with case BB scenario 5 in combination with PNS set 1.
BH + NS systems have larger space velocities than NS + BH systems because the NS is formed during the second SN explosion. Also, the space velocity acquired during the first SN explosion is much lower (10–50 km/s) due to the larger periods of the CheB + OB systems. Double BH binaries have the lowest space velocities of all (<10 km/s). As a major result we find that the adopted case BB evolutionary scenario in He + NS/BH binaries significantly influences the predicted distribution of the orbital decay timescales of double compact star binaries. 4.9.2.2. The predicted formation and merger rate. Since the galactic enrichment history of the r-process elements is expressed as a function of the metallicity, we display the evolution of the galactic merger rate of double compact star binaries vs.
[Fe/H]. The predicted evolution of the merger rate of double NS and mixed (=BH + NS and NS + BH together) pairs is given in Fig. 57 for the different case BB scenarios. Fig. 58 shows the effect of the PNS parameters on the merger rate. In Table 20 we list the predicted annual formation rate for the metallicities Z = 0.002 and 0.02. From our simulations we draw the following main conclusions. 1. The predicted present formation rate of NS + NS systems and their merger rates range within (106–104) yr1. Their formation is most influenced by the CE efficiency during the spiral-in phase after the first SN explosion (a factor of 2), the kick velocity (a factor of 20 variation), the adopted case BB RLOF scenario (a factor of 1.2–4 variation) and the amount of mass and angular momentum
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Fig. 52. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) eccentricity distribution of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.002. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 5. The red line corresponds with case BB scenario 1 in combination with PNS set 1.
lost from the system during non-conservative RLOF (a factor of 2–3 variation). These predicted rates agree with the observational estimated values (Belczynski et al., 2002). Notice that the merging of BH + NS binaries happens earlier than the one of NS + NS binaries. 2. NS + BH systems form at higher rates (up to a factor of 10–20) than NS + NS systems do because BHs form with smaller or no kicks, which means a lower chance for the binary to become disrupted. Double BH systems form the dominant class with birthrates of 104 yr1. They originate from binaries that avoid RLOF because of mass loss via SW (RSG and LBV scenario). They have large final periods and are not expected to merge within Hubble time.
4.9.3. The predicted evolution of [Eu/Fe] vs. [Fe/H] Detailed r-process calculations for merging neutron stars have been performed recently by Freiburghaus et al. (1999b). Merging neutron stars potentially can provide in a natural way the large neutron fluxes that are required for the build-up of heavy elements through rapid neutron capture. Newtonian calculations of merging NSs show that a few times 103–102 Mx of r-process matter might be ejected in one merger event. This amount is significantly larger than the typical values of 105–106 Mx of r-process material that is thought to be ejected in a SN event (e.g. Wanajo et al., 2001). However since the rate of NS mergers in the Milky Way is significantly lower than that of type II SNe, either of these two sources may account for the total amount of r-process matter in the Milky Way. We remark that recent NS merger
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Fig. 53. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) eccentricity distribution of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.02. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 1. The blue line corresponds with case BB scenario 5 in combination with PNS set 1.
computations that include general relativistic effects suggest lower amounts of ejected r-process material, i.e. 5 · 105–2 · 104 Mx (Oechslin et al., 2002). We compute the galactic evolution of the abundance of r-process elements assuming that merging NS + NS and NS + BH systems are the major rprocess source and that SNe do not contribute to the enrichment in r-process elements. Additionally we assume that the amount of matter ejected during the merger event is independent from the total mass of the system and the kind (NS or BH) of merging compact objects, all the ejected material is r-process material. Our computations are made with a constant binary fraction of 70% during galactic evolution,
the metallicity dependent SD scenario for the SNIas and with the iron yields from Woosley and Weaver (1995) reduced by a factor of two. For the solar r-abundance we take Xr,x 107 (Ka¨ppeler et al., 1989). Remark that the moment at which the merger releases its r-process yields is given by the timescale after which the original binary is transformed into a NS + NS or BH pair plus the timescale of orbital decay by GWR. In order to compare our predictions with the observations, we consider the element europium, which is almost purely (95%) produced by the r-process and therefore has a behaviour that is typical for an r-process element. Due to its numerous clean atomic lines in the visible part of the spectrum, it is one of the few elements (among the existing r-process elements) that has been observed.
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Fig. 54. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) distribution of the orbital decay timescale of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.002. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 5. The red line corresponds with case BB scenario 1 in combination with PNS set 1. N/Ntot
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Fig. 55. The normalized (respectively to the total number of NS + NS, NS + BH, BH + NS, BH + BH systems) distribution of the orbital decay timescale of NS + NS, NS + BH, BH + NS and BH + BH systems at the time of their formation for Z = 0.02. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 1. The blue line corresponds with case BB scenario 5 in combination with PNS set 1.
Fig. 59 compares the predicted [Eu/Fe] vs. [Fe/ H] relation (for the different case BB scenarios and PNS set 1) with the observational data from
Woolf et al. (1995) for the disk stars and from Burris et al. (2000) for the halo stars. The error bars (shown below in the left corner of the figure)
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Fig. 56. The normalized (respectively to the total number of NS + NS and BH + NS systems) space velocity distribution of NS + NS and BH + NS at the time of their formation for Z = 0.002 and Z = 0.02. The computations are given for the different PNS sets 1, 2, 3, 4, 5 and 6 in combination with case BB scenario 3. The red (blue) line corresponds with case BB scenario 1 (5) in combination with PNS set 1.
give the typical measurement error on the data for the disk stars. For the halo stars no errors are reported by Burris et al. (2000), though they argue that the observed spread at low Z is certainly not the result of measurements errors but due to inhomogeneous evolution. The high [Eu/ Fe] ratio (and also of other r-process elements) observed in a few halo giants (= triangles on the figure) indicates that local enrichments took place in an unmixed halo. The effect of the PNS parameters on the predicted evolution of the [Eu/Fe] ratio is given in Fig. 60. When assuming that mass transfer from the helium star to the NS always results in CE evolution and that at least 0.004 Mx of r-process material is ejected during merging, the observed [Eu/Fe] trend is well reproduced for [Fe/ H] P 2.5. Below [Fe/H] = 2.5 the predicted [Eu/Fe] ratio decreases steeply with [Fe/H] and is underproduced compared to the average ob-
served values. Larger ratios but with an overproduction at metallicities [Fe/H] > 2.5, are obtained for Mej = 0.04 Mx which is according to present hydrodynamical computations the maximum amount of r-process material that can be ejected during a merger event. If instead of CE evolution, case BB mass transfer mainly leads to the formation of a NS SW and no CE phase is encountered, the enrichment in r-process elements starts too late during galactic evolution, due to the larger orbital periods with which double NS/BH pairs are formed and consequently larger merging timescales. The [Eu/Fe] vs. [Fe/ H] relation is affected by the PNS parameters in a similar way as the merger rate of NS + NS or BH systems. A large average kick velocity ði:e: vk ¼ 450 km=sÞ and/or a small CE ejection efficiency (i.e. a = 0.5) reduces the merger rate of NS + NS or BH systems and therefore also the enrichment in r-process elements. From these
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Fig. 57. The predicted time evolution of the galactic merger rate of NS + NS (upper figure) and mixed (=BH + NS and NS + BH together) (lower figure) pairs for the different case BB scenarios in combination with PNS set 1.
main results we basically conclude that the contribution from merging NS + NS or BH pairs to the galactic enrichment in r-process elements depends critically on the effects on the orbital binary parameters of an asymmetric SN explosion, the physics of CE evolution, the adopted scenario and physics of case BB evolution,
the amount of ejected r-process material during the merger event. According to current theoretical evolutionary computations for Z = 0.02 (Dewi and Pols, 2003; Ivanova et al., 2002) the majority of the systems with a NS component and a Roche lobe filling He star, evolves through a stable mass transfer rather than through a CE phase due to a SW that is assumed to develop in the vicinity of the NS and
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Fig. 58. The predicted time evolution of the galactic merger rate of NS + NS (upper figure) and mixed (=BH + NS and NS + BH together) (lower figure) pairs for the different PNS sets in combination with case BB scenario 5.
to expel isotropically all transferred matter once the Eddington accretion limit is exceeded. This implies that the number of very short period NS + NS and NS + BH systems significantly decreases and according to our simulations, we would be inclined to conclude that merging NSs are very likely not the main contributors to the galactic enrichment in r-process elements.
However the NS SW is mainly driven by the iron lines and it is not unreasonable to think that at low metallicity the wind may not be strong enough to expel the transferred matter and a common envelope may form instead. In this case NS + NS and NS + BH systems form with merging timescales that are sufficiently short to produce significant amounts of r-elements at very low met-
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Table 20 The predicted actual (t = 15 Gyr) formation rate (in 106 yr1) of the different kinds of double compact star binaries for different PNS parameter sets combined with case BB scenario 1 (first number) and 5 (second number) Set
NS + NS
NS + BH
BH + NS
BH + BH
1 2 3 4 5 6
3.68j14.6 2.31j8.80 4.18j16.9 1.45j5.97 1.86j3.98 69.4j110
65.9j92.4 23.2j30.4 86.4j122 6.89j9.24 65.9j92.4 217j268
0.81j5.02 0.56j1.83 0.92j4.93 0.70j3.62 1.11j0 6.05j7.14
408j408 142j174 412j519 261j330 293j409 326j409
Fig. 59. The predicted [Eu/Fe] vs. [Fe/H] relation for the different case BB scenarios, PNS set 1 and Mej = 0.04 (upper figure) and 0.004 Mx (lower figure). The error bars give the typical measurement errors on the observed data for the disk stars.
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Fig. 60. The predicted [Eu/Fe] vs. [Fe/H] relation computed for the different PNS models in combination with case BB scenario 5 and Mej = 0.04 (upper figure) and 0.004 Mx (lower figure). The error bars give the typical measurement errors on the observed data for the disk stars.
allicities in the early Galaxy that correspond with the observations. 4.9.4. Comparison to other studies Accounting for the different PNS assumptions, our predicted ranges of the formation/merger rate of double compact star binaries and the distributions of their physical properties do not much differ from those of other studies (e.g. Portegies Zwart
and Yungelson, 1998; Fryer et al., 1999; Belczynski et al., 2002). The formation rate of NS + BH and BH + BH systems are often not reported in PNS studies because of their negligible small merger rate. Non-zero BH + BH merger rates of the order of the NS + NS merger rate have been predicted by (e.g. Tutukov and Yungelson, 1993; Lipunov et al., 1996; Belczynski et al., 2002). They form when mass loss by SW is strongly reduced, which results in
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smaller periods and therefore shorter merger times. This situation corresponds with our computations at low metallicity. In their model G2, which correspond with higher mass losses similar to our adopted rates at Z = 0.02, Belczynski et al. (2002) find a zero merger rate, which is in agreement with our results. Also remember that in many PNS studies the Langer SW formalism for helium stars is adopted (e.g. Portegies Zwart and Yungelson, 1998), which reduces strongly the formation rate of BHs. It is currently accepted that these rates overestimate the mass loss by a factor of 2–4. Belczynski et al. (2002) investigated the effect of hyper-critical accretion during CE phases on the NS + NS merger rate, which we have not included in our computations. In general, hyper-critical accretion leads to heavier inspiraling compact objects and larger post-CE orbital periods, because less envelope mass has to be ejected and therefore less orbital energy is needed. The merger rate of NS + NS systems is increased or decreased depending on the adopted maximum NS mass. From their results Belczynski et al. (2002) conclude that when M NS max 6 2 M , (40–80)% of the NSs in the CE phase turn into a BH and thus strongly reduces (by up to a factor of 6) the NS + NS merger rate. On the contrary for M NS max ¼ 3 M the rate increases with 20%. Accounting for inhomogeneous chemical evolution, Argast et al. (2004) computed recently the galactic enrichment in r-process elements whereby considering (separately) supernovae and double NS mergers as the production sources of r-elements. Despite they do not perform PNS modeling in order to compute in a correct way the formation and merger rate of double NSs, they come to the general conclusion that double NS mergers are probably not the dominant r-process. They argue that due to their low occurrence rate and high ejected r-process yields (compared to the SN case) merging double NSs start to enrich too late during galactic evolution and cause considerable chemical inhomogeneities in the ISM during the thin disk phase, which are not observed. In their paper Argast et al. (2004) only show the [Ba/Fe] vs. [Fe/ H] relation (whereby only accounting for the rcontribution to the Ba abundance) computed with their double NS merger model and it is therefore
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difficult to make a direct comparison with our simulations. Moreover, they adopt different iron core-collapse yields and a very simplified method to compute the SNIa rate, which results in a different [Fe/H] evolution with time. An important result from their study to mention, is that with their inhomogeneous chemical evolutionary model the observed scatter in the abundance ratios at low metallicity can be reproduced.
5. Summary and conclusions In this work, detailed binary evolution is included in chemical evolutionary computations for the first time, with the aim to study the effects that binaries may have on galactic chemical evolution. The formation of the Milky Way is simulated with the two-infall model of Chiappini et al. (1997). Although this model does not include star and gas dynamics, its main assumptions on the formation of the halo and disk (i.e. timescale and rate of infall, star formation) are supported by recent hydrodynamical simulations of the Milky Way. However, since comparisons are made mutually between model predictions with and without binaries but always with the inclusion of SNIas, any weaknesses or shortcomings in the galactic model that are linked to dynamics, affect evenly all our model predictions. Therefore our conclusions on the pure chemical effects of binaries hardly depend on them. Since we assume chemical homogeneous evolution, which is only expected during the evolution of the galactic disk and the late halo phase, our model results are averages which are compared to the observed average trends. As is usually done in chemical evolutionary computations, we computed abundance ratios rather than absolute abundances because they only depend on nucleosynthesis, stellar lifetimes and the initial mass function. To include binaries in our chemical evolutionary calculations we combined a PNS model with the galactic chemical evolutionary model. The PNS code models in detail the evolution of single and binary stars whereby accounting for all binary processes. Since several aspects of binary evolution are not yet completely understood, the modelation of the different binary interaction processes
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contains a number of free parameters which introduces some extra uncertainties. A major free parameter is the birth frequency of binaries. The physics of binary formation is not well known and we have to rely on the observations. In order to constrain the parameters and to set up a standard PNS model for our chemical evolutionary computations, we first computed with our PNS model the massive star population in the solar neighbourhood and the Magellanic Clouds. The observed average metallicity in the Magellanic Clouds is lower than the solar one and therefore the Clouds are excellent objects to measure the influence of metallicity on PNS modeling. We focused on the population of OB-type and Wolf– Rayet stars, the formation rate of SNe and the space velocity distribution of single radio pulsars. The best agreement with the observations in the solar neighbourhood is obtained when the initial fraction of interacting massive binaries is larger than 50% and when the majority of massive binaries is disrupted after the first SN explosion implying a preferred average kick velocity of 450 km/s. The required initial fraction of interacting massive binaries in order to reproduce the observed WR star population in the SMC, is much lower (20%) and could indicate that the binary formation rate is a function of metallicity. Furtheron, we find good results for the SMC when the WR p SW ffiffiffi mass loss rate relates to the metallicity Z as Z . Any modifications in our adopted RSG and WR wind mass loss formalisms leads to problems in reproducing the WC/WN number ratio. Although these comparisons with the observations can not really tell us if our adopted models for single and binary star evolution are correct, they give at least an indication that our models are reliable. Combining our PNS model with the two-infall galactic model, we simulated the chemical evolution of the solar neighbourhood for different assumptions on the binary fraction, i.e. a constant high binary fraction of 70% and one that grows linearly from zero at Z = 0 to 70% at Z = 0.02. In both cases the same binary fraction has been taken over the whole mass range. Both assumptions are inspired by our PNS predictions on the massive star population in the solar neighbourhood and Magellanic Clouds.
Binary evolution mainly affects the galactic evolution of carbon and iron. Interacting binaries of low and intermediate mass produce less carbon than their single counterparts because the majority of them avoid the dredge-up phases during late AGB evolution. In this particular evolutionary phase large amounts of carbon are transported to the surface and ejected by a superwind. This binary effect translates into a flatter evolution of the [C/Fe] abundance ratio during the thin disk phase compared to model predictions with only single stars and SNIas. A small bump in the [C/ Fe] abundance ratio during the disk evolution is also present in the averaged observations though much less pronounced than predicted by the model with exclusively single stars and SNIas. Except for the assumed binary fraction, the predicted evolution of carbon does not much depend on the adopted values for the binary parameters. The production and evolution of iron mainly depends on the production rate and formation timescale of SNIas and also on the adopted iron yields of massive stars. In this work we adopted the core-collapse iron yields from Woosley and Weaver (1995) and the single and/or double degenerate SNIa model. When combining the WW95 iron yields with the standard SNIa iron ejecta (i.e. 0.7 Mx) and an SNIa model that reproduces the observed local SNIa rate, we find [Fe/H] ratios as a function of time that are in general (for T > 1 Gyr) larger than the observed averages and [X/Fe] ratios that underproduce the observations especially during the halo, thick disk phase. The observed averaged run of the [Fe/H] and [X/Fe] ratios is better reproduced if the core-collapse iron yields are reduced with a factor of two. The use of lower iron yields was also suggested in earlier studies on chemical evolution and is supported by recent detailed spectral analysis of some well observed core-collapse SNe. The contribution of SNIas, which are formed by interacting intermediate mass binaries, to the total iron production is about 30% and increases to 50% when the WW95 iron yields are reduced with a factor of two. The precise value of these fractions depend on the used SNIa model (SD or DD model) and the related binary parameters, i.e. the common envelope ejection efficiency a and the accretion parameter bmax during RLOF.
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In order to explain the observed Ia/Ibc SN number ratio in early and late type spirals (which is too low when only the SD model or DD model is assumed) and the observed low [a/Fe] ratios in DLAs at high redshift (which requires the production of SNIas at [Fe/H] < 1), we suggest in this work that SNIas are formed by both the SD and DD model. In this case the SNIas dominate the galactic iron production by 70%. The moment at which the SNIas start to influence the iron evolution and by consequence the [X/ Fe] ratios is determined by the timescale on which SNIas are formed. We computed with our PNS model the timescales for both SNIa scenarios. The output is very important: the moment during galactic evolution at which the first SNIas start to form and the moment at which they reach a maximum sensitively depends on the adopted SNIa model and the values of the related binary parameters. Therefore we emphasize in this work that in order to compute in a correct way the time evolution of the SN rates, detailed PNS modeling must be included into a galactic chemical evolutionary model. This conclusion is enforced by computing the G-dwarf disk metallicity distribution which is one of the major constraints on CEMs and critically depends on the iron evolution. We find the best agreement with the observed distribution for the scenario in which SNIas are produced by both the SD and DD model. Binaries may provide the production of primary nitrogen which is needed to explain the observed [N/Fe] ratio during the halo, thick disk phase. We propose the following scenario: when the massive primary star explodes, carbon enriched material may be accreted by the secondary from the SN shell and consequently be CNO processed into nitrogen. Next to some single star scenarios like rotation that may offer a solution to the nitrogen discrepancy as well, the binary mechanism deserves to be further investigated. With our adopted SW mass loss rates, massive BHs (>10 Mx) are formed by stars with an initial mass above 40 Mx which explains well the observed masses of BH components in X-ray binaries. Whether these BHs are formed with or without a SN outburst is presently a matter of debate. If an explosion occurs they are thought to
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produce hypernovae which are observed as very energetic SN explosions. In order to reproduce the observed lightcurve of hypernovae with current supernova model computations large amounts of iron (10 times the amount ejected by a standard SN) are assumed to be ejected. We made some simulations with these hypernova models and find a strong underproduction of the [X/Fe] ratios during the halo evolution due to the large iron ejecta. Because of their short lifetimes stars above 40 Mx mainly affect the galactic enrichment during the halo phase. If these stars indeed explode we illustrate that mainly oxygen and only a small amount (<0.07 Mx) of iron should be ejected to recover the observations. Since we expect (from our PNS computations of the WC/ WN number ratio in the SMC) that the RSG and WR SW mass loss rates decrease with metallicity, the mass layers of the CO-core that are ejected are mainly composed of oxygen. If the SW mass loss would be independent from the metallicity we expect an overproduction of carbon during the major part of galactic evolution. The ejection of large amounts of a-elements and few iron during BH formation is in agreement with the observed large (oversolar) abundances of a-elements and a normal iron abundance (solar) in the atmosphere of the BH companion in the X-ray binary Nova Scorpii. Since the companion star is a low mass star, these extra amounts of a-elements are very likely accreted from the material ejected during the formation of the BH. Finally, we computed the galactic enrichment in r-process elements assuming that only merging NS + NS and NS + BH binaries produce r-process elements. Their contribution depends critically on the treatment of mass transfer in He star + NS binaries, the common envelope ejection efficiency, the supernova kick and the amount of ejected rprocess material during merging. Our simulations reveal that even accounting for the uncertainties in the r-process yields and the PNS parameters, binary mergers cannot reproduce the overall observed galactic evolution of r-process elements if the majority of He star + NS or BH binaries avoids CE evolution and this independent of the metallicity. However present uncertainties in the evolution of He + NS or BH systems do not allow
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to firmly conclude that NS + NS/BH mergers are not a dominant source of r-process elements. As a general conclusion, we can state that CEM predictions with binaries differ by no more than a factor of 2–3 from the predictions by models that only include single stars and SNIas. Since we computed abundance ratios, our results hardly depend on the adopted galactic model parameters and therefore should also hold for simulations with other CEMs that are different from the two-infall model.
Appendix A. Numerical solution of the isotope evolution equation
the general solution of Eq. (A.3) is Z t Gi ðr; tÞ ¼ exp gðr; t Þdt "Z Z 0 t
This general solution can be further transformed into a first order linear recurrence relation an + 1 = ban + c. This is done as follows: Z t1 Gi ðr; t1 Þ ¼ exp gðr; t Þdt
Z Wðr; tÞ dt I:F ¼ exp Gðr; tÞ Z ¼ exp gðr; tÞdt
Z
t1
gðr; t Þdt
0
t2
ðA:5Þ
gðr;t Þdt
0
exp
Z
Z ¼ exp
Z
t1 0 t1
gðr;t Þdt
t2
gðr;t Þdt
t1
ðA:1Þ þ
Z
t1
0
W i ðr;t Þexp
Z
0
þ
Z
!
t0
gðr;t Þdt
t2
0
W i ðr;t Þexp
Z
!
t0
gðr;t Þdt
0
t1
Z exp
t2
gðr;t Þdt
t1
Z þ exp
t1
0 t2
gðr;t Þdt
gðr;t Þdt
t1
t2
0
W i ðr;t Þexp
Z
Gi ðr;t1 Þ
t1
Z
dt0
0
Z þ const ¼ exp
ðA:3Þ
W i ðr;t0 Þ
gðr;t Þdt dt0 þ const
0
ðA:2Þ
t2 0
Z exp
or
Taking the integrating factor
W i ðr; t Þ exp
Z Gi ðr;t2 Þ ¼ exp
can be written as a linear differential equation:
dGi ðr; tÞ Wðr; tÞ ¼ Gi ðr; tÞ þ W i ðr; tÞ: dt Gðr; tÞ
0
and
þ Wðr; t tM 2 ÞRM 2;i ðt tM 2 Þ dP dq dM 1
dGi ðr; tÞ ¼ X i ðr; tÞWðr; tÞ þ W i ðr; tÞ dt
t1
dt þ const
Pl
dGi;inf ðr; tÞ dt
0
Z
0
PðP Þ½Wðr; t tM 1 ÞRM 1;i ðt tM 1 Þ
þ
dt0 ðA:4Þ
0
RM;i ðt tM ÞdM Z M 1;u Z qu Z P u þ fb ðt tM 1 ÞuðM 1 Þ/ðqÞ ql
gðr; t Þdt
þconst :
Ml
M 1;l
W i ðr; t Þ exp
#
For each isotope i the Eq. (3.3) dGi ðr; tÞ dt ¼ X i Wðr; tÞ Z Mu þ ð1 fb ðt tM ÞÞWðr; t tM ÞuðMÞ
!
t
0
!
t0
gðr;t Þdt
dt0 :
0
ðA:6Þ
E. De Donder, D. Vanbeveren / New Astronomy Reviews 48 (2004) 861–975
Finally, the analytical solution can be written as
Z Gi ðr; tnþ1 Þ ¼ Gi ðr; tn Þ exp Z
tnþ1
gðr; t Þdt
tn
W i ðr; t0 Þ tn Z tnþ1 exp gðr; t Þdt dt0 ; þ
with Gi(r)(k) the best estimate for Gi(r)n + 1 at the kth iteration and diðkþ1Þ ¼
tnþ1
969
Gi ðrÞðkÞ Ai bi Gi ðrÞ
ðkÞ
ðA:11Þ
;
W i ðrÞ W i ðrÞ ; Ai ðrÞ ¼ expðgðrÞDtÞ Gi ðr; tn Þ þ gðrÞ gðrÞ
t0
ðA:12Þ
ðA:7Þ
with gðr; tÞ ¼
bi ðrÞ ¼
Wðr; tÞ Gðr; tÞ
and W i ðr; tÞ ¼
Z
ðA:13Þ
Mu
ð1 fb ðt tM ÞÞWðr; t tM ÞuðMÞ
Ml
RM;i ðt tM ÞdM Z M 1;u Z qu Z P u fb ðt tM 1 Þ þ M 1;l
ql
Pl
uðM 1 Þ/ðqÞPðP Þ Wðr; t tM 1 ÞRM 1;i ðt tM 1 Þ þWðr; t tM 2 ÞRM 2;i ðt tM 2 Þ dP dq dM 1 þ
dGi;inf ðr; tÞ : dt
Gi ðr; tnþ1 Þ ¼ Gi ðr; tn Þ expðgðrÞDtÞ þ ½1 expðgðrÞDtÞ :
W i ðrÞ gðrÞ ðA:9Þ
Because g and Wi are not constant over Dt we need to iterate to obtain convergency. We perform our iteration on g. This mainly because Wi is less sensitive to the value of Gi(r,tn+1) (it is an integral over all the past values of Gi) than g. As shown by numerical computations, the improvement that is obtained by iterating on both terms g and Wi is small. The solution is now obtained by executing the following successive corrections: ðkþ1Þ
ðkÞ
¼ Gi ðrÞ ½1 þ dðkþ1Þ ; i
The iteration is stopped when diðkþ1Þ < dmax . The length of the timestep is chosen such that DGi(r)/ Gi(r) and Dr(r)/r(r) is 6 0.05. For the evaluation of the Wis we need to know which stars, previously formed, will contribute to the chemical enrichment at the considered age t of the model. Therefore we rewrite the equation for Wi(r,t) as follows:
ðA:8Þ
When integrating over a timestep Dt = tn + 1 tn we take g and Wi constant and equal to their value estimated at tn + 1/2 = tn + Dt/2 which gives
Gi ðrÞ
1 o ln gðrÞ W i ðrÞ ðexpðgðrÞDtÞ 1Þ 2 o ln GðrÞ gðrÞ W i ðrÞ gðrÞDt expðgðrÞDtÞ Gi ðtn Þ : gðrÞ
ðA:10Þ
W i ðr; tÞ Z ttðM u Þ ð1 fb ðt0 ÞÞWðr; t0 Þ ¼ 0 dM uðMÞRM;i ðt0 Þ dt0 dtM 0 Mðtt Þ Z ttmin Z qu Z P u þ /ðqÞPðP Þfb ðt0 Þ 0
ql
Pl
dM 1 Wðr; t ÞuðM 1 Þ RMi;1 ðt t Þ dtM 1 M 1 ðtt0 Þ ! dM 2 þ RMi;2 ðt t0 Þ dtM 2 M 2 ðtt0 Þ 0
dP dqdt0
dGi;inf ðtÞ : dt
0
ðA:14Þ
The upperlimit for the time integration in case of binaries depend on the fate of the considered binary system. If the primary evolves the fastest we have tmin ¼ tM 1;u , if the secondary accretes that
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much matter lost from the primary that it becomes more massive than the primary initially was it may end its evolution before the primary and is tmin ¼ tqM 1;u , and in case of an early merger the merger object may have a shorter lifetime than the original primary star, so we have then tmin ¼ tM merger . It is clear that since all the quantities which are involved in Eq. (A.14), depend on the metallicity, all the values of Gi(r,tn), Xi(r,tn) and Z(r,tn) (in a specific ring) on every past time tn of the model must be kept in memory throughout the calculation. For the integration over time we approximate the integral by a sum over series of time intervals [t(k 1), t(k)] with t0 = 0 and tðkmax Þ ¼ tnow . Within each time interval, the integrand is taken constant and estimated at the midpoint of the interval. For single stars (first integral) the integral is calculated on the path of the (M, t) plane which is fixed by the relation between mass and lifetime. For binary stars however the situation is more complicated. There we need to calculate the integral for each fixed pair of q and P, on the (M1, t) plane and (M2, t) plane. References Abt, H.A., 1983. ARA&A 21, 343. Abt, H.A., Levy, S.G., 1985. ApJSS 59, 229. Akmal, A., 1998. PhD. Thesis, University of Illinois at UrbanaChampaign. Anders, E., Grevesse, M., 1989. Geochim. Cosmochin. Acta 53, 197. Andreani, P., Vangioni-Flam, E., Audouze, J., 1988. ApJ 334, 698–706. Argast, D., Samland, M., Gerhard, O.E., Thielemann, F.-K., 2000. A&A 356, 873. Argast, D., Samland, M., Thielemann, F.-K., Gerhard, O.E., 2002. A&A 388, 842. Argast, D., Samland, D., Thielemann, F.-K., Qian, Y.-Z., 2004. A&A 416, 997. Arzoumanian, Z., Chernoff, D.F., Cordes, J.M., 2002. ApJ 568 (1), 289–301. Avila Reese, V.A., 1993. Rev. Mexicana Astron. Astrof. 25, 79– 89. Beers, T.C., Sommer-Larsen, J., 1995. ApJS 96, 175. Belczynski, K., Kalogera, V., Bulik, T., 2002. ApJ 572, 407. Benz, W., Bowers, R.L., Cameron, A.G.W., Press, W.H., 1990. ApJ 348, 647–667. Berczik, P., 1999. A&A 348, 371. Bernkopf, J., Fuhrmann, K., 1998. In: Spite, M. (Ed.), Galaxy Evolution. Editions Frontie`res, Meudon, Paris.
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