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m ratio of the baryonic matter and the black holes demography in galaxies
New Astronomy 6 (2001) 239–248 www.elsevier.nl / locate / newast
The a /m ratio of the baryonic matter and the black holes demography in galaxies a, b Anna Curir *, Paola Mazzei a b
Osservatorio Astronomico, Torino, Italy Osservatorio Astronomico, Padova, Italy
Received 17 July 2000; received in revised form 8 April 2001; accepted 10 April 2001 Communicated by K.C. Freeman
Abstract The last years have seen a big progress in establishing the existence of supermassive black holes in the centers of galaxies. There are numerous very good cases [MNRAS 291 (1997) 219] where observations require a deep potential well. These observations raise the problem of when and how they formed and eventually when they gain most of their mass. The formation of a stationary black-hole is constrained by the conditions M . 3 M ( and cJ /GM 2 ; a /m , 1, J and M being the angular momentum and the total mass of the configuration which has collapsed to the hole. In this paper we analyze the behaviour of the a /m ratio of the baryonic content in a protogalaxy, ‘‘primordial’’ scenario, and in a hot galaxy, ‘‘evolved’’ scenario, endowed with a suitable angular momentum distribution. In both the cases the baryonic matter is embedded in the gravitational potential generated by a cosmological Dark Matter (DM) halo. We deduce that the ‘‘primordial’’ scenario is less favourable to the black hole formation than the ‘‘evolved’’ one. Moreover, in the ‘‘evolved’’ scenario we find a twofold behaviour of the a /m parameter which reflects the observed bimodal distribution of the central brightness in early-type galaxies and agrees with their corresponding degree of nuclear activity. As suggested by results of our SPH simulations of barred galaxies, the treatment of the dissipative processes and the inclusion of the star formation further improve the previous framework showing that barred galaxies provide very good environment for black hole formation. 2001 Published by Elsevier Science B.V. PACS: 98.62; 98.54.k; 97.60.l; 98.62.j Keywords: Dark matter; Galaxies: halos; Galaxies: evolution; Galaxies: nuclei; Galaxies: elliptical and lenticular, cD
1. Introduction Magorrian et al. (1998) published a sample of 30 mass estimates for the putative black holes in the bulges of nearby galaxies.
*Corresponding author. Tel.: 111-810-1917; fax: 111-810-1930. E-mail addresses: [email protected] (A. Curir), [email protected] (P. Mazzei).
The formation of a massive black-hole requires a large central condensation of low angular momentum material, hence it is of interest to verify whether the process of galaxy formation leads to conditions which are favourable to the evolution of a blackhole. As is well known (de Felice and Clarke, 1990), necessary conditions for the existence of a stationary black-hole are that the mass exceeds some critical value which, depending on the equation of state, is of the order of 3M( , and also that:
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a cJ ] 5 ]]2 , 1 m GM
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(1)
where a 5 J /cM and m 5 GM /c 2 are the specific angular momentum and the mass in units of length. While the first condition is easily satisfied at galactic scales, Eq. (1) is in general a non-trivial constraint (de Felice and Yunqiang, 1982; Miller and de Felice, 1985). In Bradley et al. (1991) it was attempted by analytical calculations to get the behaviour of such a parameter inside a system having a suitable density distribution and a velocity field consistent with the observed velocity curves for bulges of spiral galaxies; in particular, in this paper, the observed high resolution velocity curve of the galaxy M31 was assumed. As density distribution was assigned an isothermal distribution having a core radius r c :
r0 r (r) 5 ]]]] (1 1 (r /r c )2 )
(2)
where r0 is the density inside the core. It was shown that the a /m parameter has a minimum near the center of the system and then diverges as r → 0. From results of cosmological simulations, a new universal density profile for cosmological DM haloes was extracted by Navarro et al. (1995, NFW in the following): kr l r (r) 5 ]]]]] (r /r s (1 1 r /r s )2 )
same density distribution as the DM, as in Eqs. (2) or (3), and we assign different velocity fields (Section 1). We find that the nuclear properties of elliptical galaxies agree with the derived behaviour of the a /m parameter (Section 2). In the second case the galactic system is composed of a stellar and gaseous disk embedded in a DM halo with different mass, initial triaxiality ratio and dynamical state: relaxed and unrelaxed (Paper 1). The unrelaxed configuration is endowed with suitable density and velocity distributions as described in Paper 2 and it is far from dynamical stability. The relaxed case corresponds to the same configuration evolved for five dynamical times (t dyn ~ r 20.5 ). The initial values of the triaxiality ratio (t 5 a 2 2 b 2 /a 2 2 c 2 where a . b . c being the axes of the ellipsoid) explored have been 0.85, 0.54 and a pure spherical case (t → `) (Paper 1). We find that bars develop in several cases, in particular if the star formation is switched on (Mazzei and Curir, 2001). Moreover barred potentials produce very efficient non-keplerian gas inflows which increase the baryonic density in the inner regions and reduce the a /m parameter of the gas below its critical value (Eq. (1)). By comparing the behaviour of the a /m parameter provided by these simulations with that derived by our analytical work, we investigate if the dissipational processes, linked with the gas and the star formation, produce more suitable conditions to reduce below the unity the a /m of the gas.
(3)
where k r l indicates the average value of the density inside the radius r s which we will define below. From the features of Eq. (3) we see that the density profile goes as r ~r 2 a , with a 5 3 in the external regions, a 5 2 in the intermediate regions and a 5 1 in the region where r , r s . In this paper we want to examine the a /m behaviour of the baryonic content in galactic systems. We follow an analytic approach for early-type galaxies, where gas dissipation is a second order effect at least for nearby galaxies, and the results of our SPH (smooth particle hydrodynamic) simulations (Curir and Mazzei, 1999a, Paper 1; Mazzei and Curir, 2001) for late-type, disk systems. In the first case we assume that the baryonic component has the
2. The a /m ratio in NFW haloes and in coreisothermal haloes In this section we deduce the behaviour of the a /m parameter for the baryonic matter inside cosmological (NFW) DM haloes assuming Eq. (3) for the density profiles of both such components. We postulate a ratio of 1:10 between the masses of such components. This amounts to suppose that the baryonic matter density field is dominated by the DM. The usual formulae for the integrated mass and angular momentum of the baryonic component, provided a velocity field is assigned, are needed to solve our point. For this assignment we have examined two different options. The first one is a
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specific angular momentum distribution suggested by cosmological simulations for primordial haloes (Barnes and Efstathiou, 1987), i.e. increasing linearly with the radius (Curir et al., 1993). We suppose that the baryonic matter is still dynamically coupled with the DM therefore, since its density distribution is the same as the DM (see Section 1), also its angular momentum distribution follows the same as the DM. We will refer to this assumption as related to ‘‘primordial’’ systems. The second one is the velocity distribution same as in Bradley et al. (1991). Since we are interested in exploring the behaviour of the a /m of the baryonic mass in the inner regions of haloes, we assume that the velocity field that contributes to the angular momentum of this component is given by a solid body rotation v 5 Kr (where K is a suitable constant, see below), as suggested by observations of galaxies of different morphological type (Davies et al., 1983; Sofue, 1996).1 We will refer to this assumption as related to ‘‘evolved’’ systems. By integration of Eq. (3), we get for the a /m parameter of the baryonic matter the behaviour plotted in Figs. 1 and 2 for the first and second option respectively. Since the a /m ratio generated by a ‘‘primordial’’ angular momentum distribution coupled with a NFW density distribution, is not monotonically decreasing toward the central regions (Fig. 1), our ‘‘primordial’’ scenario is less favourable to black hole formation than the ‘‘evolved’’ configuration (Fig. 2). In this second case the feature characterizing the a /m ratio of the baryonic matter is its constant value at r # r s . This value becomes lower than unity in suitable conditions like those we will analyze in the following. Thus at radii r , r s the enclosed a /m of the baryonic matter is given by: k r l bar cK a /m (r ,r s ) 5 ]] ]]]]]2 4p Gr s (k r l DM 1 k r l bar )
(4)
where k r l bar is just the average density of the baryonic matter enclosed inside r s , c the light velocity and G the gravitational constant. From the slopes of the observed velocity curves of Spirals and 1
This assumption translates in an upper limit for the a /m parameter in the more external regions of the galaxy, where the velocity curve is flattened to a less steep slope.
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Ellipticals (Davies et al., 1983; Sofue, 1996) we deduce the following range of values for the constant K: 10 215 # K (s 21 )#2310 214 . Here we adopt the lower value since the higher one is more suitable to spiral rotation curves (Sofue, 1996). Thus we assume K 5 10 215 s 21 . To clarify the meaning of k r l DM we follow NFW which define: k r l DM 5 dc rcrit
(5)
where rcrit is the critical density, namely rcrit 5 229 22 3 2 1.8 3 10 h gr / cm , with H0 5 100 h km / s / Mpc. The parameter dc is strongly depending on the halo concentration, C, given by the ratio r 200 /r s with r 200 the radius where the halo density becomes 200 times greater than rcrit , according to: C3 dc 5 (200 / 3) 3 ]]]]]]] ln(1 1 C) 2 C /(C 1 1)
(6)
Putting r 200 5 1 Mpc, which implies a mass of the halo equal to 4 3 10 12 M ( (NFW), we derive the values of C in Table 1 for the corresponding r s radii. In this Table we present the dependence of the a /m parameter of the baryonic matter on C for two different choices of the maximum radius of the baryonic mass, r bar , 30 kpc and 10 kpc: (a /m) 1 and (a /m) 2 are the corresponding values. We also put H0 5 50 km / s / Mpc i.e. h 5 0.5. We point out that r bar is lower than r s and therefore the baryonic matter is totally contained inside the scale radius, for C , 33.3 in the first assumption (i.e. k r l bar 52.4310 225 gr / cm 3 ), whereas is always less or equal to r s in the second one (i.e. k r l bar 5 6.5 3 10 224 gr / cm 3 ). Table 1 shows that the (a /m) 2 parameter decreases also with the decrease of the halo concentration, in particular we recover (a / m) , 1 with C 5 10 if k r l bar $ 10 222 gr / cm 3 Thus there are two possible scenarios in which the a /m of the baryonic matter becomes lower than 1 in the inner regions of our systems: a very concentrated DM halo, or, at the contrary, a very rarefied halo with an high density baryonic system (i.e. rbar . 10 222 gr / cm 3 , we remember that the total density in the solar neighbourhood is around 3.6 3 10 223 gr / cm 3 ). The higher concentration values in Table 1 are out of the range proposed by NFW even if recent studies
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Fig. 1. The behaviour of the a /m parameter inside a primordial DM halo, having an angular momentum distribution conforming to the cosmological predictions of Barnes and Efstathiou. The radius is in units of scale radius r s .
(Bullock et al., 2001) suggest that the concentration should be larger for subhaloes compared to distinct haloes and values of C higher than 80 are possible. We also stress that, if the density profile of the halo in the inner regions is proportional to r 21.5 , as is claimed in Ghigna et al. (2000), in our framework the a /m parameter of the baryonic matter becomes equal to 0 at r 5 0; in fact for r , r s is: a /m ~r 211 a with a the exponent of the density distribution. So, depending on r bar , we can derive different values of the critical radius lowering a /m below 1. In the isothermal model chosen in Bradley et al. (1991) (see Eq. (2)) the core radius, corresponding to a flattening of the density profile and no longer present with the NFW profile, translates in a steepening of the a /m curve inside such a radius. This means that the same velocity distribution used for ‘‘evolved’’ systems produces a totally different a /m
profile if coupled with different density distributions; in particular using a core density distribution as in Bradley et al., the a /m behaviour we derive (Fig. 3) well agrees with their result.
3. The a /m ratio in early-type galaxies HST observations (Faber et al., 1997) of the center of early type galaxies split the family of elliptical galaxies and spiral bulges (in the following ‘‘hot galaxies’’) in two classes, according with the behaviour of their surface brightness, I(r) ~r 2g. Luminous hot galaxies (MV , 2 22) have cores (g # 0.3), while faint hot galaxies (MV . 2 20.5) show largely featureless steep power law profiles (g 5 0.860.3) that lack cores. Inside a limiting radius of 10 pc power law galaxies are up to 1000
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Fig. 2. The behaviour of the a /m inside a galaxy having density distribution dominated by the NFW halo and velocity field describing a rigid rotation. The radius is in units of scale radius r s .
times denser in mass and luminosity than core galaxies. This dicotomy between luminous and faint hot galaxies has tempted to be explained by the same authors with two different type of mergers: an highly dissipative one for the power law and a non-dissipaTable 1 Connection between the a /m ratio and some important system parameters C
rs kpc
k rDM l gr / cm 23
(a /m) 1 r bar 530 kpc
(a /m) 2 r bar 5 10 kpc
100 80 50 40 20 10
10 12.5 20 25 50 100
8.6 3 10 223 4.7 3 10 223 1.32 3 10 223 7.3 3 10 224 1.19 3 10 224 2.1 3 10 225
0.17 0.45 1.18 2.97 35.7 130
1.27 2.9 10.9 15.8 18.9 11.4
tive one may be following to the evolution of a pair of black holes in the remnant, for the core galaxies. On the other hand, Hozumi et al. (1999) have found a connection between the density distribution of their spherical haloes, derived by high resolution N-body simulations, and the surface brightness distributions of such kinds of galaxies. They claim that a cuspy density profile as that observed in the cores of elliptical galaxies can be realized by spherical dissipational collapse provided that an anisotropic velocity dispersion is assigned. They conclude that the relaxed density distribution becomes more cuspy with increasing radial velocity anisotropy. Merritt and Quinlan (1998) and van der Marel (1999) show that a slow growth of a black hole in a pre-existing core produces a power law cusp. Therefore van der Marel suggests that the observed bimodality in cusp slopes may be due to a bimodality
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Fig. 3. The behaviour of the a /m parameter inside a galaxy having a density distribution described by an isothermal-core law and a velocity field describing a rigid rotation. The radius is in units of kpc. The base of the scaling is given by the choice of the core radius, which is chosen equal to 5 kpc.
in the ratio between the mass of the nuclear black hole and the luminosity of the galaxy. In this section we link the nuclear properties of the hot galaxies stressed by Faber et al. (1997) to the results of our analytical approach. However we point out that observations give information on the luminous matter distribution, whereas from our framework as so as from previous cited models or simulations one derives the behaviour of the DM (Kormendy, 1999). Therefore we suppose, as in the previous section, that the baryonic matter reflects closely the density distribution of DM so that haloes in the power law galaxies are the NFW haloes, whereas in core galaxies follow a core-isothermal density distribution. Our scenario is not a unique way to interpret the observed brightness properties of hot galaxies, but it
gives a possible link between the structural and dynamical properties of inner parts of these systems and their phenomenology. We find that the properties of the a /m parameter of the baryonic matter in the power law case suggest that a black hole could be formed easily in the center of an ‘‘evolved’’ baryonic system (Fig. 2). Stars in these systems would be actually very concentrated since the a /m behaviour, in this case, favours crowding, in agreement with the observed excess of stellar central luminosity. Further, in our picture the accretion on the central black hole should arise naturally without energy release, in good agreement with the quietness of these galaxies. New mechanisms for accretion with very low energy emission have been invoked (e.g. advection dominated accretion flows (Narayan and Yi, 1995)) to justify such a quietness in a general framework.
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The behaviour of a /m in a core system (Fig. 3) suggests indeed a different scenario in which the formation and the gain of the mass of the central black hole should be preexistent to the core formation. Thus we believe for the central black hole a more likely origin from a merger. This agrees with the morphological properties of the core galaxies which are boxy, slowly rotating, more massive than power law systems. Moreover as a consequence of the a /m behaviour, which avoids the direct further collapse of the baryonic matter, the formation of rotationally supported structures like disks 2 around the collapsed object is favoured. In this case, further accretion requires highly dissipational processes emitting a lot of energy. We point out that all the bright Ellipticals with AGNs in the sample of Faber et al. (1997) are core galaxies which present also much high energy activity, i.e. radio loudness, jets and dusty-gaseous central disks, than the less massive systems.
4. Insights from SPH simulations of barred galaxies Many of the galaxies which are today candidates for the presence of a massive black hole are Spirals. In this context bars should play an important role in the birth and the evolution of a central black hole since they are present in a significant fraction of Spirals and they may be efficient drivers of gas accretion. According to Martinet and Friedli (1997), in fact, triaxial deformations like stellar bars are 2
The amount of angular momentum in accretion disks is very high in proportion to their mass, so the effect of an accretion disc is typically to raise the a /m of a combined (disk1black hole) system to values much larger than one. One can get a feel for how things go by considering the behaviour of the Keplerian specific angular momentum of test particles in the Schwarzschild metric. The formula is ]] Mr 2 l K 5 ]] r 2 3M
œ
which, when evaluated at the last stable orbit r 5 3R Schw where R Schw is the Schwarzshild radius, gives ] l K 5 2Œ3M so that l K is already greater than M there. As you move out from r 5 3R Schw through a Keplerian accretion disk, l K then goes to higher and higher values.
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powerful engines for extracting angular momentum, via gravitational torques, and increasing dissipation of cold gaseous components, via shocks. The fueling of the nucleus by inflow and accretion is however a complex and controversial problem which is also connected with the fueling of AGNs (Heller and Shlosman, 1994) moreover bar dissolution can arise from nucleus accretion processes (Hasan and Norman, 1990; Friedli and Benz, 1995) or from the prolateness of the halo itself (Ideta and Hozumi, 2000). Therefore the bars we observe now need not be the ones that were responsible for creating the central black holes and systems which would develop bars in the first stages of their evolution could appear now as very nucleated systems (Sa, S0 may be E; Curir and Mazzei (1999b), Paper 2). From our SPH simulations we recognize that the a /m parameter of the gaseous component has a peculiar behaviour in the presence of a stellar bar: there is a very sharp minimum in several simulations. In Fig. 4 we present the result of simulation 5 in Table 1 (Paper 1) which refers to a baryonic disk immersed in a halo with a mass 5 times that of disk and a radius 3 times the disk radius (i.e. r bar 5 20 kpc). The inner rise of a /m is due to a shift of the center of mass of the gas with respect to the position of that of the DM system: the minimum a /m value arises just away from the center. Of course this situation is still favourable to the black hole formation. In Fig. 5 we present the a /m behaviour for the same simulation but with the star formation switched on (Mazzei and Curir, 2001). That is more enhanced than in Fig. 4, its minimum, in particular, is well below unity. This is due to the larger transfer of angular momentum from the gas to the star component in the inner regions linked both with the star formation process itself and with the growing effect of friction following the increase of dissipationless particles. As derived from SPH simulations discussed in Mazzei and Curir (2001) we point out further the key role of the halo with its initial geometry, mass and dynamical state. In particular triaxial unrelaxed haloes are very favourable environments for lowering the a /m ratio of the gas below 1. This result is
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Fig. 4. The a /m parameter of gas (circles), old stars (stars) and DM (dashed line) during the evolution of the stellar-gaseous disk described by simulation n. 5. T is the time in simulation units (1.73 3 10 8 years).
Fig. 5. Same as in Fig. 4, but with the star formation switched on.
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attained well before than in the unrelaxed spherical haloes, namely between 0.3 and 0.5 Gyr. Relaxed systems depending on their total mass, 1 or 5 times the baryonic disk respectively, require instead 0.5–1.0 Gyr or 0.3–1.1 Gyr. However as a consequence of their higher total density which enhances the friction effects in the inner regions, initial spherical relaxed haloes with the larger mass appear as the more favourable environments to attain quickly, within 0.3 Gyr, the result we are discussing. For the simulation in Fig. 5 we derive a concentration, C, equal to 44 at t510, which corresponds (Eq. (6)) to k r l DM 5 9.5 3 10 224 gr / cm 3 . However looking at Table 1, these values are not consistent with a /m less than 1, as we deduce from our simulation. Therefore the very simple scenario discussed in the previous sections and based on the strong coupling between the baryonic and DM matter (Section 1) does not account for the true distribution of these components in spiral galaxies. Both the treatment of the gas, with its dissipative properties, and its interplay with different galaxy components, as so as the properties of the DM halo, have a key role in the galaxy formation and evolution also to address the issue of black hole formation.
5. Conclusions We analyzed the behaviour of the a /m parameter of the baryonic content in galactic systems endowed in a DM halo. We follow an analytic approach for early-type systems and the results of our SPH simulations for late-type disk systems. In the first case we assume for the baryons the same density distribution as their DM halo, i.e. a NFW halo density profile, Eq. (2), or an isothermal core-like density distribution, Eq. (3). The rotation velocity is assigned following two different configurations: an ‘‘evolved’’ one corresponding to a solid body rotation as observed in the inner regions of galaxies, and a ‘‘primordial’’ one corresponding to the primordial angular momentum distribution of haloes as suggested by Barnes and Efstathiou (1987). We find that in the case of a NFW density profile coupled with an ‘‘evolved’’ velocity field, the behaviour of the a /m parameter of the baryonic matter is decreasing with the decrease of the radial distance
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until attaining a constant value inside a critical radius, r s . Such a constant value is strongly depending both on the halo concentration and on the baryonic density inside the halo scale radius, r s , (Eq. (4)) and becomes less than 1, i.e. lower than the threshold value necessary to the black hole formation, in the case of very concentrated or very rarefied haloes; in the last case a high density baryonic system is needed. A ‘‘primordial’’ halo is not so favourable to the formation and accretion of a black hole. Nuclear properties of early-type galaxies observed by Faber et al. (1997) agree with our ‘‘evolved’’ framework even if some caution is due to the fact that the link between the surface brightness and the density profile is very critical (see Section 4). We also discuss the behaviour of the a /m parameter of the gas as derived from our SPH simulations of barred galaxies. We find that the instability of a stellar 1 gaseous disk induced by a live DM halo gives rise to a bar growing which accounts for a large dissipation of the gas angular momentum in the inner region of the system. This reflects in a lowering of the a /m parameter of the gas which decreases below 1 after at least 0.3 Gyr, depending on the initial geometry, mass and dynamical state of the DM halo. We point out that the treatment of dissipative processes and the inclusion of the star formation further improve previous results showing that barred galaxies provide a very good environment for black hole formation.
Acknowledgements We are grateful to F. de Felice and to J.C. Miller for their helpful comments and to an anonymous referee for his useful criticism.
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Curir, A., Diaferio, A., de Felice, F., 1993. ApJ 413, 70. Curir, A., Mazzei, P., 1999a. A&A 352, 103, Paper 1. Curir, A., Mazzei, P., 1999b. NewA 4, 1, Paper 2 Davies, R.L., Efstathiou, G., Fall, S.M., Illinworth, G., Schechter, P.L., 1983. ApJ 266, 41. de Felice, F., Clarke, C.J.S., 1990. In: Relativity On Curved Manifolds. Cambridge University Press. de Felice, F., Yunqiang, Yu., 1982. J. Phys. A 15, 4431. Faber, S.M., Tremaine, S., Ajhar, E., Byun, Y., Dressler, A., Gebhardt, K., Grillmair, C., Kormendy, J., Lauer, T.R., Richstone, D., 1997. AJ 114, 1771. Friedli, D., Benz, W., 1995. A&A 301, 649. Genzel, R., Eckart, A., Ott, T., Eisenhauer, F., 1997. MNRAS 291, 219. Ghigna, S., Moore, B., Governato, F., Lake, G., Quinn, T., Stadel, T., 2000. ApJ 544, 616.
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The a /m ratio of the baryonic matter and the black holes demography in galaxies a, b Anna Curir *, Paola Mazzei a b
Osservatorio Astronomico, Torino, Italy Osservatorio Astronomico, Padova, Italy
Received 17 July 2000; received in revised form 8 April 2001; accepted 10 April 2001 Communicated by K.C. Freeman
Abstract The last years have seen a big progress in establishing the existence of supermassive black holes in the centers of galaxies. There are numerous very good cases [MNRAS 291 (1997) 219] where observations require a deep potential well. These observations raise the problem of when and how they formed and eventually when they gain most of their mass. The formation of a stationary black-hole is constrained by the conditions M . 3 M ( and cJ /GM 2 ; a /m , 1, J and M being the angular momentum and the total mass of the configuration which has collapsed to the hole. In this paper we analyze the behaviour of the a /m ratio of the baryonic content in a protogalaxy, ‘‘primordial’’ scenario, and in a hot galaxy, ‘‘evolved’’ scenario, endowed with a suitable angular momentum distribution. In both the cases the baryonic matter is embedded in the gravitational potential generated by a cosmological Dark Matter (DM) halo. We deduce that the ‘‘primordial’’ scenario is less favourable to the black hole formation than the ‘‘evolved’’ one. Moreover, in the ‘‘evolved’’ scenario we find a twofold behaviour of the a /m parameter which reflects the observed bimodal distribution of the central brightness in early-type galaxies and agrees with their corresponding degree of nuclear activity. As suggested by results of our SPH simulations of barred galaxies, the treatment of the dissipative processes and the inclusion of the star formation further improve the previous framework showing that barred galaxies provide very good environment for black hole formation. 2001 Published by Elsevier Science B.V. PACS: 98.62; 98.54.k; 97.60.l; 98.62.j Keywords: Dark matter; Galaxies: halos; Galaxies: evolution; Galaxies: nuclei; Galaxies: elliptical and lenticular, cD
1. Introduction Magorrian et al. (1998) published a sample of 30 mass estimates for the putative black holes in the bulges of nearby galaxies.
*Corresponding author. Tel.: 111-810-1917; fax: 111-810-1930. E-mail addresses: [email protected] (A. Curir), [email protected] (P. Mazzei).
The formation of a massive black-hole requires a large central condensation of low angular momentum material, hence it is of interest to verify whether the process of galaxy formation leads to conditions which are favourable to the evolution of a blackhole. As is well known (de Felice and Clarke, 1990), necessary conditions for the existence of a stationary black-hole are that the mass exceeds some critical value which, depending on the equation of state, is of the order of 3M( , and also that:
1384-1076 / 01 / $ – see front matter 2001 Published by Elsevier Science B.V. PII: S1384-1076( 01 )00056-2
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a cJ ] 5 ]]2 , 1 m GM
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(1)
where a 5 J /cM and m 5 GM /c 2 are the specific angular momentum and the mass in units of length. While the first condition is easily satisfied at galactic scales, Eq. (1) is in general a non-trivial constraint (de Felice and Yunqiang, 1982; Miller and de Felice, 1985). In Bradley et al. (1991) it was attempted by analytical calculations to get the behaviour of such a parameter inside a system having a suitable density distribution and a velocity field consistent with the observed velocity curves for bulges of spiral galaxies; in particular, in this paper, the observed high resolution velocity curve of the galaxy M31 was assumed. As density distribution was assigned an isothermal distribution having a core radius r c :
r0 r (r) 5 ]]]] (1 1 (r /r c )2 )
(2)
where r0 is the density inside the core. It was shown that the a /m parameter has a minimum near the center of the system and then diverges as r → 0. From results of cosmological simulations, a new universal density profile for cosmological DM haloes was extracted by Navarro et al. (1995, NFW in the following): kr l r (r) 5 ]]]]] (r /r s (1 1 r /r s )2 )
same density distribution as the DM, as in Eqs. (2) or (3), and we assign different velocity fields (Section 1). We find that the nuclear properties of elliptical galaxies agree with the derived behaviour of the a /m parameter (Section 2). In the second case the galactic system is composed of a stellar and gaseous disk embedded in a DM halo with different mass, initial triaxiality ratio and dynamical state: relaxed and unrelaxed (Paper 1). The unrelaxed configuration is endowed with suitable density and velocity distributions as described in Paper 2 and it is far from dynamical stability. The relaxed case corresponds to the same configuration evolved for five dynamical times (t dyn ~ r 20.5 ). The initial values of the triaxiality ratio (t 5 a 2 2 b 2 /a 2 2 c 2 where a . b . c being the axes of the ellipsoid) explored have been 0.85, 0.54 and a pure spherical case (t → `) (Paper 1). We find that bars develop in several cases, in particular if the star formation is switched on (Mazzei and Curir, 2001). Moreover barred potentials produce very efficient non-keplerian gas inflows which increase the baryonic density in the inner regions and reduce the a /m parameter of the gas below its critical value (Eq. (1)). By comparing the behaviour of the a /m parameter provided by these simulations with that derived by our analytical work, we investigate if the dissipational processes, linked with the gas and the star formation, produce more suitable conditions to reduce below the unity the a /m of the gas.
(3)
where k r l indicates the average value of the density inside the radius r s which we will define below. From the features of Eq. (3) we see that the density profile goes as r ~r 2 a , with a 5 3 in the external regions, a 5 2 in the intermediate regions and a 5 1 in the region where r , r s . In this paper we want to examine the a /m behaviour of the baryonic content in galactic systems. We follow an analytic approach for early-type galaxies, where gas dissipation is a second order effect at least for nearby galaxies, and the results of our SPH (smooth particle hydrodynamic) simulations (Curir and Mazzei, 1999a, Paper 1; Mazzei and Curir, 2001) for late-type, disk systems. In the first case we assume that the baryonic component has the
2. The a /m ratio in NFW haloes and in coreisothermal haloes In this section we deduce the behaviour of the a /m parameter for the baryonic matter inside cosmological (NFW) DM haloes assuming Eq. (3) for the density profiles of both such components. We postulate a ratio of 1:10 between the masses of such components. This amounts to suppose that the baryonic matter density field is dominated by the DM. The usual formulae for the integrated mass and angular momentum of the baryonic component, provided a velocity field is assigned, are needed to solve our point. For this assignment we have examined two different options. The first one is a
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specific angular momentum distribution suggested by cosmological simulations for primordial haloes (Barnes and Efstathiou, 1987), i.e. increasing linearly with the radius (Curir et al., 1993). We suppose that the baryonic matter is still dynamically coupled with the DM therefore, since its density distribution is the same as the DM (see Section 1), also its angular momentum distribution follows the same as the DM. We will refer to this assumption as related to ‘‘primordial’’ systems. The second one is the velocity distribution same as in Bradley et al. (1991). Since we are interested in exploring the behaviour of the a /m of the baryonic mass in the inner regions of haloes, we assume that the velocity field that contributes to the angular momentum of this component is given by a solid body rotation v 5 Kr (where K is a suitable constant, see below), as suggested by observations of galaxies of different morphological type (Davies et al., 1983; Sofue, 1996).1 We will refer to this assumption as related to ‘‘evolved’’ systems. By integration of Eq. (3), we get for the a /m parameter of the baryonic matter the behaviour plotted in Figs. 1 and 2 for the first and second option respectively. Since the a /m ratio generated by a ‘‘primordial’’ angular momentum distribution coupled with a NFW density distribution, is not monotonically decreasing toward the central regions (Fig. 1), our ‘‘primordial’’ scenario is less favourable to black hole formation than the ‘‘evolved’’ configuration (Fig. 2). In this second case the feature characterizing the a /m ratio of the baryonic matter is its constant value at r # r s . This value becomes lower than unity in suitable conditions like those we will analyze in the following. Thus at radii r , r s the enclosed a /m of the baryonic matter is given by: k r l bar cK a /m (r ,r s ) 5 ]] ]]]]]2 4p Gr s (k r l DM 1 k r l bar )
(4)
where k r l bar is just the average density of the baryonic matter enclosed inside r s , c the light velocity and G the gravitational constant. From the slopes of the observed velocity curves of Spirals and 1
This assumption translates in an upper limit for the a /m parameter in the more external regions of the galaxy, where the velocity curve is flattened to a less steep slope.
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Ellipticals (Davies et al., 1983; Sofue, 1996) we deduce the following range of values for the constant K: 10 215 # K (s 21 )#2310 214 . Here we adopt the lower value since the higher one is more suitable to spiral rotation curves (Sofue, 1996). Thus we assume K 5 10 215 s 21 . To clarify the meaning of k r l DM we follow NFW which define: k r l DM 5 dc rcrit
(5)
where rcrit is the critical density, namely rcrit 5 229 22 3 2 1.8 3 10 h gr / cm , with H0 5 100 h km / s / Mpc. The parameter dc is strongly depending on the halo concentration, C, given by the ratio r 200 /r s with r 200 the radius where the halo density becomes 200 times greater than rcrit , according to: C3 dc 5 (200 / 3) 3 ]]]]]]] ln(1 1 C) 2 C /(C 1 1)
(6)
Putting r 200 5 1 Mpc, which implies a mass of the halo equal to 4 3 10 12 M ( (NFW), we derive the values of C in Table 1 for the corresponding r s radii. In this Table we present the dependence of the a /m parameter of the baryonic matter on C for two different choices of the maximum radius of the baryonic mass, r bar , 30 kpc and 10 kpc: (a /m) 1 and (a /m) 2 are the corresponding values. We also put H0 5 50 km / s / Mpc i.e. h 5 0.5. We point out that r bar is lower than r s and therefore the baryonic matter is totally contained inside the scale radius, for C , 33.3 in the first assumption (i.e. k r l bar 52.4310 225 gr / cm 3 ), whereas is always less or equal to r s in the second one (i.e. k r l bar 5 6.5 3 10 224 gr / cm 3 ). Table 1 shows that the (a /m) 2 parameter decreases also with the decrease of the halo concentration, in particular we recover (a / m) , 1 with C 5 10 if k r l bar $ 10 222 gr / cm 3 Thus there are two possible scenarios in which the a /m of the baryonic matter becomes lower than 1 in the inner regions of our systems: a very concentrated DM halo, or, at the contrary, a very rarefied halo with an high density baryonic system (i.e. rbar . 10 222 gr / cm 3 , we remember that the total density in the solar neighbourhood is around 3.6 3 10 223 gr / cm 3 ). The higher concentration values in Table 1 are out of the range proposed by NFW even if recent studies
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Fig. 1. The behaviour of the a /m parameter inside a primordial DM halo, having an angular momentum distribution conforming to the cosmological predictions of Barnes and Efstathiou. The radius is in units of scale radius r s .
(Bullock et al., 2001) suggest that the concentration should be larger for subhaloes compared to distinct haloes and values of C higher than 80 are possible. We also stress that, if the density profile of the halo in the inner regions is proportional to r 21.5 , as is claimed in Ghigna et al. (2000), in our framework the a /m parameter of the baryonic matter becomes equal to 0 at r 5 0; in fact for r , r s is: a /m ~r 211 a with a the exponent of the density distribution. So, depending on r bar , we can derive different values of the critical radius lowering a /m below 1. In the isothermal model chosen in Bradley et al. (1991) (see Eq. (2)) the core radius, corresponding to a flattening of the density profile and no longer present with the NFW profile, translates in a steepening of the a /m curve inside such a radius. This means that the same velocity distribution used for ‘‘evolved’’ systems produces a totally different a /m
profile if coupled with different density distributions; in particular using a core density distribution as in Bradley et al., the a /m behaviour we derive (Fig. 3) well agrees with their result.
3. The a /m ratio in early-type galaxies HST observations (Faber et al., 1997) of the center of early type galaxies split the family of elliptical galaxies and spiral bulges (in the following ‘‘hot galaxies’’) in two classes, according with the behaviour of their surface brightness, I(r) ~r 2g. Luminous hot galaxies (MV , 2 22) have cores (g # 0.3), while faint hot galaxies (MV . 2 20.5) show largely featureless steep power law profiles (g 5 0.860.3) that lack cores. Inside a limiting radius of 10 pc power law galaxies are up to 1000
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Fig. 2. The behaviour of the a /m inside a galaxy having density distribution dominated by the NFW halo and velocity field describing a rigid rotation. The radius is in units of scale radius r s .
times denser in mass and luminosity than core galaxies. This dicotomy between luminous and faint hot galaxies has tempted to be explained by the same authors with two different type of mergers: an highly dissipative one for the power law and a non-dissipaTable 1 Connection between the a /m ratio and some important system parameters C
rs kpc
k rDM l gr / cm 23
(a /m) 1 r bar 530 kpc
(a /m) 2 r bar 5 10 kpc
100 80 50 40 20 10
10 12.5 20 25 50 100
8.6 3 10 223 4.7 3 10 223 1.32 3 10 223 7.3 3 10 224 1.19 3 10 224 2.1 3 10 225
0.17 0.45 1.18 2.97 35.7 130
1.27 2.9 10.9 15.8 18.9 11.4
tive one may be following to the evolution of a pair of black holes in the remnant, for the core galaxies. On the other hand, Hozumi et al. (1999) have found a connection between the density distribution of their spherical haloes, derived by high resolution N-body simulations, and the surface brightness distributions of such kinds of galaxies. They claim that a cuspy density profile as that observed in the cores of elliptical galaxies can be realized by spherical dissipational collapse provided that an anisotropic velocity dispersion is assigned. They conclude that the relaxed density distribution becomes more cuspy with increasing radial velocity anisotropy. Merritt and Quinlan (1998) and van der Marel (1999) show that a slow growth of a black hole in a pre-existing core produces a power law cusp. Therefore van der Marel suggests that the observed bimodality in cusp slopes may be due to a bimodality
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Fig. 3. The behaviour of the a /m parameter inside a galaxy having a density distribution described by an isothermal-core law and a velocity field describing a rigid rotation. The radius is in units of kpc. The base of the scaling is given by the choice of the core radius, which is chosen equal to 5 kpc.
in the ratio between the mass of the nuclear black hole and the luminosity of the galaxy. In this section we link the nuclear properties of the hot galaxies stressed by Faber et al. (1997) to the results of our analytical approach. However we point out that observations give information on the luminous matter distribution, whereas from our framework as so as from previous cited models or simulations one derives the behaviour of the DM (Kormendy, 1999). Therefore we suppose, as in the previous section, that the baryonic matter reflects closely the density distribution of DM so that haloes in the power law galaxies are the NFW haloes, whereas in core galaxies follow a core-isothermal density distribution. Our scenario is not a unique way to interpret the observed brightness properties of hot galaxies, but it
gives a possible link between the structural and dynamical properties of inner parts of these systems and their phenomenology. We find that the properties of the a /m parameter of the baryonic matter in the power law case suggest that a black hole could be formed easily in the center of an ‘‘evolved’’ baryonic system (Fig. 2). Stars in these systems would be actually very concentrated since the a /m behaviour, in this case, favours crowding, in agreement with the observed excess of stellar central luminosity. Further, in our picture the accretion on the central black hole should arise naturally without energy release, in good agreement with the quietness of these galaxies. New mechanisms for accretion with very low energy emission have been invoked (e.g. advection dominated accretion flows (Narayan and Yi, 1995)) to justify such a quietness in a general framework.
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The behaviour of a /m in a core system (Fig. 3) suggests indeed a different scenario in which the formation and the gain of the mass of the central black hole should be preexistent to the core formation. Thus we believe for the central black hole a more likely origin from a merger. This agrees with the morphological properties of the core galaxies which are boxy, slowly rotating, more massive than power law systems. Moreover as a consequence of the a /m behaviour, which avoids the direct further collapse of the baryonic matter, the formation of rotationally supported structures like disks 2 around the collapsed object is favoured. In this case, further accretion requires highly dissipational processes emitting a lot of energy. We point out that all the bright Ellipticals with AGNs in the sample of Faber et al. (1997) are core galaxies which present also much high energy activity, i.e. radio loudness, jets and dusty-gaseous central disks, than the less massive systems.
4. Insights from SPH simulations of barred galaxies Many of the galaxies which are today candidates for the presence of a massive black hole are Spirals. In this context bars should play an important role in the birth and the evolution of a central black hole since they are present in a significant fraction of Spirals and they may be efficient drivers of gas accretion. According to Martinet and Friedli (1997), in fact, triaxial deformations like stellar bars are 2
The amount of angular momentum in accretion disks is very high in proportion to their mass, so the effect of an accretion disc is typically to raise the a /m of a combined (disk1black hole) system to values much larger than one. One can get a feel for how things go by considering the behaviour of the Keplerian specific angular momentum of test particles in the Schwarzschild metric. The formula is ]] Mr 2 l K 5 ]] r 2 3M
œ
which, when evaluated at the last stable orbit r 5 3R Schw where R Schw is the Schwarzshild radius, gives ] l K 5 2Œ3M so that l K is already greater than M there. As you move out from r 5 3R Schw through a Keplerian accretion disk, l K then goes to higher and higher values.
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powerful engines for extracting angular momentum, via gravitational torques, and increasing dissipation of cold gaseous components, via shocks. The fueling of the nucleus by inflow and accretion is however a complex and controversial problem which is also connected with the fueling of AGNs (Heller and Shlosman, 1994) moreover bar dissolution can arise from nucleus accretion processes (Hasan and Norman, 1990; Friedli and Benz, 1995) or from the prolateness of the halo itself (Ideta and Hozumi, 2000). Therefore the bars we observe now need not be the ones that were responsible for creating the central black holes and systems which would develop bars in the first stages of their evolution could appear now as very nucleated systems (Sa, S0 may be E; Curir and Mazzei (1999b), Paper 2). From our SPH simulations we recognize that the a /m parameter of the gaseous component has a peculiar behaviour in the presence of a stellar bar: there is a very sharp minimum in several simulations. In Fig. 4 we present the result of simulation 5 in Table 1 (Paper 1) which refers to a baryonic disk immersed in a halo with a mass 5 times that of disk and a radius 3 times the disk radius (i.e. r bar 5 20 kpc). The inner rise of a /m is due to a shift of the center of mass of the gas with respect to the position of that of the DM system: the minimum a /m value arises just away from the center. Of course this situation is still favourable to the black hole formation. In Fig. 5 we present the a /m behaviour for the same simulation but with the star formation switched on (Mazzei and Curir, 2001). That is more enhanced than in Fig. 4, its minimum, in particular, is well below unity. This is due to the larger transfer of angular momentum from the gas to the star component in the inner regions linked both with the star formation process itself and with the growing effect of friction following the increase of dissipationless particles. As derived from SPH simulations discussed in Mazzei and Curir (2001) we point out further the key role of the halo with its initial geometry, mass and dynamical state. In particular triaxial unrelaxed haloes are very favourable environments for lowering the a /m ratio of the gas below 1. This result is
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Fig. 4. The a /m parameter of gas (circles), old stars (stars) and DM (dashed line) during the evolution of the stellar-gaseous disk described by simulation n. 5. T is the time in simulation units (1.73 3 10 8 years).
Fig. 5. Same as in Fig. 4, but with the star formation switched on.
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attained well before than in the unrelaxed spherical haloes, namely between 0.3 and 0.5 Gyr. Relaxed systems depending on their total mass, 1 or 5 times the baryonic disk respectively, require instead 0.5–1.0 Gyr or 0.3–1.1 Gyr. However as a consequence of their higher total density which enhances the friction effects in the inner regions, initial spherical relaxed haloes with the larger mass appear as the more favourable environments to attain quickly, within 0.3 Gyr, the result we are discussing. For the simulation in Fig. 5 we derive a concentration, C, equal to 44 at t510, which corresponds (Eq. (6)) to k r l DM 5 9.5 3 10 224 gr / cm 3 . However looking at Table 1, these values are not consistent with a /m less than 1, as we deduce from our simulation. Therefore the very simple scenario discussed in the previous sections and based on the strong coupling between the baryonic and DM matter (Section 1) does not account for the true distribution of these components in spiral galaxies. Both the treatment of the gas, with its dissipative properties, and its interplay with different galaxy components, as so as the properties of the DM halo, have a key role in the galaxy formation and evolution also to address the issue of black hole formation.
5. Conclusions We analyzed the behaviour of the a /m parameter of the baryonic content in galactic systems endowed in a DM halo. We follow an analytic approach for early-type systems and the results of our SPH simulations for late-type disk systems. In the first case we assume for the baryons the same density distribution as their DM halo, i.e. a NFW halo density profile, Eq. (2), or an isothermal core-like density distribution, Eq. (3). The rotation velocity is assigned following two different configurations: an ‘‘evolved’’ one corresponding to a solid body rotation as observed in the inner regions of galaxies, and a ‘‘primordial’’ one corresponding to the primordial angular momentum distribution of haloes as suggested by Barnes and Efstathiou (1987). We find that in the case of a NFW density profile coupled with an ‘‘evolved’’ velocity field, the behaviour of the a /m parameter of the baryonic matter is decreasing with the decrease of the radial distance
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until attaining a constant value inside a critical radius, r s . Such a constant value is strongly depending both on the halo concentration and on the baryonic density inside the halo scale radius, r s , (Eq. (4)) and becomes less than 1, i.e. lower than the threshold value necessary to the black hole formation, in the case of very concentrated or very rarefied haloes; in the last case a high density baryonic system is needed. A ‘‘primordial’’ halo is not so favourable to the formation and accretion of a black hole. Nuclear properties of early-type galaxies observed by Faber et al. (1997) agree with our ‘‘evolved’’ framework even if some caution is due to the fact that the link between the surface brightness and the density profile is very critical (see Section 4). We also discuss the behaviour of the a /m parameter of the gas as derived from our SPH simulations of barred galaxies. We find that the instability of a stellar 1 gaseous disk induced by a live DM halo gives rise to a bar growing which accounts for a large dissipation of the gas angular momentum in the inner region of the system. This reflects in a lowering of the a /m parameter of the gas which decreases below 1 after at least 0.3 Gyr, depending on the initial geometry, mass and dynamical state of the DM halo. We point out that the treatment of dissipative processes and the inclusion of the star formation further improve previous results showing that barred galaxies provide a very good environment for black hole formation.
Acknowledgements We are grateful to F. de Felice and to J.C. Miller for their helpful comments and to an anonymous referee for his useful criticism.
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