Dynamics of stellar systems: Violent relaxation and gravothermal catastrophe

Ad~. Space Res. Vol.3, No.10-12, pp.387-391, 1984 Printed in Great Britain.

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0273-I177/84 $0.00 + .50 Copyright © COSPAR

D Y N A M I C S OF STELLAR SYSTEMS: VIOLENT RELAXATION AND GRAVOTHERMAL CATASTROPHE M.

Lachi6ze-Rey

CEN-Saclay, Service d'Astrophysique, 91191 Gif-sur-Yvette Cedex, France

ABSTRACT The gravitational evolution of stellar systems exhibit collective and collisional effects. Collective effects (Violent Relaxation) seems to be unefficient to create any structure or to relax the distribution function. Gravothermal Catastrophe is shown to be a real instability driven by collisions, but its time scale is very long compared to astrophysical times. It is suggested that intermediate range perturbations may modify these conclusions.

KEYWORDS Self gravitating systems; stellar dynamics; globular clusters.

INTRODUCTION Many astrophysical systems are self gravitating, from the interstellar gas clouds to the biggest galaxy clusters and superclusters. Between these two extreme cases, stellar systems are of particular interest because gravitation is thought to be (and to have been during an important fraction of their life) the only interaction present. They constitute the application field of N body problem. These systems (globular clusters, dwarf spheroidal galaxies, bulges of spiral and lenticular galaxies, elliptical galaxies) show some similarities: a spheroidal shape, the presence of central condensations and a slow rotation rate. But despite an increasing quantity of information, many questions about their dynamical states remain unanswered. Although some simple patterns (King or Emden spheres for globular clusters, De Vaucouleur law for E galaxies,...) seem to fit well the data available, no complete and satisfactory models exist to describe them. If more observations are clearly needed (both photometric and spectroscopic), the theory must also establish which configurations are possible and from which evolution scheme they result: it is then necessary to prospect initial conditions and dynamical evolution; although non N body processes probably play some role in the origin of these systems, this paper will concentrate on pure N body evolution which certainly applies after stars are formed: the main question will be if the properties of these systems can be attributed to N body processes or not. After pointing out some characteristics of gravitational interaction (§ I), I will describe collective effects, i.e. violent relaxation (§ II) and its consequences. The next chapter will focus on the subsequent evolution.

1.1 - HOW TO DEAL WITH SELF GRAVITATING SYSTEMS Self gravitating systems are original in many senses and usual methods should not be applied without care; this makes the difficulty of N body problem. Some analogy exists with electromagnetic plasmas since the l/r- law is the same. But due to the absence of negative charge, no screening effect cuts the long range of the interaction and any local analysis is forbidden. Plasma physics will however give us a powerful tool under the form of global methods for stability analysis. On the other hand, it is tempting to use thermodynamics and fluid theory. Unfortunately the non local character of the system and the time ordering (see next page) are contrary to fluid and thermodynamic assumptions. It is not a surprise that applying such methods lead to some paradoxes as mentioned by different authors: potential energy is not proportional to the total number of particles N, but to its square; the possibility of complete collapse makes energy

387

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M. Lachi~ze-Rey

available w~thout restriction; specific heat may be negative (Hachisu and Sugimoto 1978). In dealing with such systems, the time scales must be examined carefully to establish which method must be applied; the original statistical ensembles must be used (Horwitz and Katz 1977; Kandrup 1983) without assuming a priori that thermodynamics is applicable.

1.2 - THE GRAVITATIONAL FIELD The gravitational field is given by Poisson e q u a t i o n ~ ~ = 4 ~ G = 4 ~ G l" d 3 v. It is classicaly split into two parts, the mean field < ~ > which is averaged~over a length which is a few times the mean interparticle separation, and a fluctuating p a r t : 6 ~ m ~-<~) {~is responsible for collective effects; ~ is due to the local influence of neighbour stars. Its effect is to perturb locally the trajectories and it can be treated as collisions. It has been shown (H@non 1961) that the approximation of weak binary collisions holds well for stellar systems. This kind of approach discriminates between collective and binary effects. Collective effects act with a short time scale, the dynamicalr time t D. This is the mean time necessary for a star to describe its orbit t D ~ < ~ o r b i t dr/Vr> . It is of the order of the ratio of the radius R of tne system divided by the mean stellar velocity V, which gives t D ~ 1 0 6

yrs for globular clusters,

tD

lO 7 yrs for E galaxies. Collisions act much more slowly: a binary relaxation time tBR V%

4 ~ g ~ C In £.~N~

can be defined as for usual encounters

(Spitzer and Hart 1971). It can

be shown that tBR/tD~-N/ |~ (.4N) and tBR is much longer than t D ( t B ~ l O 9 y r s

for Globular

Clusters,~lO 14 yrs for E galaxies). The two effects are clarly well separated in real systems. It must be so in numerical simulations if some ~esult is claimed to be due to collective effects. This would require tBR/tD)>I or ~ l O ~ ( L i g h t m a n and Shapiro 1978).

The time ordering t D ~ tBR is of capital importance for the choice of the kinetic equation able to describe the system. For instance, fluid theory would require tBR~ t D and must be rejected for stellar systems. We can now consider the evolution of stellar systems. For instance, Ostriker (1978) proposed a succession of five stages: 1 - Expansion with Hubble flow and apparition of perturbations. 2 - Growths of the perturbation which become non linear and decouple from Hubble flow; formation of stars. 3 - Violent relaxation of the system of just formed point masses (stars). 4 - Core contraction. S - Late evolution. The two first phases are clearly non pure N body, as stars are not formed; dissipation plays some role and the system is not isolated. The third phase,Violent Relaxation (thereafter VR), is a very quick ( t ~ t D) virialization of the system, a pure collective effect and will be studied in next chapter. The last two phases are due to the collisions but are differently cconceived by different authors. We will discuss them in chapter III.

II.i - VIOLENT RELAXATION

(VR)

VR has been the subject of many studies and mainly by H@non (1964) and gynden-Bell (1967). Some discrepancies exist however concerning its astrophysical significance. VR is a virialization of the system in a short time scale t_. u It can be analysed as the superposition of different oscillation modes which suffer some kind of Landau damping, also called dynamical phase mixing (H@non 1964). VR is an effect of the mean field only and do not involve collisions. On this short time scale, no energy can be exchanged between the particles or dissipated and the total collapse hardly exceeds a factor 2 (Lachi~ze-Rey et al. 1982). An inevitable conclusion is that the system is virialized (2 Eki n + Epo t = O) and its radius is its Jeans radius Rj. As VR does not involve collisions, it can be described by Liouville equation and presumably conserves the phase space density. This led Lynden-Bell (1967) to propose a particular statistics for its final state. It is noticeable that the latter could become similar to a maxwellian in the degenerate case which may be valid for stellar systems. This led to the term Relaxation although we will claim that VR is not a true relaxation.

Dynamics of Stellar Systems

389

11.2 - VIOLENT RELAXATION IN ASTROPHYSICS The structure of elliptical galaxies has been suggested to be related to the oceurence of VR (Lynden-Bell, 1967; Binney, 1982; Rephaeli, 1983). It has also been proposed that VR helps to form a core in stellar systems (see simulations of Gott, 1973, and Hoffmann, 1979, also discussion in Lightman Shapiro, 1978) and also that it could give a relaxed distribution: Lynden-Bell (1967) suggested that its statistics could reduce to a m a x w e l l ~ n aistribution; more recently Binney (1982) suggested for the space integrated energy distribution a Boltzmann form N(E) ~ e x p (- ~ E) which he shows to fit well elliptical galaxies. He proposed that this could be the result of VR. We first notice, as Lightman and Shapiro (1978) did, that if a core structure is created in some simulations, it does not appear in some others. The same authors pointed out that the final structure strongly depends on initial state before VR. On the other hand, if the number of particles is not very high in the system, an observed central condensation may be due to a contamination by secular effects which do not discriminate from dynamical ones. Concerning the velocity distribution, Luwel and Severne (198S) investigated Binney's (1982) suggestion: they showed that after some dynamical times, N(E) still fluctuates strongly and cannot be considered as relaxed (although the system exhibits no motion in larger scales ~ R j ) • Also simulation of Bouvier and Janin (1970) dit not show Lynden-Bell or maxwellian distribution. We are then led to the conclusion that VR does not relax the distribution and that it is not efficient in forming a core. Although this conclusion needs more support to be definitive, we will examine its consequences. It seems clear that all astrophysical systems have suffered VR and are virialized; no dynamical theory excluding VR without arguments can be easily accepted. On the other hand, there is (probably) no specific character of stellar systems due to the mechanism of VR. The peculiarities of a stellar system are therefore due either to initial conditions before VR (for instance to the profile of initial perturbations as proposed by Binney 1982) or to some evolution after VR. We must look at evolution after VR without forgetting that the system is then not relaxed and that a high level of fluctuations may be present.

III.i -

COLLISIONAL STELLAR SYSTEMS

Evolution after VR is collisional. We expect some relaxation towards a maxwellian, the formation of a core + halo structure, the time scale b e i n g ' d e f i n e d by collision. The best way to describe the system has been shown (H@non 1971) to be Fokker Planck equation df/dt = ( ~ / ~ t)coll where df/d~ ~/a~,~-~÷V~-~F/~v is the derivative along the trajectories and

(~/~aU

is the Fokker Planek the velocity, # the velocity change (ant tBR this equation can

i- ~

~

~,&v

collision operator. Here f(r,V,t) is the distribution function, v gravitational potential. < ~ v> and ~ a v a v > are the average its averaged square) during an encounter.Owing to the ordering t D ~< be averaged over the orbits.

111.2 - THE GRAVOTHERMAL CATASTROPHE (GTC) Antonov (1962) showed that the entropy of a sphere of gravitationally interacting particles is not maximized at isothermal equilibrium under certain conditions. The latter can be expressed under the form of a criterion ~ ~ @ (center)/ @ ( b o u n d a r y ~ .... ~ 700" if the density contrast between center and boundary exceeds the C i~i al cr1~ica~ value, neighbour states exist with greater entropy. As a consequence, the system may be unstable and suffers some kind of phase transition, forming a dense hot core, virialized, isothermal and ever contracting, surrounded by a tenuous halo. Lynden-Bell and Wood (1968) made a similar analysis and suggested that this so called Gravothermal Catastrophe (GTC) may occur in stellar systems. Numerical simulations with various methods (H~non, 1973, 1975; Spitzer and Hart, 1971; Spitzer and Thuan 1972; Aarseth, 1974) showed formation of an isothermal core with a time scale between 1 and i00 tBR although it was not clear which mechanism, GTC or not, was involved (see, for instance, Lynden-Bell and E ~ l e t o n , 1980). These authors, and also Hachisu and Sugimoto (1978) pointed out that GTC is linked to heat exchange between the different parts of the system. The "catastrophic" character of the system is then attributed to the negative value of some specific heat (Lynden-Bell, 1977 Hachisu and JASR ~ : l O l / ~ - z

390

M. Lachi~ze-Rey

Sugimoto, 1978). Lynden-Bell and Eggleton proposed that stellar systems (after VR) first evolve along a quasi equilibrium sequence (like King or Emden models) increasing their entropy, and that they reached the Antonov's critical point where GTC begins to occur. Numerical simulations by Cohn (1979, 1980) using Fokker Planck equation and by Bettweiser (1983) using improved fluid dynamics method confirmed this two stages scenario. GTC appears then as the possible mechanism for generating central condensation in stellar system and its theoretical study is of interest. As fluid theory and thermodynamics, which were used in most of the preceding studies, are not sufficient, we need to make a true stability analysis starting from kinetic equations. Entropy arguments are not sufficient to prove that an instability develops (Sygnet et al. 1983). III.3 -

STABILITY ANALYSIS OF GRAVOTHERMAL CATASTROPHE

Such analysis has been undertaken by different authors starting from Boltzman (Ipser and Kandrup 1980) or Fokker Planck (Inagaki 1980; Sygnet et al. 1983, hereafter SFLP) equations. A perturbation analysis is made around an initial state defined by its distribution function f ; all the results refer to isothermal maxwellian distributions, but the method used by ~ F L P is able to test any initial state, even out of equilibrium. After linearization around this initial state (f = fo + fl' # = ~o + ~i )' some quadratic forms are constructed, averaged over trajectories and integrated over phase space. Their evolution permits to deduce instability criterion and growth's rate of the perturbation. The main assuptions are: spherical symmetry, no rotation, no mass spectrum. In order to check the importance of collisions, SFLP tested an initial equilibrium state with a collisionless evolution equation. They were able to show that the system is stable, which proves that any instability will be due to the presence of collisions. As the initial state is the same as the one studied by Antonov and subsequent authors, it is interesting to compare the two results. SFLP proved the stability of the system by showing that a conserved quadratic form 6%)" , which is a Liapounov integral, is alw@ys positive. Extracting from fl an adiabatic part ~ l~Fomgand settingS" 1 ~ - ~ 4 ~ / ~ g , ~ J writes

+~(~lwhere

~2h;,

does not depend of ~ l'

W 1 appears to be the quadratic

form of Antonov (the entropy). But regardless of its sign, ~ 2h/ remains positive assuring stability. We clearly see that entropy arguments are not sufficient to conclude about instability. SFLP show that the Antonov's criterion concerns only pure static modes which have no physical significance. In the collisional case, transform: all perturbed

the quadratic form quantities ~ e x p

~ J - is no longer constant. Using Laplace ( ~ t) and applying the ordering (A<< tBR)

which will be justified at the end, SFLP derive an equation for A

~1

:

0

A2(A tBR) 2 + B 2 ~ tBR = where A 2 is a quadratic form, clearly positive. B ~ i s a quadratic form involving the collision operator. It has been shown to be positive by Inagaki (1980) in a restricted case and by SFLP for the general Fokker Planck operator. ~2~) 4 is the Antonov quadratic form and it appears that the instability condition (A ~ O) writes ~ L ~ I ~ 0 which is analog to Antonov's criterion. This latter happens instability modes. The growth rate (SFLP 1983)

there

to

be

valid

for

collisional

is greater than i / t ~ as it appears in many simulations. Near the critical point, it reduces to " & ~ 4 / ~ ~ a n d becomes null there as was suggested by Ipser and Kandrup (1980).

CONCLUSION GTC appears then to be a real instability of stellar systems. Unfortunately, due to its very long time scale, it is difficult to believe that it could have occured in astrophysical systems, at least if we adopt the assumptions used here. In this case we would be led to the conclusion that all the structures observed in stellar systems reflect the condition of their formation. It appears, however, that some globular clusters are very near to the critical point for GTC (Katz 1980). If confirmed over a big sample, this would indicate that GTC plays some role in these systems. On the other hand, it would be satisfying to show that this efficient mechanism is at the origin of the structure of some stellar systems.

Dynamics

of Stellar

Systems

There is some hope, in fact, that GTC or a similar instability could occur with a time scale short enough to be of astrophysical interest. It is known that the presence of a spectrum of difference masses in the system may accelerate the processus and this is a first possibility. On the other hand, collective effects involving the whole system (responsible for VR) and binary collisions (responsible for GTC) are the two extremes of all the possible gravitational modes for the system. The whole range of intermediate modes which is neglected in this analysis may be analog to some level of gravitational turbulence; it would probably involve the kind of oscillations present at the end of VR (Luwel and Severne, 1983). It is then possible that the coupling of these modes introduces some evolution with time scales much shorter than 30 tRR (we would a priori expect time scales between t D and tn~). As it appears very difFTcult to study theoretically this coupling of modes, an interesting possibility seems to be a stability analysis (as made in § II.3) of a state including some high level of fluctuations; this could be made with the method presented here or by simulations.

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