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ISSN 1063-7737, Astronomy Letters, 2006, Vol. 32, No. 1, pp. 14­17. c Pleiades Publishing, Inc., 2006. Original Russian Text c A.S. Rastorguev, V.N. Sementsov, 2006, published in Pis'ma v Astronomicheski Zhurnal, 2006, Vol. 32, No. 1, pp. 16­19. i

Estimating the Stochastization Time in Stellar Systems
A. S. Rastorguev
1

1*

and V. N. Sementsov

2**

Physical Faculty, Moscow State University, Vorob'evy gory, Moscow, 119992 Russia 2 Sternberg Astronomical Institute, Universitetski i pr. 13, Moscow, 119992 Russia
Received July 12, 2005

Abstract--We show that passage to a statistical description of stellar systems is possible when considering
4 the evolution on time scales longer than m 5 d c ,where d is the mean dynamical (Keplerian) time and c is the two-particle collisional relaxation time.

PACS numbers : 02.50.Ey; 04.40.-b; 05.45.-a; 95.10.Fh; 98.10.+z DOI: 10.1134/S1063773706010038 Key words: celestial mechanics, stellar dynamics, relaxation, stochastization, stellar systems.

INTRODUCTION The characteristic time scale of "forgetting" the initial conditions plays an important role in studying the dynamics of stellar systems. Indeed, if m is shorter than the evolution time scales considered in a formulated problem, then we can pass to simplifying statistical­mechanical or kinetic descriptions. Otherwise, strictly speaking, we must solve the complete celestial-mechanical problem of the motion of N gravitating bodies and solve it exactly, because there are no a priori arguments that the soughtfor physical solutions are asymptotically close to the solutions obtained numerically, most commonly in the form of moments of the distribution averaged over the entire 6N -dimensional phase space. In addition, the procedure for choosing the initial conditions that affect crucially the result becomes much more complicated. THE CHARACTERISTIC TIME SCALES OF A SYSTEM OF GRAVITATING POINTS From dimensional considerations (see Dibai and Kaplan 1976), we can obtain two time scales for a system of gravitating points of mass m distributed with a mean space density n and a velocity dispersion 2 v0 : the collisional time c =
* **

and dynamical time (the crossing time of the stellar system or the Keplerian time) d = (Gmn)-
1/2

.

(2)

The results of most studies group around these two estimates. Taking into account only the pair (twoparticle) interactions, Chandrasekhar (1948) derived a time of the order of (1), to within a factor that diverges logarithmically in impact parameter. By taking into account only the collisionless collective effects of "violent relaxation," Lynden-Bell (1967) derived a time of the order of (2), to within a factor determined by the Landau damping constant The ratio of these times can be easily shown to be given by a dimensionless combination of the scale lengths commonly considered in stellar dynamics, 1 d = c 8 p d0
3/2

.

(3)

- Here, p = 2Gmv0 2 is the impact parameter of such a close encounter at which the relative velocity vector of the two stars that had the relative velocity v0 at a (formally) infinite distance from one another turns through 90 in the frame of reference associated with their center of mass, and d0 = 0.5n-1/3 is the mean distance between the stars in the system. For convenience, we then represent the time scale of interest as

3 v0 (Gm)2 n

(1)



m

=

p d0



0 ,

(4)

E-mail: rastor@sai.msu.ru E-mail: valera@sai.msu.ru

where 0 = v0 (Gmn2/3 )-1 and is the sought-for parameter. It can be easily shown that = -1 and
14


ESTIMATING THE STOCHASTIZATION TIME

15

= 0.5 for Eqs. (1) and (2), respectively, and
3 2 0 = d c .

(5)

The classical approach to estimating m using the a priori assumption that the system relaxes to an equilibrium state, which consists in calculating the parameter in terms of a certain interaction model, involves arbitrariness in choosing the relaxation mechanism and the related parameters and contains a vicious circle (because the relaxation time to equilibrium is calculated by assuming that the equilibrium state is actually reached). SELF-CONSISTENT ESTIMATION OF THE STOCHASTIZATION TIME The methods of stochastic dynamics (see Likhtenberg and Liberman 1982), which date back to the classic work by Krylov (2003), serve as a reasonable alternative to this approach. Let us explain the essence of the method without going into mathematical details. A Hamiltonian system with 3N degrees of freedom can be represented by a point in 3N dimensional (configuration) space. The dynamical evolution of the system can be described by the motion of the representing point along a geodesic curve. Analysis of the properties of the bundle of geodesic lines emerging from a small region of close initial conditions makes it possible to determine whether the properties of stochasticity manifest themselves in the behavior of the system. Indeed, if the geodesic lines diverge rapidly (exponentially), then, provided that the volume accessible for the system is limited (in configuration space), they become greatly entangled and, at a finite observation accuracy, randomly fill the volume almost irrespective of the initial conditions. This phenomenon is called mixing and is a property strong enough to prove that the system is ergodic. The absence of mixing is indicative of a relative stability of the motion. Relaxation, in particular, results in the filling of all the phase-volume cells accessible for the system. Therefore, the rate of divergence or the rate of increase of the (coarse) phase volume filled with the representing points corresponding to different initial conditions (its logarithm is the Kolmogorov­Sinai entropy) gives an idea of the stochastization time of the system, a constructive analog of the relaxation time (Zaslavsky 1984). The Hamiltonian of a stellar system is so complex that the equations of geodesic lines cannot be derived in the most general case without using additional assumptions and simplifications. The simplest of them is to postulate a local homogeneity of the stellar system and the Poissonian nature of the appearance of the nearest neighbors of the trial star (which seems to be valid, since the stellar system is fairly
ASTRONOMY LETTERS Vol. 32 No. 1 2006

sparse). Based on this assumption, Gurzadian and Savvidy (1983, 1984, 1986) and Gurzadian (1998) derived the following formula for the stochastization time: -1 0 0.270 C -0.5 , (6) GS = 3.752/3 2 2C where C is the mean square of the dimensionless force acting on a star in a homogeneous stellar system,

max

C=
0

2 H ( )d ,

(7)

calculated from the Holtzmark distribution H ( ) (see Chandrasekhar 1948). Since the Holzmark distribution diverges at large forces max (rmin 0) -5/2 as max with GS 0, it seems natural to limit max and, accordingly, the minimum distance rmin . Gurzadian and Savvidy (1983, 1986) assumed that rmin = p and postulated C = 1; they estimated the stochastization time to be GS 0 . After correcting the obvious error in Gurzadian and Savvidy (1983) and properly integrating (7), we obtain under the same assumptions C 3e-y y
-1/3

- (2/3)
k

(8)

+y

2/3 k =0

-y , k!(k +2/3)

where y = 4 (rmin /d0 )3 /3 and (2/3) is the Gamma function, or, simplifying this expression for the realisd0 , we obtain a more accurate value tic case of rmin of integral (7), d0 . (9) C2 rmin Following Gurzadian and Savvidy (1983) and assuming that rmin = p , we obtain the corrected estimate,
GS

0.125

p d0

0.5

0 .

(10)

Under our assumptions, GS d should have been considered to be an estimate of the stochasization time. Its low value may appear surprising (10-3 0 and 10-5 0 for a globular cluster and the Galaxy, respectively). Note also that with the above constraint on rmin , the energy per unit volume of the system, its square, and other macroscopic quantities averaged over the Holzmark distribution closely match those for a continuous medium. As a matter of fact, under these conditions, the Holzmark distribution yields the regular force. The conflict between the assumptions made and the basic physical principlesisthe source of theeffects


16

RASTORGUEV AND SEMENTSOV

mentioned above. Indeed, the mixing being proved suggests the ergodicity of the system as a necessary condition, i.e, the equality between the time-averaged (i.e., measured physical) and ensemble-averaged (calculated) quantities. It is absolutely clear that encounters with impact parameters of the order of p occur on time scales of the order of the collisional relaxation time, Tr c /100, which is 1014 yr for the Galactic disk. Therefore, the assumption of rmin = p arbitrarily extends the ensemble of systems to include the events (extremely close pair interactions) that could not occur in the sought-for time m and should not have showed up when averaging over the time, but, as we see from formulas (7) and (10), make the overwhelming contribution to the estimated force. Hence the error--the implausibly short stochastization time and, as a result, the analysis of hydrodynamic phenomena. A way out of this situation could be the solution of a partially self-consistent problem, i.e., allowance for only those close interactions (encounters) that can occur on the mixing time scale with a nonnegligible probability. Therefore, we seek for an order-ofmagnitude estimate, and we can write the following equation for rmin and, hence, for the sought-for mixing time m (rmin ): m (rmin )v0 n r
2 min

using the probabilistic approach makes it possible to correct the Holzmark distribution for large dimensionless forces in formula (6) and to ensure the convergence of the integral for the second-order moment, because the corrected distribution has the asymptotics H1 ( ) exp(-1.5 2 )/ 3 at > 100QH (the dimensionless force at the mean distance). A simple estimate shows that the relative contribution from encounters with > 100QH for realistic rmin /d0 10-3 ­10-4 does not exceed 2 â 10-3 , which does not change our conclusions at all.

CONCLUSIONS Representing Eq. (13) in terms of the commonly used time scales c (1) and d (2) may prove to be more convenient. Using Eqs. (3) and (5), we obtain
m

0.224

5

4 d c .

(14)

1

(11)

In a spatially homogeneous system (which is an additional strong assumption), the ratio c : d depends on the total number N of stars in the system, which allows formula (14) to be simplified: 5 (15) m d N. Comparison of the m values in a globular cluster (m 106 ­107 yr) and the Galactic halo (m 1010 ­1011 yr) with the cosmological time (1010 yr) shows that applying the statistical­mechanical methods to stellar systems is quite justifiable from the viewpoint that an equilibrium (in the sense of filling the phase volume) is established in them in a time shorter than the age.

(clearly, using an approximate equality in this case does not introduce a serious error). The solution of the complete self-consistent problem based on the substitution of the stationary Holzmark distribution with a different, theoretically more justified, distribution of the random force appears impossible. The solution of Eq. (11) in the asymptotics (9) is r
5 min

6.5d3 p2 , 0 p d0
1/5

(12)

Note, however, that the stochastization of motions in stellar systems is not relaxation in the sense this 0 . (13) m 0.17 term is applied to ordinary gases or plasma. On time scales longer than m , a stellar system is described The latter agrees well with the result by Genkin (1972), by a number of parameters that is much smaller than who estimated m 0 by qualitatively analyzing the the number of mechanical parameters, 6N ,but larger "violent" relaxation stage, strictly speaking, based on than that in the statistical mechanics of an ideal gas, a similar idea--the attempt to take into account a especially when analyzing systems with a common wider variety of interactions. angular momentum. In connection with our estimates of the time scale, the results by Petrovskaya (1986) should be mentioned. As is well known, Agekyan (1959) suggested ACKNOWLEDGMENTS a new method to make allowance for pair interactions. It is based on the treatment of the change in the veThis work was supported in part by the Russian locity of a star as an absolutely discontinuous random process. Agekyan gave a formula for the probability Foundation for Basic Research (project no. 05-02of an encounter with a given change in the absolute 16526) and the "Program for Support of Leading velocity of a star. Petrovskaya (1986) showed that Scientific Schools" (project no. NSh-389.2.2003).
ASTRONOMY LETTERS Vol. 32 No. 1 2006

which yields the ultimate formula for m :


ESTIMATING THE STOCHASTIZATION TIME

17

REFERENCES
1. T. A. Agekyan, Astron. Zh. 36, 46 (1959) [Sov. Astron. 3, 41 (1959)]. 2. S. Chandrasekar, Foundations of Stellar Dynamics (Inostrannaya Literatura, Moscow, 1948) [in Russian]. 3. S. Chandrasekhar, Stochastic Problems in Physics and Astronomy (AIP, New York, 1943; Inostrannaya Literatura, Moscow, 1948). 4. E. A. Dibai and S. A. Kaplan, Dimensions and Similarity of Astrophysical Quantities (Nauka, Moscow, 1976) [in Russian]. 5. I. L. Genkin, Astron. Zh. 45, 1085 (1968) [Sov. Astron. 12, 858 (1968)]. 6. I. L. Genkin, Doctoral Dissertation (Alma-Ata, 1972). 7. V. G. Gurzadian, in Dynamical Studies of Star Clusters and Galaxies, Parallel Meeting P5 of Prospects of Astronomy and Astrophysics for the New Millennium, JENAM98, 7th European and Annual Czech Astronomical Society Conference, Ed. by P. Kroupa, J. Palous, and R. Spurzem (ESA, c/o ESTEC, Noordwijk, The Netherlands, 1998), p. 176.

8. V. G. Gurzadian and G. K. Savvidy, Collective Relaxation of Stellar Systems, Preprint (Yerevan Phys. Inst., Yerevan, 1983). 9. V. G. Gurzadian and G. K. Savvidy, Dokl. Akad. Nauk SSSR 277, 69 (1984) [Sov. Phys. Dokl. 29, 520 (1984)]. 10. V. G. Gurzadian and G. K. Savvidy, Astron. Astrophys. 160, 203 (1986). 11. H. E. Kandrup, Ann. N. Y. Acad. Sci. 848, 28 (1995). 12. H. E. Kandrup and C. Siopis, Mon. Not. R. Astron. Soc. 345, 727 (2003). 13. N. S. Krylov, Works on Substantiating Statistical Physics, 2nd ed. (URSS, Moscow, 2003). 14. A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1982; Mir, Moscow, 1984). 15. D. Lynden-Bell, Mon. Not. R. Astron. Soc. 136, 101 (1967). 16. I. V. Petrovskaya, Pis'ma Astron. Zh. 12, 562 (1986) [Sov. Astron. Lett. 12, 237 (1986)]. 17. G. M. Zaslavsky, Chaos in Dynamical Systems (Nauka, Moscow, 1984; Harwood, Chur, 1985).

Translated by A. Dambis

ASTRONOMY LETTERS

Vol. 32 No. 1 2006