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Astron. Astrophys. 342, 34-40 (1999)

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1. Introduction

The study of the statistics of the fluctuating gravitional force in infinite homogeneous systems was pionered by Chandrasekhar & von Neumann in two classical papers (Chandrasekhar & von Neumann 1942, 1943 hereafter CN43) and in several other papers by Chandrasekhar (1941, 1943a,b,c,d,e, 1944a, 1944b). The analysis of the fluctuating gravitional field, developed by the quoted authors, was formulated by means of a statistical treatment. In their papers Chandrasekhar & von Neumann considered a system in which the stars are distributed according to a uniform probability density, no correlation among the positions of the stars is present and where the number of stars constituting the system tends to infinity while keeping the density constant. The force F , for unit mass, acting on a star of a generical star system is given by the well known equation:


where [FORMULA] is the mass of the i-th field star, N is the total number of the stars in the system and [FORMULA] is the position vector of the i-th test star relative to the field one. The summation includes all the neighboring stars. The motion of the stars in the neighborhood of the test star produces a time variation of F . The exact dependence of F on the position and time cannot be exactly predicted, while it is possible to study the fluctuation of F from a statistical point of view.

Two distributions are fundamental for the description of the fluctuating gravitional field:

  1. [FORMULA] which gives the probability that a test star is subject to a force [FORMULA] in the range [FORMULA], [FORMULA]+ d[FORMULA];

  2. [FORMULA] which gives the joint probability that the star experiences a force F and a rate of change f , where [FORMULA].

The first distribution, known as Holtsmark's law (Holtsmark 1919), in the case of homogeneous distribution of the stars, gives information only on the number of stars experiencing a given force but it does not describe some fundamental features of the fluctuations in the gravitional field such as the speed of the fluctuations and the dynamical friction. These features can be described using the second distribution [FORMULA]. As shown by CN43, the speed of the fluctuations can be adequately expressed in terms of the mean life of a state F :


where [FORMULA] is the second moment of f . Hence, for the definition of the speed of fluctuations and of the dynamical friction one must determine the distribution [FORMULA]. For a test star moving whith velocity v in a sea of field stars characterized by a random probability distribution of the velocities, [FORMULA], we may write:


where [FORMULA] represents the velocity of a typical field star relative to the one under consideration, [FORMULA] denotes the velocity of a field star. This asymmetry of the distribution of the relative velocities produces, as shown by CN43, a deceleration of the test star in the direction of motion. This effect is known, after Chandrasekhar papers, as "dynamical friction ". Some information on dynamical friction can be obtained by means of the first moment of f . As shown by CN43:



where m is the mass of a field star, [FORMULA] is the local density, and [FORMULA] is definited in CN43 Eq. (98). These equations show that the amount of acceleration in the direction of [FORMULA], when [FORMULA], is greater than that in the direction [FORMULA], when [FORMULA]. The star suffers a deceleration being the a priori probability that [FORMULA] equal to the probability that [FORMULA].

Several authors have stressed the importance of stochastic forces and in particular dynamical friction in determining the observed properties of clusters of galaxies (White 1976; Kashlinsky 1986, 1987) while others studied the role of dynamical friction in the orbit decaying of a satellite moving around a galaxy or in the merging scenario (Bontekoe & van Albada 1987; Seguin & Dupraz 1996; Dominguez-Tenreiro & Gomez-Flechoso 1998) which is not only the framework for galaxy formation picture in hierarchical cosmological models, but also important for the study of particular aspects of the evolution of a number of astronomical systems, such as galactic nuclei, cD galaxies in rich galaxy clusters. Finally, here, we want to remember as other the statistical description of dynamical friction other works have been made on it based on different approaches for example: Fokker Planck equation based polarization cloud (Rosenblunth et al. 1957; Binney & Tremaine 1987); resonant particle interactions (Tremaine & Weinberg 1984; Weinberg 1986); fluctation dissipation (Berkenstein & Maoz 1992; Maoz 1993).

Chandrasekhar's theory (and in particular his classical formula - see Chandrasekhar 1943b) is widely employed to quantify dynamical friction in a variety of situations, even if all the theory developed by the quoted authors is based on the hypothesis that the stars are distributed uniformly and it is well known that in stellar systems, the stars are not uniformly distributed, (Elson et al. 1987; Wybo & Dejonghe 1996; Zwart et al. 1997) as well as in galactic systems, the galaxies are not uniformly distributed (Peebles 1980; Bahcall & Soneira 1983; Sarazin 1988; Liddle, & Lyth 1993; White et al. 1993; Strauss & Willick 1995). It is evident that an analysis of dynamical friction taking account of the inhomogeneity of astronomical systems can provide a more realistic representation of the evolution of these systems itself. Moreover from a pure theorethical ground we expect that inhomogeneity affects all the aspects of the fluctuating gravitational field (Antonuccio & Colafrancesco 1994; Del Popolo 1994; Del Popolo et al. 1996; Del Popolo & Gambera 1996, 1997; Gambera 1997). Firstly the Holtsmark distribution is no more correct for inhomogeneous systems. For these systems, as shown by Kandrup (1980a,b, 1983), the Holtsmark distribution must be substituted with a generalized form of the Holtsmark distribution characterized by a shift of [FORMULA] towards larger forces when inhomogeneity increases. This result was already suggested by the numerical simulations of Ahmad & Cohen (1973, 1974). Hence when the inhomogeneity increases the probability that a test particle experiences a large force increases, secondly [FORMULA] is changed by inhomogeneity. Consequently, the values of the mean life of a state, the first moment of [FORMULA] and the dynamical friction force are changed by inhomogeneity with respect to those of homogeneous systems.

This paper must be intended as the first part of a work pointed to:

  • study the effects of inhomogeneity on the distribution functions of the stochastic forces and on dynamical friction (present paper);

  • test the result against N-body simulations;

  • find a formula that describes dynamical friction in homogeneous and inhomogeneous systems only on the basis of the statistical theory.

Before continuing we want to stress that when we speak of inhomogeneity we refer to inhomogeneity in position distribution and not to that of velocity distribution. Our work follows the spirit of Kandrup's (1980a) in the sense that we are interested in the effect of a non-uniform distribution in position of stars on the distributions of the stochastic force.

The plan of the paper is the following: in Sect. 2 we sketch the calculations needed to obtain the distribution function [FORMULA] after having released the hypothesis of homogeneity. The complete calculations are developed in the appendix. Then in Sect. 3 we calculate the first moment of f and in Sect. 4 we show how dynamical friction is influenced by inhomogeneity. Finally, in Sect. 5 we draw our conclusions.

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Online publication: December 22, 1998