## 1. IntroductionThe 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 where is the mass of the i-th
field star, Two distributions are fundamental for the description of the fluctuating gravitional field: -
which gives the probability that a test star is subject to a force in the range , + d; -
which gives the joint probability that the star experiences a force **F**and a rate of change**f**, where .
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 where is the second moment of
where represents the velocity of
a typical field star relative to the one under consideration,
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 " where 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 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 is changed by inhomogeneity. Consequently, the values of the mean life of a state, the first moment of 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
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 © European Southern Observatory (ESO) 1999 Online publication: December 22, 1998 |