- Numerical tests of dynamical friction in gravitational inhomogeneous systems
- 1 Introduction
- 2 Force derivative and dynamical friction
- 3
*N*-body experiments - 4 Conclusions
- References

A&A 406, 1-5 (2003)

DOI: 10.1051/0004-6361:20030665

**A. Del Popolo**

Dipartimento di Matematica, Università Statale
di Bergamo,
via dei Caniana 2, 24129 Bergamo, Italy

Feza Gürsey Institute, PO Box 6 Çengelköy, Istanbul, Turkey

Bo
çi University, Physics Department,
80815 Bebek, Istanbul, Turkey

Received 28 January 2003 / Accepted 10 April 2003

**Abstract**

In this paper, I test by numerical simulations the results of Del Popolo & Gambera (1998),
dealing with the extension of Chandrasekhar and Von Neumann's analysis of the
statistics of the gravitational field to systems in which particles
(e.g. stars, galaxies) are inhomogeneously distributed.
The paper is an extension of that of Ahmad & Cohen (1974), in which the authors
tested some results of the stochastic theory of dynamical friction developed by
Chandrasekhar & von Neumann (1943) in the case of homogeneous gravitational systems.
It is also a continuation of the work developed in
Del Popolo (1996a,b), which extended the results of Ahmad & Cohen (1973),
(dealing with the study of the probability distribution of the stochastic force in homogeneous gravitational
systems) to inhomogeneous gravitational systems.
Similarly to what was done by Ahmad & Cohen (1974) in the case of homogeneous systems,
I test, by means of the evolution of an inhomogeneous
system of particles,
that the theoretical rate of force fluctuation
describes correctly
the experimental one.
I find that the stochastic force distribution obtained for
the evolved system is in good agreement with the Del Popolo & Gambera (1998) theory.
Moreover, in an inhomogeneous
background the friction force is actually enhanced relative to the
homogeneous case.

**Key words: **stars: statistics - celestial mechanics - methods: numerical

The study of the statistics of the fluctuating gravitational force in infinite
homogeneous systems was pioneered by Chandrasekhar & Von Neumann in two classical papers
(Chandrasekhar & Von Neumann 1942, 1943, hereafter CN43) and in several other papers by Chandrasekhar
(1941, 1943a-e, 1944a,b). The analysis of
the fluctuating gravitational field, developed by the 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.
Two distributions are fundamental for the description of the fluctuating
gravitational field:

1)
which gives the probability that a test star
is subject to a force
in the range ,
;

2)
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 a 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 gravitational field such as the *speed of the fluctuations* and
the dynamical friction. These features can be
described using the second distribution
.
Hence, for the definition of the speed of fluctuations and of the
dynamical friction one must determine the distribution
.
Information on dynamical friction can be obtained from the moments of this last distribution.
As stressed by Chandrasekhar & Von Neumann (1943), for a test star moving with velocity v in
a sea of field stars characterized by a random probability distribution
of the velocities,
,
we may write:

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, from the Chandrasekhar papers, as "

where

(4) |

where

(5) |

and where . These equations show that the amount of acceleration in the direction of when is greater than that in the direction , when : then the star suffers a deceleration, the a priori probability that being equal to the probability that .

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 orbital decay of a satellite moving around a galaxy or in the merging scenario (Bontekoe & van Albada 1987; Seguin & Dupraz 1996; Dominguez-Tenreiro & Gomez-Flechoso 1998). 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 the theory developed 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 1995; Zwart et al. 1997) in galactic systems as well, the galaxies are not uniformly distributed (Peebles 1980; Bahcall & Soneira 1982; 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. Moreover from a pure theoretical standpoint we expect that inhomogeneity affects all the aspects of the fluctuating gravitational field (Antonuccio & Colafrancesco 1994; Del Popolo & Gambera 1996, 1997; Del Popolo et al. 1996; Gambera 1997). Firstly, the Holtsmark distribution is no longer 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 continuation of Del Popolo & Gambera (1998) paper, in which the study of the effects of inhomogeneity on the distribution functions of the stochastic forces and on dynamical friction was performed.

As anticipated in Del Popolo & Gambera (1998), the next task to perform
was to test the result of the Del Popolo & Gambera (1998) paper
against *N*-body simulations, which is the object of the present paper.
The third step (to be developed in a future paper) should be that of
finding a formula that describes dynamical friction in homogeneous and
inhomogeneous systems only on the basis of
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 (1980) in the sense that we are interested in the effect of a non-uniform distribution in the position of stars on the distributions of the stochastic force.

The plan of the paper is the following: in Sect. 2 we review the calculations and formulas needed to obtain in the case of inhomogeneous systems. In Sect. 3, I show how numerical experiments are performed and they are compared with the theoretical results of Del Popolo & Gambera (1998). Finally, in Sect. 4, I draw my conclusions.

The introduction of the notion of dynamical friction is due to CN43. In the
stochastic formalism developed by CN43 the dynamical friction is discussed
in terms of :

where

As shown by CN43, the origin of dynamical friction is due to the asymmetry in the distribution of relative velocities. If a test star moves with velocity in a spherical distribution of field stars, namely , then we have that:

The asymmetry in the distribution of relative velocities is conserved in the final Eq. (6). In fact from Eq. (6) we have:

(CN43). This means that when then , while when then . As a consequence, when has a positive component in the direction of , increases on average; while if has a negative component in the direction of , decreases on average. Moreover, the star suffers a greater amount of acceleration in the direction when than in the direction when .

In other words the test star suffers, statistically, an equal number of accelerating and decelerating impulses. The modulus of deceleration being larger than that of acceleration the star slows down.

Chandrasekhar & Von Neumann's analysis was extended to inhomogeneous systems
in Del Popolo & Gambera (1998), where
and its first momentum were calculated.
Supposing that the distribution function
is given by:

where

where:

(13) |

(15) |

The results obtained by us for an inhomogeneous system are different (see Eq. (11)), as expected, from those obtained by CN43 for a homogeneous system (CN43 - Eq. (105) or Eq. (6)). At the same time it is very interesting to note that for

In order to check the validity of the quoted relation (Eq. (16)), I have performed numerical experiments. This was done similarly to Del Popolo (1996b) by evolving (now) 100 000 points (stars) acting under their mutual gravitational attraction. From the evolved positions and velocities of the stars, was computed as a function of velocity and force, similarly to Ahmad & Cohen (1974), and then compared with Eq. (16) as I shall describe in the following

To calculate the stochastic force in an inhomogeneous system,
I used an initial configuration in which particles were
distributed according
to a truncated power-law
density profile:

(17) |

(see Kandrup 1980; Del Popolo 1996a). If the velocity distribution is everywhere isotropic then the equation relating the configuration space density to the phase space density

where

(Eddington 1916; Binney & Tremaine 1987). The initial conditions were generated from the distribution function that can be obtained from Eq. (19) assuming a cut-off radius

Figure 1:
The average value of the time rate of change of the magnitude of the force as the function of the velocity.
The solid line refers to the homogeneous case (Chandrasekhar & Von Neumann 1943). The dotted and dashed line refers to the cases p=2.5 and p=4, respectively (see Eq. (26)). Crosses represent the experimental points. |

Figure 2:
The average value of the time rate of change of the magnitude of the force as the function of the force.
The solid line refers to the homogeneous case (Chandrasekhar & Von Neumann 1943). The dotted and dashed line refers to the cases p=2.5 and p=4, respectively (see Eq. (19)). Crosses represent the experimental points. |

The system of 100 000 particles was evolved over 150 dynamical times using a tree

For a general distribution
can be calculated as usual:

(20) |

and can be written, in units of , as:

In the particular case of a Maxwellian distribution for velocities:

(22) |

where , so that:

(23) |

we have that:

that expressed in units of , then Eq. (24) becomes:

Similarly to Ahamd & Cohen (1974), since to integrate out the force from Eq. (16) one has a divergent result, I consider only particles up to a certain maximum value of the force, : for example in the case

(26) |

and

we find, in units of , that:

The results of calculation and numerical experiments are plotted in Figs. 1 and 2. In Fig. 1, I plot the average value of the time rate of change of the magnitude of the force as a function of the velocity. The solid line refers to the homogeneous case while the dotted and dashed lines refer to the cases

In this paper, I tested by numerical simulations the results of the Del Popolo & Gambera (1998) paper, dealing with the average value of the time rate of change of the magnitude of the stochastic force in inhomogeneous gravitational systems. In agreement with Ahmad & Cohen (1974), the stochastic theory of dynamical friction developed by Chandrasekhar & Von Neumann (1943), in the case of homogeneous gravitational systems, gives a good description of the results of numerical experiments. The stochastic force distribution obtained for inhomogeneous systems, obtained by Del Popolo & Gambera (1998), is also in good agreement with the results of numerical experiments. Finally, in an inhomogeneous background the friction force is actually enhanced relative to the homogeneous case.

I am grateful to E. N. Ercan and G. Mamon for stimulating discussions during the period in which this work was performed. I would like to thank Bo çi University Research Foundation for the financial support through the project code 01B304.

- Ahmad, A., & Cohen, L. 1973, ApJ, 179, 885 NASA ADS
- Ahmad, A., & Cohen, L. 1974, ApJ, 188, 469 NASA ADS
- Antonuccio-Delogu, V., & Colafrancesco, S. 1994, ApJ, 427, 72 NASA ADS
- Bahcall, N. A., & Soneira, R. M. 1982, ApJ, 262, 419 NASA ADS
- Bontekoe, T. R., & van Albada, T. S. 1987, MNRAS, 224, 349 NASA ADS
- Binney, J., & Tremaine, S. 1987, Galactic Dynamics, in Princeton Series in Astrophysics (Princeton University Press)
- Chandrasekar, S. 1941, ApJ, 94, 511 NASA ADS
- Chandrasekar, S. 1943a, Rev. Mod. Phys., 15, 1 NASA ADS
- Chandrasekar, S. 1943b, ApJ, 97, 255 NASA ADS
- Chandrasekar, S. 1943c, ApJ, 97, 263 NASA ADS
- Chandrasekar, S. 1943d, ApJ, 98, 25
- Chandrasekar, S. 1943e, ApJ, 98, 47
- Chandrasekhar, S. 1944a, ApJ, 99, 47 NASA ADS
- Chandrasekhar, S. 1994b, ApJ, 99, 25
- Chandrasekhar, S., & Von Neumann, J. 1942, ApJ, 95, 489 NASA ADS
- Chandrasekhar, S., & Von Neumann, J. 1943, ApJ, 97, 1 (CN43) NASA ADS
- Del Popolo, A. 1996a, A&A, 305, 999 NASA ADS
- Del Popolo, A. 1996b, A&A, 311, 715 NASA ADS
- Del Popolo, A., & Gambera, M. 1996, A&A, 308, 373 NASA ADS
- Del Popolo, A., & Gambera, M. 1997, A&A, 321, 691 NASA ADS
- Del Popolo, A., Gambera, M., & Antonuccio-Delogu, V. 1996, Mem. Soc. Astr. It., 67, 957
- Del Popolo, A., & Gambera, M. 1998, A&A, 342, 34 NASA ADS
- Dominguez-Tenreiro, R., & Gomez-Flechoso, M. A. 1998, MNRAS, 294, 465 NASA ADS
- Eddington, A. S. 1916, MNRAS, 76, 572 NASA ADS
- Elson, R., Hut, P., & Inagaki, S. 1987, ARA&A, 25, 565 NASA ADS
- Hernquist, L. 1987, ApJSS, 64, 715
- Holtsmark, P. J. 1919, Phys. Z., 20, 162
- Kandrup, H. E. 1980a, Phys. Rep., 63, 1
- Kandrup, H. E. 1980b, ApJ, 244, 1039
- Kandrup, H. E. 1983, Ap&SS, 97, 435 NASA ADS
- Kashlinsky, A. 1986, ApJ, 306, 374 NASA ADS
- Kashlinsky, A. 1987, ApJ, 312, 497 NASA ADS
- Liddle, A. R., & Lyth, D. H. 1993, Phys. Rep., 231, 2
- Maoz, E. 1993, MNRAS, 263, 75 NASA ADS
- Peebles, P. J. E. 1980, The large scale structure of the Universe (Princeton: Priceton University Press)
- Sarazin, C. 1988, X-ray emission from Clusters of Galaxies (Cambridge: Cambridge Univ. Press)
- Seguin, P., & Dupraz, C. 1996, A&A, 310, 757 NASA ADS
- Strauss, M. A., & Willick, J. A. 1995, Phys. Rep., 261, 271 NASA ADS
- White, S. D. M. 1976, MNRAS, 174, 19 NASA ADS
- White, S. D. M., Briel, U. G., & Henry, J. P. 1993, MNRAS, 261, L8 NASA ADS
- Wybo, M., & Dejonghe, H. 1995, A&A, 295, 347 NASA ADS
- Zwart, S. F. P., Tout, C. A., & Lee, H. M. 1997, in Highlights of Astronomy, ed. J. Andersen (Kluwer Academic Publishers), 11

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