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

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2. The distribution function [FORMULA] in inhomogeneous systems

To calculate [FORMULA] in an inhomogeneous system we consider a particle moving with a velocity [FORMULA], subject to a force, per unit mass, given by Eq. (1) and to a rate of change given by

[EQUATION]

where [FORMULA] is the velocity of the field particle relative to the test one.

The expression of [FORMULA] is given following Markoff 's method by (CN43):

[EQUATION]

with [FORMULA] given by

[EQUATION]

being

[EQUATION]

where n is the average number of stars per unit volume while [FORMULA] and [FORMULA] are given by the following relations:

[EQUATION]

[EQUATION]

and [FORMULA] is the probability that a star has velocity in the range [FORMULA], positions in [FORMULA] and mass in [FORMULA].

Now we suppose that [FORMULA] is given by:

[EQUATION]

where a is a constant that can be obtained from the normalization condition for [FORMULA], j a parameter (of dimensions of velocity-1), [FORMULA] an arbitrary function, [FORMULA] the velocity of a field star. In other words we assume, according to CN43 and Chandrasekhar & von Newmann (1942), that the distribution of velocities is spherical, i.e. the distribution function is [FORMULA], but differently from the quoted papers we suppose that the positions are not equally likely for stars, that is the stars are inhomogeneously distributed in space. A lenghty calculation leads us (see Appendix for a derivation and the meaning of simbols) to find the function [FORMULA]:

[EQUATION]

This last equation introduced into Eq. (7) solves the problem of finding the distribution [FORMULA] and makes it possible to find the moments of [FORMULA] that give information regarding the dynamical friction.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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