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

## 2. The distribution function in inhomogeneous systems

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

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

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

with given by

being

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

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

Now we suppose that is given by:

where a is a constant that can be obtained from the normalization condition for , j a parameter (of dimensions of velocity-1), an arbitrary function, 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 , 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 :

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

© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998