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

## 3. Evalutation of

As we stressed in the introduction, the study of the dynamical friction is possible when we know the first moment of . This calculation can be done using the components of () in the system of coordinates previously introduced. We have that:

and similar equations for the other components of the force. The distribution function , giving the number of stars subject to a force , can be calculated as follows:

integrating we find:

This equation gives the generalized Holtsmark distribution obtained by Kandrup (1980a - Eq. 4.17) and provides the probability that a star is subject to a force in a inhomogeneous system.

As previously stressed, to calculate the first moment of we need only an approximated form for :

Using this last expression for and Eq. (A36), Eq. (A37), Eq. (14), Eq. (16), Eq. (17) and performing a calculation similar to that by CN43 the first moment of is given by:

where

and for Eq. (18) reduces to:

and consequently

this last expression can also be written as:

being

In this way we can written Eq. (20) as:

this last equation coincides with Eq. (105) by CN43.

The results obtained by us for an inhomogeneus system are different [see Eq. (18)], as expected, from that obtained by CN43 for a homogeneous system (CN43 - Eq. 105). At the same time it is very interesting to note that for (homogeneous system) our result coincides, as obvious, with the results obtained by CN43. In a inhomogeneous system, in a similar way to what happens in a homogeneus system, depends on , and (the angle between and ) while differently from homogeneous systems, is a function of the inhomogeneity parameter p. The dependence of on p is not only due to the functions , and to the density parameter but also to the parameter . In fact in inhomogeneous systems the normal field is given by , clearly dependent on p.

© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998