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

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3. Evalutation of [FORMULA]

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

[EQUATION]

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

[EQUATION]

integrating we find:

[EQUATION]

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 [FORMULA] in a inhomogeneous system.

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

[EQUATION]

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

[EQUATION]

where

[EQUATION]

and for [FORMULA] Eq. (18) reduces to:

[EQUATION]

and consequently

[EQUATION]

this last expression can also be written as:

[EQUATION]

being

[EQUATION]

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

[EQUATION]

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 [FORMULA] (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, [FORMULA] depends on [FORMULA], [FORMULA] and [FORMULA] (the angle between [FORMULA] and [FORMULA]) while differently from homogeneous systems, [FORMULA] is a function of the inhomogeneity parameter p. The dependence of [FORMULA] on p is not only due to the functions [FORMULA], [FORMULA] and to the density parameter [FORMULA] but also to the parameter [FORMULA]. In fact in inhomogeneous systems the normal field [FORMULA] is given by [FORMULA], clearly dependent on p.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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