*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
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