7. Special forms of the velocity distribution
7.1. Maxwell distribution
and for and one has from Eqs. (42) and (43)
Examples for these distributions are shown in Sect. 8, where the galactic halo is discussed.
7.2. Fixed velocity
A model with a fixed velocity corresponds to
It follows that and does not depend on l, so that
With Eq. (42), follows as
With the substitution
The argument of the -function is zero for
This yields for
Note that does not depend on l.
Since there exist (real) solutions for x only for
is restricted by for , or by for , where
Since , the critical value of G is
which is a maximum for and a minimum for .
The distribution of G can be written in terms of
which yields the probability density
and for one gets the probability density
is given by
© European Southern Observatory (ESO) 1998
Online publication: January 27, 1998