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Astron. Astrophys. 330, 963-974 (1998)

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7. Special forms of the velocity distribution [FORMULA]

7.1. Maxwell distribution

A distribution

[EQUATION]

implies

[EQUATION]

and for [FORMULA] and [FORMULA] one has from Eqs. (42) and (43)

[EQUATION]

For [FORMULA] and separation of the spatial distribution and velocity distribution, these equations read

[EQUATION]

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

[EQUATION]

It follows that [FORMULA] and [FORMULA] does not depend on l, so that

[EQUATION]

with

[EQUATION]

With Eq. (42), [FORMULA] follows as

[EQUATION]

With the substitution

[EQUATION]

one gets

[EQUATION]

The argument of the [FORMULA] -function is zero for

[EQUATION]

so that

[EQUATION]

This yields for [FORMULA]

[EQUATION]

Note that [FORMULA] does not depend on l.

Since there exist (real) solutions for x only for

[EQUATION]

[FORMULA] is restricted by [FORMULA] for [FORMULA], or by [FORMULA] for [FORMULA], where

[EQUATION]

Since [FORMULA], the critical value of G is

[EQUATION]

which is a maximum for [FORMULA] and a minimum for [FORMULA].

The distribution of G can be written in terms of

[EQUATION]

which yields the probability density

[EQUATION]

and for [FORMULA] one gets the probability density

[EQUATION]

Since

[EQUATION]

[FORMULA] is given by

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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