Plummer's Model

Consider a steady state star cluster with time-independent which implies a time-independent . Introducing the notation and , a very simple form for the distribution function is

*Eq.(1a,1b) *defines Plummer's model. Calculate the density function

If we make the substitution , , this becomes

where

In Plummer's model the density rises as the
power of
when
,
and is, of course, zero when
When we use *Eq.(3)* to eliminate
from Poisson's equation, we find

If we eliminate
and
from *Eq.(5)* in favor of the rescaled variables,

then *Eq.(5)* takes the simple form

The solution of *Eqs.(7a,7b)* is given by

The corresponding density is

Note that the density is everywhere non-zero. The total mass is finite, however, with the value