# Lecture 7 Plummer's Spherical Model continued

## 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