Lecture 7 Plummer's Spherical Model continued



Physics 141/241

Winter 2020




Plummer's Model

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

MATH

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

MATH

If we make the substitution MATH, MATH, this becomes

MATH

where

MATH

MATH

In Plummer's model the density rises as the $5^{th}$ power of $\Psi $ when $\Psi >0$, and is, of course, zero when $\Psi \leq 0.$ When we use Eq.(3) to eliminate $\rho $ from Poisson's equation, we find

MATH

If we eliminate $r$ and $\Psi $ from Eq.(5) in favor of the rescaled variables,

MATH

then Eq.(5) takes the simple form

MATH

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

MATH

The corresponding density is

MATH

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

MATH