Hydrostatic Gravitating Ensembles

Hydrostatic gravitating systems (HGE) is a name we've given to a large variety of models that describe a self gravitating blob of gas. Some models apply to stars, while others apply to star clusters and galaxies. However, since they have so much in common, we decided to unite them in a single entry. The governing equations are the hydrostatic equation

$\nabla P = - \rho \nabla \phi$

where $P$ is the pressure, $\rho$ is the density and $\phi$ is the gravitational potential. Another governing equation is the Virial theorem. According to which, the thermal velocity is the same order of magnitude as the escape velocity

$v^2 \approx |\phi|$

The gravitational potential is given by the Poisson equation

$\nabla^2 \phi = 4 \pi G \rho$

Spherical Models

Spherical symmetry immensely simplifies the governing equations. The hydrostatic equation becomes

$\frac{d P}{d r} = - \frac{G m}{r^2} \rho$

where $m$ is the mass enclosed within a radius $r$.

The Virial theorem is

$v^2 \approx \frac{G m}{r}$

Uniform Density Sphere

The density of a sphere of mass $M$ and radius $R$ is $\rho = \frac{3 M}{4 \pi R^3}$. The mass enclosed within each radius $r<R$ is

$m = M \frac{r^3}{R^3}$

The gravitational potential is

$\phi = -\frac{3}{2} \frac{G M}{R} + \frac{1}{2} \frac{G M r^2}{R^3}$

Isothermal Sphere

If the temperature is constant then

$\frac{G m}{r} \approx v^2 = const \Rightarrow m \propto r$

Hence

$\rho \propto r^{-2}$

and

$\phi \propto \int \frac{m}{r^2} dr \propto \ln r$

This divergence drove many researchers to try and find a better model.

Plummer Model

This model attempts to avoid the divergences by softening the gravitational potential

$\phi = \frac{G M}{\sqrt{r^2 + a^2}}$

where $a$ is the softening length. The corresponding density distribution is

$\rho = \frac{3 M}{4 \pi a^3} \left[ 1+\left(\frac{r}{a} \right )^2 \right ]^{-5/2}$

This model is the same as a polytrope of index 5.