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
where is the pressure, is the density and 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
The gravitational potential is given by the Poisson equation
Spherical Models Edit
Spherical symmetry immensely simplifies the governing equations. The hydrostatic equation becomes
where is the mass enclosed within a radius .
The Virial theorem is
Uniform Density Sphere Edit
The density of a sphere of mass and radius is . The mass enclosed within each radius is
The gravitational potential is
Isothermal Sphere Edit
If the temperature is constant then
This divergence drove many researchers to try and find a better model.
Plummer Model Edit
This model attempts to avoid the divergences by softening the gravitational potential
where is the softening length. The corresponding density distribution is
This model is the same as a polytrope of index 5.