thermodynamic equilibrium under self-gravitation Suppose you have a point mass M in space that is surrounded by a gas of particles mass m at temperature T. Let the gas distribution be spherically symmetric about M. What is the radial dependence of the equilibrium gas distribution? Wish to include the case were M=0.
This may be a trick question because I think the gas may evaporate over time and slowly cool. If this is so, then I want the answer to the question above for short time periods over which the evaporation is negligible.
Many thanks!
 A: You can start with the equation of hydrostatic equilibrium
$$
\frac{dp}{dr} = \frac{kT}{m}\frac{d\rho}{dr} = -\frac{d\Phi}{dr}
$$
where $T$ is the temperature, $k$ is Boltzmann's constant, $\rho$ is the density and $\Phi$ the gravitational potential (it depends on $M$). You can rearrange this equation to get
$$
\frac{d}{dr}\left(r^2\frac{d\ln\rho}{dr} \right) = -\frac{4\pi G m}{kT}r^2\rho
$$
And from here you can solve for the density profile $\rho$
A: If we take an infinitesimal volume from a sphere with radius r and center at M, we can get the infinitesimal mass. $$dm = \rho \frac43 \pi r^2dr $$
Integrate this, we will get the mass. $$\int_0^\infty dm = m$$
The force between the sheer of mass and the M can be calculated by the following. $$dF=G \frac{Mdm}{r^2}$$
Therefore the total force on the sphere will be the integral below. $$F=\int_r^\infty dF$$
The pressure can therefore be calculated. $$P=\frac{F}{\frac43 \pi r^2}$$
At the same time, using ideal gas state of equation: $$P=\rho RT$$.
Now, I stop here. After you fix it, I will make the rest up. 
A: Although detailed answers for the specific case of gas of state equation $P = \rho k_B T/m$ was given, I'll try to give a short but generic answer.
The problem can be divided into two sub-problems :


*

*Finding the equation describing the equilibrium between the gravitational force (which depends on $\rho$ via the mass) and the pressure $P$

*Finding the state equation of the considered gas, which relates $\rho$ and $P$.


The 1st problem leads to either the Newtonian hydrostatic equilibrium equation, or the Tolman–Oppenheimer–Volkoff equation in the relativistic case.
The 2nd problem depends on the physics, for stars the state equation is usually "polytropic".
