Consider a ball of gas held together by gravity, in thermal equilibrium with a constant cosmic microwave background radiation. Nothing fancy like nuclear fusion going on. Compute its radial density profile.
Unfortunately, I don't think the system is static. In a Boltzmann distribution of particle velocities, some particles will exceed gravitational escape velocity, so particles gradually evaporate away, shrinking the system over time. Compute the rate of mass loss. Normally such evaporation would cause cooling, but in our case the surrounding infinite heat reservoir of the CMBR heats it back up.
What is a simple artificial static system of gas and gravity? Maybe put it in a (spherical) box, or have periodic boundary conditions.
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