The Asymptotic Form of the Energy Density of Weakly Interacting Particles in a Static, Spherical Geometry

The asymptotic form of the energy density for a gas of particles surrounding a sphere of mass $M$ and radius $R$ is studied using Einstein's equations. It is shown that if the pressure of the gas $p$ varies linearly with the energy density $\rho$ for small $\rho$, then $\rho\sim 1/r^2$ for large $r$.