 Solution

Assume an isolated, bound, nonrotating Newtonian gas cloud, negligible surface pressure, negligible magnetic and radiation stresses, no persistent bulk flow, and a stationary or time-averaged virialized state. The stellar virial theorem then reduces towhere $\Omega <0$ is the total gravitational potential energy. Define the volume-averaged pressure $\overline{P}=V^{-1}\int PdV$ and average gravitational energy density $u_g=\Omega /V$. HenceEquivalently, for a nonrelativistic monatomic gas, the thermal internal energy $U=(3/2)\int PdV$ obeys $2U+\Omega =0$. This second derivation must not count the thermal motions again as a separate bulk kinetic energy. If the external pressure is $P_s$, the correct result is $\overline{P}-P_s=-u_g/3$; non-negligible bulk motion adds $2T_{\mathrm{bulk}}$ to the virial theorem. The initial sharp-edged $r^{-2}$ profile need not survive virialization; maintaining that profile hydrostatically to its edge would generally require a surface term.