 Stellar virial theorem

For a nonmagnetic self-gravitating fluid star, let $T=\frac{1}{2}\int \rho |\mathbf{v}{|}^{2}dV$ denote the bulk fluid kinetic energy, $\Omega$ the Newtonian gravitational potential energy, $P_s$ a uniform surface pressure, and $I=\int r^{2}dm$. The scalar virial theorem isThe pressure integral already accounts for microscopic thermal motion; it must not be counted again in the bulk $T$. A static hydrostatic equilibrium has $T=0$ and $\ddot{I}=0$. If surface pressure is negligible, it obeys $3\int PdV+\Omega =0$. For a monatomic ideal gas with internal energy $U=\frac{3}{2}\int PdV$, this becomes $2U+\Omega =0$. The negative gravitational binding term therefore balances positive thermal support. Magnetic stresses, rotation through bulk motion and nonnegligible surface terms must be retained when present.