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.

For spherical hydrostatic equilibrium with negligible surface pressure, ideal gas support and constant mean molecular weight $\mu$, the stellar virial theorem gives $3k_B\int Tdm/(\mu m_p)=\int Gmdm/r$. Since $r<R$ throughout the interior, the right side exceeds $G{M}^2/(2R)$. Division by $M$ gives the stated mass-weighted temperature bound. The central bound additionally assumes that the central temperature is maximal. A central pressure lower bound alone is insufficient without controlling central mass density.