= Stellar virial theorem
For a nonmagnetic self-gravitating fluid <star>, let $T=\frac12\int\rho|\mathbf v|^2dV$ denote the bulk fluid <kinetic energy>, $\Omega$ the <Newtonian gravitational potential energy>, $P_s$ a uniform surface <pressure>, and $I=\int r^2dm$. The scalar <virial theorem> is
$$
\frac12\ddot I=2T+3\int P\,dV-3P_sV+\Omega.
$$
The 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 $3\int P\,dV+\Omega=0$. For a monatomic <ideal gas> with internal <energy> $U=\frac32\int P\,dV$, 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.
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