## The Virial Theorem

\[dp= - \frac{Gm}4 \pi r^4} dm\]

, where \[m\]

is the mass of gas in a sphere of radius \[r\]

with \[dm\]

being the mass of gas inside a shell surrounding the sphere and \[dp\]

is the pressure difference across the shell.>/BR> Multiply both sides by \[V= \frac{4}{3} \pi r^3\]

and integrate to get\[\int^{P(R)}_0 Vdp= - \int^M_0 \frac{Gm}{3r}dm\]

Integrate the left hand side by parts to get

\[pV]^{p(R)}_{p(0)}- \int^{V(R)}_{V(0)} pdV= - \int^M_0 \frac{Gm}{3r}dm\]

The

\[[pV]^{P(R)}_{P(0)}\]

vanishes since when \[r=R\]

\[p=0\]

and when \[r=0\]

\[V=0\]

. We get \[- \int^{V(R)}_{V(0)} pdV= - \int^M_0 \frac{Gm}{3r}dm= \frac{\Omega}{3}\]

where \[\Omega\]

is the total gravitational potential energy of the star.Since

\[dV= \frac{dm}{\rho}\]

we can write \[- \int^{M}_{m(0)} \frac{p}{\rho} dm= \frac{\Omega}{3}\]

From the ideal gas equation

\[pV=nRT \rightarrow p \frac{M}{\rho} = nN_A kT \rightarrow \frac{p}{\rho}=\frac{N kT}{M}=\frac{kT}{m} = \frac{2}{3m} Kinetic Energy_{Mean}\]

.Hence the Total of the kinetic energy of the particles that male up the star is

\[U=- \frac{1}{2} \Gamma\]

.This is the virial theorem.