## The Virial Theorem

There is a relationship between the internal heat energy of a gas that exerts a pressure to resist contraction and the gravitational force that promotes contraction. Starting from the equation for hydrostatic equilibrium
$dp= - \frac{Gm}4 \pi r^4} dm$
, 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. 