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.

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