Editing Stellar dynamics (section)

=== A worked example of Virial Theorem ===

Twice kinetic energy per unit mass of the above uniform sphere is

<math display="block">\begin{align}
{2K \over M_0} & =  \overline{\langle V^2\rangle} \equiv  \langle \overline{V^2} \rangle \\
& = M_0^{-1} \int_0^{M_0} \langle V_\theta^2+V_\varphi^2+V_r^2 \rangle dM \\
& = M_0^{-1} \int_0^1 \left({V_0^2 \over 4}+{V_0^2 \over 4}+{ (1-x^2)V_0^2 \over 4}\right) d(x^3 M_0) = 0.6 V_0^2 , ~~ x \equiv {r \over r_0} = \left({M \over M_0}\right)^{1 \over 3},
\end{align}</math>
which balances the potential energy per unit mass of the uniform sphere, inside which <math>M \propto r^3 \propto x^3 </math>.

The average Virial per unit mass can be computed from averaging its local value <math> \mathbf{r} \cdot (-\mathbf{\nabla} \Phi)</math>, which yields
<math display="block">\begin{align}
 {W \over M_0} & = \overline{\mathbf{r} \cdot (-\mathbf{\nabla} \Phi)}\\
&=M_0^{-1} \int_0^{r_0}  \mathbf{r} \cdot {- G M \mathbf{r} \over |\mathbf{r}|^3} (\rho d\mathbf{r}^3) = -M_0^{-1} \int_0^{M_0} {G M \over |\mathbf{r}|} dM  \\
& = -M_0^{-1} \int_{0}^{M_0}  { G M ~ dM \over r_0~(M/M_0)^{1 \over 3} } = -{3 G M_0 \over 5 r_0} = -0.6 V_0^2,
\end{align} </math>
as required by the Virial Theorem.  For this self-gravitating sphere, we can also verify that the virial per unit mass equals the averages of half of the potential 
<math display="block"> \begin{align}
{E_\text{pot} \over M_0} & = \overline{\langle {\Phi \over 2}\rangle} \\
& = M_0^{-1} \int_{x>0}^{x<1}  {\Phi(r_0 x) \over 2} d (M_0 x^3) \\
& = {W \over M_0} = {-2K \over M_0}.\end{align}
</math>
Hence we have verified the validity of Virial Theorem for a uniform sphere under self-gravity, i.e., the gravity due to the mass density of the stars is also the gravity that stars move in self-consistently; no additional dark matter halo contributes to its potential, for example.