Editing Stellar dynamics (section)

=== A recap on worked examples on Jeans Eq., Virial and Phase space density ===

Having looking at the a few applications of Poisson Eq. and Phase space density and especially the Jeans equation, we can extract a general theme, again using the Spherical cow approach.

Jeans equation links gravity with pressure gradient, it is a generalisation of the Eq. of Motion for single particles.  While Jeans equation can be solved in disk systems, the most user-friendly version of the Jeans eq. is the spherical anisotropic version for a static <math> \langle{v_j}\rangle =0 </math> frictionless system <math> t_\text{fric} \rightarrow \infty</math>, hence the local velocity speed 
<math>  \sigma_j^2 (r) = \langle{v_j^2}\rangle(r)  - \underbrace{\langle{v_j}\rangle^2(r)}_{=0}  = { \int\limits_\infty\!\! dv_r dv_\theta dv_\varphi  ({v}_j-\overbrace{\langle{v}\rangle^p_j}^{=0})^2 f_p \over 
\int\limits_\infty\!\! dv_r dv_\theta dv_\varphi f_p},
</math> everywhere for each of the three directions <math> ~_j =~ _r, ~_\theta, ~_\varphi</math>.
One can project the phase space into these moments, which is easily if in a highly spherical system, which admits conservations of energy <math> E = </math> and angular momentum J.  The boundary of the system sets the integration range of the velocity bound in the system.

In summary, in the spherical Jeans eq., 
<math display="block"> \begin{align} {d \Phi \over d r} = & {GM(r) \over r^2}  \\
= & -{d(n \langle{v_r^2}\rangle ) \over n(r) dr}+ {\langle{v_\theta^2}\rangle  + \langle{ v_\phi^2}\rangle -2 \langle{v_r^2}\rangle \over r} ,  \\
= & -{d(n \langle{v_r^2}\rangle ) \over n(r) dr}, ~~\text{hydrostatic equilibrium if isotropic velocity } \\
= & { \langle v_t^2 \rangle \over r}, ~~\text{if purely centrifugal balancing of gravity with no radial motion}, \langle v_t^2 \rangle \equiv \langle{v_\theta^2}\rangle  + \langle{ v_\phi^2}\rangle  \end{align} </math>
which matches the expectation from the Virial theorem <math>\overline{r \partial_r \Phi} = \overline{v_\text{cir}^2} = \overline{G M \over r}= \overline{\langle v_t^2 \rangle} </math>, 
or in other words, the <math>\overline{\text{global average}}</math> kinetic energy of an equilibrium equals the average kinetic energy on circular orbits with purely transverse motion.