Editing Stellar dynamics (section)

=== Probability-weighted moments and hydrostatic equilibrium ===
Jeans computed the weighted velocity of the Boltzmann Equation after integrating over velocity space <math display="block">
 {1 \over \rho_p } \int\! \left\{\mathbf{v}_p {d [f_p m_p]\over dt}  - \langle{\mathbf{v}}\rangle_p  {d [f_p m_p]\over dt}\right\} d^3\mathbf{v} = 0, </math> and obtain the Momentum (Jeans) Eqs. of a <math>^p</math>opulation (e.g., gas, stars, dark matter):

<math display="block">\begin{align}
\overbrace{ \left({\partial \over \partial t}+\sum_{j=1}^{3} \langle{v_j^p}\rangle {\partial  \over \partial x_j}\right) \langle{v_i^p}\rangle}^{\dot{\langle{v}\rangle}_i^p} &
\underbrace{=}_{EoM} 
\overbrace{-\partial \Phi(t,\mathbf{x})\over \partial x_i}^{g_i\sim O(-GM/R^2)} ~~
\underbrace{-}^\text{pressure}_\text{balance}~~\sum_{j=1}^{3} {\partial \over \rho^p \partial x_j}
\overbrace{[\underbrace{\rho^p(t,\mathbf{x})}_{\int_\infty\!\!\!\!m_p f_p d^3\mathbf{v}} \underbrace{\sigma_{ji}^p(t,\mathbf{x})}_{O(c_s^2)}]}^{\int\limits_\infty\!\! d\mathbf{v}^3  (\mathbf{v}_j-\langle{v}\rangle^p_j) (\mathbf{v}_i-\langle{v}\rangle^p_i)m_pf_p }
- {\underbrace{\langle{v_i^p}\rangle \overbrace{[\dot{m}_p/m_p]}^{1/t|^\text{fric}_{\text{visc}~m_p=M_\text{gas}}}}_\text{snow.plough}}, \\
0& =  -{\partial \Phi(t,\mathbf{x})\over \partial x_i} -{\partial (n \sigma^2 ) \over n \partial x_i}, ~~\text{hydrostatic isotropic velocity, no flow and friction }.\end{align}
</math>

The general version of Jeans equation, involving (3 x 3) velocity moments is cumbersome.  
It only becomes useful or solvable if we could drop some of these moments, especially drop the off-diagonal cross terms for systems of high symmetry, and also drop net rotation or net inflow speed everywhere.

The isotropic version is also called 
[[Hydrostatic equilibrium]] equation where balancing pressure gradient with gravity; the isotropic version works for axisymmetric disks as well, after replacing the derivative dr with vertical coordinate dz.  It means that we could measure the gravity (of dark matter) by observing the gradients of the velocity dispersion and the number density of stars.