Editing Stellar dynamics (section)

=== A worked example of Jeans Equation in a uniform sphere ===

Jeans Equation is a relation on how the pressure gradient of a system should be balancing the potential gradient for an equilibrium galaxy.  In our uniform sphere, the potential gradient or gravity is
<math display="block"> \nabla \Phi = {d \Phi \over dr} = {\Omega^2 r} \ge 0, ~~\Omega = {V_0 \over r_0}.  </math>

The radial pressure gradient
<math display="block"> -{d (\rho \sigma_r^2) \over \rho dr} = -{d \sigma_r^2 \over dr} - {\sigma_r^2 \over r} {d \log \rho \over d\log r} = {\Omega^2 r \over 2} + 0 \ge 0.  </math>

The reason for the discrepancy is partly due to centrifugal force
<math display="block"> {\bar{V}_\varphi^2 \over r} =  {(0.8488V_0)^2 \over r} > 0, </math>
and partly due to anisotropic pressure
<math display="block"> \begin{align}
{(\sigma_\theta^2 -\sigma_r^2) \over r} &=0.25 \Omega^2 r \ge 0\\
{(\sigma_\varphi^2 -\sigma_r^2) \over r} & = 0.25\Omega^2 r - {0.1801 V_0^2 \over r} = \pm , \end{align}</math>
so <math> 0.2643V_0 = \sigma_\varphi < \sigma_r =0.5V_0</math> at the very centre, 
but the two balance at radius <math> r=0.8488r_0</math>, and then 
reverse to <math> 0.2643 V_0 = \sigma_\varphi > \sigma_r =0 </math> at the very edge.

Now we can verify that
<math display="block">\begin{align}
{\partial \langle V_r \rangle \over \partial t} & = (-\sum_{i=x,y,z} V_i \partial_i \langle V_r\rangle ) - \cancel{ \langle V_r\rangle \over t_\text{fric} } - \nabla_r \Phi + \sum_{i=x,y,z}{-\partial_i (n \sigma^2_{ir}) \over n}
\\
&={\bar{V}_\theta^2 +\bar{V}_\varphi^2 -2\bar{V}_r^2 \over r} - 0 
-{\partial \Phi \over \partial r} 
+ \left[-{d (\rho \sigma_r^2) \over \rho dr} + {\sigma_\theta^2 + \sigma_\varphi^2 -2\sigma_r^2 \over r} \right]  \\
& =  {0 + (0.4244V_0)^2 - 2 \times 0 \over r} -(\Omega^2 r) + \\
& \left[ {\Omega^2 r \over 2} + 
{(0.5V_0)^2 + (0.2643V_0)^2- 2 \times 0.25 \Omega^2 (r_0^2-r^2) \over r} \right] \\
& = 0.
\end{align} </math>
Here the 1st line above is essentially the Jeans equation in the r-direction, which reduces to the 2nd line, the Jeans equation in an anisotropic (aka <math> \beta \ne 0 </math>) rotational (aka <math>\langle V_\varphi\rangle  \ne 0</math>) axisymmetric (<math> \partial_\varphi \Phi(\mathbf{x},t) =0</math> ) sphere (aka <math> \partial_\theta n(\mathbf{x},t) =0</math>) after much coordinate manipulations of the dispersion tensor; similar equation of motion can be obtained for the two tangential direction, e.g., <math> {\partial \langle V_\varphi\rangle \over \partial t} </math>, which are useful in modelling   ocean currents on the rotating earth surface or angular momentum transfer in accretion disks, where the frictional term <math> -{ \langle V_\varphi\rangle \over t_\text{fric} } </math> is important. 
The fact that the l.h.s. <math> {\partial V_r \over \partial t} =0 </math> means that 
the force is balanced on the r.h.s. for this uniform (aka <math> \nabla_\mathbf{x} m n(\mathbf{x},t) =0</math>) spherical model of a galaxy (cluster) to stay in a steady state (aka time-independent equilibrium <math> {\partial n(\mathbf{x},t) \over \partial t} = 0 </math> everywhere) statically (aka with zero flow <math> \langle \mathbf{V}(\mathbf{x},t) \rangle =0</math> everywhere).  Note systems like accretion disk can have a steady net radial inflow <math> \langle \mathbf{V}(\mathbf{x}) \rangle < 0</math> everywhere at all time.