Editing Stellar dynamics (section)

=== A Recap on Harmonic Motions in Uniform Sphere Potential ===

Consider building a steady state model of the fore-mentioned uniform sphere of density <math> \rho_0 </math> and potential <math> \Phi(r)</math>
<math display="block"> \begin{align}
\rho(|\mathbf{r}|) &=\rho_0 \equiv M_\odot n_0, ~~ |\mathbf{r}|^2=x^2+y^2+z^2 \le r_0^2, ~~ \Omega\equiv \sqrt{4 \pi G \rho_0 \over 3} \equiv {V_0 \over r_0} \\
\Phi(|\mathbf{r}|) &= {\Omega^2 (x^2+y^2+z^2) -3 V_0^2 \over 2}= {V_e(r)^2 \over 2} - \Phi(r_0), 
\end{align}</math>
where <math> V_e(r) = V_0 \sqrt{1-{r^2 \over r_0^2}} =\sqrt{2\Phi(r_0)-2\Phi(r)}</math> is the speed to escape to the edge <math> r_0</math>.

First a recap on motion "inside" the uniform sphere potential.
Inside this constant density core region, individual stars go on resonant harmonic oscillations of angular frequency <math> \Omega </math> with <math display="block">  \begin{align}
\ddot{x} = & - \Omega^2 x =-\partial_x \Phi,  \\
\ddot{y} = & - \Omega^2 y, ~~~  {\dot{y}(t)^2 \over 2}+{\Omega^2 y(t)^2 \over 2} \equiv I_y(y,\dot{y}) ={\dot{y}(0)^2 \over 2}+ {\Omega^2 y(0)^2 \over 2} \le {(\Omega r_0)^2 \over 2} \\
\ddot{z} = & - \Omega^2 z, \rightarrow \dot{z}(t)=  \dot{z}(0) \cos (\Omega t) + \Omega z(0) \sin (\Omega t).
\end{align}</math>
Loosely speaking our goal is to put stars on a weighted distribution of orbits with various energies <math> f\left(I_x(x,\dot{x}), I_y(y,\dot{y}), I_z(z,\dot{z}\right) = DF(\mathbf{r},\mathbf{V})</math>, i.e., the phase space density or distribution function, such that their overall stellar number density reproduces the constant core, hence their collective "steady-state" potential.  Once this is reached, we call the system is a self-consistent equilibrium.