Editing Stellar dynamics (section)

=== An example of the Poisson Equation and escape speed in a uniform sphere ===

Consider an analytically smooth spherical potential
<math display="block"> \begin{align}
\Phi(r) & \equiv \left(-V_0^2\right) + \left[{r^2 -r_0^2 \over 2r_0^2}, ~~ 1 -{r_0 \over r} \right]_{\max} \!\!\!\! V_0^2  \equiv \Phi(r_0)-{V_e^2(r) \over 2},  ~~\Phi(r_0) = - V_0^2 , \\
\mathbf{g} &= -\mathbf{\nabla} \Phi(r) = -\Omega^2 r H(r_0 - r) - { G M_0 \over r^2}H(r-r_0), ~~\Omega={V_0 \over r_0}, ~~M_0 = {V_0^2 r_0 \over G},\end{align}</math>
where <math> V_e(r) </math> takes the meaning of the speed to "escape to the edge" <math> r_0</math>, and <math>\sqrt{2}V_0 </math> is the speed to "escape from the edge to infinity".  The gravity is like the restoring force of harmonic oscillator inside the sphere, and Keplerian outside as described by the Heaviside functions.

We can fix the normalisation <math> V_0 </math> by computing the corresponding density using the spherical Poisson Equation
<math display="block"> G\rho = {d \over 4 \pi r^2 dr} {r^2 d\Phi \over dr} = { d (G M) \over 4 \pi r^2 dr} = {3 V_0^2 \over 4 \pi r_0^2}H(r_0-r), </math>
where the enclosed mass 
<math display="block"> M(r) = {r^2 d\Phi \over G dr} = \int_0^{r} dr \int_0^{\pi} (r d\theta) \int_0^{2 \pi} (r \sin\theta d\varphi) \rho_0 H(r_0-r) = \left. M_0 x^3\right|_{x={r \over r_0}}.</math>

Hence the potential model corresponds to a uniform sphere of radius <math> r_0 </math>, total mass <math> M_0 </math> with
<math display="block"> {V_0 \over r_0} \equiv \sqrt{4\pi G \rho_0 \over 3} = \sqrt{G M_0 \over r_0^3}. </math>