Editing Stellar dynamics (section)

=== Example on Jeans theorem and CBE on Uniform Sphere Potential ===

Generally for a time-independent system, Jeans theorem predicts that <math> f(\mathbf{x},\mathbf{v}) </math> is an implicit function of the position and velocity through a functional dependence on "constants of motion".

For the uniform sphere, a solution for the Boltzmann Equation, written in spherical coordinates <math> (r,\theta,\phi) </math> and its velocity components <math> (V_r,V_\theta,V_\phi) </math> is
<math display="block"> f(r,\theta,\varphi,V_r,V_\theta,V_\varphi) = {C_0 \over V_0^3} \sqrt{V_0^2 \over 2Q}, </math>
where <math>C_0 = 2\pi^{-2} \rho_0 </math> is a normalisation constant, which has the dimension of (mass) density. And we define a (positive enthalpy-like dimension <math> \text{km}^2/\text{s}^2 </math>) Quantity 
<math display="block"> Q[\mathbf{x},\mathbf{v}] \equiv \left[0, \left(-V_0^2 - E \right) + {J^2 \over 2 r_0^2} \right]_\max \left[{J_z \over |J_z|}, 0\right]_\max .</math>
Clearly anti-clockwise rotating stars with <math> J_z \le 0,~~ Q=0</math> are excluded.

It is easy to see in spherical coordinates that

<math> J^2 = r^2 V_t^2 = r^2 (V_\theta^2+V_\varphi^2),  </math>

<math> J_z = V_\varphi r \sin\theta, </math>

<math> E = {V_r^2+V_t^2 \over 2} + \Phi(r), ~ V_t \equiv \sqrt{V_\theta^2+V_\varphi^2}</math>

Insert the potential and these definitions of the orbital energy E and angular momentum J and its z-component Jz along every stellar orbit, we have
<math display="block"> 2Q= \text{Heaviside}\left({V_\varphi \over |V_\varphi|}\right) \times  \left[ V_0^2 \left(1-{r^2 \over r_0^2}\right) - V_r^2  - \left(1 - {r^2 \over r_0^2}\right) {\left(V_\theta^2+V_\varphi^2\right)}, 0 \right]_\max, </math> which implies <math> |V_r| \le V_e(r)</math>, and <math> |V_\theta|, V_\varphi </math> between zero and <math> V_0 </math>.

To verify the above <math> E, ~J_z</math> being constants of motion in our spherical potential, we note
<math display="block"> dE/dt = {\partial E\over \partial t} + \mathbf{v} {\partial E \over \partial \mathbf{x}} + (\mathbf{-\nabla \Phi}) {\partial E \over \partial \mathbf{v}} </math>

<math display="block"> dE/dt 
= {\partial \Phi\over \partial t} + \mathbf{v} {\partial \Phi \over \partial \mathbf{x}} + (\mathbf{-\nabla \Phi}) \mathbf{v} = {\partial \Phi\over \partial t} =0 </math> for any "steady state" potential.

<math display="block"> dJ_z/dt
= {\partial J_z\over \partial t} + {\partial J_z \over \partial \mathbf{x}} \cdot \mathbf{v} - (\mathbf{\nabla \Phi}) \cdot {\partial J_z \over \partial \mathbf{v}}, </math> which reduces to <math> dJ_z/dt = 0 + [(V_y)V_x + (-V_x)V_y] - \left[(-y) {x \over R}{\partial \Phi(R,z) \over \partial R} + (x) {y\over R}{\partial \Phi(R,z) \over \partial R}\right]
= 0 </math> around the z-axis of any axisymmetric potential, where <math display="inline">R=\sqrt{x^2+y^2} </math>.

Likewise the x and y components of the angular momentum are also conserved for a spherical potential.  Hence <math> dJ/dt =0 </math>.

So for any time-independent spherical potential (including our uniform sphere model), 
the orbital energy E and angular momentum J and its z-component Jz along every stellar orbit satisfy
<math display="block"> dE[\mathbf{x},\mathbf{v}]/dt = dJ[\mathbf{x},\mathbf{v}]/dt= dJ_z[\mathbf{x},\mathbf{v}]/dt =0 .</math>

Hence using the chain rule, we have
<math display="block"> {d \over dt} Q(E[\mathbf{x},\mathbf{v}],J[\mathbf{x},\mathbf{v}],J_z[\mathbf{x},\mathbf{v}]) = {\partial Q \over \partial E} {dE \over dt} + {\partial Q \over \partial J_z} {dJ_z \over dt} + {\partial Q \over \partial J} {dJ \over dt} = 0 ,</math>
i.e., <math display="inline"> {d \over dt} f= f'(Q) {d Q[\mathbf{x},\mathbf{v}]\over dt}  =0 </math>, so that CBE is satisfied, i.e., our 
<math display="block"> f(\mathbf{x},\mathbf{v}) = f(E[\mathbf{x},\mathbf{v}],J[\mathbf{x},\mathbf{v}],J_z[\mathbf{x},\mathbf{v}]) </math>
is a solution to the Collisionless Boltzmann Equation for our static spherical potential.