Editing Stellar dynamics (section)

=== A unified thick disk potential ===

Consider an oblate potential in cylindrical coordinates 
<math display="block">\begin{align} \Phi(R,z) &  ={G M_0 \over 2z_0} 
 \left[2\sinh^{-1}\!\! Q - \sinh^{-1} \!\!Q_{+} - \sinh^{-1} \!\! Q_{-}\right] \\ 
&={G M_0 \over 2 z_0} \log { (\sqrt{1+  Q^2} + Q )^2 \over 
\left[\sqrt{1+ Q_{+}^2}+ Q_{+}\right] \left[\sqrt{1+Q_{-}^2} + Q_{-} \right]},\\
Q_{\pm} & \equiv {R_0 + \left|~ |z| \pm z_0~  \right| \over R}, \\
Q & \equiv {R_0 + [0, |z| - z_0 ]_\max \over R}, \\
\end{align} </math>
where <math> z_0, R_0</math> are (positive) vertical and radial length scales.  
Despite its complexity, we can easily see some limiting properties of the model.

First we can see the total mass of the system is <math>M_0 </math> because  
<math display="block"> \Phi(R,z) 
\rightarrow {G M_0 \over 2z_0} (2 Q_{-} - Q_{-} -Q_{+}) = -{G M_0 \over R} , 
</math>
when we take the large radii limit 
<math> R \rightarrow \infty, ~|z| \ge z_0, </math>, so that <math> Q = Q_{-}=Q_{+}-{2z_0 \over R} = {|z| + (R_0 - z_0)  \over R} \rightarrow 0.</math>

We can also show that some special cases of this unified potential become the potential of the Kuzmin razor-thin disk, that of the Point mass <math> M_0 </math>, and that of a uniform-Needle mass distribution: 
<math display="block"> \Phi_{KM}(R,z) = -{G M_0 \over \sqrt{ R^2 + (|z|+R_0)^2}}, ~~ z_0=0, </math>
<math display="block"> \Phi_{PT}(R,z) = -{G M_0 \over \sqrt{R^2+z^2}} , ~~ z_0=R_0=0, </math>
<math display="block"> \Phi_{UN}^{R_0=0}(R, z) = {G M_0 \over 2z_0} \left[2\sinh^{-1}\!\! {(0, |z| - z_0 )_\max \over R} - \sinh^{-1} \!\!{z_0 + |z| \over R} - \sinh^{-1} \!\!{\left|~z_0 - |z|~\right| \over R}\right]. </math>