Editing Stellar dynamics (section)

=== Oscillation frequencies in a thick disk ===

To find the vertical and radial oscillation frequencies, we do a Taylor expansion of potential around the midplane.
<math display="block"> \Phi (R_1, z) \approx \Phi(R,0) + {\omega^2 R} (R_1-R) + {\kappa^2 \over 2} (R_1-R)^2 + {\nu^2 \over 2} z^2 </math>
and we find the circular speed <math> V_\text{cir}</math> and the vertical and radial epicycle frequencies to be given by
<math display="block"> (R \omega)^2 \equiv V^2_\text{cir}  = \left[{(1+R_0/z_0) G M_0\over \sqrt{R^2+(R_0+z_0)^2} } - {(R_0/z_0) G M_0 \over \sqrt{R^2+R_0^2}} \right], </math>
<math display="block"> \nu^2 = {G M_0 (R_0/z_0 + 1) \over (R^2+(R_0+z_0)^2)^{3/2}}, </math>
<math display="block"> \kappa^2 + \nu^2 - 2 \omega^2 = 4 \pi G \rho(R,0) = {G M_0 R_0/z_0 \over (R^2+R_0^2)^{3/2}}. </math>
Interestingly the rotation curve <math> V_\text{cir} </math> is solid-body-like near the centre <math> R \ll R_0 </math>, and is Keplerian far away.

At large radii three frequencies satisfy 
<math display="inline"> \left.\left[\omega, \nu, \kappa, \sqrt{4 \pi G \rho}\right]\right|_{R\to \infty} \to [1,1+R_0/z_0,1, R_0/z_0]^{1\over 2} \sqrt{G M_0\over R^3} </math>.
E.g., in the case that <math> R \to \infty </math> and <math> R_0 / z_0 = 3</math>, the oscillations <math> \omega: \nu: \kappa = 1: 2 : 1 </math> forms a resonance.

In the case that <math> R_0 =0 </math>, the density is zero everywhere except uniform needle between <math>|z| \le z_0 </math> along the z-axis.

If we further require <math> z_0=0</math>, then we recover a well-known property for closed ellipse orbits in point mass potential,
<math display="block"> \omega: \nu: \kappa = 1: 1 : 1 .</math>