Epicyclic frequency

Remarks

Meaning: a planar orbit with angular momentum $L = R_g^2 \Omega(R_g)$ moves in the effective potential $\Phi(R, 0) + L^2 / (2 R^2)$. Its minimum is at the guiding radius $R_g$, and its second derivative there is $\kappa^2(R_g)$. So small radial deviations $x = R - R_g$ obey $\ddot x = -\kappa^2 x$ and oscillate with frequency $\kappa$.

Example: for a point mass, $\Omega^2 = GM/R^3$, so $R \, d\Omega^2/dR = -3 \Omega^2$ and $\kappa = \Omega$. For a flat rotation curve, $\Omega^2 = v_0^2 / R^2$, so $\kappa = \sqrt{2} \, \Omega$. For a rigid rotation, $\Omega$ is constant and $\kappa = 2 \Omega$.

Non-example: if $\Omega^2$ decreases faster than $R^{-4}$, then $\kappa^2 < 0$; circular orbits are then unstable to radial perturbations and no real epicyclic frequency exists. That instability criterion, $\kappa^2 < 0$, is the Rayleigh criterion for rotating flows.