= Radial frequency in a power-law central potential
{title2=$\omega_r/|\omega_\theta|=\sqrt{k+2}$}
For <central potential> $V(r)=\beta r^k$ and nonzero <angular momentum> $J$, a <circular orbit> satisfies $\beta k r_0^{k+2}=J^2/m$, so attraction requires $\beta k>0$. Twice differentiating the radial <effective potential> $V+J^2/(2mr^2)$ gives $V_{\mathrm{eff}}''(r_0)=(k+2)J^2/(mr_0^4)$. Its <harmonic approximation> therefore has radial <angular frequency> $\omega_r=\sqrt{k+2}|J|/(mr_0^2)$, whereas $|\omega_\theta|=|J|/(mr_0^2)$. This proves the ratio and shows stability precisely when $k>-2$. At $k=-2$ the tuned <effective potential> is flat; at $k<-2$ the radial disturbance grows exponentially. Rationality of this local <frequency> ratio is only a necessary condition for all nearby <orbits> to close; <Bertrand's theorem> makes a stronger finite-amplitude assertion.
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