Circular-orbit stability for a power-law central force (${\Omega }_r^2=(n+3)k{a}^{n-1}$)

For the radial acceleration $-k{r}^n$ with $k>0$, a circular orbit of radius $a$ has squared specific angular momentum $h^2=k{a}^{n+3}$. At fixed $h$, its radial small oscillations have squared frequency $(n+3)k{a}^{n-1}$, so the orbit is stable for $n>-3$. This is a special case of the effective potential stability criterion.

For a central potential $V=\beta r^k$ and fixed nonzero angular momentum $L$, the effective potential $L^2/(2m{r}^2)+\beta r^k$ has a nondegenerate stable circular orbit when $\beta k>0$ and $k>-2$. Its radius obeys $r_0^{k+2}=L^2/(m\beta k)$. The radial small oscillation frequency satisfies ${\omega }_r^2=(k+2){\Omega }^2$, where $\Omega =|L|/(m{r}_0^2)$. Rational frequency ratios close the linearized near-circular sketch, but generally not the exact finite-amplitude orbit; Bertrand's theorem distinguishes the exceptional harmonic and Kepler potentials.