= Solution
A <circular orbit> of radius $r_0>0$ is a stationary point of the <effective potential>:
$$
0=-\frac{L^2}{mr_0^3}+\beta k r_0^{k-1}.
$$
For $L\ne0$, $\beta k>0$ and $k\ne-2$, this gives
$$
\boxed{r_0=\left(\frac{L^2}{m\beta k}\right)^{1/(k+2)}.}
$$
The condition $\beta k>0$ makes the radial <force> attractive. At $k=-2$, the equation instead requires the fine tuning $L^2/m=-2\beta$; then the centrifugal and potential terms cancel at every radius and there is no isolated restoring minimum. If that equality fails, there is no positive-radius <circular orbit> for this special power.
Back to article page
