Radial stability of an axisymmetric circular orbit
= Radial stability of an axisymmetric circular orbit
{title2=$\kappa^2=r^{-3}d(j_c^2)/dr$}
At fixed <specific angular momentum> $j$, perturb the <effective potential> $\Phi(r)+j^2/(2r^2)$ about a <circular orbit>. Linear radial motion has $\ddot x+\kappa^2x=0$, where $\kappa^2=\Phi^{\prime\prime}+3\Phi^\prime/r=r^{-3}d(j_c^2)/dr$. Positive $\kappa^2$ gives stable oscillations; axial symmetry alone does not force that sign.