Radial stability of an axisymmetric circular orbit (${\kappa }^2=r^{-3}d(j_c^2)/dr$)

At fixed specific angular momentum $j$, perturb the effective potential $\Phi (r)+j^2/(2r^2)$ about a circular orbit. Linear radial motion has $\ddot{x}+{\kappa }^{2}x=0$, 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.