Radial oscillation of an elastic-string orbit (${\Omega }^2=(k/m)(4-3L/r_0)$)

For a taut elastic string with potential $k(r-L{)}^2/2$, a circular orbit has ${\omega }_0^2=k(r_0-L)/(m{r}_0)$. A radial impulse preserves angular momentum, so the curvature of the fixed-angular-momentum effective potential gives ${\Omega }^2=k/m+3{\omega }_0^2$. The ratio is three at $r_0=6L/5$ and approaches two at $r_0\gg L$. Rational linearized frequency ratios describe leading small-amplitude closed sketches without proving exact closure at finite amplitude.