Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 320 / 2 / c / Solution

During half a radial oscillation,Useand the two root productsThe stated elementary integrals then give the azimuthal advance over one radial period:Consequently the frequency ratio of the spherical isochrone model isor equivalently $T_{\varphi }/T_r=2/(1+L/\sqrt{L^2+4GMb})$.

Deep in the constant-density core, $L^2\ll 4GMb$ and ${\Omega }_{\varphi }/{\Omega }_r\to 1/2$, as for an isotropic harmonic oscillator: the radius completes two oscillations per revolution. Far outside the core, $L^2\gg 4GMb$ for the corresponding circular scale and the ratio tends to one, recovering the closed Kepler orbit. Intermediate orbits undergo apsidal precession because the ratio is generally not rational.