Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 3 / a / Solution

For a circular orbit of cylindrical radius $r$ in an axisymmetric gravitational potential,A small vertical displacement satisfiesso the vertical epicyclic frequency isFor a spherically symmetric potential $\Phi (R)$, where $R=(r^2+z^2{)}^{1/2}$,Hence $\boxed{{\Omega }_z=\Omega }$. Geometrically, a slightly tilted circular orbit remains a circular orbit in a different plane, and its height completes one oscillation per revolution.