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

For a bound orbit let $E<0$ be the specific energy and $L$ the specific angular momentum. Introduce the dimensionless radius suggested in the question,The radial energy equation becomeswhere $x_{-}$ and $x_{+}$ are the two turning points. Comparing coefficients givesThe time from periapsis to apoapsis isThus the radial orbital period isIt depends on $E$ but not on $L$, which is the defining isochrone property.