Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 320 / 1 / a / Solution

For a bound Kepler orbit about mass $M$, the specific orbital energy and specific angular momentum areThe apsidal radii areequivalently $a=(r_{-}+r_{+})/2$ and $e=(r_{+}-r_{-})/(r_{+}+r_{-})$. Combining the formulas also gives