Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 316 / 1 / iv / Solution

For the nearly edge-on orbit $I=\pi /2-I^{\prime }$ with $I^{\prime }\ll 1$, the small-angle approximation gives $\cos I=\sin I^{\prime }=I^{\prime }+O(I^{\prime 3})$. HenceAlso,Thus, on a branch where $\cos f>0$ and with separation measured in units of $a$,as stated. The absolute value is required when $R_{\mathrm{sky}}$ denotes the nonnegative separation over the whole orbit.