Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 316 / 4 / ii / Solution

In the frame rotating with the planet, put the planet at fixed radius $a_p$. At periapsis the planetesimal moves prograde faster than the planet and passes just outside it. Its relative longitude initially increases, reaches a turning point near $r=2^{1/4}{a}_p$ at a lead of about $10^{\circ }$, and then decreases because its angular speed has fallen below the frame rate. Meanwhile its radius continues to grow along the outgoing parabolic branch. A sketch should therefore show an exterior cusp-like closest passage, a short prograde sweep to maximum lead, and a long outward trail that bends retrograde in the rotating frame.