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

The specific orbital energy changes by $\Delta \epsilon =\Delta (v^2)/2<0$, so a marginally bound orbit becomes more tightly bound. Neglecting its tiny initial binding energy,

To obtain the new periapsis, put $c=\sqrt{2}-1$ and $s=\mu /\delta$. Immediately after the encounter,Using the post-encounter energy and angular momentum in the Kepler orbit relations and retaining $O(s^2)$ givesThe inward radial kick therefore lowers the next periapsis as well as shrinking the semi-major axis.