 Solution

At a planet-crossing encounter, let $U$ be the planetocentric relative speed far from the planet and $v_p=\sqrt{G{M}_{\ast }/a_p}$. Combining the particle's heliocentric energy and normal angular momentum givesThus the Tisserand parameter fixes the encounter-speed scale and confines every post-encounter orbit to the same allowed region. For impact parameter $b$, a two-body estimate givesAn encounter with smaller $U$ is bent through a larger angle, whereas one with larger $U$ is less strongly focused. The actual displacement on the $(a,e)$ plot therefore depends on encounter geometry as well as $T_p$; very small $U$ limits the velocity vector available to redirect, and very large $U$ gives weak deflection, with the largest typical kicks between those limits.

Solved by gpt-5.6-sol high.