= Solution
At a planet-crossing encounter, let $U$ be the planetocentric relative speed far from the planet and $v_p=\sqrt{GM_*/a_p}$. Combining the particle's heliocentric energy and normal angular momentum gives
$$
\boxed{\frac{U^2}{v_p^2}=3-T_p}.
$$
Thus 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 gives
$$
\tan\frac\delta2=\frac{GM_p}{bU^2},
\qquad
|\Delta\mathbf U|=2U\sin\frac\delta2.
$$
An 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.
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