= Solution
Here $a_p$ is the planet's <semi-major axis>, while $a,e,I$ are the particle's <semi-major axis>, <orbital eccentricity>, and <orbital inclination> relative to the planet's plane. The formula assumes the <circular restricted three-body problem>: the planet-to-star mass ratio is small, the particle has negligible mass, and its motion is approximately heliocentric and Keplerian away from brief encounters. The planet's own orbit is circular.
The <Jacobi constant> is exactly conserved in that ideal rotating problem. Expressing it in heliocentric <orbital elements> away from the planet gives the approximately conserved <Tisserand parameter>
$$
T_p=\frac{a_p}{a}+2\sqrt{\frac a{a_p}(1-e^2)}\cos I.
$$
The first term measures the particle's normalized binding energy. The second is twice the component of its <specific angular momentum> normal to the planet's plane, normalized by $\sqrt{GM_*a_p}$.
Solved by gpt-5.6-sol high.
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