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The body now returns on a bound, highly eccentric orbit whose periapsis crosses more deeply into the planet's neighborhood. Repeated planetary scattering produces a sequence of energy and angular-momentum kicks rather than smooth secular evolution. Possible endpoints include ejection onto an unbound orbit, collision with the planet or star, tidal disruption, or diffusion onto a detached orbit that no longer encounters the planet. Temporary protection by a mean-motion resonance is also possible. The approximate Tisserand parameter constrains weak separated encounters, but the close-encounter sequence is chaotic and does not select a unique final orbit.

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 parameterThe 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 $\sqrt{G{M}_{\ast }{a}_p}$.

For $T_p=5/2$ and $I=0$, the fixed-Tisserand parameter curve begins atIt falls smoothly toand then rises asymptotically back toward $e=1$ as $a/a_p\to \infty$. Since $T_p<3$, this coplanar locus never reaches a circular Kepler orbit.

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.