Mapping the effects of distant perturbations on particle-planet interactions

Monte-Carlo codes generally treat planestesimal-planet encounters using the two-body scattering approximation, which can be inaccurate when relative velocities are low; however, Monte-Carlo codes using the two-body approximation frequently produce results consistent with more accurate codes using numerical integration. To better understand why this breakdown occurs at low velocities, and to test a hypothesis from Greenberg et al. (1988, Icarus 75, 1-29) that may explain the unexpected accuracy of Monte-Carlo codes, we numerically integrate test body trajectories using a unique set of orbital elements defined by the geometry of the two-body approximation. This new coordinate system is ideal for examining the effects of distant planetary perturbations on particle trajectories all the way to encounter with the planet. Our results show that the failure of the two-body approximation is caused by distant planetary perturbations modifying the approach geometry of the test bodies; behavior at encounter follows two-body scattering even at very low relative velocities. By testing particle swarms encountering a planet, we found that some test bodies, whose approach orbits were shifted by distant planetary perturbations, were then replaced by similarly shifted nearby test bodies. The "particle replacement" mechanism explains why Monte-Carlo codes frequently yield outcome results comparable to numerical integration results. Moreover, we found that the relative velocity of a test body at encounter is not the critical parameter in determining the "breakdown" of two-body scattering outcome statistics; instead, we found that the semimajor axis of the test body relative to the size of the planet's Hill sphere (or the synodic period of the test body when mass is included) is much more diagnostic. Thus, our results verify that Monte-Carlo models can yield statistically accurate results, even if individual particles do not behave as assumed in those codes.