Monte-Carlo codes generally treat planetesimal-planet encounters using the
two-body scattering approximation, which becomes inaccurate for low velocity
encounters (i.e., near the planets escape velocity). However, Monte-Carlo
codes using the two-body approximation frequently produce results consistent
with more accurate codes using numerical integration (Wetherill and Cox
1985a,b). To better understand why this breakdown occurs, and to test a
hypothesis from Greenberg et al. (1988) which 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 orbit was shifted by distant planetary perturbations, were 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. Thus, our results verify that Monte-Carlo models can yield statistically accurate results, even if individual particles do not behave as assumed in those code.