ICARUS 125, 288–301 (1997) ARTICLE NO. IS965612 Mapping the Effects of Distant Perturbations on Particle–Planet Interactions WILLIAM F. BOTTKE,JR. Division of Geological & Planetary Sciences, California Institute of Technology, Mail Code 170-25, Pasadena, California 91125 E-mail: bottke@kepler.gps.caltech.edu RICHARD GREENBERG Lunar and Planetary Laboratory, University of Arizona, Tucson, Arizona AND ANDREA CARUSI AND GIOVANNI B. VALSECCHI IAS Reparto Planetologia, V. Universita 11, 00185 Rome, Italy Received December 14, 1995; revised May 30, 1996 Carlo models can yield statistically accurate results, even if individual particles do not behave as assumed in those Monte-Carlo codes generally treat planestesimal–planet en- codes.  1997 Academic Press counters using the two-body scattering approximation, which can be inaccurate when relative velocities are low; however, Monte-Carlo codes using the two-body approximation fre- I. INTRODUCTION quently produce results consistent with more accurate codes using numerical integration. To better understand why this Modeling the dynamical evolution of asteroids (or of breakdown occurs at low velocities, and to test a hypothesis planetesimals in the early solar system) with numerical from Greenberg et al. (1988, Icarus 75, 1–29) that may explain the unexpected accuracy of Monte-Carlo codes, we numerically integration techniques is often difficult, since most integra- integrate test body trajectories using a unique set of orbital tion codes are slow for the required long time scales and elements defined by the geometry of the two-body approxima- orbits can be modified by many factors: planetary (and tion. This new coordinate system is ideal for examining the asteroid) encounters, distant planetary perturbations in the effects of distant planetary perturbations on particle trajectories form of mean-motion and secular resonances, collisions, all the way to encounter with the planet. and nongravitational forces. An added complication is that Our results show that the failure of the two-body approxi- these forces work over a range of time scales; planetary mation is caused by distant planetary perturbations modifying encounters and collision effects are nearly instantaneous the approach geometry of the test bodies; behavior at encoun- when compared with the long time scales needed for reso- ter follows two-body scattering even at very low relative nances and nongravitational force effects. Even if these velocities. By testing particle swarms encountering a planet, effects could all be treated in a simple fashion, their inclu- we found that some test bodies, whose approach orbits were shifted by distant planetary perturbations, were then replaced sion into numerical integration codes would make most by similarly shifted nearby test bodies. The ‘‘particle replace- long-term dynamical modeling computationally prohib- ment’’ mechanism explains why Monte-Carlo codes frequently itive. yield outcome results comparable to numerical integration To avoid the problems associated with numerical inte- results. Moreover, we found that the relative velocity of a gration, several groups (Arnold 1965, Wetherill 1985, 1988) test body at encounter is not the critical parameter in de- have employed Monte-Carlo methods to treat these forces termining the ‘‘breakdown’’ of two-body scattering outcome in a more statistical way. These codes cut computation time statistics; instead, we found that the semimajor axis of the by assuming that asteroids always follow simple Keplerian test body relative to the size of the planet’s Hill sphere (or orbits and that planetary encounters dominate changes in the synodic period of the test body when mass is included) is much more diagnostic. Thus, our results verify that Monte- heliocentric motion. Thus, these codes are applicable only 288 0019-1035/97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.