The range of validity of the two-body approximation in models of terrestrial planet accumulation: I. Gravitational perturbations

Abstract

Gravitational perturbations in semimajor axis, eccentricity, and inclination resulting from close planetesimal encounters (near 1 AU) out to 10 Tisserand sphere of influence radii were calculated by two- and three-dimensional numerical integration. These are compared with the results of treating the encounter as a two-body problem, as is customary in Monte Carlo calculations of orbital evolution and in numerical and analytical studies of planetary accumulation. It is found that for values of ( V V e ) ⪅ 0.35 (V = relative velocity, V e = escape velocity of largest body) , the two-body body approximation fails to describe the outcome of individual encounters. In this low-velocity region, the two-body “gravitational focusing” cross section is no longer valid; “anomalous gravitational focusing” often leads to bodies on distant unperturbed trajectories becoming close encounters and vice versa. In spite of these differences, average perturbations given by the two-body approximation are valid within a factor of 2 when V V e > 0.07 . In this same velocity range the “Arnold extrapolation,” whereby a few very close encounters are used to estimate the effect of many more distant encounters, is found to be a useful approximation.