Outcomes of planetary close encounters: A systematic comparison of methodologies

Motion during a planetary close encounter can be described reasonably well by a two-body approximation, in which the effect of the Sun is temporarily neglected. The approximation is suspect for very slow encounters, such as occur in the quiescent swarm of planetesimals that encountered early planetary embryos. Öpik's formulation of the two-body approximation fails under those conditions, predicting motion different from the correct three-body behavior. The differences are generally due to small distant perturbations of heliocentric orbits during approach toward encounter, rather than due to failure of the two-body model at close encounter. This conclusion comes from intercomparison of several ways of computing and estimating the outcomes of encounters. Comparison and physical interpretation is facilitated by a strategy of studying suites of trajectories in a revealing graphical display. Reinterpretation of past Monte Carlo studies also shows that the two-body approximation can be valid even where the Öpik method is not. The approximation may apply to the quiescent planetesimal swarm if the physical effects that cause failure of the Öpik method are accounted for. The prevalent effects of distant perturbations often yield horseshoe orbits, which may substantially reduce accretion from a wide band within the feeding zone of a planetary embryo. Use of the two-body approximation in swarms will require that encounters offset by distant perturbations be statistically replaced and compensated by other perturbed trajectories. In some cases the two-body model will need to account for partial rotation of relative velocity vectors, rather than full asymptote-to-asymptote rotation. Further studies are needed to find the real limits of the two-body model. Special attention will need to be focused on close encounters of tangential and not-quite-crossing orbits, which are moderately probable and potentially deviate from two-body behavior.