A Symplectic Map for Interactions Between Planetary Embryos
Abstract
The rate of planetary accretion from planetesimals changes abruptly when the lunarsize planetary embryos produced by runaway accretion begin to perturb one another into crossing orbits. Nbody simulations of this process suggest that the damping of the embryos' orbital eccentricities by energy exchange with the remaining small planetesimals can delay this transistion, resulting in a very demanding computational problem (Kokubo and Ida 1995, 1996; Chambers and Wetherill 1996). An efficient alternative formulation of this problem would allow one to explore longer time scales and more realistic size distributions of planetesimals. To this end, a symplectic map has been derived that closely approximates the longrange gravitational interactions between planetary embryos. The map is a good approximation so long as the embryo orbits satisfy the Hill stability criteria such that they would never experience a close encounter with one another in the absence of third body perturbations. Since the map jumps from one embryo conjunction to the next, millions of orbits can be simulated in a few seconds. Initial calculations using the map have already illuminated the chaotic dynamics of the twoembryo problem that was previously described by Gladman (1993). For example, whenever the two planets become temporarily trapped in a mean motion resonance, their eccentricities execute slow, large amplitude oscillations due to the exchange of energy between the relative motion and the centerofmass motion. This occurs because the motion remains close to a lowerdimensional manifold while in resonance, as can be shown using surfaces of section. Extending the map from two embryos to three embryos is relatively straightforward and should yield insights into the interesting stability properties of these systems found by Chambers et al. (1996). The addition to the model of stochastic forcing and dynamical friction due to planetesimals is in progress.
 Publication:

AAS/Division for Planetary Sciences Meeting Abstracts #28
 Pub Date:
 September 1996
 Bibcode:
 1996DPS....28.1114S