Dynamical Configurations of Planetary Systems Arising from Energy Optimization

Observations indicate that the planet formation process often produces multiple planet systems with nearly circular orbits, regular spacing, a narrow range of inclination angles, and similar planetary masses (typically of order 10 earths). Motivated by the observational sample, this talk discusses tidal equilibrium states for forming planetary systems — subject to conservation of angular momentum, constant total mass, and fixed orbital spacing. In the low-mass limit, valid for superearth-class planets, energy optimization leads to nearly equal mass planets, with circular orbits confined to a plane. As a result, the observed systems correspond to the lowest accessible energy states. We then generalize the treatment to include the self-gravity of the planetary bodies. For systems with sufficiently large total mass in planets, the optimized energy state switches over from one with nearly equal mass planets to a configuration where one planet contains most of the material. This transition occurs for a critical mass threshold of about 40 earth masses per planetary pair (where the e value depends on the semimajor axes, stellar mass, and other system properties). These considerations of energy optimization apply over a wide range of mass scales, from binary stars to planetary systems to the moons orbiting the giant planets in our solar system.