Energy Optimization in Extrasolar Planetary Systems: The Transition from Peas-in-a-Pod to Runaway Growth

Motivated by the trends found in the observed sample of extrasolar planets, this paper determines 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 with masses of order $m_{\rm p}\sim10M_\oplus$, previous work showed that energy optimization leads to nearly equal mass planets, with circular orbits confined to a plane. The present treatment generalizes previous results by including the self-gravity of the planetary bodies. For systems with sufficiently large total mass $m_{\scriptstyle T}$ in planets, the optimized energy state switches over from the case of nearly equal mass planets to a configuration where one planet contains most of the material. This transition occurs for a critical mass threshold of approximately $m_{\scriptstyle T} \ge m_{\scriptstyle C}\sim40M_\oplus$ (where the value depends on the semimajor axes of the planetary orbits, the 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 collection of moons orbiting the giant planets in our solar system.