A Scaling Law for the Orbital Architecture of Planetary Systems Formed by Gravitational Scattering and Collisions

In the standard formation models of terrestrial planets in the solar system and close-in super-Earths in nonresonant orbits recently discovered by exoplanet observations, planets are formed by giant impacts of protoplanets or planetary embryos after the dispersal of protoplanetary disk gas in the final stage. This study aims to theoretically clarify a fundamental scaling law for the orbital architecture of planetary systems formed by giant impacts. In the giant impact stage, protoplanets gravitationally scatter and collide with one another to form planets. Using N-body simulations, we investigate the orbital architecture of planetary systems formed from protoplanet systems by giant impacts. As the orbital architecture parameters, we focus on the mean orbital separation between two adjacent planets and the mean orbital eccentricity of planets in a planetary system. We find that the orbital architecture is determined by the ratio of the two-body surface escape velocity of planets vesc to the Keplerian circular velocity vK, k = vesc/vK. The mean orbital separation and eccentricity are about 2ka and 0.3k, respectively, where a is the system semimajor axis. With this scaling, the orbital architecture parameters of planetary systems are nearly independent of their total mass and semimajor axis.