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 recently discovered by exoplanet observations, in the final stage, planets are spontaneously formed by giant impacts of protoplanets or planetary embryos after the dispersal of protoplanetary disk gas. This study aims to obtain 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 each other to complete planets. We investigate the orbital architecture of planetary systems formed from protoplanet systems by giant impacts using N-body simulations. We focus on the mean orbital separation of two adjacent planets and the mean orbital eccentricity of planets in a planetary system. We find that the orbital separation and eccentricity normalized by the Hill radius of planets are nearly independent of the total mass of the initial protoplanet systems. On the other hand, they show a positive dependence on the semimajor axis. The equilibrium random velocity of planets is responsible for this dependence. In the giant impact stage, the random velocity becomes as large as the two-body surface escape velocity of planets, v_esc; in other words, the eccentricity becomes about v_esc/v_K, where v_K is the Kepler velocity, which increases with the semimajor axis. We show that the orbital architecture normalized by the epicycle amplitude for v_esc/v_K barely depends on the semimajor axis, and this scaling includes the Hill scaling.