A Fundamental Scaling Law for the Orbital Architecture of Planetary Systems Formed by Giant Impacts

Recent exoplanet surveys revealed that the most common planets in the Galaxy are close-in planets with semimajor axes smaller than about 0.3 au and masses less than about 30 Earth masses. Most of them reside in multi-planet systems with a typical multiplicity of three. Their composition is not well constrained yet, but most may consist mainly of solid components. We call these planets close-in super-Earths.

It is widely accepted that the final stage of terrestrial planet formation in the solar system is giant impacts among protoplanets or planetary embryos after disk gas dispersal. In this stage, protoplanets gravitationally scatter and collide with one another to form planets, and then the orbital structure is determined. A similar model is proposed for the formation model of close-in super-Earths, in which planets are formed through giant impacts from a protoplanet system formed in situ or by migration from the outer disk. One of this model's crucial merits is the natural formation of planets in non-resonant orbits. In this paper, we focus on the giant-impact formation model that may be responsible for creating the solar system terrestrial planets and the majority of close-in super-Earths. 

This study aims to clarify a fundamental scaling law for the orbital architecture of planetary systems formed by giant impacts that can be applied to both the terrestrial planet system in the solar system and the close-in super-Earth systems. In the giant impact stage, protoplanets gravitationally scatter and collide with one another to form planets. We perform N-body simulations of the giant impact stage from protoplanet systems. We clarify how a protoplanet system evolves through gravitational scattering and collisions among protoplanets and what determines the final system configuration. 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. Since the system evolution in the giant impact stage is stochastic, we investigate the orbital architecture statistically with many runs. We find that the orbital architecture is determined by the ratio of the two-body surface escape velocity of planets v_esc to the Keplerian circular velocity v_K, k = v_esc/v_K. The mean orbital separation and eccentricity are about 2 ka and 0.3 k, 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.