Post-Oligarchic Evolution of Protoplanetary Embryos and the Stability of Planetary Systems

TL;DR

This paper studies the stability and evolution of protoplanetary systems, revealing how orbit-crossing times depend on initial conditions and demonstrating the stochastic growth of eccentricities, with implications for planet formation.

Contribution

It introduces an analytical model for the evolution of protoplanetary systems, linking stochastic velocity dispersion growth to system stability and planet formation potential.

Findings

Orbit-crossing time scales with initial separation and mass.

Eccentricities evolve as a random walk, following a Rayleigh distribution.

Chaotic diffusion explains observed exoplanet eccentricities.

Abstract

We investigate the orbit-crossing time (T_c) of protoplanet systems both with and without a gas-disk background. The protoplanets are initially with equal masses and separation (EMS systems) scaled by their mutual Hill's radii. In a gas-free environment, we find log (T_c/yr) = A+B \log (k_0/2.3). Through a simple analytical approach, we demonstrate that the evolution of the velocity dispersion in an EMS system follows a random walk. The stochastic nature of random-walk diffusion leads to (i) an increasing average eccentricity <e> ~ t^1/2, where t is the time; (ii) Rayleigh-distributed eccentricities (P(e,t)=e/\sigma^2 \exp(-e^2/(2\sigma^2)) of the protoplanets; (iii) a power-law dependence of T_c on planetary separation. As evidence for the chaotic diffusion, the observed eccentricities of known extra solar planets can be approximated by a Rayleigh distribution. We evaluate the isolation masses of the embryos, which determine the probability of gas giant formation, as a function of the dust and gas surface densities.