Post-Oligarchic Evolution of Protoplanetary Embryos and the Stability of Planetary Systems
Abstract
We investigate the orbit-crossing time (Tc) 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 (Tc/yr) = A+B (k0/2.3). 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> ~ t1/2, where t is the time; (ii) Rayleigh-distributed eccentricities (P(e,t)=e/σ2 (-e2/(2σ2)) of the protoplanets; (iii) a power-law dependence of Tc 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.