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

In the sequential accretion model, planets form through the sedimentation of dust, cohesive collisions of planetesimals, and coagulation of protoplanetary embryos prior to the onset of efficient gas accretion. As progenitors of terrestrial planets and the cores of gas giant planets, embryos have comparable masses and are separated by the full width of their feeding zones after the oligarchic growth. Within this context, we investigate the orbit-crossing time (T_c) of protoplanetary systems with equal planetary masses and initial separation k₀ scaled by their mutual Hill radii ("EMS systems"). In a gas-free environment, log [T_c/(1 yr)] ≃ A + B log (k₀/2.3), where A and B are functions of the planetary masses and initial eccentricities. This power law is caused by a random-walk diffusion of velocity dispersion σ(t) in the EMS systems. The diffusion also leads to (1) a Rayleigh distribution of eccentricities with probability P(t) = (e/σ²) exp [-e²/(2σ²)] at time t and (2) an evolution of average eccentricity ∝t^1/2. As evidence of this chaotic diffusion, the observed eccentricities of known extrasolar planets obey a Rayleigh distribution. In a gaseous environment, as embryos become well separated (with k₀ ≃ 6-12), their orbit-crossing tendency is suppressed by tidal drag, and their growth is stalled as their orbits are circularized. 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.