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We investigate the orbit-crossing time (T_c) of protoplanet systems both withand without a gas-disk background. The protoplanets are initially with equalmasses and separation (EMS systems) scaled by their mutual Hill's radii. In agas-free environment, we find log (T_c/yr) = A+B \log (k_0/2.3). Through asimple analytical approach, we demonstrate that the evolution of the velocitydispersion in an EMS system follows a random walk. The stochastic nature ofrandom-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) apower-law dependence of T_c on planetary separation. As evidence for thechaotic diffusion, the observed eccentricities of known extra solar planets canbe approximated by a Rayleigh distribution. We evaluate the isolation masses ofthe embryos, which determine the probability of gas giant formation, as afunction of the dust and gas surface densities.

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