The range of validity of the two-body approximation in models of terrestrial planet accumulation. II - Gravitational cross sections and runaway accretion

The validity of the two-body approximation in calculating encounters between planetesimals has been evaluated as a function of the ratio of unperturbed planetesimal velocity (with respect to a circular orbit) to mutual escape velocity when their surfaces are in contact (V/V-sub-e). Impact rates as a function of this ratio are calculated to within about 20 percent by numerical integration of the equations of motion. It is found that when the ratio is greater than 0.4 the two-body approximation is a good one. Consequences of reducing the ratio to less than 0.02 are examined. Factors leading to an optimal size for growth of planetesimals from a swarm of given eccentricity and placing a limit on the extent of runaway accretion are derived.