We calculate the rate at which planetesimals in a uniform surface density disk collide with, and are assumed to be accreted by, a massive protoplanet. The collision cross section of a protoplanet is enhanced relative to its geometric cross section due to its gravitational focusing of planetesimal trajectories. The gravitational enhancement factor of a protoplanet's cross section, Fg, increases as planetesimal random velocities (eccentricities and inclinations) decrease. For large random velocity planetesimals, encounters are sufficiently rapid ({less-than or approximate}5% of an orbital period) that Fg is well approximated by the two-body "particle in a box" formula, which neglects the gravitational effect of the Sun. As planetesimal velocities decrease, Fg increases to approximately twice the two-body value, and then rises less rapidly than the two-body value, eventually dropping below it and asymptotically approaching a constant for sufficiently small random velocities. We present a scaling argument that generalizes our results to protoplanets of arbitrary mass, radius, and orbital semimajor axis. Gravitational scatterings by a protoplanet prevent random velocities of the planetesimals within its accretion zone from becoming too small. When gravitational stirring is included, the maximum plausible value of the gravitational enhancement factor for rock protoplanets 1 AU from the Sun is Fg - 1000. If one protoplanet dominates gravitational scatterings in a given region of a protoplanetary disk, we find that planetesimal inclinations are excited much less rapidly than eccentricities, in contrast to the two-body approximation, in which energy is roughly equipartitioned between eccentric and inclined random motions. The resulting skewed velocity dispersion allows for a more rapid rate of protoplanet growth. © 1990.
