Accretion rates of protoplanets
The rate at which planetesimals in a uniform surface density disk collide with, and are (assumed to be) accreted by, a massive protoplanet is calculated. The collision cross-section of a protoplanet is enhanced relative to its geometric cross-section due to its gravitational focusing of planetesimal trajectories. The gravitation enhancement (eccentricities and inclinations) decrease. For large random velocity planetesimals, encounters are sufficiently rapid (less than or approximately = 5 percent of and orbital period) that F(sub g) is well approximated by the two-body particle in a box formula, which neglects the gravitational effect of the Sun. As planetesimal velocities decrease, F(sub g) 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. A scaling argument is presented that generalizes the 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 to small. When gravitational stirring is included, the maximum plausible value of the gravitation enhancement factor for rock protoplanets 1 Au from the Sun is F(sub g) of approximately 1000. If one protoplanet dominates gravitational scatterings in a given region of a protoplanetary disk, then it was found that the 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.