Accretion rates of protoplanets
Abstract
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, F_{g}, increases as planetesimal random velocities (eccentricities and inclinations) decrease. For large random velocity planetesimals, encounters are sufficiently rapid (⪅5% of an orbital period) that F_{g} is well approximated by the twobody "particle in a box" formula, which neglects the gravitational effect of the Sun. As planetesimal velocities decrease, F_{g} increases to approximately twice the twobody value, and then rises less rapidly than the twobody 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 F_{g}  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 twobody 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.
 Publication:

Icarus
 Pub Date:
 September 1990
 DOI:
 10.1016/00191035(90)90021Z
 Bibcode:
 1990Icar...87...40G
 Keywords:

 Gravitational Effects;
 Planetary Evolution;
 Planetary Mass;
 Protoplanets;
 Computational Astrophysics;
 Eccentricity;
 Equations Of Motion;
 Rates (Per Time);
 Three Body Problem;
 Lunar and Planetary Exploration