On the Early Evolution of Forming Jovian Planets. II. Analysis of Accretion and Gravitational Torques

We continue our numerical study of the migration of an already formed proto-Jovian companion embedded in a circumstellar disk. We first study the sensitivity of the planet's migration to its mass accretion rate, finding that the disk can supply a forming planet with mass at an essentially infinite rate (~1MJ per 25 yr) so that a gap could form very quickly via further dynamical interactions between the planet and remaining disk matter. The accreted matter has less orbital angular momentum than the planet and exerts an effective inward torque, so that inward migration is slightly accelerated. However, if a partial gap is formed prior to rapid accretion, the effective torque is small and its contribution to the migration is negligible. Although the disk can supply mass at a high rate, we show that mass accretion rates faster than ~10-4MJ yr-1 are not physically reasonable in the limit of either a thin, circumplanetary disk or of a spherical envelope. Planet growth and ultimately survival are therefore limited to the planet's ability to accept additional matter, not by the disk in which it resides. Large gravitational torques are produced both at Lindblad resonances and at corotation resonances. We compare the torques in our simulations to analytic theories at Lindblad resonances and find that common approximations to the theories predict torques that are a factor of ~10 or more larger than those obtained from the simulations. Accounting for the disk's vertical structure (crudely modeled in our simulations and the theory with a gravitational softening parameter) and small shifts in resonance positions due to pressure gradients, disk self-gravity, and inclusion of non-WKB terms in the analysis (Artymowicz) can reduce the difference to a factor of ~3-6 but do not account for the full discrepancy. Torques from the corotation resonances that are positive in sign, slowing the migration, contribute 20%-30% or more of the net torque on the planet, but are not well resolved and vary from simulation to simulation. A more precise accounting of the three-dimensional mass distribution and flow pattern near the planet will be required to accurately specify the torques from both types of resonances in the simulations. We show that the assumption of linearity underlying theoretical analyses of the interactions at Lindblad resonances is recovered in the simulations with planets with masses below 0.5MJ, but the assumption that interactions occur only at the resonances may be more difficult to support. Angular momentum transfer occurs over a region of finite width near both Lindblad and corotation resonances. The shape of the disk's response there (due, e.g., to local variations in epicyclic frequency) varies from pattern to pattern, making the true position of the resonance less clear. We speculate that the finite width allows for overlap and mixing between resonances and may be responsible for the remainder of the differences between torques from theory and simulation, but whether accounting for such overlap in a theory will improve the agreement with the simulations is not clear.