Gravitational Interaction between a Protoplanet and a Protoplanetary Disk. I. Local Three‐Dimensional Simulations

The gravitational interaction between a protoplanet and an isothermal gaseous disk is investigated through three-dimensional hydrodynamical simulations with the shearing sheet model. The torque exerted on the disk is evaluated and compared with previous estimates by the linear theory. It is found that a protoplanet mainly excites waves without vertical motion. Thus, the motion of the gas in a disk with thickness is similar to that in an infinitesimally thin disk. The angular momentum transfer is also dominated by waves without vertical motion, and the torque has a similar value to that in an infinitesimally thin disk, except for the correction factor 0.43 owing to the vertical averaging of the gravitational potential of the protoplanet. If the mass of the protoplanet is small enough, the torque increases proportionally with the square of the mass, as is predicted by the linear theory. However, for a large protoplanet whose Hill radius is larger than about the disk scale height, a nonlinear effect reduces the torque from the value proportional to the square of the mass. The torque reduction due to the nonlinearity is less significant for a disk with thickness than for an infinitesimally thin disk and is not effective for a small protoplanet with mass less than 10 times the Earth mass at 1 AU. This result suggests that during the formation of terrestrial planets and the cores of giant planets, the torque continues to increase as the protoplanets grow. The reduction in the torque due to the thickness of the disk and the nonlinearity is not large enough to solve the problem that the migration of protoplanets is too fast. Before protoplanets acquire the masses needed to suppress torque by the nonlinearity, they would experience large migration and some protoplanets would fall onto the central star in the lifetime of the gas in the protoplanetary disks.