Growth of a Migrating Protoplanet

Abstract

In the solar nebula, protoplanets may rapidly migrate to the Sun due to the tidal interaction with the nebula disk (e.g., Ward 1986, Icarus67, 164–180). We investigated the growth rate of a migrating protoplanet accreting planetesimals. The collision rate between the protoplanet and planetesimals strongly depends on the orbital elements of planetesimals (e.g., their eccentricities and inclinations). We calculated the orbital evolution of 2000 planetesimals in the vicinity of a migrating protoplanet and obtained the collision rate by directly counting the number of collisions between the protoplanet and planetesimals. In our simulation, the migration speed of the protoplanet due to the tidal interaction is given as a parameter.When the protoplanet's migration due to the tidal interaction is fast enough, the protoplanet migrates across the swarm of planetesimals and accretes a part of them. In the case of slow migration, however, the gravitational perturbation of the protoplanet opens a gap in the planetesimal disk and the migrating protoplanet shepherds planetesimals inside its orbit. Then the protoplanet cannot accrete planetesimals. We derived the criterion of the migration speed for the shepherding of planetesimals through numerical simulations and an analytical estimate. We also obtained the collision rate as a function of the migration speed in the cases of the fast migration. The obtained collision rate indicates that the migration of a protoplanet can enhance the collision rate by more than a factor 10 compared with that of the previous studies where the protoplanet's migration is neglected. We found that, during the migration of a protoplanet, the percentage of planetesimals caught by the migrating protoplanet is low (≲10%), which prevents the very rapid growth of a migrating protoplanet proposed by Ward (1986). Furthermore, using the obtained collision rate, we calculated coupled evolution of the mass and orbit of a migrating protoplanet, adopting a simple model. According to the results, in the minimum-mass solar nebula, protoplanets can grow to only a few Earth mass before falling to the Sun. The planetesimal disk is needed to be, at least, five times more massive than that of the minimum-mass solar nebula model in order to make protoplanets with the critical mass for the gas capture. The protoplanets which gained the sufficient nebular gas would open a gap in the gas disk, which can make the migration slow down enough and can save the protoplanets.