Study of type II migration under the framework of the disk instability model for giant planet formation

Context . Hydrodynamic simulations of the migration of planets formed by gravitational instability suggest that after an initial phase of fast migration, planets can open gaps and continue to migrate on a type II migration timescale. The simulation time length is typically on the order of 10 4 yr. Aims . We study the effects of the subsequent type II migration during the disk lifetime on the final orbital radii of planets. Methods . We used a numerical disk model that follows the disk formation and evolution. The disk acquires mass through the mass influx from the collapse of its parent molecular cloud core. The model reflects the influence of the properties of the parent core on the disk. Considering clumps forming at different times in a disk and also in different disks with different parent core properties, we used the type II migration rate to follow the clump migration from the formation location. We studied the dependence of the clump migration on the properties of the parent core. Results . The mass influx drag enhances the migration process. The duration and viscosity of gravitational instability, viscosity in the dead zone, and the collapse time of the parent core play important roles in planet migration. As the angular momentum and mass of the parent core increase, migration is enhanced. The final radius is sensitive to the initial radius. Clumps forming at large radii might migrate outward with the disk expansion. Conclusions . Even though type II migration is slow, clumps can migrate over significant distances. A considerable proportion of clumps migrate to the central protostar via type II migration. Our calculations support the idea that the observed pile-up of planets at <0.3 AU is explained by a scenario where planets might form at large radii, then migrate to orbits of <0.3 AU, and halt by a stopping mechanism at this location.