Orbital Migration and Mass Accretion of Protoplanets in Three‐dimensional Global Computations with Nested Grids

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Abstract
We investigate the evolution of protoplanets with different masses embedded in an accretion disk, via global fully three-dimensional hydrodynamical simulations. We consider a range of planetary masses extending from one and a half Earth's masses up to one Jupiter's mass, and we take into account physically realistic gravitational potentials of forming planets. In order to calculate accurately the gravitational torques exerted by disk material and to investigate the accretion process onto the planet, the flow dynamics has to be thoroughly resolved on long as well as short length scales. We achieve this strict resolution requirement by applying a nested-grid refinement technique which allows to greatly enhance the local resolution. Our results from altogether 51 simulations show that for large planetary masses, approximately above a tenth of the Jupiter's mass, migration rates are relatively constant, as expected in type II migration regime and in good agreement with previous two-dimensional calculations. In a range between seven and fifteen Earth's masses, we find a dependency of the migration speed on the planetary mass that yields time scales considerably longer than those predicted by linear analytical theories. This property may be important in determining the overall orbital evolution of protoplanets. The growth time scale is minimum around twenty Earth-masses, but it rapidly increases for both smaller and larger mass values. Significant differences between two- and three-dimensional calculations are found in particular for objects with masses smaller than ten Earth-masses. We also derive an analytical approximation for the numerically computed mass growth rates
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