Context. Orbital migration and gas accretion are two interdependent key processes that govern the evolution of planets in protoplanetary discs. The final planetary properties such as masses and orbital periods strongly depend on the treatment of those two processes. Aims. Our aim is to develop a simple prescription for migration and accretion in 1D disc models, calibrated with results of 3D hydrodynamic simulations. Our focus lies on non-self-gravitating discs, but we also discuss to what degree our prescription could be applied when the discs are self-gravitating. Methods. We studied migration using torque densities. Our model for the torque density is based on existing fitting formulas, which we subsequently modify to prevent premature gap-opening. At higher planetary masses, we also apply torque densities from hydrodynamic simulations directly to our 1D model. These torque densities allow us to model the orbital evolution of an initially low-mass planet that undergoes runaway-accretion to become a massive planet. The two-way exchange of angular momentum between disc and planet is included. This leads to a self-consistent treatment of gap formation that only relies on directly accessible disc parameters. We present a formula for Bondi and Hill gas accretion in the disc-limited regime. This formula is self-consistent in the sense that mass is removed from the disc in the location from where it is accreted. The prescription is appropriate when the planet is smaller than, comparable to, or larger than the disc scale height. Results. We find that the resulting evolution in mass and semi-major axis in the 1D framework is in good agreement with those from 3D hydrodynamical simulations for a range of parameters. Conclusions. Our prescription is valuable for simultaneously modelling migration and accretion in 1D models, which allows a planet's evolution to be followed over the entire lifetime of a disc. It is applicable also in situations where the surface density is significantly disturbed by multiple gap-opening planets or processes like infall. We conclude that it is appropriate and beneficial to apply torque densities from hydrodynamic simulations in 1D models, at least in the parameter space we study here. More work is needed in order to determine whether our approach is also applicable in an even wider parameter space and in situations with more complex disc thermodynamics, or when the disc is self-gravitating.
- accretion, accretion disks
- planets and satellites: formation
- protoplanetary disks