Calibrated Gas Accretion and Orbital Migration of Protoplanets in 1D Disc Models

We aim to develop a simple prescription for migration and accretion in 1D disc models, calibrated with results of 3D hydrodynamic simulations [1,2]. 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.

We study 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 [3]. These torque densities allow modelling 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. Fig. 1 shows an exampe of the time evolution of semi-major axis and mass of a growing, migrating planet. Our proposed model "High mass torque" is shown as purple dash-dotted line.

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.

Our prescription is valuable for simultaneously modelling migration and accretion in 1D-models. 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 to 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.

Fig. 1: Time evolution of an initially low-mass planet (5 M_earth), starting at 5.2 au in a disc with an initial surface density of 100 g cm^-2. Left: Semi-major axis. Right: Planetary mass. The figure shows different models for the torque density and the feeding zone radius we studied in [4].

References:
[1] D’Angelo, G. & Lubow, S. H. 2008, Astrophysical Journal, 685, 560
[2] D’Angelo, G. & Lubow, S. H. 2010, Astrophysical Journal, 724, 730
[3] Schib, O., Mordasini, C., Wenger, N., Marleau, G. D., & Helled, R. 2021, Astronomy and Astrophysics, 645, A43
[4] Schib, Mordasini & Helled 2022, Astronomy and Astrophysics accepted