Accretion of aerodynamically large pebbles

Abstract: Due to their aerodynamical coupling with gas, pebbles in protoplanetary discs can drift over large distances to support planet growth in the inner disc. In the past decade, this pebble accretion has been studied extensively for aerodynamically small pebbles (Stokes number St < 1). However, accretion can also operate in the St > 1 mode, e.g., when planetesimals collisionally fragment to smaller bodies or when the primordial gas disc disperses. This work aims to extend the study of pebble accretion to these aerodynamically loosely coupled particles. We integrate the pebble's equation of motion, accounting for gas drag, stellar and planetary gravity, in the midplane of a laminar disc. The accretion probability ($\epsilon$) is calculated as function of Stokes number, disc pressure gradient index, planet mass and eccentricity. We find that for Stokes number above unity $\epsilon$(St) first rises, due to lower drift and aided by a large atmospheric capture radius, until it reaches a plateau where the efficiency approaches 100 per cent. At high St the plateau region terminates as particles become trapped in resonance. These results are well described by a semi-analytical "kick-and-drift" model and we also provide fully analytical prescriptions for $\epsilon$. We apply our model to the accretion of $\sim 30 \mu$m dust particles in a dispersing protoplanetary and secondary (CO-rich) debris disc. It shows that physically small particles are mainly accreted as aerodynamically large Stokes number pebbles during the debris disc phase. Earth-mass planets may obtain $\sim 25$ per cent of their heavy elements through this late accretion phase.