Pebble dynamics and accretion on to rocky planets – I. Adiabatic and convective models

Abstract

We present nested-grid, high-resolution hydrodynamic simulations of gas and particle dynamics in the vicinity of Mars- to Earth-mass planetary embryos. The simulations extend from the surface of the embryos to a few vertical disk scale heights, with \rev{a spatial} dynamic range \rev{of} $\sim\! 1.4\times 10^5$. Our results confirm that "pebble"-sized particles are readily accreted, with accretion rates continuing to increase up to metre-size "boulders" for a 10\% MMSN surface density model. The gas mass flux in and out of the Hill sphere is consistent with the Hill rate, $\Sigma\Omega R_\mathrm{H}^2 = 4\, 10^{-3}$ M$_\oplus$ yr$^{-1}$. While smaller size particles mainly track the gas, a net accretion rate of $\approx 2\,10^{-5}$ M$_\oplus$ yr$^{-1}$ is reached for 0.3--1 cm particles, even though a significant fraction leaves the Hill sphere again. Effectively all pebble-sized particles that cross the Bondi sphere are accreted. The resolution of these simulations is sufficient to resolve accretion-driven convection. Convection driven by a nominal accretion rate of $10^{-6}$ M$_\oplus$ yr$^{-1}$ does not significantly alter the pebble accretion rate. We find that, due to cancellation effects, accretion rates of pebble-sized particles are nearly independent of disk surface density. As a result, we can estimate accurate growth times for specified particle sizes. For 0.3--1 cm size particles, the growth time from a small seed is $\sim$0.15 million years for an Earth mass planet at 1 AU and $\sim$0.1 million years for a Mars mass planet at 1.5 AU.