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

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 disc scale heights, with a spatial dynamic range of ˜1.4 × 105. Our results confirm that `pebble'-sized particles are readily accreted, with accretion rates continuing to increase up to metre-size `boulders' for a 10 per cent MMSN surface density model. The gas mass flux in and out of the Hill sphere is consistent with the Hill rate, Σ Ω R_H^2=4 10^{-3} M_\oplus yr^{-1}. While smaller size particles mainly track the gas, a net accretion rate of {≈ } 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 disc 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 ˜0.15 million years for an Earth-mass planet at 1 au and ˜0.1 million years for a Mars mass planet at 1.5 au.