Catching drifting pebbles

II. A stochastic equation of motion for pebbles

Anton Pannekoek Institute (API), University of Amsterdam, Science Park 904, 1090 GE Amsterdam, The Netherlands
e-mail: c.w.ormel@uva.nl; b.liu@uva.nl

Received: 29 December 2017
Accepted: 27 March 2018

Abstract

Turbulence plays a key role in the transport of pebble-sized particles. It also affects the ability of pebbles to be accreted by protoplanets because it stirs pebbles out of the disk midplane. In addition, turbulence suppresses pebble accretion once the relative velocities become too high for the settling mechanism to be viable. Following Paper I, we aim to quantify these effects by calculating the pebble accretion efficiency ε using three-body simulations. To model the effect of turbulence on the pebbles, we derive a stochastic equation of motion (SEOM) applicable to stratified disk configurations. In the strong coupling limit (ignoring particle inertia) the limiting form of this equation agrees with previous works. We conduct a parameter study and calculate ε in 3D, varying pebble and gas turbulence properties and accounting for the planet inclination. We find that strong turbulence suppresses pebble accretion through turbulent diffusion, agreeing closely with previous works. Another reduction of ε occurs when the turbulent rms motions are high and the settling mechanism fails. In terms of efficiency, the outer disk regions are more affected by turbulence than the inner regions. At the location of the H₂O iceline, planets around low-mass stars achieve much higher efficiencies. Including the results from Paper I, we present a framework to obtain ε under general circumstances.