Adding particle collisions to the formation of asteroids and Kuiper belt objects via streaming instabilities

Modelling the formation of super-km-sized planetesimals by gravitational collapse of regions overdense in small particles requires numerical algorithms capable of handling simultaneously hydrodynamics, particle dynamics and particle collisions. While the initial phases of radial contraction are dictated by drag forces and gravity, particle collisions become gradually more significant as filaments contract beyond Roche density. Here we present a new numerical algorithm for treating momentum and energy exchange in collisions between numerical superparticles representing a high number of physical particles. We adopt a Monte Carlo approach where superparticle pairs in a grid cell collide statistically on the physical collision time-scale. Collisions occur by enlarging particles until they touch and solving for the collision outcome, accounting for energy dissipation in inelastic collisions. We demonstrate that superparticle collisions can be consistently implemented at a modest computational cost. In protoplanetary disc turbulence driven by the streaming instability, we argue that the relative Keplerian shear velocity should be subtracted during the collision calculation. If it is not subtracted, density inhomogeneities are too rapidly diffused away, as bloated particles exaggerate collision speeds. Local particle densities reach several thousand times the mid-plane gas density. We find efficient formation of gravitationally bound clumps, with a range of masses corresponding to contracted radii from 100 to 400 km when applied to the asteroid belt and 150 to 730 km when applied to the Kuiper belt, extrapolated using a constant self-gravity parameter. The smaller planetesimals are not observed at low resolution, but the masses of the largest planetesimals are relatively independent of resolution and treatment of collisions.