The validity of the super-particle approximation during planetesimal formation

The formation mechanism of planetesimals in protoplanetary discs is hotly debated. Currently, the favoured model involves the accumulation of meter-sized objects within a turbulent disc, followed by a phase of gravitational instability. At best, one can simulate a few million particles numerically as opposed to the several trillion meter-sized particles expected in a real protoplanetary disc. Therefore, single particles are often used as super-particles to represent a distribution of many smaller particles. It is assumed that small-scale phenomena do not play a role and particle collisions are not modelled. The super-particle approximation is not always valid when applied to planetesimal formation because the system can be marginally collisional (of order one collision per particle per orbit). The super-particle approximation can only be valid in a collisionless or strongly collisional system, although, in many recent numerical simulations this is not the case. In this work, we present new results from numerical simulations of planetesimal formation via gravitational instability. A scaled system is studied that does not require the use of super-particles. This system is simplified for computational practicality and proper identification of important processes: 1) the evolution of particles is studied in a local shearing box; 2) the particle-particle interactions such as gravity, physical collisions, and gas drag are solved directly assuming a constant background shear flow without any feedback from the particles. We find that the scaled particles can be used to model the initial phases of clumping if the properties of the scaled particles are chosen such that all important timescales in the system are equivalent to what is expected in a real protoplanetary disc. Constraints are given for the number of particles needed in order to achieve numerical convergence. We compare this new method to the standard super-particle approach. We find that the super-particle approach produces unreliable results that depend on artifacts such as the gravitational softening in both the requirement for gravitational collapse and the resulting clump statistics. Our results show that short-range interactions (collisions) have to be modelled properly.