Relative velocities of solids in a turbulent protoplanetary disc

We use magnetohydrodynamic simulations to measure relative speeds of solids in a protoplanetary disc with turbulence generated by the magnetorotational instability. Relative velocities are calculated as functions of particle Stokes number St, which measures the aerodynamic coupling to the gas. When relative velocities V(rel) are calculated between two particles i and j such that St(i) >> St(j) and St(j) << 1, the data matches the analytical model of Ormel & Cuzzi. However, if V(rel) corresponds to two particles with the same St, only the data for the more loosely coupled solids (i. e. those with large St) follow the model. The discrepancy at the low-St end can be attributed to: (i) the numerical disc model's coarse resolution, which is unable to probe smaller turbulent eddies and, therefore, the dominant contribution to the particle relative velocities is given by the interpolation of the gas velocity inside the grid cells; (ii) the sparse particle sampling, which prevents the measurement of relative velocities between two particles in the same place at the same time. The distribution of turbulence-induced relative speeds can have a wide spread of values, which may lead to particle shattering, subject to the turbulent gas velocity. Codes such as the one used in this work, in general, underestimate relative velocities in turbulence for particles with St approximate to 1 because they lack energy on short time-scales (relative to a Kolmogorov spectrum). In making comparisons with theory, it is important to use the exact numerical energy spectrum instead of assuming a Kolmogorov inertial range.