Closed-form expressions for particle relative velocities induced by turbulence (Research Note)

In this note we present complete, closed-form expressions for random relative velocities between colliding particles of arbitrary size in nebula turbulence. These results are exact for very small particles ( those with stopping times much shorter than the large eddy overturn time) and are also surprisingly accurate in complete generality ( that is, also apply for particles with stopping times comparable to, or much longer than, the large eddy overturn time). We note that some previous studies may have adopted previous simple expressions, which we find to be in error regarding the size dependence in the large particle regime.