Turbulence-Induced Relative Velocity of Dust Particles III: The Probability Distribution

Abstract

Motivated by its important role in the collisional growth of dust particles in protoplanetary disks, we investigate the probability distribution function (PDF) of the relative velocity of inertial particles suspended in turbulent flows. Using the simulation from our previous work, we compute the relative velocity PDF as a function of the friction timescales, tau_p1 and tau_p2, of two particles of arbitrary sizes. The friction time of particles included in the simulation ranges from 0.1 tau_eta to 54T_L, with tau_eta and T_L the Kolmogorov time and the Lagrangian correlation time of the flow, respectively. The relative velocity PDF is generically non-Gaussian, exhibiting fat tails. For a fixed value of tau_p1, the PDF is the fattest for equal-size particles (tau_p2~tau_p1), and becomes thinner at both tau_p2<tau_ p1 and tau_p2>tau_p1. Defining f as the friction time ratio of the smaller particle to the larger one, we find that, at a given f in 1/2<f<1, the PDF fatness first increases with the friction time, tau_ph, of the larger particle, peaks at tau_ph~tau_eta, and then decreases as tau_ph increases further. For 0<f<1/4, the PDF shape becomes continuously thinner with increasing tau_ph. The PDF is nearly Gaussian only if tau_ph is sufficiently large (>>T_L). These features are successfully explained by the Pan & Padoan model. Using our simulation data and some simplifying assumptions, we estimated the fractions of collisions resulting in sticking, bouncing, and fragmentation as a function of the dust size in protoplanerary disks, and argued that accounting for non-Gaussianity of the collision velocity may help further alleviate the bouncing barrier problem.