Equilibrium velocities of a planetesimal population

Abstract

The random velocities v( m) of planetesimal populations specified by maximum and minimum masses and a number density n( m) ∝ m − q are calculated interatively based on two different physical models involving ratios of rates: (1) excitation of kinetic energy by gravitational perturbation and elastic collision equal to damping of kinetic energy by inelastic collisions; and (2) excitation of kinetic energy a ratio b (∼3 usually) to doubling of mass. Model (2) follows the theory of Safronov closely. Both physical models are developed approximately, using averaged factors for collision dissipation and velocity ratios, and then more precisely, allowing for reference orbit differences and for plausible variations in collisional energy dissipation with impact velocity and planetesimal mass ratios. The approximate model (2) agrees reasonably well with the results of Safronov. Both precise models are applied to populations approximating those generated by the calculations of Greenberg, Cox, and Wetherill, producing qualitatively similar velocities v( m). These results encourage analytic models of planetesimal population growth, incrementing masses and calculating velocity distributions in alternate steps. The principal improvement needed in the models is more realistic collision energy partitioning.