Self-consistent velocity dispersions and mass spectra in the protoplanetary cloud

Abstract The velocity dispersions of the planetesimals that govern the accumulation of the planets can be described by the kinetic formalism of P. Hornung, R. Pellat, and P. Barge (1985, Icarus 64, 295–307) . For a mass distribution of planetesimals, the equilibrium that results from the competition between encounters and impacts can be described by steady-state equations for the velocity dispersions. Numerical and analytical asymptotic solutions lead to the determination of three components of the velocity dispersion as a function of the mass-dependent number density. Assuming a power law for the mass spectrum, we arrive, in each direction of the disk, at two conclusions: (1) When large bodies dominate, velocity dispersion is driven by the most massive objects and is nearly the same for each population. (2) When small bodies dominate, an equipartition of the kinetic energy arises due to the dynamical friction. These conclusions support the results of numerical simulations carried out by H. Salo and J. Lukkari (1984, Earth Moon Planets 30, 229–243) and H. Salo (1985, Earth Moon Planets 33, 189–200) . For intermediate likely situations we provide the first complete description of velocity dispersions. Gravitational focusing is found to be negligible. The relation between restitution coefficient for inelastic collisions and cloud stability is also discussed. Shear viscosity is driven mainly by gravitational encounters and is nearly constant for all the populations.