Dynamical Evolution of Planetesimals in Protoplanetary Disks

The current picture of terrestrial planet formation relies heavily on our understanding of the dynamical evolution of planetesimals—asteroid-like bodies thought to be planetary building blocks. In this study we investigate the growth of eccentricities and inclinations of planetesimals in spatially homogeneous protoplanetary disks using methods of kinetic theory. Emphasis is put on clarifying the effect of gravitational scattering between planetesimals on the evolution of their random velocities. We explore disks with a realistic mass spectrum of planetesimals evolving in time, similar to that obtained in self-consistent simulations of planetesimal coagulation: the distribution scales as a power law of mass for small planetesimals and is supplemented by an extended tail of bodies at large masses representing the ongoing runaway growth in the system. We calculate the behavior of planetesimal random velocities as a function of the planetesimal mass spectrum both analytically and numerically; results obtained by the two approaches agree quite well. Scaling of random velocity with mass can always be represented as a combination of power laws corresponding to different velocity regimes (shear- or dispersion-dominated) of planetesimal gravitational interactions. For different mass spectra we calculate analytically the exponents and time-dependent normalizations of these power laws, as well as the positions of the transition regions between different regimes. It is shown that random energy equipartition between different planetesimals can only be achieved in disks with very steep mass distributions (the differential surface number density of planetesimals falling off steeper than m -4 ) or in the runaway tails. In systems with shallow mass spectra (shallower than m -3 ) the random velocities of small planetesimals turn out to be independent of their masses. We also discuss the damping effects of inelastic collisions between planetesimals and of gas drag, and their importance in modifying planetesimal random velocities.