Planetesimal Formation without Thresholds. I: Dissipative Gravitational Instabilities and Particle Stirring by Turbulence

We analyze the gravitational collapse of solids subject to gas drag in a protoplanetary disk. We also study the stirring of solids by turbulent fluctuations to determine the velocity dispersion and thickness of the midplane particle layer. The usual thresholds for determining gravitational instability in disks, Toomre's criterion and/or the Roche density, do not apply. Dissipation of angular momentum allows instability at longer wavelengths, lower densities, and higher velocity dispersions than without drag. Small solids will slowly leak into axisymmetric rings since initial collapse occurs over many orbits. Growth is fastest when particle stopping times are comparable to orbital times. Our analysis of particle stirring by turbulence is consistent with previous results for tightly coupled particles, but is generalized to loose coupling where epicyclic motions contribute to random velocities. A companion paper applies these results to turbulent protoplanetary disks.