  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.
