From Dust to Planetesimals: Criteria for Gravitational Instability of
  Small Particles in Gas
  Dust particles sediment toward the midplanes of protoplanetary disks, forming
dust-rich sublayers encased in gas. What densities must the particle sublayer
attain before it can fragment by self-gravity? We describe various candidate
threshold densities. One of these is the Roche density, which is that required
for a strengthless satellite to resist tidal disruption by its primary. Another
is the Toomre density, which is that required for de-stabilizing self-gravity
to defeat the stabilizing influences of pressure and rotation. We show that for
sublayers containing aerodynamically well-coupled dust, the Toomre density
exceeds the Roche density by many (up to about 4) orders of magnitude. We
present 3D shearing box simulations of self-gravitating, stratified, dust-gas
mixtures to test which of the candidate thresholds is relevant for collapse.
All our simulations indicate that the larger Toomre density is required for
collapse. This result is sensible because sublayers are readily stabilized by
pressure. Sound-crossing times for thin layers are easily shorter than
free-fall times, and the effective sound speed in dust-gas suspensions
decreases only weakly with the dust-to-gas ratio (as the inverse square root).
Our findings assume that particles are small enough that their stopping times
in gas are shorter than all other timescales. Relaxing this assumption may
lower the threshold for gravitational collapse back down to the Roche
criterion. In particular, if the particle stopping time becomes longer than the
sound-crossing time, sublayers may lose pressure support and become
gravitationally unstable.
