Ge/Ay133 How do small dust grains grow in protoplanetary disks?

How do we go from a well mixed gas/dust grain disk: To a mature planetary system? For solids, it is helpful to distinguish amongst several regimes:  m  cm  km  moon/Mars (oligarchs)  1-10 M Earth

Step #1: Growth from ~0.1  m to ~1 cm scales Need to think about how particles move in the sub-Keplerian field provided by the gas. First let’s look at the radial component. Drag force on particles, c = sound speed... Stopping time (shed momentum).  = particle density,  gas surface density. For small particles, well coupled to gas, radial velocity is Very slow for small particles.

Thus, let’s think about vertical motion at a given R: The mean free path is >> the particle diameter, so in the Epstein not Stokes regime (that is, the Brownian motion case). Thus… NO particle growth. Something like 1 M.Y. for 1  m grains, only 100 years for a = 1 cm. Opposite extreme: Suppose ALL collisions are sticky. As the particle settles, how large does is grow if it sweeps up all other grains that the falling particle encounters?  is the dust-to-gas ratio (not nec. 0.01).  Fast if p s = 1, details next…

Numerical simulations of coagulation/settling: If collisions are indeed sticky, then the growth and settling times are fast and largely insensitive to starting size or particle internal structure. BUT, these calculations do not allow for fragmentation during collisions!

The ultimate size distribution is sensitive to the assumptions:

If true, huge impact on SED: Disk becomes optically thin rapidly if coagulation is extensive w/o the regeneration of small dust grains. Not consistent with observations.

Suggests that small grains remain lofted, but that settling of ~cm-sized bodies should be quick. Now what?

Step #2: Growth ~1 cm to ~1 km scales. From earlier analysis, if the stopping time is long, the particles become poorly coupled to the gas. In this limit, the radial velocity is: Inbetween the small and large domain, the radial velocities approach the deviation from the Keplerian field. Growth in this regime depends critically on the physics of the collision. What determines shattering versus growth, etc. (Think about billiard balls versus snowballs…). Still, fairly slow overall, and m-sized bodies can be lost to the central star!

Are there other ways to generate planetesimals? For a geometrically thin layer of “dust bunnies”, Goldreich & Ward showed (in an analysis of planetary rings) that the layer is gravitationally unstable: Fragmentation length scale Fragmentation mass Provided the thin disk is quiescent, that is, has low velocity dispersion. As Armitage notes, the critical random speed is low, ~10 cm/s, & given by

Problems with the Goldreich-Ward instability: The required quiescent dust disk is so thin and the critical random speed so slow, the the dust disk can be stirred by Kelvin-Helmholtz turbulence. Theory suggests the z (vertical) velocity gradient should be of order (h gas /r) 2 v K /h dust This leads to a disruption of the thin dust disk, and numerical models suggest that for “normal” gas:dust ratios of ~100:1 this turbulence overwhelms the Goldreich-Ward instability. One possible way around this conclusion is to enhance the dust:gas ratio. Enhancement factors of > seem to be needed. Could local pressure maxima in a turbulent disk concentrate solids? Can highly porous solids be maintained over larger sizes than currently thought (enhancing cross sections)?

Could “dead zones” help Goldreich-Ward instability? The low random speeds of the solids does not need to be maintained over the full disk! Could dead zones near the mid-plane be the preferential sites of planetesimal formation? Armitage (2009)