How do small dust grains grow
in protoplanetary disks?
Ge/Ay133
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:
mm  cm  km  moon/Mars (oligarchs)  1-10 MEarth
Step #1: Growth from ~0.1 mm 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).
d = particle density, S 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 mm 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?
z is the dust-to-gas ratio (not nec. 0.01).
Fast if ps = 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
(hgas/r)2vK/hdust
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
>10-100 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)
Do other instabilities, manifested locally, matter?
ApJ, 620:459-469, 2005 Feb 10
Streaming Instabilities in
Protoplanetary Disks
A.N. Youdin & Jeremy Goodman
Princeton University Observatory
ABSTRACT
Interpenetrating streams of solids
and gas in a Keplerian disk produce
a local, linear instability. The two
components mutually interact via
aerodynamic drag, which generates
radial drift and triggers unstable
modes. The secular instability does
not require self-gravity, yet it
generates growing particle-density
perturbations that could seed
planetesimal formation. Growth rates
are slower than dynamical but faster
than radial drift timescales.
Growth rates and equilibrium
drift rates of solids and gas vs.
the density ratio ρp/ρg for
η = 2 × 10-3, τs = 0.01.
Do other instabilities, manifested locally, matter?
This movie shows the column density
of boulders in a protoplanetary disc.
Initially the particles have been
allowed to evolve without feeling each
other's gravity, but at the onset of the
movie self-gravity is turned on.
http://pc292.astro.lu.se/~anders/research.php
The movie at right presents a simulation
with metallicty 2x solar (Johansen, Youdin,
& Mac Low 2009, ApJ, 704, L75) . Masses
in the biggest clumps = 100-200 km bodies!