Jumping the Gap: The Formation Conditions and Mass Function of “Pebble-Pile” Planetesimals

In a turbulent proto-planetary disk, dust grains undergo large density fluctuations and under the right circumstances, these grain overdensities can overcome shear, turbulent, and gas pressure support to collapse under self-gravity (forming a “pebble pile” planetesimal). Using a simple analytic model for the fluctuations predicted in simulations, we estimate the rate-of-formation and mass function of self-gravitating, rapidly-collapsing planetesimal-mass bodies formed by this mechanism. The statistics of this process depend sensitively on the size/stopping time of the grains, disk surface density, and turbulent Mach numbers. However, when it occurs, we predict that the resulting planetesimal mass function is broad and quasi-universal, with a slope dN/dM ∝ M−(1−2), spanning a size/mass range ∼ 10− 104 km (∼ 10−5M⊕). Collapse to planetesimal through super-Earth masses is possible. The key condition is that grain density fluctuations reach large amplitudes on large scales, where gravitational instability proceeds most easily (collapse of small grains is strongly suppressed by turbulent vorticity). We show this leads to a new criterion for “pebble-pile” formation: τs & 0.05 ln(Q/Zd)/ ln(1 + 10α1/4)∼ 0.3ψ(Q, Z, α) where τs = ts Ω is the dimensionless particle stopping time. In a MMSN, this requires grains larger than a = (50, 1, 0.1)cm at r = (1, 30, 100)au. So at large radii (beyond the ice line), this may easily occur and seed core accretion. At small radii, it would depend on the existence of large “boulders.” However, because density fluctuations depend super-exponentially on τs (inversely proportional to disk surface density), lower-density disks are more unstable. In fact, we predict that cm-sized grains at ∼ 1au will form pebble piles in a disk with ∼ 10% the MMSN density, so planet formation at ∼ au may generically occur “late” as disks are evaporating. We also predict that conditions become progressively more favorable for pebble-pile formation around lower-mass, cooler stars.