Turbulent Disks Are Never Stable: Fragmentation and Turbulence-promoted Planet Formation

A fundamental assumption in our understanding of disks is that when the Toomre parameter Q 1, the disk is stable against fragmentation into smaller, self-gravitating objects (and so cannot, for example, form planets via direct collapse). However, if disks are turbulent, this criterion neglects a broad spectrum of stochastic density fluctuations that can produce rare but extremely high-density local mass concentrations that will easily collapse. We have recently developed an analytic framework to predict the statistics of these fluctuations. Here, we use these models to consider the rate of fragmentation and mass spectrum of fragments formed in a turbulent, Keplerian disk (e.g. a proto-planetary or proto-stellar disk). Turbulent disks are never completely stable: we calculate the (always finite) probability of the formation of self-gravitating structures via stochastic turbulent density fluctuations (compressions, shocks, etc.) in such a disk. Modest sub-sonic turbulence above a minimum Mach numberM& 0.1 is sufficient to produce a few stochastic fragmentation or “direct collapse” events over ∼Myr timescales, even if Q 1 and cooling is relatively “slow” (tcool torbit). In trans-sonic turbulence (M∼ 1) this extends to Q as large as ∼ 100. We derive the “true” Q criterion needed to suppress such events (over some timescale of interest), which scales exponentially with Mach number. We specify this to cases where the turbulence is powered by MRI, convection, or density/spiral waves, and derive equivalent criteria in terms of Q and/or the disk cooling time. In the latter case, cooling times as long as & 50Ω−1 may be required to completely suppress this channel for collapse. These gravo-turbulent events produce a mass spectrum concentrated near the Toomre mass ∼ (QMdisk/M∗) Mdisk (spanning rocky-to-giant planet masses, and increasing with distance from the star), with a wider mass spectrum as M increases. We apply this to plausible models of proto-planetary disks with self-consistent cooling rates and temperatures, and show that, beyond radii ∼ 1−10au, no disk temperature can fully suppress stochastic events. For theoretically expected temperature profiles, even a minimum mass solar nebula could experience stochastic collapse events, provided a source of turbulence with modest sub-sonic Mach numbers.