Philip F. Hopkins, Jessie L. Christiansen
A fundamental assumption in our understanding of disks is that when the Toomre Q>>1, the disk is stable against fragmentation into self-gravitating objects (and so cannot, e.g., form planets via direct collapse). But if disks are turbulent, this criterion neglects a broad spectrum of stochastic density fluctuations that can produce rare, high-density mass concentrations that easily collapse. Here, we use a recently-developed analytic framework to predict the statistics of these fluctuations, i.e. the rate of fragmentation and mass spectrum of fragments formed in a turbulent Keplerian (proto-planetary/stellar) disk. Turbulent disks are never completely stable: we calculate the (always finite) probability of forming self-gravitating structures via stochastic turbulent density fluctuations (compressions, shocks) in such disks. Modest sub-sonic turbulence above a Mach number ~0.1 is sufficient to produce a few stochastic fragmentation or 'direct collapse' events over ~Myr timescales, even if Q>>1 and cooling is slow (t_cool>>t_orbit). In trans-sonic turbulence (Mach~1) this extends to Q~100. We derive the true Q-criterion needed to suppress such events, which scales exponentially with Mach number. We specify to cases where turbulence is driven by MRI, convection, or spiral waves, and derive equivalent criteria in terms of Q and the disk cooling time. In the latter case, cooling times >~50*t_dyn may be required to completely suppress fragmentation. These gravoturbulent events produce a mass spectrum peaked near ~M_disk*(Q*M_disk/M_star)^2 (spanning rocky-to-giant planet masses, and increasing with distance from the star). We apply this to proto-planetary disk models and show that, at >1-10au no disk temperature can fully suppress stochastic events. For expected temperature profiles, even a minimum mass solar nebula could experience stochastic collapse events, provided a source of turbulence.
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http://arxiv.org/abs/1301.2600
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