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

Abstract

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 form planets via direct collapse). But if disks are turbulent, this neglects a spectrum of stochastic density fluctuations that can produce rare, high-density mass concentrations. 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 disk. Turbulent disks are never completely stable: we calculate the (always finite) probability of forming self-gravitating structures via stochastic turbulent density fluctuations in such disks. Modest sub-sonic turbulence above Mach number M ~ 0.1 can 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 transsonic turbulence 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 turbulence driven by magneto-rotational instability, convection, or spiral waves and derive equivalent criteria in terms of Q and the cooling time. Cooling times ≳ 50 t_dyn may be required to completely suppress fragmentation. These gravo-turbulent events produce mass spectra peaked near ~(Q M_disk/M*)^(2) M_disk (rocky-to-giant planet masses, increasing with distance from the star). We apply this to protoplanetary disk models and show that even minimum-mass solar nebulae could experience stochastic collapse events, provided a source of turbulence.

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