Turbulence in Stellar Disks is Key to Planetary Formation

TURBULENT DISKS ARE NEVER STABLE: FRAGMENTATION AND TURBULENCE-PROMOTED PLANET FORMATION

Authors:

1. Philip F. Hopkins (a,b)
2. Jessie L. Christiansen (c)

Affiliations:

a. TAPIR, Mailcode 350-17, California Institute of Technology, Pasadena, CA 91125, USA

b. Department of Astronomy and Theoretical Astrophysics Center, University of California Berkeley, Berkeley, CA 94720, USA

c. SETI Institute/NASA Ames Research Center, M/S 244-30, Moffett Field, CA 94035, USA


Abstract:

A fundamental assumption in our understanding of disks is that when the Toomre Q Gt 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 $\mathcal {M}\sim 0.1$ can produce a few stochastic fragmentation or "direct collapse" events over ~Myr timescales, even if Q Gt 1 and cooling is slow (t cool Gt 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 gsim 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.