Cooling Requirements for the Vertical Shear Instability in Protoplanetary Disks

Cooling Requirements for the Vertical Shear Instability in Protoplanetary Disks

Authors:

Lin et al

Abstract:

It is difficult to understand how cold circumstellar disks accrete onto their central stars. A hydrodynamic mechanism, the vertical shear instability (VSI), offers a means to drive angular momentum transport in cold accretion disks such as protoplanetary disks (PPDs). The VSI is driven by a weak vertical gradient in the disk's orbital motion. In order to grow, the VSI must overcome vertical buoyancy, a strongly stabilizing influence in cold disks, where heating is dominated by external irradiation. Rapid cooling, via radiative losses, reduces the effective buoyancy and allows the VSI to operate. In this paper, we quantify the cooling timescale, tc, needed for growth of the VSI. We perform a linear analysis of the VSI with cooling in vertically global and radially local disk models. For irradiated disks, we find that the VSI is most vigorous for rapid cooling with tc<Ω−1Kh|q|/(γ−1) in terms of the Keplerian orbital frequency, ΩK, the disk's aspect ratio, h≪1, the radial power-law temperature gradient, q, and the adiabatic index, γ. For longer cooling times, the VSI is much less effective because growth slows and shifts to smaller length scales, which are more prone to viscous or turbulent decay. We apply our results to PPD models where tc is determined by the opacity of dust grains. We find that the VSI is most effective at intermediate radii, from ∼5AU to ∼50AU with a characteristic growth time of ∼30 local orbital periods. Growth is suppressed by long cooling times both in the opaque inner disk and the optically thin outer disk. A reduction in the dust opacity by a factor of 10 increases cooling times enough to quench the VSI at all disk radii. Thus the formation of solid protoplanets, a sink for dust grains, can impede the VSI.