Cooling Requirements for the Vertical Shear Instability in Protoplanetary Disks

TL;DR

This paper investigates the cooling conditions necessary for the vertical shear instability (VSI) to operate in protoplanetary disks, highlighting how dust opacity and cooling times influence angular momentum transport and planet formation.

Contribution

It provides a quantitative analysis of the cooling timescale required for VSI growth in protoplanetary disks, linking thermal physics to disk stability and planet formation processes.

Findings

VSI is most effective with rapid cooling, $t_c< ext{specific threshold}$.

Longer cooling times suppress VSI growth and shift it to smaller scales.

Reducing dust opacity can quench VSI across the disk.

Abstract

The vertical shear instability (VSI) offers a potential hydrodynamic mechanism for angular momentum transport in protoplanetary disks (PPDs). The VSI is driven by a weak vertical gradient in the disk's orbital motion, but must overcome vertical buoyancy, a strongly stabilizing influence in cold disks, where heating is dominated by external irradiation. Rapid radiative cooling reduces the effective buoyancy and allows the VSI to operate. We quantify the cooling timescale $t_c$ needed for efficient VSI growth, through a linear analysis of the VSI with cooling in vertically global, radially local disk models. We find the VSI is most vigorous for rapid cooling with $t_c<\Omega_\mathrm{K}^{-1}h|q|/(\gamma -1)$ in terms of the Keplerian orbital frequency, $\Omega_\mathrm{K}$; the disk's aspect-ratio, $h\ll1$; the radial power-law temperature gradient, $q$; and the adiabatic index, $\gamma$. For longer $t_c$, 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 $t_c$ is determined by the opacity of dust grains. We find that the VSI is most effective at intermediate radii, from $\sim5$AU to $\sim50$AU with a characteristic growth time of $\sim30$ local orbital periods. Growth is suppressed by long cooling times both in the opaque inner disk and the optically thin outer disk. Reducing 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.