Cooling Requirements for the Vertical Shear Instability in Protoplanetary Disks

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 disks 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 disks aspect-ratio, $hll1$; 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.

The Runts of the Litter: Why planets formed through gravitational instability can only be failed binary stars

Recent direct imaging discoveries suggest a new class of massive, distant planets around A stars. These widely separated giants have been interpreted as signs of planet formation driven by gravitational instability, but the viability of this mechanism is not clear cut. In this paper, we first discuss the local requirements for fragmentation and the initial fragment mass scales. We then consider whether the fragments subsequent growth can be terminated within the planetary mass regime. Finally, we place disks in the larger context of star formation and disk evolution models. We find that in order for gravitational instability to produce planets, disks must be atypically cold in order to reduce the initial fragment mass. In addition, fragmentation must occur during a narrow window of disk evolution, after infall has mostly ceased, but while the disk is still sufficiently massive to undergo gravitational instability. Under more typical conditions, disk-born objects will likely grow well above the deuterium burning planetary mass limit. We conclude that if planets are formed by gravitational instability, they must be the low mass tail of the distribution of disk-born companions. To validate this theory, on-going direct imaging surveys must find a greater abundance of brown dwarf and M-star companions to A-stars. Their absence would suggest planet formation by a different mechanism such as core accretion, which is consistent with the debris disks detected in these systems.