  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.
