Gravitational Instability of Rotating, Pressure-Confined, Polytropic Gas
  Disks With Vertical Stratification
  We investigate gravitational instability (GI) of rotating,
vertically-stratified, pressure-confined, polytropic gas disks using linear
stability analysis as well as analytic approximations. The disks are initially
in vertical hydrostatic equilibrium and bounded by a constant external
pressure. We find that GI of a pressure-confined disk is in general a mixed
mode of the conventional Jeans and distortional instabilities, and is thus an
unstable version of acoustic-surface-gravity waves. The Jeans mode dominates in
weakly confined disks or disks with rigid boundaries. When the disk has free
boundaries and is strongly pressure-confined, on the other hand, the mixed GI
is dominated by the distortional mode that is surface-gravity waves driven
unstable under own gravity and thus incompressible. We demonstrate that the
Jeans mode is gravity-modified acoustic waves rather than inertial waves and
that inertial waves are almost unaffected by self-gravity. We derive an
analytic expression for the effective sound speed c_eff of
acoustic-surface-gravity waves. We also find expressions for the gravity
reduction factors relative to a razor-thin counterpart, appropriate for the
Jeans and distortional modes. The usual razor-thin dispersion relation after
correcting for c_eff and the reduction factors closely matches the numerical
results obtained by solving a full set of linearized equations. The effective
sound speed generalizes the Toomre stability parameter of the Jeans mode to
allow for the mixed GI of vertically-stratified, pressure-confined disks.
