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We investigate gravitational instability (GI) of rotating,vertically-stratified, pressure-confined, polytropic gas disks using linearstability analysis as well as analytic approximations. The disks are initiallyin vertical hydrostatic equilibrium and bounded by a constant externalpressure. We find that GI of a pressure-confined disk is in general a mixedmode of the conventional Jeans and distortional instabilities, and is thus anunstable version of acoustic-surface-gravity waves. The Jeans mode dominates inweakly confined disks or disks with rigid boundaries. When the disk has freeboundaries and is strongly pressure-confined, on the other hand, the mixed GIis dominated by the distortional mode that is surface-gravity waves drivenunstable under own gravity and thus incompressible. We demonstrate that theJeans mode is gravity-modified acoustic waves rather than inertial waves andthat inertial waves are almost unaffected by self-gravity. We derive ananalytic expression for the effective sound speed c_eff ofacoustic-surface-gravity waves. We also find expressions for the gravityreduction factors relative to a razor-thin counterpart, appropriate for theJeans and distortional modes. The usual razor-thin dispersion relation aftercorrecting for c_eff and the reduction factors closely matches the numericalresults obtained by solving a full set of linearized equations. The effectivesound speed generalizes the Toomre stability parameter of the Jeans mode toallow for the mixed GI of vertically-stratified, pressure-confined disks.

Gravitational Instability of Rotating, Pressure-Confined, Polytropic Gas Disks With Vertical Stratification