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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.