Global Axisymmetric Stability Analysis for a Composite System of Two Gravitationally Coupled Scale-free Discs

In a composite system of gravitationally coupled stellar and gaseous discs, we perform linear stability analysis for axisymmetric coplanar perturbations using the two-fluid formalism. The background stellar and gaseous discs are taken to be scale-free with all physical variables varying as powers of cylindrical radius r with compatible exponents. The unstable modes set in as neutral modes or stationary perturbation configurations with angular frequency ω = 0. The axisymmetrically stable range is bounded by two marginal stability curves derived for stationary perturbation configurations. By the gravitational coupling between the stellar and the gaseous disc components, one only needs to consider the parameter regime of the stellar disc. There exist two unstable regimes in general: the collapse regime corresponding to large-scale perturbations and the ring-fragmentation regime corresponding to short-wavelength perturbations. The composite system will collapse if it rotates too slowly and will succumb to ring-fragmentation instabilities if it rotates sufficiently fast. The overall stable range for axisymmetric perturbations is determined by a necessary D−criterion involving of the effective Mach number squared Ds (i.e., the square ratio of the stellar disc rotation speed to the stellar velocity dispersion but scaled by a numerical factor). Different mass ratio δ and sound speed ratio η of the gaseous and stellar disc components will alter the overall stability. For spiral galaxies or circumnuclear discs, we further include the dynamical effect of a massive dark matter halo. As examples, astrophysical applications to disc galaxies, proto-stellar discs and circumnuclear discs are discussed. Subject headings: hydrodynamics — ISM: general — galaxies: kinematics and dynamics — galaxies: spiral — galaxies: structure — waves.