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

For a composite system of gravitationally coupled stellar and gaseous discs, we have carried out a linear stability analysis for axisyrnmetric 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 the cylindrical radius r with compatible exponents. The unstable modes set in as neutral modes or stationary perturbation configurations with angular frequency f33 = 0. The axisyrnmetric stable range is bounded by two marginal stability curves derived from stationary perturbation configurations. Because of the gravitational coupling between the stellar arid the gaseous discs, one only needs to consider the parameter regime of the stellar disc. There exist two unstable regimes in general: a collapse regime corresponding to large-scale perturbations and a 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 against axisyrnmetric perturbations is determined by a necessary D-criterion involving the effective Mach number squared Ds2 (the squared ratio of the stellar disc rotation speed to the stellar velocity dispersion up to a numerical factor). Different mass ratio S 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. Astrophysical applications to disc galaxies, proto-stellar discs and circumnuclear discs are given as examples.