Axisymmetric stability criterion for two gravitationally coupled singular isothermal discs

Using the two-fluid formalism with the polytropic approximation, we examine the axisymmetric stability criterion for a composite system of gravitationally coupled stellar and gaseous singular isothermal discs (SIDS). Both SIDS are taken to be infinitely thin and are in a self-consistent steady background rotational equilibrium with power-law surface mass densities (proportional to r(-1)) and flat rotation curves. Recently, Lou & Shen derived exact solutions for both axisymmetric and non-axisymmetric stationary perturbations in such a composite SID system and proposed the D-s criterion of stability for axisymmetric perturbations. Here, for axisymmetric perturbations, we derive and analyse the time-dependent Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) dispersion relation to study stability properties. By introducing a dimensionless stellar SID rotation parameter D-s, defined as the ratio of the constant stellar rotation speed V-s to the constant stellar velocity dispersion a(s), one can readily determine the axisymmetric stability D-s criterion numerically by identifying a stable range of D-s. Those systems which rotate too slowly (collapse) or too rapidly (ring fragmentation) are unstable. We found that the stable range of D-s(2) depends on the mass ratio delta of the gaseous SID to the stellar SID and on the square of the ratio beta of the stellar velocity dispersion (which mimics the sound speed) to the gaseous isothermal sound speed. An increment of either delta or beta or both will diminish the stable range of D-s(2). The WKBJ results of instabilities provide physical explanations for the stationary configurations derived by Lou & Shen. It is feasible to introduce an effective Q parameter for a composite SID system. The closely relevant theoretical studies of Elmegreen, Jog and Shu et al. are discussed. A study of composite partial SID system reveals that an axisymmetric dark matter halo will promote the stability of the composite SID system against axisymmetric disturbances. Potential applications to disc galaxies, circumnuclear discs around nuclei of galaxies and protostellar discs are briefly discussed.