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 (∝ r) and flat rotation curves. Recently, Lou & Shen (2003) derived exact solutions for both axisymmetric and nonaxisymmetric stationary perturbations in such a composite SID system and proposed the Ds−criterion of stability for axisymmetric perturbations. Here for axisymmetric perturbations, we derive and analyze the time-dependent WKBJ dispersion relation to study stability properties. By introducing a dimensionless stellar SID rotation parameter Ds, defined as the ratio of the constant stellar rotation speed Vs to the constant stellar velocity dispersion as, one can readily determine the axisymmetric stability Ds−criterion numerically by identifying a stable range of Ds. Those systems which rotate too slow (collapse) or too fast (ring fragmentation) are unstable. We found that the stable range of D s depends on the mass ratio δ of the gaseous SID to the stellar SID and on the square of the ratio β of the stellar velocity dispersion (which mimics the sound speed) to the gaseous isothermal sound speed. Increment of either δ or β or both will diminish the stable range of D s . 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 (1995), Jog (1996) and Shu et al. (2000) are discussed. A study of composite partial SID system reveals that an axisymmetric dark matter halo will promote stability of composite SID system against axisymmetric disturbances. Potential applications to disc galaxies, circumnucleus discs around nuclei of galaxies, and protostellar discs are briefly discussed.