Axisymmetric stability criteria for a composite system of stellar and magnetized gaseous singular isothermal discs

Using the fluid-magnetofluid formalism, we obtain axisymmetric stability criteria for a composite disc system consisting of stellar and gaseous magnetized singular isothermal discs (MSIDs). Both (M)SIDs are presumed to be razor-thin and are gravitationally coupled in a self-consistent axisymmetric background equilibrium with power-law surface mass densities and flat rotation curves. The gaseous MSID is embedded with a non-force-free coplanar azimuthal magnetic field Bθ(r) of radial scaling r . Lou & Zou recently constructed exact global stationary configurations for both axisymmetric and nonaxisymmetric coplanar magnetohydrodynamic (MHD) perturbations in such a composite MSID system and proposed the MHD Ds−criteria for axisymmetric stability by the hydrodynamic analogy. In a different perspective, we derive and analyze here the time-dependent WKBJ dispersion relation in the low-frequency and tight-winding regime to examine axisymmetric stability properties. By introducing a rotational Mach number Ds for the ratio of the stellar rotation speed Vs to the stellar velocity dispersion as, one readily determines the stable range of D 2 s numerically to establish the Ds−criteria for axisymmetric MSID stability. Those MSID systems rotating either too fast (ring fragmentation) or too slow (Jeans collapse) are unstable. The stable range of D s depends on three dimensionless parameters: the ratio λ for the Alfvén speed to the sound speed in the gaseous MSID, the ratio β for the square of the stellar velocity dispersion to the gas sound speed and the ratio δ for the surface mass densities of the two (M)SIDs. Our WKBJ results of (M)SID instability provide physically compelling explanations for the stationary analysis of Lou & Zou. We further introduce an effective MHD Q parameter for a composite MSID system and compare with the earlier work of Elmegreen, Jog and Shen & Lou. As expected, an axisymmetric dark matter halo enhances the stability against axisymmetric disturbances in a composite partial MSID system. In terms of the global star formation rate in a disc galaxy system, it would appear physically more sensible to examine the MHD QM stability criterion against galactic observations. Relevance to large-scale structures in disc galaxies are also discussed.