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). Lou & Zou recently constructed exact global stationary configurations for both axisymmetric and nonaxisymmetric coplanar MHD perturbations in such a composite MSID system and proposed the MHD $D_s-$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 $D_s$ for the ratio of the stellar rotation speed $V_s$ to the stellar velocity dispersion $a_s$, one readily determines the stable range of $D_s^2$ numerically to establish the $D_s-$criteria for axisymmetric MSID stability. Those MSID systems rotating either too fast (ring ragmentation) or too slow (Jeans collapse) are unstable. 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.