Global structures in a composite system of two scale-free discs with a coplanar magnetic field

We investigate a theoretical magnetohydrodynamic (MHD) disc problem involving a composite disc system of gravitationally coupled stellar and gaseous discs with a coplanar magnetic field in the presence of an axisymmetric dark matter halo. The two discs are expediently approximated as razor-thin, with a barotropic equation of state, a power-law surface mass density, a ring-like magnetic field, and a power-law rotation curve in radius r. By imposing the scale-free condition, we construct analytically stationary global MHD perturbation configurations for both aligned and logarithmic spiral patterns using our composite MHD disc model. MHD perturbation configurations in a composite system of partial discs in the presence of an axisymmetric dark matter halo are also considered. Our study generalizes the previous analyses of Lou & Shen and Shen & Lou on the unmagnetized composite system of two gravitationally coupled isothermal and scale-free discs, of Lou and Shen et al. on the cases of a single coplanarly magnetized isothermal and scale-free disc, and of Lou & Zou on magnetized two coupled singular isothermal discs. We derive analytically the stationary MHD dispersion relations for both aligned and unaligned perturbation structures and analyze the corresponding phase relationships between surface mass densities and the magnetic field. Compared with earlier results, we obtain three solution branches corresponding to super fast MHD density waves (sFMDWs), fast MHD density waves (FMDWs) and slow MHD density waves (SMDWs), respectively. We examine the m = 0 cases for both aligned and unaligned MHD perturbations. By evaluating the unaligned m = 0 case, we determine the marginal stability curves where the two unstable regimes corresponding to Jeans collapse instability and ring fragmentation instability are identified. We find that the aligned m = 0 case is simply the limit of the unaligned m = 0 case with the radial wavenumber ξ → 0 (i.e., the breathing mode) which does not merely represent a rescaling of the equilibrium state. We further show that a composite system of partial discs behaves much differently from a composite system of full discs in certain aspects. We provide numerical examples by varying dimensionless parameters β (rotation velocity index), η (ratio of effective sound speed of the two discs), δ (ratio of surface mass density of the two discs), q (a measure of coplanar magnetic field strength), F (gravity potential ratio), ξ (radial wavenumber). Our formalism provides a useful theoretical framework in the study of stationary global perturbation configurations for MHD disc galaxies with bars, spirals and barred spirals.