Perturbation configurations in a two-fluid system of singular isothermal discs

Abstract

We investigate properties of stationary aligned and unaligned spiral perturbation configurations in a composite system of gravitationally coupled stellar and gaseous singular isothermal discs (SIDs) using the two-fluid formalism. Both SIDs are taken to be razor thin and are in a self-consistent background equilibrium with power-law surface mass densities and flat rotation curves. We obtain stationary perturbation solutions for aligned and unaligned spiral logarithmic configurations in such a composite SID system and derive analytically existence criteria for these solutions. In comparison with the similar problem of a single SID studied by Shu et al., there are now two possible sets of analytical solutions owing to an additional SID. For physically valid solutions, we explore parameter regimes involving the squared ratio β of velocity dispersions and the ratio δ of the surface mass densities of the two SIDs. In terms of transition criteria from axisymmetric equilibria to aligned secular and spiral dynamic bar-like instabilities, the corresponding T /∼W∼ ratio of rotation to potential energies for a composite SID system depends on β and δ, varies in a wide range, and can be considerably lower than the often-quoted value of ∼0.14. For both aligned and unaligned cases with azimuthal periodicities ∼m∼≥ 2, there exist certain parameter regimes where only one set of solutions is physically meaningful. For unaligned cases, we study marginal stabilities for axisymmetric (m= 0) and non-axisymmetric (m≠ 0) disturbances. The resulting marginal instability curves, varying with parameters, are different from those of a single SID. The case of a composite partial SID system is also studied to include the gravitational influence of a dark matter halo on the system equilibrium. For galactic applications, our model analysis contains more realistic elements and offers useful insights for the dynamics of disc galaxies consisting of stars and gas. Our analytical solutions are valuable for testing and benchmarking numerical codes. Starting from these solutions, numerical simulations are powerful tools for exploring non-linear dynamics such as large-scale spiral shocks.