Gravitationally coupled scale-free discs

In a composite fluid system of two gravitationally coupled barotropic scale-free discs bearing a rotation curve v α r^-β and a power-law surface mass density ∝₀ α r^-α with α = 1 + 2β, we construct classes of coplanar stationary aligned and logarithmic spiral perturbation configurations in the two discs. Owing to the mutual gravitational interaction, there are two independent classes of perturbation modes, with surface mass density disturbances in the two coupled discs being either in phase or out of phase. We derive analytical criteria for such perturbation modes to exist and show numerical examples. We compute the aligned and spiral perturbation modes systematically to explore the entire parameter regime. For the axisymmetric m = 0 case with radial oscillations, there are two unstable regimes of ring fragmentation and Jeans collapse corresponding to short and long radial wavelengths, respectively. Only within a certain range of the rotation parameter D_s² (square of the effective rotational Mach number for the stellar disc) can a composite disc system be stable against all axisymmetric perturbations. Compared with a single-disc system, the coupled two-disc system becomes less stable against such axisymmetric instabilities. Our investigation generalizes the previous work of Syer & Tremaine on the single-disc case and of Lou & Shen on two coupled singular isothermal discs. Non-axisymrnetric instabilities are briefly discussed. These stationary models for various largescale patterns and morphologies may be useful in the context of disc galaxies with bars and spirals or barred spirals.