Gravitationally coupled scale-free discs

In a composite fluid system of two gravitationally coupled barotropic scale-free discs bearing a rotation curve $v\propto r^{-\beta}$ and a power-law surface mass density $\Sigma_0\propto r^{-\alpha}$ with $\alpha=1+2\beta$, we construct coplanar stationary aligned and spiral perturbation configurations in the two discs. Due 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 collapse corresponding to short and long radial wavelengths, respectively. Only within a certain range of the rotation parameter $D_s^2$ (square of the effective 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.