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 .
Our investigation generalizes the previous work of Syer & Tremaine on the single-disc case and of Lou & Shen on two coupled singular isothermal discs ( SIDs ) .
Non-axisymmetric instabilities are briefly discussed .
These stationary models for various large-scale patterns and morphologies may be useful in contexts of disc galaxies .