Local gravitational instability of two-component thick discs in three dimensions

The local gravitational instability of rotating discs is believed to be an important mechanism in different astrophysical processes, including the formation of gas and stellar clumps in galaxies. We aim to study in three dimensions the local gravitational instability of two-component thick discs. We take as starting point a recently proposed analytic three-dimensional (3D) instability criterion for discs with non-negligible thickness which takes the form $Q_{\rm 3D}<1$, where $Q_{\rm 3D}$ is a 3D version of the classical 2D Toomre $Q$ parameter for razor-thin discs. Here we extend the 3D stability analysis to two-component discs, considering first the influence on $Q_{\rm 3D}$ of a second unresponsive component, and then the case in which both components are responsive. We present the application to two-component discs with isothermal vertical distributions, which can represent, for instance, galactic discs with both stellar and gaseous components. Finally, we relax the assumption of vertical isothermal distribution, by studying one-component self-gravitating discs with polytropic vertical distributions for a range of values of the polytropic index corresponding to convectively stable configurations. We find that $Q_{\rm 3D}<1$, where $Q_{\rm 3D}$ can be computed from observationally inferred quantities, is a robust indicator of local gravitational instability, depending only weakly on the presence of a second component and on the vertical gradient of temperature or velocity dispersion. We derive a sufficient condition for local gravitational instability in the midplane of two-component discs, which can be employed when both components have $Q_{\rm 3D}>1$.