Local gravitational instability of stratified rotating fluids: 3D criteria for gaseous discs

Fragmentation of rotating gaseous systems via gravitational instability is believed to be a crucial mechanism in several astrophysical processes, such as formation of planets in protostellar discs, of molecular clouds in galactic discs, and of stars in molecular clouds. Gravitational instability is fairly well understood for infinitesimally thin discs. However, the thin-disc approximation is not justified in many cases, and it is of general interest to study the gravitational instability of rotating fluids with different degrees of rotation support and stratification. We derive dispersion relations for axisymmetric perturbations, which can be used to study the local gravitational stability at any point of a rotating axisymmetric gaseous system with either barotropic or baroclinic distribution. 3D stability criteria are obtained, which generalize previous results and can be used to determine whether and where a rotating system of given 3D structure is prone to clump formation. For a vertically stratified gaseous disc of thickness $h_z$ (defined as containing $\approx$70% of the mass per unit surface), a sufficient condition for local gravitational instability is $Q_{\rm 3D}\equiv\left(\sqrt{\kappa^2+\nu^2}+c_s h_z^{-1}\right)/{\sqrt{4\pi G\rho}}<1$, where $\rho$ is the gas volume density, $\kappa$ the epicycle frequency, $c_s$ the sound speed, and $\nu^2\equiv\rho_z p_z/\rho^2$, where $\rho_z$ and $p_z$ are the vertical gradients of, respectively, gas density and pressure. The combined stabilizing effects of rotation ($\kappa^2$) and stratification ($\nu^2$) are apparent. In unstable discs, the conditions for instability are typically met close to the midplane, where the perturbations that are expected to grow have characteristic radial extent of a few $h_z$.