Local gravitational instability of stratified rotating fluids: three-dimensional criteria for gaseous discs

ABSTRACT Fragmentation of rotating gaseous systems via gravitational instability is believed to be a crucial mechanism in several astrophysical processes, such as formation planets protostellar discs, molecular clouds galactic and stars clouds. Gravitational fairly well understood for infinitesimally thin discs. However, the thin-disc approximation not justified many cases, it general interest study fluids with different degrees rotation support stratification. We derive dispersion relations axisymmetric perturbations, which can used local stability at any point system either barotropic or baroclinic distribution. Three-dimensional (3D) criteria are obtained, generalize previous results determine whether where given 3D structure prone clump formation. For vertically stratified disc thickness hz (defined containing ≈70% mass per unit surface), sufficient condition $Q_{\rm 3D}\equiv (\sqrt{\kappa ^2+\nu ^2}+c_{\rm s}h_z^{-1})/{\sqrt{4\pi G\rho }}\ \lt\ 1$, ρ gas volume density, κ epicycle frequency, cs sound speed, $\nu ^2\equiv \rho ^{\prime }_zp^{\prime }_z/\rho ^2$, $\rho }_z$ $p^{\prime vertical gradients of, respectively, density pressure. The combined stabilizing effects (κ2) stratification (ν2) apparent. In unstable conditions typically met close mid-plane, perturbations that expected grow have characteristic radial extent few hz.