A New Criterion for Bar-forming Instability in Rapidly Rotating Gaseous and Stellar Systems. II. Nonaxisymmetric Form

We have previously introduced the parameter α as an indicator of stability to m = 2 nonaxisymmetric modes in rotating, self-gravitating, axisymmetric, gaseous (α <~0.34) and stellar (α <~ 0.25) systems. This parameter can be written as α = (ft/2)^1/2^, where t = T/|W|, T is the total rotational kinetic energy, W is the total gravitational potential energy, and f is a function characteristic of the topology/connectedness and the geometric shape of a system. In this paper, we extend the stability criterion to nonaxisymmetric equilibrium systems by determining empirically the appropriate form of the function f for ellipsoids and elliptical disks and cylinders. For oblate-like ellipsoidal systems, we find that f = 2*(1-η^2^)^1/2^/η^2^*[1-e(sin^-1^(e,η^2^/e^2^)) /F(sin^-1^(e,η^2^/e^2^)] where e is the meridional eccentricity, n is the equatorial eccentricity, and F and E are the incomplete elliptic integrals of the first and second kind, respectively, with amplitude sin^-1^ e and parameter n^2^/e^2^. For prolate-ellipsoidal systems, we find an analogous expression that reduces to f = 0 in the limiting case of infinite cylinders. We test the validity of this extension of the stability indicator α by considering its predictions for previously published, gaseous and stellar, nonaxisymmetric models. The above formulation and critical values account accurately for the stability properties of m = 2 modes in gaseous Riemann S-type ellipsoids (including the Jacobi and Dedekind ellipsoids) and elliptical Riemann disks as well as in stellar elliptical Freeman disks and cylinders: all these systems are dynamically stable except the stellar elliptical Freeman disks that exhibit a relatively small region of m = 2 dynamical instability. A partial disagreement in the case of stellar Freeman ellipsoids in maximum rotation may be due to that the region of instability has not been previously deter- mined with sufficient accuracy.