A NEW CRITERION FOR BAR-FORMING INSTABILITY IN RAPIDLY ROTATING GASEOUS AND STELLAR-SYSTEMS .2. NONAXISYMMETRIC FORM

We have previously introduced the parameter ct as an indicator of stability to m = 2 nonaxisymmetric modes in rotating, self-gravitating, axisymmetric, gaseous (alpha less than or similar to 0.34) and stellar (alpha less than or similar to 0.25) systems. This parameter can be written as alpha = (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 root 1-eta(2)/eta(2)[1-e(sin(-1)e, eta(2)/e(2)/F(sin(-1)e, eta(2)/e(2))], where e is the meridional eccentricity, eta 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 eta(2)/e(2). For prolate-like 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 a 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 determined with sufficient accuracy.