A NEW CRITERION FOR BAR-FORMING INSTABILITY IN RAPIDLY ROTATING GASEOUS AND STELLAR-SYSTEMS .1. AXISYMMETRICAL FORM

We analyze previous results on the stability of uniformly and differen tially rotating, self-gravitating, gaseous and stellar, axisymmetric s ystems to derive a new stability criterion for the appearance of toroi dal, m = 2 intermediate or I-modes and bar modes. In the process, we d emonstrate that the bar modes in stellar systems and the m = 2 I-modes in gaseous systems have many common physical characteristics and only one substantial difference: because of the anisotropy of the stress t enser, dynamical instability sets in at fewer rotation in stellar syst ems. This difference is reflected also in the new stability criterion. The new stability parameter alpha = T-J/\W\ is formulated first for u niformly rotating systems and is based on the angular momentum content rather than on the energy content of a system. (T-J = L Omega(J)/2; L is the total angular momentum; Omega(J) is the Jeans frequency introd uced by self-gravity; and W is the total gravitational potential energ y.) For stability of stellar systems alpha less than or equal to 0.254 -0.258 while alpha less than or equal to 0.341-0.354 for stability of gaseous systems. For uniform rotation, one can write alpha = (ft/2)(1/ 2), where t = T/\W\, T is the total kinetic energy due to rotation, an d f is a function characteristic of the topology/connectedness and the geometric shape of a system. Equivalently, alpha = t/chi, where chi = Omega/Omega(J) and Omega is the rotation frequency. Using these forms , alpha can be extended to and calculated for a variety of differentia lly rotating, gaseous and stellar, axisymmetric disk and spheroidal mo dels whose equilibrium structures and stability characteristics are kn own. In this paper, we also estimate a for gaseous toroidal models and for stellar disk systems embedded in an inert or responsive ''halo.'' We find that the new stability criterion holds equally well for all t hese previously published axisymmetric models.