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We analyze previous results on the stability of uniformly and differentially rotating, self-gravitating, gaseous and stellar, axisymmetric systems to derive a new stability criterion for the appearance of toroidal, m = 2 intermediate or I-modes and bar modes. In the process, we demonstrate 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 tensor, dynamical instability sets in at lower rotation in stellar systems. This difference is reflected also in the new stability criterion. The new stability parameter α = TJ_/|W| is formulated first for uniformly rotating systems and is based on the angular momentum content rather than on the energy content of a system. (TJ_ = LOMEGAJ_/2; L is the total angular momentum; OMEGAJ_ is the Jeans frequency introduced by self-gravity; and W is the total gravitational potential energy. ) For stability of stellar systems α <= 0. 254-0. 258 while α <= 0. 341-0. 354 for stability of gaseous systems. For uniform rotation, one can write α = (ft/2) ¹/2^ where t = T/|W|, T is the total kinetic energy due to rotation, and f is a function characteristic of the topology/connectedness and the geometric shape of a system. Equivalently, α = t/χ, where χ = OMEGA/OMEGAJ_ and OMEGA is the rotation frequency. Using these forms, α can be extended to and calculated for a variety of differentially rotating, gaseous and stellar, axisymmetric disk and spheroidal models whose equilibrium structures and stability characteristics are known. 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 these previously published axisymmetric models.
Christodoulou et al. (Sat,) studied this question.