1997NewA....2....1C

We have investigated numerically the stability and temporal evolution of two-dimensional self-gravitating galaxy rings with flat rotation curves. The initial equilibrium models are composed of a polytropic fluid characterized by various polytropic indices (n = 0.5, 1, 1.5, 3, and 5) and embedded in a rigid scale-free logarithmic potential, but the fluid is evolved as an ideal gas with specific heat ratio gamma = 5/3. Massless and very low-mass rings are unconditionally stable against both self-gravity driven modes and convective modes. As the self-gravity is increased, one of three nonaxisymmetric modes of instability soon appears, depending on the polytropic index and the radial extent: Jeans modes (J-modes) for n = 5 rings; Intermediate modes (I-modes) for 1 ≤ n ≤ 3 and for slender I-modes, for extended n = 0.5 rings. The radial thickness of the orbiting fluid also influences the growth of these modes: more (less) extended rings require more (less) mass in order to develop an instability. If v_p is the characteristic speed dictated by the potential and v_o is the true orbital speed in the rings, then these instabilities operate for values of v_p/vo ≤ 0.95-0.99 (ring-to-galaxy mass ratios approximately ≳ 0.11-0.02). At slightly higher mass ratios, strong unstable modes commonly cause a temporary breakup of the rings and the formation of new, lumpy but long-lived, ring structures. The entire process is completed within typically 3 orbits (J-modes) to 6-10 orbits (I-modes). Thus, neither nuclear rings in barred galaxies or in active galactic nuclei, nor (co)counter-rotating "cores" or polar rings in elliptical/S0 galaxies can safely accumulate matter through accretion and survive in a smooth form. In particular, some lumpy or distorted circumnuclear rings seen in barred galaxies may be due to the acting I-modes or J-modes that set in as soon as the accreted masses outgrow the relatively small critical value given above. Furthermore, the surprising absence of unstable g-modes from many models with n ≤ 1.5 which do not satisfy the Richardson stability criterion suggests that galaxy rings exhibit dynamical properties that cannot be identified by studying infinite cylindrical/annular models. Counter to intuition, the g-modes are absent because the shear and rotation rates are high in mildly compressible planar rings with flat rotation curves.