THE BREAKUP OF SELF-GRAVITATING RINGS, TORI, AND THICK ACCRETION DISKS

Using Hachisu's self-consistent-field technique, we have constructed equilibrium sequences of self-gravitating, axisymmetric rings (or, tori) having n = 3/2 polytropic structures and radial angular velocity profiles of the form Ω ∝ R-1/2, with l = 3.0, 3.5, and 4.0. Then, using Tohline's three-dimensional hydrodynamic computer code, we have identified the position along the equilibrium sequences where the rings first become dynamically unstable toward the development of an m = 2 (ellipsoidal), nonaxisymmetric distortion. (Principally for computational convenience, we have limited this study to nonaxisymmetric distortions with even azimuthal mode numbers.) In terms of the ratio of rotational kinetic energy T to gravitational potential energy W, the instability arises in systems having T/\W\≥ 0.16. Although the dynamics of odd azimuthal modes has not been examined here, we suspect that tori having smaller values of T/\W\ are dynamically stable to all nonaxisymmetric modes because the only odd mode having a longer azimuthal wavelength - the m = 1 mode - cannot develop without requiring a shift in the center of mass of the torus. We conclude that the onset of a dynamical instability in self-gravitating rings occurs at a value of T/\W\ that is substantially lower than the value (T/\W\ ≈ 0.27) at which an analogous instability arises in centrally condensed, self-gravitating configurations. We have quantitatively measured the growth rate and the pattern speed of the m = 2 mode in eight models having values of T/\W\ in the range 0.167 ≤ T/\W\ ≤ 0.271. The character of the unstable eigenmode in these models is clearly different from the unstable, nonaxisymmetric eigenmode that was first shown by Papaloizou and Pringle to arise in zero mass, accretion tori. We are therefore convinced that the instability we have identified here is driven by the self-gravity of the torus and is distinctly different from the Papaloizou-Pringle instability. Although the early nonlinear development of this instability causes the torus to deform into an ellipsoidal configuration with noticeable density enhancements arising on opposite sides of the ring, further evolution shows that the instability does not lead to fragmentation of the torus. We therefore call into question the perceived notion that self-gravitating rings are generally unstable toward fragmentation along their length.