The ring instability in radially cold oblate galaxy models

Goodman's (1988) indicator has been employed to investigate the stability of self-consistent perfect oblate spheroids built exclusively with thin tube orbits. The results of recent N-body simulations are applied in experiments with a variety of axisymmetric perturbations which leave the flattering of the model constant, but change the radial density distribution. Goodman's indicator has been calculated by combination of the orbital indicators with the exact and unique phase-space distribution function of the thin-orbit model. This confirms that strongly flattened thin-orbit models are unstable against a ring perturbation, in agreement with the classical result for circular disks and recent N-body simulations. The transition to stability occurs at an axis ratio of 0.33 + or - 0.02. Perturbations that are concentrated in the center are most destabilizing. The limitations of the application of Goodman's indicator are discussed. 35 refs