Stability of Power-Law Disks II -- The Global Spiral Modes

Abstract

This paper reports on the in-plane normal modes in the self-consistent and the cut-out power-law disks. Although the cut-out disks are remarkably stable to bisymmetric perturbations, they are very susceptible to one-armed modes. For this harmonic, there is no inner Lindblad resonance, thus removing a powerful stabilising influence. A physical mechanism for the generation of the one-armed instabilities is put forward. Incoming trailing waves are reflected as leading waves at the inner cut-out, thus completing the feedback for the swing-amplifier. Growing three-armed and four-armed modes occur only at very low temperatures. The rotation curve index has a marked effect on stability. For all azimuthal wavenumbers, any unstable modes persist to higher temperatures and grow more vigorously if the rotation curve is rising than if the rotation curve is falling. If the central regions or outer parts of the disk are carved out more abruptly, any instabilities become more virulent. The self-consistent power-law disks possess a number of unusual stability properties. There is no natural time-scale in the self-consistent disk. If a mode is admitted at some pattern speed and growth rate, then it must be present at all pattern speeds and growth rates. Our analysis suggests that such a two-dimensional continuum of non-axisymmetric modes does not occur and that the self-consistent power-law disks admit no global non-axisymmetric normal modes whatsoever. Without reflecting boundaries or cut-outs, there is no resonant cavity and no possibility of unstable growing modes. The self-consistent power-law disks certainly admit equiangular spirals as neutral modes, together with a one-dimensional continuum of growing axisymmetric modes.