Slow Modes in Keplerian Disks

Low-mass disks orbiting a massive body can support "slow" normal modes, inwhich the eigenfrequency is much less than the orbital frequency. Slow modesare lopsided, i.e., the azimuthal wavenumber m=1. We investigate the propertiesof slow modes, using softened self-gravity as a simple model for collectiveeffects in the disk. We employ both the WKB approximation and numericalsolutions of the linear eigenvalue equation. We find that all slow modes arestable. Discrete slow modes can be divided into two types, which we labelg-modes and p-modes. The g-modes involve long leading and long trailing waves,have properties determined by the self-gravity of the disk, and are onlypresent in narrow rings or in disks where the precession rate is dominated byan external potential. In contrast, the properties of p-modes are determined bythe interplay of self-gravity and other collective effects. P-modes involveboth long and short waves, and in the WKB approximation appear in degenerateleading/trailing pairs. Disks support a finite number---sometimes zero---ofdiscrete slow modes, and a continuum of singular modes.Comment: 32 pages, 12 figures. To be published in Astronomical Journa