Slow modes in Keplerian disks

Low-mass disks orbiting a massive body can support "slow" normal modes, in which the eigenfrequency is much less than the orbital frequency. Slow mode s are lopsided, i.e., the azimuthal wavenumber m=1. We investigate the prop erties of slow modes, using softened self-gravity as a simple model for col lective effects in the disk. We employ both the WKB approximation and numer ical solutions of the linear eigenvalue equation. We find that all slow mod es are stable. Discrete slow modes can be divided into two types, which we label g-modes and p-modes. The g-modes involve long leading and long traili ng waves, have properties determined by the self-gravity of the disk, and a re only present in narrow rings or in disks where the precession rate is do minated by an external potential. In contrast, the properties of p-modes ar e determined by the interplay of self-gravity and other collective effects. P-modes involve both long and short waves, and in the WKB approximation ap pear in degenerate leading and trailing pairs. Disks support a finite numbe r - sometimes zero - of discrete slow modes and a continuum of singular mod es.