TIDALLY DRIVEN INCLINATION INSTABILITY IN KEPLERIAN DISKS

We consider a disk or ring that orbits about a central object and is tidally perturbed by a low-mass companion in circular orbit. We investigate the vertical (out of the orbit plane) stability of the disk. We first consider the case that the disk consists of free (noninteracting) ballistic particles. Near the locations of parametric resonances, where OMEGA(r) = mOMEGA(p)/(m +/- 2) for companion angular speed OMEGA(p) and integer m, vertical oscillations of free particles are unstable with growth rate that is linear in the tidal potential. However, the horizontal parametric instability of free particles is stronger by a factor of several at the 3:1 resonance. For disks in which the collective effects are important, disks are shown to be tilt-unstable. Let integer pairs (k, l) label the modal numbers of a quantity that varies periodically in ktheta - lOMEGA(p)t. Inclination Lindblad resonances that lie within a disk cause inclination growth by the following process. A tilt perturbation of the disk gives rise to a (1, 0) mode for the vertical displacement. Under the influence of the tidal potential component phi(m) with modal numbers (m, m), these vertical motions couple to give rise to a resonant vertical displacement with modal numbers (m +/- 1, m) at locations where OMEGA(r) = mOMEGA(p)/(m +/- 2). This resonant response then couples back with the (m, m) tidal field to give rise to a (1, 0) vertical stress that acts to increase the tilt. The tilt growth rate is quadratic in the tidal potential. However, the tilt growth rate is numerically substantially smaller than the eccentricity growth rate. Several factors contribute to the tilt instability. In particular, tidally induced horizontal motions are very important in destabilizing the tilt for low m. We discuss these results mainly in the context of superhump binary accretion disks and planetary rings.