Spiral Bending Waves Launched at a Vertical Secular Resonance

Abstract

The excitation of spiral bending waves at a secular vertical resonance in a particle disk is examined. These waves are one-armed spirals of very long wavelength that are launched at sites where a secondary's nodal regression rate matches that of the disk. Nodal bending waves usually propagate radially outward as leading waves from a secular resonance exterior to the perturber, and inward as trailing waves from a secular resonance that lies interior. Their pattern speed is negative, so the spiral pattern rotates in a retrograde sense. The waves carry negative angular momentum but very little energy, and their excitation can damp the inclination of the secondary. Here we apply this theory to the case of two mutually precessing planets orbiting in a particle disk and compare their damping rate with the more familiar inclination excitation due to mean motion vertical resonances. We suggest that under certain circumstances, nodal wave damping may be an important element in maintaining planetary and/or embryo orbits in a nearly coplanar state.