The Response of Accretion Disks to Bending Waves: Angular Momentum Transport and Resonances

Abstract

We investigate the linear tidal perturbation of a viscous Keplerian disk by a companion star orbiting in a plane inclined to the disk. We consider m = 1 perturbations with odd symmetry with respect to the z = 0 midplane. The response frequency may be either finite or vanishing. These long-wavelength perturbations produce a well-defined warp. Since the response of a viscous disk is not in phase with the perturbing potential, a tidal torque is exerted on the disk. When the perturber rotates outside the disk, this torque results in a decrease of the disk angular momentum and thus in an increase of its accretion rate. We show that this tidal torque is comparable to the horizontal viscous stress acting on the background flow when the perturbed velocities in the disk are on the order of the sound speed. If these velocities remain subsonic, the tidal torque can exceed the horizontal viscous stress only if the viscous parameter αv, which couples to the vertical shear, is larger than the parameter αh coupled to the horizontal shear. In protostellar disks, bending waves, which are predominantly excited in the outer regions, are found to propagate and transport a significant fraction of the negative angular momentum they carry deep into the disk inner parts. If the waves are reflected at the center, resonances occur when the frequency of the tidal waves is equal to that of some free normal global bending mode of the disk. If such resonances exist, tidal interactions may then be important even when the binary separation is large. Out of resonance, the torque associated with the secular perturbation, which is proportional to αv, is generally much larger than that associated with the finite-frequency perturbations. As long as the waves are damped before they reach the center, the torque associated with the finite-frequency perturbations does not depend on the viscosity, in agreement with theoretical expectation. These calculations are relevant to disks around young stars and maybe also to disks in X-ray binary systems.