Torques as the Viscosity of Protoplanetary Disks

We revisit the idea that density-wave wakes of planets drive accretion in protostel-lar disks. The effects of many small planets can be represented as a viscosity if the wakes damp locally, but the viscosity is proportional to the damping length. Damping occurs mainly by shocks even for earth-mass planets. The excitation of the wake follows from standard linear theory including the torque cutoff. We use this as input to an approximate but quantitative nonlinear theory based on Burger's equation for the subsequent propagation and shock. Shock damping is indeed local but weakly so. If all metals in a minimum-mass solar nebula are invested in planets of a few earth masses each, dimensionless viscosities [alpha] of order dex(-4) to dex(-3) result. We compare this with observational constraints. Such small planets would have escaped detection in radial-velocity surveys and could be ubiquitous. If so, then the similarity of the observed lifetime of T Tauri disks to the theoretical timescale for assembling a rocky planet may be fate rather than coincidence.