Planetary torques as the viscosity of protoplanetary disks

We revisit the idea that density wave wakes of planets drive accretion in p rotostellar 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 because of shocks even for Earth- mass planets. The excitation of the wake follows from standard linear theor y including the torque cutoff. We use this as input to an approximate but q uantitative 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 Ear th masses each, dimensionless viscosities (alpha) of the order of -4 dex to -3 dex result. We compare this with observational constraints. Such small planets would have escaped detection in radial velocity surveys and could b e ubiquitous. If so, then the similarity of the observed lifetime of T Taur i disks to the theoretical timescale for assembling a rocky planet may be f ate rather than coincidence.