  We investigate how planets interact with viscous accretion disks, in the
limit that the disk is sufficiently low mass that the planet migrates more
slowly than the disk material. In that case, the disk's surface density profile
is determined by the disk being in viscous steady state (VSS), while
overflowing the planet's orbit. We compute the VSS profiles with 2D
hydrodynamical simulations, and show that disk material piles up behind the
planet, with the planet effectively acting as a leaky dam. Previous 2D
hydrodynamical simulations missed the pileup effect because of incorrect
boundary conditions, while previous 1D models greatly overpredicted the pileup
due to the neglect of non-local deposition. Our simulations quantify the
magnitude of the pileup for a variety of planet masses and disk viscosities. We
also calculate theoretically the magnitude of the pileup for moderately deep
gaps, showing good agreement with simulations. For very deep gaps, current
theory is inadequate, and we show why and what must be understood better. The
pileup is important for two reasons. First, it is observable in directly imaged
protoplanetary disks, and hence can be used to diagnose the mass of a planet
that causes it or the viscosity within the disk. And second, it determines the
planet's migration rate. Our simulations determine a new Type-II migration rate
(valid for low mass disks), and show how it connects continuously with the
well-verified Type-I rate.
