The Evolution of Protoplanetary Disk Edges

We investigate gap formation in gaseous protostellar disks by a planet in a circular orbit in the limit of low disk viscosity. This regime may be appropriate to an aging disk after the epoch of planet formation. We find that the distance of planet to the gap outer boundary can be between the location of the $m=2$ and $m=1$ outer Lindblad resonances. This distance is weakly dependent upon both the planet's mass and disk viscosity. We find that the evolution of the disk edge takes place on two timescales. The first timescale is set by the spiral density waves driven by the planet. The second timescale depends on the viscosity of the disk. The disk approaches a state where the outward angular momentum flux caused by the disk viscosity is balanced by the dissipation of spiral density waves which are driven at the Lindblad resonances. This occurs inefficiently however because of the extremely low gas density near the planet. We find that the distance between the planet and the peak density at the disk outer edge is only weakly dependent on the viscosity and planet mass, however the ratio of the gas density near the planet to that in the disk (or the slope of density along the disk edge) is strongly dependent upon both quantities. We find that the disk density profile along the edge scales approximately with disk viscosity divided by the square of the planet mass. We account for this behavior with a simple scenario in which the dissipation of angular momentum from the spiral density waves is balanced against diffusion in the steep edge of the disk.