An empirically-derived formula for the shape of planet-induced gaps in protoplanetary disks

Abstract: This study uses numerical hydrodynamics calculations and a novel method for densely sampling parameter space to measure the precise shape of a gap opened by a planet in a gaseous disk, as a function of planet-to-star mass ratio, disk Mach number, and disk viscosity. Formulas for gap depth and width are determined, which are combined to form a complete formula for surface density as a function of radius in the disk. This new analytical formula is compared with numerically-derived gaps opened by planets ranging from very low masses up to a few times Jupiter's mass, and excellent agreement is found over a wide range of parameter space relevant to planet-disk interactions. A simple-to-use code is presented to rapidly generate synthetic disk profiles.