A Simple Analytical Model for Gaps in Protoplanetary Disks

An analytical model is presented for calculating the surface density as a function of radius $\Sigma(r)$ in protoplanetary disks in which a planet has opened a gap. This model is also applicable to circumbinary disks with extreme binary mass ratios. The gap profile can be solved for algebraically, without performing any numerical integrals. In contrast with previous one-dimensional gap models, this model correctly predicts that low-mass (sub-Jupiter) planets can open gaps in sufficiently low-viscosity disks, and it correctly recovers the power-law dependence of gap depth on planet-to-star mass ratio $q$, disk aspect ratio $h/r$, and dimensionless viscosity $\alpha$ found in previous numerical studies. Analytical gap profiles are compared with numerical calculations over a range of parameter space in $q$, $h/r$, and $\alpha$, demonstrating accurate reproduction of the "partial gap" regime, and general agreement over a wide range of parameter space.