A Simple Analytical Model for Gaps in Protoplanetary Disks

TL;DR

This paper introduces a simple analytical model for calculating the surface density profiles of gaps in protoplanetary disks caused by planets, accurately matching numerical results and extending applicability to low-viscosity disks and circumbinary systems.

Contribution

The paper presents a novel algebraic analytical model for disk gaps that accurately predicts gap profiles and the conditions for gap opening by low-mass planets, improving upon previous models.

Findings

Model accurately predicts gap profiles across parameter space

Low-mass planets can open gaps in low-viscosity disks

Analytical results agree with numerical simulations

Abstract

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.