On the width and shape of the corotation region for low-mass planets

TL;DR

This paper develops a semi-analytic model for the corotation region of low-mass planets, accurately predicting the horseshoe region width and highlighting its significance for planetary torque estimates.

Contribution

It introduces a simple model for the corotation region that matches simulations and clarifies the impact of phenomena like Lindblad wakes on horseshoe width.

Findings

Derived an expression for horseshoe region width, x_s, in terms of planet and disc parameters.

Found that horseshoe drag can be an order of magnitude larger than linear corotation torque.

Showed that Lindblad wakes can influence the horseshoe width despite being separated by a pressure scale height.

Abstract

We study the coorbital flow for embedded, low mass planets. We provide a simple semi-analytic model for the corotation region, which is subsequently compared to high resolution numerical simulations. The model is used to derive an expression for the half-width of the horseshoe region, x_s, which in the limit of zero softening is given by x_s/r_p = 1.68(q/h)^(1/2), where q is the planet to central star mass ratio, h is the disc aspect ratio and r_p the orbital radius. This is in very good agreement with the same quantity measured from simulations. This result is used to show that horseshoe drag is about an order of magnitude larger than the linear corotation torque in the zero softening limit. Thus the horseshoe drag, the sign of which depends on the gradient of specific vorticity, is important for estimates of the total torque acting on the planet. We further show that phenomena, such as the Lindblad wakes, with a radial separation from corotation of ~ a pressure scale height H can affect x_s, even though for low-mass planets x_s << H. The effect is to distort streamlines and to reduce x_s through the action of a back pressure. This effect is reduced for smaller gravitational softening parameters and planets of higher mass, for which x_s becomes comparable to H.