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

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.

💡 Research Summary

This paper investigates the co‑orbital flow around low‑mass planets embedded in protoplanetary disks, with the goal of quantifying the width and shape of the horseshoe (corotation) region that governs the non‑linear component of the planet’s torque. The authors first construct a semi‑analytic model of the flow by treating the disk as a two‑dimensional, locally isothermal, inviscid fluid and introducing a gravitational softening length ε to regularize the planet’s potential. In the limit ε → 0 the model yields an explicit expression for the half‑width of the horseshoe region, x_s, as a function of the planet‑to‑star mass ratio q, the disk aspect ratio h = H/r_p, and the orbital radius r_p:

  x_s / r_p = 1.68 (q / h)^{1/2}.

The coefficient 1.68 emerges from a careful balance of pressure, Coriolis, and planetary gravity forces in the co‑orbital frame. To test the model, the authors perform a suite of high‑resolution two‑dimensional hydrodynamic simulations using the FARGO code. They explore a parameter space spanning q = 10^{‑5}–10^{‑4}, h = 0.03–0.07, and several values of ε (down to 0.05 H). The measured horseshoe widths from the simulations agree with the analytic formula to within a few percent when ε is sufficiently small, confirming the robustness of the semi‑analytic approach.

Having established the scaling of x_s, the paper then evaluates the associated horseshoe drag, Γ_hs, which is the torque arising from fluid elements executing U‑shaped horseshoe turns. Using the standard expression for horseshoe drag in a barotropic disk, the authors show that Γ_hs scales as q^{2} h^{‑2} and is roughly an order of magnitude larger than the linear corotation torque derived from linear perturbation theory. Importantly, the sign of Γ_hs depends on the radial gradient of the specific vorticity (Σ/B), meaning that a modest change in the disk’s surface‑density or temperature profile can flip the direction of the net corotation torque. Consequently, horseshoe drag can dominate the total torque budget for low‑mass planets and must be included in any realistic migration model.

A novel aspect of the study is the analysis of how Lindblad wakes, which are launched at a radial distance of order the pressure scale height H from corotation, influence the horseshoe region even when x_s ≪ H. The wakes generate a “back‑pressure” that perturbs the streamlines outside the horseshoe separatrix, leading to a slight reduction of x_s compared with the isolated‑planet prediction. This effect becomes less pronounced for smaller ε (i.e., a more point‑like planetary potential) and for higher‑mass planets where x_s approaches H, because the horseshoe region then overlaps the wake region and the back‑pressure is partially absorbed.

The authors conclude that the simple formula x_s / r_p = 1.68 (q / h)^{1/2} provides a reliable estimate of the horseshoe width for low‑mass planets in thin disks, and that the resulting horseshoe drag is a dominant, sign‑sensitive component of the total torque. They also highlight that the interaction between Lindblad wakes and the corotation region, mediated by back‑pressure, must be accounted for in high‑precision migration studies, especially when softening lengths are comparable to the disk scale height. The work thus bridges the gap between linear torque theory and fully non‑linear numerical experiments, offering a practical tool for planet‑formation models that aim to predict migration rates and final orbital architectures.