On the horseshoe drag of a low-mass planet. II Migration in adiabatic disks

We evaluate the horseshoe drag exerted on a low-mass planet embedded in a gaseous disk, assuming the disk’s flow in the coorbital region to be adiabatic. We restrict this analysis to the case of a planet on a circular orbit, and we assume a steady flow in the corotating frame. We also assume that the corotational flow upstream of the U-turns is unperturbed, so that we discard saturation effects. In addition to the classical expression for the horseshoe drag in barotropic disks, which features the vortensity gradient across corotation, we find an additional term which scales with the entropy gradient, and whose amplitude depends on the perturbed pressure at the stagnation point of the horseshoe separatrices. This additional torque is exerted by evanescent waves launched at the horseshoe separatrices, as a consequence of an asymmetry of the horseshoe region. It has a steep dependence on the potential’s softening length, suggesting that the effect can be extremely strong in the three dimensional case. We describe the main properties of the coorbital region (the production of vortensity during the U-turns, the appearance of vorticity sheets at the downstream separatrices, and the pressure response), and we give torque expressions suitable to this regime of migration. Side results include a weak, negative feed back on migration, due to the dependence of the location of the stagnation point on the migration rate, and a mild enhancement of the vortensity related torque at large entropy gradient.

💡 Research Summary

This paper presents a comprehensive analytical treatment of the corotation (horseshoe) torque acting on a low‑mass planet embedded in a gaseous protoplanetary disk, under the explicit assumption that the flow in the planet’s co‑orbital region is adiabatic. The authors restrict themselves to a planet on a circular orbit, adopt a steady‑state flow in the rotating frame, and deliberately neglect any saturation of the corotation region by assuming that the upstream flow before the horseshoe U‑turns remains unperturbed. Within this framework they recover the classic barotropic horseshoe drag, which is proportional to the radial gradient of vortensity (potential vorticity). In addition, they identify a second, previously unrecognised contribution that scales with the radial entropy gradient.

The entropy‑related term originates from evanescent pressure waves launched at the separatrices of the horseshoe region. Because the adiabatic flow makes the horseshoe region intrinsically asymmetric when an entropy gradient is present, the pressure perturbation at the stagnation point (the point where the separatrices intersect) is no longer symmetric. This asymmetry produces a net pressure excess on one side of the planet, which translates into an extra torque. The magnitude of this torque depends sensitively on the softening length used to regularise the planetary potential; as the softening length is reduced, the pressure perturbation grows sharply, implying that in a fully three‑dimensional disk—where the effective softening is naturally small—the entropy‑driven torque can dominate the total corotation torque.

The paper also details several ancillary dynamical features of the co‑orbital flow. During each U‑turn, fluid parcels conserve both vortensity and entropy, but experience a change in pressure and temperature that generates a modest amount of vorticity. Downstream of the U‑turns, thin vorticity sheets (or “vorticity sheets”) develop along the separatrices, altering the local shear and influencing the subsequent evolution of the horseshoe region. These sheets act to inhibit rapid mixing and therefore delay the saturation of the corotation torque.

A further subtle effect is the feedback of the migration rate on the torque itself. As the planet migrates, the stagnation point shifts radially, which modifies the pressure perturbation and consequently the entropy‑related torque. The authors find this feedback to be weakly negative: faster inward migration slightly reduces the extra torque, providing a modest self‑regulating mechanism.

Putting all these ingredients together, the authors derive a compact torque formula that includes both the vortensity‑gradient term and the entropy‑gradient term, with explicit dependence on the softening length and on the pressure response at the stagnation point. This formula reduces to the classic barotropic result when the entropy gradient vanishes, but predicts a substantially larger (often positive) corotation torque in disks with realistic temperature and density gradients.

The implications are significant for planet formation theory. In disks where the temperature drops steeply outward—a common situation in young, massive disks—the entropy‑driven horseshoe drag can counteract, or even outweigh, the usual Lindblad (wave) torque that drives inward migration. Consequently, low‑mass planets may experience slowed, stalled, or even outward migration, helping to resolve the long‑standing “type‑I migration problem”. Moreover, because the effect is amplified in three dimensions, the results suggest that fully 3‑D hydrodynamic simulations should exhibit even stronger entropy‑related torques than the 2‑D models presented here.

Overall, the paper extends the theoretical foundation of type‑I migration by incorporating adiabatic thermodynamics, clarifying the role of entropy gradients, and providing a practical torque prescription that can be implemented in population synthesis models and numerical simulations of planet‑disk interaction.