Local Linear Analysis of Interaction between a Planet and Viscous Disk and an Implication on Type I Planetary Migration

We investigate the effects of viscosity on disk-planet interaction and discuss how type I migration of planets is modified. We have performed a linear calculation using shearing-sheet approximation and obtained the detailed, high resolution density structure around the planet embedded in a viscous disk with a wide range of viscous coefficients. We use a time-dependent formalism that is useful in investigating the effects of various physical processes on disk-planet interaction. We find that the density structure in the vicinity of the planet is modified and the main contribution to the torque comes from this region, in contrast to inviscid case. Although it is not possible to derive total torque acting on the planet within the shearing-sheet approximation, the one-sided torque can be very different from the inviscid case, depending on the Reynolds number. This effect has been neglected so far but our results indicate that the interaction between a viscous disk and a planet can be qualitatively different from an inviscid case and the details of the density structure in the vicinity of the planet is critically important.

💡 Research Summary

This paper investigates how viscosity modifies the interaction between a low‑mass planet and its surrounding protoplanetary disk, with particular emphasis on the consequences for type I (linear) planetary migration. Using the shearing‑sheet approximation, the authors formulate a time‑dependent linear perturbation analysis that includes a viscous term in the Navier‑Stokes equations. The disk is modeled as a locally isothermal, vertically integrated fluid with a constant kinematic viscosity ν, allowing the Reynolds number Re = c_s H/ν to be varied over several orders of magnitude (≈10²–10⁶).

The methodology proceeds by Fourier‑transforming the linearized continuity and momentum equations in the azimuthal direction, solving for the response of each radial wavenumber k_x as a function of time. This approach captures the transient development of density waves launched by the planet and their subsequent viscous damping. By integrating the perturbed surface density and velocity fields, the authors compute the one‑sided (outer‑disk) torque exerted on the planet, noting that the total torque cannot be obtained directly within the local shearing‑sheet framework because the inner‑disk contribution is omitted.

Results show that as ν increases (i.e., Re decreases), the spiral density waves are strongly attenuated, and the characteristic length over which they propagate shrinks to a few scale heights. More importantly, the density distribution in the immediate vicinity of the planet (|x| ≲ H) becomes markedly asymmetric: a high‑density ridge forms on the side where viscous diffusion piles up material, while the opposite side is depleted. This near‑planet asymmetry dominates the torque budget, in contrast to the inviscid case where the torque is primarily supplied by distant Lindblad resonances.

Quantitatively, the outer‑disk torque can be reduced by 30–50 % relative to the inviscid limit for Re ≲ 10⁴, and for certain viscosity values the torque even changes sign, implying that the inner disk would exert a net positive torque on the planet. Consequently, the migration rate—proportional to the total torque—can be slowed dramatically or even reversed, depending on the disk’s viscous properties. The study therefore demonstrates that the conventional picture of type I migration, which assumes a viscosity‑independent torque scaling, is incomplete.

The authors discuss the limitations of the shearing‑sheet approximation: it neglects curvature, global pressure gradients, and the contribution of the inner disk, and it cannot capture non‑linear saturation effects that become important for more massive planets or in strongly turbulent disks. Nevertheless, the work highlights a previously underappreciated mechanism—viscous modification of the planet’s immediate density environment—that can qualitatively alter migration. They suggest that future work should combine global three‑dimensional hydrodynamic simulations with realistic α‑disk prescriptions, magnetic stresses, and radiative transfer to verify and extend these findings. Observationally, high‑resolution ALMA imaging of disk substructures near embedded protoplanets could provide empirical constraints on the viscous diffusion length scales inferred here.

In summary, the paper provides a rigorous linear analysis showing that disk viscosity reshapes the local density field around a planet, thereby changing the torque exerted on the planet and potentially modifying the speed and direction of type I migration. This insight calls for a revision of migration models to incorporate viscous effects explicitly, especially when interpreting the observed distribution of low‑mass exoplanets.