A recipe for orbital eccentricity damping in the type-I regime for low-viscosity 2D discs

¹ Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
e-mail: pichierri@mpia.de
² Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange UMR7293, Boulevard de l’Observatoire, 06304 Nice Cedex 4, France

Received: 12 October 2022
Accepted: 12 December 2022

Abstract

Context. It is well known that partial and deep gap opening depends on a disc’s viscosity; however, damping formulas for orbital eccentricities have only been derived at high viscosities, ignoring partial gap opening.

Aims. In this work, we aim to obtain a simple formula to model eccentricity damping of the type-I regime in low-viscosity discs, where even small planets of a few to a few tens of Earth masses may start opening partial gaps in the gas surface density around their orbit.

Methods. We performed high-resolution, 2D, locally isothermal hydrodynamical simulations of planets with varying masses on fixed orbits in discs with varying aspect ratios and viscosities. We determined the torque and power felt by the planet to ultimately derive migration and eccentricity damping timescales.

Results. We first find a lower limit to the gap depths below which vortices appear; this happens roughly at the transition between type-I and classical type-II migration regimes. For the simulations that remain stable, we obtain a fit to the observed gap depth in the limit of vanishing eccentricities that is similar to the one currently used in the literature but accurate down to α = 3.16 × 10⁻⁵. We then record the eccentricity damping efficiency as a function of the observed gap depth and the initial eccentricity. When the planet has opened a deep enough gap such that the surface density is less than ~80% of the unperturbed disc surface density, a clear linear trend is observed independently of the planet’s eccentricity; at shallower gaps, this linear trend is preserved at low eccentricities, while it deviates to more efficient damping when e is comparable to the disc’s scale height. Both trends can be understood on theoretical grounds and are reproduced by a simple fitting formula.

Conclusions. Our combined fits for the gap depth and eccentricity damping efficiency yield a simple recipe to implement type-I eccentricity damping in N-body codes in the case of partial gap opening planets that is consistent with high-resolution 2D hydrodynamical simulations. The typical error of the final fit is of the order of a few percent, and at most ~20%, which is the error of type-I torque formulas widely used in the literature. This will allow a more self-consistent treatment of planet-disc interactions of the type-I regime for population synthesis models at low viscosities.