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

It is known that gap opening depends on the disc's viscosity; however, eccentricity damping formulas have only been derived at high viscosities, ignoring partial gap opening. We aim at obtaining a simple formula to model $e$-damping of the type-I regime in low viscosity discs, where even small planets may start opening partial. We perform high resolution 2D locally isothermal hydrodynamical simulations of planets with varying masses on fixed orbits in discs with varying aspect ratios and viscosities. We determine the torque and power felt by the planet to derive migration and eccentricity damping timescales. We first find a lower limit to the gap depths below which vortices appear; this happens roughly at the transition between type-I and type-II 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 is accurate down to $\alpha=3.16\times 10^{-5}$. We record the $e$-damping efficiency as a function of the observed gap depth and $e$: when the planet has opened a deep enough gap, a linear trend is observed independently of $e$; at shallower gaps this linear trend is preserved at low $e$, 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. Our combined fits yield a simple recipe to implement type-I $e$-damping in $N$-body for partial gap opening planets that is consistent with high-resolution 2D hydro-simulations. The typical error of the fit is of the order of a few percent, and lower than 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.