On the corotation torque for low-mass eccentric planets

We present the results of high-resolution 2D simulations of low-mass planets on fixed eccentric orbits embedded in protoplanetary discs. The aim of this study is to determine how the strength of the sustained, non-linear corotation torque experienced by embedded planets varies as a function of orbital eccentricity, disc parameters and planetary mass. In agreement with previous work we find that the corotation torque diminishes as orbital eccentricity, e, increases. Analysis of the time-averaged streamlines in the disc demonstrates that the width of the horseshoe region narrows as the eccentricity increases, and we suggest that this narrowing largely explains the observed decrease in the corotation torque. We employ three distinct methods for estimating the strength of the unsaturated corotation torque from our simulations, and provide an empirical fit to these results. We find that a simple model where the corotation torque, Gamma(C), decreases exponentially with increasing eccentricity [i.e. Gamma(C) alpha exp (-e/e(f))] provides a good global fit to the data with an e-folding eccentricity, e(f), that scales linearly with the disc scale height at the planet location. We confirm that this model provides a good fit for planet masses of 5 and 10 M-circle plus in our simulations. The formation of planetary systems is likely to involve significant planet-planet interactions that will excite eccentric orbits, and this is likely to influence disc driven planetary migration through modification of the corotation torque. Our results suggest that high fidelity models of planetary formation should account for these effects.