  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. $\Gamma_C \propto \exp{(-e/e_f)}$) 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 $\Me$ 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.
