  Gravitational coupling between a protoplanetary disc and an embedded
eccentric planet is an important, long-standing problem, which has been not yet
been conclusively explored. Here we study the torque and associated orbital
evolution of an eccentric planet in a two-dimensional disc via the
semi-analytical, fully global linear approach. Our methodology has the
advantage that the spatial structure of the density waves launched by the
planet is solved for fully. This allows us to account for the possibility of
torque excitation over an extended radial interval for each Fourier harmonic of
the perturbation, as opposed to earlier approximate treatments localized around
Lindblad and corotation resonances. We systematically explore the torque
behaviour across the space of disc properties (assuming power law profiles for
the disc surface density and temperature), including the aspect ratio.
Crucially, we examine the torque variation as the orbital eccentricity becomes
supersonic relative to the gas motion (when planetary eccentricity is of order
the disc aspect ratio), finding that the torque robustly reverses its sign near
this transition. We also find that for shallow surface density gradients
planetary migration may become outwards beyond this transition, although the
rapid eccentricity damping (which is typically $\sim 10^2$ times faster than
the orbital migration rate) would quickly restore inwards migration as the
planet circularizes. Our self-consistently computed torques are in qualitative
agreement with past numerical studies of eccentric planet-disc coupling. We
provide our torque data for different disc parameters to the community for
future testing and implementation in population synthesis studies.
