Compact binary systems in Einstein-{AE}ther gravity. II. Radiation reaction to 2.5 post-Newtonian order

We obtain the equations of motion for compact binary systems (black holes or neutron stars) in an alternative theory of gravity known as Einstein-AEther theory, which supplements the standard spacetime metric with a timelike four-vector (the AEther field) that is constrained to have unit norm. The equations make use of solutions obtained in Paper I for the gravitational and AEther field potentials within the near zone of the system, evaluated to 2.5 post-Newtonian (PN) order ($O(v/c)^5$ beyond Newtonian gravity), sufficient to obtain the effects of gravitational radiation reaction to the same order as the quadrupole approximation of general relativity. Those potentials were derived by applying the post-Minkowskian method to the field equations of the theory. Using a modified geodesic equation that is a consequence of the effects of the interaction between the AEther field and the internal strong-gravity fields of the compact bodies, we obtain explicit equations of motion in terms of the positions and velocities of the bodies, focussing on the radiation-reaction terms that contribute at 1.5PN and 2.5PN orders. We obtain the rate of energy loss by the system, including the effects of dipole gravitational radiation (conventionally denoted $-1$PN order) and the analogue of quadrupole radiation (denoted $0$PN order). We find significant disagreements with published results, based on calculating the energy flux in the far zone using a ``Noether current'' construction.