Non-thermal emission from mildly relativistic dynamical ejecta of neutron star mergers

Binary neutron star mergers are expected to produce fast dynamical ejecta, with mildly relativistic velocities extending to $\beta=v/c>0.6$. We consider the radio to X-ray synchrotron emission produced by collisionless shocks driven by such fast ejecta into the interstellar medium. Analytic expressions are given for spherical ejecta with broken power-law mass (or energy) distributions, $M(>\gamma\beta)\propto(\gamma\beta)^{-s}$ with $s=s_{\rm KN}$ at $\gamma\beta<\gamma_0\beta_0$ and $s=s_{\rm ft}$ at $\gamma\beta>\gamma_0\beta_0$ (where $\gamma$ is the Lorentz factor). For parameter values characteristic of merger calculation results -- a "shallow" mass distribution, $1<s_{\rm KN}<3$, for the bulk of the ejecta (at $\gamma\beta\approx 0.2$), and a steep, $s_{\rm ft}>5$, "fast tail" mass distribution -- our model provides an accurate (to 10's of percent) description of the evolution of the flux, including at the phase of deceleration to sub-relativistic expansion. This is a significant improvement over earlier results, based on extrapolations of results valid for $\gamma\beta\gg1$ or $\ll1$ to $\gamma\beta\approx1$, which overestimate the flux by an order of magnitude for typical parameter values. It will enable a more reliable inference of ejecta parameters from future measurements of the non-thermal emission. For the merger event GW170817, the existence of a "fast tail" is expected to produce detectable radio and X-ray fluxes over a time scale of $\sim10^4$days.