Non-thermal emission from mildly relativistic dynamical ejecta of neutron star mergers: spectrum and sky image

Binary neutron star mergers are expected to produce fast dynamical ejecta, with mildly relativistic velocities extending to $\beta=v/c>0.6$. In a preceding paper, we derived an analytic description of the time-dependent radio to X-ray synchrotron flux produced by collisionless shocks driven by such fast ejecta into the interstellar medium, for spherical ejecta with broken power-law mass (or energy) distributions, $M(>\gamma\beta)\propto(\gamma\beta)^{-s}$ with $s=s_\text{KN}$ at $\gamma\beta<\gamma_0\beta_0$ and $s=s_\text{ft}$ at $\gamma\beta>\gamma_0\beta_0$ (where $\gamma$ is the Lorentz factor). Here, we extend our analysis and provide analytic expressions for the self-absorption frequency, the cooling frequency, and the observed angular size of the emitting region (which appears as a ring in the sky). For parameter values characteristic of merger calculation results -- a "shallow" mass distribution, $1<s_\text{KN}<3$, for the bulk of the ejecta (at $\gamma\beta\approx0.2$), and a steep, $s_\text{ft}>5$, "fast tail" mass distribution -- the analytic results reproduce well (to tens of percent accuracy) the results of detailed numeric calculations, a significant improvement over earlier order-of-magnitude estimates (based on extrapolations of results valid for $\gamma\beta\ll1$).