Late-time non-thermal emission from mildly relativistic tidal ejecta of compact objects merger

Mergers of compact objects (binary neutron stars, BNS, or neutron star-black hole, NSBH) with a substantial mass ratio ($q>1.5$) are expected to produce a mildly relativistic ejecta within $\sim20^\circ$ from the equatorial plane. We present a semi-analytic approach to calculate the expected synchrotron emission observed from various viewing angles, along with the corresponding radio maps, that are produced by a collisionless shock driven by such ejecta into the interstellar medium. This method reproduces well (up to $\sim30\%$ deviations) the observed emission produced by 2D numerical calculations of the full relativistic hydrodynamics. We consider a toroidal ejecta with an opening angle of $15^\circ\leq\theta_ \text{open}\leq30^\circ$ and broken power-law mass distribution, $M(>\gamma\beta)\propto(\gamma\beta)^{-s}$ 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). The parameter values are chosen to characterize 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. While the peak flux is dimmer by a factor of $\sim$2-3, and the peak time remains roughly the same (within $20\%$), for various viewing angles compared to isotropic equivalent ejecta ($\theta_\text{open}=90^\circ$) considered in preceding papers, the radio maps are significantly different from the spherical case. The semi-analytic method can provide information on the ejecta geometry and viewing angle from future radio map observations and, consequently, constrain the ejection mechanism. For NSBH mergers with a significant mass ejection ($\sim0.1M_\odot$), this late non-thermal signal can be observed to distances of $\lesssim 200$Mpc for typical parameter values.