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

Abstract: Mergers of compact objects (binary neutron stars, BNS, or neutron star-black hole, NSBH) with a substantial mass ratio ($q&gt;1.5$) are expected to produce a mildly relativistic ejecta within $\sim20<sup>\circ$ 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 $$15^{\circ\leq\theta_} \text{open}\leq30<sup>\circ$$ and broken power-law mass distribution, $M(&gt;\gamma\beta)\propto(\gamma\beta)<sup>{-s}$ with $s=s_{\rm KN}$ at $\gamma\beta&lt;\gamma_0\beta_0$ and $s=s_{\rm ft}$ at $\gamma\beta&gt;\gamma_0\beta_0$ (where $\gamma$ is the Lorentz factor). The parameter values are chosen to characterize merger calculation results -- a "shallow" mass distribution, $1&lt;s_{\rm KN}\&lt;3$, for the bulk of the ejecta (at $\gamma\beta\approx 0.2$), and a steep, $$s_{\rm ft}\&gt;5$$, "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 ($\theta_\text{open}=90<sup>\circ$) 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.