Synchrotron break frequencies of mildly-to-highly relativistic outflows observed off-axis

Abstract

We consider the synchrotron spectrum produced by mildly-to-highly relativistic collisionless shocks. Simple analytic formulae are derived for the break frequencies (peak frequency, self-absorption frequency, synchrotron and inverse Compton cooling frequencies) of the emission produced by post-shock plasma elements propagating at an angle $θ_e$ relative to the observer's line of sight. These formulae reproduce well the results of earlier exact analytic calculations valid for ultra-relativistic shocks and also hold for $γ<10$ and for "off-axis" propagation (deviating from the ultra-relativistic results by approximately an order of magnitude). Our results will improve parameter estimation accuracy from future observations of synchrotron emission produced by collisionless shocks driven by the relativistic ejected material from compact objects mergers and jetted tidal disruption events. The improved accuracy for mildly relativistic velocities is essential since most events will be observed off-axis, with $γ<10$ outflows dominating the synchrotron emission (due to relativistic beaming). For GW170817, our results imply that (i) the Lorentz factor of the plasma emitting the observed radiation is bounded by $2.6<γ$ at $t\sim10$ days and by $γ<12$ at $t>16$ days, (ii) the interstellar medium (ISM) density, $n$, and the fraction of internal energy density held by magnetic fields, $\varepsilon_B$, are bounded by $n\cdot\varepsilon_B\lesssim 3\times10^{-7}$cm$^{-3}$. In future merger events in higher-density ISM, the peak and cooling frequencies may be identified in the radio and X-ray bands; consequently, $γ,n\cdot\varepsilon_B$ could be measured as opposed to the case of GW170817, where these frequencies are out of the observable range.