The Peak Frequency and Luminosity of Synchrotron Emitting Shocks: from Non-Relativistic to Ultra-Relativistic Explosions

Synchrotron emission is ubiquitous in explosive astrophysical events -- it is a natural byproduct of shocks formed when matter expelled by the explosion collides with ambient material. This emission is well-observed in various classes of transients, and is often interpreted within a canonical `equipartition' framework that allows physical properties of the shock to be inferred from the frequency and luminosity at which the observed spectral energy distribution (SED) peaks. This framework has been remarkably successful in explaining observations of radio supernovae. It has also been used for trans-relativistic explosions, where the shock velocities approach the speed of light. However, the conventional framework does not incorporate relativistic effects. Neither does it account for thermal electrons, which have been shown to be important for high-velocity shocks. In this paper we describe a revised framework that accounts for these two effects, and is applicable to non-relativistic, trans-relativistic, and ultra-relativistic explosions. We show that accounting for these effects can dramatically change the inferred parameters of high-velocity shocks, and in particular -- that the shock velocity, ambient density, and total energy are overestimated by the conventional non-relativistic framework. We delineate the phase-space where such modifications are important in terms of observationally measurable parameters. We also find a novel upper limit on the peak synchrotron luminosity of shock-powered transients, which is remarkably consistent with existing observations. Finally, we discuss a prediction of the model -- that the SED will qualitatively change as a function of shock velocity -- and show that this is broadly consistent with data for representative events (e.g., SN1998bw, AT2018cow, CSS161010, AT2020xnd).