Synchrotron Emission from Cooled Particle Distributions

Synchrotron emitting electrons can lose energy (`cool') through various processes including radiative losses (e.g., synchrotron or inverse-Compton cooling) and adiabatic expansion. Such cooling will shift electrons in energy-space and therefore change the electron distribution function. This in turn alters the nature of synchrotron emission and absorption from these electrons. In past literature these effects have typically been considered using either simplified one-zone frameworks, or using numerical methods as part of more accurate local modeling. In this work we extend the latter `local' treatment by deriving analytic expressions that are both accurate and more computationally efficient than previous numerical approaches. Considering two concrete cases of injected power-law and thermal electron distribution functions, we derive analytic fitting functions for the resulting emission and absorption coefficients including the effects of cooling. These fitting functions can be applied to synchrotron afterglow modeling from a variety of astrophysical sources, such as gamma-ray bursts (GRBs), luminous fast blue optical transients (LFBOTs), and jetted tidal disruption events (TDEs).