A Late-time Flattening of Light Curves in Gamma-Ray Burst Afterglows

The afterglow emission from gamma-ray bursts (GRBs) is usually interpreted as synchrotron radiation from relativistic electrons accelerated at the GRB external shock. We investigate the temporal decay of the afterglow emission at late times, when the bulk of the shock-accelerated electrons are non-relativistic (the "deep Newtonian phase," as denoted by Huang and Cheng). We assume that the electron spectrum in the deep Newtonian phase is a power-law distribution in momentum with slope p, as dictated by the theory of Fermi acceleration in non-relativistic shocks. For a uniform circumburst medium, the deep Newtonian phase begins at t{_{\scriptsize {DN}}}\sim 3\,\epsilon _{e,-1}^{5/6}t{_{\scriptsize {ST}}}, where t _ST marks the transition of the blast wave to the non-relativistic, spherically symmetric Sedov-Taylor (ST) solution, and epsilon _e = 0.1 epsilon _{e, -1} quantifies the amount of shock energy transferred to the electrons. For typical parameters, the deep Newtonian stage starts ~0.5 to several years after the GRB. The radio flux in this phase decays as F _νvpropt ^{-3(p + 1)/10}vpropt ^-(0.9÷1.2), for a power-law slope 2 < p < 3. This is shallower than the scaling F _νvpropt ^{-3(5p - 7)/10}vpropt ^-(0.9÷2.4) derived by Frail et al., which only applies if the GRB shock is non-relativistic, but the electron distribution still peaks at ultra-relativistic energies (a regime that is relevant for a narrow time interval, and only if t _DN >~ t _ST, namely, epsilon _e >~ 0.03). We discuss how the deep Newtonian phase can be reliably used for GRB calorimetry, and we comment on the good detection prospects of trans-relativistic blast waves at 0.1÷10 GHz with the Karl G. Jansky Very Large Array and LOw-Frequency ARray.