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, that decelerates from ultra-relativistic to non-relativistic
speeds as it sweeps up the surrounding medium. We investigate the temporal
decay of the emission from GRB afterglows at late times, when the bulk of the
shock-accelerated electrons are non-relativistic. For a uniform circumburst
medium, we show that such "deep Newtonian phase" begins at t_{DN} ~ 3
epsilon_{e,-1}^{5/6} t_{ST}, where t_{ST} marks the transition of the blast
wave to the non-relativistic spherically-symmetric Sedov-Taylor solution, and
epsilon_e = 0.1 epsilon_{e,-1} quantifies the amount of shock energy
transferred to the post-shock electrons. For typical parameters, the deep
Newtonian stage starts ~ 0.5-several years after the GRB. The radio flux in
this phase decays as F_nu ~ t^{-3(p+1)/10}, with decay slope between -0.9 and
-1.2, for a power-law distribution of shock-accelerated electrons with index 2
< p < 3. This is shallower than the commonly assumed scaling F_nu ~
t^{-3(5p-7)/10} (with decay slope between -0.9 and -2.4) derived by Frail et
al. (2000), 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}, or
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 EVLA and LOFAR.
