GRB afterglows in the deep Newtonian phase

In many GRBs, afterglows have been observed for months or even years. It deserves noting that at such late stages, the remnants should have entered the deep Newtonian phase, during which the majority of shock-accelerated electrons will no longer be highly relativistic. However, a small portion of electrons are still ultra-relativistic and capable of emitting synchrotron radiation. Under the assumption that the electrons obey a power-law distribution according to their kinetic energy (not simply the Lorentz factor), we calculate optical afterglows from both isotropic fireballs and beamed ejecta, paying special attention to the late stages. In the beamed cases, it is found that the light curves are universally characterized by a flattening during the deep Newtonian phase. Implication of our results on orphan afterglows is also addressed.