Non-equilibrium effects in steady relativistic $e^+e^-\gamma$ winds
  We consider an ultra-relativistic wind consisting of electron-positron pairs and photons with the principal goal of finding the asymptotic Lorentz factor $\gamma_{\infty}$ for zero baryon number. The wind is assumed to originate at radius $r_i$ where it has a Lorentz factor $\gamma_i$ and a temperature $T_i$ sufficiently high to maintain pair equilibrium. As $r$ increases, $T$ decreases and becomes less than the temperature corresponding to the electron mass $m_e$, after which non-equilibrium effects become important. Further out in the flow the optical depth $\tau$ drops below one, but the pairs may still be accelerated by the photons until $\tau$ falls below $\sim 2\times10^{-5} \gamma_{i}^{3/4}$. Radiative transfer calculations show that only at this point do the radiation flux and pressure start to deviate significantly from their blackbody values. The acceleration of the pairs increases $\gamma$ by a factor $\sim 45$ as compared to its value at the photosphere; it is shown to approach $\gamma_{\infty} \sim 1.4\times 10^3 (r_i/10^6\mbox{cm})^{1/4} \gamma_{i}^{3/4} T_i/m_e$. 
Ole M. Grimsrud and Ira Wasserman