Steady State Equilibrium Condition of $npe^{\pm}$ Gas and Its Application to Astrophysics

The steady equilibrium conditions for a mixed gas of neutrons, protons, electrons, positrons and radiation field (abbreviated as $npe^{\pm}$ gas) with/without external neutrino flux are investigated, and a general chemical potential equilibrium equation $\mu_n=\mu_p+C\mu_e$ is obtained to describe the steady equilibrium at high temperatures ($T>10^9$K). An analytic fitting formula of coefficient $C$ is presented for the sake of simplicity as the neutrino and antineutrino are transparent. It is a simple method to estimate the electron fraction for the steady equilibrium $npe^{\pm}$ gas that using the corresponding equilibrium condition. As an example, we apply this method to the GRB accretion disk and approve the composition in the inner region is approximate equilibrium as the accretion rate is low. For the case with external neutrino flux, we calculate the initial electron fraction of neutrino-driven wind from proto-neutron star model M15-l1-r1. The results show that the improved equilibrium condition makes the electron fraction decrease significantly than the case $\mu_n=\mu_p+\mu_e$ when the time is less than 5 seconds post bounce, which may be useful for the r-process nucleosynthesis