Precise Synchrotron Radiation Spectrum of Fast-cooling Electrons and the Prompt GRB Emission

It is generally believed that the synchrotron radiation of electrons from the internal shock is the main radiation mechanism of the prompt GRB (gamma-ray burst) emission. However, what this model predicts can not explain observations well. In this paper, we confirm that electrons are quickly cooled due to radiation losses and also point out that the synchrotron radiation spectrum presented in previous papers is a relatively rough estimation. We get the precise synchrotron radiation spectrum of fast-cooling electrons by carrying out a numerical calculation, and thereby reasonably explain the observed distribution of low-energy spectral index (α) of long GRBs based on a unified model. In addition, we fit the correlation between α and the peak energy of the νFν spectrum (Ep).