We report a correlation based on a spectral simulation study of the prompt emission spectra of gamma-ray bursts (GRBs) detected by the Swift Burst Alert Telescope (BAT). The correlation is between the Epeak energy, which is the peak energy in the u F_ u spectrum, and the photon index (Gamma) derived from a simple power-law model. The Epeak - Gamma relation, assuming the typical smoothly broken power-law spectrum of GRBs, is log Epeak = 3.258 - 0.829Gamma (1.3 < Gamma < 2.3). We take into account not only a range of Epeak energies and fluences, but also distributions for both the low-energy photon index and the high-energy photon index in the smoothly broken power-law model. The distribution of burst durations in the BAT GRB sample is also included in the simulation. Our correlation is consistent with the index observed by BAT and Epeak measured by the BAT, and by other GRB instruments. Since about 85% of GRBs observed by the BAT are acceptably fit with the simple power-law model because of the relatively narrow energy range of the BAT, this relationship can be used to estimate Epeak when it is located within the BAT energy range.
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