A connection between spectral sharpness and energies as well as flux in fermi gamma-ray bursts

We revisit the sharpness angle (θ) under the peak or break of gamma-ray burst (GRB) spectra with the best peak flux P and time-integrated F spectral data provided by the Fermi GBM Burst Catalogue. We compute the sharpness angles of best-fit model spectra and check some interesting relations between θ and physics quantities. It is found that (i) a positive correlations between θ and the observed fluence as well as the isotropic radiated energy holds among GRBs, especially for the F spectra; (ii) when checking the correlation between θ and energy flux a weaker anti-correlation holds among GRBs and a tighter anti-correlation holds within GRBs, especially in single pulses. Our results further show that the spectral shape is related to the energy and flux by cross-checking other measures of spectral curvature. The correlated relationship between spectral sharpness and energy flux can be well explained as a thermal origin for GRB prompt emission: A large entropy around the peak of the light curve makes the photosphere approach the saturation radius, resulting in an intense emission with a narrow spectrum; as the entropy decreases, the photosphere deviates from the saturation radius, resulting in weaker emission with a broader spectrum.