Spectral Evolution of the 2010 September Gamma-ray Flare from the Crab Nebula

Strong gamma-ray flares from the Crab Nebula have been recently discovered by AGILE and confirmed by Fermi-LAT. We study here the spectral evolution in the gamma-ray energy range above 50 MeV of the 2010 September flare that was simultaneously detected by AGILE and Fermi-LAT. We revisit the AGILE spectral data and present an emission model based on rapid (within 1 day) acceleration followed by synchrotron cooling. We show that this model successfully explains both the published AGILE and Fermi-LAT spectral data showing a rapid rise and a decay within 2 and 3 days. Our analysis constrains the acceleration timescale and mechanism, the properties of the particle distribution function, and the local magnetic field. The combination of very rapid acceleration, emission well above 100 MeV, and the spectral evolution consistent with synchrotron cooling contradicts the idealized scenario predicting an exponential cutoff at photon energies above 100 MeV. We also consider a variation of our model based on even shorter acceleration and decay timescales, which can be consistent with the published averaged properties.