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In 2007 September, the Crab Nebula exhibited a bright gamma-ray flare in theGeV energy range that was detected by AGILE. The observed emission at >160 MeVindicates that the radiating electrons had energies above the classicalsynchrotron radiation-reaction limit, thus presenting a serious challenge toclassical models for electron acceleration in astrophysical environments. Inthis paper, we apply our recently developed time-dependent self-similaranalytical model describing electrostatic acceleration in the explosivereconnection region around the pulsar wind termination shock to the 2007September flare. This event was unique in that it displayed both long-duration"wave" and short-duration "sub-flare" features. The unusual temporal variationmakes this flare an especially interesting test for our model. We demonstratethat our model can reproduce the time-dependent gamma-ray spectrum for thisevent, as well as the associated gamma-ray light curve, obtained by integratingthe spectrum for photon energies >100 MeV. This establishes that ourtime-dependent electrostatic acceleration model can explain both wave andsub-flare transients, which lends further support to the theoretical frameworkwe have developed. We also further examine the validity of the self-similarelectric and magnetic field evolution implied by our model. We conclude thatstrong electrostatic acceleration driven by shock-induced magnetic reconnectionis able to power the Crab Nebula gamma-ray flares by energizing the electronson sub-Larmor timescales.

Time-Dependent Electron Acceleration in Pulsar Wind Termination Shocks: Application to the 2007 September Crab Nebula Gamma-Ray Flare