  In 2007 September, the Crab Nebula exhibited a bright gamma-ray flare in the
GeV energy range that was detected by AGILE. The observed emission at >160 MeV
indicates that the radiating electrons had energies above the classical
synchrotron radiation-reaction limit, thus presenting a serious challenge to
classical models for electron acceleration in astrophysical environments. In
this paper, we apply our recently developed time-dependent self-similar
analytical model describing electrostatic acceleration in the explosive
reconnection region around the pulsar wind termination shock to the 2007
September flare. This event was unique in that it displayed both long-duration
"wave" and short-duration "sub-flare" features. The unusual temporal variation
makes this flare an especially interesting test for our model. We demonstrate
that our model can reproduce the time-dependent gamma-ray spectrum for this
event, as well as the associated gamma-ray light curve, obtained by integrating
the spectrum for photon energies >100 MeV. This establishes that our
time-dependent electrostatic acceleration model can explain both wave and
sub-flare transients, which lends further support to the theoretical framework
we have developed. We also further examine the validity of the self-similar
electric and magnetic field evolution implied by our model. We conclude that
strong electrostatic acceleration driven by shock-induced magnetic reconnection
is able to power the Crab Nebula gamma-ray flares by energizing the electrons
on sub-Larmor timescales.
