  The gamma-ray space telescopes AGILE and Fermi detected short and bright
synchrotron gamma-ray flares at photon energies above 100 MeV in the Crab
Nebula. This discovery suggests that electron-positron pairs in the nebula are
accelerated to PeV energies in a milliGauss magnetic field, which is difficult
to explain with classical models of particle acceleration and pulsar wind
nebulae. We investigate whether particle acceleration in a magnetic
reconnection layer can account for the puzzling properties of the flares. We
numerically integrate relativistic test-particle orbits in the vicinity of the
layer, including the radiation reaction force, and using analytical expressions
for the large-scale electromagnetic fields. As they get accelerated by the
reconnection electric field, the particles are focused deep inside the current
layer where the magnetic field is small. The electrons suffer less from
synchrotron losses and are accelerated to extremely high energies. Population
studies show that, at the end of the layer, the particle distribution piles up
at the maximum energy given by the electric potential drop and is focused into
a thin fan beam. Applying this model to the Crab Nebula, we find that the
emerging synchrotron emission spectrum peaks above 100 MeV and is close to the
spectral shape of a single electron. The flare inverse Compton emission is
negligible and no detectable emission is expected at other wavelengths. This
mechanism provides a plausible explanation for the gamma-ray flares in the Crab
Nebula and could be at work in other astrophysical objects such as relativistic
jets in active galactic nuclei.
