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Energetic Constraints on a Rapid Gamma-Ray Flare in PKS 1222+216

We study theoretical implications of a rapid Very-High-Energy (VHE) flare detected by MAGIC in the Flat-Spectrum Radio Quasar PKS 1222+216. The minimum distance from the jet origin at which this flare could be produced is 0.5 pc. A moderate Doppler factor of the VHE source, D_{VHE} ~ 20, is allowed by all opacity constraints. The concurrent High-Energy (HE) emission observed by Fermi provides estimates of the total jet power and the jet magnetic field strength. Energetic constraints for the VHE flare are extremely tight: for an isotropic particle distribution they require a huge co-moving energy density in the emitting region and a very efficient radiative process. We disfavor hadronic processes due to their low radiative efficiency, as well as the synchrotron scenario recently proposed for the case of HE flares in the Crab Nebula, since the parameters needed to overcome the radiative losses are quite extreme. The VHE emission can be explained by the Synchrotron Self-Compton (SSC) mechanism for D_{VHE} ~ 20 or by the External Radiation Compton (ERC) mechanism involving the infrared radiation of the dusty torus for D_{VHE} ~ 50. After discussing several alternative scenarios, we propose that the extreme energy density constraint can be satisfied when the emission comes from highly anisotropic short-lived bunches of particles formed by the kinetic beaming mechanism in magnetic reconnection sites. By focusing the emitting particles into very narrow beams, this mechanism allows one to relax the causality constraint on the source size, decreasing the required energy density by 4 orders of magnitude.