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A statistical model for the $γ$-ray variability of the Crab nebula

A statistical scenario is proposed to explain the $γ$-ray variability and flares of the Crab nebula, which were observed recently by the Fermi/LAT. In this scenario electrons are accelerated in a series of knots, whose sizes follow a power-law distribution. These knots presumably move outwards from the pulsar and have a distribution in the Doppler boost factor. The maximal electron energy is assumed to be proportional to the size of the knot. Fluctuations at the highest energy end of the overall electron distribution will result in variable $γ$-ray emission via the synchrotron process in the $\sim 100$ MeV range. Since highly boosted larger knots are rarer than smaller knots, the model predicts that the variability of the synchrotron emission increases with the photon energy. We realize such a scenario with a Monte-Carlo simulation and find that the model can reproduce both the two $γ$-ray flares over a period of $\sim$ year and the monthly scale $γ$-ray flux fluctuations as observed by the Fermi/LAT. The observed $γ$-ray spectra in both the steady and flaring states are also well reproduced.