The angular resolution of gamma-ray detectors does not allow one to separate the nebula from the pulsar in the Crab. It is generally assumed that the steady emission of gamma rays comes from the nebula. Using the 'outer magnetospheric gap' model, an alternative mechanism in which the steady emission of gamma rays could come from a compact region, a couple of light cylinder radii beyond the pulsar.
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Kwok et al. (1991) studied this question.