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Hoshino et al. (1992, Ap.J. 390, 454) showed that perpendicular shocks in a relativistic pulsar wind composed of electrons, positrons and ions can account for the acceleration to a power-law distribution of the synchrotron-emitting $e^\pm$ in the Crab Nebula. The thickness scale of such shocks is the ions' Larmor radius, which for a wind with $\gamma \sim 10^6$ and $B \sim 10^{-4} \mbox{G}$ is of order $0.01 \mbox{pc}$, observationally resolvable and comparable with the thickness of the Nebula's synchrotron emission ``wisps''.
Here we present a model of the pulsar wind shock structure that accounts remarkably well for the optical brightness profile of the wisps. This model computes the self-consistent orbits of the ions in a background of magnetized, relativistically hot $e^\pm$, and explains the synchrotron emission enhancements as compressions of the $e^\pm$ in the resulting magnetic field structure. The wind geometry is assumed to be a radial outflow concentrated around the rotational equatorial plane of the pulsar, in accordance with the X-ray morphology of the Nebula, and accounts for the ellipsoidal arc shape of the wisps. The beaming and boosting effects due to the mildly relativistic bulk motion of the $e^\pm$ are incorporated in order to account for the brightness discrepancy between the NW and SE sides of the Nebula. The synchrotron emission spectrum of the first wisp is modelled as that from a relativistic Maxwellian distribution, as the acceleration time scale is thought to be longer than the flow time through this initial part of the shock, yielding predictions for the spectral shape of the brightest wisp. We examine the implications of our model for the composition of the pulsar wind, and discuss the possibility of observations in other wavebands.
