An axisymmetric magnetohydrodynamic model for the Crab pulsar wind bubble

We extend Kennel and Coroniti's (1984) spherical magnetohydrodynamic models for the Crab Nebula to include the pinching effect of the toroidal magnetic field. Since the bulk nebular flow is likely to be very submagnetosonic, a quasi-static treatment is possible. We show that the pinching effect can be responsible for the observed elongation of the pulsar wind bubble, as indicated by the surface brightness contours of optical synchrotron radiation. From the observed elongation we estimate a value for sigma, the ratio of Poynting flux to plasma kinetic energy flux in the free pulsar wind, which is consistent with previous results from spherical models. Using the inferred magnetic field configuration inside the pulsar wind bubble, combined with the observed dimensions of the X-ray nebula, we are able to constrain the particle distribution function. We conclude that, for a power-law injection function, the maximum energy has to be much larger in the pulsar equatorial region than in the polar region.