Polarization of the Crab Nebula with disordered magnetic components

In this paper, we present an expanding disc model to derive the polarization properties of the Crab nebula. The distribution function of the plasma and the energy density of the magnetic field are prescribed as functions of the distance from the pulsar using the model derived by Kennel and Coroniti with sigma = 0.003, where sigma is the ratio of the Poynting flux to the kinetic energy flux in the bulk motion just before the termination shock. Unlike the case for previous models, we introduce a disordered magnetic field, which is parametrized by the fractional energy density of the disordered component. The flow dynamics are not solved, and the mean field is toroidal. The averaged degree of polarization over the disc is obtained as a function of the inclination angle and fractional energy density of the disordered magnetic field. It is found for the Crab Nebula that the disordered component contains about 60 per cent of the magnetic field energy. This value is supported by the facts that the disc appears not as 'lip-shaped' but as 'rings' in the observed intensity map, and that the highest degree of polarization of similar to 40 per cent is reproduced for rings, which is consistent with the observations. We suggest that, because the disordered field contributes to the pressure rather than to the tension, the pinch force may have been overestimated in previous relativistic magnetohydrodynamic simulations. The disruption of the oppositely directed magnetic fields, which is proposed by Lyuvarsky, may actually take place. The relativistic flow speed, which is indicated by the front-back contrast, can be detected in the asymmetry of distributions of the position angle and depolarization.