Polarization of the Crab Nebula with disordered magnetic components

Abstract

In this paper, we present an expanding disc model to derive polarization properties of the Crab nebula. The distribution function of the plasma and the energy density of the magnetic field are prescribed as function of the distance from the pulsar by using the model by Kennel and Coroniti (1984) with $σ= 0.003$, where $σ$ is the ratio of Poynting flux to the kinetic energy flux in the bulk motion just before the termination shock. Unlike previous models, we introduce disordered magnetic field, which is parameterized by the fractional energy density of the disordered component. Flow dynamics is not solved. The mean field is toroidal. Averaged polarization degree over the disc is obtained as a function of inclination angle and fractional energy density of the disordered magnetic field. It is found for the Crab that the disordered component has about 60 percent of the magnetic field energy. This value is also supported by the facts that the disc appears not `lip-shape' but as `rings' in the intensity map as was observed, and that the highest polarization degree of $\sim 40$ percent is reproduced for rings, being consistent with the observation. We suggest that because the disordered field contributes rather pressure than tension, the pinch force may be over-estimated in previous relativistic magnetohydrodynamic simulations. Disruption of the toroidal magnetic field with alternating direction, which is proposed by Lyubarski (2003), may actually takes place. The relativistic flow speed, which is indicated by the front-back contrast, can be detected in asymmetry in distributions of the position angle and depolarization.