The integrated non-thermal spectrum of plerionic supernova remnants is well accounted for within the framework of the homogeneous model by Pacini & Salvati (1973, hereafter PS), who regard the plerion as an adiabatically expanding spherical bubble of magnetized relativistic fluid. The bubble is replenished by the continuous conversion of the pulsar's energy outflow into magnetic energy and relativistic particles. The energy balance of the plerion is determined by the competition between the decreasing pulsar's input and the losses the fluid undergoes because of radiation and expansion. Both particles and magnetic field are assumed to be distributed uniformly through the bubble.
When the synchrotron emission of the Crab Nebula is calculated on the basis of the PS model, both the decline of luminosity with time (Véron-Cetty & Woltjer 1991) and the frequency of the low energy spectral break (Marsden et al. 1984) are consistent with a magnetic field strength around G, in agreement with estimates based on equipartition arguments (Woltjer 1958) and also with recent measurements of the nebular inverse Compton radiation flux (De Jager & Harding 1992). Other applications of the PS model include the interpretation of the overall spectral and evolutionary properties of other plerionic supernova remnants, as 3C58 and G11.2-0.3, which seem to be very different from the Crab Nebula (Woltjer et al. 1997).
Despite the success of the model at describing the integrated properties of plerions, the assumption of homogeneity is unsatisfactory. If as assumed the particles are accelerated in the general environment of the pulsar and if loops of magnetic field are created by its rotation, important gradients could be expected in the particle and field distributions. Also observationally such gradients are in evidence. The Crab Nebula does not have a uniform emissivity at any wavelength: it is concentrated towards the center, the more strongly so at the higher frequencies.
We shall here consider a modification of the PS model in which loops of magnetic field are continuously injected with the relativistic electrons attached to the field lines. Since little is known about the detailed conditions around the pulsar we shall assume that the injection takes place at a certain distance from the center. In the case of the Crab Nebula the moving wisps do in fact suggest that the injection occurs at some from the pulsar. Since the propagation velocity of magnetic disturbancies is likely to be much larger than the expansion velocity of the bubble, we shall assume that the field adjusts itself to constant energy density, an assumption that is strictly justified only if the particle pressure is below the magnetic pressure and the field is chaotic.
While in the case of the Crab Nebula the polarization studies at radio and optical wavelengths (Wilson 1972a, Woltjer 1958) indicate a predominance of azimuthal magnetic field, the modest degree of polarization shows that there are different field directions along the line of sight. Such a less regular field (and the effect of reconnection) may well cause the particles to move more freely through parts of the nebula. Thus the present model, which includes only convection, is likely to be a limiting case. In fact we will find that the observed emissivity distribution is intermediate between the one computed here and the homogeneous case.
© European Southern Observatory (ESO) 2000
Online publication: July 13, 2000