Phenomenological modelling of the Crab Nebula's broadband energy spectrum and its apparent extension

Context. The Crab Nebula emits exceptionally bright non-thermal radiation across the entire wavelength range from the radio to the most energetic photons. So far, the underlying physical model of a relativistic wind from the pulsar terminating in a hydrodynamic standing shock has remained fairly unchanged since the early 1970s when it was first introduced. One of the predictions of this model is an increase in the toroidal magnetic field downstream from the shock where the flow velocity drops quickly with increasing distance until it reaches its asymptotic value, matching the expansion velocity of the nebula.Aims. The magnetic field strength in the nebula is poorly known. Using the recent measurements of the spatial extension and improved spectroscopy of the gamma-ray nebula, it has become -for the first time - feasible to determine in a robust way both the strength as well as the radial dependence of the magnetic field in the downstream flow.Methods. In this work, we introduce a detailed radiative model which was used to calculate the emission from non-thermal electrons (synchrotron and inverse Compton) as well as from thermal dust present in the Crab Nebula in a self-consistent way to compare it quantitatively with observational data. Special care was given to the radial dependence of the electron and seed field density.Results. The radiative model was used to estimate the parameters related to the electron populations responsible for radio and optical/X-ray synchrotron emission. In this context, the mass of cold and warm dust was determined. A combined fit based upon a chi(2) minimisation successfully reproduced the complete data set used. For the best-fitting model, the energy density of the magnetic field dominates over the particle energy density up to a distance of approximate to 1.3  r(s) (r(s): distance of the termination shock from the pulsar). The very high energy (VHE: E  > 100 GeV) and ultra-high energy (UHE: E  > 100 TeV) gamma-ray spectra set the strongest constraints on the radial dependence of the magnetic field, favouring a model where B(r) = (264 +/- 9) mu G(r/r(s))(-0.51 +/- 0.03). For a collection of VHE measurements during epochs of higher hard X-ray emission, a significantly different solution B(r) = (167 +/- 5) mu G(r/r(s))(-0.29(+0.03, -0.06)) is found.Conclusions. The high energy (HE: E  > 100 MeV) and VHE gamma-ray observations of the Crab Nebula lift the degeneracy of the synchrotron emission between particle and magnetic field energy density. The reconstructed magnetic field and its radial dependence indicates a ratio of Poynting to kinetic energy flux sigma  approximate to  0.1 at the termination shock, which is approximate to 30 times larger than estimated up to now. Consequently, the confinement of the nebula would require additional mechanisms to slow the flow down through, for example, excitation of small-scale turbulence with possible dissipation of the magnetic field.