Synchrotron and inverse-Compton emission from radio galaxies with non-uniform magnetic field and electron distributions

I investigate the effect of non-uniform magnetic fields in the extended structures of radio galaxies on the observed synchrotron and inverse-Compton emission. On the assumption of an isotropic field, with a given power spectrum and a Gaussian distribution of the Cartesian components of the magnetic field strength, I derive a simple integral that can be used numerically to calculate the synchrotron emissivity from any electron population. In the case of power-law spectra, I show that it is possible to estimate the difference between the synchrotron emissivity from a region with such a field and that from the commonly assumed arrangement where $B$ is constant everywhere, though fully tangled, and that this difference is small, though it increases if the electron energy density scales with the field. An aged electron spectrum in such a field produces a characteristic curved synchrotron spectrum which differs significantly from the classical Jaffe-Perola spectrum, and I discuss some effects that this might have on standard spectral age fitting. Finally, I show that inverse-Compton scattering of the cosmic microwave background is only moderately affected by such a field structure, with the effects becoming more important if the electrons follow the field. Magnetic-field estimates in the literature from combined synchrotron and inverse-Compton modelling will give reasonable estimates of the mean magnetic field energy density if the field is non-uniform but isotropic.