Two models for the evolution of an initially pure power-law electron energy spectrum can be distinguished (Myers & Spangler 1985). In the Kardashev-Pacholczyk (KP) model, the pitch angle of an individual electron is constant, and as the electrons whose motion is more perpendicular to the field lose energy faster than electrons travelling nearly parallel to the field, there are always some high energy electrons left to radiate at high frequencies. In the Jaffe-Perola (JP) model, the pitch angle distribution is rapidly isotropized, with individual electrons sampling all pitch angles. The energy losses are independent of an electron's initial pitch angle, leading to a sharp break in the energy spectrum.
In the conventional approach, one averages over all orientations of a constant field (Myers & Spangler 1985). This is not strictly correct. Random orientations also imply random field strengths, and the field strength is normally assumed to be constant. In addition, the large observed polarizations rule out a truly random field, although large polarizations can be given by a sheared random field (Laing 1980).
It should be emphasized that of these two models, the JP model has a far more secure physical basis, and would be expected to give the best description of reality. An anisotropic pitch-angle distribution excites Alfvén waves which scatter the electrons in pitch angle (Wentzel 1969; Melrose 1980). Even if this does not occur, the pitch-angle changes as an electron moves between regions of differing field-strength. It is surprising, therefore, that detailed observations of Cygnus A (Carilli et al. 1991) are well fit by the KP model and strongly rule out the JP model spectrum.
In this paper I examine the radio spectrum allowing for the random nature of the magnetic field. In Section 2 I first consider the spectra from a volume containing a random magnetic field with the KP and JP prescriptions for the evolution of the electron energy spectrum. In Section 3 I go on to consider a realistic non-scattering model in which the pitch angles change as the relativistic electrons move along the varying field. A Jaffe-Perola type model allowing for diffusion between regions of different field strength and the effects of inverse Compton losses is presented in Section 4. I consider some effects relevant to real sources in Section 5, and give my conclusions in Section 6.