Introduction

The papers cited above concentrated on the synchrotron emission ensemble averaged over a volume. The standard models for the evolution of an initially power-law radio spectrum are the Kardashev-Pacholczyk or KP model (Kardashev 1962; Pacholczyk 1970), and the Jaffe-Perola or JP model (Jaffe & Perola 1973). In the KP model, the pitch angle of an individual electron remains constant, so that those electrons travelling nearly parallel to the field suffer small losses. In the JP model, scattering randomizes the electron pitch angles so that every electron samples all pitch angles, and all electrons have the same losses. This leads, in the JP model, to a spectrum that falls exponentially above the break frequency, whereas in the KP model there always remains a population of high energy electrons (those travelling nearly parallel to the field) which can radiate at high frequencies, so that the spectrum does not fall exponentially but simply steepens to another power law of larger spectral index.

In a previous paper (Tribble 1993) I considered the effects of a random rather than uniform magnetic field. The synchrotron losses now vary with position, so that the electron energy spectrum is no longer the same everywhere. With efficient pitch-angle scattering, although the spectrum from any small part of the emitting volume has an exponential break, the spectra from different parts of the volume have breaks at different frequencies. The superposition of these individual spectra, having a range of break frequencies, gives a total spectrum with a much more gentle break. The resultant spectra can then vary between the two extremes of the JP and KP models, depending on the range of break frequencies in the volume. If inverse Compton losses dominate, or if diffusion is very efficient, then the electron energy spectrum is almost independent of position, and while the magnetic field and hence break frequency (which in this case is proportional to the field strength B) do vary with position, the sharp spectral break is only slightly smoothed. If inverse Compton losses are small and electrons do not diffuse efficiently, then electrons in high-field regions lose energy much more rapidly than those in low field regions, the break frequency is therefore much lower in the high field regions, the variations in break frequency from place to place are large (proportional in this case to 1/B^3), and the spectral break of the integrated spectrum is rather gentle. An analogy can be drawn with the KP model - the electrons in the weak field regions retain their energy, rather like the electrons travelling parallel to the field in the KP model.

Further effects complicating spectral ageing have been considered. Alexander (1987) allowed for adiabatic expansion of the synchrotron plasma, and showed that estimates of the spectral age were reasonably accurate if the initial magnetic field strength is used to calculate the synchrotron losses. Wiita & Gopal-Krishna (1990) allowed for the large-scale variation of field strength, while Siah & Wiita (1990) looked at the variation of age with the filling factor of an inhomogeneous field. More recently, Katz-Stone, Rudnick & Anderson (1993) have shown that the radio colour-colour plot can be a useful tool to study the shape of the radio spectrum.

While the work just mentioned has given a good understanding of how the integrated spectrum of a source is determined by synchrotron emission, little work has been done on the evolution of radio images as the electron population ages. The theoretical work on the structure of radio images has generally been focussed on two areas.

The first concentrates on models of jets propagating in an ambient medium (e.g. Clarke, Norman & Burns 1989; Matthews & Scheuer 1990a,b). These simulations can reproduce many of the features of extragalactic radio sources. The emissivity in these models is simply a function of the local field strength. Although Matthews & Scheuer included synchrotron losses, primarily suppressing bright points where the field is very strong, this was secondary to the dynamical effects they were interested in.

The second area of theoretical study is the formation of filaments in extended radio sources. The stability of the synchrotron plasma has been studied by several authors, for example Simon & Axford (1967), Eilek & Caroff (1979), Achterberg (1989), Gouveia Dal Pino & Opher (1989), Bodo et al. (1990). The plasma is unstable under certain conditions: a region more strongly magnetized than average will radiate more strongly, and the relativistic particle pressure falls, encouraging the collapse of the region. The timescale for this instability is of order the synchrotron loss time, which is the time scale on which the relativistic particle pressure declines.

In this paper I concentrate solely on spectral ageing in a random magnetic field, ignoring all of the (important) dynamical evolution. The numerical simulations and instabilities generate an inhomogeneous magnetic field. I then take a random magnetic field as a base, and consider only the effect of spectral ageing. Spectral ageing is then treated in isolation, removed from the complexities of the source dynamics. This approach should be valid provided that particle reacceleration is unimportant.

I therefore consider the evolution of the radio emission from a volume containing a random magnetic field that is uniformly filled with relativistic electrons. In Section 2 I describe how the models are constructed. I show some example images in Section 3, and describe how the images change as the source ages. In Section 4 I consider the model spectra and intensity-spectral index correlations. The polarization properties are studied in Section 5, filamentation and instabilities in Section 6, with the conclusions presented in Section 7.

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