In this paper we have presented a simple one-parameter method of modelling the effects of anomalous transport on energetic electrons. Anomalous transport is likely to occur wherever magnetised electrons (i.e., electrons which are tied to field lines) move in a magnetic field which has a stochastic component. In astrophysics, this is the rule rather than the exception and anomalous transport can be expected, for example, in the interstellar medium, as well as in the intra-cluster medium of clusters of galaxies. The primary observational diagnostic is the intensity and spectrum of the synchrotron radiation emitted by the transported particles. For the simplest case in which the magnetic field is of constant magnitude and of random orientation, we have presented general expressions for the surface brightness as a function from position of injection, and also for the spatial variation of the spectral index.
Applying these to the diffuse emission observed from the outer parts of the Coma cluster, we note that it is not possible to distinguish between the various forms of transport merely from the profile of the surface brightness. However, the expected spatial dependence of the spectral index is very sensitive to the type of transport. Assuming that the effects of particle acceleration are negligible in the outer parts of the cluster, and that the electron distribution has achieved a steady state, we find that standard diffusive transport cannot produce the observed rapid softening of the spectrum with radius. Under these assumptions, the type of transport indicated is supra-diffusion, in which particles move almost ballistically in a field configuration which has an ordered radial component.
The computations we have presented contain several major simplifications. In addition to the assumption of constant, randomly orientated magnetic field, and the simple planar or spherical geometry, we have assumed that the parameters governing the transport are independent of the particle's energy. In reality, the type of transport itself (i.e., the value of ) will change depending on the energy range and timescales considered. Thus, at very large times (which may exceed the synchrotron lifetime), a particle can be expected to decorrelate from the magnetic field and perform diffusion (e.g., Duffy et al., 1995 ). We do not model the situation in which a significant change in occurs within the synchrotron lifetime of an electron. Finally, in order to apply such models to well-observed objects such as spiral galaxies, it will be necessary to include additional effects such as that of a galactic wind, as well as bremsstrahlung and ionisation loss processes. However, these processes will not change our basic conclusion that it is the spatial dependence of the synchrotron spectral index which provides the most sensitive measure of the transport properties of the emitting electrons.
© European Southern Observatory (ESO) 1997
Online publication: April 8, 1998