The power law radio spectrum shows that the emission is synchrotron radiation, so the cluster contains both relativistic electrons and a magnetic field. Lack of associated X-rays from inverse Compton radiation sets a lower limit of approximately 0.1 on the field strength (Gursky & Schwarz 1977; Rephaeli & Gruber 1988). The combination of hot gas and magnetic fields leads to an observable Faraday rotation measure (RM) in galaxy clusters which has been detected (Kim, Tribble & Kronberg 1991), although the RMs are smaller than expected if the field known to be present were uniform. The magnetic field is therefore tangled, and resolved RM fluctuations of sources at the centres of cooling flows (Dreher, Carilli & Perley 1987; Taylor et al. 1990) indicate that the tangling scale of the field is of order 10 kpc, although it must be emphasized that this value is uncertain. It is also not clear that the magnetic field at cooling flow centres is typical of the general intracluster medium.
The origin of the relativistic electrons and magnetic fields remains unknown. Jaffe (1977) proposed that the relativistic electrons came from radio sources in the cluster and diffused out to fill the cluster volume. Unfortunately the electrons lose energy rather rapidly and have difficulty reaching the outer reaches of the halo if they diffuse at the Alfvén speed, although Holman, Ionson & Scott (1979) showed that the ion sound speed was more appropriate, rescuing the model. Alternatively, Dennison (1980) and Vestrand (1982) proposed that relativistic protons diffused throughout the cluster and produced a secondary population of relativistic electrons in collisions with thermal protons. In this paper I do not address the formation of the radio halo, but rather consider what the present state of the halo can tell us about the magnetic field structure.
I first consider the properties of radio emission from a volume containing a random magnetic field. I then present simulations of the emission from such a volume and evaluate the variations in intensity. These models are then compared with observations of the radio halo in the Coma cluster and used to constrain the properties of the magnetic field there.
Throughout this paper I assume H_0 = 50 km/s/Mpc.