Introduction
Some of the richest and most luminous X-ray clusters of galaxies
contain radio halos---diffuse cluster wide radio emission not
associated with individual galaxies. The most prominent and well
studied example is the Coma cluster (Willson 1970; Jaffe, Perola &
Valentijn 1976; Valentijn 1978; Hanisch 1980; Cordey 1985;
Schlickeiser, Sievers & Thiemann 1987; Henning 1989; Kim et al.
1989; Kim et al. 1990).
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.
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Peter Tribble, peter.tribble@gmail.com