It is probably more valid to conserve energy than magnetic flux, as the source achieves rough pressure equilibrium with its surroundings and evolves at roughly constant pressure thereafter. Radio sources are confined by an external medium that prevents them from expanding and suffering catastrophic adiabatic expansion losses (Gull & Northover 1973; Longair, Ryle & Scheuer 1973). The pressure in the source is similar to that in the external medium (Arnaud et al. 1984), so that the magnetic energy does not expand into free space to fill the volume of the cluster but must rather be mixed into the medium. We can then fill a volume of size D where
B_0^2 D_0^3 = B^2D^3 --> D = D_0(B_0/B)^2/3. (1)
More details of how the magnetic energy contained in the lobes is distributed into the intracluster medium are discussed in Section 6.
As a very rough calculation, we can consider a ``typical'' high
redshift radio source (Daly 1992), with a 35 field in a
cylindrical volume of length 150 kpc and radius 15 kpc. Assuming
conservation of magnetic flux, this can give a 1 field over a
cube 280 kpc on a side. There is sufficient magnetic energy to give a
1 field over a
volume a little over 500 kpc on a side. To put this another way, this
is an energy of 1.5
x10^52 J which is a fairly typical
energy for a luminous radio source.
For a specific example, consider the radio galaxy Cygnus A. Although very luminous for a nearby source, Cygnus A may not be atypical of high redshift radio sources. The two lobes of Cygnus A are about 30 kpc across and contain a field (from equipartition arguments) of about 50. The magnetic energy in these lobes is equivalent to a 1 field in a volume 500 kpc on a side. The point is clear - a single radio source can contain sufficient magnetic energy to magnetize the entire core of a galaxy cluster.
These large energy contents are not restricted to the ultra-luminous sources. More extended lower luminosity sources can contain as much energy as Cygnus A, because although more diffuse they are substantially larger. For example, the radio galaxy 3C234 (Alexander 1987) has a field strength (from minimum energy arguments) of between 5 and 10, in a cylinder almost 500 kpc long and about 40 kpc in diameter. This gives enough energy to fill a volume of between 250 and 400 kpc on a side with a 1 field, depending on which end of the range of minimum energy fields is taken.
Similar calculations can be performed for other radio sources, leading to similar results. During the lifetime of a cluster, it will have contained several radio sources, not all as luminous as the powerful high redshift radio quasars. We must now integrate over all the sources a cluster and its predecessors has contained throughout its history.