. 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 luminous radio
source contains sufficient magnetic energy to magnetize the entire core
of a galaxy cluster.
The main point I wish to emphasize is that a powerful radio source
contains an enormous amount of magnetic energy and flux. Consider as an
example the radio source 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 luminous radio
source contains sufficient magnetic energy to magnetize the entire core
of a galaxy cluster.
Similar calculations can be performed for other bright radio sources, leading to similar results. During the lifetime of a cluster, it will have contained several radio sources. At high redshifts, most radio sources appear to be in rich environments.
To calculate the total magnetic field left over from all relic sources in the cluster's history we must integrate over the luminosity function, and this also requires a relation to be found between source luminosity and magnetic energy. I will sketch the calculation here, just to give a rough estimate of the expected magnetic energy per cluster.
I approach the second point first. The luminosity of a source in terms of the volume V, particle density N_r, and magnetic field B is
L ~ N_r B^1+
V.
Assuming equipartition of field and particle energy densities, N_r ~ B^2, and the magnetic energy content is E_B = B^2V. Then
L ~ E_B^(3+)/2
V^-(1+)/2 ~
E_B^(3+)/2
d^-k(1+)/2,
where the volume is taken to scale as V ~ d^k, where k=3 for a spherically expanding source, and k=2 for a lengthening cylinder. This can be rewritten as
E_B ~ L^2/(3+) d^k(1+)/(3+).
There is of course a lot of freedom in this expression for
E_B, but I take values for k in the range 2 to 3,
in the
range 0.5 to 1, m around unity. The final exponent is quite
uncertain as a result, but I get
E_B ~ d L^1/2,
with quite large errors on the exponents.
There is no strong dependence of physical size on redshift (Masson 1980), so I will crudely average over the size of the sources. I take the luminosity function to be of the form
dN / d ln L ~ 1 / L^2,
following Dunlop & Peacock (1990). We can then integrate over the current source density, weighted by magnetic energy per source, to get
_B = 
E_B (dN/dL) dL
which is
_B = 2
_c E_0
(L_c/L_0)^{1/2}/3
where E_0 is the magnetic energy of a source of luminosity
L_0, and _c is the
density of sources of luminosity L_c. If the reference energy
E_0 is that of Cygnus A, and the integration includes sources
10^4 times fainter, then
_B 
10^-8 E_0
Mpc^-3.
This is the present density, we must now integrate over the cluster's past history. The source lifetime is about 10^7 yr, so there have been of order a thousand generations of sources. This is an overestimate, but the space density was somewhat higher and the radio lifetimes somewhat shorter in the past, partially compensating. The lifetime cannot exceed 10^8 yr without reacceleration, so the error isn't that large. This gives an accumulated energy density at the present day of
_{B,tot} 10^-5 E_0
Mpc^-3.
This is a lot of energy - it corresponds to ten thousand Cygnus A units
within z=0.1, or several per rich cluster. As radio sources at
higher redshifts were typically in clusters (Hintzen, Ulvestad &
Owen 1983; Yee & Green 1987; Yates, Miller & Peacock 1989; Hill
& Lilly 1991), much of the magnetic energy is now associated with
present day clusters. The total accumulated energy is easily sufficient
to account for a 1 field in
clusters today.