Model uncertainties

The derived energy density will be in error by whatever factor the equipartition estimate is, and it is difficult to assess how large this uncertainty is likely to be. This problem will afflict any calculation of the magnetic energy content of radio sources - it is not peculiar to the model presented in this paper.

Another uncertainty is the assumption that all of the original energy in relativistic electrons is radiated in observable radio emission and is included in the integral over the luminosity function. Van der Laan & Perola (1969) argued that the lack of a large population of steep spectrum sources meant that sources must die quickly rather than simply fade away by radiative or adiabatic losses, and suggested a model where the relativistic electrons were lost by diffusion out of the source. The sources also lose energy by inverse Compton scattering. In both cases the estimate of the energy density I have given must be revised upwards to allow for the fraction of the particle energy that is lost to inverse Compton radiation, and the fraction of energy lost by particles leaving the source and no longer contributing to the observed luminosity.

The magnetic energy density could be underestimated by a factor of a few in this way. While very luminous sources have field strengths greater than the inverse Compton equivalent of = 3.18(1+z)^2, increases from 12 to 50 in the redshift range of interest (z=1--3), so that the intermediate luminosity sources which dominate the integral (see Fig. 1) lose a substantial fraction of their energy to inverse Compton radiation. This effect is even more important if the magnetic field strength is lower than the equipartition estimate. Lowering the magnetic field strength of a source reduces its contribution to the magnetic energy density, but this is compensated for by the fact that the energy density is underestimated because a smaller fraction of the total energy is emitted in the radio waveband and thus included in the integral over source luminosity. As a result, the actual energy density could be substantially higher than the energy density I have derived, but is unlikely to be substantially less.

The energy density could also be underestimated due to a new population of low luminosity sources. As discussed in Section 5.1, the derived energy density does not rely on extrapolation of the luminosity function to low luminosities. Therefore, changing the luminosity function cannot decrease the energy density. It is always possible that a new population of sources could lead to a substantial upturn in the luminosity function at low luminosities and increase the energy density somewhat. I will briefly mention two possible classes of object that might be present at low luminosities.

The interstellar medium of cluster galaxies could be stripped and mixed into the intracluster medium. While little is known about magnetic fields in elliptical galaxies, the interstellar medium in spiral galaxies contains magnetic fields of strength (Wielebinski & Krause 1993). As the galaxy passes through the intracluster medium the interstellar medium is stripped by ram pressure, depositing enriched gas and magnetic fields into the cluster gas. If a spiral galaxy has a 3 field in a volume of 1000 kpc^3 then the magnetic energy is ~10^48 J. A thousand such galaxies gives a total of 10^51 J. While this is a very substantial energy content, it is only a very small fraction of the energy input by powerful radio sources.

Another possible class of radio sources which might not have been included are the giant or relic sources. An example is 0917+75, which has been considered by Harris et al. (1993). Harris et al. show that this relic, of relatively low luminosity, has a similar energy content to the Coma radio halo, so that this source alone has sufficient magnetic energy to magnetize a significant fraction of a cluster. Due to the low field strength, these giant sources are of low luminosity. Due to their low surface brightness, they might be missed in interferometric surveys.

While both the stripped interstellar media of galaxies and giant radio sources can clearly contribute to an intracluster magnetic field, it is clear that their contribution is small compared to that of powerful radio galaxies, and that their is in fact no shortage of magnetic energy. Even if the equipartition estimate is in error, this does not affect the fundamental conclusion that powerful radio sources can give strength fields in clusters at the present time. The estimate of accumulated magnetic energy would have to be reduced by more than an order of magnitude before it would have any problem giving the observed fields.

The results are also insensitive to the choice of cosmological model. Changing the Hubble constant H_0 changes the space density H_0^3, the real luminosity L H_0^-2 and the age of the universe t_0 H_0^-1, leading to an energy density independent of H_0. The energy per cluster is likewise independent of H_0 as, while the space density of clusters changes, so does the physical size of each cluster. Therefore the derived magnetic field strengths are independent of H_0.

• On to section 6