The energy available per cluster

I have calculated that the accumulated present day energy density in magnetic remnants is ~10^48 J per cubic Mpc. As I argued in Section 2, most of the energy will end up in clusters of galaxies. To calculate the energy available per cluster, we need only find the space density of clusters. This is given by X-ray surveys (Piccinotti et al. 1982; Edge et al. 1990; Gioia et al. 1990). Edge et al. (1990) give the cluster density in two forms, the luminosity function and the temperature function. The luminosity function is

dN / dL_x = 10^{-6.65+/-0.11} (L_x / 10^44 erg/s)^{-2.17+/-0.15} (10^44 erg/s)^-1 Mpc^-3, (21)

and the temperature function is

dN / dT = 10^{-3.96+/-0.26} T^{-4.93+/-0.37} Mpc^-3 keV^-1. (22)

The integrated space densities of clusters with luminosities exceeding L_min or temperatures exceeding T_min are

N(>L_min) = {10^{-6.65+/-0.11} / 1.17+/-0.15} (L_min / 10^44 erg/s)^{-1.17+/-0.15} Mpc^{-3}, (23)


N(>T_min) = {10^{-3.96+/-0.26} / 3.93+/-0.37} T_min^{-3.93+/-0.37} Mpc^{-3}. (24)

In both cases, integrating down to fairly poor clusters (L_min ~ a few x10^43 erg/s or T_min~2 keV) gives a total space density of 10^-6 per cubic Mpc. A similar space density is obtained by simply taking Abell's (1958) statistical sample of galaxy clusters. Bahcall (1979) gives the space density of Abell R=1 clusters to be 1.2x10^-6 per cubic Mpc. The space density of rich clusters of galaxies is therefore about 10^-6 per cubic Mpc, and I will use this value in the following discussion.

As emphasized in Section 2, most of the magnetic remnants will be incorporated in clusters. With a cluster space density of 10^-6 per cubic Mpc and a magnetic energy density of 10^48 J per cubic Mpc, this gives about 10^54 J for the magnetic energy in a rich cluster. The important point to note is that this energy is more than adequate to give a 1microGauss field in a galaxy cluster. We could consider two possible models for the intracluster magnetic field. A uniform 1microGauss field in a sphere of radius 1 Mpc gives a total energy of almost 5x10^53 J, while if the magnetic field declines with radius as B^2 ~ [1.0 + (r/r_0)^2]^{-3/2} from a central value of 1microGauss with r_0 = 250 kpc then the magnetic energy out to large radii is still less than 10^53 J.

The figure of 10^54 J per cluster is an average value. In particular, it is an average over clusters of a variety of richnesses. Richer clusters are expected to accumulate more magnetic energy, simply because they are more massive and thus will have accumulated more magnetic remnants. Clusters as rich as the Coma cluster would be expected to have accumulated rather more than the average 10^54 J of magnetic energy.

Peter Tribble,