If the wake does no more than generate turbulence in the gas which winds a preexisting field then the field will straighten and return to its original configuration on timescales of only a few times 10^8 yr (Jaffe 1980). If the induced turbulence drives a dynamo then the effect is completely different as an irreversible change in the field's topology occurs. In either case, the filling factor of the wakes is rather small so that a filamentary network of enhanced field is set up throughout the cluster. One should see, at high enough resolution, a patchy halo with holes (Roland 1981). Furthermore, because the density of galaxies falls with increasing radius, the halo should become more ragged at the edges.
The characteristics of a halo from the wakes proposed by Roland (1981) will now be examined. The halo contains approximately 200 cells at 1 arcmin resolution, and each of 150 galaxies has a tail about 200 kpc or 5 resolution elements long. Each beam contains a contribution from 42 wakes, leading to a predicted halo contrast of 50%. This is very much larger than the granularity observed, so that the Coma radio halo cannot be made up solely of wakes behind galaxies. Using the upper limit of 5% for the contrast derived in the previous section and adding a uniform background to reduce the contrast implies that the wakes can be responsible for no more than 10% of the total halo emission.
This last conclusion does not mean that wakes are ruled out or even unimportant. The emission contrast between the wakes and the general cluster could still be large if the filling factor of the wakes is correspondingly small. If d is the average diameter of the wake then the filling factor is
f (d/50 kpc)^2. (13)
For example, if d = 5 kpc then the wakes only fill 1% of the cluster and could have volume emissivities 10 times greater than the more diffuse halo material while still being responsible for only 10% of the total emission, so that it is rather difficult to set limits on the field or emission enhancement in galactic wakes. It is clear, however, that galactic wakes as envisaged by Roland (1981) are responsible for only a small fraction of the observed emission. In a similar fashion, the limits on the contribution of discrete cluster sources are even stronger, in agreement with Hanisch (1980) who found that less than 1 or 2% of the halo emission could be due to cluster galaxies.
This last conclusion, that at least 90% of the halo is due to truly diffuse emission, has important consequences. It shows that the relativistic electrons and magnetic fields are truly a cluster wide phenomenon, not just local to specific regions. The inverse Compton lower limit on the field strength applies throughout the cluster volume. The field is not just localized to specific regions such as wakes or galactic debris. In addition, it constrains models of the origin of the relativistic electrons to allow only those which would fill the cluster. This agrees with the results of Valtaoja (1984) who found that diffusion models best fitted the extent of the halo, albeit with a large diffusion coefficient. As noted by Hanisch (1982), the electrons are guided by the field and must also diffuse through the tangled field structure in a similar way to thermal electrons (Tribble 1989). In the context of the present results, a large diffusion coefficient would help smooth out a set of initially localised sources of relativistic electrons.
The dynamo model of Ruzmaikin et al. (1989) leads to a set of turbulence scales. The constraint derived in the previous section cannot be directly applied to a field tangled on several different scales, because although the field might be correlated on scales of 20 kpc the dominant field inhomogeneities are those on much smaller scales. In a similar manner, the observations cannot rule out a uniform component of the field, as long as small scale structure is present. The dynamo model of Ruzmaikin et al. goes further and requires the field to be concentrated in intense ropes with a small filling factor. The emission contrast between the ropes and the background is much stronger than the contrast between strong and weak field regions for a Gaussian field, although the filling factor is small so that the bulk of the emission might still come from the low field regions. The important factor here is that the small scale structure is not random but is strongly correlated on larger scales, so that the constraint derived on the coherent structure applies in this case also and limits the outer scale of the turbulence. Indeed, the constraint is even stronger because the intrinsic contrast given by this kind of structure is greater than for a Gaussian field.
The electrons in the intense ropes, with field strengths of perhaps 10, will lose their energy much faster than the electrons outside the ropes which have their energy losses dominated by inverse Compton losses. Thus the ropes will fade and all the emission will come from the low field strength regions. The constraint on the cell size still applies, however, because the field in the cells will be correlated on the same scale as the ropes.