Fig 3. A simulation of the Coma radio halo at 1 arcmin=40 kpc resolution, for a magnetic field scale length of 16 kpc.
The noise level of the Coma radio halo has been calculated from the map of Kim et al. (1990). Regions near strong radio sources were avoided, and the contrast calculated after fitting to the overall profile of the halo. As the Coma cluster has two strong sources (5C4.81 and 5C4.85) at its centre, the contrast was calculated in regions about 8 arcmin from the cluster centre where the halo flux at 1.4 GHz is approximately 2 mJy per 1 arcmin beam. The contrast was found to be similar in all the regions examined, and shows structure to be present at a contrast level of approximately 5%.
The measured contrast does not give the intrinsic fluctuations in the halo, as thermal noise, errors from the discrete source subtraction, and poisson noise from unremoved cluster and background sources will be present. The residual noise level due to spurious features is quoted as 30Jy per beam (Kim et al. 1990), close to the thermal noise level. The actual noise level on the map (outside of the halo and away from strong sources) is reasonably uniform at approximately 70Jy per beam. The 5% contrast represents a dispersion of 100Jy on a mean level of 2 mJy per beam, so that some of the observed contrast is probably real. Source subtraction errors should not be a problem as affected areas of the map were not used, although one or two sources with fluxes of 1 mJy or less might be present. The observed low level of fluctuations rules out a significant contribution from individual sources to the halo. According to Hanisch (1980) no more than 10 mJy of the total halo flux is due to cluster sources, which would lead to fluctuations smaller than the thermal noise level. Weak background sources could also lead to small fluctuations.
As some of the observed contrast in the image might be due to other causes than granularity in the halo, I take the observed 5% contrast level as an upper limit and ask what field configurations are consistent with this limit. Generally, the number of cells must be large enough to reduce the noise below this level, so that an upper limit may be set on the field scale size.
The simplest possible model is that in which the field is uniform over a cell of size r_0 with random strength and direction. The observed fluctuations will be the cell to cell fluctuations reduced by averaging over the number of cells in the beam, both along the line of sight and in the plane of the sky. The intrinsic fluctuation level for a spectral index of unity is 1.1, so that for a depth of 640 kpc, the field scale size must be less than 16 kpc. This simple minded estimate agrees with that obtained by taking the modelled emission in a box smoothed to the required resolution, and is shown in Fig. 4.
Fig 4. The predicted halo contrast as a function of field scale length r_0 smoothed to 1 arcmin resolution from the models discussed in the text (solid line). The short dashed line shows the intrinsic contrast (unaffected by smoothing). Error bars are the fluctuation levels calculated from the simulated halos. The long dashed line shows the upper limit on the contrast in the Coma radio halo, leading to an upper limit on the field scale size.
A second method is to calculate the fluctuations in my models with the same profile as the Coma radio halo, in the same way as for the observational data. I show, in Fig. 4, the fractional fluctuations of the simulations as a function of the scale size. Again, the scale length of the field must be less than about 15 kpc. This conclusion is also obtained by comparing the simulations presented in this paper with the map of Kim et al. (1990). The simulations with the larger scale lengths are much less smooth than the observations, and those simulations with r_0 > 30 kpc tend to be multiply peaked. The emission in the two simulations with magnetic field scale lengths of 40 kpc and 25 kpc is clearly far more clumped than the Coma radio halo.