Comparison with the Coma radio halo
The most detailed observations (in the sense of having the greatest
resolution) are the 1.4 GHz observations by Kim et al. (1990), and
these will be considered here. Lower frequency maps cover a larger
area, because the halo is effectively larger at lower frequency, and
possibly indicate some structure at large radii (Cordey 1985; Hanisch
1987). This structure might not, however, be indicative of small scale
structure in the magnetic field but could simply be large scale
features related to the spatial distribution of the relativistic
electrons, which is not what I am considering in this paper. In
addition, the Coma cluster will not have reached equilibrium at large
radii so there will still be traces of the cluster formation
process.
\centerline{\hbox{\psfig{figure=/hemlock/tribble/papers/halo/fig3.ps}}}
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
30
Jy 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.
\centerline{\hbox{\psfig{figure=/hemlock/tribble/papers/halo/fig4.ps}}}
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.
Up to:
___________________________________
Peter Tribble, peter.tribble@gmail.com