=
(B sin
)^{n+1}
^{-n},
(1) from a point with
magnetic field B at angle to the line of
sight is proportional to
=
(B sin)^{n+1}^{-n},
(1)
where the spectral index n corresponds to a power law electron energy
distribution with power law index 
= 2n+1. As the field
strength and direction vary from cell to cell of the random field, so
will the emitted radio intensity. I assume throughout that the electron
pitch angle distribution is isotropic, that the electron energy
spectrum is independent of position, and that there is no spectral
curvature.
I assume the magnetic field to be turbulent, with the turbulence being homogeneous and isotropic. No mean field is present. I further assume that the magnetic field is a Gaussian random field. This makes it easy to construct examples from a power spectrum. Also, the angles are isotropic and B is drawn from a Maxwellian distribution. The mean emissivity averaging over all angles and field strengths is
<> =
[2(3/2)^{-(n+1)/2} / (n+3)] {
[(n+4)/2] / (3/2)}^-n (2)
and the mean square emissivity is
<^2> =
[(3/2)^{-(n+1)} / (n+2)] {[(2n+4)/2] /
(3/2)}^-2n,
(3)
where I have set the rms field strength equal to unity. I define the
contrast 
as the
dispersion divided by the mean,
= 
[(<^2>/<>^2)-1].
(4)
This is of order unity showing that the possible emissivities of a coherent magnetic cell cover a wide range, the width of the distribution being larger for steeper spectra.
If an emitting region is made up of a number of cells along the line of sight each with weight W_i then the total contrast is easily shown to be
_T^2 = 
_i
W_i^2 ^2 /(_i
W_i)^2, (5)
with the contrast
for each cell. For N equal cells the contrast is reduced by a
factor 1/N, as
expected. For a Gaussian emissivity profile of equal FWHM the contrast
is further reduced by a factor [(2 ln2)/
]^1/4 = 0.81. If
one could measure the contrast in a radio halo then the cell size could
be estimated. One complicating factor is that radio halos are observed
with only finite resolution so that if the cells are small the contrast
is reduced by averaging within the observing beam.
For such a structured magnetic field the emissivity differs from that expected if the field strength is everywhere the same. The ratio is simply the average of the appropriate power of the field,
<B^(n+1)> = (3/2)^{-(n+1)/2} [(n+4)/2]/(3/2). (6)
Differences from the uniform field strength case are rather small, so that the mean emissivity gives a good estimate of the rms field strength.
The relativistic electrons lose energy by inverse Compton scattering
and synchrotron emission. The inverse Compton scattering is equivalent
to synchrotron losses in a 3.2
field and will
dominate if the magnetic field is weaker than this. This paper
considers radio halos with field strengths of order 1, so that
electrons lose energy at a rate approximately independent of the local
field strength. I therefore ignore changes in energy losses from cell
to cell. If the field is stronger then electrons in the most luminous
regions lose energy faster and the emission fades relative to other
cells, reducing the intensity contrast. I only consider situations
where the field is sufficiently weak that this effect can be neglected,
which would not be true for powerful radio sources.