B ^2/3. (5)
The emission depends on the field strength as
I B^1+, (6)
where is the radio spectral index.
I therefore take a simple model in which a constant magnetic field is simply compressed while being flux frozen. I also take the high energy particle density to be constant, assuming diffusion to be efficient. I take a realistic density profile of the form
= 1 / ((r + ar_c)(r + r_c)), (7)
which varies as r^-2 at large radii, and has a central cooling flow where the density varies as 1/r. The density has a small central core (I take a=0.1) to avoid singularities.
Fig 1. The predicted synchrotron surface brightness as a function of radius (in units of the core radius) for the simple flux-frozen halo models. The solid line has radio spectral index =1, the dotted line is for =0.7, and the dashed line is for =1.3.
Fig. 1 shows the synchrotron profiles for three values of the spectral index. Steeper spectra give smaller halos, as the dependence of emissivity on field strength is stronger.
The resulting profiles, especially if a realistic spectral index is chosen, look very similar to the radial profile of the Perseus cluster given by Burns et al. (1992), and in fact appear to give a better fit to the data than the models presented in that paper. In the case of the Perseus cluster, Burns et al. (1992) assumed that =0.5 in constructing their models. In reality, =1 or somewhat greater would be more realistic (for the Coma cluster halo, =1.34 [Kim et al. 1990]). Note, however, that a realistic spectral index will also improve the fit of the models of Burns et al. to the data.