Models

I have simulated magnetic fields in a cubical box. The field is constructed by selecting a power spectrum for the vector potential A and choosing components Ã(k) accordingly. The transform Btilde(k) = ik Ã(k) is then transformed back to the real field using an FFT routine (Press et al. 1986). This automatically ensures the magnetic field is divergence free.

The Ã(k) are chosen according to the prescription

P(A_i) = [1 / (2 ) (k)] exp[-A_i^2 / 2 ^2(k)], (9)

where A_i represents either the real or imaginary part of a component of Ã. It is convenient to change to polar coordinates and look at the amplitude and phase of A,

P(A, ) dA d = [A / ^2(k)] exp[-A^2 / 2 ^2(k)] dA {d / 2 }. (10)

Therefore is uniformly distributed between 0 and 2 and A is drawn from a Rayleigh distribution. A Gaussian power spectrum was used, ^2(k) = k^2 exp(-k^2/k_0^2). (11)

This is equivalent to a Gaussian longitudinal magnetic field autocorrelation function f = exp(-r^2/2r_0^2), which defines the field scale length r_0. With this definition the RM autocorrelation function varies as 1-r^2/r_0^2 near the origin. The scale k_0 is adjusted to vary the field correlation length relative to that of the grid. Because the grid is only of finite size only a small range of scale sizes can be investigated while keeping a reasonable number of grid points per cell and a reasonable number of cells in the box. The results were found to be similar for a range of power spectra, although none of the power spectra had power on a large range of scales.

I have smoothed the maps with a Gaussian observing beam of FWHM w. A simple analytic approximation to the reduction in contrast is _S = _0/ (1+w^2/r_0^2), (12)

where r_0 is the field scale length, _0 the map's intrinsic contrast and _S the observed contrast after smoothing. For the observations of Kim et al. (1990) analysed below, w = 1 arcmin 40 kpc.

\centerline{\hbox{\psfig{figure=/hemlock/tribble/papers/halo/fig1.ps}}}

Fig 1. A simulation of the Coma radio halo at 1 arcmin=40 kpc resolution, for a magnetic field scale length of 40 kpc. The scale is in arcmin, the solid contours are the higher values and the dashed contours the lower values.

I present models that have the same overall profile as the Coma radio halo, approximated as a circular Gaussian of FWHM 16 arcmin (Kim et al. 1990), and smoothed to 1 arcmin resolution, for a variety of field scale sizes. These models assume a spectral index n=1 which is a compromise between the slightly flatter spectral index of n 0.7 at the centre of the halo and the value of n 1.3 for the halo as a whole (Kim et al. 1987). This approximation isn't a serious concern as Section 2 shows that the variation of contrast with spectral index is reasonably small over the range of interest. Examples are shown in Figs. 1, 2 and 3. These maps are constructed by periodically extending a half size emission map, shaping to the desired profile and smoothing to 1 arcmin resolution. The emission contrast is then corrected from a rectangular to a Gaussian emission distribution along the line of sight.

\centerline{\hbox{\psfig{figure=/hemlock/tribble/papers/halo/fig2.ps}}}

Fig 2. A simulation of the Coma radio halo at 1 arcmin=40 kpc resolution, for a magnetic field scale length of 25 kpc.

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Peter Tribble, peter.tribble@gmail.com