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Appendix A: : Rotation measure and depolarization in a foreground Faraday screen

Under the assumption that the RM spatial variations seen in the presented work are fully explained by external patchy screens, we investigated if these RM structural information are consistent with the observed depolarization levels at longer wavelengths. Indeed, the most direct consequence arising from RM structures which fluctuate on small scales is the beam depolarization effect.

We performed this analysis through the software FARADAY (Murgia et al. 2004). In particular we performed 2D Montecarlo simulations by assuming for the rotation measure a simple power law power spectrum of the type:

For each source we simulated RM images by keeping fixed n = 11 / 3, corresponding to the Kolmogorov power law index for the parent magnetic field power spectrum, and by varying the minimum and maximum scale of the RM fluctuations (Λ_min, Λ_max). The computational grid was 1024  ×  1024 pixel² and the pixel-size was fixed to 1 kpc. The resulting field of view was then  ~ 1024  ×  1024 kpc², that was enough to recover the projected size of the sources and to properly sample the RM power spectrum scales. We then added to the simulated RM images a Gaussian noise having σ_noise = Err_fit, in order to mimic the effect of the noise in the observed RM images and we then added an RM offset corresponding to the RM Galactic contribution in the sources location. Finally, the simulated RM images were convolved with a Gaussian function having FWHM equal to the beam of the observations, and blanked as the observations.

We performed several 2D simulations of synthetic RM images to evaluate the range of Λ_min − Λ_max which could explain the observed RM images. Figure A.1 (left panels) shows some examples of simulated RM images, which we found to reproduce quite well the observed RM distributions.

To quantify the two dimensional fluctuations of RM and to compare the simulations with the observations, we used the structure function as a statistical tool. The RM structure function is defined by

where  =  ⟨  ⟩ _(x,y) indicates that the average is taken over all the positions (x,y) in the RM image. Blank pixels were not considered in the statistics. The structure function S(r) was then computed by radially averaging S(dx,dy) over regions of increasing size of radius . In the middle panels of Fig. A.1 we compare the observed (black points) and the simulated (grey-dashed) RM structure function.

In the case of an external Faraday screen, a reduction of the fractional polarization at longer wavelengths is expected if the minimum scale of the RM fluctuation (Λ_min) extend to small scales (beam depolarization). Once a good representation of the RM structures have been found, we investigated the expected beam depolarization to check the consistency with the observed polarization at different wavelengths. In particular, we produced, at each observing frequency, the expected Q and U images corresponding to the simulated RM, and we taken into account the beam depolarization effects by convolving Q and U to the appropriate resolution. The predicted and observed mean degrees of polarization are then compared.

For each source we found a combination of model parameters Λ_min, Λ_max that gives a reasonable representation of the RM structure across the source as well as of the depolarization at longer wavelengths. The values of Λ_min and Λ_max we found for these sources are consistent with those found in the literature for a few other galaxy clusters (e.g. Murgia et al. 2004; Govoni et al. 2006; Guidetti et al. 2008; Bonafede et al. 2010). Therefore, even if we cannot exclude a possible presence of an internal contribution to the observed RM images, the 2D Montecarlo simulations indicate that both RM and polarization data can be explained by the presence of a foreground Faraday screen.

The different magnetic field scales Λ_B of the clusters can be compared by calculating the magnetic field autocorrelation length, which takes into account the minimun and maximum scale of fluctuations (Λ_min, Λ_max), and the index of the power spectrum (n). Following the magnetic field autocorrelation length formula presented by Enßlin & Vogt (2003), for the new data analyzed here we found a magnetic field autocorrelation length variation of a factor of 9. This difference may influence the σ_RM values of about factor of 3 (). However, we note that the exact determination of the magnetic field power spectrum is beyond the scope of this paper, and the numbers indicated in the caption of Fig. A.1 should be considered as indicative.

© ESO, 2010