Measuring the cluster magnetic field power spectra from Faraday rotation maps of Abell 400, Abell 2634 and Hydra A

We apply a novel technique of Faraday Rotation measure (RM) map analysis to three galaxy clusters, Abell 400, Abell 2634 and Hydra A, in order to estimate cluster magnetic field strengths, length scales and power spectra. This analysis essentially a correlation analysis - is based on the assumption that the magnetic fields are statistically isotropically distributed across the Faraday screen. We investigate the difficulties involved in the application of the analysis to observational data. We derive magnetic power spectra for three clusters, i.e. Abell 400, Abell 2634 and Hydra A, and discuss influences on their shapes caused by the observational nature of the data such as limited source size and resolution. We successfully apply various tests to validate our assumptions. We show that magnetic fluctuations are probed on length scales ranging over at least one order of magnitude. Using this range for the determination of magnetic field strength of the central cluster gas yields 3 muG in Abell 2634, 6 muG in Abell 400 and 12 muG in Hydra A as conservative estimates. The magnetic field autocorrelation length lambda(B) was determined to be 4.9 kpc for Abell 2634, 3.6 kpc for Abell 400 and 0.9 kpc for Hydra A. We show that the RM autocorrelation length lambda(RM) is larger than the magnetic field autocorrelation length lambda(B) - for the three clusters studied, we found lambda(RM) similar or equal to 2...4lambda(B) - and thus, they are not equal as often assumed in the literature. Furthermore, we investigate in a response analysis if it is possible to determine spectral slopes of the power spectra. We find that integrated numbers can be reliably determined from this analysis but differential parameters such as spectral slopes have to be treated differently. However, our response analysis results in spectral slopes of the power spectra of spectral indices alpha = 1.6 to 2.0 suggesting that Kolmogorov spectra (alpha = 5/3) are possible but flatter spectral slopes than alpha = 1.3 can be excluded.