Investigations of Faraday Rotation Maps of Extended Radio Sources ...

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xii ABSTRACT correlation function and similarly the Fourier power spectrum of an RM map of an extended background radio source can be used to measure the components of the magnetic autocorrelation and power spectrum tensor within a foreground Faraday screen. It is possible to reconstruct the full non-helical part of this tensor in the case of an isotropic magnetic field distribution statistics. The magnetic field strength, the energy spectrum and the autocorrelation length λ B can be obtained from this non-helical part alone. It is demonstrated that λ B can differ substantially from λ RM , which is the characteristical scale of an RM map. In typical astrophysical situations λ RM > λ B . Strategies to analyse observational data are discussed, taking into account – with the help of a window function – the limited extent of the polarised radio source, the spatial distribution of the electron density and the average magnetic energy density in the screen, and allowing for noise reducing data weighting. The effects of possible observational artefacts, and strategies to avoid them are briefly discussed. This technique of Faraday rotation measure map analysis is applied to three galaxy clusters, Abell 400, Abell 2634 and Hydra A, in order to estimate cluster magnetic field strengths, length scales and power spectra under the assumption that typical field values scale linearly with the electron density. The difficulties involved in the application of the analysis to observational data are investigated. Magnetic power spectra are derived for the three clusters and influences on their shapes caused by the observational nature of the data such as limited source size and resolution are discussed. Various tests are successfully applied to validate the assumptions of the analysis. It is shown that magnetic fluctuations are probed on length scales ranging over at least one order of magnitude. Using this range for the determination of the magnetic field strengths of the central cluster gas yields 3 µG in Abell 2634, 6 µG in Abell 400 and 12 µG in Hydra A as conservative estimates. The magnetic field autocorrelation length λ B is determined to be 4.9 kpc for Abell 2634, 3.6 kpc for Abell 400 and 0.9 kpc for Hydra A. For the three clusters studied, it is found that λ RM ≃ 2...4λ B . Furthermore, in a response analysis it is investigated if it is possible to determine the spectral slopes of the power spectra. It is found that the integrated numbers can be reliably determined from this analysis but differential parameters such as spectral slopes have to be treated differently. Realising that such a statistical analysis can be corrupted by map making artefacts, a new method to calculate Faraday rotation measure maps from multi-frequency polarisation angle data sets is proposed. In order to solve the so called nπ-ambiguity problem which arises from the observational ambiguity of the polarisation angle determined only up to additions of ±nπ, where n is an integer, it is suggested to use a global scheme. Instead of solving the nπ-ambiguity for each data point independently, the proposed algorithm, which was chosen to be called Pacman (Polarisation Angle Correcting rotation Measure ANalysis), solves the nπ-ambiguity for a high signal-tonoise region “democratically” and uses this information to assist the computations in adjacent low signal-to-noise areas. The Pacman algorithm is tested on artifically generated RM maps and applied to two polarisation data sets of extended radio sources located in the Abell 2255 and Hydra A cluster. The RM maps obtained using Pacman are compared to RM maps obtained employing already existing methods. The reliability and the robustness of Pacman is demonstrated. In order to study the influence of map making artefacts, which are imprinted by wrong solutions to the nπ-ambiguities, and of the error treatment of
the data, magnetic field power spectra were calculated from various RM maps. The sensitivity of the statistical analysis to artefacts and noise in the RM maps is demonstrated and thus, the importance of an unambiguous determination of RM values and of an understanding of the nature of the noise in the data. Statistical tests are presented and performed in order to estimate the quality of the derived RM maps, which demonstrate the quality improvements due to Pacman. In order to take the window function properly into account in the measurement of the magnetic power spectrum of clusters of galaxies, a Bayesian maximum likelihood analysis of RM maps of extended radio sources is developed. Using this approach, it is also possible to determine the uncertainties in the measurements. This approach is applied to the RM map of Hydra A North and the power spectrum of the cluster magnetic field is derived. For Hydra A, a spectral index of a Kolmogorov type turbulence is observed over at least one order of magnitude in k-space. A dominant scale at about 3 kpc is found on which the magnetic power is concentrated. Furthermore, the influence of the assumptions about the sampling volume (described by a window function) on the magnetic power spectrum is investigated and it is found that the typical magnetic energy density scales approximately linearly with the electron density (and not with the field strength as assumed before). The central magnetic field strength is determined to be 7 ± 2 µG for the most likely geometries. xiii
Page 1 and 2: Investigations of Faraday Rotation
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Page 6 and 7: vi CONTENTS 3.4.2 Real Space Analys
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72 CHAPTER 4. PACMAN - A NEW RM MAP
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118 CONCLUSIONS tually statisticall
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120 BIBLIOGRAPHY Conway, R. G. & St
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122 BIBLIOGRAPHY Ikebe, Y., Makishi
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124 BIBLIOGRAPHY Simard-Normandin,
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126 ACKNOWLEDGEMENTS in his long em
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Publications Journals 1. A Bayesian
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