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

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108 CHAPTER 5. A BAYESIAN VIEW ON FARADAY ROTATION MAPS The maximum likelihood method is numerically limited by computational power since it involves matrix multiplication and inversion, where the latter is a N 3 process. Thus, not all points of the many which are defined in the maps can be used. However, it is desirable to use as much information as possible from the original map. Therefore, it was chose to randomly average neighbouring points with a scheme which let to a map with spatially inhomogeneously resolved cells. The resulting map is highly resolved on top and lowest on the bottom with some random deviations which makes it similarly to the error weighting of the observed data. N = 1500 independent points were used for the analysis. On the left panel of Fig. 5.1, the averaged RM map is shown which was used for the test. As a first guess for the maximum likelihood estimator, the power spectra derived by the Fourier analysis was used. The resulting power spectrum is shown as filled circles with 1-σ error bars in Fig. 5.2. As can be seen from this figure, the input power spectrum and the power spectrum derived by the maximum likelihood estimator agree well within the one σ level. Integration over this power spectrum results in a field strength of (4.7 ± 0.3)µG in agreement with the input magnetic field strength of 5µG. 5.4 Application to Hydra A 5.4.1 The Data ∆ The maximum likelihood estimator which was introduced and tested in the last sections is applied in the following to the Faraday rotation map of the north lobe of the radio source Hydra A (Taylor & Perley 1993). The data were kindly provided by Greg Taylor. For this purpose, a high fidelity RM map was used which was presented in Chapter 4 and was generated by the developed algorithm Pacman using the original polarisation data. Pacman also provides error maps σ i by error propagation of the instrumental uncertainties of polarisation angles. The Pacman map which was used is shown in the right panel of Fig. 5.3. For the same reasons as mentioned in Sect. 5.3, the data were averaged. An appropriate averaging procedure using error weighting was applied such that RM i = ∑ j RM j/σj 2 ∑ , (5.32) j 1/σ2 j and the error calculates as σ 2 RM i = ∑ ( ) j 1/σj 2 ( ∑ ) 2 = j 1/σ2 j 1 ∑ j 1/σj 2 . (5.33) Here, the sum goes over the set of old pixels {j} which form the new pixels {i}. The corresponding pixel coordinates {i} were also determined by applying an error weighting scheme x i = ∑ j x j/σ 2 j ∑ j 1/σ2 j and y i = ∑ j y j/σj 2 ∑ . (5.34) j 1/σ2 j
5.4. APPLICATION TO HYDRA A 109 DEC−−SIN −11.876 −11.878 −11.88 −11.882 −11.884 138.928 138.926 138.924 RA−−−SIN 138.922 −4000 −2000 0 2000 4000 RM rad/m2 −4000 −2000 0 2000 4000 RM rad/m2 Figure 5.3: The final RM map from the north lobe of Hydra A which was analysed with the maximum likelihood estimator; right: original Pacman map, left: error weighted map. Note that the small scale noise for the diffuse part of the lobe is averaged out and only the large scale information carried by this region is maintained. Furthermore, note that each pixel has also a (not displayed) error weighted position. The analysed RM map was determined by a gridding procedure. The original RM map was divided into four equally sized cells. In each of these the original data were averaged as described above. Then the cell with the smallest error was chosen and again divided into four equally sized cells and the original data contained in the so determined cell were averaged. The last step was repeated until the number of cells reached a defined value N. It was decided to use N = 1500. This is partly due to the limitation by computational power but also partly because of the desired suppression of small scale noise by a strong averaging of the noisy regions. The final RM map which was analysed is shown in the left panel of Fig. 5.3. The most noisy regions in Hydra A are located in the coarsely resolved northernmost part of the lobe. It was chosen not to resolve this region any further but to keep the large-scale information which is carried by this region. 5.4.2 The Window Function As mentioned in Sect. 5.2.1, the window function describes the sampling volume and, thus, one has to find a suitable description for it based on Eq. (5.6). Hydra A (or 3C218) is located at a redshift of 0.0538 (de Vaucouleurs et al. 1991). For the derivation of the electron density profile parameter, the work by Mohr et al. (1999) was used which was done for ROSAT PSPC data while using the deprojection of X-ray surface brightness profiles as described in the Appendix A of Pfrommer & Enßlin (2004). Since Hydra A is known to exhibit a strong cooling flow as observed in the X-ray studies, a double β-
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Investigations of Faraday Rotation
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”Vom Himmel lächelte der Herr zu
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vi CONTENTS 3.4.2 Real Space Analys
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List of Figures 1.1 nπ-ambiguity .
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xii ABSTRACT correlation function a
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