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

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62 CHAPTER 3. MEASURING MAGNETIC FIELD POWER SPECTRA FFT_data_A400.FITS_0 FFT_Data_hydra.FITS_0 1000 1000 800 800 600 600 400 400 200 200 100 200 300 400 500 100 200 300 400 500 Figure 3.6: The power |〈 ˆ RM(k)〉| 2 of the Fourier transformed RM map of Abell 400 on the left side and Hydra A on the right side. The image for Abell 2634 looks similar. Only half of the Fourier plane is exhibited the other half is point symmetric to the one shown suppressed by the limited window size for small k-vectors in Fourier space. Therefore, a response analysis is necessary in order to understand the influence of the window on the shape of the magnetic energy spectrum. For this purpose, the projected window function was Fourier transformed by employing a FFT algorithm and inserted into Eq. (3.46). Then the response functions ŵ exp (k) obtained were compared to the observed power spectrum ŵ obs (k) as shown in Fig. 3.7 for the case of Abell 2634, where for the normalisation a magnetic field strength B of 5 µG was chosen. The corresponding figures look very similar for Abell 400 and Hydra A. From Fig. 3.7, one can clearly see that the response to delta function like input power spectra on small p scales in Fourier space (i.e. large r in real space) is a smeared out function as one would expect. The response for larger p becomes more strongly peaked suggesting that at k-scales larger than 0.3 kpc −1 for Abell 2634 the influence of the window function becomes negligible. From similar plots for the other two clusters under consideration, scales of about 0.4 kpc −1 for Abell 400 and 1.0 kpc −1 for Hydra A are found. Thus, one would use this value as a lower cutoff k min for the determination of the magnetic field strength. From the response power spectra ŵ p (k ⊥ ) calculated for different p requiring a
3.4. APPLICATION 63 w exp p (k)/[G2 cm 3 ] 1e+59 1e+58 1e+57 1e+56 1e+55 1e+54 w obs (k) p = 0.04 kpc -1 p = 0.11 kpc -1 p = 0.30 kpc -1 p = 0.81 kpc -1 p = 2.22 kpc -1 1e+53 1e+52 0.01 0.1 1 10 k [kpc -1 ] Figure 3.7: Responses at various scales p to the window function of Abell 2634 assuming a B = 5µG in comparison to the observed magnetic autocorrelation function ŵ(k). magnetic field strength B, one can derive a magnetic field strength B exp by integration. In Fig. 3.8, the comparison between these two field strengths is plotted for the different p-scales. It can be seen that for the smallest p’s the deviation between expected field strength and actual observed one is significantly but for larger p they are almost equal suggesting that on these scales the influence of the window is negligible. Taking Fig. 3.8 into account, the values determined above as lower cutoff k min can be confirmed. Another possibility to assess the influence of the window on the power spectra is to match the added response functions and the actual observed power spectra ŵ obs (k). This was done by applying Eq. (3.48) to a set of response functions generated for closely spaced p-scales. The resulting function ŵ exp (k) was matched to the actual observed power spectrum ŵ obs (k) for each of the three clusters by varying B 0 , the spectral index α of the power law in Eq. (3.48) to match the slope of the functions and p min as lower cutoff to fit the function at its turnover for small k-scales. The resulting functions are shown in Fig. 3.9 in comparison to the respective observed power spectra. The shape of the power spectra was matched for Abell 2634 for a spectral index of α = 1.6, for Abell 400 for an α = 1.8 and for Hydra A for an α = 2.0 whereas a lower k-cutoff of 0.09 kpc −1 was used for Abell 2634, 0.08 kpc −1 for Abell 400 and 0.3 kpc −1 for Hydra A. The spectral index α can also be used to plot the respective power law as represented by the straight lines in Fig. 3.9. The slopes of these lines clearly deviate
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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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x LIST OF FIGURES
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xii ABSTRACT correlation function a
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xiv ABSTRACT
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2 CHAPTER 1. INTRODUCTION in heat c
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4 CHAPTER 1. INTRODUCTION Maxwell
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6 CHAPTER 1. INTRODUCTION broadenin
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8 CHAPTER 1. INTRODUCTION This sync
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10 CHAPTER 1. INTRODUCTION fluctuat
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112 CHAPTER 5. A BAYESIAN VIEW ON F
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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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