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

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112 CHAPTER 5. A BAYESIAN VIEW ON FARADAY ROTATION MAPS ε B (k)*k [erg cm -3 ] 1e-11 1e-12 1e-13 α B = 1.0 α B = 0.67 α B = 0.5 α B = 0.3 α B = 0.1 k -2/3 1e-14 0.1 1 10 100 k [kpc -1 ] Figure 5.5: Power spectra for N = 1500 and n l = 5 are presented. Different exponents α B in the relation B(r) ∼ n e (r) α B of the window function were used. The inclination angle of the source was chosen to be θ = 45 ◦ . model used as already deduced in Sect. 5.4.2. The sudden decrease for α B < 0.1 might be due to non-Gaussian effects. The magnetic field strength derived for this plateau region ranges from 9 µG to 5 µG. The correlation length of the magnetic field λ B was determined to range between 2.5 kpc and 3.0 kpc whereas the RM correlation length was determined to be in the range of 4.5 . . . 5.0 kpc. These ranges have to be considered as a systematic uncertainty since one is not yet able to distinguish between these scenarios observationally. Another systematic effect might be given by uncertainties in the electron density itself. Varying the electron density normalisation parameters (n e1 (0) and n e2 (0)) leads to a vertical displacement of the power spectrum while keeping the same shape. In order to study the influence of the inclination angle on the power spectrum, an α B = 0.5 was used being in the middle of the most likely region derived. Furthermore for this calculations, smaller bins were used and thus, the number of bins was increased to n l = 8. The power spectrum was calculated for two different inclination angles θ = 30 ◦ and θ = 45 ◦ . The results are shown in Fig. 5.7 in comparison with a Kolmogorov like power spectrum. As can be seen from Fig. 5.7, the power spectra derived agree well with a Kolmogorov like power spectrum over at least one order of magnitude. For the inclination angle of θ = 30 ◦ , the following field and map properties were derived B = 5.7 ± 0.1 µG, λ B = 3.1 ± 0.3 kpc and λ RM = 6.7 ± 0.7 kpc. For θ = 45 ◦ , it was calculated B = 7.3 ± 0.2 µG, λ B = 2.8 ± 0.2 kpc and λ RM = 5.2 ± 0.5 kpc. The value of the log likelihood ln L was determined to be slightly higher for the inclination angle of θ = 30 ◦ . The flattening of the power spectra for large k’s can be explained by
5.5. RESULTS AND DISCUSSION 113 4100 4050 4000 3950 ln L ∆ (a) 3900 3850 3800 3750 3700 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Figure 5.6: The log likelihood ln L ∆ (⃗a) of various power spectra assuming different α B while using a constant inclination angle θ = 45 ◦ is shown. One can clearly see that α B = 0.1 . . . 0.8 are in the plateau of maximum likelihood. The sudden decrease for α B < 0.1 in the likelihood might be due to non-Gaussian effects becoming too strong. α B small scale noise which was not modelled separately. Although the central magnetic field strength decreases with decreasing scaling parameter α B , the volume integrated magnetic field energy E B within the cluster core radius r c2 increases. The volume integrated magnetic field energy E B calculates as follows ∫ rc2 ∫ E B = 4π dr r 2 B2 (r) 0 8π = B2 rc2 ( ) 0 dr r 2 ne (r) 2αB , (5.36) 2 0 n e0 where it is integrated from the cluster centre to the core radius r c2 of the second, the non-cooling flow, component of the electron density distribution. The magnetic field profile was integrated for the various scaling parameters and the corresponding field strengths which were determined by the maximum likelihood estimator. The result is plotted in Fig. 5.8. The higher magnetic energies for the smaller scaling parameters which correspond to smaller central magnetic field strengths are due to the higher field strengths in the outer parts of the cool cluster core. This effect would be much more drastic if one had extrapolated the scaling B(r) ∝ n e (r) α B to larger cluster radii and integrated over a larger volume.
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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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8 CHAPTER 1. INTRODUCTION This sync
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10 CHAPTER 1. INTRODUCTION fluctuat
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12 CHAPTER 1. INTRODUCTION where a
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14 CHAPTER 1. INTRODUCTION where L
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20 CHAPTER 1. INTRODUCTION Figure 1
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22 CHAPTER 1. INTRODUCTION standard
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24 CHAPTER 1. INTRODUCTION Consider
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26 CHAPTER 1. INTRODUCTION • Prob
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28 CHAPTER 1. INTRODUCTION where n
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30 CHAPTER 2. IS THE RM SOURCE-INTR
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