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

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106 CHAPTER 5. A BAYESIAN VIEW ON FARADAY ROTATION MAPS 5.3 Testing the Algorithm In order to test the algorithm, the maximum likelihood estimator was applied to generated RM maps with a known magnetic power spectrum ε B (k). Eq. (3.31) gives a prescription for the relation between the amplitude of RM, | RM(k ˆ ⊥ )| 2 , and the magnetic power spectrum in Fourier space. or ε obs B (k) = k 2 a 1 A Ω (2π) 4 ∫ 2π 0 k 2 dφ | ˆ RM( ⃗ k ⊥ )| 2 . (5.28) | RM(k ˆ ⊥ )| 2 = a 1 A Ω (2π) 3 ε obs B (k), (5.29) where A Ω is the area Ω for which RM’s are actually measured and a 1 = a 2 0 n2 e0 L, where L is the characteristic depth of the Faraday screen. −200 −100 0 100 200 RM rad/m2 −200 −100 0 100 200 RM rad/m2 Figure 5.1: On the right panel, a small part (37 × 37 kpc) of a typical realisation of a RM map which is produced by a Kolmogorov like magnetic field power spectrum for k ≥ k c = 0.8 kpc −1 and a magnetic field strength of 5 µG. On the left panel, the RM data used for the data matrix ∆ i is shown where arbitrary neighbouring points were averaged in order to reduce the number of independent points in a similar way as it is done later with the observational data. As Faraday screen, a box with sides being 150 kpc long and a depth of L = 300 kpc is assumed. For the sake of simplicity, a uniform electron density profile is assumed with a density of n e0 = 0.001 cm −3 . The magnetic field power spectrum used is expressed by ⎧ ⎨ ε obs B (k) = ⎩ ε B k 2 k 1−α 0 kc 2+α ε B ∀k ≤ k c k0 ( k k 0 ) −5/3 ∀k ≥ k c . (5.30)
5.3. TESTING THE ALGORITHM 107 where the spectral index was set to mimic Kolmogorov turbulence with energy injection at k = k c , and ∫ ε B = 〈B2 〉 kmax 8π = dk ε obs B (k), (5.31) 0 where k max = π/∆r is determined by the pixel size (∆r) of the used RM map. The latter equation combined with Eq. (5.30) gives the normalisation k 0 in such a way that the integration over the accessible power spectrum will result in a magnetic field strength of B for which 5 µG was used. Furthermore, a k c = 0.8 kpc −1 was used. In order to generate a RM map with the magnetic power spectrum ε B (k) for the chosen Faraday screen, the real and imaginary part of the Fourier space was filled independently with Gaussian deviates. Then these values were multiplied by the appropriate values given by Eq. (5.29) corresponding to their place in k-space. As a last step, an inverse Fourier transformation was performed. A typical realisation of such a generated RM map is shown in the right panel of Fig. 5.1. 1e-12 maximised lnL input power spectrum Fourier analysis ε B (k)*k [erg cm -3 ] 1e-13 1e-14 1e-15 0.1 1 10 100 k [kpc -1 ] Figure 5.2: Power spectra for a simulated RM map as shown in Fig. 5.1. The input power spectra is shown in comparison to the one found by the Fourier analysis as described in Chapter 3 and the one which was derived by the maximum likelihood estimator developed here. One can clearly see the good agreement within one σ between input power spectrum and the power spectrum derived by the maximum likelihood method. For the analysis of the resulting RM map only a small part of the initial map was used in order to reproduce the influence of the limited emission region of a radio source. A Fourier analysis as described in Chapter 3 was applied to this part. The resulting power spectrum is shown in Fig. 5.2 as dashed line in comparison with the input power spectrum as dotted line.
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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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2 CHAPTER 1. INTRODUCTION in heat c
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