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

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88 CHAPTER 4. PACMAN – A NEW RM MAP MAKING ALGORITHM tion of an error weighted least square fit in the RM calculation leads to a reduction of noise in the power spectra. Note that in these two power spectra, the influence of the nπ-artefacts are still present even in the error weighted standard fit RM map. The two remaining power spectra in Fig. 4.5 allow to investigate the influence of the nπ-artefacts. The dashed-dotted line represents the power spectra as calculated from a Pacman RM map allowing σk max = 20 ◦ performing an error weighted fit. Again, no window weighting was applied to this calculation. One can clearly see that there is one order of magnitude difference between the error weighted standard fit power spectra and the Pacman one at large k-values. Since also higher noise levels are allowed for the polarisation angles, this drop can only be explained by the removal of the nπ-artefacts in the Pacman map. That there is still a lot of noise in the map on small real space scales (large k’s), which governs the power spectra on these scales, can be seen from the dotted power spectrum, which was determined as above while applying a simple window weighting in the calculation. There is an additional difference of about one order of magnitude between the two power spectra at large k-scales determined for the two Pacman scenarios. For comparison the power spectrum of pure white noise is plotted as straight dashed line in Fig. 4.5. It can be shown analytically, that white noise as observed through an arbitrary window results in a spectrum of ε B (k) ∝ k 2 . As another independent statistical test, the gradient vector alignment statistic Ṽ was applied to the RM and ϕ 0 maps calculated by the Pacman and the standard fit algorithm. The gradients of RM and ϕ 0 were calculated using the scheme as described in Sect. 2.3. For the two RM maps shown in Fig. 4.3, a ratio of Ṽstdfit/Ṽpacman = 20 was found, which indicates a significant improvement mostly resulting from removing the nπ-artefacts by the Pacman algorithm. The calculation of the normalised quantity V yields V stdfit = −0.75 and V pacman = −0.87. Smoothing the Pacman RM map σk max slightly leads to a drastic decrease of the quantities Ṽ and V . This indicates that the statistic is still governed by small scale noise. This can be understood by looking at the RM maps of Abell 2255E in Fig. 4.3. The extreme RM values are situated at the margin of the source which is, however, also the noisiest region of the source. Calculating the normalised quantity for the high signal-to-noise region yields V pacman = −0.47. 4.5.2 Hydra North The polarised radio source Hydra A, which is in the centre of the Abell cluster 780, also known as the Hydra A cluster, is located at a redshift of 0.0538 (de Vaucouleurs et al. 1991). The source Hydra A shows an extended, two-sided radio lobe. Detailed X-ray studies have been performed on this cluster (e.g. Ikebe et al. 1997; Peres et al. 1998; David et al. 2001). They revealed a strong cooling core in the cluster centre. The Faraday rotation structure was observed and analysed by Taylor & Perley (1993). They reported RM values ranging between −1 000 rad m −2 and +3 300 rad m −2 in the north lobe and values down to −12 000 rad m −2 in the south lobe. Polarisation angle maps and their error maps for frequencies at 7 815, 8 051, 8 165, 8 885 and 14 915 MHz resulting from observation with the Very Large Array were kindly provided by Greg Taylor. A detailed description of the radio data reduction can be found in Taylor et al. (1990) and Taylor & Perley (1993). This section concentrates on the north lobe of Hydra A whereas the south lobe is
4.5. APPLICATION TO DATA 89 separately discussed in Sect. 4.5.3. The south lobe is more depolarised, leading to a lower signal-to-noise than in the north lobe. The north lobe is a good example of how to treat noise in the data, while the south lobe gives a good opportunity to discuss the limitations and the strength of the algorithm Pacman. Figure 4.6: A pixel-by-pixel comparison for the Hydra north lobe of RM stdfit values obtained using the standard fit plotted on the y-axis versus the RM Pacman values calculated by employing the new algorithm Pacman. The scattered points at ± 3000-4000 rad m −2 and ± 10 000 rad m −2 are a result of spurious solutions to the nπ-ambiguity obtained by the standard fit algorithm. For the calculation of the RM maps, all five frequencies were used. Using the polarisation data for the four frequencies at around 8 GHz, the standard fit RM map was calculated which is shown in the upper middle panel of Fig. 4.3. The maximal allowed error in polarisation angle was chosen to be σk max = 25 ◦ and RM max = 15 000 rad m −2 . A Pacman RM map is shown in the middle of the middle panel in Fig. 4.3 below the standard fit map. This map was calculated using the four frequencies at around 8 GHz with a σk=8GHz max = 30◦ and additionally where possible
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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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10 CHAPTER 1. INTRODUCTION fluctuat
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12 CHAPTER 1. INTRODUCTION where a
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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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