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

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30 CHAPTER 2. IS THE RM SOURCE-INTRINSIC OR NOT? Figure 2.1: This plot is adapted from Rudnick & Blundell (2003) and shows polarisation structures of the source PKS 1246-410. The source is ≈ 30 ” long. Left column: Original data from Taylor et al. (2002) after applying a median weight filter. Top: polarised intensity at 8 GHz, peak flux density 4 mJy/beam (red); middle rotation measures ≈ − 1300 (blue) to 1300 (red) rad/m 2 ; bottom: intrinsic polarisation angle ϕ 0 − 90 ◦ (blue) to 90 ◦ (red). Right Column: three different simulations of rotation measures having the same colour coding as the actual rotation measures have. Note the different patchiness of the simulated rotation measure images compared to the original data although they share the same RM power spectrum. Taylor et al. (2002), which is shown in the middle left panel of Fig. 2.1 after applying a smoothing algorithm. If the RM is generated in a thin dense mixing layer enclosing the radio source, co-spatial structures in the ϕ 0 and RM distributions are expected. On the other hand, if the Faraday rotation is generated in an external intra-cluster medium than there should be no correlation observed between RM and ϕ 0 . Rudnick & Blundell (2003) search for co-spatial structures in the distribution of RM and ϕ 0 . For that, they derive RM-ϕ 0 scatter plots in which they compare these two quantities at each point of the image. For the source PKS 1246-410, this scatter plot is shown in the upper middle panel of Fig. 2.2. Rudnick & Blundell argue that local co-alignment of ϕ 0 and RM should lead to strongly clustered point distributions in such a scatter plot. Since also statistically independent RM and ϕ 0 distributions may produce such clustering Rudnick & Blundell generate synthetic RM maps having the same power spectrum as the observed one, but random phases. They repeat their analysis with these simulated images and find that the scatter plots for the simulated maps show less clustering. They conclude that the scenario of a mixing layer is the most likely one.
2.2. THE ARGUMENT RECONSIDERED 31 2.2 The Argument Reconsidered The conclusion of Rudnick & Blundell (2003) is statistically questionable since the human eye will detect in any data set apparent correlations. Furthermore, ϕ 0 and RM carry correlated noise as noted by Rudnick & Blundell since these two quantities are generated from the same set of polarisation angle distributions. In the worst case, this correlation will produce step-function like artefacts in the distributions at the same location. These artefacts are present in the original data. The analysis performed by Rudnick & Blundell is designed to have the same RM power spectrum as the observation but higher order correlations are neglected by the random phase realisation. Unfortunately, the statistic used by Rudnick & Blundell is very sensitive to higher order correlations. Any clustering in the RM-ϕ 0 scatter plot is a result of patches of nearly constant values in the RM and ϕ 0 images. A special relation in the Fourier phases (or in higher order correlations) is required in order to recover the appearance of patches in the RM and ϕ 0 distributions. The strength of the clustering in the scatter plot is indeed strongest if the ϕ 0 and RM features are correlated. However, the clustering does not disappear if the RM and ϕ 0 patches have independent distributions, since every ϕ 0 patch is still overlaid by a small number of RM patches, so that the associated clustering in the scatter plot only gets split into a corresponding number of smaller clusters. These clusters happen to be co-aligned on a vertical constant ϕ 0 line in the plot, since they all belong to the same ϕ 0 coherence patch. A corresponding mechanism splits the pixels of an RM cell into a horizontal constant RM line. Such vertical and horizontal features can be indeed seen in the scatter plot of PKS 1246-410 (upper middle panel of Fig. 2.2). In order to perform an experiment which maintains also higher order statistics, one can simply exchange subregions of the RM image from one lobe to the other. This should keep the same RM correlation functions but it will destroy any real correlation between ϕ 0 and RM, since the different regions of the source are independent. This simple experiment was done by diving the source in roughly two equal regions about the centre and shifting the coordinates in right ascension such that two subregions overlap. The ϕ 0 distribution of the east (west) lobe is plotted against the RM data of the west (east) lobe. The shifting of lobes is preferable to a reflection about the centre eliminating possible radial influences on any correlation. The result of this experiment is shown in the upper right panel of Fig. 2.2. The clustering of points does not vanish even though the distributions should be independent. By going a step further, one can generate patchy RM and ϕ 0 maps consisting of patches with nearly constant values but which are statistically independent from each other. For both maps similar recipes were used. First a number N of random seed points ⃗ X i (i ∈ {1...N}) within a square area is drawn, and then the area is split into cells around the seed points by means of a Voronoi-tessellation: Each point ⃗x of the area belongs to the cell of its nearest seed ⃗ X i . Then each seed is attributed a random value ψ i (ψ stands in the following for both RM and ϕ 0 ) and a small two-dimensional random vector ⃗ ∇ψ i (= auxiliary RM or ϕ 0 gradients within the patches, only used for the map construction). Each pixel ⃗x within the cell of seed i gets a value ψ(⃗x) = ψ i + ⃗ ∇ψ i · (⃗x − ⃗ X i ) + σ(⃗x) , (2.1) where σ(⃗x) is a small random noise term. The resulting map consists of patches with
Page 1 and 2: Investigations of Faraday Rotation
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Page 6 and 7: vi CONTENTS 3.4.2 Real Space Analys
Page 8 and 9: List of Figures 1.1 nπ-ambiguity .
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80 CHAPTER 4. PACMAN - A NEW RM MAP
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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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