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

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82 CHAPTER 4. PACMAN – A NEW RM MAP MAKING ALGORITHM where ⃗ ∇RM(⃗x) and ⃗ ∇ϕ 0 (⃗x) are the gradients of RM and ϕ 0 . A map pair, which was constructed from a set of independent random polarisation angle maps ϕ(k), will give a V ≥ −1. A map pair without any correlated noise will give a V ≈ 0. Hence, the statistic V is suitable to detect especially correlated small scale pixel noise. The denominator in Eq. (4.9) is the normalisation and enables the comparison of V for RM maps of different sources, since V is proportional to the fraction of gradients which are artefacts. Since, the comparision of the quality between maps calculated for the same source using the two different algorithms is desired, it is useful to introduce the unnormalised quantity ∫ Ṽ = d 2 x ∇RM(⃗x) ⃗ · ⃗∇ϕ 0 (⃗x). (4.10) The quantity Ṽ gives the absolute measure of correlation between the gradient alignments of RM and ϕ 0 . The smaller this value is the smaller is the total level of correlated noise in the maps. 4.4.2 Error Under or Over Estimation One problem, one is faced with is the possibility that the measurement errors are under or overestimated. Such a hypothesis can be tested by performing a reduced χ 2 ν test which is considered to be a measure for the goodness of each least squares fit and calculates as follows ] 2 χ 2 ν ij = 1 f∑ [ϕ obsij (k) − (RM ij λ 2 k + ϕ0 ij ) ν σ 2 = s2 ij , (4.11) k=1 k ij 〈σ kij 〉 k where ν = f −n c is the number of degrees of freedom and f is the number of frequencies used and n c is the number of model parameters (here n c = 2). The parameter s 2 is the variance of the fit and 〈σ kij 〉 k is the weighted average of the individual variances: ⎡ 〈σ kij 〉 k = ⎣ 1 f f∑ k=1 ⎤−1 1 ⎦ σk 2 ij (4.12) If one believes in the assumption of Gaussian noise, that the data are not corrupted and that the linear fit is an appropriate model then any statistical deviation from unity of the map average χ 2 ν of this value indicates an under or over error estimation. For a χ 2 ν > 1, the errors have been underestimated and for a χ 2 ν < 1, they have been overestimated. Unfortunately this analysis does not reveal in its simple form at which frequency the errors might be under or overestimated. However, one can test for the influence of single frequencies by leaving out the appropriate frequency for the calculation of the RM map and comparing the resulting χ 2 ν ij values with the original one. It is advisable to evaluate the reduced χ 2 ν ij -maps in order to locate regions of too high or too low values for χ 2 ν by eye.
4.5. APPLICATION TO DATA 83 4.4.3 Magnetic Power Spectra The Pacman algorithm is especially useful for the determination of RM maps of extended radio sources. RM maps of extended radio sources are frequently analysed in terms of correlation length and the RM producing magnetic field strength. One analysis relying on statistical methods in order to derive magnetic field power spectra was developed in Chapter 3. It was realised that the method is sensitive to map making artefacts and pixel noise, which can lead to misinterpretation of the data at hand. Therefore this method is a good opportunity to study the influences of noise and artifacts on the resulting power spectra. The observational nature of the data is taken into account by introducing a window function, which can be interpreted as the sampling volume. Especially, a noise reducing data weighting scheme can be introduced to account for observational noise. One reasonable choice of weighting is to introduce a factor σ0 RM /σij RM , which is called simple window weighting in the following. Another possible weighting scheme is a thresholding scheme described by 1/(1 + σ RM ), which means that the noise 0 /σij RM below a certain threshold σ0 RM is acceptable and areas of higher noise are downweighted. Note, that magnetic power spectra, which are calculated using the approach explained above, are used as a valuable estimator for magnetic field strengths and correlation lengths. However, since the spectra are shaped by and are very sensitive to small scale noise and map making artefacts, they are used here to quantify the influence of noise and artefacts rather than as an estimator for characteristic properties of the RM producing magnetic fields. 4.5 Application to Data For the calculation of what is called further on the standard fit RM maps, the least squares method was used as suggested by Vallée & Kronberg (1975) and Haves (1975). Also for the individual RM fits, this method was adopted. As an upper limit for the RM values |RM| ij = RM max ± 3σij RM was used, where σij RM was calculated following Eq. (4.5). The least-squares fit for the individual data points were always performed as error weighted fits as described by Eq. (4.3) and (4.4), if not stated otherwise. For the Pacman calculations, α = β = 1 was used for the free parameters in Eq. (4.7). However, the method suggested by Ruzmaikin & Sokoloff (1979) was also used as standard fit algorithm. The results did not change substantially. 4.5.1 Abell 2255E The Abell cluster 2255, which has a redshift of 0.0806 (Struble & Rood 1999), has been studied by Burns et al. (1995) and Feretti et al. (1997). The polarised radio source B1713+641, which is called hereafter A2255E, has a two sided radio lobe structure and is not directly located in the cluster centre. Polarisation observations were performed using the Very Large Array (VLA) at 4535, 4885, 8085, and 8465 MHz. The data reduction was done with standard AIPS (Astronomical Imaging Processing Software) routines (Govoni et al., in prep.). The polarisation angle maps and their standard error
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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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4 CHAPTER 1. INTRODUCTION Maxwell
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10 CHAPTER 1. INTRODUCTION fluctuat
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30 CHAPTER 2. IS THE RM SOURCE-INTR
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