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

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28 CHAPTER 1. INTRODUCTION where n is the dimension of θ and det V indicates the determinant. It is noteworthy that Eq. (1.88) is exact when y(µ xi , θ) depends linearly on the various θ i . The likelihood function can also be used as a power spectrum estimator since such an estimator has to minimise the variance. If one identifies the covariance matrix as defined by Eq. (1.60) and assuming the mean to be zero 〈y i 〉 = 0, the likelihood function changes L(θ; x) = [ 1 (2π) N/2 (det C) 1/2 exp − 1 ] 2 ∆T C −1 ∆ , (1.89) where C is the covariance matrix, which expresses the theoretical expectations and dependents on the model parameters, ∆ i are the data and N is the dimension of the covariance matrix. 1.4.4 Structure of the Thesis The interpretation of the Faraday rotation data as being associated with an external Faraday screen has been challenged several times and is subject to a broad discussion. Therefore, Chapter 2 is devoted to this question. After being confident of the interpretation as Faraday rotation being external to the source, Chapter 3 describes the application of autocorrelation and power spectrum analysis to the Faraday rotation data. However, it was realised that the map making algorithm might produce artefacts related to the mentioned nπ-ambiguity influencing this analysis significantly. Therefore, in Chapter 4 a new RM map-making algorithm called Pacman is introduced and compared to the standard algorithms. Finally, motivated by the recent and great success of maximum likelihood methods in the determination of power spectra, this method is applied to Faraday rotation data in Chapter 5.
Chapter 2 Is the Faraday Rotation Source-Intrinsic or not? For clusters of galaxies, the interpretation of Faraday rotation data relies on the assumption that the Faraday rotation medium and the source of the linearly polarised emission are well separated. This assumption has been challenged by Bicknell et al. (1990) and more recently by Rudnick & Blundell (2003). This chapter is aimed to give evidence in favour for the validity of the assumption of an external Faraday screen which is presumed in the following chapters. This work is published in Enßlin et al. (2003). My part in this work was the application of the statistics to several observational and generated data sets. 2.1 A Long Lasting Debate As already mentioned in Sect. 1.2.4, it is difficult to distinguish between source intrinsic and source external Faraday depolarisation. Most authors argue in favour for the Faraday screen being external to an extragalactic extended radio source and identify the Faraday screen with the surrounding intra-cluster medium. Another scenario has also been widely discussed. In this scenario, the dominating Faraday rotating region is located in a dense, thin mixing layer around the radio source. The consequence of this scenario is that the estimated field strengths for the intra-cluster medium would be lower by at least one order of magnitude. Bicknell et al. (1990) argue in favour of the latter scenario. They performed a threedimensional smoothed particle hydrodynamical simulation of a transonic shear layer so that large-scale nonlinear surface waves can form on the lobes of extragalactic radio sources through the successive merging of smaller waves generated by the Kelvin- Helmholtz instability. They propose a theory which relates the surface waves developing to substantial variations in rotation measures. They argue, that such a model being a surface effect would also give rise to a dependence of the polarisation angle ϕ on the wavelength squared as observed. Recently, Rudnick & Blundell (2003) proposed a different method in order to find arguments for the scenario of the Faraday rotating region being a thin dense layer around the radio source. They compare the intrinsic polarisation angle ϕ 0 with the RM at each point of the source PKS 1246-410 in the Centaurus cluster observed by 29
Page 1 and 2: Investigations of Faraday Rotation
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78 CHAPTER 4. PACMAN - A NEW RM MAP
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100 CHAPTER 5. A BAYESIAN VIEW ON F
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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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