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

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72 CHAPTER 4. PACMAN – A NEW RM MAP MAKING ALGORITHM is then used to constrain the possible nπ-combinations for the least squares fit for the subsequent observed frequencies. To these methods is refered as the “standard fit” algorithms, as they are currently the most widely used methods. However, these methods might still give multiple acceptable solutions for data with low signal-tonoise, requiring the solution to be flagged and all the information carried by these data is lost. Furthermore, it still can happen that the algorithm chooses a wrong RM and imprints spurious artefacts on the RM and the ϕ 0 maps. Recently, a completely different approach was proposed by Sarala & Jain (2001) which takes the circular nature of the polarisation angle into account. The authors apply a maximum likelihood method to spectral polarisation data. Although this approach is not biased towards any RM value, it is rather designed for a large number of observed wavelengths. Similarly, de Bruyn (1996) and Brentjens & de Bruyn (2004) propose an RM-synthesis via wide-band low-frequency polarimetry. However, typically the observations are only performed at three or four wavelengths especially for extended (diffuse) radio sources. For a new approach for the unambiguous determination of RM and ϕ 0 , it is assumed that if regions exhibit small polarisation angle gradients between neighbouring pixels in all observed frequencies simultaneously, then these pixels can be considered as connected. Note, that the gradient is calculated modulo π, which implies that polarisation angles of 0 and π are regarded as having the same orientation and thus, the cyclic nature of polarisation angles is reflected. Information about one pixel can be used for neighbouring ones and especially the solution to the nπ-ambiguity should be the same. In cases of small gradients, assuming continuity in polarisation angles allows to assign an absolute polarisation angle for each pixel with respect to a reference pixel within each observed frequency. This assignment process has to be done for each spatially independent patch of polarisation data separately, such as each side of a double-lobed radio source. The reference pixel is defined to have a unique absolute polarisation angle and the algorithm will start from this pixel to assign absolute polarisation angles with respect to the reference pixel while going from one pixel to its neighbours. Figuratively, the algorithm eats its way through the set of available data pixel. It might become clear now why the algorithm was named Pacman 1 (Polarisation Angle Correcting rotation Measure ANalysis). Pacman reduces the number of least squares fits in order to solve for the nπambiguity. Preferably, the reference point is chosen to have a high signal-to-noise ratio so that in many areas, it is sufficient to solve the nπ-ambiguity only for a small number of neighbouring pixels simultaneously and to use this solution for all spatially connected pixels. Pixels with low signal-to-noise will profit from their neighbouring pixels allowing a reliable determination of the RM and ϕ 0 . Faraday rotation measure maps are analysed in order to get insight into the properties of the RM producing magnetic fields such as field strengths and correlation lengths. Artefacts in RM maps which result from nπ-ambiguities can lead to misinterpretation of the data. In Chapter 3, a method was proposed to calculate magnetic power spectra from RM maps and to estimate magnetic field properties. It was realised 1 The computer code for Pacman is publicly available at http://dipastro.pd.astro.it/˜cosmo/Pacman
4.2. THE NEW PACMAN ALGORITHM 73 that map making artefacts and small scale pixel noise have a noticeable influence on the shape of the magnetic power spectra on large Fourier scales – small real space scales. Thus, for the application of these kind of statistical methods it is desirable to produce unambiguous RM maps with as little noise as possible. This and similar applications are motivations for designing Pacman and to quantify its performance. In order to detect and to estimate the correlated noise level in RM and ϕ 0 maps, a gradient vector product statistic V was proposed in Chapter 2. It compares RM gradients and intrinsic polarisation angle gradients and aims to detect correlated fluctuations on small scales, since RM and ϕ 0 are both derived from the same set of polarisation angle maps. In Sect. 4.2, the idea and the implementation of the Pacman algorithm is described in detail. In Sect. 4.3, Pacman is tested on artificially generated RM maps and its ability to solve the nπ-ambiguity properly is demonstrated. In Sect. 4.5, this algorithm is applied to polarisation observation data sets of two extended polarised radio sources located in the Abell 2255 (Govoni et al. 2002) cluster and the Hydra cluster (Taylor & Perley 1993). Using these polarisation data, the stability of the Pacman algorithm is demonstrated and the resulting RM maps are compared to RM maps obtained by a standard fit algorithm. The importance of the unambiguous determination of RM’s for a statistical analysis is demonstrated by applying the statistical approach developed in Chapter 3 to the RM maps in order to derive the power spectra and strength of the magnetic fields in the intra-cluster medium. The influence of error treatment in the analysis is also discussed. The philosophy of this statistical analysis and the calculation of the power spectra is briefly outlined in Sect. 4.4.3 whereas the results of the application to the RM maps are presented in Sect. 4.5. In addition, the gradient vector product statistic V is applied as proposed in Chapter 2 in order to detect map making artefacts and correlated noise. The concept of this statistic is briefly explained in Sect. 4.4.1 and in Sect. 4.5, it is applied to the data. After the discussion of the results, the conclusions and lessons learned from the data during the course of this work are given in Sect. 4.6. Throughout the rest of the chapter, a Hubble constant of H 0 = 70 km s −1 Mpc −1 , Ω m = 0.3 and Ω Λ = 0.7 in a flat universe is assumed. 4.2 The new Pacman algorithm 4.2.1 The Idea As described in the introduction, the Faraday rotation measure RM ij at each point with map pixel coordinate (ij) of the source, is usually calculated by applying a least squares fit to measured polarisation angles ϕ ij (k) observed at frequency k ∈ 1...f such that ϕ ij (k) = RM ij λ 2 k + ϕ 0 ij, (4.1) where ϕ 0 ij is the intrinsic polarisation angle at the polarised source. Since every measured polarisation angle is observationally constrained only to a value between 0 and π, one has to replace ϕ ij (k) in the equation above by ˜ϕ ij (k) = ϕ ij (k) ± n ij (k) π, where n ij (k) is an integer, leading to the so called nπ-ambiguity.
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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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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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