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

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42 CHAPTER 3. MEASURING MAGNETIC FIELD POWER SPECTRA Another important issue in previous analyses is the assumption that the RM ordering scale read off RM maps is equivalent to the magnetic field’s characteristic length scale, the autocorrelation length λ B . This might have lead to underestimations of field strengths in the past since another simpler equation than Eq. (1.47) often used for the derivation of the magnetic field strength is 〈B 2 z〉 = 〈RM 2 〉 a 2 0 n2 e L λ z , (3.1) where 〈RM 2 〉 is the RM dispersion, L is the depth of the Faraday screen, a 0 = e 3 /(2πm 2 e c4 ), n e is the electron density and λ z is a characteristic length scale of the fields. At this point it becomes clear that the definition of the characteristic length scale λ z is crucial for the derivation of magnetic field strength. The assumption of a constant magnetic field throughout the cluster leads to a definition of λ z = L and thus, 〈B 2 z〉 ∝ L −2 . The characteristic length scale for the cell model would be the size of each cell λ z = l cell . Another definition one could think of is the RM ordering scale, λ z = λ RM . However, the correct length scale is the autocorrelation length λ z of the z-component of the magnetic field measured along the line of sight. In order to overcome such limitations, a purely statistical approach which incorporates the vanishing divergence of the magnetic field is developed in Sect. 3.3 whereas its philosophy is outlined in Sect. 3.2. This approach relies on the assumption that the fields are statistically isotropically distributed in Faraday screens such that the z-component is representative for all components. Starting from this assumption, a relation between the observationally accessible RM autocorrelation function and the magnetic autocorrelation tensor is established in real and in Fourier space such that one gains access to the power spectrum of the magnetic field inhabited by the Faraday screen. In Sect. 3.4, this statistical analysis is applied to observational data by reanalysing the Faraday rotation measure maps of three extended extragalactic radio sources: Hydra A (Taylor & Perley 1993), 3C75 and 3C465 (Eilek & Owen 2002) which were kindly provided by Greg Taylor, Jean Eilek and Frazer Owen, respectively. The results are presented and discussed in Sect. 3.5. Throughout the rest of this 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. 3.2 The Philosophy If magnetic fields are sampled in a sufficiently large volume, they can hopefully be regarded to be statistically homogeneous and statistically isotropic. This means that any statistical average of a quantity depending on the magnetic fields does not depend on the exact location, shape, orientation and size of the used sampling volume. The quantity which is of interest is the autocorrelation (or two-point-correlation) function (more exactly: tensor) of the magnetic fields. The information contained in the autocorrelation function is equivalent to the information stored in the power spectrum, as stated by the Wiener-Khinchin Theorem (WKT) (compare Sect. 1.4.2). Therefore, two equivalent approaches are presented, one based in real space, and one based in Fourier space. The advantage of this redundancy is that some quantities are
3.2. THE PHILOSOPHY 43 easier accessible in one, and others in the other space. Further, this allows to crosscheck computer algorithms based on this work by comparing results gained by the different approaches. Faraday rotation maps of extended polarised radio sources located behind or embedded in a Faraday screen are the observables which can be used to access the magnetic fields. Since an RM map shows basically the line-of-sight projected magnetic field distribution, the RM autocorrelation function is mainly given by the projected magnetic field autocorrelation function. Therefore, measuring the RM autocorrelation allows to measure the magnetic autocorrelation, and thus, provides a tool to estimate magnetic field strength and correlation length. The situation is a bit more complicated than described above, due to the vector nature of the magnetic fields. This implies that there is an autocorrelation tensor instead of a function, which contains nine numbers corresponding to the correlations of the different magnetic components against each other, which in general can all be different. The RM autocorrelation function contains only information about one of these values, the autocorrelations of the magnetic field component parallel to the lineof-sight. However, in many instances the important symmetric part of the tensor can be reconstructed and using this information the magnetic field strength and correlation length can be obtained. This is possible due to three observations: 1. Magnetic isotropy: If the sampling volume is sufficiently large, so that the local anisotropic nature of magnetic field distributions is averaged out, the (volume averaged) magnetic autocorrelation tensor is isotropic. This means, that the diagonal elements of the tensor are all the same, and that the off-diagonal elements are described by two numbers, one giving their symmetric, and one giving their anti-symmetric (helical) contribution. 2. Divergence-freeness of magnetic fields: The condition ∇ ⃗ · ⃗B = 0 couples the diagonal and off-diagonal components of the symmetric part of the autocorrelation tensor. Knowledge of one diagonal element (e.g. from an RM measurement) therefore specifies fully the symmetric part of the tensor. The trace of the autocorrelation tensor, which can be called scalar magnetic autocorrelation function w(r), contains all the information required to measure the average magnetic energy density ε B = w(0)/(8 π) or the magnetic correlation length λ B = ∫ ∞ −∞ dr w(r)/w(0). 3. Unimportance of helicity: Although helicity is a crucial quantity for the dynamics of magnetic fields, it does not enter any estimate of the average magnetic energy density, or magnetic correlation length, because helicity only affects offdiagonal terms of the autocorrelation tensor. The named quantities depend only on the trace of the tensor and are therefore unaffected by helicity. One cannot measure helicity from a Faraday rotation map alone, since it requires the comparison of two different components of the magnetic fields, whereas the RM map contains information on only one component. In a realistic situation, the sampling volume is determined by the shape of the polarised radio emitter and the geometry of the Faraday screen, as given by the electron density and the magnetic field energy density profile. The sampling volume can
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Investigations of Faraday Rotation
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”Vom Himmel lächelte der Herr zu
Page 6 and 7: vi CONTENTS 3.4.2 Real Space Analys
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92 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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