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38 3. Intergalactic magnetic fields or in the MHD cascade investigated by Goldreich & Sridhar 1997. The analysis of Vogt & Enßlin (2003, 2005) suggests that the magnetic field power spectrum has a power law form with the slope appropriate for Kolmogorov turbulence and that the autocorrelation length of the magnetic field fluctuations is a few kpc. Also Adaptive Mesh Refinement (AMR) simulations by Brüggen et al. (2005) are consistent with the Kolmogorov slope. However, the Kolmogorov theory (1941), which assumes homogeneous and incompressible turbulence, cannot be rigorously applied because of the evidence in the intergalactic medium of gas radial scaling and in some cases of shells of compressed gas. Moreover, the deduction of a Kolmogorov slope could be premature: there is a degeneracy between the slope and the outer scale, which is difficult to resolve with current Faraday rotation data (Murgia et al. 2004; Guidetti et al. 2008; Laing et al. 2008). Indeed, Murgia et al. (2004) pointed out that shallower magnetic field power spectra are possible if the magnetic field fluctuations have structure on scales of several tens of kpc. Guidetti et al. (2008) showed that a power-law power spectrum with a Kolmogorov slope, and an abrupt long-wavelength cut-off at 35 kpc gave a very good fit to their Faraday rotation and depolarization data for the radio galaxies in A 2382, although a shallower slope extending to longer wavelengths was not ruled out. In the next section I will describe the method used in this thesis to derive the magnetic power spectrum from our RM maps. 3.7.1 Magnetic field power spectrum from two-dimensional analysis The relation between the magnetic field power spectrum and the observed RM distribution is in general quite complicated, depending on the fluctuations in the thermal gas density, the geometry of the source and the surrounding medium, and the effects of incomplete sampling. In order to derive the magnetic field power spectrum, one must make the following assumptions (e.g. Guidetti et al. 2008; Laing et al. 2008): 1. The observed Faraday rotation is due entirely to a foreground ionized medium. This can be addressed by lack of deviation fromλ 2 rotation over a wide range of polarization position angle and the lack of associated depolarization (Sec 3.2.1) 2. The magnetic field is an isotropic, Gaussian random variable 3. The form of the magnetic field power spectrum is independent of position 4. The magnetic field is distributed throughout the Faraday-rotating medium, whose density is a smooth, spherically symmetric function 38
3.7. The magnetic field power spectrum 39 5. The amplitude of ŵ( f ) is spatially variable, but is a function only of the thermal electron density. These assumptions guarantee that the spatial distribution of the magnetic field can be described entirely by its power spectrum ŵ( f ) and that for a medium of constant depth and density, the power spectra of magnetic field and RM are proportional (Enßlin & Vogt 2003). If the RM fluctuations are isotropic, the RM power spectrum Ĉ( f ⊥ ) is the Hankel transform of the auto-correlation function C(r ⊥ ), defined as C(r ⊥ )=〈RM(r ⊥ + r ′ ⊥)RM(r ′ ⊥)〉, (3.12) where r ⊥ and r ′ ⊥ are vectors in the plane of the sky and〈〉 is an average over r′ ⊥ . One could think of obtaining the magnetic field power spectrum directly by Fourier transforming Eq. 3.12. In reality, the observations are affected first by the effects of convolution with the beam, which modify the spatial statistics of RM, and secondly, by the limited size and irregular shape of the sampling region (the region of the source over which the RM has been derived), which results in a complicated window function (Enßlin & Vogt 2003) for computational work in the frequency space. Finally, most of the useful properties of the auto-correlation function are related to the outer scale at C(r ⊥ ) approaches the zero-level, which in most cases is uncertain in the presence of large-scale fluctuations. The alternative strategy proposed by Laing et al. (2008), and applied in the work of this thesis, first estimates the power spectrum of the RM, Ĉ(f ⊥ ), where Ĉ(f ⊥ )d f x d f y is the power in the area d f x d f y , and derives that of the three-dimensional magnetic-field ŵ(f). Laing et al. (2008) demonstrated a procedure that takes into account the convolution effects and minimises the effects of uncertainties in the zero-level. In particular, they showed that 1. in the short-wavelength limit (meaning that changes in Faraday rotation across the beam are adequately represented as a linear gradient), the measured RM distribution is closely approximated by the convolution of the true RM distribution with the observing beam 2. the structure function is a powerful and reliable statistical tool to quantify the two dimensional fluctuations of RM, given that it is independent of the zero-level and structure on scales larger than the area under investigation. The structure function is defined by S (r ⊥ )= (3.13) 39
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Alma Mater Studiorum Università de
Page 5: To Hypatia from Alexandria in Egypt
Page 9 and 10: Contents Abstract i 1 Physics of ra
Page 11 and 12: 4.6.3 The outer scale of the magnet
Page 13 and 14: Abstract The existence of diffuse m
Page 15 and 16: 2. Two-dimensional Monte Carlo simu
Page 17 and 18: Chapter 1 Physics of radio galaxies
Page 19 and 20: 1.2. Non-thermal emission mechanism
Page 21 and 22: 1.2. Non-thermal emission mechanism
Page 23 and 24: 1.3. Radio galaxies 7 FR II sources
Page 25 and 26: 1.3. Radio galaxies 9 (a) (b) Figur
Page 27 and 28: 1.3. Radio galaxies 11 pressure, th
Page 29 and 30: Chapter 2 The X-ray environment of
Page 31 and 32: 2.2. The thermal component 15 Figur
Page 33 and 34: 2.3. Radio source - environment int
Page 35 and 36: 2.3. Radio source - environment int
Page 37 and 38: Chapter 3 Magnetic fields in the ho
Page 39 and 40: 3.1. Introduction 23 Figure 3.1: Ex
Page 41 and 42: 3.1. Introduction 25 to the relativ
Page 43 and 44: 3.2. Faraday Rotation 27 The RM can
Page 45 and 46: 3.2. Faraday Rotation 29 Figure 3.4
Page 47 and 48: 3.3. Depolarization 31 3.3 Depolari
Page 49 and 50: 3.5. The magnetic field profile 33
Page 51 and 52: 3.6. Tangled magnetic field 35 rad
Page 53: 3.7. The magnetic field power spect
Page 57 and 58: 3.7. The magnetic field power spect
Page 59 and 60: Chapter 4 Structure of the magneto-
Page 61 and 62: 4.1. The radio source 3C 449: gener
Page 63 and 64: 4.3. The Faraday rotation in 3C 449
Page 65 and 66: 4.3. The Faraday rotation in 3C 449
Page 67 and 68: 4.4. Depolarization 51 Figure 4.4:
Page 69 and 70: 4.4. Depolarization 53 1.25 arcsec
Page 71 and 72: 4.5. Two dimensional analysis 55 sm
Page 73 and 74: 4.5. Two dimensional analysis 57 Fi
Page 75 and 76: 4.6. Three-dimensional analysis 59
Page 83 and 84: 4.7. Summary and comparison with ot
Page 89 and 90: Chapter 5 Ordered magnetic fields a
Page 91 and 92: 5.1. The Sample 75 35 49 00 ROSAT P
Page 93 and 94: 5.1. The Sample 77 5.1.1 0206+35 02
Page 95 and 96: 5.2. Analysis of RM and depolarizat
Page 97 and 98: 5.3. Rotation measure images 81 IMA
Page 99 and 100: 5.3. Rotation measure images 83 cor
Page 101 and 102: 5.4. Depolarization 85 Therefore, a
Page 103 and 104: 5.5. Rotation measure structure fun
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5.6. Rotation-measure bands from co
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5.7. Rotation-measure bands from a
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5.7. Rotation-measure bands from a
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5.8. Discussion 99 5.8 Discussion 5
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5.8. Discussion 101 line−of−sig
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5.8. Discussion 103 flat slopes. I
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5.9. Conclusions and outstanding qu
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5.9. Conclusions and outstanding qu
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Chapter 6 Faraday rotation in two e
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6.2. Two-dimensional analysis: rota
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6.3. Two-dimensional analysis: stru
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6.4. Preliminary conclusions 121 Ta
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6.4. Preliminary conclusions 123
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Summary and conclusions The goal of
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6.5. Summary 127 responsible for th
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6.6. General conclusions 129 radio
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6.7. Future prospects 131 and is th
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Appendix 133
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136 Data reduction and imaging of l
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144 BIBLIOGRAPHY Bîrzan, L., Raffe
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146 BIBLIOGRAPHY Ensslin, T.A. & Go
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148 BIBLIOGRAPHY Jeltema, T.E. & Pr
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150 BIBLIOGRAPHY Owen, F.N., 1989,
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152 BIBLIOGRAPHY 152
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