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36 3. Intergalactic magnetic fields 3.7 The magnetic field power spectrum In order to derive the three-dimensional magnetic field power spectrum from the analysis of the RM images, it is necessary to assume statistical isotropy for the field, since only the component of the magnetic field along the line-of-sight contributes to the observed RM. It is important to underline that the turbulence in the intergalactic medium can be locally inhomogeneous, as expected in the MHD regime, particularly regarding small scales fluctuations. However, it is likely that the local anisotropies are isotropically distributed, so that whenever the volume sampled is large enough these anisotropies tend to average out along any line-of-sight. Therefore, the assumption of magnetic-field isotropy must be taken in the sense that the field has no preferred direction when averaged over a sufficiently large volume. If the intergalactic magnetic field can be approximated as Gaussian random variable, then its spatial distribution can be described by the power spectrum of the component along the lineof-sight, or its Fourier transform (the autocorrelation function, Wiener-Khinchin Theorem). For intergalactic magnetic fields, these assumptions are justified by: • the patchiness of most of the RM images across radio galaxies, consistent with isotropic random magnetic fields • the fact that a sum of a large number of independent and identically-distributed random variables (i.e., the field components) approaches a Gaussian distribution (Central Limit Theorem,Rice et al. 1955). Following Laing et al. (2008), the three-dimensional magnetic field power spectrum can be expressed as ŵ( f ) such that ŵ( f )d f x d f y d f z is the power in a volume d f x d f y d f z of frequency space, with f (magnitude f ) representing a vector in the frequency space with the f z coordinate along the line-of-sight. The other interesting property of the magnetic field is its auto-correlation length which in term of ŵ( f ) is given by: Λ B = 1 4 ∫ ∞ 0 ∫ ∞ 0 ŵ( f ) f d f ŵ( f ) f 2 d f (3.10) The constant 1 4 differs by a factor of 4π from that in the equivalent expressions in Enßlin & Vogt (2003, equation 39): a factor of 2π is due to the use of spatial frequency f rather than wave-number k (k=2π f ) and a factor of 2 is due to the different integration limits. 1 Enßlin & Vogt (2003, 2005) have shown that typicallyΛ RM >Λ B since the former gives more weight to the largest spatial scales. In the literature, these scales have often been assumed to 1 Enßlin & Vogt (2003) integrate from−∞ to+∞ in r and r ⊥ . 36
3.7. The magnetic field power spectrum 37 Figure 3.6: Comparison of theσ RM profiles obtained from simulations of a Gaussian and random magnetic field (solid line) and the expectations from the analytical model of Eq. 3.9 by assuming three differentΛ c :Λ min =6 kpc,Λ B =Λ Bz = 16 kpc,Λ RM =69 kpc (dashed and dotted lines). The best agreement is given byΛ c =Λ B . The simulated magnetic field power spectrum has slope n=2. The source is assumed to be halfway through the cluster and the magnetic field strength and gas density have the same radial profiles in both the simulations and the analytical formulation. Taken from Murgia et al. (2004). be identical and the approximationΛ c =Λ RM has been used in the single-scale model (Eq. 3.9), leading to underestimates of the magnetic field strength. Murgia et al. (2004) have shown thatΛ B is a good approximation forΛ c in Eq. 3.9 and therefore that this length must be used to compare the analytical predictions from the single-scale model with RM analyses assuming a multi-scale field (Fig. 3.6). SinceΛ B depends on the underlying magnetic field power spectrum (Eq. 3.10), the latter must be estimated first. Multiple different estimators of spatial statistics of the RM distributions such as structure and auto-correlation functions or a multi-scale statistic have been used to derive the field strength, its relation to the gas density and its power spectrum (Murgia et al. 2004; Govoni et al. 2006; Guidetti et al. 2008; Laing et al. 2008). The technique of Bayesian maximum likelihood has also been used for this purpose (Enßlin & Vogt 2005; Kuchar & Enßlin 2009). These studies have shown that the magnetic field power spectrum is well approximated by a power-law ŵ( f )∝ f −q (3.11) over a range of spatial frequencies, corresponding respectively to the outer and inner fluctuation scales of the magnetic field 2 . Although the theory of intergalactic magnetic field fluctuations is still debated, a power-law functional form is expected if the turbulence is mostly hydrodynamic, 2 Spatial frequency and scale are related in the sense: f= 1/Λ 37
Page 1: 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: 3.6. Tangled magnetic field 35 rad
Page 55 and 56: 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
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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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