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58 4. The magneto-ionic medium around 3C 449 Table 4.2: CPL power spectrum parameters for the six individual sub-regions of 3C 449. Region FWHM CPL (arcsec ) Best Fit Min Slope Max Slope q f max q − f max q + f max N SPUR 5.50 2.53 1.96 1.58 0.23 3.44 ∞ N LOBE 1.25 2.87 1.60 2.31 0.55 3.35 ∞ N JET 1.25 3.15 1.21 2.29 0.30 4.27 ∞ S JET 1.25 2.76 1.95 2.02 0.3 3.69 ∞ S LOBE 1.25 2.71 1.68 2.36 0.65 3.05 ∞ S SPUR 5.50 2.17 1.53 0.20 0.12 3.95 0.12 Lower and upper limits are quoted at∼90% confidence. Table 4.3: Best-fitting parameters for the joint CPL and BPL fits to all six sub-regions of 3C449. Best Fit Min Slope Max Slope q f max χ 2 q − f max q + f max joint CPL 2.68 1.67 33.5 2.55 1.30 2.81 2.00 Best Fit Min Slope Max Slope q l f b q h χ 2 q − l f b q − h q + l f b q + l joint BPL 2.07 0.031 2.97 17.7 1.99 0.044 2.91 2.17 0.021 3.09 Lower and upper limits are quoted at∼90% confidence. The values of q and f max for the joint CPL fit and q h , q l and f b for the joint BPL fit are the same for all sub-regions, while the normalizations are varied to minimize the overallχ 2 . In the joint BPL fit, the maximum frequency is fixed at f max = 1.67 arcsec −1 . the free parameters of the fit are the six normalizations, D 0 , one for each sub-region, the high and low-frequency slopes, q h and q l , and the break and maximum spatial frequencies f b and f max . I found best fitting parameters of q l = 2.07, q h =2.98, f b =0.031 arcsec −1 . As noted earlier, I also fixed f max = 1.67 arcsec −1 to ensure consistency with the observed depolarizations at 1.25- arcsec resolution. The corresponding structure functions are plotted in Fig. 4.7(g)–(l) and the normalizations for the individual regions are given in Table 4.4. As for the CPL fits, the errors bars are derived from the rms scatter of the structure functions of multiple convolved RM realizations. It is evident from Fig. 4.7 that the structure functions corresponding to the BPL power spectrum, which gives less power on large spatial scales, agree much better with the data. The joint BPL fit has aχ 2 of 17.7, compared with 33.5 for the joint CPL fit (the former has only two extra parameters), which confirms this result. I have so far ignored the effects of any outer scale of the magnetic-field fluctuations. This is justified because the structure functions for the spurs continue to rise at the largest observed separations, indicating that the outer scale must be> ∼ 10 arcsec (≃30 kpc). The model structure 58
4.6. Three-dimensional analysis 59 Table 4.4: Normalizations and expected depolarization for the individual CPL, joint CPL and joint BPL fit parameters at 1.25 arcsec. Region Observed< k> CPL JOINT CPL JOINT BPL C 0 f max < k> C 0 < k> D 0 < k> [rad 2 m −4 ] [arcsec −1 ] [rad 2 m −4 ] [rad 2 m −4 ] [rad 2 m −4 )] N LOBE 61±6 0.96 1.60 63±3 1.52 66±3 1.91 52±4 N JET 106±12 1.34 1.21 106±5 4.76 110±5 0.5 109±5 S JET 91±11 1.50 1.95 70±5 1.94 73±5 1.52 65±4 S LOBE 50±5 1.18 1.68 53±2 1.28 50±2 2.20 45±3 Col.1: region; Col. 2: observed Burn law< k>. Col. 3, 4 and 5: normalization constant C 0 , fitted maximum spatial frequency f max for the best CPL power spectrum of each region and the predicted for each power spectrum. Col. 6, 7 as Col. 3 and 5 but for the joint fit to each CPL power spectrum; Col. 8 and 9 as Col. 3, and 5 but for the joint BPL power spectrum. For both the joint CPL and BPL fits, the maximum frequency is fixed at f max = 1.67 arcsec −1 . In calculating each value of< k> only data with p>4σ p are included. functions fit to the observations assume that the outer scale is infinite and the realizations are generated on sufficiently large grids in Fourier space that the effects of the implicit outer scale are negligible over the range of scales here sampled. I used structure-function data for the entire source to determine an approximate value for the outer scale in Sec. 4.6.3. I now adopt the BPL power spectrum with these parameters and investigate the spatial variations of the RM fluctuation amplitude using three-dimensional simulations. 4.6 Three-dimensional analysis 4.6.1 Models I used the software packageFARADAY (Murgia et al. 2004) to compare the observed RM with simulated images derived from three-dimensional multi-scale magnetic-field models. Given a field model and the density distribution of the thermal gas,FARADAY calculates an RM image by integrating Eq. 3.2 numerically. As in Sec. 4.5, the fluctuations of RM were modeled on the assumption that the magnetic field responsible for the foreground rotation is an isotropic, Gaussian random variable and therefore characterized entirely by its power spectrum. Each point in a cube in Fourier space was first assigned components of the magnetic vector potential. The amplitudes were selected from a Rayleigh distribution of unit variance, and the phases were random in [0, 2π]. The amplitudes were then multiplied by the square root of the power spectrum of the vector potential, which is simply related to that of the magnetic field. The corresponding components of the magnetic field along the line-of-sight were then calculated and transformed to real space. This procedure ensured that the magnetic field was divergence-free. The field components in real 59
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Alma Mater Studiorum Università de
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To Hypatia from Alexandria in Egypt
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Contents Abstract i 1 Physics of ra
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4.6.3 The outer scale of the magnet
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Abstract The existence of diffuse m
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2. Two-dimensional Monte Carlo simu
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Chapter 1 Physics of radio galaxies
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1.2. Non-thermal emission mechanism
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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 and 54: 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: 4.5. Two dimensional analysis 57 Fi
Page 77 and 78: 4.6. Three-dimensional analysis 61
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
Page 105 and 106: 5.6. Rotation-measure bands from co
Page 111 and 112: 5.7. Rotation-measure bands from a
Page 113 and 114: 5.7. Rotation-measure bands from a
Page 115 and 116: 5.8. Discussion 99 5.8 Discussion 5
Page 117 and 118: 5.8. Discussion 101 line−of−sig
Page 119 and 120: 5.8. Discussion 103 flat slopes. I
Page 121 and 122: 5.9. Conclusions and outstanding qu
Page 123 and 124: 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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