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66 4. The magneto-ionic medium around 3C 449 (a) (c) (b) (d) Figure 4.11: Comparison between observed and synthetic profiles of rms and mean Faraday rotation at resolutions of 5.5 arcsec (a and b) and 1.25 arcsec (c and d). The synthetic profiles are derived from the best-fitting model withΛ max = 65 kpc. The black points represent the data with vertical bars corresponding to the rms fitting error. (a) and (c): Profiles ofσ RM . The magenta crosses represent the mean values from 35 simulations at 5.5-arcsec resolution, and the vertical bars are the rms scatter in these profiles due to sampling. (b) and (d): Profiles of〈RM〉 derived from single example realizations at 5.5 and 1.25-arcsec resolutions. scale. The model withΛ max =65 kpc gives the best representation of the data. The fit is within the estimated errors except for a marginal discrepancy at the very largest (and therefore poorly sampled) separations. That withΛ max =20 kpc is inconsistent with the observed pseudo-structure function for any separation> ∼ 20 arcsec (≃6 kpc) where the sampling is still very good, and is firmly excluded. The model withΛ max =205 kpc has slightly, but significantly too much power on large scales. I emphasize that such estimate of the outer scale of the RM fluctuations is essentially independent of the functional form assumed for the variation of field strength with radius in the central region, which affects the structure function only for small separations. The results are almost identical if I fit the field-strength variation with either the profile of Eq. 4.5 (withη≈0) or that of Eq. 4.6. As for the structure functions of individual regions, the pseudo-structure function at large 66
4.7. Summary and comparison with other sources 67 separations is clearly affected by poor sampling: this increases the errors, but does not produce any bias in the derived values. At large radii, however, the integral in Eq. 4.7 becomes small, so the noise on the RM image is amplified. This is a potential source of error, and I therefore checked the results using numerical simulations. I calculated the mean and rms structure functions for sets of realizations of RM images generated with theFARADAY code for different values ofΛ max . These structure functions are plotted in Figs. 4.12(b) and (c). The main difference between the model structure functions derived from simulations and the pseudo-structure functions described earlier is that the former show a steep decline in power on large scales in place of a plateau. This occurs because the smooth fall-off in density and magnetic field strength with distance from the nucleus suppresses the fluctuations in RM on large scales. The results of the simulations confirm the analysis using the pseudo-structure function. The mean model structure function withΛ max = 65 kpc again fits the data very well, except for a marginal discrepancy at the largest scales. In view of the deviations from spherical symmetry on large scales evident in the X-ray emission surrounding 3C 449 (Fig. 4.1), I do not regard this as a significant effect. In order to check that the best-fitting density and field model also reproduces the data at small separations, I repeated the analysis at 1.25-arcsec resolution. The observed pseudo-structure function is shown in Fig. 4.12(d), together with the the predictions for the BPL power spectrum withΛ max = 205, 65 and 20 kpc. The observed structure function is compared with the mean from 35 simulations withΛ max = 65 kpc in Fig. 4.12(e). In both cases, the agreement forΛ max = 65 kpc is excellent. In Fig. 4.13, example realizations of this model with the best-fitting field variation are shown for resolutions of 1.25 and 5.5 arcsec alongside the observed RM images. 4.7 Summary and comparison with other sources 4.7.1 Summary In this work I have studied the structure of the magnetic field associated with the ionized medium around the radio galaxy 3C 449. I have analysed images of linearly polarized emission with resolutions of 1.25 arcsec and 5.5 arcsec FHWM at seven frequencies between 1.365 and 8.385 GHz, and produced images of degree of polarization and rotation measure. The RM images at both the angular resolutions show patchy and random structures. In order to study the spatial statistics of the magnetic field, I used a structure-function analysis and performed two- and threedimensional RM simulations. I can summarize the results as follows. 67
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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
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1.3. Radio galaxies 7 FR II sources
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1.3. Radio galaxies 9 (a) (b) Figur
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1.3. Radio galaxies 11 pressure, th
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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 and 74: 4.5. Two dimensional analysis 57 Fi
Page 75 and 76: 4.6. Three-dimensional analysis 59
Page 81: 4.6. Three-dimensional analysis 65
Page 85 and 86: 4.7. Summary and comparison with ot
Page 87 and 88: 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
Page 125 and 126: Chapter 6 Faraday rotation in two e
Page 127 and 128: 6.2. Two-dimensional analysis: rota
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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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