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64 4. The magneto-ionic medium around 3C 449 Table 4.6: Results from the three-dimensional fits at both the angular resolutions of 1.25 and 5.5 arcsec. FWHM Λ max singleη brokenη (arcsec) (kpc) B 0 (µG) η χ 2 red B 0 (µG) r m (kpc) χ 2 red 1.25 205 4.1±1.1 0.8±0.4 0.48 - - - 5.50 205 2.8±0.5 0.1±0.1 2.2 3.5±0.7 17±9 1.9 5.50 65 - - - 3.5±1.2 16±11 1.8 5.50 20 - - - 3.5±0.8 11±8 2.1 I then made 35 simulations with the best-fitting values of B 0 andηfor each outer scale and evaluated the weightedχ 2 ’s for the resultingσ RM profiles. All three values ofΛ max under investigation give reasonable fits to the observedσ RM profile along the whole radio source. The fit forΛ max = 65 kpc is marginally better than for the other two values (χ 2 red = 1.8), consistent with the results of Sec. 4.6.3 below. In this case the central magnetic field strength is 3.5±1.2µG and the break radius is 16±11 kpc. For the power spectrum withΛ max = 65 kpc and these best-fitting parameters, I also produced three-dimensional simulations at a resolution of 1.25 arcsec. Even though the fitting procedure is based only on the low-resolution data, this model also reproduces the 1.25-arcsec profile very well (χ 2 red =0.7). Combining the values ofχ2 for the two resolutions, using the 1.25-arcsec profile close to the core and the 5.5-profile at larger distances, I findχ 2 red =1.8. Figure 4.11 shows a comparison of the observed radial profiles for rms Faradayσ RM and 〈RM〉 with the synthetic ones derived for this model. The syntheticσ RM profile plotted in Fig. 4.11 is the mean over 35 simulations and may be compared directly with the observations. In contrast, the〈RM〉 profile is derived from a single example realization. It is important to emphasize that the latter is one example of a random process, and is not expected to fit the observations; rather, I aim to compare the fluctuation amplitude as a function of position. The values of B 0 andχ 2 red for all of the three-dimensional simulations, together withηand r m for the single and double power-law profiles, respectively, are summarized in Table 4.6. 4.6.3 The outer scale of the magnetic-field fluctuations The theoretical RM structure function for uniform field strength, density and path length and a power spectrum with a low-frequency cut-off should asymptotically approach a constant value (2σ 2 RM for a large enough averaging region) at separations > ∼ Λ max . The observed RM structure function of the whole source is heavily modified from this theoretical one by the scaling of the electron gas density and magnetic field at large separations, which acts to suppress power on large spatial scales. In Sec. 4.5 I therefore limited the study of the structure function to sub-regions of 3C 449 where uniformity of field strength, density and path length (and therefore of the power- 64
4.6. Three-dimensional analysis 65 Figure 4.10: Distribution of the bestfitting values of B 0 and r m from 35 sets of simulations, each covering ranges of 0.5 – 10µG in B 0 and 0 – 50 kpc in r m . The sizes of the circles are proportional toχ 2 and the blue cross represents the means of the distribution weighted by 1/χ 2 . The plot shows the degeneracy between B 0 and r m described in the text. spectrum amplitude) is a reasonable assumption, inevitably limiting our ability to constrain the power spectrum on the largest scales. Now that I have an adequate model for the variation of n e (r)〈B 2 (r)〉 1/2 with radius (Eq. 4.6), I can correct for it to derive what I call the pseudo-structure function – that is the structure function for a power-spectrum amplitude, which is constant over the source. This can be compared directly with the structure functions derived from the Hankel transform of the power spectrum. To evaluate the pseudo-structure function, I divided the observed 5.5-arcsec RM image by the function [∫ L 1/2 n e (r) 2 B(r) dl] 2 (4.7) 0 (Enßlin & Vogt 2003), where the radial variations of n e and B are those of the best-fitting model (Sec. 4.6) and the upper integration limit L has a length of 10 times the core radius. The integral was normalized to unity at the position of the radio core: this is equivalent to fixing the field and gas density at their maximum values and holding them constant everywhere in the group. The normalization of the pseudo-structure function should then be quite close to that of the two central jet regions. The pseudo-structure function is shown in Fig. 4.12(a) together with the predictions for the BPL power spectra withΛ max =205, 65 and 20 kpc. As expected, the normalization of the pseudostructure function is consistent with that of the jets (Fig. 4.7a and b). The comparison between the synthetic and observed pseudo-structure functions indicates firstly that they agree very well at small separations, independent of the value ofΛ max . This confirms that the best BPL power spectrum found from a combined fit to all six sub-regions is a very good fit over the entire source. Secondly, despite the poor sampling on very large scales, the asymptotic values of the predicted structure functions for the three values ofΛ max are sufficiently different from each other that it is possible to determine an approximate outer 65
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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
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 and 74: 4.5. Two dimensional analysis 57 Fi
Page 75 and 76: 4.6. Three-dimensional analysis 59
Page 77 and 78: 4.6. Three-dimensional analysis 61
Page 79: 4.6. Three-dimensional analysis 63
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
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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6.2. 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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