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120 6. Faraday rotation in two extreme environments Figure 6.10: RM structure functions for the sub-regions of M 87 indicated in the panels at 0.4 arcsec FWHM. The horizontal bars represent the bin widths and the crosses the mean separation for data included in the bins. The red curves are the best fits including the effects of the convolving beam. The error bars represent the rms variations for structure functions derived for multiple realizations of the indicated cut-off power-law power spectra on the observed grid for each region. Structure functions have been computed at the resolutions of the RM images and binned by separation over four areas of 0755+37 (the two jets and the two lobes), and in three areas of M 87 (the inner and outer E lobe and the W lobe). The reason for splitting the E lobe of M 87 into two sub-regions was that the structure function derived for the whole lobe showed a decrease at large separation which probably indicates the presence of significant spatial variations in the field and/or density (cf. 3C 449, Chapter 4). All of these sub-regions have enough data points to ensure that the structure functions are robustly determined. A first-order correction for uncorrelated random noise,σ fit , evaluated in each sub-region, has been applied to all of the structure functions. A cut-off power law (CPL) power spectrum (see also Eq. 4.3) Ĉ( f ⊥ ) = 0 f ⊥ < f min = C 0 f −q ⊥ f ⊥ ≤ f max = 0 f ⊥ > f max . (6.1) gives model structure functions in very good agreement with the data for both sources (Figs.6.9 and 6.10). The values of f max have also been estimated by fitting the mean values of k for the sub-regions. I produced multiple RM realisations corresponding to the CPL model power spectrum on the observed grid, including the effects of the convolving beam, to evaluate the rms deviations of their structure functions. These take into account the effects of undersampling and are plotted as error bars attached to the observed points. The best-fitting parameters are quoted for each sub-region separately in Table 6.4, together with the observed and predicted values of 120
6.4. Preliminary conclusions 121 Table 6.4: Best-fitting power spectrum parameters for the individual sub-regions of 0755+37 and M 87. Source Region FWHM q f max f min k obs k syn [arcsec] [arcsec −1 ] [arcsec −1 ] [rad 2 m −4 ] [rad 2 m −4 ] 0755+37 E LOBE 4.0 2.48 8.33 ∞ 64±3 69±3 E JET 2.45 ” 0.017 112±5 119±4 W JET 2.20 ” 0.052 156±7 164±5 W LOBE 2.19 0.69 0.028 109±5 104±5 E LOBE 1.3 2.52 8.33 ∞ 41±3 37±4 E JET 2.51 4.16 0.017 63±4 57±5 W JET 2.21 8.33 0.052 58±5 55±5 W LOBE 2.19 0.69 0.028 38±6 40±4 M 87 E out LOBE 0.4 3.05 2.6 ∞ 23470±600 34350±150 E in LOBE ” 2.87 2.6 ” 35310±800 59540±200 W LOBE ” 2.85 2.6 ” 15178±500 14491±100 Burn law k. In M 87, the slopes and cut-off frequencies derived for the individual sub-regions are mutually consistent. A combined fit, weighting all of the sub-regions equally and allowing their normalizations to vary, gives q=2.95 and f max = 2.75 arcsec −1 . The slopes for the different parts of 0755+37 differ significantly from each other, and no combined fit has been attempted. The structure functions for the lobes of M 87 show little firm evidence for any levelling off at large separations (the points at> ∼ 15 arcsec in the East lobe have very large errors). In the W lobe, the field outer scale must be> ∼ 30 arcsec (≃2.7 kpc). The fits therefore assume f min =∞ for this source. In contrast, the structure functions for both of the jets and the W lobe of 0755+37 level off and clearly show the suppression of the RM fluctuations above some outer scale. Better fits to these structure functions are provided by setting a finite value for f min of 0.017-0.052 arcsec −1 , corresponding to a maximum field fluctuations on scalesΛ max ≈ 15−50 kpc. The situation is more ambiguous in the E lobe of 0755+37, because of the large errors, and in this case the fit with f min =∞ is shown. Further work on the outer scale will require a three-dimensional analysis which takes into account the spatial variations of density and path length across the sources. 6.4 Preliminary conclusions I have studied in detail the images of linearly polarized emission from the FR I radio galaxies 0755+37, in a sparse group, and M 87 at the centre of the cool-core Virgo cluster. I have produced images of RM and Burn law k with resolutions of 4.0 and 1.3 arcsec FWHM for 0755+37, and 0.4 arcsec FWHM for M 87. The spatial variations of the RM distributions in both sources have been investigated using a structure-function technique. 121
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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
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2.2. The thermal component 15 Figur
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2.3. Radio source - environment int
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2.3. Radio source - environment int
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Chapter 3 Magnetic fields in the ho
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3.1. Introduction 23 Figure 3.1: Ex
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3.1. Introduction 25 to the relativ
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3.2. Faraday Rotation 27 The RM can
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3.2. Faraday Rotation 29 Figure 3.4
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3.3. Depolarization 31 3.3 Depolari
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3.5. The magnetic field profile 33
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3.6. Tangled magnetic field 35 rad
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3.7. The magnetic field power spect
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Chapter 4 Structure of the magneto-
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4.1. The radio source 3C 449: gener
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4.3. The Faraday rotation in 3C 449
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4.3. The Faraday rotation in 3C 449
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4.4. Depolarization 51 Figure 4.4:
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4.4. Depolarization 53 1.25 arcsec
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4.5. Two dimensional analysis 55 sm
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4.5. Two dimensional analysis 57 Fi
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4.6. Three-dimensional analysis 59
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4.7. Summary and comparison with ot
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Page 97 and 98: 5.3. Rotation measure images 81 IMA
Page 99 and 100: 5.3. Rotation measure images 83 cor
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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
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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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Page 139 and 140: 6.4. Preliminary conclusions 123
Page 141 and 142: Summary and conclusions The goal of
Page 143 and 144: 6.5. Summary 127 responsible for th
Page 145 and 146: 6.6. General conclusions 129 radio
Page 147 and 148: 6.7. Future prospects 131 and is th
Page 149: Appendix 133
Page 152 and 153: 136 Data reduction and imaging of l
Page 160 and 161: 144 BIBLIOGRAPHY Bîrzan, L., Raffe
Page 162 and 163: 146 BIBLIOGRAPHY Ensslin, T.A. & Go
Page 164 and 165: 148 BIBLIOGRAPHY Jeltema, T.E. & Pr
Page 166 and 167: 150 BIBLIOGRAPHY Owen, F.N., 1989,
Page 168 and 169: 152 BIBLIOGRAPHY 152
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Alma Mater Studiorum Universit`a degli Studi di Bologna ... - Inaf

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