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54 4. The magneto-ionic medium around 3C 449 4.5 Two dimensional analysis 4.5.1 General considerations In order to interpret the fluctuations of the magnetic field responsible for the observed RM and depolarization of 3C 449, I first discuss the statistics of the RM fluctuations in two dimensions. I use the notation of Laing et al. (2008) in which f= ( f x , f y , f z ) is a vector in the spatial frequency domain, corresponding to the position vector r=(x, y, z). The z-axis is taken to be along the line-of-sight, so that the vector r ⊥ = (x, y) is in the plane of the sky and f ⊥ = ( f x , f y ) is the corresponding spatial frequency vector. The goal is to estimate the RM power spectrum Ĉ(f ⊥ ), where Ĉ(f ⊥ )d f x d f y is the power in the area d f x d f y and in turn to derive the three-dimensional magnetic-field power spectrum ŵ(f), defined so 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. In order to derive the magnetic-field power spectrum, I make the simplifying assumptions discussed in Sec. 3.7.1, which can be summarized as follows: the observed Faraday rotation occurs in a foreground screen (in agreement with the results in Secs. 4.3 and 4.4) whose density is a spherically symmetric function; the magnetic field is distributed throughout the Faraday-rotating medium and it is an isotropic, Gaussian random variable, whose power spectrum is independent of position; the power spectrum amplitude is a function only of the thermal electron density. These assumptions guarantee that the spatial distribution of the magnetic field can be described entirely by its power spectrum ŵ( f ) and that for a medium of constant depth and density, the power spectra of magnetic field and RM are proportional (Enßlin & Vogt 2003) In Sec. 4.3.2, I showed that the Galactic contribution to the 3C 449 RM is substantial and argued that a constant value of−160.7 rad m −2 is the best estimate for its value. Fluctuations in the Galactic magnetic field on scales comparable with the size of the radio sources could be present; conversely, the local environment of the source might make a significant contribution to the mean RM. Both of these possibilities lead to difficulties in the use of the autocorrelation function. Therefore, following the approach of Laing et al. (2008), I initially used the RM structure function (Eq. 3.13) to determine the form of field power spectrum (Sec. 4.5.2), while for its normalization (determined by global variations of density and magnetic field strength), I made use of three-dimensional simulations (Sec. 4.6). 4.5.2 Structure functions I calculated the structure function for discrete regions of 3C 449, over which the spatial variations of thermal gas density, rms magnetic field strength and path length are expected to be reasonably 54
4.5. Two dimensional analysis 55 small. For each of these regions, I first made unweighted fits of model structure functions derived from power spectra with simple, parametrized functional forms, accounting for convolution with the observing beam. I then generated multiple realizations of a Gaussian, isotropic, random RM field, with the best-fitting power spectrum on the observed grids, again taking into account the effects of the convolving beam. Finally, I made a weighted fit using the dispersion of the synthetic structure functions as estimates of the statistical errors for the observed structure functions, which are impossible to quantify analytically (Laing et al. 2008). These errors, which result from incomplete sampling, are much larger than those due to noise, but depend only weakly on the precise form of the underlying power spectrum. The measure of the goodness of fit isχ 2 , summed over a range of separations from r ⊥ = FWHM to roughly half of the size of the region: there is no information in the structure function for scales smaller than the beam, and the upper limit is set by sampling. The errors are, of course, much higher at the large spatial scales, which are less well sampled. Note, however, that estimates of the structure function from neighbouring bins are not statistically independent, so it is not straightforward to define the effective number of degrees of freedom. I selected six regions for the structure-function analysis, as shown in Fig. 4.2. These are symmetrically placed about the nucleus, consistent with the orientation of the radio jets close to the plane of the sky. For the north and south jets, I derived the structure functions only at 1.25- arcsec resolution, as the low-resolution RM image shows no additional structure and has poorer sampling. For the north and south lobes, I computed the structure functions at both resolutions over identical areas and compared them. The agreement is very good, and the low-resolution RM images do not sample significantly larger spatial scales, so I show only the 1.25-arcsec results. Finally, I used the 5.5-arcsec RM images to compute the structure functions for the north and south spurs, which are not detected at the higher resolution. The structure function has a positive bias given by 2σ 2 noise , whereσ noise is the uncorrelated random noise in the RM image (Simonetti, Cordes & Spangler 1984). The mean noise of the 1.25 and 5.5-arcsec RM maps is
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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
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 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: 4.4. Depolarization 53 1.25 arcsec
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
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
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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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