Investigations of Faraday Rotation Maps of Extended Radio Sources ...

More documents

Recommendations

Info

44 CHAPTER 3. MEASURING MAGNETIC FIELD POWER SPECTRA be described by a window function, through which an underlying virtually statistical homogeneous magnetic field is observed. The window function is zero outside the probed volume, e.g. for locations which are not located in front of the radio source. Inside the volume, the window function scales with the electron density (known from X-ray observations), with the average magnetic energy profile (guessed from reasonable scaling relations, but testable within the approach), and – if wanted – with a noise reducing data weighting scheme. The effect of a finite window function is to smear out the power in Fourier-space. Since this is an unwanted effect one either has to find systems which provide a sufficiently large window or one has to account for this bias. The effect of a too small window on the results depends strongly on the shape and the size of the window. However, one can assess the influence of a finite window but it has to be done for each application at hand separately. In general, the analysis is sensitive to magnetic power on scales below a typical window size, and insensitive to scales above. The same magnetic power spectrum can have very different realisations, since all the phase information is lost in measuring the power spectrum (for an instructive visualisation of this see Maron & Goldreich 2001). Since the presented approach relies on the power spectrum only, it is not important if the magnetic fields are highly organised in structures like flux-ropes, or magnetic sheets, or if they are relatively featureless random-phase fields, as long as their power spectrum is the same. The autocorrelation analysis is fully applicable in all such situations, as long as the fields are sampled with sufficient statistics. The fact that this analysis is insensitive to different realisations of the same power spectrum indicates that the method is not able to extract all the information which may be contained in the map. Additional information is stored in higher order correlation functions, and such can in principle be used to make statements about whether the fields are ordered or purely random (chaotic). The information on the magnetic field strength ( B ⃗ 2 , which is the value at origin of the autocorrelation function), and correlation length (an integral over the autocorrelation function) does only depend on the autocorrelation function and not on the higher order correlations. The presented analysis relies on having a statistically isotropic sample of magnetic fields, whereas MHD turbulence seems to be locally inhomogeneous, which means that small scale fluctuations are anisotropic with respect to the local mean field. However, whenever the observing window is much larger than the correlation length of the local mean field the autocorrelation tensor should be isotropic due to averaging over an isotropic distribution of locally anisotropic subvolumes. This works if not a preferred direction is superposed by other physics, e.g. a galaxy cluster wide orientation of field lines along a preferred axis. However, even this case can in principle be treated by co-adding the RM signal from a sample of clusters, for which a random distribution of such hypothetical axes can be assumed. In any case, it is likely that magnetic anisotropy also manifests itself in the Faraday rotation maps, since the projection connecting magnetic field configurations and RM maps will conserve anisotropy in most cases, except alignments by chance of the direction of anisotropy and the line-of-sight. There are cases where already an inspection by eye seems to reveal the existence of magnetic structures like flux ropes or magnetic sheets. As already stated, the presence of such structures does not limit the analysis, as long as they are sufficiently sampled. Otherwise, one has to replace e.g. the isotropy assumption by a suitable generalisation.
3.3. THE METHOD 45 In many cases, this will allow an analysis similar to the one described here. 3.3 The Method 3.3.1 Real Space Formulation For a line of sight parallel to the z-axis and displaced by ⃗x ⊥ from it, the Faraday rotation measure arising from polarised emission passing from the source at z s (⃗x ⊥ ) through a magnetised medium to the observer located at infinity is expressed by RM(⃗x ⊥ ) = a 0 ∫ ∞ z s(⃗x ⊥ ) dz n e (⃗x) B z (⃗x), (3.2) where a 0 = e 3 /(2πm 2 e c4 ), ⃗x = (⃗x ⊥ , z), n e (⃗x) is the electron density and B z (⃗x) is the magnetic field component parallel to the line of sight. In the following, any redshift effects are neglected, which can be included by inserting the factor (1+z redshift (z)) −2 into the integrand. The focus of this work is on the statistical expectation of the two-point, or autocorrelation function of Faraday rotation maps which can be defined as C RM (⃗r ⊥ ) = 〈RM(⃗x ⊥ )RM(⃗x ⊥ + ⃗r ⊥ )〉 ⃗x⊥ , (3.3) where the brackets indicate the map average with respect to ⃗x ⊥ . However, an observed RM map is limited in size which leads to noise in the calculation of the correlation function on large scales since less pixel pairs contribute to the correlation function for larger pixel separations. In order to suppress this under-sampling, the observable correlation function can be defined as CRM(r obs ⊥ ) = 1 ∫ dx 2 ⊥ RM(⃗x ⊥ )RM(⃗x ⊥ + ⃗r ⊥ ), (3.4) A Ω where the area A Ω of the region Ω for which RM’s are actually measured is used as normalisation assuming that RM(⃗x ⊥ ) = 0 for ⃗x ⊥ /∈ Ω. Furthermore, it is required that the mean RM is zero since statistically isotropic divergence free fields have a mean RM of zero to high accuracy. A non vanishing mean RM in the observational data stems from foregrounds (e.g. the galaxy) which are of no interest here or has its origin in large scale fields to which the approach is insensitive by construction in order to suppress statistically under-sampled length scales. As stated in Sect. 3.2, the virtually homogeneous magnetic field component can be thought to be observed through a window function f(⃗x) which describes the sampling volume. One would chose a typical position in the cluster ⃗x ref (e.g. its centre) and define n e0 = n e (⃗x ref ) and B ⃗ 0 = 〈 B ⃗ 2 (⃗x ref )〉 1/2 . The window function can then be expressed as f(⃗x) = 1 {⃗x⊥ ∈Ω} 1 {z≥zs(⃗x ⊥ )} h(⃗x ⊥ ) g(⃗x) n e (⃗x)/n e0 , (3.5) where 1 {condition} is equal to one if the condition is true and zero if not. The dimensionless average magnetic field profile g(⃗x) = 〈 ⃗ B 2 (⃗x)〉 1/2 /B 0 is assumed to scale with the density profile such that g(⃗x) = (n e (⃗x)/n e0 ) α B . Reasonable values for the exponent α B range between 0.5 and 1 (see Sect. 1.3.2).
Page 1 and 2:
Investigations of Faraday Rotation
Page 3:
”Vom Himmel lächelte der Herr zu
Page 6 and 7:
vi CONTENTS 3.4.2 Real Space Analys
Page 8 and 9: List of Figures 1.1 nπ-ambiguity .
Page 10 and 11: x LIST OF FIGURES
Page 12 and 13: xii ABSTRACT correlation function a
Page 14 and 15: xiv ABSTRACT
Page 16 and 17: 2 CHAPTER 1. INTRODUCTION in heat c
Page 18 and 19: 4 CHAPTER 1. INTRODUCTION Maxwell
Page 20 and 21: 6 CHAPTER 1. INTRODUCTION broadenin
Page 22 and 23: 8 CHAPTER 1. INTRODUCTION This sync
Page 24 and 25: 10 CHAPTER 1. INTRODUCTION fluctuat
Page 26 and 27: 12 CHAPTER 1. INTRODUCTION where a
Page 28 and 29: 14 CHAPTER 1. INTRODUCTION where L
Page 30 and 31: 16 CHAPTER 1. INTRODUCTION 1.3 The
Page 32 and 33: 18 CHAPTER 1. INTRODUCTION In gener
Page 34 and 35: 20 CHAPTER 1. INTRODUCTION Figure 1
Page 36 and 37: 22 CHAPTER 1. INTRODUCTION standard
Page 38 and 39: 24 CHAPTER 1. INTRODUCTION Consider
Page 40 and 41: 26 CHAPTER 1. INTRODUCTION • Prob
Page 42 and 43: 28 CHAPTER 1. INTRODUCTION where n
Page 44 and 45: 30 CHAPTER 2. IS THE RM SOURCE-INTR
Page 56 and 57: 42 CHAPTER 3. MEASURING MAGNETIC FI
Page 86 and 87: 72 CHAPTER 4. PACMAN - A NEW RM MAP
Page 108 and 109:
94 CHAPTER 4. PACMAN - A NEW RM MAP
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
100 CHAPTER 5. A BAYESIAN VIEW ON F
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
118 CONCLUSIONS tually statisticall
Page 134 and 135:
120 BIBLIOGRAPHY Conway, R. G. & St
Page 136 and 137:
122 BIBLIOGRAPHY Ikebe, Y., Makishi
Page 138 and 139:
124 BIBLIOGRAPHY Simard-Normandin,
Page 140 and 141:
126 ACKNOWLEDGEMENTS in his long em
Page 142:
Publications Journals 1. A Bayesian
show all

Investigations of Faraday Rotation Maps of Extended Radio Sources ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?