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

More documents

Recommendations

Info

60 CHAPTER 3. MEASURING MAGNETIC FIELD POWER SPECTRA 100 A 2634 A 400 Hydra A (8π) 1/2 [µG] 10 1 0.1 0.01 0.1 1 10 100 k c [kpc -1 ] Figure 3.4: Magnetic field strength B dependent on the upper k-cutoff in the integration of Eq. (3.29). The equivalent beamsizes k beam are represented as vertical lines. displays the magnetic field strength B = √ 8π〈ε B 〉 estimated from ε B (k < k c ) = ∫ kc 0 dk ε B (k). (3.55) 3.4.4 Test the model and assess the influence of the window function The various possibilities to test the window function and to assess its influences on the results outlined in Sects. 3.3.3 and 3.3.4 were also applied to the data. The χ 2 -function was derived by radially averaging the χ 2 (⃗x ⊥ ) distribution resulting from Eq. (3.38), where for the magnetic autocorrelation length λ B the value derived in the Fourier analysis was used since this approach seems to be more accurate. The resulting radially binned distributions are shown in Fig. 3.5 determined for the three clusters. There is no apparent spatial large scale trend visible for Abell 400 and Abell 2634 which indicates a reasonable model for the window function. In the case of Hydra A, there appears to be a trend of higher values for χ 2 (x ⊥ ) towards larger r, comparable to small scale trends seen in the χ 2 (x ⊥ ) distribution of Abell 2634 and Abell 400. Furthermore, the χ 2 (⃗x ⊥ ) distribution was integrated following Eq. (3.40) and χ 2 av was calculated to be 1.0 for Abell 2634, 1.2 for Abell 400 and 1.6 for Hydra A. A refinement of the model describing parameter does not seem to be required for Abell 2634 and Abell 400. Since the value for χ 2 av of 1.6 for Hydra A is deviating from 1, the geometry of the source was varied, i.e. the inclination angle θ between source and observer plane was
3.4. APPLICATION 61 10 1 χ 2 (r) 0.1 0.01 0 20 40 60 80 100 120 140 160 r [kpc] A2634 A400 Hydra A Figure 3.5: Testing the window function by calculating the χ 2 (x ⊥ ) distribution for all three clusters. varied from 30 degree up to 60 degree resulting in field strength ranging from 17 µG to 33 µG and values for χ 2 av of 1.4 to 1.8, respectively. These values were derived by integrating over the full accessible unrestricted (without cutoffs) k-space. Furthermore, the scaling parameter α B in the scaling relation of the electron density to the magnetic energy density was varied. A value for α B of 0.5 is still in the limit of reasonable values (see Sect. 1.3.2). Furthermore, such a scaling parameter of α B = 0.5 means that the magnetic energy ε B (x) ∝ n e (⃗x) and thus, the magnetic field would be proportional to the thermal energy of the cluster gas assuming approximately isothermal conditions. However, in this case (α B = 0.5), one obtains for χ 2 av a value of 1.2 and the magnetic field strength is reduced to 17 µG by integrating over the full accessible k-space. One can not be sure that the cause of the trend in the χ 2 (x ⊥ ) distribution is due to a geometry other than assumed because it could also be explained by a fluctuation similar to the one seen in the χ 2 (x ⊥ ) distributions of Abell 2634 and Abell 400. There is no reason to change the initial assumption for the geometry of Hydra A. However, one should keep in mind that the central field strength given could be slightly overestimated for Hydra A but it is not clear to what extent if at all. A good test for the validity of the isotropy assumption is the inspection of the Fourier transformed RM data | RM(k)| ˆ 2 as shown in Fig. 3.6 for Abell 400 and Hydra A. The FFT data look similar for Abell 2634. In this figure half of the Fourier plane is shown since the other half is symmetric to the one exhibited. No apparent anisotropy is present in this figure. It was mentioned already that the magnetic energy spectrum ε obs B (k) in Fig. 3.3 is
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 114 and 115: 100 CHAPTER 5. A BAYESIAN VIEW ON F
Page 124 and 125:
110 CHAPTER 5. A BAYESIAN VIEW ON F
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 ...

Create successful ePaper yourself

Delete template?

Save as template?