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

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52 CHAPTER 3. MEASURING MAGNETIC FIELD POWER SPECTRA smearing, to unavoidable observational noise and RM steps due to the nπ-ambiguity explained in Sect. 1.2.4. These artefacts are described in more detail in the following. Beam Smearing: The finite size l beam of a synthesised beam of a radio interferometer should smear out RM structures below the beam size, and therefore can lead to a smooth behaviour of the measured RM autocorrelation function at the origin, even if the true autocorrelation function has a cusp there. Substantial changes of the RM on the scale of the beam can lead to beam depolarisation, due to the differentially rotated polarisation vectors within the beam area (Conway & Strom 1985; Laing 1988; Garrington et al. 1988). Since beam depolarisation is in principal detectable by its frequency dependence, the presence of sub-beam structure can be noticed, even if not resolved (Tribble 1991; Melrose & Macquart 1998). The magnetic power spectrum derived from a beam smeared RM map should cut-off at large k ∼ π/l beam . Noise: Instrumental noise can be correlated on several scales, since radio interferometers sample the sky in Fourier space, where each antenna pair baseline measures a different k ⊥ -vector. It is difficult to understand to which extend noise on a telescope antenna baseline pair will produce correlated noise in the RM map, since several independent polarisation maps at different frequencies are combined in the map making process. Therefore only the case of spatially uncorrelated noise is discussed, as it may result from a pixel-by-pixel RM fitting routine. This adds to the RM autocorrelation function C obs noise (⃗x ⊥, ⃗r ⊥ ) = σ 2 RM,noise (⃗x ⊥) δ 2 (⃗r ⊥ ) . (3.41) In Fourier space, this leads to a constant error for ŵ(k) ŵ noise (k) = 2 〈σ 2 RM,noise 〉 (3.42) and therefore to an artificial component in the magnetic power spectrum ε obs B (k) increasing by k 2 . If it turns out that for an RM map with an inhomogeneous noise map (if provided by an RM map construction software) the noise affects the small-scale power spectrum too severely, one can try to reduce this by down-weighting noisy regions with a suitable choice of the data weighting function h(⃗x ⊥ ) which was introduced in Sect. 3.3.1 for this purpose. RM steps due to the nπ-ambiguity: An RM map is often derived by fitting the wavelength-square behaviour of the measured polarisation angles. Since the polarisation angle is only determined up to an ambiguity of nπ (where n is an integer), there is the risk of getting a fitted RM value which is off by m ∆RM from the true one, where m is an integer, and ∆RM = π (λ 2 min − λ2 max) −1 is a constant depending on the used wavelength range from λ min to λ max . This can lead to artifical jumps in RM maps, which will affect the RM autocorrelation function and therefore any derived magnetic power spectrum. In order to get an idea of its influences, the possible error by an additional component in the derived RM map can be modelled by RM amb (⃗x ⊥ ) = ∑ i m i ∆RM ⃗1 {⃗x⊥ ∈Ω i } , (3.43) where Ω i is the area of the i-th RM patch, and m i is an integer, mostly +1 or -1. Assuming that different patches are uncorrelated, the measured RM autocorrelation
3.3. THE METHOD 53 function is changed by an additional component, which should be asymptotically for small r ⊥ ( CRM amb (r ⊥) = ∆RM 2 η amb 1 − r ) ⊥ , (3.44) l amb where η amb is the area filling factor of the ambiguity patches in the RM map, and l amb a typical patch diameter. Thus, the artificial power induced by the nπ-ambiguity mimics a turbulence energy spectrum with slope s = 1, which would have equal power on all scales. A steep magnetic power spectrum can therefore possibly be masked by such artefacts. Fortunately, for a given observation the value of ∆RM is known and one can search an RM map for the occurrence of steps by ∆RM over a short distance (not necessarily one pixel) in order to detect such artefacts. 3.3.4 Assessing the Influence of the Window Function A completely different problem is the assessment of the influence of the window function on the shape of the magnetic power spectrum and thus, on the measured magnetic energy spectrum. As stated above, the introduction of a finite window function has the effect to smear out the power spectrum and Eq. (3.26) becomes Ĉ exp ⊥ (⃗ k ⊥ ) = 1 ∫ d 3 q ŵ(q) q2 ⊥ | ˆf( ⃗ k ⊥ − ⃗q)| 2 2 q 2 (2π) 3 . (3.45) V [f] The term q⊥ 2 /q2 ≤ 1 in the expression above states some loss of magnetic power. Without this term, the expression would describe the redistribution of magnetic power within Fourier space, where in this case the magnetic energy would be conserved. However, the expression can be employed as an estimator of the response of the observation to the magnetic power on a given scale p by setting ŵ(q) = δ(q −p) in the above equation. For an approximate treatment of a realistic window and a spherical data average, one can derive ŵp exp (k ⊥ ) = 2p ∫ k⊥ +q π |k ⊥ −q| Here the projected Fourier window was introduced dq q ∫ 2π √ 0 dφW ⊥ (q(cos φ, sin φ)) 4q 2 p 2 − (q 2 + p 2 − k 2 ⊥ )2 . (3.46) W ⊥ ( ⃗ k ⊥ ) = | ˆf ⊥ ( ⃗ k ⊥ )| 2 (2π) 2 , (3.47) A [f⊥ ] where A [f⊥ ] = ∫ d 2 x ⊥ f 2 ⊥ (⃗x ⊥) while defining the projected window function as f ⊥ (⃗x ⊥ ) = ∫ dzf 2 (⃗x)V [f] /A Ω . One can also require for the input power spectrum a field strength by setting ŵ(q) = 2π 2 (B 2 /p 2 ) δ(q − p) in Eq. (3.45). The resulting power spectrum can be treated as an observed power spectrum and thus, a magnetic field strength B exp can be derived by integration following Eq. (3.29). The comparison of B and B exp for the different p-scales yields a further estimate for reliable ranges in k-space. So far the response power spectrum was derived for an underlying magnetic spectrum located only on scales p. One could also add the particular response functions
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Investigations of Faraday Rotation
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”Vom Himmel lächelte der Herr zu
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vi CONTENTS 3.4.2 Real Space Analys
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List of Figures 1.1 nπ-ambiguity .
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x LIST OF FIGURES
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xii ABSTRACT correlation function a
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xiv ABSTRACT
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102 CHAPTER 5. A BAYESIAN VIEW ON F
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118 CONCLUSIONS tually statisticall
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120 BIBLIOGRAPHY Conway, R. G. & St
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122 BIBLIOGRAPHY Ikebe, Y., Makishi
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124 BIBLIOGRAPHY Simard-Normandin,
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126 ACKNOWLEDGEMENTS in his long em
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Publications Journals 1. A Bayesian
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