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

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10 CHAPTER 1. INTRODUCTION fluctuating magnetic field component along the line of sight, inhomogeneities in the electron distribution and the magnetised medium, Faraday depolarisation (see next section) and beam smearing (e.g. Sokoloff et al. 1998). Based on heuristic arguments, Burn (1966) proposed a simple expression for the depolarisation by a fluctuating magnetic field component p = p max B 2 B 2 , (1.27) where B 2 is the energy density in the regular field component and B 2 is the energy density in the total field. This expression is to be treated in a statistical sense. Ignoring for the moment other depolarising effects, this means that only about 20% of the total magnetic field in spirals is associated with the large scale/regular component of the field. This ratio would be higher if other depolarising effects were included. 1.2.4 Faraday Rotation When linearly polarised radio emission propagates through a magnetised plasma, the polarisation plane of the radiation is rotated. This phenomenon is called Faraday rotation effect. Its detection is one of the most powerful methods to study magnetic field structure and strength. An electromagnetic wave is usually described as ⃗E(⃗x, t) = ⃗ E 0 exp i( ⃗ k⃗x − ωt), (1.28) where ⃗ k is the wave vector and ⃗ E 0 is the amplitude of the wave. For the study of wave propagation in a uniform cold magnetised plasma, where for convenience ⃗ B = (0, 0, B), one has to evaluate the following wave equation ⃗ k × ( ⃗ k × ⃗ E) + ω 2 c 2 K · ⃗E = 0, (1.29) where K is the dielectric tensor (see e.g. chap. 6, Sturrock 1994). If the wave propagation along the magnetic field in a plasma is considered the dispersion relation is different for right and left hand circularly polarised waves introducing a phase velocity difference as the waves pass through it. A linearly polarised wave propagating along the magnetic field in a magnetised plasma can be decomposed into opposite-handed circularly polarised waves which are then characterised by different wave vectors k R,L = ω c [ 1 − ωp 2 ] 1/2 , (1.30) ω(ω ± ω g ) where R and + corresponds to right hand, L and − corresponds to left hand polarisation, respectively, (as seen by the observer receiving the waves), ω p is the plasma frequency (Eq. 1.5) and ω g is the gyrofrequency ω g = eB/m e c. Assuming the same amplitude E 0 , their wave description is ( ) ( ) kL + k R kL − k R E 1 = 2E 0 cos z − ωt cos z (1.31) 2 2 ( ) ( ) kL + k R kL − k R E 2 = 2E 0 sin z − ωt sin z . (1.32) 2 2
1.2. OBSERVING ASTROPHYSICAL MAGNETIC FIELDS 11 The combined wave will be still linearly polarised but the direction of the electric field vector will change with distance z. If ϕ denotes the angle between electric field vector and the x-axis so that tan ϕ = E 2 /E 1 , one can derive ϕ = k L − k R z, (1.33) 2 and thus, dϕ dz = 1 2 (k L − k R ). (1.34) In radio astronomy, propagation over large distances are considered over which ω p and ω g are slowly varying, i.e. k R,L are also changing slowly with distance d. Therefore, ϕ can be expressed by ϕ = ∫ d 0 dl dϕ ∫ d dl = dl 1 2 (k L − k R ). (1.35) In the intra-cluster space, magnetic fields of the order of ∼ µG are observed leading to gyro-frequencies of about ω g ∼ Hz. Furthermore, the electron density in the plasma is about n e ∼ 10 −3 cm −3 causing plasma frequencies of a few ω p ∼ kHz. Radio astronomy detects radiation at frequencies from 10 MHz to 10 GHz. Thus, the wave vector k R,L can be approximated (using ω g , ω p ≪ ω) by k R,L = ω [ 1 − ω2 ( p c 2ω 2 1 ∓ ω ) ] g . (1.36) ω Inserting this expression into Eq. (1.34) yields and substituting the expressions for ω g and ω p yields 0 dϕ dl = ω2 pω g 2c ω 2 (1.37) dϕ dl = 2πn ee 3 B m 2 ec 2 ω 2 . (1.38) Hence, integrating this expression gives the total rotation of the plane of polarisation over the path of propagation of radio waves ϕ = e 3 ∫ 1 d 2πm 2 e c2 ν 2 0 dl n e B ‖ + ϕ 0 , (1.39) where the frequency of the wave is expressed as ν = ω/2π and ϕ 0 is the intrinsic polarisation angle. So far it was assumed that the electromagnetic wave propagates parallel (or anti-parallel) to the magnetic field. If one considers a more general direction of propagation and still uses the approximation ω ≫ ω g , ω p , one would find that the rotation of the plane of polarisation is still expressed by Eq. (1.39), where B ‖ represents then the magnetic field component parallel to the wave vector ⃗ k. Equation (1.39) can be written in a more convenient way ∫ d ϕ = a 0 λ 2 dl n e B ‖ + ϕ 0 , (1.40) 0
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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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