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

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4 CHAPTER 1. INTRODUCTION Maxwell’s equations take the following pleasant form ∇ · ⃗E = 4πϱ, (1.7) ∇ · ⃗B = 0, (1.8) ∇ × ⃗ E = − 1 c ∂ ⃗ B ∂t , (1.9) ∇ × ⃗ B = 4π c ⃗ j, (1.10) where ⃗ E is the electrical field strength, ⃗ B is the magnetic field strength, ϱ is the charge density and ⃗j is the current density. In this description, Ohm’s law is given by ⃗j ′ = σ ⃗ E ′ , (1.11) where σ is the conductivity and the primed quantities refer to the rest frame of the fluid. Since most astrophysical fluids are electrically neutral and nonrelativistic, ⃗j ′ = ⃗j and ⃗E ′ = E ⃗ + (⃗v × B)/c. ⃗ Thus, Ohm’s law becomes ( ⃗j = σ ⃗E + 1 ) c ⃗v × B ⃗ . (1.12) Multiplying this expression by ∇× and substituting Maxwell’s Eq. (1.9) and Eq. (1.10) into Ohm’s law 1 results in the magnetic diffusivity equation ∂B ⃗ ( ∂t = ∇ × ⃗v × B ⃗ ) ( + η ∇ 2) B, ⃗ (1.13) where η = c 2 /σ4π is the magnetic diffusion ( coefficient as approximated here to be spatially constant. The first term ∇ × ⃗v × B ⃗ ) in the equation above is the dynamo or convective term. It describes the tendency of magnetic fields lines to be frozen in and the amplification of magnetic fields. The second term η ( ∇ 2) B ⃗ on the right hand side of Eq. (1.13) is the diffusive term which represents the resistive leakage of magnetic field lines across the conducting fluid. Thereby, it describes the dissipation of magnetic fields since it allows dilution and cancellation of magnetic fields. Since the diffusive and the convective term are competing, it is useful to introduce a dimensionless parameter, the magnetic Reynolds number R M , R M ≃ |convective term| |diffusive term| = L−1 vB ηL −2 B , (1.14) so that R M ≃ Lv/η, where L is a length characteristic of the spatial variation of the magnetic field. Note that this equation only refers to orders of magnitude. Depending on whether R M ≪ 1 or R M ≫ 1 the convective or the diffusive term, respectively, dominates. In typical astrophysical situations, R M is a large number and thus, the diffusive term is negligible. An interesting regime to note is an infinite magnetic Reynolds number being equivalent to infinite conductivity (ideal MHD). In this regime, the magnetic field lines are frozen in the plasma and thus, follow the movements of the fluid. 1 while using the equality for (∇ 2 ) ⃗ B = ∇(∇· ⃗B)−∇×(∇× ⃗ B) and Maxwell’s Equation ∇· ⃗B = 0.
1.2. OBSERVING ASTROPHYSICAL MAGNETIC FIELDS 5 1.2 Observing Astrophysical Magnetic Fields There are several methods in order to measure strength and structure of astrophysical magnetic fields (e.g. Ruzmaikin et al. 1988; Kronberg 1994; Beck et al. 1996; Widrow 2002, for reviews). The four widely used methods are the observation of the Zeeman effect, the analysis of the observed polarisation of optical starlight scattered on dust particles, the observation of synchrotron emission and, the analysis of multiwavelength studies of linearly polarised radio sources in order to measure the Faraday rotation effect. Their use for different scales and objects in the universe is strongly dependent on the sensitivity of the signal which is aimed to be detected by these methods as briefly described in the following. 1.2.1 The Zeeman Effect One of the most obvious possibilities is the effect of magnetic fields on spectral lines named after and discovered by Pieter Zeeman in 1896. Twelve years after its discovery, it was used by Hale (1908) to detect magnetic fields in sun spots which was the first observations of extraterrestrial magnetic fields. Bolton & Wild (1957) proposed to use the Zeeman splitting of the 21 cm line of neutral hydrogen to detect magnetic fields in the ISM. Ten years later, this measurement was realised by Verschuur (1968). In the absence of external fields, the electronic energy levels of atoms are independent on the direction of the total angular momentum J (orbital angular momentum L plus spin S) of electrons. The energy levels are degenerate. An external magnetic field removes the degeneracy and leads to a dependence of the energy levels on the orientation of the angular momentum with respect to the magnetic field. The energy levels are split into 2j + 1 equidistant levels, where j is the quantum number associated with the total angular momentum J. The energy difference between neighbouring levels which are split by the influence of the magnetic field is ∆E = gµB, (1.15) where µ = e¯h/2m e c = 9.3 × 10 −21 erg G −1 is Bohr’s magneton and g is the Lande factor, which relates the angular momentum of an atom to its magnetic moment. Once this energy difference ∆E is measured, the magnetic field can be determined without the need of any further assumption. Furthermore, Zeeman splitting is sensitive to the total magnetic field strength in contrast to other methods such as synchrotron emission and Faraday rotation which are sensitive to the magnetic field component perpendicular and parallel to the line of sight, respectively. Thus, the observation of the Zeeman effect is the most direct way to detect astrophysical magnetic fields strength. Unfortunately, the Zeeman effect is extremely difficult to detect. The line shift resulting from the energy splitting is ∆ν ν = 1.4 g B ν Hz µG . (1.16) For the two most commonly used spectral lines for Zeeman splitting observations in astrophysics, which are the 21 cm neutral hydrogen absorption line and the 18 cm OH molecule line, ∆ν/ν ≃ 10 −9 g(B/µG) and thus, assuming a magnetic field of 10 µG, the line shift is only about 30 MHz. The line shift due to thermal Doppler line
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72 CHAPTER 4. PACMAN - A NEW RM MAP
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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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