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

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