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

1.2. OBSERVING ASTROPHYSICAL MAGNETIC FIELDS 15
distribution, the so called β-profile (Cavaliere & Fusco-Femiano 1976):
n e (r) = n e0 (1 + r 2 /r 2 c )−3β/2 , (1.46)
where n 0 is a characteristic electron density (i.e. at the cluster centre n e0 = n e (0)), r c
is the cluster radius, and the β parameter fit to the X-ray observations. Assuming this
density profile in the calculation of the RM dispersion, the following relation can be
derived by integrating Eq. (1.45) (Felten 1996; Feretti et al. 1995)
√〈RM 2 (r ⊥ )〉 = KBn 0rc 1/2 Λc
1/2 √
Γ(3β − 0.5)
(1 + r⊥ 2 , (1.47)
/r2 c )(6β−1)/4 Γ(3β)
where r ⊥ is the projected distance from the cluster centre, Γ is the Gamma function,
the constant K = 624 if the source lies completely beyond the cluster and K = 441
if the source is halfway embedded in the cluster. Usually, the cell size λ c and the RM
dispersion is estimated from the observed RM distribution and the electron density is
given by X-ray measurements. The only unknown in Eq. (1.47) is the magnetic field
strength B which is then easy to determine.
Crusius-Waetzel et al. (1990) and Goldshmidt & Rephaeli (1993) propose a similar
method based on a cell model to evaluate Eq. (1.45) directly. They assume that the field
correlation length l 0 ≪ r c and that the characteristic scale for variations of the electron
density is small compared to the field correlation length l 0 resulting in
〈
RM 2〉 ∫
= a 2 l 0
0
3
〈 〉
ds 1 n 2 (s 1 ) B 2 (s 1 ) , (1.48)
where s 1 is some location at the RM distribution. Their analysis also employs a single
correlation scale.
This analysis of RM distributions is very oversimplified. A Faraday screen which
is build up of uniform cells with uniform fields but random field directions violates
Maxwell’s equation ∇ · ⃗B = 0. Furthermore, it is most likely that a distribution of
length scales exist. In that case, it is also not clear how the cell size λ c is defined.
It might differ significantly from the coherence length of the RM distribution λ RM
which in turn is not necessarily equivalent to the coherence length of the magnetic
field λ B .
A completely different method compared to the ones mentioned above was suggested
by Kolatt (1998) who proposes to apply a maximum likelihood method to analyse
Faraday rotation correlation in order to derive the primordial magnetic field.
In this work, a statistical analysis of RM distributions is proposed. First, the
concept of RM autocorrelation and its relation to magnetic field autocorrelation and
power spectrum are derived. This analysis relies on the condition ∇ · ⃗B = 0 and
thus, incorporates one important property of magnetic fields directly. Furthermore,
a definition of field and RM correlation length is given. This approach has been
successfully applied to data and the results are discussed in Chapter 3. In Chapter
5, a Bayesian maximum likelihood approach is suggested in order to determine the
power spectrum of the magnetic field fluctuations from observed RM distributions.
This method also allows the calculation of error bars.