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

3.3. THE METHOD 45
In many cases, this will allow an analysis similar to the one described here.
3.3 The Method
3.3.1 Real Space Formulation
For a line of sight parallel to the z-axis and displaced by ⃗x ⊥ from it, the Faraday
rotation measure arising from polarised emission passing from the source at z s (⃗x ⊥ )
through a magnetised medium to the observer located at infinity is expressed by
RM(⃗x ⊥ ) = a 0
∫ ∞
z s(⃗x ⊥ )
dz n e (⃗x) B z (⃗x), (3.2)
where a 0 = e 3 /(2πm 2 e c4 ), ⃗x = (⃗x ⊥ , z), n e (⃗x) is the electron density and B z (⃗x) is
the magnetic field component parallel to the line of sight. In the following, any redshift
effects are neglected, which can be included by inserting the factor (1+z redshift (z)) −2
into the integrand.
The focus of this work is on the statistical expectation of the two-point, or autocorrelation
function of Faraday rotation maps which can be defined as
C RM (⃗r ⊥ ) = 〈RM(⃗x ⊥ )RM(⃗x ⊥ + ⃗r ⊥ )〉 ⃗x⊥ , (3.3)
where the brackets indicate the map average with respect to ⃗x ⊥ . However, an observed
RM map is limited in size which leads to noise in the calculation of the correlation
function on large scales since less pixel pairs contribute to the correlation function
for larger pixel separations. In order to suppress this under-sampling, the observable
correlation function can be defined as
CRM(r obs ⊥ ) = 1 ∫
dx 2 ⊥ RM(⃗x ⊥ )RM(⃗x ⊥ + ⃗r ⊥ ), (3.4)
A Ω
where the area A Ω of the region Ω for which RM’s are actually measured is used as
normalisation assuming that RM(⃗x ⊥ ) = 0 for ⃗x ⊥ /∈ Ω. Furthermore, it is required
that the mean RM is zero since statistically isotropic divergence free fields have a
mean RM of zero to high accuracy. A non vanishing mean RM in the observational
data stems from foregrounds (e.g. the galaxy) which are of no interest here or has its
origin in large scale fields to which the approach is insensitive by construction in order
to suppress statistically under-sampled length scales.
As stated in Sect. 3.2, the virtually homogeneous magnetic field component can be
thought to be observed through a window function f(⃗x) which describes the sampling
volume. One would chose a typical position in the cluster ⃗x ref (e.g. its centre) and
define n e0 = n e (⃗x ref ) and B ⃗ 0 = 〈 B ⃗ 2 (⃗x ref )〉 1/2 . The window function can then be
expressed as
f(⃗x) = 1 {⃗x⊥ ∈Ω} 1 {z≥zs(⃗x ⊥ )} h(⃗x ⊥ ) g(⃗x) n e (⃗x)/n e0 , (3.5)
where 1 {condition} is equal to one if the condition is true and zero if not. The dimensionless
average magnetic field profile g(⃗x) = 〈 ⃗ B 2 (⃗x)〉 1/2 /B 0 is assumed to scale
with the density profile such that g(⃗x) = (n e (⃗x)/n e0 ) α B
. Reasonable values for the
exponent α B range between 0.5 and 1 (see Sect. 1.3.2).