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

3.3. THE METHOD 47
(e.g. Subramanian 1999) where the longitudinal, normal, and helical autocorrelation
functions, M L (r), M N (r), and M H (r), respectively, only depend on the distance, not
on the direction. The condition ∇ ⃗ · ⃗B = 0 leads to ∂/∂r i M ij (⃗r) = 0 (here and
below the sum convention is used). This allows to connect the non-helical correlation
functions by
d
dr (r2 M L (r)) (3.10)
M N (r) = 1 2r
(Subramanian 1999). The zz-component of the magnetic autocorrelation tensor depends
only on the longitudinal and normal correlations, and not on the helical part:
M zz (⃗r) = M L (r) r2 z
r 2 + M N(r) r2 ⊥
r 2 with ⃗r = (⃗r ⊥, r z ) , (3.11)
which implies that Faraday rotation is insensitive to magnetic helicity. It is also useful
to introduce the magnetic autocorrelation function
w(⃗r) = 〈 ⃗ B(⃗x) · ⃗B(⃗x + ⃗r)〉 = M ii (⃗r) , (3.12)
which is the trace of the autocorrelation tensor, and depends only on r (in the case
of a statistically isotropic magnetic field distribution, in the following called briefly
isotropic turbulence):
w(⃗r) = w(r) = 2M N (r) + M L (r) = 1 r 2 d
dr (r3 M L (r)) . (3.13)
In the last step, Eq. (3.10) was used. Since the average magnetic energy density is given
by 〈ε B 〉 = w(0)/(8π) the magnetic field strength can be determined by measuring the
zero-point of w(r). This can be done by Faraday rotation measurements: The RM
autocorrelation can be written as
C ⊥ (r ⊥ ) = 1 2
∫ ∞
−∞
√
dr z w( r⊥ 2 + r2 z) =
∫ ∞
r ⊥
dr
r w(r)
√ . (3.14)
r 2 − r⊥
2
and is therefore just a line-of-sight projection of the magnetic autocorrelations. Thus,
the magnetic autocorrelations w(r) can be derived from C ⊥ (r ⊥ ) by inverting an Abel
integral equation:
w(r) = − 2
π r
= − 2 π
d
dr
∫ ∞
r
∫ ∞
r
dy
dy y C ⊥(y)
√
y 2 − r 2 (3.15)
C′ ⊥ (y) √
y 2 − r 2 , (3.16)
where the prime denotes a derivative. For the second equation, it was used that w(r)
stays bounded for r → ∞.
Now, an observational program to measure magnetic fields is obvious: From a
high quality Faraday rotation map of a homogeneous, (hopefully) isotropic medium of
known geometry and electron density (e.g. derived from X-ray maps) the RM autocorrelation
has to be calculated (Eq. (3.8)). From this an Abel integration (Eq. (3.15) or
(3.16)) leads to the magnetic autocorrelation function, which gives 〈B 2 〉 at its origin.
Formally,
〈B 2 〉 = w(0) = − 2 ∫ ∞
dy C′ ⊥ (y) , (3.17)
π y
0