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

3.3. THE METHOD 49
space counterparts as:
ˆM N (k) =
∫
∫ ∞
d 3 rM N (r) e i⃗k·⃗r = 4π dr r 2 sin(k r)
M N (r)
0
k r
∫
d 3 rM H (r) e i⃗k·⃗r ,
Ĥ(k) = d dk ˆM H (k) = d dk
= 4π k
∫ ∞
0
dr r 2 M H (r)
(3.23)
k r cos(k r) − sin(k r)
. (3.24)
k r
One can also introduce the Fourier transformed trace of the autocorrelation tensor
ŵ( ⃗ k) = ˆM ii ( ⃗ k) = 2 ˆM N (k). A comparison with the transformed zz-component of
the autocorrelation tensor
ˆM zz ( ⃗ k) = ˆM
(
N (k) 1 − kz/k 2 2) (3.25)
reveals that in the k z = 0 plane, these two functions are identical (up to a constant
factor 2). Since the 2-d Fourier transformed normalised RM map is also identical to
this, as a transformation of Eq. (3.8) shows, one can state
Ĉ ⊥ ( ⃗ k ⊥ ) = ˆM zz ( ⃗ k ⊥ , 0) = 1 2 ŵ(⃗ k ⊥ , 0). (3.26)
This Fourier-space version of Eq. (3.14) says, that the 2-d transformed RM map reveals
the k z = 0 plane of ˆM zz ( ⃗ k), which in the isotropic case is all what is required to
reconstruct the full magnetic autocorrelation ŵ(k) = 2 Ĉ⊥(k).
A power spectrum P [F ] ( ⃗ k) of a function F (⃗x) is given by the absolute-square
of its Fourier transformation P [F ] ( ⃗ k) = | ˆF ( ⃗ k)| 2 . The WKT states that the Fourier
transformation of an autocorrelation function C [F ] (⃗r), estimated within a window with
volume V n (as in Eq. (3.4)), gives the (windowed) power spectrum of this function, and
vice versa:
P [F ] ( ⃗ k) = V n Ĉ [F ] ( ⃗ k) . (3.27)
The WKT allows to write the Fourier transformed autocorrelation tensor as
ˆM ij ( ⃗ k) = 1 V 〈 ˆB i ( ⃗ k) ˆB j ( ⃗ k)〉 , (3.28)
where V denotes the volume of the window function, which is for practical work with
RM maps often the probed effective volume V = V [f] as defined in Sect. 3.3.1.
Thus, the 3-d magnetic power spectrum (the Fourier transformed magnetic autocorrelation
function w(⃗r)) can be directly connected to the one-dimensional magnetic
energy spectrum in the case of isotropic turbulence:
ε B (k) dk = 4 π k2
(2π) 3
ŵ(k)
8 π dk = k2 ŵ(k)
dk , (3.29)
2 (2π)
3
where ŵ( ⃗ k) = ŵ(k) is due to isotropy. The WKT also connects the 2-dimensional
Fourier transformed RM map with the Fourier transformed RM autocorrelation function:
Ĉ ⊥ (k ⊥ ) = 〈| RM(k ˆ ⊥ )| 2 〉
. (3.30)
a 1 A Ω