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

34 CHAPTER 2. IS THE RM SOURCE-INTRINSIC OR NOT?
This alignment product has the following properties
〈⃗p, α⃗p〉 = |α|p 2 , (2.4)
〈⃗p, ⃗q〉 = −p q, if ⃗p ⊥ ⃗q (2.5)
where α is a real (positive or negative) number. An isotropic average of the alignment
product of two 2-dimensional vectors leads to a zero signal, since the positive (aligned)
and negative (orthogonal) contributions cancel each other out.
Then the alignment statistics of the vector fields ⃗p(⃗x) = ⃗ ∇ RM(⃗x) and ⃗q =
⃗∇ ϕ 0 (⃗x) can be defined as
A = A[⃗p, ⃗q] =
∫ d 2 x 〈⃗p(⃗x), ⃗q(⃗x)〉
∫ d 2 x |⃗p(⃗x)| |⃗q(⃗x)| , (2.6)
which fulfils the required properties mentioned above:
No. 1: The statistic does not depend on any global relation between ϕ 0 and RM.
This can be demonstrated by changing a potential functional dependence using nonlinear
data transformations. Any non-pathological 1 , piecewise continuous and piecewise
monotonic pair of transformations RM ∗ = S(RM) and ϕ ∗ 0 = T (ϕ 0) do not
destroy the alignment signal due to the identity
〈 ⃗ ∇RM ∗ , ⃗ ∇ϕ 0 ∗ 〉 = ∣∣ S ′ (RM) T ′ (ϕ 0 ) ∣∣ 〈 ⃗ ∇RM, ⃗ ∇ϕ 0 〉. (2.7)
Inserted into Eq. (2.6), one finds that the weights of the different contributions to the
alignment signal might be changed by the transformation, but except for pathological
cases any existing alignment signal survives the transformation and no spurious signal
is produced in the case of uncorrelated maps.
No. 2 & 3: A simple calculation shows that the expectation value for A is zero for
independent maps and it is unity for aligned maps. For illustration, the simulated pair
of independent maps has A = −0.03, which can be regarded as a test of the statistic
with a case where the statistic of Rudnick & Blundell incorrectly detects co-alignment.
The simulated pair of co-aligned maps has A = 0.89 illustrating A’s ability to detect
correlations. Property No. 3 implies requirement No. 2.
No. 4: From the discussion so far, it should be obvious that many essential properties
of the alignment statistics can be derived analytically. However, as an additional
useful example the effect of a small amount of noise present in both maps can be estimated.
Each noise component is assumed to be uncorrelated with RM, to ϕ 0 , and
also to the other noise component. For small noise levels, it can be found that
A[⃗p + δ⃗p, ⃗q + δ⃗q] ≈
A[⃗p, ⃗q]
(1 + δp 2 /p 2 ) (1 + δq 2 /q 2 ) , (2.8)
where the bar denotes the statistical average. This relation holds only approximately,
since the non-linearity of A prevents exact estimates without specifying the full probability
distribution of the fluctuations. As can be seen from Eq. (2.8), uncorrelated
noise reduces the alignment signal, but does not produce a spurious alignment signal,
in contrast to correlated noise, which usually does.
1 A pathological transformation would e.g. split the RM or ϕ 0 value range into tiny intervals, and
randomly exchange them or map them all onto the same interval.