Alma Mater Studiorum Universit`a degli Studi di Bologna ... - Inaf

3.7. The magnetic field power spectrum 39
5. The amplitude of ŵ( f ) is spatially variable, but is a function only of the thermal electron
density.
These assumptions guarantee that the spatial distribution of the magnetic field can be described
entirely by its power spectrum ŵ( f ) and that for a medium of constant depth and density, the power
spectra of magnetic field and RM are proportional (Enßlin & Vogt 2003).
If the RM fluctuations are isotropic, the RM power spectrum Ĉ( f ⊥ ) is the Hankel transform
of the auto-correlation function C(r ⊥ ), defined as
C(r ⊥ )=〈RM(r ⊥ + r ′ ⊥)RM(r ′ ⊥)〉, (3.12)
where r ⊥ and r ′ ⊥ are vectors in the plane of the sky and〈〉 is an average over r′ ⊥ . One could
think of obtaining the magnetic field power spectrum directly by Fourier transforming Eq. 3.12.
In reality, the observations are affected first by the effects of convolution with the beam, which
modify the spatial statistics of RM, and secondly, by the limited size and irregular shape of the
sampling region (the region of the source over which the RM has been derived), which results in
a complicated window function (Enßlin & Vogt 2003) for computational work in the frequency
space. Finally, most of the useful properties of the auto-correlation function are related to the
outer scale at C(r ⊥ ) approaches the zero-level, which in most cases is uncertain in the presence of
large-scale fluctuations.
The alternative strategy proposed by Laing et al. (2008), and applied in the work of this
thesis, first estimates the power spectrum of the RM, Ĉ(f ⊥ ), where Ĉ(f ⊥ )d f x d f y is the power in
the area d f x d f y , and derives that of the three-dimensional magnetic-field ŵ(f). Laing et al. (2008)
demonstrated a procedure that takes into account the convolution effects and minimises the effects
of uncertainties in the zero-level. In particular, they showed that
1. in the short-wavelength limit (meaning that changes in Faraday rotation across the beam
are adequately represented as a linear gradient), the measured RM distribution is closely
approximated by the convolution of the true RM distribution with the observing beam
2. the structure function is a powerful and reliable statistical tool to quantify the two
dimensional fluctuations of RM, given that it is independent of the zero-level and structure
on scales larger than the area under investigation.
The structure function is defined by
S (r ⊥ )= (3.13)
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