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3.7. The magnetic field power spectrum 37
Figure 3.6: Comparison of theσ RM profiles obtained from simulations of a Gaussian and random
magnetic field (solid line) and the expectations from the analytical model of Eq. 3.9 by assuming
three differentΛ c :Λ min =6 kpc,Λ B =Λ Bz = 16 kpc,Λ RM =69 kpc (dashed and dotted lines). The
best agreement is given byΛ c =Λ B . The simulated magnetic field power spectrum has slope n=2.
The source is assumed to be halfway through the cluster and the magnetic field strength and gas
density have the same radial profiles in both the simulations and the analytical formulation. Taken
from Murgia et al. (2004).
be identical and the approximationΛ c =Λ RM has been used in the single-scale model (Eq. 3.9),
leading to underestimates of the magnetic field strength. Murgia et al. (2004) have shown thatΛ B
is a good approximation forΛ c in Eq. 3.9 and therefore that this length must be used to compare
the analytical predictions from the single-scale model with RM analyses assuming a multi-scale
field (Fig. 3.6). SinceΛ B depends on the underlying magnetic field power spectrum (Eq. 3.10), the
latter must be estimated first.
Multiple different estimators of spatial statistics of the RM distributions such as structure and
auto-correlation functions or a multi-scale statistic have been used to derive the field strength, its
relation to the gas density and its power spectrum (Murgia et al. 2004; Govoni et al. 2006; Guidetti
et al. 2008; Laing et al. 2008). The technique of Bayesian maximum likelihood has also been used
for this purpose (Enßlin & Vogt 2005; Kuchar & Enßlin 2009). These studies have shown that the
magnetic field power spectrum is well approximated by a power-law
ŵ( f )∝ f −q (3.11)
over a range of spatial frequencies, corresponding respectively to the outer and inner fluctuation
scales of the magnetic field 2 . Although the theory of intergalactic magnetic field fluctuations is
still debated, a power-law functional form is expected if the turbulence is mostly hydrodynamic,
2 Spatial frequency and scale are related in the sense: f= 1/Λ
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