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

4.6. Three-dimensional analysis 65
Figure 4.10: Distribution of the bestfitting
values of B 0 and r m from 35 sets
of simulations, each covering ranges of
0.5 – 10µG in B 0 and 0 – 50 kpc in r m .
The sizes of the circles are proportional
toχ 2 and the blue cross represents the
means of the distribution weighted by
1/χ 2 . The plot shows the degeneracy
between B 0 and r m described in the text.
spectrum amplitude) is a reasonable assumption, inevitably limiting our ability to constrain the
power spectrum on the largest scales.
Now that I have an adequate model for the variation of n e (r)〈B 2 (r)〉 1/2 with radius (Eq. 4.6), I
can correct for it to derive what I call the pseudo-structure function – that is the structure function
for a power-spectrum amplitude, which is constant over the source. This can be compared directly
with the structure functions derived from the Hankel transform of the power spectrum. To evaluate
the pseudo-structure function, I divided the observed 5.5-arcsec RM image by the function
[∫ L
1/2
n e (r) 2 B(r) dl] 2 (4.7)
0
(Enßlin & Vogt 2003), where the radial variations of n e and B are those of the best-fitting model
(Sec. 4.6) and the upper integration limit L has a length of 10 times the core radius. The integral
was normalized to unity at the position of the radio core: this is equivalent to fixing the field and
gas density at their maximum values and holding them constant everywhere in the group. The
normalization of the pseudo-structure function should then be quite close to that of the two central
jet regions.
The pseudo-structure function is shown in Fig. 4.12(a) together with the predictions for the
BPL power spectra withΛ max =205, 65 and 20 kpc. As expected, the normalization of the pseudostructure
function is consistent with that of the jets (Fig. 4.7a and b).
The comparison between the synthetic and observed pseudo-structure functions indicates
firstly that they agree very well at small separations, independent of the value ofΛ max . This
confirms that the best BPL power spectrum found from a combined fit to all six sub-regions
is a very good fit over the entire source. Secondly, despite the poor sampling on very large
scales, the asymptotic values of the predicted structure functions for the three values ofΛ max
are sufficiently different from each other that it is possible to determine an approximate outer
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