Esclavitud, Trabajo y tolerancia en la provincia de Cartago

5.4.1 The Data∆

The maximum likelihood estimator which was introduced and tested in the last sections is applied in the following to the Faraday rotation map of the north lobe of the radio source Hydra A (Taylor & Perley 1993). The data were kindly provided by Greg Taylor.

For this purpose, a high fidelityRM map was used which was presented in Chapter 4 and was generated by the developed algorithmPacmanusing the original polarisation data. Pacmanalso provides error maps σi by error propagation of the instrumental

uncertainties of polarisation angles. ThePacmanmap which was used is shown in the right panel of Fig. 5.3.

For the same reasons as mentioned in Sect. 5.3, the data were averaged. An appro- priate averaging procedure using error weighting was applied such that

RMi = P jRMj/σj2 P j1/σ2j , (5.32)

and the error calculates as

σ2_RM i = P j 1/σ_j2 P j1/σ2j 2 = 1 P j1/σj2 . (5.33)

Here, the sum goes over the set of old pixels {j_} which form the new pixels _{i_}. The corresponding pixel coordinates {i_} were also determined by applying an error weighting scheme xi = P jxj/σj2 P j1/σj2 and y_i = P jyj/σj2 P j1/σ2j . (5.34)

−4000 −2000 0 2000 4000 RM rad/m2 −4000 −2000 0 2000 4000 RM rad/m2 138.922 138.924 138.926 138.928 −11.884 −11.882 −11.88 −11.878 −11.876 RA−−−SIN DEC−−SIN

Figure 5.3: The final RM map from the north lobe of Hydra A which was anal- ysed with the maximum likelihood estimator; right: originalPacmanmap, left: error weighted map. Note that the small scale noise for the diffuse part of the lobe is av- eraged out and only the large scale information carried by this region is maintained. Furthermore, note that each pixel has also a (not displayed) error weighted position.

The analysed RM map was determined by a gridding procedure. The original RM map was divided into four equally sized cells. In each of these the original data were averaged as described above. Then the cell with the smallest error was chosen and again divided into four equally sized cells and the original data contained in the so determined cell were averaged. The last step was repeated until the number of cells reached a defined valueN. It was decided to useN = 1500. This is partly due to the

limitation by computational power but also partly because of the desired suppression of small scale noise by a strong averaging of the noisy regions.

The final RM map which was analysed is shown in the left panel of Fig. 5.3. The most noisy regions in Hydra A are located in the coarsely resolved northernmost part of the lobe. It was chosen not to resolve this region any further but to keep the large-scale information which is carried by this region.

5.4.2 The Window Function

As mentioned in Sect. 5.2.1, the window function describes the sampling volume and, thus, one has to find a suitable description for it based on Eq. (5.6). Hydra A (or 3C218) is located at a redshift of 0.0538 (de Vaucouleurs et al. 1991). For the derivation of the electron density profile parameter, the work by Mohr et al. (1999) was used which was done for ROSAT PSPC data while using the deprojection of X-ray surface brightness profiles as described in the Appendix A of Pfrommer & Enßlin (2004). Since Hydra A is known to exhibit a strong cooling flow as observed in the X-ray studies, a doubleβ-

10000 100000 1e+06 1e+07 1e+08 0 5 10 15 20 25 30 35 40 <RM 2> - <RM> 2 and f 2(r)* 7e17 r [kpc] αB = 1.0; θ = 60o αB = 1.0; θ = 45o αB = 1.0; θ = 30o αB_α = 1.0; θ = 10o B = 1.0; θ = 0o 10000 100000 1e+06 1e+07 1e+08 0 5 10 15 20 25 30 35 40 <RM 2> - <RM> 2 and f 2(r)* 4e17 r [kpc] αB = 0.5; θ = 60o αB = 0.5; θ = 45o αB = 0.5; θ = 30o αB_α = 0.5; θ = 10o B = 0.5; θ = 0o 10000 100000 1e+06 1e+07 1e+08 0 5 10 15 20 25 30 35 40 <RM 2> - <RM> 2 and f 2(r)* 3e17 r [kpc] αB = 0.1; θ = 60o αB = 0.1; θ = 45o αB = 0.1; θ = 30o αB = 0.1; θ = 10o αB = 0.1; θ = 0o (a) (c) (b)

Figure 5.4: The comparison between the integrated squared window function f2(r)

(lines) with the RM dispersion function _hRM2(r)_i (open circles) and _hRM2_{i −} hRM(r)_i2 _{(filled circles). Different models for the window function were assumed.}

In (a)αB= 1.0, in (b)αB= 0.5and in (c)αB= 0.1were used, where the inclination

angleθof the source was varied. It can be seen that models for the window function withαB = 0.1. . .0.5andθ= 10◦. . .50◦ match the shape of the dispersion function

very well.

profile was assumed1_{and for the inner profile n}

e1(0) = 0.056cm−3 andrc1 = 0.53

arcmin was used while for the outer profile ne2(0) = 0.0063 cm−3 and rc2 = 2.7

arcmin and aβ= 0.77was applied.

Assuming this electron density profile to be accurately determined, there are two other parameters which enter in the window function. The first one is related to the source geometry. For Hydra A, a clear depolarisation asymmetry between the two lobes is observed known as the Laing-Garrington effect (Garrington et al. 1988; Laing 1988) suggesting that the source is tilted against thexy-plane (Taylor & Perley 1993). In fact, the north lobe points towards the observer. In order to take this into account, an angleθwas introduced which describes the angle between the source and thexy-plane such that the north lobe points towards the observer. Taylor & Perley (1993) determine an inclination angle ofθ= 45◦.

The other parameter is related to the global magnetic field distribution which is assumed to scale with the electron density profile B(r) _∝ ne(r)αB. In a scenario

in which an originally statistically homogeneously magnetic energy density gets adi- abatically compressed, one expects αB = 2/3. If the ratio of magnetic and thermal

1_{defined asn} e(r) = [n 2 e1(0)(1 + (r/rc1) 2 )−3β_+n2 e2(0)(1 + (r/rc2) 2 )−3β_]1/2 .

pressure is constant throughout the cluster thenαB = 0.5. However,αB might have

any other value. Dolag et al. (2001) determined anαB = 0.9for the outer regions of

the cluster Abell 119.

In order to constrain the applicable ranges of these quantities, one can compare the integrated squared window function with theRM dispersion function _hRM(r⊥)2iof

theRM map used since

hRM2(r⊥)i ∝

Z ∞

−∞dz f

2_(r

⊥, z), (5.35)

as stated by Eq. (3.39). Therefore, the shape of the two functions was compared. The result is shown in Fig. 5.4. For the window function, three differentαB = 0.1,0.5,1.0

were used and for each of these, five different inclination anglesθ= 0◦,10◦,30◦,45◦

and 60◦ _{were employed, although the θ = 0}◦ _{is not very likely considering the ob-}

servational evidence of the Laing-Garrington effect as observed in Hydra A by Tay- lor & Perley (1993). The different results are plotted as lines of different style in Fig. 5.4. The filled and open dots represent theRM dispersion function. The solid circles indicate the binned hRM2_i function. The open circles represent the binned hRM2_{i − h}RM_i2 function, which is cleaned from any foregroundRM signals.

From Fig. 5.4, it can be seen that models withαB = 1.0or θ > 50◦ are not able

to recover the shape of theRM dispersion function and, thus, one expectsαB <1.0

andθ <50◦ _{to be more likely.}