2. Marco teórico
2.8 Secuencia didáctica para la producción de textos escritos
The conclusion of Rudnick & Blundell (2003) is statistically questionable since the human eye will detect in any data set apparent correlations. Furthermore,ϕ0andRM
carry correlated noise as noted by Rudnick & Blundell since these two quantities are generated from the same set of polarisation angle distributions. In the worst case, this correlation will produce step-function like artefacts in the distributions at the same location. These artefacts are present in the original data.
The analysis performed by Rudnick & Blundell is designed to have the sameRM power spectrum as the observation but higher order correlations are neglected by the random phase realisation. Unfortunately, the statistic used by Rudnick & Blundell is very sensitive to higher order correlations. Any clustering in theRM-ϕ0 scatter plot
is a result of patches of nearly constant values in theRM andϕ0 images. A special
relation in the Fourier phases (or in higher order correlations) is required in order to recover the appearance of patches in theRM andϕ0distributions.
The strength of the clustering in the scatter plot is indeed strongest if theϕ0 and RM features are correlated. However, the clustering does not disappear if the RM and ϕ0 patches have independent distributions, since everyϕ0 patch is still overlaid
by a small number ofRM patches, so that the associated clustering in the scatter plot only gets split into a corresponding number of smaller clusters. These clusters happen to be co-aligned on a vertical constantϕ0 line in the plot, since they all belong to the
same ϕ0 coherence patch. A corresponding mechanism splits the pixels of an RM
cell into a horizontal constantRM line. Such vertical and horizontal features can be indeed seen in the scatter plot of PKS 1246-410 (upper middle panel of Fig. 2.2).
In order to perform an experiment which maintains also higher order statistics, one can simply exchange subregions of theRM image from one lobe to the other. This should keep the sameRM correlation functions but it will destroy any real correlation betweenϕ0 and RM, since the different regions of the source are independent. This
simple experiment was done by diving the source in roughly two equal regions about the centre and shifting the coordinates in right ascension such that two subregions overlap. Theϕ0 distribution of the east (west) lobe is plotted against theRM data of
the west (east) lobe. The shifting of lobes is preferable to a reflection about the centre eliminating possible radial influences on any correlation. The result of this experiment is shown in the upper right panel of Fig. 2.2. The clustering of points does not vanish even though the distributions should be independent.
By going a step further, one can generate patchy RM andϕ0 maps consisting of
patches with nearly constant values but which are statistically independent from each other. For both maps similar recipes were used. First a number N of random seed pointsX~i (i ∈ {1...N}) within a square area is drawn, and then the area is split into cells around the seed points by means of a Voronoi-tessellation: Each point~xof the area belongs to the cell of its nearest seed X~i. Then each seed is attributed a random valueψi(ψstands in the following for bothRM andϕ0) and a small two-dimensional
random vector∇~ψi(= auxiliaryRM orϕ0gradients within the patches, only used for the map construction). Each pixel~xwithin the cell of seedigets a value
ψ(~x) =ψi+∇~ψi·(~x−X~i) +σ(~x), (2.1)
-1500 -1000 -500 0 500 1000 1500 -80 -60 -40 -20 0 20 40 60 80 RM PA0 -1500 -1000 -500 0 500 1000 1500 -80 -60 -40 -20 0 20 40 60 80 RM PA0 RM Kilo rad/m/m -150 -100 -50 0 50 100 150 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 RM Kilo rad/m/m PA Degrees -150 -100 -50 0 50 100 150 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 RM Kilo rad/m/m PA0 Degrees -150 -100 -50 0 50 100 150 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -150 -100 -50 0 50 100 150 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 RM Kilo rad/m/m PA0 Degrees -150 -100 -50 0 50 100 150 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -150 -100 -50 0 50 100 150 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5
Figure 2.2: This plot is adapted from Enßlin et al. (2003). Upper left panel: Uncor- rectedϕ-RM scatter plot of PKS 1246-410. The underlying maps were smoothed by a median weight filter. Upper middle panel: ϕ0-RM scatter plot of PKS 1246-410.
Upper right panel: ϕ0-RM scatter plot of PKS 1246-410, but with theRM maps of
the Eastern and Western radio lobe exchanged as an independent experiment. Lower left panel: Simulated ϕ0 (white lines) and RM (grey scale) maps with independent
coherence patches. Lower middle panel: ϕ0-RM scatter plot of the simulated inde-
pendentϕ0and RM maps shown on the left. Lower right panel: ϕ0-RM scatter plot
of simulated maps with co-alignedϕ0 andRM coherence patches.
nearly constant values, but which exhibit some internal trends and noise.
Theϕ0andRM maps were slightly smoothed, a20%border region was cut away
in order to suppress edge effects, and a ϕ0-RM scatter plot is generated. A typical
realisation of such a map and its scatter plot is shown in the lower left and lower middle panel of Fig. 2.2, respectively. A strongly clustered distribution is visible even though the individual ϕ0 and RM maps were completely independent. Furthermore, nearly
horizontal and vertical chains of clusters are visible for the reasons given above. The map smoothing produces bridges between these clusters, since it gives intermediate values to pixels which are at the boundaries ofϕ0andRM cells.
The deviations from the strict horizontal and vertical directions visible in the scat- ter plot of PKS 1246-410 should be caused by trends within the coherence cells. Note that such structures are also visible in the simulated scatter plots of Rudnick & Blun- dell, although the smooth realisations of theirRM maps have smeared them out (see their Fig. 2 for a comparison of the patchiness of observed and simulatedRM maps). Therefore, one can conclude that the data of PKS 1246-410 favours statistically independentϕ0 andRM maps. The occurrence of vertical and horizontal lines in the
indeed misaligned.
In order to further investigate this, a model is constructed in which theϕ0andRM
patch positions are absolutely identical. This is constructed by moving the Voronoi- tessellation seed points of theRM map to the locations of nearby seed points of theϕ0
map, thereby assuring that there is a one-to-one mapping. All other variables (RMi,
ϕ0ietc) were kept as before. The recomputedRMmap has therefore exactly the same
patch locations as the ϕ0 map. The horizontal and vertical cluster alignments and
stripes are absent there (see the lower right panel of Fig. 2.2). There are now several stripes with diagonal orientations due to pixels at theϕ0-RM cell boundaries which
received simultaneously intermediate values inϕ0andRM by the smoothing.