On the Importance of Coherence Effects in High Order Statistics: a Comment on “lowering Inferred Cluster Magnetic Field Strengths – the Radio Galaxy Contributions”

Rudnick & Blundell (2003, hereafter RB) claimed to have found a relationship between the intrinsic polarization angle (PA) of the radio source PKS 1246-410 and its Faraday rotation measure (RM) by testing the clustering strength of the PA-RM scatter plot. They argue that this would imply that most of the measured RM is due to the source itself and not due to magnetic fields in the foreground intracluster medium (ICM). This would imply that former ICM magnetic field estimates have to be strongly reduced. However, we show that the claimed relationship is an artifact of an improperly performed Null-experiment. We demonstrate that strong clustering in the PA-RM scatter plot naturally arises even if two completely independent PA and RM maps are compared, with the only requirement that these maps have patchy structures. The Null-experiment of RB was performed with a smooth, non-patchy RM map, whereas the observed RM map clearly shows distinct individual patches. This fundamental difference of the observed and simulated data explains why RB found a stronger clustering signal in the observed data than in their Null-experiment. Further, we argue, that an inspection of the PA-RM scatter plot of the observed data gives strong support for the PA and RM being independent in PKS 1246-410, indicating that the measured RM values are mostly from the foreground ICM magnetic field and are not source-intrinsic. Therefore the claim of RB that RM based ICM magnetic field estimates have to be strongly reduced is premature at best. Subject headings: galaxies: clusters: general — magnetic fields — polarization — radio continuum: galaxies 1. pa-rm scatter plots A summary of the observational evidence for the presence of magnetic fields embedded within the thermal gas in clusters of galaxies is presented in two recent review papers (Carilli & Taylor 2002, andWidrow 2002). Rudnick & Blundell (2003, hereafter RB) use Faraday rotation measure observations of PKS 1246-410 by Taylor, Fabian & Allen (2002) to attempt to address an important question on ICM magnetic fields: What fraction of the measured RM signal of a radio galaxy embedded in the ICM is internal to the radio source or in a local mixing layer with the thermal ICM, as opposed to the foreground intracluster medium? If most of the RM signal is intrinsic to the radio source then the ICM magnetic field strength would be significantly lower than if all of the RM is extrinsic to the source. RB’s approach is to search for co-aligned structures in the polarization angle (PA) and rotation measure (RM) maps of the radio galaxy PKS 1246-410. Such a coalignment would be a strong signature of a source intrinsic RM origin. Since no direct correlation between the PA and RM values can be expected even for co-aligned distributions, RB analyze the PA-RM scatter plot, which is generated by plotting for each map pixel the PA and the RM value in the same diagram. They argue that local coalignment of PA and RM values should lead to a strongly clustered point distribution in the scatter plot. Since also statistically independent PA and RM maps may produce such clusters they perform a Null-experiment. They generate synthetic RM maps, which have the same powerspectrum as the observed RM map, but random phases. With these simulated maps they repeat their analysis and find much less clustering in the scatter plot. From this they conclude that there is a strong co-alignment in the PA-RM maps of PKS 1246-410, and therefore most of the RM should be local to the radio source. If true, this would imply that published ICM magnetic field strengths are overestimates. In the following we explain why RB’s Null-experiment is set up improperly, so that the strong conclusions drawn from it are at least premature. We do this first through a general argument, then by an illustrative numerical example. Finally we give independent reasons why the ICM magnetic field strength should be much higher than RB claim. 2. patches and high-order statistics The statistics of a map can be described by its N-point autocorrelation function, or equivalently by the powerspectrum and relation of the Fourier phases. Fixing the 2-point autocorrelation function is equivalent to specifying the power spectrum. RB designed their Null-experiment to be similar to the observational data on the 2-point level, but higher order correlations or phase coherence are neglected by their random phase realization. Unfortunately, the chosen statistic is very sensitive to such higher order correlations. Any clustering in the PARM scatter plot is caused by having patches of nearly con1 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str.1, 85740 Garching, Germany 2 Department of Astronomy, University of Virginia, P.O. Box 3818, Charlottesville, VA 22903-0818 3 National Radio Astronomy Observatory, Socorro, NM 87801, USA 1