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Are the Faraday Rotating Magnetic Fields Local to Intracluster Radio Galaxies

Are the Faraday Rotating Magnetic Fields Local to Intracluster Radio Galaxies
Are the Faraday Rotating Magnetic Fields Local to Intracluster Radio Galaxies

a r X i v :a s t r o -p h /0301552v 4 16 J u l 2003

Draft version February 2,2008

Preprint typeset using L A T E X style emulateapj v.11/12/01

ARE THE FARADAY ROTATING MAGNETIC FIELDS LOCAL TO INTRACLUSTER RADIO

GALAXIES?

Torsten A.En?lin 1,Corina Vogt 1,T.E.Clarke 2,3,Greg B.Taylor 3

Draft version February 2,2008

ABSTRACT

We investigate the origin of the high Faraday rotation measures (RMs)found for polarized radio galaxies in clusters.The two most likely origins are,magnetic ?elds local to the source,or magnetic ?elds located in the foreground intra-cluster medium (ICM).The latter is identi?ed as the null hypothesis.Rudnick &Blundell (2003)have recently suggested that the presence of magnetic ?elds local to the source may be revealed in correlations of the position angles (PAs)of the source intrinsic linear polarization and the RMs.We investigate the claim of Rudnick &Blundell to have found a relationship between the intrinsic PA 0of the radio source PKS 1246-410and its RM,by testing the clustering strength of the PA 0-RM scatter plot.We show that the claimed relationship is an artifact of an improperly performed null-experiment.We describe a gradient alignment statistic aimed at ?nding correlations between PA 0and RM maps.This statistic does not require any null-experiment since it gives a unique (zero)result in the case of uncorrelated maps.We apply it to a number of extended radio sources in galaxy clusters (PKS 1246-410,Cygnus A,Hydra A,and 3C465).In no case is a signi?cant large-scale alignment of PA 0and RM maps detected.We ?nd signi?cant small-scale co-alignment in all cases,but we are able to fully identify this with map making artifacts through a suitable statistical test.We conclude that there is presently no existing evidence for Faraday rotation local to radio lobes.Given the existing independent pieces of evidence,we favor the null hypothesis that the observed Faraday screens are produced by intracluster magnetic ?elds.

Subject headings:galaxies:clusters:general —magnetic ?elds —polarization —radio continuum:

galaxies

1.introduction

A detailed summary of the observational evidence for the presence of magnetic ?elds embedded within the ther-mal gas in clusters of galaxies is presented in two recent review papers (Carilli &Taylor 2002,and Widrow 2002)and is only brie?y summarized here.

Magnetic ?elds in galaxy clusters are known to exist due to the detection of di?use cluster wide synchrotron emis-sion (Willson 1970)in a number of clusters.This emission is not associated with individual galaxies in the cluster and is observationally classi?ed as halo or relic emission (Feretti 1999).Although initially thought to be relatively rare objects,recent surveys have signi?cantly increased the number of radio halos and relics (Giovannini,Tordi,&Fer-etti 1999;Giovannini &Feretti 2000;Kempner &Sarazin 2001;Bacchi et al.2003).The presence of this di?use synchrotron emission reveals the widespread distribution of magnetic ?elds within the intracluster medium (ICM)in these clusters.Equipartition assumptions provide min-imum energy magnetic ?eld estimates in radio halos of 0.1-1μG (Feretti 1999;Bacchi et al.2003)and 0.4-2.7μG in radio relics (En?lin et al.1998).

Estimates of volume averaged intracluster magnetic ?eld strengths can be obtained by comparing the synchrotron and inverse Compton emission (e.g.,Harris &Grindlay 1979,Rephaeli et al.1987).The same relativistic particle population which produces the di?use synchrotron emis-sion is expected to up-scatter the ambient photon ?eld in the ICM to produce inverse Compton X-rays.Typical esti-mates yield a volume-averaged intracluster magnetic ?eld in the range of 0.2-1μG (see e.g.Carilli &Taylor 2002and references therein).We emphasize that inverse Comp-ton based estimates have to be regarded as lower limits.Faraday rotation measure (RM)studies of extended ra-dio sources embedded in clusters probe the magnetic ?eld component integrated along the line-of-sight,weighted by the plasma electron density.Faraday rotation is the ro-tation of the position angle (PA)of the plane of linear polarization of radio waves,which traverse a magnetized plasma.The RM is the proportionality constant of this λ2-dependent e?ect:PA(λ)=PA 0+λ2RM,where PA 0is the source intrinsic PA,only directly observable at short-est wavelength.The RM and PA 0can be obtained by multi-frequency measurements of a polarized radio source.Carilli &Taylor (2002)review the generally large RM values obtained from extended radio sources embedded in clusters,and summarize the evidence that indicates that the observed RM of embedded sources are indeed probing a foreground ICM,with conservative estimates of mag-netic ?eld strengths between a few μG and 10s of μG.Similar magnetic ?eld strengths are determined from sta-tistical RM studies by Kim,Tribble &Kronberg (1991),and Clarke,Kronberg &B¨o hringer (2001).

Finally,the asymmetric depolarization of double radio lobes embedded in galaxy clusters can be understood as resulting from a di?erence in the Faraday depth of the two lobes (Laing 1988;Garrington et al.1988)which strongly supports the association of RMs with ICM magnetic ?elds.

1Max-Planck-Institut f¨u r Astrophysik,Karl-Schwarzschild-Str.1,85740Garching,Germany

2Department of Astronomy,University of Virginia,P.O.Box 3818,Charlottesville,VA 22903-08183

National Radio Astronomy Observatory,Socorro,NM 87801,USA

1

2

We note that such a di?erence in Faraday depth might also be explained by local e?ects near the source,if there is a di?erence in the mixing layers of the magnetized radio plasma with the dense environmental gas between the ra-dio lobe head side and the back-?ow side of an FR II radio galaxy.In contrast,this scenario would have di?culties to explain the observed strong RM and depolarization asym-metry of FR I radio galaxies like Hydra A(Taylor&Perley 1993),which are not believed to have back-?ows.However, this discussion shows that the association of the RM with the intra-cluster medium–in the following regarded as the null hypothesis–is not unambiguous,since it could also be produced in a magnetized plasma skin or mixing layer of the observed radio galaxy(Bicknell et al.1990). For that reason Rudnick&Blundell(2003,hereafter RB) used Faraday rotation measure observations of PKS1246-410by Taylor,Fabian&Allen(2002)in an attempt to estimate the fraction of the measured RM signal which is local to the radio source.If most of the RM signal is local to the radio source then the derived ICM magnetic?eld strength should be signi?cantly lower than if all of the RM originates in the ICM.

In the following we examine RB’s claim for evidence of source-local magnetic?elds.We do this?rst through a general argument(Sect.2)followed by an illustrative numerical example(Sect.3).Then,in order to have an unbiased tool to measure cross correlations of RM and PA maps,we specify the mathematical requirements of a suit-able statistic(Sect.4),and construct a gradient alignment statistic which meets our quality criteria(Sect.5).This statistic is also sensitive to correlated noise resulting from the map making imperfections(Appendix B).We there-fore design a gradient vector product statistic,which is only sensitive to such imperfections,in order to separate spuri-ous signals from any astrophysical signal(Sect.6).Appli-cation of our approach to several datasets(PKS1246-410, Cygnus A,Hydra A,and3C465)does not reveal any evi-dence for source-local magnetic?elds(Sect.7).We con-clude(Sect.8)that the null hypothesis that the RMs are connected to the ICM is not only consistent with,but also strongly favored by present data.The basic assumption for Faraday rotation based ICM magnetic?eld estimates of RM being mostly generated by the ICM magnetic?elds seems therefore to be valid.

2.pa0-rm scatter plots

If the Faraday rotation observed in cluster radio galaxies is produced by a mixing layer between the radio lobe and the ICM gas then it is possible that the magnetic struc-tures within the Faraday screen are somehow related to the magnetic?eld orientation within the radio lobes.In such a case,co-spatial structures in RM and PA0maps are possible since both quantities contain geometrical informa-tion about the source local magnetic?eld geometry:The PA0gives the plane-of-the-sky direction perpendicular to the sky-projected magnetic?eld within the radio lobe,and the RM the line-of-sight component of the magnetic?eld within the Faraday rotating medium.

The idea of RB is to test the above scenario by searching for the expected co-spatial structures in the PA0and RM maps of the radio galaxy PKS1246-410.One di?culty with such an analysis is that no direct correlation between the PA0and RM values can be expected even for co-spatial structures,since there is no generic reference point for PA0s.Therefore RB analyzed the PA0-RM scatter plot generated by plotting for each map pixel the PA0and the RM value in the same diagram.They argue that local co-alignment of PA0and RM values should lead to a strongly clustered point distribution in the scatter plot.Since sta-tistically independent PA0and RM maps may also pro-duce such clusters they perform a null-experiment.They generate synthetic RM maps,which have the same power-spectrum as the observed RM map,but random phases. With these simulated maps they repeat their analysis and ?nd much less clustering in the scatter plot.From this they conclude that there is a strong co-alignment in the PA0-RM maps of PKS1246-410,and therefore most of the RM should be local to the radio source.In order to give further support to their claim,they report a number of re-gions in a larger sample of radio galaxy maps where they see corresponding structures in the RM and PA0maps. This latter?nding is statistically very questionable, since within any two large datasets the human eye of-ten?nds apparent correlations within some sub-regions, even if the two datasets are uncorrelated.Furthermore, observed RM and PA0always carry correlated noise,as recognized by RB and analytically demonstrated in Ap-pendix B,since they are generated from the same set of PA maps.In the worst case the noise produces step-function like artifacts at the same location in RM and PA0maps. We note that correlated steps are present within the afore-mentioned regions of RM and PA0maps and give–at least to our eyes–the dominant contribution to the PA0-RM correlation impression reported by RB.

However,the claim of a statistical detection of a PA0-RM correlation can be rigorously investigated.The null-experiment applied by RB is designed to have the same two-point autocorrelation function as the observational data,but higher order correlations are neglected by the random phase realization.Unfortunately,the chosen statistic is very sensitive to such higher order correlations. Any clustering in the PA0-RM scatter plot is caused by having patches of nearly constant values in the PA0and RM maps.The appearance of patches requires special re-lations of the Fourier phases,or–equivalently–non-trivial higher order correlation functions to be speci?ed.

The strength of the clustering in the scatter plot is in-deed strongest if the PA0and RM structures are corre-lated.However,the clustering does not disappear if the RM and PA0patches have independent distributions,since every PA0patch is still overlaid by a small number of RM 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 ver-tical constant PA0line in the plot,since they all belong to the same PA0coherence patch.A corresponding e?ect splits the pixels of an RM cell into a horizontal line of clusters.Such horizontal and vertical structures are in-deed visible in the scatter plot of PKS1246-410(Fig.B1, also Fig.4of RB).

We note that a proper null-experiment,which would maintain the higher-order correlations,can also be con-structed empirically by exchanging subregions of the ob-served RM image from one radio lobe to the other.This

Are the Faraday Rotating Magnetic Fields Local to Intracluster Radio Galaxies?3

should keep the same RM correlation functions,but will destroy any real PA0-RM alignment since the di?erent regions of the source are independent.We have per-formed this lobe switching by simply dividing the source in roughly two equal regions about the center and shift-ing the coordinates in right ascension such that the two subregions overlap.We have then plotted the PA0of the east(west)lobe against the RM of the west(east)lobe. This source division is preferable to a re?ection about the declination axis through the core as our lobe shifting re-sults in a correlation of central source region PA0with the outer lobe region of RM thus eliminating possible radial in?uences on the correlation.The results of such an exper-iment are also presented in Fig.B1.Clustering of points in the PA0-RM scatter plots are clearly seen in both plots in contrast to RB’s claim that clustering results from a relation between PA0and RM.

While RB noted that higher order correlations in inde-pendent RM and PA0maps can systematically produce spurious signals in their statistic.It seems to follow that this implies that their statistic does not allow any conclu-sions about the existence of PA0-RM co-alignment.They argue that the presentation of a counter example,as we provide in the next section,does not invalidate their ap-proach,since it cannot be demonstrated that the example used is realized in nature.However,we note that RBs approach will not be useful if patchiness is important.Al-ready an inspection by eye of the maps of PKS1246-410 in comparison to RB’s random phase maps shows that na-ture(or the map making software)produces patchy maps which must exhibit clustering in PA0-RM scatter plots, independent of whether PA0-RM alignment is present or not.

In the next section,we provide a patchy map generating algorithm,which illustrates our general argument.More importantly,it allows us to test the gradient alignment statistic,introduced in Sect.5.

3.synthetic patchy maps

For illustration purposes we construct a simple exam-ple,which captures the main e?ects,but is not meant to be exact in all respects.We construct patchy RM and PA0 maps,which are statistically independent from each other. For both maps we use a similar recipe4.

First a number N of random seed points X 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 x of 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 both RM and PA0)and a small two-dimensional random vector ?ψi (=auxiliary RM or PA0gradients within the patches,only used for the map construction).Each pixel x within the cell of seed i gets a value

ψ( x)=ψi+ ?ψi·( x? X i)+σ( x),(1) whereσ( x)is a small random noise term.The resulting map consists of patches with nearly constant values,but which exhibit some internal trends and noise.The adopted parameters are described in Appendix A.

The PA0and RM maps are slightly smoothed,a20% border region is cut away in order to suppress edge e?ects, and a PA0-RM scatter plot is generated.A typical realiza-tion of such a scatter plot is shown in Fig.B1.A strongly clustered distribution is visible even though the individual PA0and RM maps were completely independent.Further-more,nearly horizontal and vertical chains of clusters are visible for the reason given in Sect.2.The map smooth-ing produces bridges between these clusters,since it gives intermediate values to pixels which are at boundaries of PA0or RM cells.

The deviations from the strict horizontal and vertical di-rections visible in the scatter plot of PKS1246-410should be caused by trends within the coherence cells.We note that such structures are also visible in the simulated scat-ter plots of RB,although the smooth realizations of their RM maps has smeared them out(see their Fig.2for a comparison of the patchiness of observed and simulated RM maps).

We therefore conclude that the data of PKS1246-410 favors statistically independent PA0and RM maps.The occurrence of vertical and horizontal lines in the scatter plot of the observational data demonstrates that the PA0 and RM patches are indeed misaligned.

In order to further investigate this,we construct a model in which the PA0and RM patch positions are absolutely identical.We construct this by moving the Voronoi-tessellation seed points of the RM map to the locations of nearby seed points of the PA0map,thereby assuring that there is a one-to-one mapping.All other variables (RM i,PA0,i etc)were kept as before.The recomputed RM map has therefore exactly the same patch locations as the PA0map.The horizontal and vertical cluster align-ments and stripes are absent there(see Fig.B1).There are now several stripes with diagonal orientations due to pixels at the PA0-RM cell boundaries which received si-multaneously intermediate values in PA0and RM by the smoothing.

4.requirements for suitable statistics

In order to quantitatively investigate the location of the Faraday rotation measure screen we must develop a statis-tical basis for distinguishing local and foreground e?ects. Therefore,we have to construct a statistic which allows us to search for local correlations between PA0and RM without requiring a global relation between these quanti-ties.In order to design a proper statistic one should?rst specify mathematical requirements in order to avoid po-tential pitfalls.We require any suitable statistic A to ful?ll the following conditions:

1.The statistic should be sensitive to the presence

of correlated spatial changes of PA0and RM,in-

dependent of the local values of these quantities.

Correlations in PA0and RM maps can not be ex-

pected to follow a functional form like RM( x)=

f(PA0( x))+δf( x).

2.The statistic A should not require a null experi-

ment,which is always problematic since the con-

struction of a synthetic dataset with all the impor-

4The computer code can be requested from TAE(ensslin@mpa-garching.mpg.de).

4

tant statistical properties of the real data repro-

duced is very di?cult.

3.The statistic has to provide unique expectation val-

ues in case of uncorrelated(A=0)and in case of

fully correlated(A=1)maps.

4.The statistic should be analytic and su?ciently sim-

ple,allowing that basic properties can be derived

and understood analytically.

5.The signi?cance of the statistic should increase

monotonically with the map size.

The approach of RB to calculate the autocorrelation of PA0-RM scatter plots ful?lls our requirement No.1,or –more exactly–requirement No.1is inspired by their work.However,their approach requires a null experiment (violating No.2),since it is not clear a-priori what the meaning of the derived correlation in the scatter plot is (violating No.3).Although it is possible to derive analytic equations describing the autocorrelation of a scatter plot of two quantities,RB’s application of a median weight?lter to the RM and PA maps is a strongly non-linear operation which makes it very hard to understand the mathemati-cal properties of their data treatment(violating No.4). Finally,if one imagines the map size growing to in?nity, one easily realizes that the scatter plot will be smoothly ?lled with points,independent if there were local correla-tions between PA0-RM or not.Therefore the signi?cance of the method of RB starts to decrease when the clumps are so densely spaced that they start to merge,and van-ishes in the limit of an in?nitely large map(violating No.

5).Applying a median weight?lter to maps–as RB did–sharpens the clumps in the scatter plot considerably,thus suppresses the clump merging and therefore enhances the autocorrelation signal.

5.gradient alignment statistics

In order to construct a statistic which ful?lls our re-quirements,we introduce the gradient alignment statistic A of di?erent maps by comparing the gradient p= ?RM and q= ?PA0.5The idea is to check for alignment of p and q,indicating correlated changes in PA0and RM,thus ful?lling condition No.1.Since the absolute values of RM and PA0are not of any signi?cance for the question of co-alignment,the comparison should give the same signal for parallel and anti-parallel gradients.Therefore,instead of the scalar product

p· q=p q(cosφp cosφq+sinφp sinφq)=p x q x+p y q y(2) of p=(p x,p y)=p(cosφp,sinφp)and q=(q x,q y)= q(cosφq,sinφq)we construct an alignment product p, q =p q(cos2φp cos2φq+sin2φp sin2φq)

=(p2x?p2y)(q2x?q2y)+4p x p y q x q y

p2x+p2y

d2x| p( x)|| q( x)|,(4)

which has all required properties:

No.1:The statistic does not depend on any global

relation between PA0and RM.This can be demonstrated

by changing a potential functional dependence using non-

linear data transformations.Any non-pathological6,piece-

wise continuous and piecewise monotonic pair of transfor-

mations RM?=S(RM)and PA?0=T(PA0)do not destroy

the alignment signal due to the identity

?RM?, ?PA0? =|S′(RM)T′(PA0)| ?RM, ?PA0 .

(5)

Inserted in Eq.4one?nds that the weights of the di?erent

contributions to the alignment signal might be changed by

the transformation,but except for pathological cases any

existing alignment signal survives the transformation and

no spurious signal is produced in the case of the null hy-

pothesis of uncorrelated maps.

No.2&3:A simple calculation shows that the ex-

pectation value for A is zero for independent maps and it

is unity for aligned maps.For illustration,the simulated

pair of independent maps has A=?0.03,which can be

regarded as a null-experiment testing our statistic with a

case where RB’s statistic incorrectly detects co-alignment.

The simulated pair of co-aligned maps has A=0.89illus-

trating A’s ability to detect correlations.Property No.3

implies requirement No.2.

No.4:From the discussion so far it should be obvious

that many essential properties of the alignment statistics

can be derived analytically.However,as an additional

useful example we estimate the e?ect of a small amount

of noise present in both maps.Each noise component is

assumed to be uncorrelated with RM,to PA0,and also to

the other noise component.We?nd for small noise levels

A[ p+δ p, q+δ q]≈

A[ p, q]

δp2/δq2/

Are the Faraday Rotating Magnetic Fields Local to Intracluster Radio Galaxies?5 No.5:If the RM and PA0maps are enlarged by an

additional region with the same statistical properties the

expectation value of A is unchanged,due to the averaging

property of Eq.4,and?uctuations in A decrease due to

the central limit theorem.

6.detecting map-making artifacts

Before the gradient alignment statistic can be applied

to real datasets,it has to be noted that it is very sensitive

to correlations on small scales,since it is a gradient square

statistic.Observed RM and PA0maps will always have

some correlated?uctuations on small scales,since they

are both derived from the same set of radio maps,so that

any imperfection in the map making process leads to cor-

related?uctuations in both maps.Therefore the strength

of signal contamination by such correlated errors has to be

estimated,and–if possible–minimized before any reliable

statement about possible intrinsic correlations of PA0and

RM can be made.This point can not be overemphasized!

The level of expected correlated noise is investigated in

Appendix B.Fortunately,the noise correlation is of known

functional shape,which is a linear anti-correlation of the

PA0and RM errors.This allows us to detect this noise

via a gradient vector-product statistics.

V=V[ p, q]= d2x p( x)· q( x)

6

map making artifacts (see Appendix B).Thus it is uncer-tain if this dataset has su?ciently high signal-to-noise to make it suitable for a PA 0-RM alignment analysis.

In summary,we do not ?nd any evidence for a signi?cant large-scale co-alignment of PA 0and RM maps of polarized radio sources in galaxy clusters.

8.summary

We investigated if there is evidence for co-aligned struc-tures in RM and PA 0maps of extended radio sources in galaxy clusters as claimed by RB in order to argue for source-local RM generating magnetic ?elds.

First,we have demonstrated that the null-experiment performed by RB was poorly designed for testing the corre-lation between PA 0and RM in PKS 1246-410.The lack of phase coherence in the simulated data resulted in less clus-tering in the simulated PA 0-RM scatter plots compared to the observational scatter https://www.wendangku.net/doc/d32325050.html,ing independent,patchy distributions of PA 0and RM we show that the correlations due to the mutual overlap of the RM and PA 0patches pro-duce horizontal and vertical chains of clusters as seen in the PKS 1246-410data,whereas co-aligned PA 0and RM patches produce diagonal stripes.The observed cluster-ing therefore favors independent PA 0and RM maps as expected from foreground intracluster magnetic ?elds.Second,we introduce and apply a novel gradient align-ment statistic A .This statistic reveals PA 0and RM cor-relations regardless of whether they are source intrinsic or due to artifacts in the observation or RM map making pro-cess.Applying this statistic to a number of radio galaxies (PKS 1246-410,Cygnus A,Hydra A,and 3C465)does not reveal any signi?cant large-scale co-alignment of PA 0and

RM maps.We ?nd signi?cant small-scale co-alignment in all observed map pairs,but we are able to fully iden-tify these with map making artifacts by another new suit-able statistical test,the gradient vector product statistic V .Thus,we introduced two new tools to analyze data of Faraday rotation studies of extended radio sources.They are powerful in revealing and discriminating observational or map making artifacts (by V ),and source-intrinsic PA 0-RM correlations (by A +V ),both indicators of potential problems for RM based ICM magnetic ?eld estimates.Future,sensitive searches for potential,weak source-intrinsic PA 0-RM correlations with our,or similar statis-tics would require observational datasets with a much higher signal to noise ratio,and a very well de?ned obser-vational (dirty )beam.We add that such datasets would also be crucial for detailed measurements of the magnetic power spectra of the ICM,as proposed by En?lin &Vogt (2003).

We conclude that the observed RM signals of radio galaxies embedded in galaxy clusters seems to be domi-nated by ~1?10μG ICM magnetic ?elds in accordance with independent evidence.

We thank Chris Carilli for the data of Cygnus A and for his suggestion of exchanging areas of the RM map to get uncorrelated maps with conserved higher order statis-tics.Data for 3C465was kindly provided by Jean Eilek and Frazer Owen.We acknowledge lively discussion with Larry Rudnick.The presentation bene?ted strongly from comments of two anonymous referees,from Larry Rudnick and Katherine Blundell,and from Matthias Bartelmann.

APPENDIX

parameters used for the artificial rm and pa 0maps

The parameters to generate the arti?cial RM and PA 0maps are:N =22for both the RM and the PA 0maps.The ?nal

map size is N pxl ×N pxl =50×50pixels.The seed points are distributed by assigning x RM

/PA

i

=([?1.25,1.25],[?1.25,1.25])L,where [a,b ]denotes a random number,uniformly drawn from the interval from a to b ,and L denotes some arbitrary length unit so that the ?nal map has coordinates running from ?L to +L .Further,RM i =[?1600,1600][0,1]1/4rad m ?2, ?RM i =320[0,1]1/2(cos θRM i

,sin θRM i )rad m ?2L ?1,with θRM i =[0,2π],and σRM ( x )=[?48,48]rad m ?2.Similarly,PA i =[?90,90]deg, ?PA i =18[0,1]1/2(cos θPA i

,sin θPA i )deg L ?1,with θPA i =[0,2π],and σPA ( x )=[?2.7,2.7]deg.Terms like [0,1]α

lead to non-uniform distributions between 0and 1,which for 0<α<1are biased towards smaller values.

avoidable and unavoidable correlated noise in rm and pa 0maps

RM and PA 0maps are generated from the same set of radio maps being subject to observational noise.Therefore the noise of the RM and PA 0maps should be expected to be correlated.In order to understand this correlation one has to investigate the map making procedure.

In order to derive the RM maps,one requires a number n of PA maps at di?erent wavelength λi ,which we denote by PA i ( x ).The ?rst crucial step is to solve the so called ‘n-π’ambiguity,which arises from the fact that the PA is only de?ned modulo π,whereas the determination of RM requires absolute PA values,since

PA i ( x )=PA 0( x )+RM( x )λ2i

(B1)allows PA i values which deviate more than πfrom PA 0.Any mistake in solving this ambiguity leads to strong step-function like artifacts in RM,and correlated with these,steps in PA 0maps.The impact of such steps on any sensitive correlation statistics can be disastrous.Fortunately,such n-πambiguities can be strongly suppressed by using map-global algorithms to assign absolute PAs (Dolag et al.,in prep.).

However,even if the n -πambiguity is solved,observational noise δPA i in the individual maps,which we assume to be independent from map to map,leads to correlated noise in RM and PA 0,as we see in the following analysis.Typical RM mapping algorithms use a χ2statistic,giving

RM obs =(λ2λ4?PA ?RM

Are the Faraday Rotating Magnetic Fields Local to Intracluster Radio Galaxies?7 where we used

λ4?λ4 δRM2 ,and δRMδPA0 =?

δRM2 δRM2 =?

λ2/(n(λ22)).(B5) Similarly,one?nds

| ?δRM|2 = | ?δPA|2 /(n(λ22)),and | ?δPA0|2 = | ?δPA|2 λ4?

λ2/ λ4.In the case of non-constant astrophysical RM and PA0values,|V|will be smaller.However,since noise is usually strongest on the smallest scales,the gradient of the noise can easily be stronger than the gradient of the astrophysical signal,leading to V~r.

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Widrow,L.M.2002,Reviews of Modern Physics,74,775 Willson,M.A.G.,1970,MNRAS,151,1

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-1.5Fig.B1.—Upper left panel:Uncorrected PA-RM scatter plot of PKS 1246-410(similar to Figure 4of RB).The underlying maps were smoothed by a median weight ?lter in the same way as RB smoothed the data in order to amplify the clustering.Upper middle panel :Similar to the upper left panel,but now corrected PA 0-RM scatter plot of PKS 1246-410.Note the appearance of diagonal stripes,which are arti?cial correlations due to errors in the RM maps,which imprint themselves onto the PA 0in the Faraday de-rotation step.If they were true correlations,no speci?c direction within the scatter plot should be favored.The decreasing trend from left to right indicates actually an anti-correlation,as predicted in Appendix B as a signature of observational noise which propagates through the map making process.Upper right panel:PA 0-RM scatter plot of PKS 1246-410,but with the RM maps of the Eastern and Western radio lobe exchanged as a null-experiment.The clustering remains despite the fact that PA 0and RM maps are now independent by construction.Lower left panel:Simulated PA 0(white lines)and RM (greyscale)maps with independent coherence patches.Lower middle panel:PA 0-RM scatter plot of the simulated independent PA 0and RM maps shown on the left.Lower right panel:PA 0-RM scatter plot of simulated maps with co-aligned PA 0and RM coherence patches.Note the horizontal and vertical patches in the plot of independent PA 0and RM.Similar structures are clearly visible in the scatter plots of PKS 1246-410but are absent in the scatter plots of co-aligned PA 0and RM.We note that these qualitative impressions about the bridges between clusters in the scatter plot are re?ected in the results of our two gradient statistics introduced in Sects.5and 6:The gradient vector product statistic V is sensitive to an imbalance between the diagonally elongated features (increasing versus decreasing diagonals which correspond to correlated versus anti-correlated changes in RM and PA 0).The gradient alignment statistic A measures if both variables RM and PA 0change simultaneously as one moves in the maps from patch to patch (and therefore in the scatter plot from one cluster to another,as in the lower right panel),or if these changes happen sequentially (e.g.?rst RM changes rapidly,and later PA 0changes rapidly,as in all upper and in the lower middle panels).

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各种开关电源变压器各种高频变压器参数EEEEEEEIEI等等的参数

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PQ型磁芯规格及参数 EE型磁芯规格及参数 EC、EER型磁芯规格及参数

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