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Mon. Not. R. Astron. Soc.000, 1–10 (0000) Printed 26 October 2018 (MN LATEX style file v2.2)

Pacman (II): Application and Statistical Characterisation ofImproved RM Maps

C. Vogt1⋆, K. Dolag2 and T. A. Enßlin11Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str.1, Postfach 1317, 85741 Garching, Germany2Dipartimento di Astronomia, Universita di Padova, vicolo dell’Osservatorio 2, 35122 Padova, Italy

Accepted ???. Received ???; in original form ???

ABSTRACTWe proposed a new method –Pacman– to calculate Faraday rotation measure (RM) mapsfrom multi-frequency polarisation angle data (Dolag et al.) in order to avoid the so-callednπ-ambiguity. Here, we apply ourPacmanalgorithm to two polarisation data sets of extendedradio sources in the Abell 2255 and the Hydra A cluster, and compare the RM maps obtainedusingPacmanto RM maps obtained employing already existing methods. Thereby, we pro-vide a new high quality RM map of the Hydra north lobe which is in a good agreement withthe existing one but find significant differences in the case of the south lobe of Hydra A. Wedemonstrate the reliability and the robustness ofPacman. In order to study the influence ofmap making artefacts, which are imprinted by wrong solutions to thenπ-ambiguities, andof the error treatment of the data, we calculated and compared magnetic field power spectrafrom various RM maps. The power spectra were derived using the method recently proposedby Enßlin & Vogt (2003). We demonstrate the sensitivity of statistical analysis to artefacts andnoise in the RM maps and thus, we demonstrate the importance of an unambiguous determina-tion of RM maps and an understanding of the nature of the noisein the data. We introduce andperform statistical tests to estimate the quality of the derived RM maps, which demonstratethe quality improvements due toPacman.

Key words: Intergalactic medium – Galaxies: cluster: general

1 INTRODUCTION

Magnetic fields are ubiquitous throughout the universe. Onemethod to study them is the analysis of the Faraday rotation ef-fect. This effect describes that if polarised radiation traverses amagnetised plasma the plane of polarisation of the radiation isrotated. The change in polarisation angle is proportional to thesquared wavelengthλ2. The proportionality constant is called rota-tion measure (RM). Various methods have been proposed in orderto calculate RM maps from multi-frequency polarisation data sets(e.g. Vallee & Kronberg 1975; Haves 1975; Ruzmaikin & Sokoloff1979; Sarala & Jain 2001). The difficulty involved in this calcula-tion arises from the fact that observations constrain the polarisationangleϕ only up to additions of±nπ. This leads to the so-callednπ-ambiguity.

In a previous paper (Dolag et al. 2003, hereafter Paper I), weproposed a new approach for the unambiguous determination ofRM and of the intrinsic polarisation angleϕ0 at the observed, po-larised radio source from multi-frequency polarisation data sets.The proposed algorithm uses a global scheme in order to solvethenπ-ambiguity. It assumes that if regions exhibit small polari-

⋆ E-mail: cvogt@mpa-garching.mpg.de (CV); kdolag@pd.astro.it (KD);ensslin@mpa-garching.mpg.de (TAE)

sation angle gradients between neighbouring pixels in all observedfrequencies simultaneously then these pixels can be considered asconnected. Information about one pixel can be used for neighbour-ing ones and especially the solution to thenπ-ambiguity should bethe same.

Faraday rotation measure maps are analysed in order to get in-sight into the properties of the RM producing magnetic fieldssuchas field strengths and correlation lengths. Artefacts in RM mapswhich result fromnπ-ambiguities can lead to misinterpretation ofthe data. Recently, Enßlin & Vogt (2003) proposed a method tocalculate magnetic power spectra from RM maps and to estimatemagnetic field properties. They successfully applied theirmethodto RM maps (Vogt & Enßlin 2003) and realised that map makingartefacts and small scale pixel noise have a tremendous influenceon the shape of the magnetic power spectra on large Fourier scales– small real space scales. Thus, for the application of thesekindof statistical methods it is desirable to produce unambiguous RMmaps with as little noise as possible. This and similar applicationsare our motivations for designingPacman(Paper I) and to quantifyits performance (this Paper).

In order to detect and to estimate the correlated noise levelinRM andϕ0 maps, Enßlin et al. (2003) recently proposed a gradientvector product statistic. It compares RM gradients and intrinsic po-larisation angle gradients and aims to detect correlated fluctuations

c© 0000 RAS

2 C. Vogt, K. Dolag, T. A. Enßlin

on small scales, since RM andϕ0 are both derived from the sameset of polarisation angle maps.

In Paper I, we describe in detail the idea and the implemen-tation of thePacman1 algorithm. Furthermore, we presented a testof the algorithm on artificially generated maps where we demon-strated thatPacmanyields the right solution to thenπ-ambiguity.Here, we want to apply this algorithm to polarisation observa-tion data sets of two extended polarised radio sources located inthe Abell 2255 (Govoni et al. 2002) cluster and the Hydra cluster(Taylor & Perley 1993). This is described in Sect. 3. Using thesepolarisation data, we demonstrate the stability of ourPacmanal-gorithm and compare RM maps obtained from a standard fit al-gorithm. A “standard fit” algorithm is considered to be the algo-rithm which performs a pixel by pixel least squares fit using themethods suggested by Vallee & Kronberg (1975); Haves (1975);Ruzmaikin & Sokoloff (1979).

We demonstrate the importance of the unambiguous determi-nation of RMs for a statistical analysis by applying the statisticalapproach developed by Enßlin & Vogt (2003) to the RM maps inorder to derive the power spectra and strength of the magnetic fieldsin the intra-cluster medium. We also discuss the influence oferrortreatment in the analysis. The philosophy of this statistical analy-sis and the calculation of the power spectra is briefly outlined inSect. 2.3 whereas the results of the application to the RM maps arepresented in Sect. 3.

In addition, we apply thegradient vector product statistic Vas proposed by Enßlin et al. (2003) in order to detect map makingartefacts and correlated noise. The concept of this statistic is brieflyexplained in Sect. 2.1 and in Sect. 3, it is applied to the data. Afterthe discussion of our results, we give our conclusions and lessonslearned from the data during the course of this work in Sect. 4.

Throughout the rest of the paper, we assume a Hubble constantof H0 = 70 km s−1 Mpc−1, Ωm = 0.3 andΩΛ = 0.7 in a flatuniverse. The notation used follows Paper I.

2 STATISTICAL ANALYSIS OF RM MAPS

2.1 Gradient Vector Product Statistic

Enßlin et al. (2003) introduced agradient vector product statistic Vto reveal correlated noise in the data. Observed RM andϕ0 mapswill always have some correlated fluctuations on small scales, sincethey are both calculated from the same set of polarisation anglemaps, leading to correlated fluctuation in both maps. The noisecorrelation betweenϕ0 and RM errors is an anti-correlation of lin-ear shape and can be detected by comparing the gradients of thesequantities. A suitable quantity to detect correlated noisebetweenϕ0 and RM is therefore

V =

∫

d2x∇RM(x) ·∇ϕ0(x)∫

d2x |∇RM(x)||∇ϕ0(x)|, (1)

where∇RM(x) and∇ϕ0(x) are the gradients of RM andϕ0. Amap pair, which was constructed from a set of independent randompolarisation angle mapsϕ(k), will give a V & −1. A map pairwithout any correlated noise will give aV ≈ 0. Hence, the statisticV is suitable to detect especially correlated small scale pixel noise.

The denominator in Eq. (1) is the normalisation and enablesthe comparison ofV for RM maps of different sources, sinceV is

1 The computer code forPacman will be publicly available athttp://dipastro.pd.astro.it/˜cosmo/Pacman

proportional to the fraction of gradients which are artefacts. Since,we especially want to compare the quality between maps of thesame source calculated using the two different algorithms,it is use-ful to introduce the unnormalised quantity

V =

∫

d2x∇RM(x) ·∇ϕ0(x). (2)

The quantityV gives the absolute measure of correlation betweenthe gradient alignments of RM andϕ0. The smaller this value is thesmaller is the total level of correlated noise in the maps.

2.2 Error Under or Over Estimation

One problem, we were faced with during the course of our statisti-cal analysis is the possibility that the measurement errorsare underor overestimated. Such a hypothesis can be tested by performing areducedχ2

ν test which is considered to be a measure for the good-ness of each least squares fit and calculates as follows

χ2νij

=1

ν

f∑

k=1

[

ϕobsij (k)− (RMij λ2k + ϕ0

ij)]2

σ2kij

=s2ij

〈σkij〉k

, (3)

whereν = f−nc is the number of degrees of freedom andf is thenumber of frequencies used andnc is the number of model param-eters in our case,nc = 2. The parameters2 is the variance of thefit and〈σkij

〉k is the weighted average of the individual variances:

〈σkij〉k =

[

1

f

f∑

k=1

1

σ2kij

]

−1

(4)

If one believes in the assumption of Gaussian noise, that thedata are not corrupted and that the linear fit is an appropriate modelthen any statistical deviation from unity of the map averageχ2

ν ofthis value indicates an under or over error estimation. For aχ2

ν > 1,the errors have been underestimated and for aχ2

ν < 1, they havebeen overestimated.

Unfortunately this analysis does not reveal in its simple format which frequency the errors might be under or overestimated.However, one can test for the influence of single frequenciesbyleaving out the appropriate frequency for the calculation of the RMmap and comparing the resultingχ2

νijvalues with the original one.

It is advisable to evaluate theχ2νij

-maps in order to locate re-gions of too high or too low values forχ2

ν by eye.

2.3 Magnetic Power Spectra

OurPacmanalgorithm is especially useful for the determination ofRM maps of extended radio sources. RM maps of extended radiosources are frequently analysed in terms of correlation length andthe RM producing magnetic field strength. One analysis relying onstatistical methods in order to derive magnetic field power spectrawas recently developed by Enßlin & Vogt (2003). They success-fully applied their method to existing RM maps of radio sources lo-cated in Abell 2634, Abell 400 and Hydra A (Vogt & Enßlin 2003).In the course of their work, they realised that their method is sen-sitive to map making artefacts and pixel noise, which can lead tomisinterpretation of the data at hand. Therefore this method is agood opportunity to study the influences of noise and artifacts onthe resulting power spectra.

The observational nature of the data is taken into accountby introducing a window function, which can be interpreted as

c© 0000 RAS, MNRAS000, 1–10

Pacman II: Application 3

the sampling volume. Especially, a noise reducing data weight-ing scheme can be introduced to account for observational noise.One reasonable choice of weighting is to introduce a factorσRM0 /σRM

ij , which we call simple window weighting. Another pos-sible weighting scheme is a thresholding scheme described by1/(1 + σRM

0 /σRMij ), which means that the noise below a certain

thresholdσRM0 is acceptable and areas of higher noise are down-

weighted.Note, that magnetic power spectra, which are calculated using

the approach explained above, are used as valuable estimator formagnetic field strengths and correlation lengths. However,since thespectra are shaped by and are very sensitive to small scale noise andmap making artefacts, we use them here to quantify the influenceof noise and artefacts rather than as an estimator for characteristicproperties of the RM producing magnetic fields.

3 APPLICATION TO DATA

For the calculation of what we call further on the standard fitRMmaps, we used theχ2 method suggested by Vallee & Kronberg(1975) and Haves (1975). Also for the individual RM fits, weadopted this method. We used as an upper limit for the RM values|RM|ij = RMmax ± 3σRM

ij , whereσRMij was calculated follow-

ing Eq.(5) in Paper I. The least-squares fit for the individual datapoints were always performed as error weighted fits as describedby Eq. (3) and (4) in Paper I, if not stated otherwise. For thePac-mancalculations, we used for the free parametersα = β = 1 inEq. (7) in Paper I.

However, we also used the method suggested byRuzmaikin & Sokoloff (1979) as standard fit algorithm. Theresults did not change substantially.

3.1 Abell 2255E

The Abell cluster 2255, which has a redshift of 0.0806(Struble & Rood 1999), has been studied by Burns et al. (1995)andFeretti et al. (1997). The polarised radio source B1713+641, whichwe call hereafter A2255E, has a two sided radio lobe structure andis not directly located in the cluster centre. Polarisationobserva-tions were performed using the Very Large Array (VLA) at 4535,4885, 8085, and 8465 MHz. The data reduction was done with stan-dard AIPS (Astronomical Imaging Processing Software) routines(Govoni et al., in prep.). The polarisation angle maps and their stan-dard error maps for the four frequencies were kindly provided to usby Federica Govoni. A preliminary analysis of an RM map of thissource in order to determine the properties of the magnetic fieldin the intra-cluster gas in Abell 2255 is presented in Govoniet al.(2002).

Using the polarisation angle data, an RM map employing thestandard fit algorithm was calculated, where the maximal allowederror in polarisation angle was chosen to beσmax

k = 25. The re-sulting RM map is shown in the upper left panel of Fig 1. Theoverlaid contour indicates the area which would be covered if po-larisation angle errors were limited byσmax

k = 10. For these cal-culations, we assumedRMmax = 1500 rad m−2.

As can be seen from this map, typical RM values range be-tween -100 rad m−2 and +210 rad m−2. Since this cluster is notknown to inhabit a cooling core in its centre (Feretti et al. 1997),these are expected values. However, the occurrence of RM valuesaround 1000 rad m−2, which can be seen as grey areas in the stan-dard fit map of A2255E, might indicate that for these areas the

nπ-ambiguity was not properly solved. One reason for this sus-picion is found in the rapid change of RM values occurring be-tween 1 or 2 pixel from 100 to 1000 rad m−2. All these jumpshave a∆RM of about 1000 rad m−2 indicatingnπ-ambiguitiesbetween 4 GHz and 8 GHz which can be theoretically calculatedfrom ∆RM = π/(λ2

2 − λ21). Another important point to note is

that parts of the grey areas lie well within theσmaxk = 10 con-

tour, and hence, contribute to the results of any statistical analysisof such a parametrised RM map.

Therefore, the polarisation data of this source provide a goodpossibility to demonstrate the robustness and the reliability of ourproposed algorithm. The RM map calculated byPacmanis shownin the middle left panel. The same numbers for the correspondingparameters were used;σmax

k = 25, RMmax = 1500 rad m−2.An initial comparison by eye reveals that the method used in

the standard fit procedure produces spurious RMs in noisy regions(manifested as grey areas in the upper left panel of Fig. 1) aspre-viously mentioned while the RM map calculated byPacmanshowsno such grey areas. The apparent jumps of about 1000 rad m−2

observed in the standard fit map have disappeared in thePacmanmap. Note, that in principle by realising that these steps are due tonπ-ambiguities, one could re-run the standard fit algorithm with alowerRMmax and this wrong solutions would disappear. However,by doing this a very strong bias is introduced since there might alsobe large RM values which are real. This bias can be easily relaxedby usingPacman.

A pixel-by-pixel comparison is shown in Fig. 2. In this fig-ure theRMstdfit values obtained for pixels using a standard fit areplotted on they-axis against theRMPacman values obtained at thesame corresponding pixel locations using thePacmanalgorithm onthex-axis. The black points represent the error weighted standardfit whereas the red points represent the non-error weighted standardfit. Note thatPacmanapplied only an error weighted least-squaresfit to the data. From this scatter plot, one can clearly see that bothmethods yield same RM values for most of the points which isexpected from the visual comparison of both maps. However, thepoints forRMstdfit values at around± 1000 rad m−2 are due tothe wrong solution of thenπ-ambiguity found by the standard fitalgorithm.

For the demonstration of the effect ofnπ-ambiguity arte-facts and pixel noise on statistical analysis, we determined thecluster magnetic field power spectra by employing the approachbriefly described in Sect. 2.3. For detailed discussion and descrip-tion of the application of the method to data, we refer the readerto Vogt & Enßlin (2003). In the calculation, we assumed that thesource plane is parallel to the observer plane. Please note,that theapplication of an RM data filter in order to remove bad data or datawhich might suffer from a wrong solution to thenπ-ambiguity asdescribed by Vogt & Enßlin (2003) is here not necessary and notdesirable. This is on the one hand due to the use of an error weight-ing scheme in the window function to suppress bad data and on theother hand due to the wish to study the influence of noise and mapmaking artefacts on the power spectra.

The RM area used for the calculation of any power spectrumis the RM area which would be covered by thePacmanalgorithmwhile using the same set of parameterσmax

k , RMmax. This en-sures that pixels on the border or noisy regions of the image notassociated with the source – as seen in the standard fit RM map ofA2255E in the upper left panel of Fig. 1 – are not considered forthe analysis. (The same philosophy also applies for the discussionof the Hydra data in Sect. 3.2 and Sect. 3.3).

The power spectra calculated for various map making sce-

c© 0000 RAS, MNRAS000, 1–10

4 C. Vogt, K. Dolag, T. A. Enßlin

A2255E Hydra (north) Hydra (south)

Faraday Rotation Faraday Rotation Faraday Rotation

Error RMError RM Error RM

Pacman

stdfit

Figure 1. Comparison of the standard fit RM maps (upper panels) and thePacmanRM maps (middle panels) for A2255E (left column), the north (middlecolumn) and the south lobe (right column) of Hydra A. RM values not represented by the colour bar are coloured in grey. Panels at the bottom exhibit therespectiveσRM-maps. For the standard fit RM map of A2255E aσmax

k= 25 andkmin = f = 4 was used, contours indicate the area covered by a standard

fits RM map withσmaxk

= 10. The parameters used for thePacmanRM map of A2255E areσmaxk

= 25, σ∆max = 25, g = 1.2 andkmin = f = 4;

for the standard fit RM maps of Hydra A areσmaxk

= 25 andkmin = f = 4 (at 8 GHz); for thePacmanRM map of Hydra (north) areσmaxk

= 30,kmin = 4, f = 5, σ∆

max = 25 andg = 1.5; for thePacmanRM map of Hydra (south) areσmaxk

= 35, kmin = f = 5, σ∆max = 35 andg = 2.0.

c© 0000 RAS, MNRAS000, 1–10

Pacman II: Application 5

Figure 2. A pixel-by-pixel scatter plot ofRMstdfit values obtained usingthe standard fit on they-axis versus theRMPacman values calculated fromthe polarisation data of Abell 2255 by employing our new algorithm Pac-man. Black points represent results from an error weighted standard least-squares fit and red points from a non-error weighted standardfit. The pointsfor RMstdfit values of around± 1000radm

−2 are artefacts of a wrongsolution of thenπ-ambiguity.

narios are shown in Fig. 3. The solid and dashed line representpower spectra calculated from standard fit RM maps. The solidline was calculated from an RM map which was obtained by usingσmaxk = 20 where no error weighting was applied to the standard

fit. The possible window weighting as introduced in Sect. 2.3wasalso not applied. This scenario is therefore considered as aworstcase. The dashed line was calculated from an RM map employingan error weighted standard fit while only allowing errors in the po-larisation angle ofσmax

k = 10. Again no window weighting wasapplied. From the comparison of these two power spectra alone,one can see that the largek-scales – and thus small real space scales– are sensitive to pixel-noise. Therefore, the applicationof an errorweighted least square fit in the RM calculation leads to a reductionof noise in the power spectra. Note that in these two power spectra,the influence of thenπ-artefacts are still present even in the errorweighted standard fit RM map.

The two remaining power spectra in Fig. 3 allow us to investi-gate the influence of thenπ-artefacts. The dashed-dotted line rep-resents the power spectra as calculated from aPacmanRM mapallowingσmax

k = 20 performing an error weighted fit. Again, nowindow weighting was applied to this calculation. One can clearlysee that there is one order of magnitude difference between the er-ror weighted standard fit power spectra and thePacmanone at largek-values. Since we also allow higher noise levelsσmax

k for the po-larisation angles, this drop can only be explained by the removingof thenπ-artefacts in thePacmanmap. That there is still a lot ofnoise in the map on small real space scales (largek’s), which gov-erns the power spectra on these scales, can be seen from the dottedpower spectrum, which was determined as above while applying asimple window weighting in the calculation. There is an additional

10000

100000

1e+06

1e+07

1e+08

1e+09

1e+10

1e+11

0.01 0.1 1 10

ε B(k

) [G

2 cm]

k [kpc-1]

stdfit: (σ = 20 degree)stdfit: err weight (σ= 10 degree)pac: err weight (σ= 20 degree)

pac: err and window weight (σ= 20 degree)

Figure 3. Various power spectra for the cluster Abell 2255 determinedfromdifferent RM map making scenarios are shown. The solid line represents apower spectrum calculated from a standard fit RM map withσmax

k= 20

where no error weighting was applied. The dashed power spectrum was ob-tained from a standard fit RM map withσmax

k= 10 where error weight-

ing was applied to the standard fit. The dashed-dotted power spectrum wasdetermined from aPacmanRM map withσmax

k= 20 and error weight-

ing was applied to the RM fits. The dotted line represents a power spectrumcalculated from aPacmanRM map as above, but additionally a simple win-dow weighting was applied for the determination of the powerspectra. Forcomparison, the power spectra of pure white noise is plottedas a straightdashed line.

difference of about one order of magnitude between the two powerspectra at largek-scales determined for the twoPacmanscenarios.

For comparison the power spectrum of pure white noise isplotted as straight dashed line in Fig. 3. It can be shown analyt-ically, that white noise as observed through an arbitrary windowresults in a spectrum ofεB(k) ∝ k2.

As another independent statistical test, we applied thegradi-ent vector alignment statisticV to the RM andϕ0 maps calculatedby thePacmanand the standard fit algorithm. The gradients of RMandϕ0 were calculated using the scheme as described in footnote5 in Enßlin et al. (2003). For the two RM maps shown in Fig. 1, wefind a ratio ofVstdfit/Vpacman = 20, which indicates a significantimprovement mostly resulting from removing thenπ-artefacts bythePacmanalgorithm. The calculation of the normalised quantityV yieldedVstdfit = −0.75 andVpacman = −0.87. Smoothing thePacmanRM map slightly leads to a drastic decrease of the quan-tities V andV . This indicates that the statistic is still governed bysmall scale noise. This can be understood by looking at the RMmaps of Abell 2255 in Fig. 1. The extreme RM values are situatedat the margin of the source which is, however, also the noisiest re-gion of the source. Calculating the normalised quantity forthe highsignal-to-noise region yieldsVpacman = −0.56. For comparison,the normalised quantity for the standard fit of the same high signal-to-noise region isVstdfit = −0.94.

3.2 Hydra North

The polarised radio source Hydra A, which is in the centre of theAbell cluster 780, also known as the Hydra A cluster, is located ata redshift of 0.0538 (de Vaucouleurs et al. 1991). The sourceHy-dra A shows an extended, two-sided radio lobe. Detailed X-raystudies have been performed on this cluster (e.g. Ikebe et al. 1997;

c© 0000 RAS, MNRAS000, 1–10

6 C. Vogt, K. Dolag, T. A. Enßlin

Figure 4. A pixel-by-pixel comparison for the Hydra north lobe ofRMstdfit values obtained using the standard fit plotted on they-axis versusthe RMPacman values calculated by employing our new algorithmPac-man. The parallel scattered points at± 3000-4000 rad m−2 and± 10 000

rad m−2 are a result of spurious solutions to thenπ-ambiguity obtainedby the standard fit algorithm. For the calculation of the RM maps, all fivefrequencies were used.

Peres et al. 1998; David et al. 2001). They revealed a strong cool-ing core in the cluster centre. The Faraday rotation structure wasobserved and analysed by Taylor & Perley (1993). They reportedRM values ranging between−1 000 rad m−2 and+3300 rad m−2

in the north lobe and values down to−12 000 rad m−2 in the southlobe.

Polarisation angle maps and their error maps for frequenciesat 7 815, 8 051, 8 165, 8 885 and14 915 MHz resulting from ob-servation with the Very Large Array were kindly provided to us byGreg Taylor. A detailed description of the radio data reduction canbe found in Taylor et al. (1990) and Taylor & Perley (1993).

In this section, we concentrate on the north lobe of Hydra Aand discuss the south lobe separately in Sect. 3.3. The southlobeis more depolarised, leading to a lower signal-to-noise than in thenorth lobe. The north lobe is a good example of how to treat noisein the data, while the south lobe gives a good opportunity to discussthe limitations and the strength of our algorithmPacman.

Using the polarisation data for the four frequencies at around8 GHz, we calculated the standard fit RM map which is shownin the upper middle panel of Fig. 1. The maximal allowed er-ror in polarisation angle was chosen to beσmax

k = 25 andRMmax = 15 000 rad m−2. A PacmanRM map is shown in themiddle of the middle panel in Fig. 1 below the standard fit map.This map was calculated using the four frequencies at around8GHz with aσmax

k=8GHz = 30 and additionally where possible thefifth frequency at 15 GHz was used with aσmax

k=15GHz = 35.RMmax was chosen to be15 000 rad m−2. From a visual compar-ison of the two maps, one can conclude that the standard fit maplooks more structured on smaller scales and less smooth thanthePacmanmap. This structure on small scales might be misinter-

1e+07

1e+08

1e+09

1e+10

0.01 0.1 1 10 100

ε B(k

) [G

2 cm]

k [kpc-1]

stdfit; no weight; kmin=f=4stdfit; thresholding; kmin=f=4

pacman; no weight; kmin=4; f=5pacman; threshholding; kmin=4; f=5

Figure 5. Various power spectra calculated for the RM maps for the northlobe of Hydra A are shown. The dashed dotted line and the dotted line rep-resent the power spectra calculated from the standard fit RM map and thePacmanRM map, respectively, while no window weighting was applied.A threshold window weighting assuming aσRM

0 = 75 rad m−2 yield thedashed line power spectrum for the standard fit map and the solid line for thePacmanmap. The standard fit maps were calculated from the four frequen-cies at 8 GHz whereas thePacmanmaps were determined using additionallythe fifth frequency when possible. For both algorithms,σmax

k= 35 was

used. The influence of the small–scale pixel noise on the power spectra canclearly be seen in this figure at largek-values.

preted as small scale structure of the RM producing magneticfield.Note that the difference in the RM maps is mainly due to the us-age of the fifth frequency for thePacmanmap demonstrating thatall information available should be used for the calculation of RMmaps in order to avoid misinterpretation of the data.

Since the four frequencies around 8 GHz are close together, awrong solution of thenπ-ambiguity would manifest itself by dif-ferences of about∆RM = 10 000 rad m−2. Such jumps are notobserved in the main patches of the standard fit RM map shownin Fig. 1. However, including the information contained in the po-larisation angle map of the 5th frequency at 15 GHz, which is de-sirable as explained above, one introduces the possibilityof nπ-ambiguities resulting in∆RM = 3 000...4 000 rad m−2. There-fore, a scatter plot of a pixel-by-pixel comparison betweena stan-dard fit and aPacmanmap, calculated both using the additionalavailable information on the fifth frequency, is shown in Fig. 4.The parallel scattered points are spurious solutions foundby thestandard fit algorithm. One can clearly see that they developat ±3000-4000 rad m−2 and less pronounced at± 10 000 rad m−2.

As we will discuss below, the 15 GHz data set may have alower signal–to–noise ratio than the 8 GHz data and thus, a lowererror thresholdσmax

k will eliminate most of the wrong fits. How-ever, it is a particular strength ofPacmanthat it yields still reliableresults even if choosing larger error thresholds.

In order to study the influence of the noise on small scales,we calculated the power spectra from RM maps obtained using dif-ferent parameter sets for thePacmanand the standard fit algorithm.As for Abell 2255E, all calculations of power spectra were done forRM areas which would be covered by thePacmanfit if the sameparameters were used. Again this leads to exclusion of pixels in thestandard fit RM map which are not associated with the source.

There is a clear depolarisation asymmetry of the two lobesof Hydra A observed as described by the Laing–Garrington effect

c© 0000 RAS, MNRAS000, 1–10

Pacman II: Application 7

(Taylor & Perley 1993; Lane et al. 2004). Therefore, we assume forthe calculation of any power spectra for the Hydra source, that thesource plane is tilted by an inclination angle of 45 where the northlobe points towards the observer and the south lobe away fromtheobserver.

For a first comparison, we calculated the power spectra fortwo RM maps similar to the one shown in Fig. 1. They were ob-tained using the four frequencies at 8 GHz for the standard fital-gorithm and using additionally the fifth frequency when possiblefor the Pacmanalgorithm. The other parameters were chosen tobe σmax

k = 35 andRMmax = 15 000 rad m−2. For these twoRM maps, we determined power spectra applying firstly no win-dow weighting at all and secondly threshold window weighting asdescribed in the end of Sec. 2.3. We choose the threshold to beσRM0 = 75 rad m−2 which represents the high signal-to-noise re-

gion. The respective spectra are exhibited in Fig. 5.For the power spectra,nπ-artefacts should play only a mi-

nor role in the calculation of the power spectra as explainedabove.Therefore, any differences arising in these spectra are caused by thevarying treatment of noise in the map making process or in theanal-ysis. Comparing the power spectra without window weighting, theone obtained from thePacmanRM map lies below the one from thestandard fit RM map. The difference is of the order of half a magni-tude for largek’s – small real space scales. The standard fit powerspectrum seems to increase with largerk but thePacmanone seemsto decrease over a greater range ink. The reason for this differenceis the same one which was responsible for the different smoothnessin the two RM maps shown in Fig. 1, namely that only four fre-quencies were used for the standard fit RM map. Even introducinga window weighting scheme, which down-weights noisy region,cannot account entirely for that difference as can be seen from thecomparison of the window weighted power spectra. This is espe-cially true because the small-scale spatial structures arefound evenin the high signal-to-noise regions when using only four frequen-cies for the RM map calculation. This demonstrates again howim-portant it is to include all available information in the mapmakingprocess.

In order to study the influence of the maximal allowed mea-surement error in the polarisation angleσmax

k , we calculated thepower spectra for a series ofPacmanRM maps obtained forσmax

k

ranging from 5 to 35. The RM maps were derived using allfive frequencieskmin= 5, not allowing any points to be consid-ered for which only four frequencies fulfilσkij

< σmaxk . Again an

RMmax = 15 000 rad m−2 was used. The resulting power spectrausing a threshold window weighting (σRM

0 = 50 rad m−2) are ex-hibited in Fig. 6. One can see that they are stable and do not differsubstantially from each other even though the noise level increasesslightly when increasing theσmax

k . This demonstrates the robust-ness of ourPacmanalgorithm.

For comparison, we plotted two more power spectra in Fig. 6.For the solid red line spectrum, no window weighting was appliedto an RM map which was calculated as above havingσmax

k = 35.One can clearly see that the spectrum is governed by noise on largek-scales – small real space scales. Thus, some form of windowweighting in the calculation of the power spectra seems to benec-essary in order to suppress a large amount of noise.

Another aspect arises in the noise treatment if we considerthe power spectra represented by the blue stars which was calcu-lated using a threshold window weighting (σ0 = 50 rad m−2). Thispower spectrum represents the analysis for an RM map obtainedusing only the four frequencies at 8 GHz whileσmax

k = 35. It isstriking that the noise level on largek-scales (on small real space

1e+07

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0.1 1 10 100

ε B(k

) [G

2 cm]

k [kpc-1]

σmax k = 5o

σmax k = 10o

σmax k = 15o

σmax k = 20o

σmax k = 25o

σmax k = 30o

σmax k = 35o

no weighting; σmaxk = 35o

kmin=f=4; σmax k = 35o

Figure 6. Power spectra for the north lobe of Hydra A calculated from a se-ries ofPacmanRM maps are shown. The various coloured dashed and dot-ted lines show power spectra from RM maps calculated forσmax

kranging

from 5 to 35 using all five frequencieskmin = 5. The spectra were de-termined applying threshold window weighting (σRM

0 = 50 rad m−2). Forcomparison the solid red line power spectrum, which is the largest spectrumat k > 10 kpc−1, is not window weighted forσmax

k= 35. The power

spectrum represented by the blue stars was derived from anPacmanRMmap obtained using the four frequencies at 8 GHz and aσmax

k= 35.

scales) is lower than for the RM maps obtained for the five fre-quencies, one would expect it to be the other way round. Appar-ently the fifth frequency has a lot of weight in the determination ofσRMij (see Eq. (3) in paper I) since this frequency is almost twice

as large as the other four. If the measurement errors of the polarisa-tion angles are underestimated this can lead to an underestimationof the uncertaintyσRM

ij in the final RM value due to Gaussian er-ror propagation. A more accurate error estimate could be achievedby considering the error treatment as described by Johnson et al.(1995). They state in their Appendix that errors are not uniformacross synthesis images and propose a different error algorithm.

One can test the hypothesis of underestimation of errors byperforming aχ2

ν test as described in Sect. 2.2. Theχ2νij

map forthe PacmanRM map of Hydra North derived by using a multi-frequency fit (see middle upper panel of Fig. 1) is exhibited in theleft panel of Fig. 7. The red values are close to unity. However theyellow and blue regions in this map are values larger than unity in-dicating that the errors are underestimated in these regions. Notethat in these regions there is also the fifth frequency used for theRM fit which might indicate that the errors for this frequencyareunderestimated by a larger factor than for the other four frequen-cies.

We also applied thegradient vector product statisticV to theRM andϕ0 maps obtained by using the RM maps as shown in themiddle panel of Fig. 1 and the respectiveϕ0 maps. The calculationyielded for the ratioVstdfit/Vpacman = 1.2. This corresponds to aslight decrease of correlated noise in thePacmanmaps. The calcu-lation of the normalised quantity resulted inVstdfit = −0.85 forthe standard fit andVpacman = −0.83 for thePacmanmaps. Thisis expected since the low signal-to-noise areas of the northlobe ofHydra A will govern this gradient statistic and for both RM mapsthese areas look very similar. In order to visualise this, wecalcu-lated aV map which is exhibited in Fig. 7. The extreme negativevalues observed for the low signal-to-noise regions (red regions) in-dicate strong anti-correlation and thus, anti-correlatedfluctuations

c© 0000 RAS, MNRAS000, 1–10

8 C. Vogt, K. Dolag, T. A. Enßlin

0 5 10 15 20

reduced chi2

−50000 −40000 −30000 −20000 −10000 0

arbitrary units

Figure 7. On the left hand side aχ2νij is shown. The yellow and green regions have values larger than unity indicating underestimation of errors. Note that

for these regions the RM fit used five frequencies whereas in the outer parts of the source only four frequencies were used. AV map for the north lobe ofHydra A is exhibited on the right panel. The extreme negativevalues (red and black) correspond to strong anti-correlation of RM andϕ0 fluctuations, whichare produced by noise in the observed maps. However, the green and blue coloured regions have values around zero and are therefore high signal-to-noiseregions (compare with theσRM map in the middle panel of Fig. 1).

of RM andϕ0 in these regions. For the high signal-to-noise regions(blue and green) moderate values varying around zero are observed.Note the striking morphological similarity of the RM error (σRM)map in the lower middle panel of Fig. 1 and of theV map as shownin Fig. 7. These two independent approaches to measure the RMmaps accuracy give a basically identical picture. These approachesare complementary since they use independent information.TheRM error map is calculated solely fromσk maps, which are basedon the absolute polarisation errors, whereas theV map is basedsolely on the RM derived from the polarisation angleϕ(k) maps.

However, we note that the results for magnetic field strengthsand correlation lengths presented by Vogt & Enßlin (2003) are notchanged substantially. This is owing to the fact that Vogt & Enßlin(2003) considered for the calculation of the magnetic field proper-ties only the power on scales which were larger than the resolutionelement (beam). Therefore the small scale noise was not consid-ered for this calculation and thus, no change in the results givenis expected. The central magnetic field strength for this cluster asderived during the course of this work is about 10µG and the fieldcorrelation length is about 1 kpc being consistent with results pre-sented in Vogt & Enßlin (2003).

3.3 The Quest for Hydra South

As already mentioned, the southern part of the Hydra source has tobe addressed separately, as it differs from the northern part in manyrespects. One of them is that the depolarisation is higher inthesouth lobe which especially complicates the analysis of this part.

An RM map obtained employing the standard fit algorithm us-ing the four frequencies at 8 GHz is shown in the upper right panelof Fig. 1. This RM map exhibits many jumps in the RM distributionwhich seems to be split in lots of small patches having similar RMs.Furthermore, RMs of−12 000 rad m−2 are detected (indicated asred regions in the map). If one includes the fifth frequency at15GHz in the standard fit algorithm, the appearance of the RM mapdoes not change significantly, and most importantly, the extremevalues of about−12 000 rad m−2 do not vanish.

The application of thePacmanalgorithm with conservativesettings for the parametersσmax

k , σ∆max and g, leads to a split-

ting of the RM distribution into many small, spatially disconnectedpatches. Such a map does look like a standard fit RM map and thereare still the RM jumps and the extreme RM values present. How-ever, if one lowers the restrictions for the construction ofpatches,thePacmanalgorithm starts to connect patches to the patch with thebest quality data available in the south lobe using its informationon the global solution of thenπ-ambiguity. An RM map obtainedpushingPacmanto such limits is exhibited in the middle right panelof Fig. 1. Note that the best quality area is covered by the bright

c© 0000 RAS, MNRAS000, 1–10

Pacman II: Application 9

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) [G

2 cm]

k [kpc-1]

south - pacman - simplesouth - pacman - thresh

south - sdtfit - simplesouth - sdtfit - thresh

north - pacman - thresh

Figure 9. Various power spectra for the RM maps of the south lobe of theHydra source are compared with the power spectrum of the north lobe cal-culated from thePacmanRM map (kmin = 5, σmax

k= 35) employing a

threshold weighting withσRM0 = 50 rad m−2 represented as filled circles.

blue regions and a zoom-in for this region is also exhibited in thesmall box to the lower left of the source. In the lowest right panelof Fig. 1, aσRM -map is shown indicating the high quality regionsby the red colour.

The RM distribution in thisPacmanmap exhibits clearlyfewer jumps than the standard fit map although the jumps do notvanish entirely. Another feature that almost vanishes in thePacmanmap are the extreme RMs of about−12 000 rad m−2. The RM dis-tribution of thePacmanmap seems to be smoother than the one ofthe standard fit map. However, if the individual RM fits for points,which deviate in their RM values depending on the algorithm used,are compared in aϕ(k)-λ2

k-diagram, the standard fit seems to bethe one which would have to be preferred since it does fit better tothe data at hand.Pacmanhas some resistance to pick this smallestχ2 solution if it does not make sense in the context of neighbour-ing pixel information. As an example, we plotted individualRMfits which are observed across the RM jumps in Fig. 8. Howeveras shown above, one has to consider that the measurement errorsmight be underestimated which gives much tighter constraints onthe fit than it would otherwise be.

In order to investigate these maps further, we also calculatedthe threshold weighted (σRM

0 = 50 rad m−2) and simple weightedpower spectra from thePacmanand the standard fit RM maps ofthe south lobe and compared them to the one from the north lobe.The RM maps used for the comparison were all calculated employ-ingσmax

k = 35 andkmin = f = 5. The various power spectra areshown in Fig. 9. One can clearly see that the power spectra calcu-lated for the south lobe lie well above the power spectra fromthenorth lobe, which is represented as filled circles.

Concentrating on the comparison of the simple windowweighted power spectra of the south lobe in Fig. 9 (solid linefromPacmanRM map and dashed line from standard fit RM map) to theone of the north lobe, one finds that the power spectrum calculatedfrom thePacmanRM map is closer to the one from the north lobethan the power spectrum derived from the standard fit RM map.This indicates thatPacmanRM map might be the right solution tothe RM determination problem for this part of the source. However,this difference vanishes if a different window weighting scheme isapplied to the calculation of the power spectrum. This can beseen

by comparing the threshold window weighted power spectra (dot-ted line for the power spectra of thePacmanRM map and dasheddotted line for the power spectra from the standard fit RM map)inFig. 9. Since the choice of the window weighting scheme seemstohave also an influence on the result the situation is still inconclu-sive.

An indication for the right solution of thenπ-ambiguity prob-lem may be found in the application of ourgradient vector productstatistic V to thePacmanmaps and the standard fit maps of thesouth lobe (as shown in Fig. 1). The calculations yielded a ratioVstdfit/Vpacman = 12. This is a difference of one order of magni-tude and represents a substantial decrease in correlated noise in thePacmanmaps. The calculation of the normalised quantity yieldedVstdfit = −0.69 for the standard fit maps andVpacman = −0.47for thePacmanmaps. This result strongly indicates that thePacmanmap should be the preferred one.

The final answer to the question about the right solution ofthe RM distribution problem for the south lobe of Hydra A has tobe postponed until observations of even higher sensitivity, higherspatial resolutions and preferably also at different frequencies areavailable.

4 CONCLUSIONS & LESSONS

We demonstrated the robustness of ourPacmanalgorithm whichis especially useful for the calculation of RM andϕ0 maps of ex-tended radio sources. It dramatically reducesnπ-artefacts in noisyregions and makes the unambiguous determination of RMs in theseregions possible. Any statistical analysis of the RM maps will profitfrom this improvement.

In the course of this work the observational data taught us thefollowing lessons:

(i) It is important to use all available information obtained by theobservation. It seems especially desirable to have a good frequencycoverage. The result is a smoother, less noisy map.

(ii) For the individual least-squares fits of the RM, it is advisableto use an error weighting scheme in order to reduce the noise.

(iii) Global algorithms are preferable, if reliable RM values areneeded from low signal-to-noise regions at the edge of the source.

(iv) Sometimes it is necessary to test many values of the parame-tersRMmax, σmax

k andσ∆max, which govern thePacmanalgorithm,

in order to investigate the influence on the resulting RM maps.(v) One has to keep in mind that whichever RM map seems to

be most believable, it might still contain artefacts. One should doa careful analysis by looking on individual RM pixel fits but onealso should consider the global RM distribution. We presented thegradient vector product statisticV as a useful tool to estimate thelevel of reduction of cross-correlated noise in the calculated RMandϕ0 maps.

(vi) For the calculation of power spectra it is always preferableto use a window weighting scheme which has to be carefully se-lected.

(vii) Finally, under or overestimation of the measurement errorsof polarisation angles (which will propagate through all calcula-tions due to error propagation) could influence the final results. Weperformed a reducedχ2

ν test in order to test if the errors are over orunder estimated.

We conclude that the calculation of RM maps is a very dif-ficult task requiring a critical view to the data and a carefulnoise

c© 0000 RAS, MNRAS000, 1–10

10 C. Vogt, K. Dolag, T. A. Enßlin

Figure 8. An example for individual RM fits across an RM jump observed inthe RM maps of the south lobe of Hydra A. The solid line indicates the standardfit solution and the dashed one represents thePacmansolution. The error bars in these plots indicate the polarisation errors which were multiplied by 3. Notethat this is an extreme example, wherePacmanchooses a solution to the individual fits in noisy regions which has not necessarily the minimalχ2 value byconstruction. However, the global statistical tests applied to the whole RM map of Hydra South indicate that thePacmanmap has less artefacts in comparisonto the standard fit RM map. We note, that for this particular local example it is not trivial to decide which algorithm is giving the right solution.

analysis. However, we are confident thatPacmanoffers a good op-portunity to calculate reliable RM maps and to understand the dataand its limitations better. Furthermore, such improved RM mapsand the according error analysis for Hydra North and Abell 2255were presented in this paper.

ACKNOWLEDGEMENTS

We want to thank Federica Govoni and Greg Taylor for allowingusto use their data on Abell 2255 and Hydra to test the new algorithm.We like to thank Tracy Clarke, James Anderson, Phil Kronbergand an anonymous referee for useful comments on the manuscript.K. D. acknowledges support by a Marie Curie Fellowship of theEuropean Community program ”Human Potential” under contractnumber MCFI-2001-01227.

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