Downstream depolarization in the Sausage relic: a 1-4 GHz Very Large Array study
G. Di Gennaro, R.J. van Weeren, L. Rudnick, M. Hoeft, M. Brüggen, D. Ryu, H.J.A. Röttgering, W. Forman, A. Stroe, T.W. Shimwell, R.P. Kraft, C. Jones, D.N. Hoang
DDraft version February 15, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Downstream depolarization in the Sausage relic: a 1–4 GHz Very Large Array study
G. Di Gennaro,
1, 2
R.J. van Weeren,
1, 2
L. Rudnick, M. Hoeft, M. Br¨uggen, D. Ryu, H.J.A. R¨ottgering, W. Forman, A. Stroe, ∗ T.W. Shimwell,
1, 7
R.P. Kraft, C. Jones, and D.N. Hoang Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA Minnesota Institute for Astrophysics, University of Minnesota, 116 Church St. S.E., Minneapolis, MN 55455, USA Th¨uringer Landessternwarte, Sternwarte 5, 07778 Tautenburg, Germany Hamburger Sternwarte, Universit¨at Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany Department of Physics, School of Natural Sciences UNIST, Ulsan 44919, Korea ASTRON, The Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA, Dwingeloo, The Netherlands (Received 17 June 2020; Accepted 11 February 2021)
ABSTRACTRadio relics are elongated sources related to shocks driven by galaxy cluster merger events. Althoughthese objects are highly polarized at GHz frequencies ( (cid:38) QU -fitting approach to model the Stokes I , Q and U emission, obtaining best-fit intrinsic polarizationfraction ( p ), intrinsic polarization angle ( χ ), Rotation Measure (RM) and wavelength-dependentdepolarization ( σ RM ) maps of the cluster. Our analysis focuses on the northern relic (RN). For thefirst time in a radio relic, we observe a decreasing polarization fraction in the downstream region.Our findings are possibly explained by geometrical projections and/or by decreasing of the magneticfield anisotropy towards the cluster center. From the amount of depolarization of the only detectedbackground radio galaxy, we estimate a turbulent magnetic field strength of B turb ∼ . µ Gauss in therelic. Finally, we observe Rotation Measure fluctuations of about 30 rad m − around at the medianvalue of 140.8 rad m − at the relic position. Keywords: galaxies: clusters: individual (CIZA J2242.8+5301) – galaxies: clusters: intra-clustermedium – large-scale structure of Universe – magnetic field – polarization – radiationmechanisms: non-thermal – diffuse radiation – shock waves INTRODUCTIONRadio relics are synchrotron sources generally locatedin the outskirts of merging galaxy clusters. They areelongated, often arc-shaped, and not associated withany optical counterparts. It is now accepted that thesesources trace particles (re)accelerated due to the prop-agation of shock waves generated by a cluster-clustermerger event (see Brunetti & Jones 2014; van Weerenet al. 2019, for a theoretical and observational review).Being synchrotron sources, radio relics are also tracers of
Corresponding author: Gabriella Di [email protected] ∗ Clay Fellow the magnetic field in cluster outskirts. Numerical sim-ulations (e.g. Dolag et al. 1999; Br¨uggen et al. 2005;Vazza et al. 2018), as well as observations (e.g. Govoni &Feretti 2004; Bonafede et al. 2010a), show that the mag-netic field intensity declines with radius (and hence withparticle density) in clusters, with central values of a few µ Gauss (Bonafede et al. 2010a). On the other hand, it isexpected that, during a cluster merger, the un-orderedmagnetic fields in the intracluster medium (ICM) arecompressed, amplified and aligned with the propagatingshock plane, generating strongly linearly polarized emis-sion ( (cid:38) a r X i v : . [ a s t r o - ph . GA ] F e b G. Di Gennaro et al. of cluster merger shocks ( M = 1 − z = 0 .
192 (Kocevski et al.2007).The cluster is the result of the collision of two equal-mass sub-clusters (Dawson et al. 2015; Jee et al. 2015),with a small inclination of the merger axis to the planeof the sky (i.e. | i | (cid:46) ◦ , van Weeren et al. 2011). Thecluster hosts two main radio relics, in the north andin the south, several tailed radio galaxies and severalpatches of diffuse emission (see Di Gennaro et al. 2018).High-frequency studies, up to 30 GHz, showed a possi-ble steepening in the integrated radio spectrum from ∼ − . ∼ − . ν > . The radio spectrum is defined as S ν ∝ ν α , with α the spectralindex. Zel’dovich (SZ) effect (also supported by single-dish ob-servations, see Loi et al. 2017). Single-dish observationsrevealed that this relic is strongly polarized (up to 60%at 8.35 GHz, Kierdorf et al. 2017), although the poor res-olution (i.e. 90 (cid:48)(cid:48) ) strongly limited their analysis. Fromthe relic width (55 kpc) and X-ray downtream velocity(about 1000 km s − ), van Weeren et al. (2010) estimatedmagnetic field strengths of 5 or 1.2 µ Gauss.The paper is organized as follows: in Sect. 2 we de-scribe the data reduction and the imaging procedures; inSect. 3 we present the QU fitting approach; we highlightthe effect of the Galactic Rotation Measure in Sect. 4;the results and discussion are given in Sect. 5 and 6; weend with the conclusion in Sect. 7. Throughout the pa-per, we assume a flat ΛCDM cosmology with H = 70km s − Mpc − , Ω m = 0 . Λ = 0 .
7, which givesa conversion factor of 3.22 kpc/ (cid:48)(cid:48) and a luminosity dis-tance of ≈
944 Mpc, at the cluster’s redshift ( z = 0 . OBSERVATIONS AND DATA REDUCTIONWe made use of the same 1–4 GHz VLA observationspresented in Di Gennaro et al. (2018), to which we re-fer for a detailed description of the data reduction. Theobservations were made with all the four array config-urations (namely, A, B, C and D), some of them splitinto sub-datasets (see Table 1 in Di Gennaro et al. 2018).Due to the large angular size of the cluster, and the lim-ited field of view (FOV) at 2–4 GHz, we observed threeseparate pointings in this frequency range. We brieflysummarize the data reduction strategy below.First, we Hanning smoothed the data, and removedradio frequency interference (RFI) with the tfcrop modefrom the flagdata task in
CASA . Then, we calibrated theantenna delays, bandpass, cross-hand delays, and polar-ization leakage and angles using the primary calibrators3C138, 3C147, and/or 3C48. For the polarization leak-age calibration, we can only make use of an unpolarizedsource , hence we discarded all the sub-datasets where3C48 was the only calibrator (for further details, see DiGennaro et al. 2018). We determined the global cross-hand delay solutions ( gaintype=‘KCROSS’ ) from the po-larized calibrator 3C138, taking a RL-phase difference of − ◦ (both L- and S-band) and polarization fractionsof 7.5% and 10.7% (L- and S-band respectively). Weused 3C147 to calibrate the polarization leakage terms( poltype=‘Df’ ), and 3C138 to calibrate the polariza-tion angle ( poltype=‘Xf’ ). The solution tables were In principle, a calibrator with enough parallactic angle cover-age can also be used for the leakage calibration. This kind ofcalibrator was not available in our observations. olarization study of CIZA J2242.8+5301 Table 1.
Datacube information. Columns 1 to 3: Gaussian uv-taper, weighting and robust parameters for the imaging. Column4: final resolution of the datacubes. Column 5: total number of channels in the 1–2 and 2–4 GHz bands. Column 6: channelwidth in MHz in the 1–2 and 2–4 GHz bands. Column 7: noise map for the Stokes I , Q and U datacubes.uv-taper weighting robust resolution ν σ rms[1 . − . [ (cid:48)(cid:48) ] [ (cid:48)(cid:48) × (cid:48)(cid:48) ] [MHz] [ µ Jy beam − ]1–2 GHz 2–4 GHz 1–2 GHz 2–4 GHz I Q U . × . . × .
55 104 75 4 16 8.9 10.1 10.05 Briggs 0 7 × ×
13 104 136 4 8 18.2 5.1 5.4Note: The noise levels in the last column have been calculated as standard deviation of the datacube, in a central, “empty”region of the cluster. For the 2 . (cid:48)(cid:48) -tapered images, we only produced stamps of the single sources, hence we report the mapnoise locally to RN. applied on the fly to determine the complex gain so-lution for the secondary calibrator J2202+4216. Addi-tional RFI removal was performed, using the tfcrop and rflag modes (in CASA ) and
AOFlagger (Offringa et al.2010), before and after applying the calibration tablesto the target field, respectively. The data were aver-aged by a factor of two in time and a factor of fourin frequency. This reflects a frequency resolution (i.e.channel width) of ∆ ν = 4 and ∆ ν = 8 MHz, at 1–2and 2–4 GHz respectively. The only exception is the2 . (cid:48)(cid:48) -tapered dataset at 2–4 GHz, for which we averageby a factor of eight, i.e. ∆ ν = 16 MHz. Finally, self-calibration was performed to refine the amplitude andphase calibration on the target.To retrieve the images for all the Stokes parameters(i.e., I , Q and U ) at each channel ∆ ν , as requiredfor a detailed polarization analysis, we employed the WSClean (Offringa et al. 2014). Images were producedwith different weightings (i.e.
Briggs and uniform ),and uv-tapers (i.e., 2 . (cid:48)(cid:48) , 5 (cid:48)(cid:48) and 10 (cid:48)(cid:48) ). Bad spectral win-dows and channels were discarded from the final analy-sis. For the Stokes- Q and - U images, we also used theoptions -join-channels , -join-polarizations and -squared-channel-joining , which prevent the Q -, U -flux to be averaged out to zero . After imaging, chan-nel images that where too noisy or low-quality were re-moved. In the end, a total of 240 channels, for the 5 (cid:48)(cid:48) -and 10 (cid:48)(cid:48) -tapered images, and 179 channels, for the 2 . (cid:48)(cid:48) -tapered images, were used. This results in a final fre-quency coverage of 1.26–3.60 GHz. The single-channelimages were re-gridded to the same pixel grid and con-volved to the same resolution (see Tab 1). Finally, allthe single images were primary-beam corrected, by tak-ing the beam variation with the frequency taken into https://sourceforge.net/p/wsclean/wiki/RMSynthesis/ account , and merged into a single datacube for eachStokes parameter. Errors in the single channel imageswere estimated using the rms noise level from a central,empty, region of the cluster (at 7 (cid:48)(cid:48) and 13 (cid:48)(cid:48) resolution)or locally for the sources of interest (at 4 . (cid:48)(cid:48) and 2 . (cid:48)(cid:48) resolution). POLARIZATION THEORY AND MODELLINGAPPROACHThe linear polarization emission can be described interms of Stokes parameters for the total intensity, I , andthe orthogonal components, Q and U : P ( λ ) = p ( λ ) I ( λ ) exp[2 iχ ( λ )] = Q ( λ ) + iU ( λ ) , (1)and λ is the observing wavelength. Here, p ( λ ) is thefractional (or degree of) polarization and χ ( λ ) is the po-larization angle, which are wavelength-dependent quan-tities that can be written as: p ( λ ) = P ( λ ) I ( λ ) = (cid:112) Q ( λ ) + U ( λ ) I ( λ ) (2)and χ ( λ ) = 12 arctan (cid:18) U ( λ ) Q ( λ ) (cid:19) . (3)The passage of the polarized radiation through a fore-ground magneto-ionic medium, such as the ICM, resultsin a rotation of polarization plane via the Faraday effectaccording to χ ( λ ) = χ + RM λ , (4)where χ is the intrinsic polarization angle and RM isthe Faraday rotation measure. This is defined as: The beam shapes have been obtained with
CASA v. 5.3.
G. Di Gennaro et al.
Figure 1.
Result of the QU fit assuming the external depolarization model (EDF, Eq. 7) on a single pixel of the northern relic.Left panel: Fits on Stokes I , Q and U fluxes. Central panel: Resulting fractional polarization, p ( λ ), and polarization angle, χ ( λ ), estimated from Eqs. 2 and 3, respectively. Right panel: Corner plot for the distribution of the uncertainties in the fittedpolarization parameters (i.e. p , χ , RM and σ ); contour levels are drawn at [0 . , . , . , . σ , with σ the 68% statisticaluncertainty (see dashed lines in the 1D histogram). RM = 0 . (cid:90) observersource n e B (cid:107) dl [rad m − ] , (5)where n e is the electron density (in cm − ), B (cid:107) the mag-netic field (in µ Gauss) along the line of sight, l the pathlength through the magneto-ionic medium (in pc), andwith the sign of the equation defined positive for a mag-netic field pointing towards the observer.The traditional way to retrieve the intrinsic polariza-tion angle χ is to observe χ at several wavelengths,and linearly fit Eq. 4. The long-standing problem ofthis approach is the lack of a sufficient number of χ ( λ )measurements. In this work, this issue is overcome bythe large number of channel images with high signal-to-noise (S/N) of our wide-band observations (see Sect.3.1).Several models of the polarized signal, in the presenceof Faraday rotation, are known. In the simplest scenario,Eq. 1 can be written as: P ( λ ) = p I exp[2 i ( χ + RM λ )] , (6)with p the intrinsic polarization fraction. This cor-responds to the physical situation of a single Faradayscreen in the foreground. In this case, dχ/dλ and p ( λ )are constant.Observations have shown that radio relics depolarizeat frequencies (cid:46) P ( λ ) = p I exp( − σ λ ) exp[2 i ( χ + RM λ )] , (7)where σ RM is the dispersion about the mean RM acrossthe beam on the sky.On the other hand, IFD occurs when the emittingsource and the Faraday screen (i.e. the rotating layer)are mixed. In this case, depolarization is due to therandom direction of the plane of polarization throughthe emitting region, and it can be parametrized as: P ( λ ) = p I (cid:20) − exp( − ς λ )2 ς λ (cid:21) exp[2 i ( χ + RM λ )] , (8)where ς RM is the internal dispersion of the random field.3.1. QU-modelling approach
Stokes Q ( λ ) and U ( λ ) fitting has been used in lit-erature to determine the polarization properties of amagneto-ionic layer (e.g. O’Sullivan et al. 2012; Ozawaet al. 2015; Anderson et al. 2016). In this approach, Q ( λ ) and U ( λ ) were fitted simultaneously with co-sine and sine models, while I ( λ ) was fitted with a log-parabolic model (see also Massaro et al. 2004), which olarization study of CIZA J2242.8+5301 I ν = I ν a + b log( ν/ν ref ) , (9)where we fixed the reference frequency ν ref to 1 GHz.In this model, b is the curvature parameter and thespectral index is calculated as the log-derivative, i.e. α = a + 2 b log( ν/ν ref ). For each channel image inthe I ( λ ), Q ( λ ) and U ( λ ) datacubes, the uncertain-ties were computed by adding in quadrature the relative(spatial) map noise and 5% of the Stokes I , Q and U fluxin each channel. Here, the 5% represents a spatially-independent intrinsic scatter which takes into accountthe flux variations between the single-frequency channelmaps. The origin of this scatter is not fully clear, but itis probably related to bandpass calibration and/or de-convolution uncertainties.We fitted our data with the Markov Chain MonteCarlo (MCMC) method (Foreman-Mackey et al. 2013)to explore the best-set of model parameters (Ozawaet al. 2015). During the fitting procedure, all the pa-rameters (i.e. I , a and b for Stokes I , and p , χ , RMand σ for the combined Stokes Q and U ) were leftfree to vary through the full parameter space. In thefitting, we constrained p , χ and σ (or ς ) to thefollowing physical conditions: ≤ p ≤ ≤ χ < πσ ≥ ς ≥ , (10)and we assumed a single-RM component model (see alsoAppendix A)). The upper limit for the polarization an-gle is set to π because the polarization vectors have nopreferred direction. In this convention, χ = 0 and χ = π/ λ . It is worth noting that the p value obtained from the MCMC fit could be an under-estimation of the intrinsic polarization fraction, becauseof the limited λ coverage, and possible misalignment ofthe intrinsic polarization angle χ from different emit-ting sites along the line of sight. Hereafter, we refer to p as the best-fit intrinsic polarization fraction. The uncer-tainties on the best-fitting parameters were determinedwith the MCMC analysis. The results of the fitting pro-cedure using the EFD model on a representative single The initial guesses for the parameters were obtained with theleast square method ( scipy.optimize.leastsq in Python).
Table 2.
Averaged RM values of the sources labelled inFig. 2 observed in the 1–2 GHz frequency range. The “un-certainty” on RM is represented by the standard deviationof the RM pixel distribution within the source.Source RA
J2000
DEC
J2000 (cid:104) RM (cid:105) ± std (RM)[ h m s ] [ ◦ (cid:48) (cid:48)(cid:48) ] [rad m − ]1 22 44 31.5 +53 00 39.0 − . ± .
42 22 42 12.4 +52 47 56.5 − . ± .
63 22 42 05.2 +52 59 32.0 +1 . ± .
14 22 41 22.1 +53 02 15.5 − . ± .
25 22 41 00.1 +53 04 15.7 − . ± .
16 22 41 33.1 +53 11 07.7 − . ± .
47 22 43 02.2 +53 19 42.2 − . ± .
58 22 43 37.5 +53 09 15.5 − . ± .
09 22 41 22.9 +52 52 54.3 − . ± .
710 22 43 05.2 +53 17 33.8 − . ± . pixel in the cluster northern relic are displayed in Fig.1. Similar result were found using the IDF model (Eq.8), except for ς RM which is higher due to the differentfunctional way it describes the depolarization. ROTATION MEASURE FROM OUR GALAXYThe best-fit Rotation Measure value obtained could,in principle, give information on the magnetic fieldstructure of the diffuse radio emission in the cluster (Eq.5). However, in order to have a reliable estimation ofthe RM associated with the ICM, the contribution ofthe foreground Galactic RM needs to be estimated andremoved from the calculations.The Galactic coordinates of CIZAJ2242 are l = 104 ◦ and b = − ◦ , meaning that the cluster lies on closeto the Galactic plane. Hence, the RMs of the clustersources are strongly affected by the Faraday rotationfrom our Galaxy. Using the map of the Galactic con-tribution to Faraday rotation provided by Oppermannet al. (2015) , we found an average contribution of about − ±
57 rad m − in a region of 20 (cid:48) around the clus-ter center coordinates. However, the current availableGalactic RM map is affected by very poor angular res-olution (i.e. ∼ (cid:48) / pixel), which is comparable with thecluster size ( ∼ (cid:48) ). For this reason, we lack detailedinformation on the RM variations on the cluster/sub-cluster scale. G. Di Gennaro et al. ′ ) Figure 2.
Total polarized emission of the 1–2 GHz field of view (FOV ∼ (cid:48) in radius) to search for polarized radio galaxiesoutside CIZAJ2242. A zoom on those sources is shown in the insets, where the Rotation Measure and the total intensity aredisplayed in the left and right panel, respectively. The RM colorscale is fixed for all the sources. The averaged RM values ofthose sources are listed in Table 2. We investigated the RM values of compact sourceswithin the field of view of our observations, but outsidethe cluster region. In this way, we exclude the contri-bution of the ICM on the RM estimation. Since thesize of the primary beam depends on the frequency asFOV ∝ ν − , and we want to maximize the area wherewe search for polarized sources, we only used the 1–2GHz observations. We found a total of 10 sources inthe 1–2 GHz FOV ( ∼ (cid:48) , see Fig. 2). Their RotationMeasure values, listed in Table 2, are consistent with theaverage Galactic RM value found by Oppermann et al.(2015), with a median value of about −
80 rad m − andstandard deviation of about 42 rad m − . Moreover, wefound that sources close to each other (i.e., sources 4and 5, and sources 7 and 10) have similar RM, suggest-ing that the Galactic foreground might remain approx-imately constant in that region, on those spatial scales(3 (cid:48) − (cid:48) , i.e. few hundreds of kpc, at the cluster distance).However, we find a strong variation from in RM north tosouth and east to west, although without a clear trend.It remains therefore difficult to quantify a unique Rota-tion Measure value from the Galactic foreground, and tosubtract it from our measured RM values for the clustersources. For this reason, in the following maps and plots we report the best-fit RM value, including the Galacticcontribution. RESULTS5.1.
Polarized flux densities and fractions
We obtained the total averaged polarization imagesin the 1.26–3.60 GHz band by means of the RM-Synthesis technique (Brentjens & de Bruyn 2005), us-ing the pyrmsynth tool . In Fig. 3 and in the toppanel of Fig. 4, we show the total averaged polariza-tion images of the entire cluster at 7 (cid:48)(cid:48) resolution andof the northern relic at 2 . (cid:48)(cid:48) resolution, at the effectivefrequencies of 2.3 and 2.0 GHz, respectively. We re-trieve the polarized intensity at the canonical frequen-cies, i.e. 1.5 and 3.0 GHz (i.e. at wavelength of 0.2 and0.1 m, respectively), using the fit results of Eq. 7 asdescribed in Section 3.1. In Table 3 we report the polar-ized and total flux densities, the correspondent factionalpolarization (Eq. 3), and the amount of depolarization https://github.com/mrbell/pyrmsynth olarization study of CIZA J2242.8+5301 RN R3R1 R2 RS1 RS2RS3 RS4R5R4 I AG BCD E FH L MJ1JN OK1K2 K3K4
Figure 3.
Total averaged polarized emission for CIZAJ2242 in the 1.26–3.60 GHz band (effective frequency of 2.3 GHz) at 7 (cid:48)(cid:48) resolution. This image is not corrected for the Ricean bias. The radio contours are from the averaged total intensity image,in the same frequency band and at the same resolution, with contours drawn at levels of 3 σ rms × (cid:112) [1 , , , , , . . . ], with σ rms = 4 . µ Jy beam − . Sources are labelled following Fig. 2 in Di Gennaro et al. (2018). DP . . = 1 − ( p . /p . ) , for the diffuse radiosources in the cluster.We detect significant polarized emission both fromthe numerous radio galaxies and from the diffuse radio In this convention, DP . . = 0, i.e. p . = p . ,means no depolarization, while DP . . = 1, i.e. p . ∼ sources. The brightest polarized structure of the clusteris the northern relic (RN), with integrated polarized fluxdensities of P . = 17 . ± . P . = 19 . ± . . (cid:48)(cid:48) resolution (i.e. the highestresolution available in our observations), the polarizedemission traces the relic’s filamentary structure observedalready in the total intensity (see top panel in Fig. 4 in G. Di Gennaro et al.
RN1 RN2 RN3 RN4 RN5RN1 RN2 RN3 RN4 RN5
Figure 4.
Top panel: High-resolution (2 . (cid:48)(cid:48) × . (cid:48)(cid:48) ) total averaged polarized image in the 1.26–3.60 GHz band (effectivefrequency of 2.0 GHz) zoomed on the northern relic ( σ Q , rms[1 . − . = 11 . σ U , rms[1 . − . = 11 . µ Jy beam − ).As for Fig. 3, this image is not corrected for the Ricean bias. Bottom panel: High-resolution (2 . (cid:48)(cid:48) × . (cid:48)(cid:48) ) Stokes I observationin the 1–2 GHz band (Di Gennaro et al. 2018) with the polarization electric field vectors at 2 . (cid:48)(cid:48) resolution, corrected for FaradayRotation, displayed in red; the length of the vectors is proportional to the intrinsic polarization fraction (scale in the bottomright corner). White and black arrows in the two panels indicate the points where the relic breaks into separate filaments,following Fig. 7 in Di Gennaro et al. (2018). this manuscript and Fig. 7 in Di Gennaro et al. 2018).Hints of polarized emission at 13 (cid:48)(cid:48) resolution are seenalso from the very faint relic northward of RN, i.e. R5,with high degree of polarization at both 3.0 and 1.5 GHz(i.e. about 35% and 30%).Particularly bright in polarization is also the relic lo-cated eastward of RN, i.e. R1 ( P . = 1 . ± . P . = 2 . ± . ∼ . . ∼ (cid:48)(cid:48) resolution. Here, the emission onlycomes from two out of the five “arms” that were detectedin Di Gennaro et al. (2018), i.e. only RS1 and RS2.This is not completely a surprise, as these two “arms”are also the brightest in total intensity (see Di Gennaroet al. 2018).No polarized emission is detected for the diffusesources R2 and I. Finally, we detect polarized emission olarization study of CIZA J2242.8+5301 Table 3.
Polarized ( P ν ) and total intensity ( I ν ) flux densities, and integrated polarization fraction ( p ν ) for the diffuse radiosources labelled in Fig. 3 at ν = 1 . . P (a)3 . I . p (b)3 . P a1 . I . p (b)1 . DP . . [ (cid:48)(cid:48) × (cid:48)(cid:48) ] [mJy] [mJy] [mJy] [mJy]RN 7 × . ± . . ± . .
37 19 . ± . . ± . .
18 0 . ×
13 1 . ± . . ± . .
22 2 . ± . . ± . .
20 0 . × . ± . . ± . .
28 2 . ± . . ± . .
19 0 . ×
13 – 3 . ± . . ± . × . ± .
05 3 . ± . .
23 0 . ± .
02 10 . ± . .
05 0 . × . ± .
04 1 . ± . .
47 1 . ± . . ± . .
46 0 . ×
13 0 . ± .
03 1 . ± . .
35 1 . ± . . ± . .
31 0 .
12I 13 ×
13 – 1 . ± . . ± . (a) Uncertainties are of the same order of those on the total intensity which are given by (cid:113) ( ζI λ ) + σ ,I N beam ( ζ =0 .
05 is the calibration uncertainty, σ rms ,I is the Stokes I noise map and N beam = A source /A beam is the number of beam inthe source where we measure the flux). (b) Uncertainties are dominated by the precision on the leakage calibration (0.5%,https://science.nrao.edu/facilities/vla/docs/manuals/obsguide/modes/pol). from the radio galaxies in and around the cluster (i.e. A,B, C, D, E, F, H, J, K1, M, N and O), whose degree ofpolarization at 1.5 and 3.0 GHz ranges between 1–10%,consistently with other similar objects (e.g. O’Sullivanet al. 2012).5.2.
Intrinsic fractional polarization, intrinsicpolarization angle, RM and depolarization maps
In Fig. 5 we show a comparison between the totalintensity and total averaged polarization maps of thenorthern relic at 7 (cid:48)(cid:48) resolution (panels (a) and (b), re-spectively), best-fit intrinsic and 1.5 GHz polarizationfractions ( p and p . , panels (c) and (d) respec-tively), Rotation Measure (RM, panel (e)) and exter-nal wavelength-dependent depolarization ( σ RM , panel(f)) maps. The polarization best-fit parameter mapsof the full cluster at 13 (cid:48)(cid:48) resolution is shown in Fig.6. These result from the QU -fitting approach for thecase of the External depolarization (Eq. 7) for eachpixel with averaged polarized emission above f × σ rms ,P .Here, σ rms ,P is obtained at the given resolution as theroot mean squared level of the averaged polarized emis-sion measured in a central, “empty” region of the clus-ter. We use f = 2 for the 2 . (cid:48)(cid:48) -tapered images with weighting=‘uniform’ and f = 3 for all the other res-olutions and weighting=‘Briggs’ . The correspondinguncertainty maps are displayed in Appendix B.The northern relic (RN) shows very high best-fit in-trinsic polarization fraction values at the outermostedge, with the eastern side up to 60% and the west-ern side up to 40% polarized. We also note a radialdecreasing of p towards the cluster center. The intrin-sic polarization angles approximately follow the shock normal, which is assumed to be perpendicular to theStokes I edge, supporting the scenario where the mag-netic field is aligned after the shock passage (see alsobottom panel in Fig. 4). The angles remain alignedalso in the downstream region. The Rotation Measurevalue is not constant along the relic, it spans east towest from RM ∼ −
150 rad m − to RM ∼ −
130 radm − , respectively, with median value of about −
141 radm − . Given the large distance from the cluster center(i.e. ∼ . ∼ − , have a dominant scale of ∼ (cid:48)(cid:48) − (cid:48)(cid:48) , andwe cannot distinguish, with the available data, whetherthis is due to fluctuations in our Galaxy or in the ICM(see Sect. 6.5). Similar east-west RM and p vari-ations were reported with Effelsberg observations at4.85 and 8.35 GHz (Kierdorf et al. 2017). To the con-trary, the RM value measured in the western side ofthe relic (RM ∼ −
130 rad m − ) differs from what hasbeen found by the Sardina Radio Telescope at 6.6 GHz(RM ∼ −
400 rad m − , Loi et al. 2017). No north-south best-fit intrinsic polarization gradient across therelic’s width was found by either Kierdorf et al. (2017)or Loi et al. (2017), although their observations sufferfrom much lower resolution (i.e., 90 (cid:48)(cid:48) and 2 . (cid:48) , respec-tively) which smoothed out any possible downstreamgradient. Interestingly, we measure RM values of about −
100 rad m − where the relic breaks in the RN1-RN2and RN3-RN4 filaments (see panel (e) in Fig. 5). Fi-nally, we do not find any particular east-west trend inthe σ RM behavior, with an overall value of σ RM ∼ − G. Di Gennaro et al. (a)(c)(e) (b)(d)(f)
RN1 RN2 RN3 RN4 RN5R5 R3
Figure 5.
Panels (a) and (b): 1–4 GHz Stokes I emission of the northern relic (Di Gennaro et al. 2018) and correspondent1.26–3.60 GHz averaged polarized emission (not corrected for the Ricean bias) at ∼ (cid:48)(cid:48) resolution. Panels (c), (d), (e), and(f): intrinsic polarization fraction, polarization fraction at 1.5 GHz, Rotation Measure and External Depolarization maps at 7 (cid:48)(cid:48) resolution. Black arrows in the plots are located at same physical coordinates, and indicate the points where the relic breaks intoseparate filaments (see also Fig. 4 in this manuscript and Fig. 7 in Di Gennaro et al. 2018). Uncertainty maps correspondingto panels (c) to (f) are displayed in Appendix B. rad m − (see panel (f) in Fig. 5). These values differfrom the high-frequency observations, as Kierdorf et al.(2017) did not measure any depolarization for the north-ern relic.The radio relic R4 is characterized by a very highbest-fit intrinsic polarization fraction ( ∼ ∼ p towardsthe cluster center. The RM values are rather constantacross R1 and R4, RM ∼ −
142 rad m − , consistentwith the one found for source N: since this radio galaxyis located outside of the cluster, its Rotation Measureis likely associated with the screen of our Galaxy ratherthan the ICM. Also, R1 and R4 have a very small val-ues of σ RM , again consistent with their spatial positionin the cluster, in a region of low ICM density.In the southern relic (RS), we measure a relativelylow best-fit intrinsic polarization fraction of ∼ − ∼ −
90 to ∼ −
80 rad m − . As for the northern relic,since RS is located in the cluster outskirts, we speculatethat most of its RM is due to the Galaxy. The discrep-ancy between RM RN and RM RS can be either due to ourGalaxy, whose RM variation is very uncertain (Sect. 4),or to a different combination of n e B (cid:107) along the line ofsight northward and southward the cluster ICM (see Eq.5).Finally, the polarized radio galaxies in the cluster fieldpresent different values of Rotation Measure. This possi-bly reflects the combination of their different position inthe ICM with the Galactic contribution, although theirintrinsic RM cannot be fully excluded. Among them,sources D and C are particularly interesting. Theyare located, in projection, in the cluster center and wemeasure a large difference in RM in the source’s lobes,with the northwestern being negative (i.e. ∼ −
600 and ∼ −
200 rad m − , for source D and C respectively) olarization study of CIZA J2242.8+5301 +52°56'+53°00'04'08' D e c li n a t i o n ( J ) Intrinsic Polarization Fraction
External Polarization Angle [rad]
Right Ascension (J2000) +52°56'+53°00'04'08' D e c li n a t i o n ( J )
150 145 140 135 130 125 120
Rotation Measure [rad m ] Right Ascension (J2000)
External Depolarization [rad m ] Figure 6.
From top left to bottom right: intrinsic polarization fraction ( p ), intrinsic angle ( χ ), Rotation Measure (RM)and depolarization ( σ RM ) maps of CIZAJ2242 at 13 (cid:48)(cid:48) resolution. Stokes I radio contours at the same resolution are drawn inblack at levels of 3 σ rms × (cid:112) [1 , , , , , . . . ], with σ rms = 6 . µ Jy beam − (Di Gennaro et al. 2018). Negative and positiveuncertainty maps are displayed in Appendix B. G. Di Gennaro et al. and the southeastern being positive (i.e. ∼ +300 and ∼ +250 rad m − , for source D and C respectively). Suchan extreme variation of RM in the lobes of the two radiogalaxies probably originates in the radio galaxies them-selves, although some effects might also be associatedwith the large amount of ICM traversed by the polar-ized emission. However, for these sources we find thata single-RM model does not properly fit the data, evenwithin a single resolution element (i.e. a single pixel,see Appendix A). We therefore suggest the presence ofa complex RM structure, as is observed also in otherradio galaxies (e.g. O’Sullivan et al. 2012). This studyis, however, beyond the scope of this paper. DISCUSSIONRadio relics are thought to trace merger-induced shockwaves which (re-)accelerate electrons and compress andamplify the cluster magnetic fields (e.g., Enßlin et al.1998). While several studies have been performedto investigate the mechanism to produce the highly-relativistic electrons in radio relics (e.g. Brunetti &Jones 2014; Fujita et al. 2015; Donnert et al. 2016; Kanget al. 2017), studies of their magnetic field propertieshave been challenging, mostly because depolarization ef-fects are stronger at low frequencies (i.e. (cid:46) ∼ −
40 kpc). Additionally, we investigate possible cor-relations between the polarization parameters and lookfor the presence of possible underlying trends amongthem by calculating the running median along the x -axis, with moving boxes of 20 windows. The uncertain-ties are calculated as σ ± / √ N , with σ + = y . − y . and σ − = y . − y . (with y . , y . and y . the16%, 50%, i.e. the median, and 84% of the distribution,respectively), and N the number of windows (Lameeet al. 2016). The existence of a correlation was then eval-uated by means of the Pearson coefficient, r p (Pearson1895), where we define | r p | ≤ . . < | r p | ≤ . | r p | > . r s , which assesses whether the re-lationship is monotonic (i.e. | r s | ≤ .
3: no/very weaklymonotonic ; 0 . < | r s | ≤ .
7: weakly/moderately mono-tonic ; | r s | > .
7: strongly monotonic).The following discussion is focused on the Sausagerelic. In Sect. 6.1 we present the radial profiles of thebest-fit polarization parameters; in Sect. 6.2 we discuss possible explanation for the profile found for the best-fit p ; in Sect. 6.3 we look at the contribution of the tur-bulent magnetic field in the post-shock region; in Sect.6.4 we investigate the limitation of the observing band-width coverage; finally, in Sect. 6.5 we look at the RMfluctuation in the relic.6.1. Polarization parameters radial profiles
We repeated the QU fit using Eq. 7 in beam-sizedboxes (i.e. 7 (cid:48)(cid:48) , resulting in a linear size of about 20 kpcat the cluster redshift, see legend in Figs. 7 and 8, andFig. C.1) covering the filament RN3, which we considerto be representative part of the relic (see Fig. 5). Foreach single radial annulus (i.e. same-colored markersin Figs. 7 and 8), the polarization parameters have asimilar trend along the filament (i.e. east to west, Fig.7), with the exception for the Rotation Measure whichshows a variation of about 30 rad m − . On the otherhand, a clear north-south trend is visible for the best-fitintrinsic polarization fraction. It drops about 35–40%,from an average value of (cid:104) p (cid:105) d =0kpc = 0 . ± .
04 at theshock position to (cid:104) p (cid:105) d =66kpc = 0 . ± .
06 in the inner-most downstream annulus (top panel in Fig. 7). Thesame trend is also observed for the polarization fractionat 1.5 GHz (Fig. 8). At this wavelength, the drop is evenlarger, about 60% (from (cid:104) p . (cid:105) d =0kpc = 0 . ± .
04 to (cid:104) p . (cid:105) d =66kpc = 0 . ± . (cid:104) σ RM (cid:105) d =0kpc = 10 . ± . (cid:104) σ RM (cid:105) d =66kpc = 13 . ± . − , bottom panel in Fig.7). Hints of these radial trends are also seen in the entirerelic (Fig. 9; see Appendix C for a view on the beam-sized boxes where we performed the QU fit). In thiscase, the radial information is obtained by looking at thespectral index, α . , since steeper values are locatedfurther in the downstream region where synchrotron andInverse Compton energy losses increase (e.g., Di Gen-naro et al. 2018). We calculated α . using the LO-FAR (150 MHz), GMRT (610 MHz) and VLA (1.5 and3.0 GHz) maps described in Hoang et al. (2017), vanWeeren et al. (2010) and Di Gennaro et al. (2018), re-spectively. We found Pearson and Spearman rank coeffi-cients of r p = − .
28 and r s = − .
28 for the p – α . distribution, and r p = 0 .
16 and r s = 0 .
24 for the σ RM – α . distribution. These measurements show, for thefirst time, that the northern relic in CIZAJ2242 suffersfrom both wavelength- and radial-dependent depolariza-tion.Finally, no clear downstream variations are seen forthe intrinsic polarization angle corrected for the shock olarization study of CIZA J2242.8+5301 p d shock = 0 kpcd shock = 22 kpcd shock = 44 kpcd shock = 66 kpc , c o rr [ r a d ] R M [ r a d m ] distance East-West [kpc] R M [ r a d m ] Figure 7.
From top to bottom: East-West profiles on theRN3 filament for the best-fit intrinsic polarization fraction( p ), intrinsic polarization angle corrected for the shock nor-mal ( χ , corr ), Rotation Measure (RM) and depolarization( σ RM ) using the External Faraday Rotation dispersion model(Eq. 7). Different colors represent different distances fromthe shock ( d shock , see legend), being the shock located at theoutermost edge of the relic, and the correspondent shadedareas show the uncertainties on the measurements. distance East-West [kpc] p . G H z d shock = 0 kpcd shock = 22 kpcd shock = 44 kpcd shock = 66 kpc Figure 8.
As the top panel in Fig. 7, but for the polarizationfraction at 1.5 GHz.
Table 4.
Pearson ( r p ) and Spearman ( r s ) rank correlationcoefficients of the running median in Figs. 9 and 10.Parameters r p r s p –RM − . − . p – σ RM − . − . p . –RM − . − . p . – σ RM − . − . p – α . − . − . σ RM – α . .
16 0 . normal in the plane of the sky ( χ , corr = χ − n , secondpanel in Fig. 7) and for the Rotation Measure (thirdpanel in Fig. 7; see also Sect. 6.5).6.2. On the downstream depolarization
In the following sections, we discuss two possible ex-planations for the observed radial profile of the polar-ization fraction. In particular, we investigate the role ofwavelength-dependent depolarization and Faraday Ro-tation (Sect. 6.2.1) and include a three-dimensionalmodelling of the relic (Sect. 6.2.2).6.2.1.
Wavelength-dependent depolarization and FaradayRotation effects
A naive explanation for the downstream depolariza-tion is the effect of a complex magneto-ionic layer thatmight differently rotate the polarization vectors in dif-ferent parts of the relic. According to this scenario, thebottom panel in Fig. 7 and the right panel in Fig. 9 Uncertainties on χ , corr are determined included the uncertain-ties on χ ( ∼ .
01 rad, from the fitting procedure using MCMC)and on n within the beam region ( ∼ .
02 rad at 7 (cid:48)(cid:48) resolution). G. Di Gennaro et al. p raw data pointsRunning median(± uncertainties) R M [ r a d m ] raw data pointsRunning median(± uncertainties) Figure 9.
Distributions of the intrinsic polarization fraction and external wavelength-dependent depolarization as a functionof the spectral index (grey circles in the left and right panel, respectively). The grey histograms show the projected distributionof the y - and x -axis quantities along each axis. The black solid line shows the running median of p and σ RM calculated using20 windows in the α . space, while the yellow area represents the correspondent uncertainties. both suggest a mild increasing contribution of the exter-nal wavelength-dependent depolarization in the down-stream region.We investigated the relation between the best-fit in-trinsic polarization fraction and the measured RotationMeasure and external wavelength-dependent depolariza-tion (left column in Fig. 10). In both cases, we do notsee particular trends, nor underlying fluctuations fromthe analysis of the running median. Both the Pearsonand Spearman rank coefficients confirm the visual in-spection, being r p = − .
06 and r s = − .
09 for the p –RM distribution and r p = − .
06 and r s = − . p – σ RM one (see Table 4). We therefore con-clude that our best-fit intrinsic polarization fraction isindependent from external factors, as the Faraday Rota-tion and the wavelength-dependent depolarization. Onthe other hand, an anti-correlation in the p . – σ RM distribution is observed ( r p = − .
73 and r s = − . p . –RM one( r p = − .
07 and r s = − . Relic three-dimensional shape
For a power law electron energy distribution withslope δ = 1 − α , i.e. dN ( E ) /dE ∝ E − δ , in a regionwith homogeneous magnetic field the intrinsic polarisa-tion amounts to (Rybicki & Lightman 1986): p = 3 δ + 33 δ + 7 . (11)Therefore, if the slope of the electron distribution variesacross the relic the intrinsic polarisation will also vary.According to the standard scenario for relic formation, electrons are (re-)accelerated at the shock front, witha power law energy distribution, and cool subsequentlydue to synchrotron and Inverse Compton energy losses.Locally, the resulting electron spectrum may show abreak, even if the sum of all these spectra is a powerlaw again, (see Di Gennaro et al. 2018, for a detailedspectral analysis of the relic). The locally curved spec-tra thus show a different intrinsic degree of polarizationthan the overall relic. From Eq. 11, the downstreamregion with the aged electron population would have ahigher intrinsic polarisation fraction (orange line in Fig.11).Although the decreasing radial profile of the best-fitpolarization degree seems to be in contrast with theabove description, the complex shape of the shock frontand the downstream region may impact the polarization,for instance by an inhomogeneous intrinsic polarisationfractions and by large differences in the path through themagnetized ICM from the emission to the observer. Inthis context, to reproduce a correct projected intrinsicpolarization profile, it is necessary to take into accounta realistic shape of the shock front, which has to includethe contribution of its inclination with respect to theline of sight (M. Hoeft et al. in prep.).Following Di Gennaro et al. (2018), we created a toymodel assuming that the shock front is a sphericallysymmetric cap in the plane determined by the line ofsight and the cluster center, with a curvature radius of1.5 Mpc and opening angle of 2 ψ = 36 ◦ (see also Fig.10 in Kierdorf et al. 2017). The alignment of electricfield vectors with the shock normal (bottom panel inFig. 4) implies that the magnetic field is dominantlytangled on scales smaller than the resolution of the ob- olarization study of CIZA J2242.8+5301
180 160 140 120 100
RM [rad m ] p raw data pointsRunning median(± uncertainties)
180 160 140 120 100
RM [rad m ] p . G H z raw data pointsRunning median(± uncertainties)
10 20 30 40 50 60 RM [rad m ] p raw data pointsRunning median(± uncertainties)
10 20 30 40 50 60 RM [rad m ] p . G H z raw data pointsRunning median(± uncertainties) Figure 10.
Distributions of the intrinsic and 1.5 GHz polarization fractions (left and right column respectively) as a functionof the absolute relative Rotation Measure and external wavelength-dependent depolarization (grey circles in the top and bottompanels, respectively). The grey histograms show the projected distribution of the y - and x -axis quantities along each axis. Forboth columns, the solid black line represents the running median of the y -axis variable (i.e. p and p . ) calculated using20 windows in the space of the x -axis variable (i.e. RM and σ RM ). The yellow shaded area represents the uncertainty on therunning median. servations (i.e. 2 . (cid:48)(cid:48) ). If the polarization angle reflectsthe structure of the magnetic field, we can assume ashock-compression scenario to explain the polarizationproperties of the relic (Enßlin et al. 1998). In this sce-nario, an upstream isotropically tangled magnetic fieldis compressed by the shock front resulting in a down-stream anisotropically tangled field, causing polarizedsynchrotron emission. In the specific case of RN, weadopt a shock Mach number of 3.7 which correspondsto an intrinsic polarization fraction of 58%, when theshock is observed perfectly edge on. This value matchesthe maximum p we estimated in the relic (see panel(c) in Fig. 5). The emission of different parts of theshock front is summed up, taking into account the an-gle between the shock normal and the line of sight,90 ◦ − ψ . The more this angle deviates from 90 ◦ the lowerthe intrinsic polarization becomes. Since those parts of the shock which deviate more from 90 ◦ are shifted fur-ther downstream with respect to the outermost edge ofthe relic, the intrinsic polarization fraction decreases to-wards the downstream. For our model parameters, thesetwo effects, namely the downstream increase in polar-ization due to the aging of the electrons population andthe decrease due to the shift of those parts of the shockwhich are not seen perfectly edge on, cancel out, result-ing in an almost constant theoretical p profile. This,however, still deviates from our observations (see blueline and black squares in Fig. 11).It is worth noting that we have used here a very sim-plified geometrical model that, for instance, does notexplain the east-west p variation we observed in therelic. Moreover, it does not include the effect of emittingregions at different Faraday depths in the relic down-stream. According to the spherical model described6 G. Di Gennaro et al. distance from the shock [kpc] p = 0°= 18° p obs,0 (best-fit) Figure 11.
Theoretical profiles of the intrinsic polariza-tion fraction in the post-shock region assuming a shock waveperfectly aligned with the plane of the sky (i.e. ψ = 0 ◦ ,orange line) and assuming an opening angle for the relic of ψ = 18 ◦ (Di Gennaro et al. 2018, blue line). Black squaresrepresent the best-fit intrinsic polarization fraction valuesobtained from a smaller sector of RN3 (i.e. where we couldassume constant polarization parameters in the east-west di-rection). above, at a distance of 60 kpc of the outer edge, the emis-sion from the “back side” of the cluster travels about 800kpc through the magnetised ICM, which causes addi-tional downstream depolarization. Interestingly, no ev-idence of multiple-RM components in the downstreamregion are observed in our data (see Appendix A). Thissuggests either that the relic cannot be described simplyby a smooth spherical cap (e.g. overlapping filamentarystructures) or we might be actually observing only thefront/back side of the radio relic. On the other hand, thegeometrical projections involve a number of adjustableparameters (see, e.g. Kang et al. 2012). Hence, a de-tailed modeling, which should include the shock shape,its downstream spectral and polarized characteristicsand its physical properties (such as the Mach numberdistribution, e.g. Ha et al. 2018; Botteon et al. 2020), iscomplicated and needs to be further examined.6.3. Turbulent magnetic field in the post-shock region
In the presence of both ordered and random magneticfield, Eq. 11 can be written as (Sokoloff et al. 1998;Govoni & Feretti 2004): p = 3 δ + 33 δ + 7 11 + (cid:18) B rand B ord (cid:19) , (12)where B ord represents the magnetic field componentthat is aligned with the shock surface and B rand rep-resents the isotropic magnetic field component. Thus, Right Ascension (J2000) +53°08' D e c li n a t i o n ( J ) source O Figure 12.
Subaru g - gi - i optical image of source O (Dawsonet al. 2015; Jee et al. 2015). 1–4 GHz total intensity radiocontours at 2 . (cid:48)(cid:48) resolution are overlaid at levels of 3 σ rms = (cid:112) (1 , , , . . . ), with σ rms = 5 . µ Jy beam − the map noise(Di Gennaro et al. 2018). the ratio B rand /B ord describes the order of isotropy ofthe magnetic field distribution.In the northern relic of CIZAJ2242, the polarizationangle seems to follow well the shock normal (see bot-tom panel in Fig. 4), and no change is observed in thedownstream region (second panel in Fig. 7). This sug-gests that the component of the magnetic field parallelto the polarization angle is approximately constant inthe downstream region. However, our measurements arelimited by the observing resolution, which can hide thepresence of tangled magnetic field on smaller scales andlead to a decreasing polarization fraction. If this is thecase, from Eq. 12, we can relate the radial decrease of p with the decrease of the degree of anisotropy in thedownstream region (i.e. the ratio B rand /B ord increases).Given the averaged values found in the RN3 filament, i.e. (cid:104) p (cid:105) d =0kpc ∼ .
49 and (cid:104) p (cid:105) d =66kpc ∼ .
28, and assuming δ = 3 (i.e. α = −
1) we find that the ratio B rand /B ord should increase of about 40% in the downstream region.Shock propagation in the ICM generates vorticity whichboosts turbulence and amplify the magnetic field (e.g.,Ryu et al. 2008). Behind the shock, turbulence behavesmore or less as a “decaying” turbulence, (see, e.g., Porteret al. 2015; Donnert et al. 2018), which might lead tothe decreasing degree of anisotropy. Further studies areneeded, however, upon this point. olarization study of CIZA J2242.8+5301 B turb is related tothe wavelength-dependent depolarization, according to(Sokoloff et al. 1998; Kierdorf et al. 2017): σ RM = 0 . (cid:114) (cid:104) n e (cid:105) B turb (cid:115) L Λ f , (13)where (cid:104) n e (cid:105) is the average electron density in cm − , f isthe volume filling factor of the Faraday-rotating gas, L is the path length through the thermal gas and Λ is theturbulence scale, both in pc unit. In the cluster area,only source O is a background polarized radio galaxy(see Fig. 12). From our QU fit, we found that theamount of the external depolarization for this source isvery similar to that in RN, i.e. σ RM ∼
22 rad m − (seepanel (f) in Fig. 5 and bottom left panel in Fig. 6).Given the proximity of source O and RN and assumingthat there is no contribution to the depolarization fromsource O itself and from the Galactic plane, we can usethis σ RM in Eq. 13 to obtain an approximate estimationof the tangled magnetic field in the northern relic, being B turb ∼ . µ Gauss. Here, we used (cid:104) n e (cid:105) = 10 − cm − (Ogrean et al. 2014), L = 350 kpc , f = 0 . ,i.e. the linear scale of our best resolution observation(i.e. 2 . (cid:48)(cid:48) ). Note that the estimated B turb is consistentwith the upper value of the total magnetic field strengthquoted by van Weeren et al. (2010), leading to a ratioof magnetic and the thermal pressures P mag /P th ∼ . Effect of the limited frequency-band coverage
The basic assumption of the QU -fitting approach isthat, given observations in a wide band ∆ λ = λ − λ and assuming a theoretical model, one can extrap-olate the intrinsic polarization parameters, p and χ ,at the ideal wavelength λ → λ and lower λ the betterone can validate the theoretical model. However, dueto the lack of high-resolution information at higher fre-quencies we cannot exclude the possibility of the exis-tence of a more complex model to describe the polar-ized emission in RN. For example, Ozawa et al. (2015)found a step-like fractional polarization profile in the The path length of the magnetized plasma crossed by the po-larized emission is L ≈ √ d s r s , where d s = 10 kpc and r s = 1 . This is about one order of magnitude smaller than what iscommonly used for galaxy clusters (i.e. 100 kpc, see Iapichino &Br¨uggen 2012). R M [ r a d m ] raw data pointsRunning median(± uncertainties) Figure 13.
Distributions of the absolute relative RotationMeasure as a function of the spectral index (grey circles).The grey histograms show the projected distribution of the y - and x -axis quantities along each axis. The black solid lineshows the running median of RM in the α . space using20 windows. The yellow area represents the uncertainties onthe running median. radio relic in Abell 2256, with the fractional polariza-tion increase occurring above 3.0 GHz. However, it isimportant to note that the presence of more complexmodels would result in a strong deviation from the Burnmodel in the downstream region, where a larger amountof magnetized plasma (i.e. the ICM) is crossed. Despitethe low S/N, however, we see that the Burn approxi-mation still holds in this region. Finally, ∆ λ also setsthe amount of wavelength-dependent depolarization de-tectable. Given our observing band, it would be ratherdifficult to determine p ( λ ) if σ RM ≥
100 rad m − .Interestingly, if we extract the profiles of the polar-ization parameters using an Internal Faraday RotationDispersion model (i.e. Eq. 8), we found consistent p , χ and RM profiles as those we found using the ExternalDepolarization model, and a larger amount of internaldepolarization ς RM , in agreement with the mathemat-ical differences of the two formulas. This means that,with the current data in hand, we cannot distinguishbetween an External or Internal depolarization modelfor the northern relic in CIZAJ2242. Lower-wavelengthwide-band observations (i.e. C- and X-band, 4–8 and8–12 GHz respectively) might then help to infer the na-ture of the polarized emission of the northern relic inCIZAJ2242.6.5. Investigation for intrinsic RM fluctuations
We found very weak/no correlations between RM andthe spectral index and between RM and the externalwavelength-dependent depolarization (Figs. 13 and 14,8
G. Di Gennaro et al.
180 160 140 120 100
RM [rad m ] R M [ r a d m ] raw data pointsRunning median(± uncertainties) Figure 14.
Distribution of the external wavelength-dependent depolarization as a function of the absolute rel-ative Rotation Measure (grey circles). The grey histogramsshow the projected distribution of the y - and x -axis quanti-ties along each axis. The black solid line shows the runningmedian of σ RM in the RM space calculated using 20 windows.The yellow area represents the uncertainties on the runningmedian. respectively). The absence of correlation in the lattercase is expected in case of external beam depolarization(Govoni & Feretti 2004).In Sect. 4, we show evidence for strong Rotation Mea-sure variation of the Galactic foreground, over angularscales of 3 (cid:48) − (cid:48) , by investigating the RM values in radiogalaxies outside the cluster. Along the northern relic,a variation of 30 rad m − around the median value of140.8 rad m − is also found on much smaller scales (i.e.15 (cid:48)(cid:48) − (cid:48)(cid:48) , see Fig. 5). At the cluster position ( l = 104 ◦ and b = − ◦ ), strong variation from the Galactic planeis expected (van Eck, priv. comm.), although detailedstudies are still missing. If the detected RM variation isentirely due to the Galactic plane, this would show forthe first time that Galactic RM variation is also presenton relatively small scales.Alternatively, this variation could be due to the ICM,and to the magnetic field close to the relic. As shownin Figs. 5 and 6, the strongest RM fluctuations aremeasured at the connection of two pairs of filaments, i.e.RN1–RN2 and RN3–RN4, where we measure on average∆RM ∼
30 rad m − (see panel (e) in Fig. 5). If thisis entirely due to the ICM, given the relation betweenRM and B (cid:107) (Eq. 5), we can constrain the magnetic fieldvariation in the relic, being ∆ B (cid:107) ∼ µ Gauss, where wehave used n e = 10 − cm − and L = 350 kpc. Assuminga global value of 5 µ Gauss (van Weeren et al. 2010),we obtain a magnetic field variation of roughly 20%. In
Table 5.
Pearson ( r p ) and Spearman ( r s ) rank correlationcoefficients of the running median in Figs. 13 and 14.Parameters r p r s RM– α . − . − . σ RM –RM 0 . − . case of weaker global magnetic field, i.e. 1.2 µ Gauss(van Weeren et al. 2010), variations increase up to 80%. CONCLUSIONSIn this work, we have presented a polarimetric studyof the merging galaxy cluster CIZA J2242.8+5301 ( z =0 . QU -fitting approach toobtain information on the polarization parameters, i.e.intrinsic polarization fraction ( p ), intrinsic polarizationangle ( χ ), Rotation Measure (RM) and depolarization( σ RM ), for the full cluster at 2 . (cid:48)(cid:48) , 4 . (cid:48)(cid:48) , 7 (cid:48)(cid:48) and 13 (cid:48)(cid:48) res-olution. This work mainly focused on the most promi-nent source in CIZA J2242.8+5301, i.e., the northernradio relic (RN). Below, we summarize the main resultsof our work: • CIZA J2242.8+5301 is bright in polarized light,with the emission coming from several sources,both diffuse and associated with radio galaxies.In particular, at the highest resolution available(i.e. 2 . (cid:48)(cid:48) ) the northern relic mimics the filamen-tary structure seen in total intensity emission (DiGennaro et al. 2018). • In agreement with previous studies (van Weerenet al. 2010; Kierdorf et al. 2017), we found a highdegree of intrinsic polarization in RN, with theeastern side having a higher value than the westernone (i.e. p , east ∼ .
55 and p , west ∼ .
35, with p the best-fit values from the QU -fit). • The polarization vectors strongly align with theshock surface also in high resolution observation(i.e. 2 . (cid:48)(cid:48) ), implying that the magnetic field isdominantly tangled on scales smaller than ∼ • For the first time we were able to investigate thepolarization parameters in the relic post-shock re-gion on ten-kpc scales. We found that both thebest-fit intrinsic and 1.5 GHz polarization frac-tions (i.e. p and p . ) decrease towards thecluster center. While, for the latter, a strongcontribution of the external wavelength-dependent olarization study of CIZA J2242.8+5301 p does not correlate with RMand σ RM . • We speculate that complex geometrical projec-tions and/or relic shape could possibly explain the p downstream depolarization, although detailedmodelings should be further worked. We also notethat the decrease of the degree of magnetic fieldanisotropies (i.e. B ord /B rand ) by about 40% mightexplain the depolarization. • We detect only one polarized background radiogalaxy, i.e. source O. Its σ RM is similar to theaverage value in the northern relic, and allows usto set an approximate value on the turbulent clus-ter magnetic field of about 5 . µ Gauss. • Different Rotation Measures are observed in thenorthern and southern relics (RM RN ∼ −
140 andRM RS ∼ −
80 rad m − , respectively). This couldbe either due to variation of the foreground Galac-tic Faraday Rotation or to a different contributionof n e B (cid:107) in the ICM along the line of sight. • Rotation Measure fluctuations of about 30 radm − on physical scales of about 3 (cid:48) − (cid:48) are observedat the location of the northern relic. With the cur-rent data in hand we cannot determine whetherthis is due to Galactic plane or to magnetic fieldlocal to the relic. In the former case, this will bethe first evidence of small-scale Galactic RM fluc-tuations. In the latter case, we estimate a mag-netic field variation of about 1 µ Gauss.Recently, the polarization properties of radio relicswere investigated by Wittor et al. (2019) and Roh et al.(2019) using numerical simulations. Although they were able to reproduce some properties of observed relics,such as the global observed degree of polarization, theyfound that it is difficult to explain the high degree polar-ization (up to ∼
60 %) and the uniformity of the intrin-sic polarization angle of the Sausage relic. Incorporatingrealistic modelings, as well as matching the spatial reso-lution for simulations and observations, would be crucialsteps for the understanding of the observed polarizationproperties of relics and the connection to the underlyingmagnetic field. ACKNOWLEDGMENTSWe thank the anonymous referee for useful commentswhich have improved the quality of the manuscript.GDG and RJvW acknowledge support from the ERCStarting Grant ClusterWeb 804208. HJAR acknowl-edge support from the ERC Advanced Investigator pro-gramme NewClusters 321271. RJvW acknowledges sup-port of the VIDI research programme with projectnumber 639.042.729, which is financed by the Nether-lands Organisation for Scientific Research (NWO). Par-tial support for LR comes from U.S. National ScienceFoundation grant AST 17-14205 to the University ofMinnesota. DR acknowledges support from the Na-tional Research Foundation of Korea through grants2016R1A5A1013277 and 2020R1A2C2102800. AS ac-knowledges support through a Clay Fellowship admin-istered by the Smithsonian Astrophysical Observatory.WF, CJ and RPK acknowledge support from the Smith-sonian Institution and the Chandra High ResolutionCamera Project through NASA contract NAS8-03060.This research made use of APLpy, an open-source plot-ting package for Python (Robitaille & Bressert 2012).APPENDIX A. QU -FIT PLOTSIn Fig. 1 we show an example of the QU -fitting results on a single pixel with high S/N at the shock location. In Fig.A.1, we show the same results but applied on a pixel in the relic downstream. Despite the lower S/N, a single-RMcomponent QU fit still provides a good match to our data. In Figs. A.2, we show the Faraday spectrum on these twopixels, obtained with pyrmsynth . The RM cube ranges from − − , with a FWHM of 60 rad m − .The two symmetric side-lobes we see next to each peak are likely due to interference in the Faraday spectra, as we donot use the RM-CLEAN option (see footnote 2 in Brentjens 2011). B. UNCERTAINTY MAPS ON THE POLARIZATION PARAMETERSIn this section, we show the p , RM and σ RM negative and positive uncertainty maps correspondent to Fig. 5(d), (e)and (f) (right and left column in B.1), and the p . uncertainty maps (Fig. B.2). We also present the polarizationparameter uncertainty (negative and positive) maps of the full cluster at 13 (cid:48)(cid:48) resolution (Figs. B.3 and B.4). The map0 G. Di Gennaro et al.
Figure A.1.
As Fig. 1 but for a pixel further in the RN downstream region.
RM [rad m ] R M S F [ a r b i t r a r y un i t s ] high S/N pixel on RN
490 140 210
RM [rad m ] R M S F [ a r b i t r a r y un i t s ] low S/N pixel on RN
560 160 240
RM [rad m ] R M S F [ a r b i t r a r y un i t s ] pixel source D Figure A.2.
Faraday spectrum on the pixels displayed in Figs. 1 (left panel) and A.1 (central panel). In the right panel, theFaraday spectrum of a high S/N pixel in source D is shown. The inset in the two plots shows the zoom on the Faraday peak. of the polarization fraction at 1.5 GHz and its correspondent uncertainty map of the full cluster at 7 (cid:48)(cid:48) resolution isdisplayed in Fig. B.5). C. ANNULI ON RN3 AND GRID USED FOR THE CORRELATION ANALYSISHere, we display the regions where we performed the QU -fit. The boxes shown in Fig. C.1 generate the profiles inFigures 7 and 8. The boxes shown in Fig. C.2 generate Figures 9, 10, 13 and 14. Each box has the same size of therestoring beam, i.e. 7 (cid:48)(cid:48) × (cid:48)(cid:48) (about 22 ×
22 kpc at the cluster redshift). The polarized flux in each box is above athreshold of 3 σ rms ,P (see Sect. 3). REFERENCES Akamatsu, H., van Weeren, R. J., Ogrean, G. A., et al.2015, A&A, 582, A87Anderson, C. S., Gaensler, B. M., & Feain, I. J. 2016, ApJ,825, 59Basu, K., Vazza, F., Erler, J., & Sommer, M. 2016, A&A,591, A142Bicknell, G. V., Cameron, R. A., & Gingold, R. A. 1990,ApJ, 357, 373 Bonafede, A., Feretti, L., Murgia, M., et al. 2010a, A&A,513, A30—. 2010b, arXiv e-prints, arXiv:1009.1233Bonafede, A., Vazza, F., Br¨uggen, M., et al. 2013, MNRAS,433, 3208Botteon, A., Brunetti, G., Ryu, D., & Roh, S. 2020, A&A,634, A64Brentjens, M. A. 2011, A&A, 526, A9Brentjens, M. A., & de Bruyn, A. G. 2005, A&A, 441, 1217 olarization study of CIZA J2242.8+5301 +53°07'08'09'10'11' D e c li n a t i o n ( J ) Positive Uncertainties p Negative Uncertainties p +53°07'08'09'10'11' D e c li n a t i o n ( J ) R M [ r a d m ] R M [ r a d m ] Right Ascension (J2000) +53°07'08'09'10'11' D e c li n a t i o n ( J ) R M [ r a d m ] Right Ascension (J2000) R M [ r a d m ] Figure B.1.
Positive (left column) and negative (right column) uncertainty maps corresponding to panels (c), (e) and (f) inFig. 5.
Figure B.2. G. Di Gennaro et al. +52°56'+53°00'04'08' D e c li n a t i o n ( J ) Intrinsic Polarization Fraction Neg Uncertainties
Intrinsic Polarization Angle Neg Uncertainties [rad]
Right Ascension (J2000) +52°56'+53°00'04'08' D e c li n a t i o n ( J ) Rotation Measure Neg Uncertainties [rad m ] Right Ascension (J2000)
External Depolarization Neg Uncertainties [rad m ] Figure B.3.
The negative uncertainty maps corresponding to Fig. 6. olarization study of CIZA J2242.8+5301 +52°56'+53°00'04'08' D e c li n a t i o n ( J ) Intrinsic Polarization Fraction Pos Uncertainties
Intrinsic Polarization Angle Pos Uncertainties [rad]
Right Ascension (J2000) +52°56'+53°00'04'08' D e c li n a t i o n ( J ) Rotation Measure Pos Uncertainties [rad m ] Right Ascension (J2000)
External Depolarization Pos Uncertainties [rad m ] Figure B.4.
The positive (bottom panel) uncertainty maps corresponding to Fig. 6. G. Di Gennaro et al.
Right Ascension (J2000) +52°56'+53°00'04'08' D e c li n a t i o n ( J ) p Right Ascension (J2000)
Uncertainties
Figure B.5.
Polarization fraction map at 1.5 GHz (left panel) and correspondent error map (right panel) of CIZAJ2242 at 7 (cid:48)(cid:48) resolution. Stokes I radio contours at the same resolution are drawn in black at level of 3 σ rms √ , , , , . . . , with σ rms = 4 . µ Jybeam − (Di Gennaro et al. 2018). Figure C.1.
Total averaged polarization image at 7 (cid:48)(cid:48) resolution of the northern relic with the boxes used to investigate thepresence correlation among the polarization parameters in Figs. 7 and 8. The position of the shock (i.e. d shock = 0 kpc) isdisplayed by the black dashed line.Donnert, J. M. F., Stroe, A., Brunetti, G., Hoang, D., &Roettgering, H. 2016, MNRAS, 462, 2014Enßlin, T. A., Biermann, P. L., Klein, U., & Kohle, S. 1998,A&A, 332, 395Farnsworth, D., Rudnick, L., & Brown, S. 2011, AJ, 141,191 Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman,J. 2013, PASP, 125, 306Frick, P., Sokoloff, D., Stepanov, R., & Beck, R. 2011,MNRAS, 414, 2540Fujita, Y., Takizawa, M., Yamazaki, R., Akamatsu, H., &Ohno, H. 2015, ApJ, 815, 116 olarization study of CIZA J2242.8+5301 Right Ascension (J2000) +53°08'09'10' D e c li n a t i o n ( J ) Figure C.2.