What can we really learn about Magnetic Fields in Galaxy Clusters from Faraday Rotation observations?
aa r X i v : . [ a s t r o - ph . H E ] S e p Draft version June 7, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
WHAT CAN WE REALLY LEARN ABOUT MAGNETIC FIELDS IN GALAXY CLUSTERS FROM FARADAYROTATION OBSERVATIONS?
Gilad Rave , Doron Kushnir , and Eli Waxman Draft version June 7, 2018
ABSTRACTWe construct a simple and robust approach for deriving constraints on magnetic fields in galaxyclusters from rotation measure (RM) maps. Relaxing the commonly used assumptions of a correlationbetween the magnetic field strength and the plasma density and of a power-law (in wave number)magnetic field power spectrum, and using an efficient numerical analysis method, we test the consis-tency of a wide range of magnetic field models with RM maps of 11 extended sources in 5 clusters,for which the data were made available to us. We show that the data reveal no indication for a radialdependence of the average magnetic field strength, and in particular no indication for a correlationbetween the gas density and the field strength. The RM maps of a considerable fraction of the sourceseither require or are consistent with the presence of a spatially uniform magnetic field of a relativelysmall strength, 0 . . µ G, which contributes significantly to the RM. The RM maps of all but onesource do not require a power-law magnetic field power spectrum, and most are consistent with apower spectrum dominated by a single wave length. The uncertainties in the magnetic field strengths(and spatial correlation lengths) derived from RM maps exceed an order of magnitude (and oftenmore). These uncertainties imply, in particular, that there is no indication in current RM data fora systematic difference between the magnetic field strengths in radio-halo clusters and in radio-quietclusters. With the improvement expected in the near future of the quality and quantity of RM data,our analysis method will enable one to derive more accurate constraints on magnetic fields in galaxyclusters.
Subject headings: extragalactic magnetic fields, galaxy clusters INTRODUCTIONThe strength and spatial structure of magnetic fieldsin the intra-cluster medium (ICM) of galaxy clustershave been estimated using a wide range of methods.These methods include analyses of the properties of ra-dio relic and halo sources, of cluster non-thermal X-rayemission, of cluster cold fronts (appearing in X-ray im-ages), and of Faraday rotation measure maps of polar-ized radio sources (for reviews, see Carilli & Taylor 2002;Govoni & Feretti 2004). The results of these analyses im-ply that the ICM of all galaxy clusters are permeated by ∼ µ G magnetic fields. Such fields play a critical rolein determining the energy balance in the ICM plasmathrough their effects on heat conduction and cosmic raypropagation, with implications to many cluster phenom-ena (for reviews, see Feretti et al. 2012; Ferrari et al.2008).In this paper we discuss the constraints that are im-plied by Faraday rotation measure (RM) data on theproperties of the ICM magnetic fields. The Faraday ef-fect causes a rotation of the plane of polarization of anelectromagnetic wave, which is linearly proportional tothe component of the magnetic field in the direction ofpropagation, chosen henceforth as z . The RM is definedasRM ≡ d ψ d λ = RM Z (cid:18) B z µ G (cid:19) (cid:16) n e − cm − (cid:17) (cid:18) d z (cid:19) , (1) Department of Particle Physics and Astrophysics, WeizmannInstitute of Science, Rehovot 76100, Israel Institute for Advanced Study, Einstein Drive, Princeton,New Jersey, 08540, USA where ψ is the angle of rotation, λ is the (radio) wave-length, RM ∼ = 0 .
812 rad m − , n e is the (free) electronnumber density, and the integral is along the line of sightto the source (Burn 1966). Cluster RM maps are ob-tained by fitting the wavelength dependence of the po-larization angles measured along each line of sight to the λ law of Eq. (1).The two dimensional RM maps do not allow one touniquely determine the three dimensional magnetic fieldstructure. Moreover, RM maps are available over onlya small fraction of the angular extent of the clusters.Thus, RM constraints on the ICM magnetic fields aretypically derived under simplifying assumptions regrad-ing the magnetic field structure. Commonly used as-sumptions are that the magnetic field strength is corre-lated with the plasma density, and that the power spec-trum of the spatial distribution of the magnetic field isa power-law in wave numbers. Furthermore, the rangeof model parameters (e.g. the range of wave numbersor the range of power-law indices) explored is limited inmost derivations of magnetic field constraints, either byassumption or by some qualitative analysis of the twodimensional RM maps, in order to reduce the requiredcomputational resources.The analysis presented here of the constraints imposedon the ICM magnetic fields improves on earlier work inseveral ways. First, we relax the commonly used as-sumptions of a correlation between the magnetic fieldstrength and the plasma density, and of a power-law (inwave number) power spectrum. Indeed, we find that theRM maps show no evidence for a correlation betweenthe field strength and the plasma density (as previouslyshown, e.g., by Eilek & Owen 2002), and that a power-law power spectrum is generally not require to accountfor the observations. Second, we use a more efficient nu-merical analysis method, that allows us to explore a widerange of model parameters. Finally, we apply a uniformanalysis to sources observed in several clusters, which al-lows us to draw conclusions applicable to the populationof clusters (rather than to a single cluster or source).RM maps have been derived and analyzed for radiosources in 16 nearby galaxy clusters based on observa-tions of more than 40 radio sources (see Table 1). TheRM data available and earlier analyses of this data aredescribed in §
2. Our analysis method is described in § §
4. We have applied ouranalysis method to all the data that were made availableto us, 11 sources within 5 clusters, by F. Govoni, A.Bonafede, V. Vacca, and their collaborators. We thankthem for making the data available, and note that ouranalysis method may be straightforwardly applied to allexisting data. Our results are summarized and their im-plications are discussed in §
5. In particular, we addressthe issue of magnetic field strength bi-modality betweenmerging and non-merging galaxy clusters.In our analysis we assume that the observed RM areentirely due to the ICM magnetic fields and not intrinsicto the sources (e.g. Pizzo et al. 2011), and that the con-tribution of the magnetic field of our own Galaxy to theRM is smaller than the fit errors (e.g. Taylor & Perley1993). We adopt a ΛCDM cosmology with H =72 km s − Mpc − , Ω m = 0 .
27, and Ω Λ = 0 . EXISTING DATA & EARLIER ANALYSES2.1.
Existing RM maps
RM maps have been derived and analyzed for radiosources in 16 nearby galaxy clusters (Table 1). De-tailed RM maps were constructed for more than 40 radiosources (Table 2) at different impact parameters (angulardistance between the line of sight and the cluster X-raycenter), lying within or behind the ICM. As can be seenin Figure 1, the radio sources observed in these studiesare distributed over a wide range of distances from theirclusters’ centers, with roughly half of the sources within2 r c ( r c as defined by Eq. (5)).The sources included in our study are indicated in Ta-ble 2. More than half (7/11) of our analyzed sources arewithin Coma, a prominent radio halo (RH) cluster, and4 of these sources are coincident with the RH. We alsohave one source in each of the clusters A2065, A2142,A2199 and Ophiuchus, none of which are hosting a RH.2.2. Earlier analyses
For five of the galaxy clusters listed in Table 1(Coma, A2199, Hydra A, A2255, and A2382), de-tailed models of the ICM magnetic fields havebeen constructed and constrained by the observedRM maps (Taylor & Perley 1993; Govoni et al. 2006;Guidetti et al. 2008; Bonafede et al. 2010; Vacca et al.2012). For the rest of the sources only a limited anal-ysis was carried out (Bonafede et al. 2011; Govoni et al.2010). We briefly summarize below the main results ofboth types of studies.2.2.1.
Clusters for which detailed analyses were carried out
For Coma, Bonafede et al. (2010) assumed that thespatial distribution of the magnetic field is given by arealization of a Gaussian random field with a Fourierpower spectrum that is a power-law of the wave num-bers. They have assumed a Kolmogorov spectral index, n = 11 /
3, and by comparing various statistical propertiesof the observed RM images and those simulated, the spa-tial wavelength range was constrained to Λ min = 2 kpcand Λ max = 34 kpc.In order to generate a correlation of the field strengthwith the plasma density, the (real space) magnetic fieldobtained in the realizations was multiplied by an r -dependent factor to generate h B ( r ) i = B [ n e ( r ) /n ] η , (2)where the average is over all realizations. The RM mapsderived from the magnetic field realizations were com-pared to the observed RM maps, to constrain the centralmagnetic field strength B to be in the range 3 . . µ G,and the correlation index η to be in the range 0 . . σ c.l.).A similar analysis has been carried out for A2199 byVacca et al. (2012). In their analysis, Λ max = 35 kpc,Λ min = 0 . n = 2 . max = 35 ±
28 kpc, Λ min = 0 . ± . n =2 . ± . B = 11 . ± . µ G and η = 0 . ± . σ c.l.).Besides Coma and Abell 2199, the ICM mag-netic fields of only three other clusters – Hydra A(Taylor & Perley 1993), A2255 (Govoni et al. 2006), andA2382 (Guidetti et al. 2008) – have been investigatedwith a similar approach, i.e. by simulating random mag-netic field models and constraining their properties bycomparison to the RM data.For Hydra A, several models were explored for the ICMfields. Taylor & Perley (1993) find ∼ µ G to ∼ µ Gmagnetic field strength assuming spatial variability on ∼ ∼ . µ G.They try several models with a few different values forthe power-spectrum power-law index, keeping the fieldstrength at the cluster center constant at 2 − . µ G, andconclude that it is necessary to use a (spatially) variablepower index (in the range n = 2 − B = 1–13 µ G (1 σ confidence region) for models with different correlationindex between the average magnetic field strength andthe gas density. Note that all their models produce sim-ilar magnetic field strengths when averaged over a largevolume (1 Mpc ).2.2.2. Clusters with limited analyses
Comparing the standard deviation of each RM source, σ RM (averaging the observed values over all valid pixelsin each source), to the distance from the cluster center,not normalized to account for the different cluster sizes(e.g. to r c , the core radius of the β -model), and assuming TABLE 1Galaxy Clusters used in RM studies (1)Cluster (2) z (3)Mrg (4)Radio (5)CC (6) M [10 M ⊙ ] (7) β (8) r c [kpc] (9) n [10 − cm − ]3C 129 0 . · · · · · · · · · . +9 . − . . +0 . − . +125 − . +0 . − . A0119 0 . X · · · · · · . +1 . − . . +0 . − . +19 − . +0 . − . A0400 0 . X · · · · · · . +0 . − . . +0 . − . +6 − . +0 . − . A0401 0 . X G · · · . +1 . − . . +0 . − . +8 − . +0 . − . A0514 † . X · · · · · · . +1 . − . . +0 . − . +144 − . +0 . − . Hydra A 0 . · · · · · · W 5 . +0 . − . . +0 . − . +1 − . +4 . − . Coma 0 . X G+R · · · . +2 . − . . +0 . − . +15 − . +0 . − . A2065 0 . X · · · W 23 . +29 . − . . +0 . − . +260 − . +0 . − . A2142 0 . · · · M W 21 . +5 . − . . +0 . − . +4 − . +0 . − . A2199 0 . · · · · · · S 6 . +0 . − . . +0 . − . +7 − . +0 . − . A2255 0 . X G+R · · · . +2 . − . . +0 . − . +25 − . +0 . − . A2382 ‡ . · · · · · · · · · . +1 . − . . +0 . − . +250 − . +0 . − . A2634 0 . · · · · · · W 5 . +1 . − . . +0 . − . +30 − . +0 . − . Centaurus 0 . · · · · · · S 3 . +0 . − . . +0 . − . +3 − . +1 . − . Ophiuchus 0 . X M · · · . +4 . − . . +0 . − . +16 − . +0 . − . † Mass ref.: Girardi et al. 1998, β -model ref.: Govoni et al. 2001. ‡ Mass ref.: Demarco et al. 2003, β -model ref.: Guidetti et al. 2008. Tab . 1.— Galaxy Clusters used in RM studies, including some relevant cluster properties. (1) Cluster name; (2) Redshift (
Nasa/ipacExtragalactic Database ); (3) Merger activity (Bonafede et al. 2011, and references therein); (4) Diffuse radio emission: G = gianthalo, M = mini halo, R = relic (Feretti et al. 2012); (5) Cool-Core: W = weak, S = strong (Hudson et al. 2010); (6) Virial mass(Reiprich & B¨ohringer 2002, unless otherwise noted); (7) (8) (9) β -model (Eq. 5) parameters with 1- σ fit errors adapted according to ouradopted cosmology (Chen et al. 2007, unless otherwise noted). −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8−4−3−2−101234 Right Ascension [ r/r c ] D ec li n a t i o n [ r / r c ]
3C 129A0119A0400A0401A0514Hydra AComaA2065A2142A2199A2255A2382A2634CentaurusOphiuchus
Fig. 1.—
Distribution of RM sources. Each marker represents the mean position of an extended RM source, relative to its host galaxycluster (in the sense that the radio emission passes through its ICM) X-ray center. Note that a few of the central sources cover a relativelylarge area of their respective cores. a ”Gaussian cell model ”, Govoni et al. (2010) find thesegalaxy clusters are consistent with B ∼ µ G. METHOD an analytical formulation based on the approximation that themagnetic field strength is constant throughout the cluster and thatits direction varies over a single length scale We assume that the cluster’s magnetic field may bedescribed as a realization of a random process, the sta-tistical properties of which we seek to constrain usingthe measured RM map. We consider a statistical modelacceptable if the power spectrum of the observed (twodimensional) RM map is consistent with being drawnfrom the distribution of power spectra predicted by themodel. To determine whether or not this is the case, we
TABLE 2Radio sources analyzed in previous work (1)Cluster (2)Source (3)Morph. (4)FR (5)Class (6) r ⊥ /r c (7)LOS (8)Ref.3C 129 3C 129 HT · · · galaxy 1.73 0 113C 129.1 · · · I galaxy 0.01 0 11A0119 0053-015 NAT · · ·
E 0.15 0 30053-016 NAT · · · galaxy 0.90 0 33C 029 · · ·
I S 2.83 0 3A0400 3C 075 T · · ·
E0 0.24 0 2A0401 A401A NAT · · · galaxy 2.93 0 6A401B HT · · · galaxy 4.20 0 6A0514 A514A · · · II · · · ∞ · · · D 1.48 0 4A514C · · · II · · · ∞ · · · · · · QSO 5.32 ∞ · · · · · · galaxy 20.0 0 10 Coma
5C 04.042 · · ·
II galaxy 5.77 ∞
15C 04.074 ‡ · · · II · · · ∞
15C 04.081 † NAT I cD 0.87 0 15C 04.085 † WAT I-II cD 0.40 0 15C 04.114 ‡ · · · I · · · ∞
15C 04.127 · · · · · ·
QSO 3.66 ∞
15C 04.152 · · ·
II galaxy 6.68 ∞ A2065
A2065A · · ·
II galaxy 1.96 ∞ A2142
A2142A HT I galaxy 2.66 0 6
A2199
3C 338 · · ·
I cD2 0.05 ∞ ‡ HT · · · galaxy 2.98 · · · † NAT · · · galaxy 1.72 · · · † NAT · · · galaxy 1.03 · · · · · · galaxy 11.6 · · · · · · · · · · · · galaxy 4.33 · · · † · · · · · · filaments · · · † · · · · · · · · · · · · · · · · · · · · · · · · ‡ · · · · · · · · · · · · · · · · · · · · · · · · · · · I E 0.44 0 7PKS 2149-158 B · · ·
I E 0.41 0 7PKS 2149-158 C · · ·
I E 0.15 0 7A2634 3C 465 T I cD 0.39 0 2Centaurus PKS 1246-410 · · ·
I cD1 0.34 0 12
Ophiuchus
OPHIB HT I galaxy 2.50 0 6 † The radio source is coincident with the cluster RH. ‡ The radio source might be coincident with the cluster RH (edge of diffuse emission).
Tab . 2.— The radio sources for which RM maps have been derived and analyzed in previous works, along with some of their relevantcharacteristics. In this work we analyzed the sources in Coma, A2065, A2142, A2199, and Ophiuchus (italicized). (1) Host GC name;(2) Source name; (3) Radio morphology: (H)T = (Head) Tail, (N/W)AT = (Narrow/Wide) Angle Tail; (4) Fanaroff-Riley classification;(5) Source object type; (6) Impact parameter relative to X-ray center; (7) Line of sight position: ∞ = background source, 0 = clustermember; (8) RM reference: 1. Bonafede et al. 2010; 2. Eilek & Owen 2002; 3. Feretti et al. 1999; 4. Govoni et al. 2001; 5. Govoni et al.2006; 6. Govoni et al. 2010; 7. Guidetti et al. 2008; 8. Perley & Taylor 1991; 9. Pizzo et al. 2011; 10. Taylor & Perley 1993; 11. Taylor et al.2001; 12. Taylor et al. 2002; 13. Vacca et al. 2010; 14. Vacca et al. 2012. compare the distribution of some measure of the differ-ences between the RM power spectra of different real-izations of the model with the distribution of the samemeasure of differences between the model’s power spec-tra and the observed power spectrum. We define the RMpower spectrum asRM( k ) = D | RM( k x , k y ) | E | ( k x ,k y ) | = k , (3)where ( k x , k y ) is the 2D Fourier k -vector correspondingto the projected distance on the plane of the sky, andthe average on the rhs is over all vectors with magnitude k . For our analysis we use two measures of the distancebetween two power spectra, RM ( k ) and RM ( k ): thesum and the maximum of the absolute value of the log- arithm of the ratio of powers in the two power spectraat all wave numbers, i.e. P k | log(RM ( k ) / RM ( k )) | andmax k | log(RM ( k ) / RM ( k )) | , respectively. For each sta-tistical model we generated ∼
500 realizations and com-pared the resulting distance measure distributions usingthe Kolmogorov-Smirnov (KS) test. Similar results areobtained for the two distance measures used.We write the magnetic field as a sum of Fourier com-ponents, P k ˜ B ( k ) exp( i k · x ) with˜ B ( k ) = B k e iβ ˆΩ h ˆ k cos (cid:0) α ˆΩ (cid:1) + ˆ k sin (cid:0) α ˆΩ (cid:1)i , (4)where ˆ k and ˆ k are two unit vectors orthogonal to ˆ k (chosen to align along the z -direction of Cartesian coor-dinates) and to each other (ensuring ∇ · B = 0), and B k is real. Since the magnetic field is assumed to bestatistically homogeneous and isotropic, which choosethe phases α ˆΩ and β ˆΩ to be uniformly distributed in[ − π, + π ]. The k -space resolution is chosen as ∼ π/r c .We consider two types of power spectra for the am-plitude B k : a single characteristic wave length Λ, with B k = 0 for k = 2 π/ Λ, and a power-law spectrum withindex n , with k B k ∝ k − n for k in some fixed range[ k min , k max ] (corresponding to Λ min = 1 kpc , Λ max =32 kpc) and zero elsewhere. In both cases, we allow thepresence of a uniform ( k = 0) component (as proposedby Taylor & Perley 1993), with amplitude B hom . Theaverage magnetic field strength, B ≡ (3 (cid:10) B z (cid:11) ) / , is as-sumed to be constant throughout the cluster (i.e., withno radial dependance). As described in §
4, the simplemodels we consider are sufficient to account for the vastmajority of observations, and hence more general modelsare not motivated by the data.For any realization of the magnetic field model, theFaraday RM map is obtained using Eq. (1) and a β -model density profile, n e = n (cid:0) r /r c (cid:1) − β , (5)from the literature (as listed in Table 1). We treat clus-ter members as situated precisely halfway through theICM, and other sources as lying at infinity. For thesetwo cases, the integral of Eq. (1) may be solved analyt-ically, as shown in Appendix A. Note also that due tothe linear nature of Eq. 1, it is possible to explore asmany spectral index values as desired, without having tosimulate the magnetic field more than once at each wavenumber k .In order to account for the finite angular resolution ofthe observations, RM values at each pixel are obtainedby properly averaging the values obtained in a band ofsurrounding pixels corresponding to the reported angu-lar beam size. Our results are not sensitive to ∼ ∼ RESULTSIn this section we present the results of our analysis,which was described in §
3. For every source we sim-ulated five types of magnetic field models: single-scale,power-law, both with and without a homogeneous com-ponent, and a simple homogeneous field. Figures 2 and 3show the RM power spectra of 5C4.114 and A2199A, andcompare them to those obtained for two types of mod-els: a homogeneous field model for 5C4.114 and a singlescale model for A2199A. The results are summarized inTable 3, where 95% c.l. ranges are shown for acceptedmodel parameters (as is the convention also in what fol-lows). For some of the sources, most notably 5C4.114 andA2142A, the RM power-spectra resemble closely thoseobtained for a homogeneous magnetic field, reflecting thefact that the power spectrum is determined by the shapeof the sources. Our realizations show that a homoge-neous sub µ G magnetic field is sufficient in order to gen-erate their observed RM power-spectra: 0 . . µ Gfor 5C4.114 and 0 . . µ G for A2142A.Figure 4 shows the minimal magnitude of the mag-netic field implied by observations of each source, whichis the minimal magnitude of a homogeneous field requiredto generate the maximal (absolute) RM value of eachsource. Using Eqs. (1) and (5), this minimal field is B ≥ c LOS √ x,y | RM( x, y ) | (cid:16) x + y r c (cid:17) β − Γ (cid:0) β (cid:1) RM n r c √ π Γ (cid:0) β − (cid:1) , (6)where c LOS = 1 for a background source or c LOS = 2 for acluster member, and ( x, y ) correspond to right ascensionand declination relative to the cluster center. The lowerlimits are are ∼
30% higher than the acceptable ranges ofvalues inferred above for 5C4.114 and A2142A. The smalloffset supports the validity of our simulations’ results.It is due to the fact that the lower limits are derivedbased on the RM of a single pixel, in contrast with the(weighted) averaging over many pixels inherent to ourstatistical analysis.For most sources, a single-scale model is consistentwith the data. All accepted ranges of field strengths ofour single-scale models, with and without a homogeneouscomponent, are plotted in Figure 5 as a function of theprojected distance from the cluster center. We also indi-cate B CMB ≡ (cid:0) πaT (cid:1) / ≃ µ G (7)as a dashed gray line, where relevant, and notice thataccepted field strengths are distributed around this line.
TABLE 3Simulation results – accepted models (1)Source (2)Comp. (3) B hom [ µ G] (4) B [ µ G] (5)Λ[kpc]5C4.042 U+P 0 . .
025 3 . . † ≤ .
57 0 . . ≤ .
28 0 . . ≤ .
46 0 . . . ≤ . ≤ . ≤ .
41 1 . . ≤ . . ≤ . . . ≤ .
23 0 . . ≤ .
19 0 . . † Acceptable power-law indices for the source 5C4.042 rangefrom n = 0 to n = 2 . Tab . 3.— (1) Source name; (2) Acceptable models (95% c.l.):U=uniform, S=single-scale, P=power-law, ’/’=or, ’+’=and. Notethat a uniform component may always be added to a varyingone; (3) Uniform component magnetic field strength; (4) Averagemagnitude of the spatially varying magnetic field component;(5) Allowed spatial variation scales of the single-scale model.
Λ [kpc] | R M ( k ) | [ k p c r a d / m ]
20 40 60 80 100 120 140 160Data 50 100 150 200 250 300 350 00.060.120.180.24 sum k | log(RM model ( k ) / RM ∗ ( k )) | P D F C D F model−modelmodel−data Fig. 2.—
A homogeneous field model for 5C4.114. Left panel: a histogram of realization power-spectra, for a homogeneous field modelwith B hom = 26 . p KS = 0 . Λ [kpc] | R M ( k ) | [ k p c r a d / m ] sum k | log(RM model ( k ) / RM ∗ ( k )) | P D F C D F model−modelmodel−data Fig. 3.—
Same as Figure 2, but for the single-scale field model with Λ = 16 kpc and B = ( (cid:10) B z (cid:11) ) / = 0 . µ G for A2199A (withouta homogeneous component). The maximized KS test statistic is p KS = 0 . All the radio sources analyzed in this work, exceptA2065A (the only source for which no model was ac-cepted and may thus require a more complicated model),are consistent with some power-law magnetic field in therange of scales of Λ min = 1 kpc to Λ max = 32 kpc, chosento accommodate scales down to the angular resolutionand up to the typical source size. In Figure 6 we showthe ranges of acceptable magnetic field strengths, for ourpower-spectrum models, along with the analytical lowerbounds found using Eq. (6). Again, there is no apparentradial trend of magnetic field strength, and the valuesare scattered around the B CMB line.We note that although there is a correlation betweenthe spectral index of the (3D) magnetic field power-spectrum and the resulting shape of the (2D) RM power-spectrum (e.g. the average RM power spectrum of asingle-scale model peaks at the same scale as that of themagnetic field power spectrum), the spectral index of themagnetic field power spectrum is not given by a power-law fit to the RM power spectrum. DISCUSSIONWe presented a simple and robust approach for de-riving constraints on magnetic fields in galaxy clustersfrom rotation measure (RM) maps ( § −2 −1 Impact Parameter [ r/r c ] B c o n s t [ µ G ] A2065A2142ComaOphiuchusA2199
Fig. 4.—
The lower-bounds on the magnetic field, as derivedusing Eq. (6) (arrows), and the ranges of magnetic field strengthsin the homogeneous field models accounting for the A2142A and5C4.114 RM data (95% c.l. error bars). The small offset betweenthe bounds and allowed ranges is due to the fact that the lowerbounds are derived based on the RM of a single pixel, in contrastwith the (weighted) averaging over many pixels inherent to ourstatistical analysis. ters (Coma, A2065, A2142, A2199, and Ophiuchus), forwhich the data were made available to us ( § § . . µ G, which contributes significantly tothe RM. The presence of such spatially uniform fieldshas already been suggested by Taylor & Perley (1993) intheir analysis of Hydra A, but was not taken into ac-count in recent studies (Taylor & Perley 1993, infer fieldvalues which are ∼
10 times larger than our estimate ofthe uniform components in Coma, but the RM values inHydra A are also an order of magnitude larger).The RM maps of all but one source do not require apower-law magnetic field power spectrum, and most areconsistent with a power spectrum dominated by a singlewave length (see table 3). The magnetic field values in-ferred by earlier analyses ( § B CMB = 3 µ G. In suchmodels, the suppression observed in radio quiet galaxyclusters of the synchrotron luminosity, by a factor of ∼ B . µ G.Therefore, accurate measurements of the magnetic fieldsin radio emitting and radio quiet galaxy clusters may dis- −2 −1 Impact Parameter [ r/r c ] B [ µ G ] µ GA2065A2142ComaOphiuchusA2199 −5 −4 −3 −2 −1 Impact Parameter [ r/r c ] B [ µ G ] µ GA2065A2142ComaOphiuchusA2199 −1 Impact Parameter [ r/r c ] B [ µ G ] p Λ [ k p c ] A2142ComaOphiuchusA2199 −1 Impact Parameter [ r/r c ] B [ µ G ] p Λ [ k p c ] A2142ComaOphiuchusA2199
Fig. 5.—
The accepted ranges of magnetic field strengths of the varying component, B = (3 (cid:10) B z (cid:11) ) / , for single-scale models. Thetop-left and top-right panels depict the accepted ranges of magnetic field strengths as a function of the impact parameter for the single-scalemodels without and with a homogeneous component respectively. Note that the y-axis scale differs between panels. In the bottom-left andbottom-right panels we show the corresponding B Λ / values. The error bars represent the range of accepted values (95% c.l.), and thearrows indicate the lower-bounds as derived in eq. 6. criminate between models (see, however, Basu 2012, forlack of such bi-modality between RH and radio quitedclusters in Planck data).The range of acceptable magnetic field strengths de-rived in this work implies that it is difficult to test thehypothesis of magnetic field bi-modality using RM maps(see the scatter around B = B CMB in figs. 5 & 6). Note,in particular, that we have shown here that there is noevidence to support some of the assumptions used in ear- lier works (discussed in § −2 −1 Impact Parameter [ r/r c ] B [ µ G ] µ GA2065A2142ComaOphiuchusA2199 −5 −4 −3 −2 −1 Impact Parameter [ r/r c ] B [ µ G ] µ GA2065A2142ComaOphiuchusA2199
Fig. 6.—
The accepted ranges of magnetic field strengths of the varying component, B = (3 (cid:10) B z (cid:11) ) / , for power-law models. The leftand right panels depict the accepted ranges of magnetic field strengths for power-law models without and with a homogeneous component,respectively. Note that the y-axis scale differs between the panels. The error bars represent the ranges of accepted values (95% c.l.), andthe arrows indicate the lower-bounds as derived in Eq. (6). the values of depolarization for both populations (withand without a RH) are different realizations of the samedistribution is acceptable. However, one should note thattheir analysis is based on RM measurements of sourcesthat lie at distances from the cluster center that extendto ∼ r c , while radio halos typically extend only up to ∼ r c . It is unclear what the conclusion of the analysiswould be if only sources up to this shorter distance areconsidered.Accurate measurements of the magnetic fields ingalaxy clusters might be possible in the near future,when the quality and quantity of RM data is improved.Higher sensitivity, resolution, and imaging capability willbe provided by the EVLA extension of the VLA. Mostimportant for RM maps, the resolution will be improvedto 0 . . ∼ µ Jy r.m.s.) radio continuum surveycovering the entire Southern Sky. It will have about 45times the sensitivity of the NVSS, and an angular res-olution 4.5 times better (Cassano et al. 2012). Becauseof the excellent short-spacing UV coverage of ASKAP, EMU will also have higher sensitivity to extended struc-tures such as cluster halos (Cassano et al. 2012). In POS-SUM, the plan is to use ASKAP’s unique survey capa-bilities to measure the Faraday rotation of three millionextragalactic radio sources over 30,000 square degrees(”POSSUM Wide”).APERTIF, the new Phased Array Feed system thatwill be installed on WSRT, will increase by about a factor30 the observed area on the sky, at frequencies of 1 . . . µ Jy r.m.s. and ∼ down to 5 µ Jy(Cassano et al. 2012).Other instruments, such as the LWA, MeerKAT, andthe African SKA, will also contribute to increasing thequantity and quality of RM observations.As RM data becomes more abundant, with moresources being observed, closer to and within galaxy clus-ter cores, our proposed analysis method will providehigher quality estimates of the magnetic field strength.Furthermore, it may prove useful to exploit the extra in-formation encoded in the polarization maps themselves(from which the RM maps are derived), as has recentlybeen done for A2199 (Vacca et al. 2012), in order to bet-ter constrain magnetic field models.The authors thank F. Govoni, A. Bonafede, V. Vaccaand their collaborators for the contribution of the RMdata and useful discussions. D. K is supported by NSFgrant AST-0807444. GR & EW are partially supported0 by a UPBC grant.APPENDIX A. FROM ICM MAGNETIC FIELDS TO RM MAPS
We derive below an analytic expression for the RM, eq. (1), obtained for a magnetic field of the form given by eq. (4), B ( r ) = N ∆ k ∆ u ∆ φB k max X k = k min k − n F k , (A1)with F k = +1 X u = − π X φ = − π e iβ ˆΩ h ˆ k cos (cid:0) α ˆΩ (cid:1) + ˆ k sin (cid:0) α ˆΩ (cid:1)i e i k · r . (A2)Here, ∆ φ, ∆ u, ∆ k are the k -space resolution of the azimuthal, (cosine of the) polar, and radial (spherical) coordinates,respectively, and N is a normalization factor to be determined below by requiring B = (3 (cid:10) B z (cid:11) ) / . Writing F k = +1 X u =0 + π X φ = − π h e iβ ˆΩ (cid:16) ˆ k cos (cid:0) α ˆΩ (cid:1) + ˆ k sin (cid:0) α ˆΩ (cid:1)(cid:17) e + i k · r + e iβ − k (cid:16) − ˆ k cos( α − k ) − ˆ k sin( α − k ) (cid:17) e − i k · r i (A3)and choosing α − ˆΩ = α ˆΩ and β − ˆΩ + β ˆΩ = π , so that B ( r ) is real, we have F k = +1 X u =0 + π X φ = − π (cid:16) ˆ k cos (cid:0) α ˆΩ (cid:1) + ˆ k sin (cid:0) α ˆΩ (cid:1)(cid:17) h e + i ( β ˆΩ + k · r ) + e − i ( β ˆΩ + k · r ) i = 2 +1 X u =0 + π X φ = − π (cid:16) ˆ k cos (cid:0) α ˆΩ (cid:1) + ˆ k sin (cid:0) α ˆΩ (cid:1)(cid:17) cos (cid:0) β ˆΩ + k · r (cid:1) (A4)and B z ( r ) = 2 N B X k cos (cid:0) α ˆΩ (cid:1) cos (cid:0) β ˆΩ + k · r (cid:1) k − n ∆ k ∆ u ∆ φ p − u , (A5)where the factor √ − u arises from the z component of ˆ k (the z -component of ˆ k vanishes). The average magneticfield strength is then given by D ( B z ( r )) E = 4 N B X k , k D cos (cid:16) α ˆΩ (cid:17) cos (cid:16) α ˆΩ (cid:17)E| {z } = δ k , k D cos (cid:16) β ˆΩ + k · r (cid:17) cos (cid:16) β ˆΩ + k · r (cid:17)E| {z } = δ k , k Y i =1 , k − n i ∆ k i ∆ u i ∆ φ i q − u i = N B X k,u,φ k − n (∆ k ∆ u ∆ φ ) (cid:0) − u (cid:1) , (A6)which is independent on r . Defining N = N ′ / (∆ k ∆ u ∆ φ ), (cid:10) B z (cid:11) = N ′ B X k,u,φ k − n ∆ k ∆ u ∆ φ (cid:0) − u (cid:1) ≈ N ′ B Z k max k min k − n d k Z (cid:0) − u (cid:1) d u Z π d φ = N ′ B k − n max − k − n min − n
23 2 π, (A7)we find N ′ = 14 π − nk − n max − k − n min , (A8)1and the final expression for the magnetic field is B z ( r ) = B √ π k max X k = k min X u =0 2 π X φ =0 N k N Ω p − u cos (cid:0) α ˆΩ (cid:1) cos (cid:0) β ˆΩ + k · r (cid:1) . (A9)Here N k = (5 − n ) k − n ∆ kk − n max − k − n min , N = ∆ u ∆ φ. (A10)The rotation measure, RM, is now given byRM( x, y ) = RM Z B z n e d z (A11)= RM B n r c X k,u,φ N k N Ω p (1 − u ) cos (cid:0) α ˆΩ (cid:1) × √ √ πr c Z d z cos (cid:0) β ˆΩ + χ + kuz (cid:1) (cid:18) x + y + z r c (cid:19) − b | {z } ≡ I , (A12)where b = 32 β , χ = k p − u ( x cos φ + y sin φ ) . (A13)One may write I = I ℜ (cid:20) e i ( β ˆΩ + χ ) 1 √ π Z (cid:0) ζ (cid:1) − b e iωζ d ζ (cid:21) , (A14)where I = ( r/r c ) − β , ζ = z/r , ω = kur , r = p r c + x + y . (A15)For a background source, the integral over the line of sight ( R + ∞−∞ ) is a Fourier transform, F ζ n(cid:0) ζ (cid:1) − b o ( ω ) = 2 − b Γ( b ) | ω | b − K b − ( | ω | ) , (A16)where K is the modified Bessel function of the second kind, and the final expression for the rotation measure isRM( x, y ) = RM B n r c X k,u,φ N k N Ω √ − u Γ (cid:0) β (cid:1) (cid:18) kur c r (cid:19) β − K β − ( kur ) × cos (cid:0) α ˆΩ (cid:1) cos (cid:16) β ˆΩ + k p − u ( x cos φ + y sin φ ) (cid:17) . (A17)For a cluster member the integral over the line of sight ( R ∞ ) is a Fourier Cosine/Sine transform,12 [ FC + i FS ] ζ n(cid:0) ζ (cid:1) − b o ( ω )= 12 2 − b Γ( b ) ω b − " K b − ( ω ) + iπ I b − ( ω ) − L − b ( ω )2 sin( bπ ) , (A18)where I is the modified Bessel function of the first kind, and L is the modified Struve function, and the final expressionfor the rotation measure of a cluster member isRM( x, y ) = RM B n r c X k,u,φ N k N Ω √ − u Γ (cid:0) β (cid:1) (cid:18) kur c r (cid:19) β − cos (cid:0) α ˆΩ (cid:1) × ( cos (cid:0) β ˆΩ + χ (cid:1) K β − ( kur ) + sin (cid:0) β ˆΩ + χ (cid:1) π L − β ( kur ) − I β − ( kur )2 sin (cid:0) βπ (cid:1) ) . (A19) REFERENCESBasu, K. 2012, MNRAS, 421, L112 Bonafede, A., Feretti, L., Murgia, M., Govoni, F., Giovannini, G.,Dallacasa, D., Dolag, K., & Taylor, G. B. 2010, A&A, 513, A302