Mem. S.A.It. Vol. 82, 658c© SAIt 2011 Memorie della

The intra-cluster magnetic �eld power spectrumin Abell 2199

V. Vacca1,2, M. Murgia2, F. Govoni2, L. Feretti3, G. Giovannini3,4,

R. A. Perley5, and G. B. Taylor5,6

1 Dip. di Fisica, Univ. Cagliari, Cittadella Universitaria, I–09042 Monserrato (CA), Italy2 INAF - OAC, Poggio dei Pini, Strada 54, I–09012 Capoterra (CA), Italy

e-mail: vvacca@oa-cagliari.inaf.it3 INAF - IRA, Via Gobetti 101, I–40129 Bologna, Italy4 Dipartimento di Astronomia, Univ. Bologna, Via Ranzani 1, I–40127 Bologna, Italy5 Adjunct Astronomer at the NRAO, Socorro, NM 87801 USA6 Dep. of Physics and Astronomy, Univ. of New Mexico, Albuquerque NM, 87131, USA

Abstract. We investigate the magnetic field power spectrum in the cool core galaxy clusterA2199 by analyzing the polarized emission of the central radio source 3C 338. We use VeryLarge Array observations at 1.4, 5, and 8 GHz to produce detailed Faraday rotation measureand fractional polarization images of the radio galaxy. By comparing the observations andthe predictions of 3D magnetic field models with a Bayesian approach, we constrain thestrength and structure of the magnetic field associated with the intra-cluster medium.

Key words. Galaxies: cluster: general – Galaxies: cluster: individual: A2199 – Magneticfields – Polarization – Cosmology: large-scale structure of Universe

1. Introduction

Linearly polarized radiation propagatingthrough a magnetized plasma experiences arotation of the plane of polarization which isproportional to the thermal gas density and themagnetic field strength along the line-of-sight.Indeed, it is possible to obtain importantinformation about the intra-cluster magneticfields by combining polarization images ofradio sources located inside or behind galaxyclusters with X-ray observations of the thermalgas.

In this contribution, we investigate themagnetic field power spectrum in the nearby

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galaxy cluster A21991 by analyzing multi-frequency polarization observations of the cen-tral radio source 3C 338. A2199 is an inter-esting target for Faraday rotation studies be-cause of the presence of X-ray cavities asso-ciated with the radio galaxy lobes (Johnstoneet al. 2002), indicating that the rotation of thepolarization plane is likely to occur entirelyin the intra-cluster medium (ICM), since com-paratively little thermal gas should be presentinside the radio emitting plasma. A previousrotation measure (RM) study has been doneby Ge & Owen (1994) based on 5 GHz VeryLarge Array (VLA) data. By combining theinformation from this RM image with de-

1 z=0.0311 (Smith et al. 1997), 1 ′′=0.61 kpc

Vacca et al.: A2199 659

10 kpc

-1000 -500 0 500

RAD/M/M

16 28 3316 28 3616 28 3916 28 42

39 32 20

39 32 40

39 33 00

39 33 20

39 33 40

RIGHT ASCENSION (J2000)

DECLINATION (J2000)

Fig. 1. Total intensity radio contours of 3C 338at 8 GHz overlaid to the RM image of the radiogalaxy 3C 338. The angular resolution is 2.5′′×2.5′′.Contour level are drawn at: 0.06, 0.12, 0.24, and0.96 mJy/beam.

projected ROSAT data, and assuming a verysimple magnetic field model, Eilek & Owen(2002) inferred an averaged magnetic field

value along the line of sight of 15 µG.In this work we try to improve upon the

previous estimate by analyzing additional dataand by performing a numerical modeling ofthe intra-cluster magnetic field fluctuations.Following Murgia et al. (2004), we simulateGaussian random 3D magnetic field modelswith different power law power spectra and wecompare the synthetic and the observed imagesin order to constrain the strength and structureof the magnetic field associated with the ICM.

2. Faraday rotation analysis

The presence of a magnetic field in an ionizedplasma creates a difference in the phase veloci-ties for left versus right circularly polarized ra-diation. As a consequence, the polarized emis-sion from a radio source propagating throughthe plasma experiences a phase shift betweenthe two components. This corresponds to a ro-tation in the polarization angle. For a com-pletely foreground screen, as we expect here,the rotation is

Ψobs(λ) = Ψint + λ2RM, (1)

where Ψobs(λ) is the observed position angleat wavelength λ, and Ψint is the intrinsic po-

larization angle of the polarized emission. Byconsidering an electron density ne, a magneticfield B, and a path length l, the Faraday RM is:

RM = 812∫ l

0ne[cm−3]B[µG] ·dl[kpc] rad m−2(2)

We produced the Faraday RM image by run-ning the FARADAY code (Murgia et al. 2004).The RM image is created by fitting pixel bypixel the observed polarization angle Ψobs ver-sus the squared wavelength λ2 (see Eq. 1) forall the frequencies.

The final RM image of 3C 338 is shownin Fig. 1 with total intensity contours at 8 GHzoverlaid. The image has a resolution of 1.5 kpc.The RM has a patchy structure, and its dis-tribution is characterized by a mean valueof 〈RM〉 =-54 rad/m2 and a standard devia-tion σRM =460 rad/m2. Unresolved RM struc-tures in the foreground screen cause a progres-sive reduction of degree of polarization of thesource (i.e. depolarization, see § 4), because ofan incoherent sum of the radio signal.

3. Magnetic field modeling

We choose to model a power law power spec-trum with index n of the form

|Bk |2 ∝ k−n (3)

in the wave number range from kmin to kmaxand 0 outside. Moreover, we suppose that thepower spectrum normalization varies with thedistance from the cluster center such that theaverage magnetic field strength scales as afunction of the thermal gas density accordingto

〈B(r)〉 = 〈B0〉[ne(r)

n0

]η(4)

where 〈B0〉 is the average magnetic fieldstrength at the center of the cluster and ne(r)is the thermal electron gas density, taken fromJohnstone et al. (2002).

Overall, our magnetic field model dependson five parameters: the strength at the clustercenter 〈B0〉, the radial slope η, the power spec-trum index n, and finally the minimum and themaximum scale of fluctuation, Λmin = 2π/kmaxand Λmax = 2π/kmin, respectively.

660 Vacca et al.: A2199

Fig. 2. Top left panel: Bayesian analysis of the RMstructure function. The dots represent the data (errorbars are comparable to the size of the symbols). Theshaded area represents the population of syntheticRM structure functions from the posterior distribu-tion. The dashed line corresponds to the most prob-able value for the model parameters. Top right, andbottom panels: 1-dimensional (histograms) and 2-dimensional (colors and contours) marginalizationof the posterior for the model parameters. The con-tours are traced at 0.9, 0.75, and 0.5 the peak value.

4. Results from 2D and 3D analysis

To constrain the magnetic field strength andstructure, we proceeded in two steps. First, weperformed a 2D analysis of the RM fluctua-tions and of the source depolarization to con-strain the slope n and the range of scales ofthe power spectrum. Second, we performed 3Dnumerical simulations to constrain the strengthof the field and its scaling with the gas density.In both cases, we made use of the FARADAYcode to produce synthetic polarization imagesof 3C 338 and to compare them to the ob-served ones by means of the Bayesian infer-ence, whose use has been first introduced inthe RM analysis by Enßlin & Vogt (2003).The synthetic images are gridded to the samegeometry as the data and are convolved to thesame angular resolution. Moreover, we maskedthe synthetic images using the observations inorder to reproduce the window function im-posed by the shape of 3C 338, and we added

Fig. 3. Top left panel: Bayesian analysis of thesource depolarization. The dots represent the data.The shaded area represents the population of syn-thetic polarization from the posterior distribution.The dashed line corresponds to the most proba-ble value for the model parameters. Top right, andbottom panels: 1-dimensional (histograms) and 2-dimensional (colors and contours) marginalizationof the posterior for the model parameters. The con-tours are traced at 0.9, 0.75, and 0.5 the peak value.

Gaussian noise, in order to mimic the noise ofthe observed RM image.

In particular, we characterize the RM im-age by its structure function S RM(r⊥), whichis obtained by averaging RM values corre-sponding to pixels located at the scale distancer⊥. We then applied the Bayesian method bychoosing uniform priors for the four free powerspectrum parameters: the normalization norm,the minimum and the maximum scale of fluctu-ation Λmin, and Λmax, and the slope n. In Fig. 2we show the results of the Bayesian analysisof the RM structure function. The maximumscale of fluctuation, the normalization, and theslope of the power spectrum appear to be char-acterized by a peak that corresponds to themaximum posterior probability for that config-uration of parameters, while for the minimumscale of fluctuation we have just an upper limit.

The power spectrum used to model the RMimage should also be consistent with the ob-served depolarization of the radio source at in-

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Fig. 4. Bayesian 3D analysis of the RM struc-ture function for the model with (right) and with-out (left) cavities. Top panels: The dots representthe data (error bars are comparable to the size of thesymbols). The shaded area represents the populationof synthetic RM structure functions from the pos-terior distribution. The dashed line corresponds tothe most probable value for the model parameters.Bottom panels: 1-dimensional (histograms) and 2-dimensional (colors and contours) marginalizationof the posterior for the model parameters. The con-tours are traced at 0.9, 0.75, and 0.5 the peak value.

creasing wavelengths. In Fig. 3 the posteriorfrom the depolarization analysis is shown forthe five free parameters: the intrinsic degree ofpolarization FPOL0, the normalization norm,the slope n, the minimum, and the maximumscale of fluctuation Λmin, and Λmax of the mag-netic field power spectrum. All the model pa-rameters appear to be well constrained, andtheir values are consistent with the structurefunction analysis.

Overall, the combined 2D analysis,RM image and depolarization, allowed usto constrain the power spectrum index ton=(2.8±1.3), and the minimum and maximumscale on the range from Λmin =(0.7±0.1) kpcto Λmax = (35±28) kpc. In the next step we fixthese values and we constrain the strength ofthe magnetic field and its scaling with the gasdensity with the aid of 3D simulations.

The result of the Bayesian analysis for3D magnetic field models is shown in Fig. 4(left panels) for the two free parameters: thestrength at the cluster center 〈B0〉, and the ra-dial slope η. The two parameters appear wellconstrained. We found a magnetic field with acentral strength 〈B0〉=(9.6±2.3) µG, and a ra-dial slope η=(0.7±0.2).

Hence, we tried to improve our analy-sis by including the X-ray cavities in the3D modeling. In particular, we removed fromthe 3D simulations the gas density insidetwo ellipsoidal regions centered on the ra-dio lobes. We repeated the Bayesian analy-sis including the cavity information and wefound 〈B0〉=(10.1±2.6)µG, and η=(0.8±0.2)(see Fig. 4, right panels).

Acknowledgements. This work makes use ofresults produced by the Cybersar Project man-aged by the Consorzio COSMOLAB, a projectco-funded by the Italian Ministry of University andResearch (MIUR) within the Programma OperativoNazionale 2000-2006 “Ricerca Scientifica,Sviluppo Tecnologico, Alta Formazione” perle Regioni Italiane dell’Obiettivo 1 (Campania,Calabria, Puglia, Basilicata, Sicilia, Sardegna)Asse II, Misura II.2 “Societa dell’Informazione”,Azione a “Sistemi di calcolo e simulazione adalte prestazioni”. More information is available athttp://www.cybersar.it. This research was partiallysupported by ASI-INAF I/088/06/0 - High EnergyAstrophysics and PRIN-INAF2005. The NationalRadio Astronomy Observatory (NRAO) is a facilityof the National Science Foundation, operated undercooperative agreement by Associated Universities,Inc. We are grateful to Antonella Fara and RiccardoPittau for the assistance with the Cybersar-OACcomputer cluster.

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