JOINT ANALYSIS OF CLUSTER OBSERVATIONS: I. MASS PROFILE OF ABELL 478 FROM COMBINED X-RAY, SUNYAEV-ZEL’DOVICH, AND WEAK LENSING DATA Andisheh Mahdavi, Henk Hoekstra and Arif Babul DepartmentofPhysicsandAstronomy,UniversityofVictoria,Victoria,BCV8W3P6,Canada Jonathan Sievers CanadianInstitute forTheoreticalAstrophysics,60St. GeorgeSt.,Toronto, ONM5S3H8,Canada Steven T. Myers NationalRadioAstronomyObservatory,1003LopezvilleRd.,Socorro,NM87801 and 7 J. Patrick Henry 0 InstituteforAstronomy,2680WoodlawnDrive,HonoluluHI96822,U.S.A. 0 SubmittedMay 16, 2006; Accepted March 2, 2007 for publication in TheAstrophysicalJournal 2 r ABSTRACT a Weprovideanewframeworkforthejointanalysisofclusterobservations(JACO)usingsimultaneous M fits to X-ray, Sunyaev-Zel’dovich (SZ), and weak lensing data. Our method fits the mass models simultaneously to all data, provides explicit separation of the gaseous, dark, and stellar components, 4 and—forthefirsttime—allowsjointconstraintsonallmeasurablephysicalparameters. JACOincludes 1 additionalimprovementstopreviousX-raytechniques,suchasthetreatmentoftheclustertermination shock and explicit inclusion of the BCG’s stellar mass profile. An application of JACO to the rich 1 galaxy cluster Abell 478 shows excellent agreement among the X-ray, lensing, and SZ data. We find v that Abell 478 is consistent with a cuspy dark matter profile with inner slope n=1. Accounting for 2 the stellar mass profile of the BCG allows us to rule out inner dark matter slopes n>1.1 at the 99% 7 confidence level. At large radii, an r−3 asymptotic slope is preferred over an r−4 behavior. All single 3 power law dark matter models are ruled out at greater than the 99% confidence level. JACO shows 3 that self-consistent modeling of multiwavelength data can provide powerful constraints on the shape 0 7 of the dark profile. 0 Subject headings: Dark matter—X-rays: galaxies: clusters—gravitational lensing—galaxies: clusters: / individual (Abell 478) h p - o 1. INTRODUCTION berg, Shapiro, & Inagaki 2002; Ahn & Shapiro 2005) r N-body simulations of our Cold Dark Matter (CDM) or whether the cores quickly collapse, forming an r−2 t isothermal cusp (Kochanek & White 2000). If the cores s dominated universe suggest that evolved galaxies and a arestable,measurementsoff(x)canprovideconstraints clusters of galaxies are contained in dark halos with : on the interaction cross section and hence on the nature v self-similar density profiles. In the collisionless limit, of dark matter itself. i ρ (r/r )/δ = f(x). The characteristic density δ and X raddmius rd depend on the halo mass and the cosmological Rich, & 1014M⊙ clusters of galaxies are excellent lab- d r parameters, but the normalized profile f(x) itself does oratories for the measurement of f(x) because of the a large number of independent techniques available for not (Navarro, Frenk, & White 1997; Moore et al. 1998; mass measurement. In this series of papers, we exam- Ghignaetal. 2000;Subramanian,Cen,&Ostriker2000; inethecomplextaskofconstrainingf(x)simultaneously Fukushige & Makino 2001; Williams, Babul, & Dalcan- usingtheX-rayemittingintraclustermedium(ICM),the ton 2004; Merritt et al. 2005). The robustness with Sunyaev-Zel’dovich decrement, and weak gravitational which collisionless N-body simulations predict a univer- lensing maps. For the joint analysis of cluster observa- sal profile makes the observationalmeasurement of f(x) tions (JACO), we always begin with separate luminous particularlyimportant. If we are to be satisfiedthat our and dark mass profiles, and calculate the observables— understanding of hierarchicalstructure formationis cor- the X-ray spectrum, the SZ decrement, or the tangen- rect, we need to use the best available data to test the tial lensing shear—directly from the underlying physical theoretical dark matter profiles in detail. model. In other words, our fits always take place in the Observations of f(x) also have implications for self- data space, rather than in mass, density or temperature interacting dark matter (SIDM) as an alternative to the space. This approach allows for (1) deformation of the collisionless models. If the weak interaction cross sec- theoreticalprofilestoaccountfortheappropriateinstru- tion of the particles in a halo is large enough, substan- mental degradationat eachwavelengthand (2) the abil- tial effects on the shape of the density profile should ity to combine lensing, SZ, and X-ray data in a single appear within a Hubble time. Initially cuspy profiles grand total fit. should evolve constant-density cores with a size 3% ∼ Other works have examined the multiwavelength ap- of the virial radius (Spergel & Steinhardt 2000; Burkert proach. Miralda-Escude & Babul (1995) carried out 2000). It is an open question whether the SIDM cores the first comparison of physical X-ray and strong lens- are stable and long-lived enough to be observable (Bal- ing models for a sample of three rich clusters; Squires 2 Mahdavi et al. et al. (1996) extended the comparison to weak lensing. Pointecouteau, Arnaud, & Pratt 2005; Vikhlinin et al. Zaroubi et al. (1998) and Zaroubi et al. (2001), using 2006). However, in a few cases, evidence for a signif- idealized X-ray and SZ observations of simulated clus- icantly shallower n . 0.5 (Ettori et al. 2002; Sander- ters, showed that the unique axisymmetric deprojection son, Finoguenov, & Mohr 2005; Voigt & Fabian 2006) of the gas density profile is possible by combining the exists, echoing similar results in several strong gravita- two types of data. Doré et al. (2001), also using sim- tionallensingstudies(Tyson,Kochanski,&dell’Antonio ulated clusters, argued that the simultaneous deprojec- 1998;Sand, Treu, & Ellis 2002;Sand et al. 2004). tionofSZandweaklensingdata issensitive tothe noise The substantial progress allowed by Chandra and properties of the data. De Filippis et al. (2005) and XMM-Newtondataleadsustoconsiderwhetherthecon- Sereno et al. (2006) use combined X-ray and SZ data straints on the dark matter structure could benefit from to place constraints on the triaxiality and inclination of a refinementof the mass fitting technique and the use of the intracluster medium assuming that they are single- optical and radio data in a simultaneous fit. In this pa- component, isothermal systems; they do not attempt to per we show that through the use of a multiwavelength constrain the dark profile. Assuming a Navarro et al. modelingtechniquespecificallyaimedatconstrainingthe (1997) mass profile, LaRoque et al. (2006) use joint X- the dark matter (as opposed to the total) mass profile, ray and SZ data from 38 clusters to constrain the gas enhanced constraints on the slope of the dark mass dis- mass fraction. tribution in clusters are possible. In 2 we motivate the § JACO is the first method to use X-ray, SZ, and weak technique and describe it in detail. We apply the tech- lensingdatasimultaneouslytoconstraintheshapeofthe nique to the rich cluster Abell 478 in 3, and compare § dark matter profile in clusters. This paper describes a theresultswithpreviousworkandN-bodymodelsin 4. § testofthismethodontheclusterAbell478. Chiefamong We conclude in 5. § theassumptionsallowingtheextractionofthedarkmat- 2. METHOD terprofileis thatthe gasinhydrostaticequilibriumwith 2.1. General Principles the overall gravitational potential. If hydrostatic equilibrium is strongly violated, then determination of the The purpose of JACO is to constrain the shape of structure ofthe dark halo fromthe X-raydata alone be- the dark matter profile in clusters of galaxies via a sin- comes difficult. Because ongoing or recent cluster merg- gle multi-parameter fit to X-ray, lensing, and Sunyaev- ers arecommon,the identificationof clusters thatare as Zel’dovichdata. Thebroadfeaturesofourtechniqueare: closetoprototypical(or“boring”)aspossibleisvaluable Separation of the mass model into the gaseous, stellar, for detailed mass measurement. The difficulty of find- and dark components. Rather than fitting for the total ingsystemssuitableforhydrostaticanalysisisevidentin gravitating mass, JACO splits the potential into three the latestChandraandXMM-Newtonstudies ofrelaxed separatelymodeledcomponents,thusguaranteeingtheir clusters. positivity. As a result JACO is incapable of generating Clusters that appeared smooth and relaxed before unphysicalmassprofilesasissometimesthecasewithpa- Chandra and XMM-Newton observations often contain rameterized X-ray temperature profiles (Pizzolato et al. features that make them unfit for equilibrium analy- 2003;Buote&Lewis2004). Thestellarmasscontributes sis. Examplesincludecoldfronts(Vikhlinin,Markevitch, a substantial part of the gravitating mass within 3% of & Murray 2001; Bialek, Evrard, & Mohr 2002; Dupke the cluster virial radius (Sand et al. 2004), an impor- & White 2003; Hallman & Markevitch 2004) or shocks tant but sometimes neglected effect when testing for the (Markevitch & Vikhlinin 2001; Markevitch et al. 2002; presence of a SIDM constant density core. Fabian et al. 2003; Fujita et al. 2004) in recent merg- Direct constraints from uncorrelated data. Weconduct ers. In clusters with X-ray substructure, the use of hy- minimal data processing. We use the mass models to drostatic equilibrium may yield incorrect results (Poole calculate, project, and PSF-distort theoretical spectra, et al. 2006). In many otherwise seemingly relaxed clus- and then fit these to the measured X-ray count spectra. ters, heating by a central AGN is required to reconcile Weavoiddeprojectingthedata,thusguaranteeingfitsto the short cooling times with the deficit of kT . 1 keV uncorrelateddata. Thegastemperatureishandledasan gas in the quantities predicted by quasihydrostatic cool- internalvariable,so thatwe skipthe temperaturefitting ing flow models (Tamura etal. 2001;Fabianet al. 2001; stage altogether. For Sunyaev-Zel’dovichmeasurements, Matsushita et al. 2002; Peterson et al. 2003; Kaastra we calculate and fit the uncorrelated Fourier modes of et al. 2004; Voit & Donahue 2005). Gas in the region the decrement directly. directly affected by the AGN is unlikely to be in hydro- No X-ray temperature weighting. Because the mass static equilibrium, and the equilibrium equations may models are fit directly to the spectra there is no need to yield incorrect results. calculate emission- or otherwise-weighted temperatures. Themostrecentworkshows,however,thatmanyclus- Hydrodynamic N-body simulations show that emission- ters can be successfully modeled as equilibrium systems weightedtemperaturesdonotaccuratelyreflectthemea- outside the region of influence of the central AGN. In sured spectroscopic temperature (Mazzotta et al. 2004; these regions, the total density distribution frequently Rasia et al. 2005); recent methods for getting around resembles a pure n = 1 Navarro et al. (1997) profile, thisproblemrequirethecalculationoftheoreticalweights andeitherasingleordoubleβ-modelprovidesasuccess- that depend on the X-ray calibration files (Vikhlinin ful descriptionof the X-ray surface brightness (Schmidt, 2006). JACO calculates 2D spectra directly from the Allen, & Fabian 2001; Arabadjis, Bautz, & Garmire massmodels. Temperatureanddensityvariationswithin 2002; Allen, Schmidt, & Fabian 2002; Lewis, Buote, & anygivenannulusarereflectedintheprojected2DX-ray Stocke 2003; Buote & Lewis 2004; Pointecouteau et al. spectra without any emission weighting. 2004; Buote, Humphrey, & Stocke 2005; Gavazzi 2005; What is truly new here is (1) the direct calculation of JACO 3 X-ray spectra from the gravitational potential, (2) the Thus the spectrum is expressed purely as a function of use of this potential to model Sunyaev-Zel’dovich and massandmetallicity. Notemperaturefittingisrequired. weak lensing observations simultaneously, and (3) the Once we know the pressure, the Sunyaev-Z’eldovich ability to conduct a full, covariant error analysis on all decrement follows directly from an integral of the pres- the physicalparameters. There are alsootherimportant sure along the line of sight (Birkinshaw 1999): features that have not been implemented in any prior Pσ T method, e.g. the strict calculation of the mean molec- ySZ =Z m c2dz (5) ular weight from the metallicity profile ( 2.2) and the e § whereσ is theelectronscatteringcrosssection,andm formal modeling for the termination shock as the clus- T e ter boundary( 2.3). Someofthe otheraspects ofJACO is the electron mass. Similarly, the reduced weak gravi- § tational lensing shear in the tangential direction may be have been discussed before. For example, Markevitch & Vikhlinin(1997)separatethegravitationalpotentialinto calculated directly from the total mass model (Miralda- Escude 1991): gaseousanddarkcomponentswhenmodelingAbell2256. Fukazawa et al. (2006) and Humphrey et al. (2006)em- ploy the stellar profile of the BCG in the X-ray fit. And κ¯( R n = 4 gives a de Vaucouleurs (1953) profile. The shape q − i−1−p − i i parameters of the this profile are measured from high  1 Note that Pizzolatoetal. (2003) have anextra4π infrontof quality optical photometry, while the normalization is their equation (3) because they take a different definition of the allowedto vary freely in the fit. The BCG profile allows emissivity. JACO 5 TABLE 1 List ofParameters in theCluster Model Parameter SherpaName Units Canbefit? Comments ∆ contrast ··· N Overdensityformassesandconcentrations ··· precision ··· N Precisiongoal ρd darkmodel ··· N Integer specifyingdarkmassmodel ρ∗ starmodel ··· N Integer specifyingstellarmassmodel z redshift ··· Y Clusterredshift Mg Mg 1014M⊙ Y Gasmasswithinr∆ rx1 rx1 Mpc Y Coreradiusoffirstβ-model rx2 rx2 Mpc Y Coreradiusofsecond β-model rx3 rx3 Mpc Y Coreradiusofthirdβ-model β1 b1 ··· Y Slopeoffirstβ-model β2 b2 ··· Y Slopeofsecondβ-model β3 b3 ··· Y Slopeofthirdβ-model f2 xfrac2 ··· Y Normalizationofsecond β-modelrelativetothe1st f3 xfrac3 ··· Y Normalizationofthirdβ-modelrelativetothe1st Md Md 1014M⊙ Y Darkmasswithinr∆ n ndark ··· Y Slopeofdarkmatterdensityprofile r0 r0 Mpc Y Sizeofinnerconstant-density darkmatter core rd/c rdm1 ··· Y Darkmatter concentrationa Z0 z0 ··· Y Metallicityatr=0 Z∞ zinf ··· Y Metallicityatlargeradii MBCG Mstar 1014M⊙ Y TotalstellarmassofBCG αs alpha ··· Y Indexof3Dstellarlightprofile,exp−(r/rs)αs rs rs Mpc Y Scaleradiusofstellarprofile rshock rshock Mpc Y Radiusofterminationshock Note. —AdditionalparametersrepresentingtheresidualsoftX-raybackgroundmayalsobefittothedata. aTheusercandecidetofittheconcentration c≡r∆/rd,orelsetofitrd itself. 2.6. PSF Distortion TABLE 2 A crucial step in comparing the projected model to EnergyDependence of thePSF the data is to account for PSF distortions. The JACO approach is to manipulate the data as little as possible, Coefficient ACIS MOS1 MOS2 pn soweapplythedistortiontothemodelspectraF (i)via ν c1 1.137 5.074 4.759 6.636 a convolution matrix: c2 -0.054 -0.236 -0.203 -0.305 N c3 0.076 0.002 0.014 -0.175 S (i)= Π (i,j)F (j) (21) c4 0.034 -0.018 -0.023 -0.007 ν ν ν c5 6.119 1.472 1.411 1.525 Xj=1 c6 -0.254 -0.010 -0.005 -0.015 where S (i) is the final model spectrum for compar- c7 -0.652 -0.001 -0.001 -0.012 ν c8 0.051 -0.002 0.000 -0.001 ison with the data, and Π (i,j) contains the energy- ν dependent contribution of each annulus to itself and every other annulus. Given any set of annuli it is possible to calculate Πν(i,j) and thus convolve the model with ζ =c5+c6E+c7θ+c8Eθ (24) the PSF. Here θ is measured in arcminutes and E is measured In general, the PSF of Chandra and XMM-Newton is in keV. The XMM-Newton calibration team has derived a function of energy and position. JACO makes an al- lowance for this. However, the shape of the PSF is not (Ghizzardi2001a,2001b)valuesofc1−8fromarchivalob- servationaldata. WhileextensivetreatmentoftheChan- strictlycircularorevenellipsoidal. Wetakenoaccountof dra PSF exists, no functional fits to the azimuthally av- this anisotropy, instead treating the PSF as azimuthally eraged Chandra PSF have been published. For this rea- symmetric at every point. In-flight XMM-Newton cal- son, we undertake ray-tracing MARX (Wise 1997) sim- ibration studies have found that this treatment of the ulations ofbrightpointsourcesat variousoff-axisangles PSF yields sufficiently accurate encircled energy profiles and energies. except for very bright point sources (Ghizzardi 2001a, We find that equation (22) provides an acceptable de- 2001b). The model PSF is given by scription of the Chandra PSF as well. We measure c1−8 pν(φ)∝(1+φ2/φ20)ζ (22) inasimilarmannertoGhizzardi(2001a),exceptthatwe where φ is the angular distance in arcseconds between useray-tracedratherthanin-orbitdata. Thepolynomial thesourcecenterandthepositionbeingconsidered. The coefficients for all instruments are shown in Table 2. shape parameters φ and ζ are functions of energy and UsingthisfunctionalformforthePSFwecancalculate 0 position across the detector, and are empirically fit as the PSF convolution matrix Πν(i,j). The details of the polynomials in photon energy E = hν and source posi- calculation are given in Appendix B. tion θ: 2.7. Software Interface φ =c +c E+c θ+c Eθ (23) 0 1 2 3 4 6 Mahdavi et al. Fig. 1.— (left) HST WFPC2 image of the core of Abell 478, with superposed X-ray contours from the Chandra ACIS-S image. right Profileofthesurfacebrightnessplusbackground fortheBCG,aswellasthebest-fitSérsicmodel(α=0.289). TheJACOsoftwarepackagewillsoonbepubliclyavail- a rich cluster at z=0.088. The most recent available able at http://jacocluster.sourceforge.net. It consists of studies with Chandra (Sanderson et al. 2005; Voigt & the following components: Fabian 2006) and XMM-Newton (Pointecouteau et al. 2004) suggest a peak ICM temperature of of 7 9 1. A core C language library which calculates the ∼ − keV,making this cluster one of the more massive known observed X-ray spectrum, tangential shear, and within z <0.1. For this analysis we assume H =70 km Compton y parameter from the input dark, gas, s−1 Mpc−1, Ω =0.3, and Ω =0.7. With the0se values, and stellar mass distributions; 1′ =98.7 kpc a0t the distanceΛof Abell 478. 2. An interface which links the core library to the This cluster is an ideal test case for JACO, and high- Sherpa data analysis package (Freeman, Doe, & lights the advantages of direct spectral fitting for hydro- Siemiginowska 2001) as a standard user-defined static mass determination. The Chandra and XMM- model, and provides the necessary scripts for Newton observations record a total of over 1,000,000 graphical viewing and fitting; sourcephotons;yetPointecouteauet al. (2004),Sander- son et al. (2005), and Voigt & Fabian (2006) show that 3. A parallelized interface, Hrothgar, which runs temperature profiles of this cluster disagree when calcu- the core JACO routine on multiprocessor facili- latedwithsingletemperatureemission-weightedplasma. ties such as Beowulf clusters. Hrothgar is a gen- We will show that the addition of optical data (for the eral purpose package and will soon be available at stellar mass profile and the weak gravitational lensing http://hrothgar.sourceforge.net. shear)andradiodata(for the Sunyaev-Zel’dovichdecre- ment)improvestheconstraintsonthedarkmatterdistri- 4. A set ofoptionaldata reduction scripts whichpro- butioninthe cluster,andbringthe ChandraandXMM- cess standard archival X-ray data releases accord- Newton data into closer agreement. ing to the procedures described in Appendix C. Weak lensing and SZ data reduction is up to the 3.1. X-ray Data user. On request, the authors can also provide a routine for transforming JACO Compton y maps To model Abell 478, we fit JACO X-ray models to into interferometer observables. The existing rou- available XMM-Newton and Chandra archival. There tines are customized for the Cosmic Background are four instruments in all: EPIC pn, EPIC MOS 1 and Imager (Padin et al. 2002), but other arrays are 2, and Chandra ACIS-S. We used XMM-Newton obser- easily accommodated. vation sequence 109880101 (126ks, of which 43ks were useful) andChandra ObsID 1669(42ks,of which allwas JACO offers a choice of numerical integration meth- useful). For the Chandra data, we use CIAO 3.3 and ods. ThefastermethodusesanadaptiveGauss-Legendre CALDB 3.2.1. For the XMM-Newton data, we use the quadrature with 10 or 20 abscissae depending on the latest CCF calibration release before August 2006. complexity of the integral. JACO also offers adaptive We applythe genericreductiondescribedinAppendix Gauss-Kronod quadrature with convergence checking. C. We extract spectra in circular regions around the The latter method is slower but is guaranteed to con- X-ray centroid, which coincides in all datasets with vergetoagivenaccuracy. Wefindthatthefastermethod RA(J2000) = 04:13:25.3, DEC(J2000) = +10:27:54. A producesmodelspectrawithamedianaccuracyof0.1%. totalof76annuliareused,ofwhichChandracoversonly theinnermost40. The ACISdataareusedtochoosethe 3. APPLICATIONTOABELL478 sizesofthe innermost40annuli; werequireatleast2500 Because the JACO technique is new, we apply it to background-subtracted counts per annulus. A similar a well-known, relaxed cluster of galaxies. Abell 478 is requirement is used to choose the sizes of the outer 36 JACO 7 Fig. 2.—(left)SZdataandresidualsforthebest-fitJACOmodel;(right)weaklensingdataandresidualsforthebest-fitJACOmodel annuli—2500background-subtractedcountsintheEPIC by integrating the best-fit Sérsic model to infinity. The pn camera. The innermost annulus is 13′′ (22 kpc) in galaxy is in a high-extinction region; E(B-V) = 0.589 radius; the annuli increase regular by 5′′ (8 kpc) until a (Schlegel, Finkbeiner, & Davis 1998). After applying distanceof4.6′(450kpc),atwhichpointtheyincreaseby the extinction correction of 1.7 mag and a k-correction 14′′-40′′ until the final annulus at 9′ (890 kpc). The ex- 0.14mag(Poggianti1997),wefindthatthetotalF606W tractedspectra were fit overthe 0.6-6keV energyrange. luminosity of the galaxy is 4.9 0.3 1011 L⊙. This ± × The JACO plasma code was set to MEKAL. value is important in constraining the average stellar The galactic absorption column varies with position mass-to-light ratio of the BCG, Υ∗ . For an evolved BCG on the sky (Sun et al. 2003; Pointecouteau et al. 2004; stellar population with mean age & 3Gyr in the wave- Vikhlinin et al. 2005). This effect is more important for length regime of the F606W filter, it is expected that the XMM-Newton data, which cover a wider field than 1 < Υ∗BCG < 4M⊙/L⊙ depending on the stellar initial Chandra; we take the gradient into account by allowing mass function (IMF) (Maraston 1998). The lower limit thehydrogencolumndensitytovarylinearlywithradius. of 1 applies regardless of the choice of IMF (Maraston Takingthevariationofn withpositionintoaccountstill 1998). H does not resolve the temperature discrepancies between the Chandra and XMM-Newton temperatures at inter- 3.3. Weak Lensing Data mediateradii(Pointecouteauetal. 2004;Vikhlininetal. 2005). The weak lensing measurements for A478 are based on archival data taken with the CFH12k camera on the Canada-France-Hawaii-Telescope (CFHT). The camera 3.2. Hubble Space Telescope Data consists of an array of 6 by 2 CCDs, each 2048 by 4096 Figure 1 shows a 300s archival Hubble Space Tele- pixels. The pixel scale is 0.206′.′, which ensures good scope WPFC2 image of the central 60 kpc diameter sampling for the sub-arcsecond imaging data used here. ∼ region. The corresponding HST observation number is The resulting field ofview is about42 by 28 arcminutes, U5A40901R, taken with the 2000˚A bandpass F606W whichis ofgreatimportancewhenstudying nearbyclus- ∼ filter. Superposed arethe ChandraACIS surface bright- ters, such as A478. ness contours. The weak lensing analysisrequiresgoodimage quality The stellar mass of the BCG makes a non-negligible and therefore we selected only data with seeing better contribution to the matter distribution within 50 kpc than 0.8”. This selection yielded 9 exposures with 605s (Sandetal. 2002,2004;Vikhlininetal. 2006),andhence of integration each, resulting in a total integration time contributestothe equationofhydrostaticequilibriumat of 5445s. Unfortunately only R-band observations were all radii. To model the stellar mass profile of the BCG, available. This in principle complicates removal of clus- wefititsHST surfacebrightnessprofilewitha3DSérsic ter members. However, our study of a large suite of R- (1968) model (equation 16). Figure 1 shows the best band observations of other clusters (Hoekstra 2007) has fit model: α = 0.289 0.002 and r = 0.36 0.02′′. shownthat we can readily correctfor this source of con- BCG 0 ± ± Thismodelisaddedtothegaseousanddarkcomponents tamination using the excess galaxy counts as a function when conducting the grand total fit. of radius. Using the photometric solutionforthe HST image,we Detrended data (de-biased and flatfielded) are pro- calculateanuncorrectedF606Wmagnitudeof15.1 0.1 videdtothecommunitythroughCADC.Weprocessthis ± 8 Mahdavi et al. Fig. 3.— Spectra and best-fit models for Chandra ACIS-S(as), XMM-Newton MOS1(m1), MOS2 (m2), and pn cameras. Only72 of the268totalspectraareshown;theinnerandouter radiioftheextracted annuli(inarcminutes)appearnexttotheinstrumentname. . datathroughthe analysispipeline describedinHoekstra for the size, apparent magnitude and shape parameters et al. (1998), Hoekstra, Franx, & Kuijken (2000), and andtheirmeasurementerrors. Theimageisinspectedby Hoekstra(2007). Firstweusethehierarchicalpeakfind- eye,inordertoremoveareasthatwouldleadtospurious ingalgorithmfromKaiser,Squires,&Broadhurst(1995) detections. tofindobjectswithasignificance>5σoverthelocalsky. To measure the small, lensing induced distortions it The peak finder gives estimates for the object size, and is important to accurately correct the shapes for PSF werejectallobjectssmallerthanthesizeofthePSF.The anisotropy,as wellas forthe diluting effect ofseeing. To remaining objects are analyzed, which yields estimates characterize the spatial variation of the PSF we select a JACO 9 Fig. 4.—Twovisualizationsofthebest-fituniversalmodels(equation14)forAbell478. (top)Thespectraarecollapsedintothreecolor regimes: 0.6-1.5keV(upperline),1.5-2.0keV(middleline),and2.0-6.0keV(lowerline). Thecountratesareoffset0,-1,and-2ordersof magnitude, respectively,forclarity. Datatomodelratiosarealsoshown. (bottom) Thespectraandmodelsareshownascolorratios. We showtheratioofcountratesinthethreebandstothetotalcountrate: 0.6-1.5keV(lowerline),1.5-2.0keV(middleline),and2.0-6.0keV (upperline). Eacherrorbarhererepresents∼50–70ofthefitspectraldatapoints,eachofwhichinturnrepresents ∼50photons. sample of moderately bright stars from our images (e.g. an accuracy of 2% (Heymans et al. 2006). ∼ Hoekstra et al. 1998). Seeing circularizes the images, The catalog of objects with corrected shapes is used thus lowering the raw lensing signal. To correct for the for the weak lensing analysis. In the analysis presented seeing,weneedtorescalethepolarizationstotheir“pre- here, we consider the tangential distortion as a func- seeing”value,as outlinedin Hoekstraetal. (1998). Our tion of radius from the cluster center. The resulting pipeline has been tested on simulated images in great measurements,using galaxieswith apparentmagnitudes detail (e.g. Hoekstra et al. 1998; Heymans et al. 2006). 22

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