toappearinAstrophysicalJournalv538Aug1,2000 PreprinttypesetusingLATEXstyleemulateapjv.26/01/00 THE SUNYAEV-ZEL’DOVICH EFFECT IN ABELL 370 Laura Grego1, John E. Carlstrom2, Marshall K. Joy3, Erik D. Reese2, Gilbert P. Holder2, Sandeep Patel4, Asantha R. Cooray2, and William L. Holzapfel5 to appear in Astrophysical Journal v538 Aug 1, 2000 ABSTRACT We present interferometric measurements of the Sunyaev-Zel’dovich (SZ) effect towards the galaxy 0 cluster Abell 370. These measurements, which directly probe the pressure of the cluster’s gas, show the 0 gas distribution to be strongly aspherical, as do the x-ray and gravitational lensing observations. We 0 calculatethecluster’sgasmassfractionintwoways. WefirstcomparethegasmassderivedfromtheSZ 2 measurementstothelensing-derivedgravitationalmassnearthecriticallensingradius. Wealsocalculate r the gas mass fraction from the SZ data by deprojecting the three-dimensional gas density distribution a M and deriving the total mass under the assumption that the gas is in hydrostatic equilibrium (HSE). We testtheassumptionsintheHSEmethodbycomparingthetotalclustermassimpliedbythetwomethods 6 andfindthattheyagreewithintheerrorsofthemeasurement. Wediscussthepossiblesystematicerrors in the gas mass fraction measurement and the constraints it places on the matter density parameter, 1 Ω . M v Subject headings: cosmic background radiation—cosmology: observations, galaxies: 5 clusters:individual(Abell 370)–techniques: interferometric 8 0 3 1. INTRODUCTION cm−3; and cools slowly (tcool > tHubble), mainly via ther- 0 mal Bremsstrahlung in the x-ray band. The x-ray sur- 0 Clustersofgalaxies,byvirtueofbeingthelargestknown face brightness is proportional to the emission measure, 0 virialized objects, are important probes of large scale S n2Λ(T )dl, where the integration is along the line / structure and can be used to test cosmological models. x ∝ e e h of sight, and so, under simplifying assumptions, the gas p Rich clusters are extremely massive, ∼ 1015M⊙, as indi- mass cRan be calculated from an x-ray image deprojection cated by the presence of strongly gravitationally lensed - andthemeasuredgastemperature. Sincethesoundcross- o backgroundgalaxies and by the deep gravitationalpoten- ingtimeoftheclustergasismuchlessthanthedynamical r tialnecessarytoexplainboththe largevelocitydispersion st (> 1000 km s−1) in the member galaxies and the high time,onemayreasonablyassumethat,intheabsenceofa a recentmerger,theclustergasisrelaxedinthecluster’spo- measured temperature (>5 keV) of the ionized intraclus- : tential. The totalbinding mass canbe extractedfromthe v ter gas. Dynamical mechanisms for segregating baryonic gas density and temperature distribution under this as- i matter fromdark matter onthese mass scalesaredifficult X sumption. A significant body of work exists in which the to reconcile with observations and standard cosmological r models, and so within the virial radius the mass compo- gasmassfraction,fg,ismeasuredinthisway,withfg mea- a surementsoutto radiiof1 Mpc ormore (White & Fabian sition of clusters is expected to reflect the universal mass 1995; David et al. 1995; Neumann & Bohringer, 1997; composition. Underthefairsamplehypothesis,acluster’s Squires et al. 1997; Mohr, Mathiesen & Evrard 1999). gasmassfraction,whichisalowerlimittotheitsbaryonic The mean cluster gas mass fraction within approximately massfraction,isthenalowerlimittotheuniversalbaryon the virial radius was calculated in Mohr et al. (1999) to fraction, i.e., f f . gas ≤ B be (0.0749 0.0005)h−3/2. Here, and throughout the pa- The luminous baryonic content of galaxy clusters is ± per, we assume the value of the Hubble constant to be mainly contained in the gaseous intracluster medium (ICM).Thegasmassisnearlyanorderofmagnitudelarger H◦ =100h km s−1 Mpc−1. Toderivethegasmassfractionfromx-rayimagingdata, than the mass in optically observed galaxies (e.g., White one is required to deproject the surface brightness into a et al. 1993, Forman & Jones 1982). Hence, the gas mass model for the density distribution. As the x-ray emis- is not only a lower limit to the cluster’s baryonic mass, it sion is proportional to the square of the gas density, the is a reasonable estimate of it. gas mass measurement can be biased by clumped, multi- The intracluster medium has largely been studied phase gas, should it be present. Also, the emission from through observations of its x-ray emission. The ICM is the cores of relaxed clusters may be dominated by cool- hot, with electron temperatures, T , from 5 to 15 keV; e rarefied,withpeakelectronnumberdensitie∼sofn 10−3 ingflows,whichcomplicatethe interpretationofthex-ray e ≃ 1Harvard-SmithsonianCenter forAstrophysics,60GardenSt.,Cambridge,MA02138 2DepartmentofAstronomy&Astrophysics,5640S.EllisAve.,UniversityofChicago,Chicago,IL60637 3SpaceScienceLaboratory, ES84,NASAMarshallSpaceFlightCenter,Huntsville,AL35812 4DepartmentofAstronomy,UniversityofAlabama,Huntsville,AL35899 5DepartmentofPhysics,UniversityofCalifornia,Berkeley,CA94720 1 2 data and may bias the result strongly if not taken into gas mass fractions from interferometric SZ observations. account (Allen 1998; Mohr et al. 1999). In addition, the Here, the shape parameters are derived directly from the x-ray surface brightness is diminished in proportion to its SZdatasetratherthanfromanx-rayimage. Weapplythis distance; S 1/(1+z)4, where z is the redshift of the methodtothe SZeffectmeasurementstowardsthecluster x ∝ cluster, neglecting experiment-specific K-corrections, and Abell 370, which were made as part of an SZ survey of so itbecomesincreasinglydifficult to makesensitivex-ray distantclusters,the firstresults of whichwerereportedin measurementsoftheICMastheclusterredshiftincreases. Carlstrom, Joy & Grego (1996), and are further reported We presentascheme for measuringthe gasmass fractions in Carlstrom et al. (1997). We choose this primarily be- with the Sunyaev-Zel’dovicheffect which is different from causeithasbeenstudiedatopticalandx-raywavelengths, the x-ray method in a number of ways, and also provides and so allows a comparison of the SZ data with other ob- an independent measurement of f . servations. Upondetailedinvestigation,itisapparentthat g The Sunyaev-Zel’dovich(SZ) effect is a spectral distor- thisclusterisoneofthemostdifficultinoursampletoan- tion of Cosmic Microwave Background (CMB) radiation alyze, with significant ellipticity and complicated optical duetoscatteringofCMBphotonsbyhotplasma(Sunyaev and x-ray structure; as such, it serves as a test of the gas & Zel’dovich 1970). The SZ effect can be detected signifi- mass fraction analysis method, which we plan to use on a cantlyingalaxyclusters,wheretheionizedintraclustergas large sample of clusters. serves as the scattering medium. A small fraction, 1%, With the interferometric SZ measurements, comple- ≤ of CMB photons are inverse-Compton scattered and, on mented by observationsat other wavelengths,we measure average, gain energy. At frequencies less than about 218 the cluster’s gas mass fraction in two ways. First, we cal- GHz, the intensity of the CMB radiation is diminished culate the surface gas mass from the SZ measurements ascomparedto the unscatteredCMB,andthe SZeffectis and the surface total mass from strong gravitationallens- manifestedasabrightnesstemperaturedecrementtowards ing observations and models. Second, we measure the gas the cluster. This decrement, ∆T , has a magnitude pro- mass fraction in a manner similar to x-ray analyses. The SZ portional to the total number of scatterers, weighted by gas mass is inferred from a deprojected model; the total their associatedtemperature, ∆TSZ n T dl, where n mass is determined from the spatial distribution of the is the number density of electrToCnMs,BT∝is thee eelectron teme- gas under the isothermal hydrostatic equilibrium (HSE) e R perature, T is the temperature of the CMB, and the assumption. We test the assumptions made in the HSE CMB integration is again along the line of sight. Note that this analysis by comparing the total cluster masses derived in is simply proportionalto the integrated electronpressure. the two methods. Also,themagnitudeoftheSZdecrementisindependentof The optical and x-ray observations of this cluster are redshift,soasthelongastheclusterisresolved(theexper- discussed in Section 2, and the SZ observations are dis- iment’scharacteristicbeamsizeisnotlargerthantheangle cussed in Section 3. The method for modeling the SZ subtended by the cluster), the SZ effect can be measured data is presented in Section 4, and the gas mass fraction towards arbitrarily distant clusters. results and the systematic uncertainties are discussed in A cluster’s gas mass is directly proportional to its in- Section 5. The cosmological implications of the results tegrated SZ effect if the gas is isothermal. So under and plans for future work are discussed in Section 6. the isothermality condition, an image deprojection is not 2. OPTICALANDX-RAYOBSERVATIONS strictly requiredto obtain the gas mass. The cluster’s gas The spatial distribution and redshifts of a number of mass fraction can be calculated by comparing the inte- Abell 370’s constituent galaxies have been measured op- grated SZ decrement, in effect a surface gas mass, to the tically. Mellier et al. (1988), using their 29 best galaxy totalclustermassinthesamevolume. Thetotalmasscan spectra, find Abell 370 has a mean redshift z = 0.374 bemeasuredwithstrongorweakgravitationallensing,for and velocity dispersion σ = 1340+230 km s−1. These example. The SZ images may also be deprojected to infer v −150 observations also show that the cluster is dominated by the three-dimensional gas mass and the HSE mass. Since two giant elliptical galaxies, at (α = 02h 39m 52.7s, the SZ decrement is directly propotional to the electron J2000 δ = 01◦ 34′ 20.3′′) and (α = 02h 39m 53.2s, density, the SZ image deprojection will not be affected J2000 J2000 δ = −01◦ 34′ 57.6′′), with a projected separation of strongly by clumped gas. Thus, the cluster’s gas mass J2000 about 40′−′ in the north-south direction. The major axes fraction can then be measured as a function of cluster ra- of these two dominant galaxies are also oriented nearly dius as well. north–south. TheMellieretal.opticalisoplethmapshows RecentclustergasmassfractionmeasurementsfromSZ twodensitypeaks,withsimilarnorth-southorientation,al- effect observations are presented in Myers et al. (1997). thoughthesearenotcenteredonthetwoellipticalgalaxies. In this work, the integrated SZ effect is measured using The29galaxyvelocitiesdonotappeartobedistributedin a single radio dish operating at centimeter wavelengths. a Gaussian manner, although the statistics on this mod- The integrated SZ effect is used to normalize a model for est sample are not conclusive. Should this be significant, the gas density from published x-ray analyses, and this its may suggest that the cluster is bimodal or that it is gas massis comparedto the published total massesto de- not yet fully relaxed, and the assumption of a simple one- termine the gas mass fraction. For three nearby clusters, componentmodelinhydrostaticequilibriumwouldnotbe A2142, A2256 and the Coma cluster, Myers et al. find a gas mass fraction of (0.061 0.011)h−1 at radii of 1-1.5 appropriate. Themeanvelocitiesinthetwoisoplethpeaks h−1 Mpc;fortheclusterAbel±l478,theyreportagasmass are not found to differ significantly, however, nor are the fraction of (0.16 0.014)h−1. calculated barycenters of the two peaks, and so it is diffi- ± cult to make a firm conclusion on the state of the cluster In this work, we describe a method to calculate cluster based on these data alone. 3 Abell 370 was the first cluster found to have an asso- extended emission,andso may not be resolvedin this im- ciated strongly gravitationally lensed arc (z = 0.725; age. It may also be explained by source variation; if the arc Soucail et al. 1988). In addition to this giant arc, which point source is an AGN, its surface brightness may have apparently consists of multiple images of a single source, significantlyvariedbetweenthe1979IPCobservationand threefaintpairsofimages,eachpairpresumedtobelensed the 1994-1995HRI observations. The image looks less el- images of a single galaxy(Kneib et al. 1993),anda radial liptical than does the HRI image, but because the point arc (Smail et al. 1996) have been found to be associated sourceemissioncannotbe excludedfromthe cluster emis- with this cluster. Gravitational lens models have been sion in the IPC data, it is not useful for constraining the used to map the totalmass distribution in the coreregion spatial distribution of the cluster gas. of Abell 370. Kneib et al. (1993) find the position and Abell 370 has also been observed with the ASCA x- elongation of the giant arc and the positions of the multi- ray satellite. Using 37.5 kiloseconds of ASCA GIS/SIS ple images are best reproduced by a bimodal distribution observations, Mushotzky & Scharf (1997) determined the of matter, with the two centers of mass nearly coincident emission-weighted average temperature of the cluster to with the cluster’s two large elliptical galaxies. A subse- be7.13+1.05 keV.Analyzingthesamedata,Ota,Mitsuda, −0.83 quently discoveredradialarcis consistentwith this model & Fukazawa (1998) find the emission-weighted tempera- without modifications (Smail et al. 1996). ture to be 6.6+1.1 keV. Both of these values are system- −0.9 Abell370wasobservedwiththeHighResolutionImager atically lower than the value obtained in the first analysis (HRI) ofthe ROSATx-raysatellite. Inthe ROSATband- of these data by Bautz et al. (1994), who obtain a value pass (0.1-2.4 keV), x-ray Bremsstrahlung emission from of 8.8 0.8 keV. The discrepancy is presumably because a plasma at gas temperatures typical of clusters depends the Mu±shotzky et al.and Ota et al.analysesused the val- onlyweaklyonthegastemperature. InFigure1,weshow ues of the response matrices of the ASCA telescope and an HRI image of A370, the result of 30 kiloseconds x-raydetectorswhichhavebeenrefinedsincetheBautzet ∼ of observation time. The bright source to the north, at al. analysis (M. Bautz, private communication). For our αJ2000 = 02h 39m 55s, δJ2000 = −01◦ 32′ 33′′, is coin- analysis,we adopt the Ota et al. value of 6.6+−10..19 keV, be- cident with what appears in the Digital Sky Survey to cause this published analysis includes detailed discussion be a nearby elliptical galaxy, NPM1G -01.0096 (Klemola, of the spectral fitting procedure. Jones & Hanson 1987), and does not appear to be asso- The brightpointsourceevidentintheROSATHRI im- ciated with A370. The image indicates that the intra- age has not been removed from the ASCA spectra in any cluster medium also is elongatedin the north-southdirec- ofthe threepublishedanalyses. Ifthis emissionoriginates tion, and the emission is strongly peaked at the position in Bremsstrahlung emission from hot gas in the ellipti- of the southern dominant elliptical galaxy, numbered 35 calgalaxy,itscharacteristictemperatureisexpectedtobe inMellieretal.(1988),andisoffsetfromthe centerofthe lessthan1or2keV(Sarazin&White 1988andreferences largescaleemission. The clustergasdistributiondoes not therein). Since the sensitivity of ASCA is optimized for appear clearly bimodal at this resolution. higher energies than this, we would not expect the mea- We determine the x-ray surface brightness from the sured temperature to be greatly contaminated. We assess ROSAT HRI image, which has been filtered to include whetherthesourceisextended,andfindthatitispointlike only PHA channels 1-7 in orderto reduce the background attheresolutionofthe HRIwitha2.4σ certainty. Asthe level (David et al. 1997) and blocked into 8′′ 8′′ pixels. source is unresolved, this emission is also consistent with A circular region with a radius of 24′′ cent×ered on the a galactic cooling flow; if this were the case, the tempera- pointlike x-ray source to the north of the cluster was ex- ture again shouldn’t be greatly contaminated. Should the cluded from our modeling. We fit a β-model to the data; emission originate from an AGN in the galaxy, though, this model described in more detail in Section 4, which the effect on the measured temperature is more difficult parametrizes the x-ray surface brightness with a core ra- to predict, and will depend strongly on the index of the dius,θc,andapowerlawindex,β. Thesurfacebrightness power law of the AGN spectrum. of a spheroidally symmetric, isothermal gas distribution Withoutfurthermeasurements,wecannotruleoutthat will have ellipticalcontours,andso we also fit for the axis the cluster’s spectrum is contaminated by emission from ratio. an AGN. It is also possible that if the cluster is in We find the best-fit axis ratio of the x-ray image to fact bimodal with two distinct subclusters, the measured be 0.73 0.03, with major axis exactly north-south, and emission-weighted average T will be a value between the the cent±er position to be αJ2000 = 02h 39m 53s, δJ2000 = temperaturesof the two subcelusters. Lackingmoredefini- 01◦ 34′ 33′′. The best fitting shape parameters for the tive measurements,we continue with ourassumptionthat −elliptical model are β= 0.70±0.03 and θc= 85.0±3.5′′. If the gas is isothermal at kT =6.6+−10..19 keV. The estimated the axis ratio is set to equal 1.0, the best fitting shape effectsofthedescribeduncertaintiesontheresultsaredis- parameters are β= 0.72 0.03 , θc= (70.1 3.2)′′ (90% cussed in Section 5.3. ± ± confidence on a single parameter). We compare A370’s measured line-of-sight velocity dis- Abell 370 was also observed with the Einstein IPC for persion with our adopted gas temperature, and find 4000seconds. IntheIPCimage,thenorthernpointsource β µm σ2/kT = 1.87+1.02. If the energy per unit is not apparent. This may be because the FWHM of the msapsecs c≡ontainpedvin tehe galaxi−e0s.5is9equal to that in the gas, instrument pointspreadfunction is about90′′ atthe mid- and the cluster is spherical, we expect β to be 1. If spec dle of the 0.4-4.0 keV spectral range of the instrument. the temperature measurement is biased low, e.g., fr∼om an (It improves to about 60′′ at higher energies.) The point AGN, this will contribute to the β discrepancy. Elon- source is offset by less than 120′′ from the center of the gation of the cluster potential alonsgpetche line of sight or a 4 alignment of two (or more) subclusters will also enhance 0.725 0.07mJy. Thepointsourcemodelisremovedfrom ± the measured β . theu-v orspatialfrequency,datasetinordertoconstruct spec Theopticalandx-rayobservationssuggestthatA370is the SZ decrementmap. A Gaussiantaper withFWHM of not a relaxed cluster, and may be heavily substructured, 1.2kλ is appliedin theu-v plane to the combineddataset but due to insufficient signal-to-noise ratios and contam- and map was CLEANed, restricting the CLEAN compo- inating sources, these observations still leave some ambi- nents to the central200′′, about the size of the decrement guity. in the primary. The map presented in Figure 2a is made 3. SZOBSERVATIONS witharestoringbeamofGaussianFWHM59′′ x86′′,with contourintervalsof1.5σ. The rms noiseinthe mapis50 We observedAbell 370for 50hoursovernine days with µ Jy beam−1, or 14.9 µK, and the integrated flux density the Owens Valley Radio Observatory (OVRO) Millimeter of the source is 1.54 0.17 mJy. Arrayin 1996August and43hours oversix days with the We employed−nine±of the 6.1 meter telescopes of Berkeley-Illinois-Maryland Association (BIMA) Millime- the BIMA array for SZ observations, using the same terObservatoryin1997June-August. Wesetourpointing centimeter-wave receivers we used at OVRO. We center center to αJ2000 =02h 39m 52.5s, δJ2000 = 01◦ 34′ 20′′, the 800 MHz bandwidth of the BIMA digital correlator − the position of the northern dominant elliptical galaxy. at 28.5 GHz. We achieved system temperatures of 30-55 We outfitted the millimeter telescopes with centimeter- K, scaled to above the atmosphere, depending on eleva- wavelength receivers to broaden the instrument’s angu- tionandatmosphericwatercontent. Holographicantenna lar resolution to the large angular scales typical of clus- pattern measurements were also made with this system. ters. The receivers are based on low noise High Electron The primary beam at BIMA is nearly Gaussian with a Mobility Transistor (HEMT) amplifiers (see Pospieszalski 396′′ FWHM. The data were edited and reduced using 1995), operating from 26 to 36 GHz. The single-mode re- the MIRIAD data reduction program (Sault, Teuben & ceiversareconstructedtorespondonlytocircularlypolar- Wright 1995), taking care to the remove any spurious in- ized light. The receivers interface with each array’s delay terferencefromthespectralchannelsanddroppingthelow lines and correlator. The standard observational scheme signal-to-noise channels at the spectral filter edges. entailed interleaving 20 minute cluster observations with Again using DIFMAP, we find the same point source observations of a strong reference source near the clus- at about 45′′ to the east of the pointing center in a high ter,whichallowsmonitoringoftheinstrumentalamplitude resolution (projected baselines longer than 1.4 kλ) image. and phase gain. The amplitude gain drifts were minimal; ThepointsourcemeasuredatBIMAin1997hasfluxden- thegainchangeswerelessthanapercentoveramanyhour sity of +0.70 0.17 mJy. The primary beam attenuation track, and the average gains were quite stable from day at the point s±ource position in the BIMA system is 6% to day. The amplitude gains were calibrated by compar- less than the attenuation at this position at OVRO∼, and ing the measured flux of Mars with that predicted by the so the point source observations are consistent with the Rudy model for Mars’ whole disk brightness temperature flux being constant in time. We model and subtract the (Rudy, 1987) at the observed frequency and the apparent point source from the data set, apply a Gaussian taper size of Mars as indicated by the Astronomical Almanac. with FWHM of 0.8 kλ in the u-v plane and construct a The Rudy model is a radiative transfer model with an es- map of the decrement. The map was CLEANed, restrict- timated accuracy of 4% (90% confidence) at centimeter ing the CLEAN components to the central 300′′ of the wavelengths. Data taken when the projectedbaseline was image, about the size of the primary beam. ∼The restoring within 3% of the shadowing limit were not used. beam used to make the map in Figure 2b. is a Gaussian The OVRO Millimeter array consists of six 10.4 me- withFWHM 91′′ x95′′,andthe contourspresentedareat ter telescopes. The OVROcontinuumcorrelatorwasused intervals of 1.5 σ. The rms noise in the map is 180 µJy to correlate two 1 GHz bands centered at 28.5 GHz and beam−1,or31.4µK,andtheintegratedfluxdensityofthe 30 GHz. The system temperatures ranged from 40-60 K, source is 4.34 0.52 mJy. depending on elevation and atmospheric water content. Itisins−tructiv±etocomparethegeneralcharacteristicsof The primarybeams of the telescopes were measuredholo- theSZ imagestothex-raymap. Atthe resolutionofboth graphically and can be approximated as Gaussian with a the OVRO and BIMA instruments, the gas is extended in full width at half maximum (FWHM) of 252′′. The data the north-south direction, though less markedly than in werecalibratedandeditedusingtheMMAdatareduction the x-raymap. The substructuresuggestedby the lensing package (Scoville et al. 1993) taking care to remove data and velocity dispersion data is not evident here. We pro- taken during poor weather or which show any anomalous ducemapsfromeachofthedatasets,increasingtheimage phase jumps. resolution, but the gas still does not show significant sub- We image the data using the DIFMAP package (Shep- structure orbimodality. The SZ decrementdistributionis ard, Pearson,& Taylor 1994). Examining images of A370 not peaked at the pointing center, but south of it, at a madewithprojectedbaselinesgreaterthan2.0kλ,wefind position about halfway between the two elliptical galaxies apointsource45′′ totheeastofthemapcenter(mapcen- (this will be quantified in the next section). The detailed ter is the pointing center) in both continuum bands. Its appearance of the SZ map, especially the shapes of the measuredfluxdensityinthe28.5GHzband,attenuatedby least significant contours, depends on the specific method the instrumental primary beam pattern, is +0.69 0.10 of point source removal and the CLEANing of the data, ± mJy and is +0.84 0.10 mJy in the 30 GHz band. This andsoitisnotusefultocomparethespatiallyfilteredand ± source, 0237-0147, is discussed further in Cooray et al. CLEANed SZ and x-ray images at such a level of detail. 1998. We combine the two continuum channels, and find Aswewilldiscussinthefollowingsection,thequantitative thebestfittingmodelisapointsourcewithfluxdensityof 5 analysisoftheSZdataisdoneusingtheu-v datadirectly. If the cluster gas is isothermaland its symmetry axis is in theplaneofthesky,thiselectrondensitydistributionleads 4. MODELING to the following two-dimensional SZ temperature decre- In order to assess the data more quantitatively, we fit ment: a parametrized model to the SZ brightness distribution. 1−3β θ2 2 2 Since the SZ data are taken in the spatial frequency, or η ∆T (θ)=∆T(0) 1+ , (4) u-v, domain, we do the model comparisons in u-v domain θc2! as well. We measure the SZ temperature decrement in units of where θ = η/D , D is the angular diameter distance, η A A antenna temperature, T , which relates an observed in- θ = r /D , and ∆T(0) is the temperature decrement at a c c A tensity change to a Rayleigh-Jeans temperature change, zero projected radius, ∆Iν = 2kcν22∆Ta. In order to recover the true blackbody 2 −23β temperaturedecrement,the∆T measuredandreported l RJ hneortesstrhiocutllydibnetmheulRtiapylileedighb-yJeaanfascltiomrito.f 1T.h02e1S,Zastewmepaerre- ∆T(0)∝Tene◦Z 1+(cid:18)rc(cid:19) ! dl, (5) ature decrement, ∆TSZ, is proportional to the Compton where the integral, dl is along the line of sight. Formally, y-parameter, TCMB this integral extends from the observer along the line of kσ sight through the cluster infinitely; in practice, a cutoff T y = n (l)T (l)dl, (1) m c2 e e radius for the cluster is used. The β-model distribution e Z of the electron density projects to a β-model distribution wherek isBoltzmann’sconstant,σ istheThomsonscat- T ofthe SZ decrementif the systemobeys sphericalorellip- teringcrosssection,m istheelectronmass,n istheelec- e e soidal symmetry. tron density, T is the electron gas temperature, and the e We perform a χ2 analysis of the β-model, by compar- integral extends along the line of sight (dl). The propor- ingβ-modelstothecombinedBIMAandOVROdatasets. tionality depends on the observing frequency; it depends We vary the following parameters: centroid position, β, also on the electron temperature when relativistic correc- θ , axis ratio (defined to be < 1), position angle (defined tions are included6 (Rephaeli 1995, Challinor & Lasenby c counter clockwise from north), and ∆T(0). The position 1998). and flux density of the radio-bright point source are also At 28.5 GHz, ∆TRJ = 1.92 y in the non-relativistic TCMB − fit. The fitting procedure is conducted in several steps. Rayleigh-Jeans approximation, where we adopt the value The model corresponding to each set of the seven clus- oftheCMBofFixsenetal.(1996)derivedfromtheCOBE ter fit parameters and the three point source parameters FIRASmeasurements,T =2.728K.Includingtherel- CMB is multiplied by the primary beam response. The Fourier ativistic corrections for kTe =6.6 keV, T∆CTMSZB =−1.86 y. transform of the result is compared directly with the in- We fit a simple model to the SZ data. The β-model terferometerdata. Weusethe holographicallydetermined (Cavaliere&Fusco-Femiano1976,1978)isfrequentlyused primary beams when modeling the data, and the entire to fit the density profiles of galaxy clusters. The spheri- datasets are used to do the analysis. The inner u-v ra- callysymmetric,isothermalβ-modeldescribesne,theden- diuscutoffisdeterminedbythe shadowinglimit,the limit sity distribution of the gas, as varying as a function of where one telescope would partially block another, i.e., cluster radius, r: when the projected baseline is less than the diameter of a r2 −3β/2 telescope dish. For the BIMA data this limit is 0.58 kλ ne(r)=ne◦ 1+ r2 , (2) and for the OVRO data it is 1 kλ. The χ2 statistic is (cid:18) c(cid:19) minimized, and the best fit values are determined using a wherene◦ isthenumberdensityofthegasatthecenterof downhill simplex method. the spheroidally symmetric gas distribution; rc, the core Performing the fitting procedure, we find the best fit radius, is a characteristic size of the cluster; and β is the parametervaluesforA370are: θ =93′′,β =1.77,∆T(0) c power law index. = 609µK,andaxisratio=0.64withmajoraxisexactly Sincetheclusterappearsellipticalinprojection,wegen- no−rth-south. The best fit central position is 20′′ to the eralize the spherically-symmetric β-model and allow the south of the pointing center, at α = 02h 39m 53.2s, J2000 gas density profileto be spheroidallysymmetric, i.e., with δ = 01◦ 34′ 40.4′′, about halfway between the two J2000 biaxial symmetry. In the spheroidally symmetric model, giantellip−ticalgalaxies,and7′′ northofthefittedcentroid theelectronnumberdensityinaprolatespheroidisafunc- forthex-rayimage. Theχ2 statisticforthebestfitvalues tion of η2 = r2 +a2z2, where r and z are the radial and is135798for136136degreesoffreedom,yieldingareduced height coordinates in the cylindrical coordinate system, χ2 of 0.9975. As points of comparison, we fit to the data and a < 1 is the axis ratio of the two unique axes. If anullmodelforthe cluster (withthe pointsourcecompo- the spheroid is oblate, then the density is a function of nent), which gives a reduced χ2 of 0.9986; we also fit the η2 = a2r2 +z2, with a < 1. The electron density then cluster to a negative point source, finding a reduced χ2 of follows the distribution: 0.9986. η2 −3β/2 In order to determine the uncertainties in these fitted ne(η)=ne◦ 1+ r2 , (3) parameters, the χ2 statistic is derived for a large range (cid:18) c(cid:19) 6The change in spectral intensity due to the Sunyaev-Zel’dovich effect is calculated in Equation (4-8) Challinor & Lasenby (1998): ∆(TSZ/T)RJ = (eyxx−2e1x)2[xcoth(x/2)−4+θef(x)],wherex= khTνe andθe= mkeTce2. Thelasttermcorrectsforrelativisticeffects. At28.5GHz, f(x)=3.58. 6 of β, θ , and ∆T(0), keeping centroid position, position lowing the axis ratios of the two components to vary does c angle, andpoint sourceposition andflux fixed to the best not improve the fit. fit values. Presented in Figure 3a is a contour plot of the β and θ fit results, where ∆T(0) is allowed to assume its 5. ANALYSIS c best fit value at every pair of β and θ . The axis ratio is c 5.1. Comparison of SZ Gas Mass to the Strong Lensing left to assume its best fit value at each set of β, θ , and c Mass Estimate ∆T(0); it varies from 0.60 to 0.68 for points with ∆χ2 < 5 fromthe best fit point. The centroidposition, when left As indicated by Equation 1, the SZ brightness at any to assume its best fit value at each point, does not ap- point in the two-dimensional projected image is simply preciably change, varying less than 5′′. Figure 3b shows proportional to the integrated electron density along the the contours derived when fitting the data to a spherical lineofsight,ifthegasisisothermal. Undertheisothermal model for the gas distribution, i.e., the axis ratio is fixed assumption, we can directly measure the total number of toavalueof1.0;inthiscase,thebestfitparametervalues electrons in the gas contained in the cylindrical volume of are θ = 34′′, β = 0.86, ∆T(0) = 785 µK, the reduced a chosen radius, the long axis of which is defined by the c χ2 statistic was 0.9976. − line of sight. The total mass in the ionized phase can be The full line contoursare markedfor ∆χ2 = 2.3,4.61, calculatedfromthisassumingavalueforthenumberofnu- and 6.17 which indicate 68.3%, 90.0%, and 95.4% confi- cleonsperelectron. Ifthegashassolarmetallicity,asmea- dence, respectively,for the two-parameterfit. The dashed sured by Anders & Grevesse (1989), the nucleon/electron lines indicate ∆χ2 = 1.0, 2.71, and 6.63; the projection ratiois1.16. Thenucleon/electronratiochangeslessthan onto the β or θ axis of the interval contained by these 1% for values of the metallicity from 0.1 to 1.0. c contours indicate the 68.3%, 90% and 99% confidence in- The interferometric measurements recover much of the terval on the single parameter. The values of ∆T(0) for total SZ decrement on the angular scales measured, i.e. which∆χ2 <1ateach(β,θ )pointrangeabout15%from the integrated flux from the data comprises 40-50% (de- c the best fitvalue. This illustratesthat β and θ arecorre- pendingontheinstrument)oftheintegratedfluxinabest c lated strongly and are not individually well constrained fitmodeltothesedata. Andsotheunmodeledinterferom- by these data. This poor constraint is at least partly eter data provides a strong lower limit to the integrated due tothe lowdeclinationofA370,whichmakessampling SZdecrementatthe angularscalesofinterest. The model non-redundant spatial frequencies difficult at OVRO and is fitted to the u-v data in order to estimate the full, two- BIMA. The shape parameters fit from the x-ray data are dimensional decrement, and thence the surface gas mass. notconsistentwiththosefromthe SZdata. This couldbe This method does not assumeanything aboutthe state of a result of shocks and complexity in the gas phase which the ICM other than that it is isothermal. affect the x-ray emission and SZ effect differently. This We fit the SZ data in the three-dimensional parameter discrepancy is difficult to resolve with the information at space 0.4 <β < 4.0,10′′

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