Parametric Strong Gravitational Lensing Analysis of Abell 1689

43 Pages·2008·1.36 MB·English

Mon.Not.R.Astron.Soc.000,1–43(????) Printed4February2008 (MNLATEXstylefilev2.2)
Parametric Strong Gravitational Lensing Analysis of Abell 1689
A. Halkola1⋆, S. Seitz1 and M. Pannella2
1Universita¨ts-SternwarteMu¨nchen,Scheinerstraße1,D-81679Mu¨nchen,Germany
2Max-Planck-Institutfu¨rextraterrestrischePhysik,Giessenbachstraße,Postfach1312,D-85741Garching,Germany
6
0
0
Accepted????.Received????;inoriginalform????
2

y
a ABSTRACT
M
WemeasurethemassdistributionofgalaxyclusterAbell1689within0.3Mpc/h ofthe
70
9 cluster centre using its strong lensing effect on 32 backgroundgalaxies, which are mapped
1
in altogether 107 multiple images. The multiple images are based on those of Broadhurst

etal.(2005a)withmodificationstobothincludenewandexcludesomeoftheoriginalimage
1
systems.Theclusterprofileisexploredfurtheroutto 2.5Mpc/h withweaklensingshear
v ∼ 70
measurementsfromBroadhurstetal.(2005b).
0
7 The masses of 200 cluster galaxies are measuredwith FundamentalPlane in order to
∼
4 accuratelymodelthesmallscalemassstructureinthecluster.Theclustergalaxiesaremod-
5 elled as elliptical truncated isothermal spheres. The scaling of the truncation radii with the
0 velocity dispersions of galaxies are assumed to match those of i) field galaxies (Hoekstra
6 etal.2004)andii)theoreticalexpectationsforgalaxiesindenseenvironments(Merritt1983).
0 The darkmatter componentof the cluster is describedby either non-singularisothermalel-
/
h lipsoids(NSIE)orellipticalversionsoftheuniversaldarkmatterprofile(ENFW).Toaccount
p forsubstructureinthedarkmatterweallowfortwodarkmatterhaloes.
- ThefittingofasingleisothermalspheretothesmoothDMcomponentresultsinavelocity
o dispersionof1450+39km/sandacoreradiusof77+10kpc/hwhileanNFWprofilehasaavirial
r 31 8
t radiusof2.86 0.1−6Mpc/h andaconcentration−of4.7+0.6.
s ± 70 0.5
a Thetotalmassprofileiswelldescribedbyeithera−nNSISprofilewithσ=1514+18 km/s
17
: and core radius of r =71 5 kpc/h , or an NFW profile with C=6.0 0−.5 and
v c 70
± ±
i r200=2.82 0.11Mpc/h70. The errorsare assumed to be due to the errorin assigningmasses
X totheindi±vidualgalaxiesinthegalaxycomponent.Theirsmallsizeisduetotheverystrong
r constraintsimposedbymultipleimagesandtheabilityofthesmoothdarkmattercomponent
a
toadjusttouncertaintiesin thegalaxymasses. Thederivedtotalmassisin goodagreement
withthemassprofileofBroadhurstetal.(2005a)despitetheconsiderabledifferencesinthe
methodologyused.
Using also weak lensing shear measurements from Broadhurst et al. (2005b) we
can constrain the profile further out to r 2.5 Mpc/h . The best fit parameters change
70
∼
to σ=1499 15 km/s and r =66 5 kpc/h for the NSIS profile and C=7.6 0.5 and
c 70
± ± ±
r =2.55 0.07Mpc/h fortheNFWprofile.
200 70
±
UsingthesameimageconfigurationasBroadhurstetal.(2005a)weobtainastronglens-
ingmodelthatissuperiortothatofBroadhurstetal.(2005a)(rmsof2.7”comparedto3.2”).
Thisisverysurprisingconsideringthelargerfreedominthesurfacemassprofileintheirgrid
modelling.
Key words: gravitational lensing – cosmology:dark matter – galax-
ies:clusters:individual:Abell1689
1 INTRODUCTION a straightforward mass determination using the gravitational lens
effect on background galaxies, the dynamics of cluster members
Abell 1689 at a redshift of 0.18 is one of the richest clusters
andtheX-rayemissionoftheintra-clustergas.Nevertheless,these
of galaxies on the sky. Its closeness and richness should allow
methodshavecomeupwithstrikinglydifferentresultsinthepast.
ObservationswiththeChandra(Xue&Wu2002)andXMM-
Newton (Andersson & Madejski 2004) satellites yield masses
⋆ E-mail:[email protected] roughly a factor 2 lower than strong lensing measurements (e.g.
2 A. Halkola,S.Seitzand M. Pannella
Dye et al. 2001). The first line-of-sight velocity measurements Broadhurst etal.(2005b) alsocarriedoutacombinedstrong
of cluster members (Teague et al. 1990) had resulted in a veloc- andweaklensinganalysis.Theyruleoutasoftenedisothermalpro-
ity dispersion of σ 2355 km/s, compared to a value of only fileata10-σlevel.Accordingtotheirwork,auniversaldarkmatter
≈
σ 1028 km/s for an isothermal fit to weak lensing measure- profile(NFW)withaconcentrationofC =13.7+1.4 andavirialra-
≈ 1.1
ments by, for example, King et al. (2002). Therefore the isother- diusofr =2.04 0.07Mpc/h fitstheshear−andmagnification
vir ± 100
malspherevelocitydispersionestimatesoftheclusterfromstrong based on weak and strong lensing data well. They point out that
lensing,X-rayandweaklensinganalysisoriginallyimpliedamass the surprisingly large concentration for A1689 together with re-
estimatedifferentatafactorofupto5. sultsfromotherclusterscouldpointtoanunknownmechanismfor
Theapparentlyincompatibleweakandstronglensingresults theformationofgalaxyclusters.Thestrongrejectionofanisother-
for an isothermal sphere are most puzzling since both methods malspheretypeprofilehasalsobeenreportedingalaxyclusterCl
measure the (same) line-of-sight projected two dimensional sur- 0024+1654byKneibetal.(2003)whoprobedtheclusterprofileto
facemassdensityofthecluster.Ifparametersobtainedwiththese verylargeclustercentricradius(5Mpc).
two methods on different angular scales do not agree for agiven Oguri et al. (2005) however have demonstrated that halo
massprofile,itimpliesthati)theassumedmassprofiledoesnotde- triaxiality can lead to large apparent central concentrations, if
scribethetruemassdistributionatall,orii)thatoneanalysis(more these haloes are analysed assuming spherical symmetry. This is
likelytheweaklensinganalysis)suffersfromunderestimatedsys- because a highly elongated structure along the LOS imitates a
tematicerrors.Broadhurstetal.(2005a)haveshownthatthisisthe highcentraldensityifinvestigatedinprojectiononly.Allowingfor
caseforA1689,i.e.thatinpreviousanalysesthecontaminationof triaxialitythecentralconcentrationislesswellconstrainedandnot
the’backgroundgalaxies’withclustermemberscouldhavebiased indisagreementwithresultsfromnumericalsimulationsofcluster
the lensing signal of the background galaxies. Their background massprofiles.
galaxiesshow(comparedtoClowe&Schneider(2001);Kingetal.
(2002))afactorofroughlytwohigherlensingsignalonlargescales Inourworkwewanttoaddressthefollowingpoints:
and make the order of magnitude mass estimate in the weak and
Doesoneobtainthesamemassprofilewithananalysisofthe
stronglensinganalysisagree. •
stronglensingeffectusingadifferentmethod?Weconcentrateon
ThediscrepantresultsfromtheclusterdynamicsandtheX-ray
differencesbetweengridmethodusedbyBroadhurstetal.(2005a)
data relative to the strong lensing analysis can potentially be ex-
andourparametricmethod.
plained,ifsomeassumptionsintheinterpretationofgalaxies’dy-
Are mass profiles from weak (WL) and strong (SL) lensing
namics and the gas X-ray emissivity, i.e. having one spherically •
atallcompatiblewitheachother,ordoesthecombinationofboth
symmetric isothermal structure indynamical equilibrium, arenot
observationsalreadyruleoutanNFWoranisothermalprofile?
valid. Indeed, Girardi et al. (1997) identified three substructures
Howmuchdoesthelargeconcentrationandthelevelofcom-
(usingspectroscopicdatafromTeagueetal.(1990))inAbell1689 •
patibility of the WL and SL results depend on the values of two
whicharewellseparatedinvelocitybutoverlapalongthelineof
outermostWLsheardatapoints?
sight.Thisreducedthepreviousvalueofthecluster’svelocitydis-
Howgoodistherelativeperformanceofanisothermalsphere
persion from 2355 km/s by Teague et al. (1990) to 1429 km/s, a •
vs. an NFW profile, based on both the strong and weak lensing
valueingoodagreementwithstronglensingresults.Evidencefor
analyses?
substructureandmergingwasalsofoundinvelocitydifferencesof
X-rayemissionlinesandinX-raytemperaturemapsbyAndersson Weaimtoinvestigateallthesepointswithparametriccluster
&Madejski(2004). Theypointedoutthatthe(byafactoroftwo models.Ourbasicassumptionsarethatsubstructurefollowsgalax-
low) X-ray mass estimatewould double if two equal mass struc- iesandtheclustermasscanbedescribedbymassassociatedwith
turesalongtheline-of-sightareresponsiblefortheX-rayemission thegalaxies (both luminous and dark) plus asmooth component.
in stead of just one structure. The X-ray surface brightness map, The multiple image configurations determine if there are one or
however, and the weak lensing data of A1689 indicate an almost more of these smooth components necessary, e.g. two haloes of
circular(2Dprojected)massdistributioncentredonthecDgalaxy. similarmasslikeinA370(seee.g.Kneibetal.(1993);Abdelsalam
Thisisnotnecessarilycontradictingthesubstructureresultssum- etal.(1998)),oramassivehalopluslessmassive,group-likecom-
marisedabove,aslongasthetwomajorcontributionsinmassare ponents.
onthesamelineofsight,andeachofthemisafairlyrelaxedstruc- Wedescribethesmoothcomponentby2parametrichalopro-
ture. The issue of relax/unrelaxed systems and the cluster X-ray files. Deviations from symmetry are accounted for by elliptical
temperature-massrelationisstudiedin10X-rayluminousgalax- deflectionangles (ENFW)or potentials(NSIE)depending on the
iesbySmithetal.(2005)usingChandraX-rayandweakandstrong model.Thefirsthaloprofileisthesocalleduniversaldarkmatter
lensing.Theyfindthatalargefractionoftheirclustersareexperi- profile(hereafterNFWprofile),anoutcomeofnumericalsimula-
encing, or recovering from, a cluster-cluster merger and that the tions of cold dark matter cosmologies (Navarro et al. 1996). The
scatter in thecluster X-ray temperature - mass relation issignifi- second isanon-singular isothermal ellipsoid(NSIE)which natu-
cantlylargerthanexpectedfromtheory. rallyreproducestheobservedflatrotationcurvesofbothlateand
Thecurrentstatusofthestrongandweaklensing,thedynam- early-typegalaxies.
ical and the X-ray mass estimates for A1689 are defined by the Theprofilesofclustergalaxiesaredescribedwithanelliptical
worksofBroadhurstetal.(2005a),Broadhurstetal.(2005b),Gi- truncated isothermal sphere profile (Blandford & Kochanek
rardietal.(1997)andAndersson&Madejski(2004)respectively. 1987) whose velocity dispersions are determined using both
Themassestimatesagreewellwiththeexception of thestilllow Fundamental-PlaneandFaber-Jacksonrelations.
X-raymassestimate.TheX-raymasscanbebrought inlinewith
the masses from other methods if two equal mass substructures Provided a range of plausible radial mass profiles are tested
alongtheLOSareresponsiblefortheX-rayemission(Andersson with parametric halo profiles significantly better performance of
&Madejski2004). gridmethods,liketheoneusedbyBroadhurstetal.(2005a)(also
ParametricStrongGravitationalLensing Analysisof Abell1689 3
Diego et al. (2005a,b)), can give clues to the existence of dark
Table1.Asummaryofthedatausedinthisstudy.
matter substructure not traced by galaxies (dark mini haloes) if
these dark haloes are numerous/massive enough to influence the Filter t(ks) #ofexposures psf(′′)
lensing observables on a relevant level. The difference can also F555W(WFPC2) 44.2 17 0.17
resultfromthedetailsofaparticularmodelling,e.g.thetreatment F814W(WFPC2) 5.0 5 0.20
of the cluster galaxy component and in the dark matter profiles F475W(ACS) 9.5 8 0.11
F625W(ACS) 9.5 8 0.10
usedinmodellingthesmoothDMofthecluster.
F775W(ACS) 9.5 8 0.10
F850LP(ACS) 16.6 14 0.11
In section 2 we give a brief summary of the data and data
analysis used in this paper. Section 3 describes our method
to obtain the best fitting lensing models, results are given and 2.1.2 HST-ACS
discussedinsection4.Wedrawconclusionsinsection5.
Theadvanced CameraforSurveyshasalargerfieldofview than
WFPC2atasimilarresolution.ThedatainfiltersF475W,F625W,
The cosmology used throughout this paper is Ω =0.30,
m F775WandF850LPcomefromHSTproposal9289byFord.The
Ω =0.70andH =70km/s/Mpc,unlessotherwisestated.
Λ 0 ”on the fly re-processing” (OTFR) provides flatfielded and cali-
brateddata.Theindividualexposuresweretransformedtoacom-
moncoordinatesystemusingPyDrizzleinpyraf.Thecosmicrays
wereagainremovedin2stagesaswiththeWFPC2data.
2 DATAANDDATAANALYSIS
We have used archived optical HST (WFPC2 and ACS) data in 2.2 ObjectCatalogues
filtersF555WandF814W(WFPC2)andF475W,F625W,F775W
The object catalogue was obtained with SExtractor (Bertin &
andF850LP(ACS).AsummaryofthedatacanbeseeninTable1.
Arnouts1996).TooptimisetheextractionparametersofSextractor
Therelativelywidefield-of-view(202”x202”)andhighresolution
-detectionthresholdandnumberofcontiguouspixels-aprocedure
(pixelscale0.05”) of ACSallow ustoprobe A1689over thearea
similartothatinHeidtetal.(2003)wasfollowed.Sourceswerede-
where most of the multiple images are formed. WFPC2 data are
tectedfromasignal-to-noiseweightedsumofthefourACSfilters.
used to further constrain the photometric redshifts in the central
Thephotometryforthedetectedsourceswasthendoneinallfilters.
regionofA1689coveredbytheobservations.
OutputofSExtractorusedlaterinthelensingmodelincludetotal
andaperturephotometryaswellassourceellipticitiesandtheirpo-
sitionangles.
2.1 DataReduction
2.1.1 HST-WFPC2
2.3 Spectroscopicredshifts
The WFPC2 data in filters F555W and F814W come from HST The redshift data are collected from several published studies of
proposal6004byTyson.Weusepipelineflatfieldedimageswhich Abell1689(Girardietal.1997,Baloghetal.2002,Ducetal.2002,
werecombinedusingiraf1tasksincombinationwithpsffittingcos- Golse20022).Intotal84spectroscopicredshiftswereavailablein
micrayrejectionalgorithmsdevelopedinhouse(Go¨ssl&Riffeser theACSfieldofA1689.ExceptGolse,thesestudiesconcentrateon
2002). The steps of data reduction were as follows. First all fea- theline-of-sight velocitiesofthecluster members inorder toob-
tureswithFWHMlessthan1pixelandahighsignaltonoisewere tainthevelocitydispersionandhenceanestimateofthedynamical
markedascosmicraysandnotusedinanyfurtheranalysis.Inthe massofthecluster.Theredshiftinformationisusedtogetsecure
second step the four chips of each WFPC2exposure were trans- clustermembersforthelensinganalysis,toexcludegalaxieswhich
formedtoasinglecoordinatesystem.Inthisstepboththegeomet- arenotpartoftheclusterandtocomparephotometricredshiftswith
ricaldistortionsoftheWFPC2chipsaswellastranslationandrota- spectroscopicones.
tionbetweenthedifferentCCDsandexposuresweretakencareoff.
ThedescriptionofHoltzmanetal.(1995)wasusedtoremovethe
geometricaldistortions.Thedifferentchipshaveslightlydifferent
2.4 Photometricredshifts
photometriczeropointsandbeforetheimageswerestackedallthe
imageswerenormalisedtothezeropointoftheplanetarycamera. We have calculated photometric redshifts using the method de-
Thestackingwasdone by takingakappa-sigma clipped meanof scribed in Bender et al. (2001). For the central regions we were
eachpixel.Itwasfoundduringthereductionprocessthatthetwo abletouseall6filters,otherwiseonlythe4availableACSfilters
stagecosmicrayrejectionwasnecessaryinordertoremoveallthe were used. Comparison between photometric redshifts and avail-
cosmicraysefficiently.Mostofthecosmicrayswereremovedby ablespectroscopiconesisshowninFigure1.Unfortunatelyspec-
psf-fittinginthefirststageandlargerpointlikecosmicrayswere troscopicredshiftsarenotnumerousandthosethatexistaremainly
removedinthestackingstagebythekappa-sigmaclipping. forclustermembers.ThemodelellipticalgalaxyintheSEDlibrary
usedtocalculatethephotometricredshiftsisbluerthanthereddest
clustermembersandhencetheclustergalaxiesarepushedtored-
shiftsslightlyhigherthanthatofthecluster.Theslopeofthecluster
1 IRAF is distributed bytheNational Optical Astronomy Observatories, redsequencecausesthelargespreadinphotometricredshiftsofthe
which are operated by the Association of Universities for Research in
Astronomy,Inc., undercooperative agreement with theNational Science
Foundation. 2 Availableathttp://tel.ccsd.cnrs.fr/documents/archives0/00/00/22/79/
4 A. Halkola,S.Seitzand M. Pannella
tometric redshifts are used to constrain redshifts of the multiple
images. For these the low redshift peak can be excluded because
gravitationallensingisinefficientifthebackgroundsourceisclose
tothelens.
3 LENSINGMODELS
InthissectionwedescribehowthelensingmodelsofA1689were
constructed;thedifferentmasscomponentsofthecluster,themul-
tipleimagesystemsandtheoptimisationofmodelparameters.The
lensingprofilesaredescribedindetailinappendixA.
3.1 ClusterGalaxyComponent
Thecluster galaxies wereselected usingthe redsequence method
supportedbyspectroscopicredshiftswhereavailable.Thespectro-
scopic data are taken from Girardi et al. (1997) (using data from
Teagueetal.(1990)),Baloghetal.(2002)andDucetal.(2002).We
haveexcluded 6objectsfrom thecluster catalogue obtained with
Figure 1. Comparison between spectroscopic and photometric redshifts. theredsequencemethodbasedonfore/backgroundobjectslistedin
Noticehowthereddestclusterellipticals havebeenpushedtohigherred- Baloghetal.(2002)andDucetal.(2002).Fig.3showsacolour-
shiftstocompensatefortherelativebluenessofthemodelellipticalSEDin magnitudediagrammeofthecluster.WechosetousefiltersF475W
theSEDlibraryusedtocomputethephotometricredshifts.Filledsymbols andF775WfromtheACSobservationsastheclusterredsequence
showthemostlikelyredshift,opensymbolsthesecondmostlikelyforthe isseen particularly clearly in these two filters. Galaxies included
objectswhereitisclosertothetrueredshift.
inthelensinganalysis aremarked by trianglesinFigure3.Solid
trianglesshowthegalaxieswhichhavebeenspectroscopicallycon-
firmedtobeclustermembersinoneormoreofthereferredpapers.
Tofindmore members a redsequence inthe CMdiagramme was
determined by fittingaline to thebright end of the redsequence.
ThosegalaxieswhoseF475W-F775Wcolourdeviatedbylessthan
0.3magnitudes fromthe fittedsequence wereincluded ascluster
members(regionbetweenthetwoinclineddashedlinesinFigure
3).
Figure4shows thepositionsof cluster members inthefield
ofA1689(usingthesamesymbolsasinFigure3).
The cluster galaxies were modelled with an elliptical BBS
profile (see appendix A for further details). We have treated the
twoparameterprofileasaoneparameterprofilebyassumingthat
s = s(σ). We find the fit to multiple images with a galaxy com-
ponent described by the values for s (=185 h 1 kpc) and σ
−
(=136km/s) andthescaling(s = s ∗(σ/σ )2)found for galax∗-
ies inthe Red-Sequence Cluster Sur∗v×ey (Hoe∗kstra et al. 2004) is
very poor. This means that the galaxy haloes in A1689 must be
significantlystripped.Thestrippingofgalaxiesinclusterenviron-
menthasbeenreportedearlierbye.g.Natarajanetal.(1998,2002)
whohaveusedgalaxy–galaxylensinginclusterstostudytheprop-
Figure2.PhotometricredshiftdistributionofobjectsinthefieldofA1689. ertiasofgalaxyhaloesin6clustersatredshiftsZ=0.17-0.58.They
Theclusterisclearly visibleasapeakatz 0.2.Theinlaid figureshows
∼ foundstrongevidencefortidallytruncatedhaloesaroundthegalax-
objects considered as cluster galaxies. In a number of cases the 4000 Å
iescomparedtogalaxiesinthefield.
breakofclustergalaxieswasmisinterpreted asLymanbreakandhencea
Webaseourmodelsforthetidalstrippingofgalaxiesonob-
redshiftz 3wasassigned(seetextfordetails).
∼ servationalworkbyHoekstraetal.(2004)forgalaxiesinthefield
(s σ2)andtheoreticalexpectationsforgalaxiesinclusterenvi-
∝
cluster members. The photometric redshift distribution of all ob- ronment (s σ) (Merritt 1983). Weonly take the scaling of the
∝
jectsisshowninFigure2.Theclusterappearsasanarrowpeakat truncationradiiwiththevelocitydispersionsofthegalaxiesfrom
z 0.2ontheredshifthistogram.Objectsatz 3-4areeithergrav- theaforementioned works and findthenormalisation of thetrun-
∼ ∼
itationally lensed background galaxies or cluster galaxies whose cationradius, s ,tofitthemultipleimages.Thetwoscalinglaws
4000ÅbreakwasmisidentifiedasLymanbreak.Thisplacesthem adoptedinthep∗aperarethen1) s = s (σ /136km/s)2 and2)
toredshifts 3.Toclearlydiscriminatebetweenthetwobreakswe s= s (σ /136km/s),whereforbot∗h×scalignagl lawsandallMod-
∼ ∗× gal
would need both redder and especially bluer filtersthan the ones elswehavefoundthes thatbestreproducestheobservedmultiple
available. This confusion is not important in our case since pho- images. The scaling la∗w for the truncation of galaxies in cluster
ParametricStrongGravitationalLensing Analysisof Abell1689 5
environmentwillbetreatedinmoredetailinaforthcomingpubli-
cation(Halkolaetal.,2006inpreparation).
Thepositions,ellipticities(ofsurfacebrightness)andposition
anglesweretakenfromSExtractoroutputparameters.Thevelocity
dispersions of cluster galaxies were determined mostly using the
FundamentalPlane.ForasmallnumberofgalaxiesalsotheFaber-
Jacksonrelationwasused.
3.1.1 CentralVelocityDispersionsofClusterGalaxies&Halo
VelocityDispersions
The Fundamental Plane (hereafter FP) links together, in a tight
way,kinematic(velocitydispersion),photometric(effectivesurface
brightness) and morphological (half light radius) galaxy proper-
ties(Dressleretal.1987;Djorgovski&Davis1987;Benderetal.
1992).Weassumethatthecentralvelocitydispersionsofagalaxy,
asderivedfromtheFP,isequaltothehalovelocitydispersion,and
thatmassindiskcanbeneglected.
TheFPrelationallowsustoestimatethevelocitydispersionof
galaxiesmoreaccuratelythanthestandardFaber-Jacksonrelation
approach (Faber & Jackson 1976). We model the 2–dimensional
lightprofilesofclustergalaxieswithPSF–convolvedSersic(Sersic
Figure3.Cluster redsequence inACS F475W/F775W colour-magnitude 1968)profilesusingtwopackages,GALFIT(Pengetal.2002)and
diagramme. The solid triangles represent cluster galaxies with spectro- GIM2D (Simard et al. 1999), to have a better handle on the sys-
scopicredshifts,opentrianglesallotherobjectswhichwereconsideredas tematics.TheanalysiswasperformedontheF775WACSimage.
clustermembersinthelensinganalysis.Theremainingobjectsarerepre- 176objectswithABmagnitudesbrighterthan22werefitted.The
sentedbydots.Aclearclusterredsequencecanbeseen.Seesection3.1for pointspreadfunctionusedtoconvolvethemodelswasderivedby
detailsontheselectioncriterionforclustermembership.
stackingstarsidentifiedinthefield.Theresultscoming out from
thetwocompletelydifferentsoftwaresagreeverywell.
In order to be able to use a FP determination for cluster
galaxies at redshift 0.2 in restframe Gunn r filter (Jørgensen
∼
et al. 1996; Ziegler et al. 2001; Fritz et al. 2005), all the ob-
served F775W surface brightnesses (extinction corrected) were
AB
converted to restframe Gunn r ones and corrected for the
GT
cosmological dimming. Since the observed F775W passband is
closetorestframeGunnrattheredshiftofA1689,theconversion
factorbetweenobservedF775WandrestframeGunnrissmall.
Themeanobservedsurfacebrightnesswithinr is:
e
µ =F775W +2.5log(2π)+5log(r ) 10log(1+z),
h eiF775W observed e −
(1)
wherethelasttermcorrectsforthedimmingduetotheexpansion
oftheUniverse.ItisthenconvertedtorestframeGunnr by:
GT
µ = µ A +K(r,F775W,z)+GT , (2)
h eir h eiF775W− F775W corr
TheGalacticextinctioncorrection A iscalculated from
F775W
thelistofA/E(B-V)inTable6ofSchlegeletal.(1998),alongwith
theirestimateofE(B-V)calculatedfromCOBEandIRASmapsas
well asthe Leiden-Dwingeloo maps of HI emission. Weadopted
forA avalueof0.06.
F775W
Figure4.PositionsofclustergalaxiesinthefieldofAbell1689.Thesym- The”k-correctioncolour”,K(r,F775W,z),isthedifferencebe-
bolscorrespondtothoseinFig.3.Originofthecoordinate systemison tweenrestframeGunnrandobserved F775Wmagnitudeandin-
thecentralcDgalaxy.Theboxwidthis200”(correspondingto 625kpcat cludes also the 2.5log(1+z) term. It was obtained by using an
∼
z=0.18).
ellipticaltemplatefromCWW(Colemanetal.1980)andsynthetic
SEDsobtainedforoldstellarpopulations(10Gyr,i.e.z =5ob-
f
served at z=0.2) with the BC2003 Bruzual and Charlot models
(Bruzual&Charlot2003).Allmodelsgiveaconversionfactorof
approximately0.174. Thecorrectionneeded topassfromtheAB
photometricsystemtotheGunn&ThuansystemisGT 0.17.
corr≈
WeusedtheFPcoefficientsfromFritzetal..FortheGunnr
6 A. Halkola,S.Seitzand M. Pannella
TheEinstein-radiusofanisothermalspherecanthenbewrit-
tenasθ = 1.4 (σ/220km/s)2 D,whereDisageometricalfactor
E ′′
oforderunitydependingonredshiftsoftheobjectsandcosmology
(0.78<D<0.92for1<z <6andouradoptedcosmology).Clus-
s
termemberswithσ>24km/swereincludedinthegalaxycompo-
nentofthecluster.Thislimitissomewhatarbitraryandbelowthe
luminositylimitwhereFPandFJaredetermined.AnEinsteinra-
diussmallerthanthepixelsizeoftheACSensuresthatallgalaxies
whichcouldsignificantlyaffectimagemorphologieslocallyarein-
cludedwhenexternalshearandconvergencefromtheothercluster
galaxiesandtheclusterhaloarepresent.
3.1.2 EllipticitiesofClusterGalaxiesandTheirHaloes
Blandford&Kochanek (1987), Kormannetal.(1994) andothers
havenotedthat forellipticalpotentialstheaccompanying surface
massdensitycanhavenegativevalues.Wehaveusedellipticalpo-
tential for our NSIE profile since it is straightforward to imple-
ment (all parameters of interest can be calculated from the ana-
lyticderivativesofthepotential).Analternativeapproachistohave
anellipticalmassdistributionasdemonstrated by Kormann et al.
(1994) but the expressions for α,κandγ are considerably more
Figure5.Totalobservedmagnitude-σrelation forthreez 0.2clusters. complicated.
∼
RedpointsaremeasuredvaluestakenfromZiegleretal.(2001)forA2218 For the ENFW and BBS profiles we have introduced the
(z=0.18),greenpointsarevaluestakenfromFritzetal.(2005)forA2390 ellipticity to the deflection angle, see appendix A for details
(z=0.23)andfull(empty)blackpointsrefertothevelocitydispersionesti-
and references. The effect of using an elliptical deflection angle
matesobtainedinthispaperforA1689usingtheGIM2D(GALFIT)mor-
insteadofanellipticalmassdistributionisshowninFigure6.The
phology. The literature values have been transformed to F775WAB mag- ellipticitiesof the mass distributions wereestimated by fittingan
nitudes byapplying relatively smallcolour terms (0.04,-0.4)andtheAB
ellipsetoκ=0.2isodensitycontoursforbothBBS(top)andENFW
correction(0.4).
(bottom) haloes with 6 different ellipticities (0, 0.05, 0.1, 0.15,
0.2and0.25).Ontherightpanelaquarteroftheκ=0.2isodensity
bandthen contourfordifferentprofileellipticitiesaredrawninsolid.Dashed
linesshowthebestfitellipses.FortheBBSmodel theisodensity
1.048 logR =1.24 logσ 0.82 I +ZP , (3)
∗ e ∗ − ∗h ie FPr contoursstarttodeviatefromanellipseatǫdefl >0.15,wherethe
contours first appear boxy before turning peanut shaped. For the
where the I term, i.e. the mean surface brightness in units of
L /pc2,isghivieenfortheGunnrbandbytheequation: ENFW profile the contours are only slightly peanut shaped at
⊙ ǫpot=0.25.LeftpanelsofFigure6showhowtheellipticityofmass
log I = 0.4( µ 26.4). (4) deviates from ǫ =2ǫ and ǫ =3ǫ lines shown dashed.
h ie − h rie− kappa defl kappa defl
We have assumed that light and mass have the same ellipticity
The zero–point of the FPZP isaquantity changing with
FPr and used the relation in Figure 6 to convert the measured galaxy
both the cluster peculiarity and, mainly, with the cluster redshift.
ellipticities(ǫ )toBBSmodelellipticities(ǫ ).Ahistogram
WeusedforZP thevaluepublishedinFritzetal.(2005).Their kappa defl
FPr of the ellipticities of the included cluster galaxies are shown in
studywasfocusedonA2218andA2390,twomassiveclustersatal-
Figure7.
mostthesameredshiftasA1689.Theyappliedabootstrapbisector
methodinestimatingtheZP andrelativeuncertainties,finding
FPr
avalueof0.055 0.022.
±
Finally,weinsertedthevaluesderivedfromourmorphologi-
3.2 DarkMatterNotAssociatedwithGalaxies
calfittingproceduresintotheFPrelation.Theuncertaintiesonthe
derivedvelocitydispersionswereestimatedbytakingintoaccount DifferentstudieshaveshownthatAbell1689isnotasimplestruc-
theerrorsonthemorphologicalparameters,thepropagatedphoto- turewithonlyonecomponent.Inanearlystronglensinganalysisof
metricuncertainties,theerrorontheZP valueandtheintrinsic A1689Miralda-Escude&Babul(1995)assumedtwohalocompo-
FPr
scatter of the FP relation, which gives the main contribution. We nentsbasedonthedistributionofgalaxies.Girardietal.(1997),on
foundthatanestimateof0.1inlog(σ)isagoodvalueforthetotal theotherhand,spectroscopicallyidentifiedthreedistinctgroupsin
uncertaintyinvelocitydispersionforobjectshavingavelocitydis- Abell1689.MorerecentlyAndersson&Madejski(2004)havealso
persiongreaterthan70km/s.Forlowervelocitydispersionsdown foundevidenceforsubstructureandpossiblemergerinX-raydata.
to24km/s,weassumedanoveralluncertaintyof0.2dex.Thefitted ThispromptedustomodeltheclusterdarkmatterinA1689with
parametersforthe80mostmassivegalaxiesaretabulatedinTable twodarkmatterhaloes.Theuseofmorehaloesinthemodellingis
B1. notdesiredsincethisincreasesproblemsrelatedtoalargenumber-
Additionally,wehaveobtainedthevelocitydispersionsof26 offreeparameters;largerparameterspacetoexplore,increaseddif-
galaxies using the Faber-Jackson relations derived using the 176 ficulty of finding global minimum and degeneracies between the
galaxies for which we have obtained the velocity dispersions via freeparameters.Bothhaloeshave6freeparameters:position(x,y),
FP.Theseareallfaintgalaxieswithσ<60km/S. ellipticity,positionangleandinthecaseofNSIEprofilevelocity
ParametricStrongGravitationalLensing Analysisof Abell1689 7
dispersionσandcoreradiusr andfortheNFWprofilevirialra-
c
diusr andconcentrationparameterC.
200
Forallthefollowingmodellingwehaveconstrainedthefirst
halotoresidewithin50”inxandyfromthecDgalaxyinorderto
reducethevolumeoftheparameterspaceandtoreducethedegen-
eracybetweentheparametersofthetwohaloes.Wedonotwantto
tightlyconnectthehalowithanyofthegalaxiesbutusetheposition
ofthecDasafirstguessforthepositionoftheclustercentre.Thisis
supportedbyX-raymapsofAbell1689(Xue&Wu2002;Ander-
sson&Madejski2004)aswellasweaklensingstudies(Kingetal.
2002)whichplacethecentreofthemassdistributionverynearthe
cDgalaxy.Thepositionofthesecondhalowasinitiallysettoco-
incidewiththevisuallyidentifiedsubstructuretothenorth-eastof
theclustercentrebutwasleftunconstrainedintheoptimisation.
3.3 MultipleImages&Arcs
Itisevidentlyofgreatimportanceforthemodellingtofindasmany
multipleimagesaspossible.Thecolourandsurfacebrightnessof
anobjectareunaffectedbygravitationallensingandwehavehence
usedthecolour,surfacebrightnessandthemorphologyoftheim-
ages to identify multiple image systems. We have first identified
Figure6.Ellipticityofsurfacemassdensityvs.profileellipticity(indeflec-
tionangle)forthetwoprofiles(BBSisonthetoptwopanelsandENFW arcs and obvious multiple images which were then used to find
onthebottom two). Theellipticity ofκ isestimated byfitting anellipse aninitialsetofhaloparameters.Initialconstraintsincludeimages
tolinesofconstantsurfacemassdensity.Rightpanelsshowκ=0.2isoden- fromimagesystems1,3,4,5,6,12andarcsthatcontainimages
sitycontoursinsolidandfittedellipsesindashedlines.Profileellipticities systems 8, 14, 20 and 32. A model based on these images could
are0.0,0.05,0.10,0.15,0.20,0.25andincreasefrombottomtotop.The nowbeusedtosearchformoreimagesfortheexistingimagesys-
curveshaveanoffsetof0.25inbforclarity.Leftpanelsshowǫkappaasa tems as well as new image systems which could be included in
functionofprofileellipticityǫdefl.Thedashedlinesareǫkappa=3ǫdefland themodelandconstrainthemodelparametersfurther.Newimages
ǫkappa=2ǫdefllines. weresearchedforbylookingforimagepositionswhosesourcelies
withinaspecifieddistance,e.g.5”,fromthesourceofanexisting
image.ThismethodisbasicallythesameasdescribedinSchramm
&Kayser(1987)andKayser&Schramm(1988).
The images we have found are with a few exceptions also
identifiedinthepioneeringworkofBroadhurstetal.(2005a)and
essentiallyformasubsetoftheirimages.Forouranalysiswehave
mergedthetwoimagecataloguestoobtainacatalogueof107mul-
tipleimagesin32imagesystemsoneof whichisanlong arc.In
the merging we have split the image system 12 from Broadhurst
etal.(2005a)totwoseparatesystems(12and13)with2additional
images fromour catalogue. The splittingincludes separating two
imageswiththesamespectroscopicredshiftintotwodifferentim-
age systems. We have done this based on the morphology of the
images and our lensing models and we believe these to originate
from2differentsources. AlsoSeitzetal.(1998) intheiranalysis
of cluster MS-1512 reported two sources at the same redshift. In
thecaseofMS-1512Teplitzetal.(2004)usednear-infraredspec-
troscopy to confirm that the sources were indeed separate with a
differenceofonly400km/sinvelocity(0.0013inredshift).Topos-
itivelyidentifyasetofimagestooriginatefromasinglesourceis
verydifficultwithoutextremelyaccuratespectroscopyorobviously
the same (complex) morphology. The field of A1689 has a vast
number ofimagesthatcanpotentiallybeerroneously assignedto
anymultipleimagesystem.Inourworkwehaveratherexcludedan
Figure7.Ahistogramoftheellipticities, ǫdefl,fortheBBSprofilesused imagethanincludeitinanimagesystem.Forthisreasonwehave
tomodeltheclustergalaxies.Mostofthegalaxieshaveanellipticitywell
alsoexcludedimagesystem20fromBroadhurstetal.(2005a)from
below0.15wheretheisodensitycontoursofsurfacemassdensitystartto
ouranalysis.ImagesystemsnotusedbyBroadhurstetal.(2005a)
appearpeanutshaped.
aresystems31and32,asystemwith2imagesandalongarcre-
spectively.
Oftheimagessystemsused8haveanevennumberofimages.
The missing images in these cases are always demagnified and
8 A. Halkola,S.Seitzand M. Pannella
Figure9.Acomparisonofphotometricredshiftsofmultipleimagesfrom
this work to those from Broadhurst et al. (2005a). The correspondence
isvery goodandforall objects within the errors. Theoneobject witha
zphot 1fromBroadhurstetal.(2005a)andzphot 3.4fromthisworkhas
∼ ∼
aspectroscopicredshiftofzspec=3.04.
basedonthelensingmodelsmostlyexpectedtolienear agalaxy
makingtheiridentificationverydifficult.
InFigure8weshowallthemultipleimagesusedinthisstudy.
Moredetails,suchaspositionsandredshifts,oftheimagesystems,
arcsandimagescanbefoundinAppendixCTablesC1andC2as
wellasBroadhurstetal.(2005a).
Acomparison betweenphotometricredshiftsfromthiswork
and those of Broadhurst et al. (2005a) isshown inFigure 9. The
overall agreement is very good. The one object with a z 1 from
∼
Broadhurstetal.(2005a)andz 3.4fromthisworkbelongstoim-
∼
age system 1 and is one of the few objects with a spectroscopic
redshift(z =3.0).
spec
In Figure 10 we show the photometric redshift probability
density of the 5 multiple image systems with a spectroscopically
known redshift. In the figures the different colours represent the
probability densitiesof individual multipleimages of thesystem.
In most of the cases the spectroscopic and photometric redshifts
agreeverywell.Onlyimagesystems10and12haveabroadpho-
tometricredshiftprobabilitydensitydistribution.
Thelensingpower ofaclusterdepends ontheratio D /D ,
ds s
whereD istheangulardiameterdistancebetweentheclusterand
ds
thesource and D isthe angular diameter distanceof thecluster.
d
InFigure11weshowthepowerofalensatredshift0.18fordif-
Figure10.Photometricredshiftprobabilitydensityforthe5imagesystems
ferentsourceredshifts.VerticallinesinFigure11showtherange
wherespectroscopicredshiftisknownforatleastoneoftheimagesinthe
of allowed D /D ratiosfor the image systems withphotometric
ds s system.Ineachpanelthecolouredthinlinesrepresentindividualredshift
redshifts. The five squares mark the image systems with known
probabilitydensitiesfortheimagesinthesystem.Onlyimagesystems10
redshifts(twohavethesameredshift).TheD /D ratiocanbewell
ds s and12haveabroadphotometricredshiftprobability densitydistribution,
constrained by photometric redshifts alone. With the help of the theotherspectroscopicredshiftsarerecoveredwell.
fivespectroscopicredshiftswecanveryaccuratelyseparatethege-
ometric factor from the deflection angle allowing us to constrain
theclustermasstightly.
ParametricStrongGravitationalLensing Analysisof Abell1689 9
Figure8.Positionsofmultipleimagesusedinconstrainingthemodelparameters.Theimagesfromeachmultipleimagesystemhavethesamenumber,the
imageswithinanimagesystemarecodedwithletters.Theboxsideis 600kpcwithourcosmology.
∼
3.4 FindingOptimalModelParameters where ~θ is the position of image i in image system k and
k,i
~θ(<~β >)isthepredictedimagepositioncorrespondingtoimage
Goodnessoffitinstronggravitationallensingcanbequantifiedin ait~θ kf,riommeansourceofsystemkat< ~β >andσ istheer-
twoways.Theproperwayistocalculateaχ2 intheimageplane, k,i k,i k
rorinimagepositionsforsystemk(estimatedtobe1pixelforall
i.e. how far an image predicted by a model isfrom the observed
images).
one. Incalculating thepositionsof predicted images of animage
Calculating image plane χ2 is unfortunately very time con-
systemweassumethattheimagesofasystemoriginatefromthe
suming since the lens equation needs to be inverted numerically.
averagesource of theimages. Theexpression for animage plane
An additional complication is that for some values of the model
χ2isthen
parametersnot all observed images necessarily exist. Thismeans
thatanimageplaneχ2doesnotnecessarilyconvergetotheoptimal
parametersbutistrappedinalocalminimum.
~θ ~θ(<~β >)2 Goodnessoffitcanalsobeestimatedbyrequiringthatallim-
χ2 = | k,i− i k,i | , (5) agesofanimagesystemoriginatefromthesamesourceandhence
σ2
Xk Xi k minimise the dispersion of the source positions. The problem in
10 A. Halkola,S. Seitzand M.Pannella
if a given surface mass density satisfies image constraints then a
newsurfacemassdensitycanbefound,bysuitablyrescalingthis
surfacemassdensityandbyadding aconstant masssheet, which
satisfiesimagepositionsaswellasrelativemagnificationsequally
well.
For haloes with variable mass profile this can also create a
degeneracy between the parameters of the profile. For the NSIE
modelahighcoreradiuscanbecompensatedforbyalargerveloc-
itydispersionandforNFWahigherscaleradiusdemandsalower
concentrationparameter.
Thesedegeneraciescanbebrokenifmultipleimagesystemsat
differentredshiftsandatdifferentradiicanbefound.Positionofa
radialcriticalline,andsoradialarcs,dependscriticallyonthemass
distributioninthecentralregionsandhencethecoreradius.Onthe
other hand tangential arcsgive strong constraints on the mass on
largerscales. Welldefinedhalo parameterscan bedetermined by
havingradialarcs, alargenumber ofmultipleimagesatdifferent
redshiftsandbyminimisingintheimageplane.
4 CONSTRUCTINGLENSINGMODELS
Wehaveconstructedintotal4stronglensingmodelsfortheclus-
Figure11.Thelensing power ofacluster depends ontheratio Dds/Ds,
where Dds is the angular diameter distance between the cluster and the ter.Themassdistributionsarecomposedoftwocomponentsasde-
source and Dd is the angular diameter distance of the cluster. This ratio scribedintheprevious section:thecluster galaxiesand asmooth
flattensrapidlyforredshiftslargerthan0.5(foraclusteratredshift0.18). dark matter component. They give us the best fitting NSIE and
Inthefigureweshowtheallowed Dds/Ds ratiosofthedifferentmultiple ENFWparametersforthesmoothdarkmattercomponentandthe
imagesystems.Thesquaresindicate objectswithspectroscopic redshifts. massprofileofthedifferentmasscomponentsaswellthetotalmass
SinceDds/Ds variesonlylittleforz>2theredshiftsofsourcesarenotof profileof thecluster. WithModels I andII weaimto establisha
great importance. Thespectroscopic redshifts are important infixing the
well defined total mass profile for the cluster using the multiple
overallmassscale.
imagepositionsandthephotometricredshiftsofthesources.The
differencebetweenModelsIandIIisinthescalinglawusedforthe
this case is that the errors are measured in the image plane and clustergalaxies.ModelIassumesas σ2lawwhereasforModel
∝
donotnecessarilyrepresenttheerrorsinsourcepositions.Wetake IIweuses σlaw.ModelsIbandIIbreplicatethesetupofBroad-
∝
account of this by rescaling errors in the image plane with local hurstetal.(2005a)andwiththesewewanttocompareresultswith
magnification. Rescaling by magnification largely avoids bias to- our parametric models to the their more flexible kappa-in-a-grid
wardsclusterparameterswithhighmagnification(largecoreradius model. For Models Ib and IIbwehave used imagesfrom Broad-
fortheNSIEmodelorsmallconcentrationforENFWmodel).The hurstetal.(2005a)onlyandhaveleftthephotometricredshiftsof
sourceplaneχ2,χ2,canbewritteninthefollowingway, thesourcesfreeaswasdoneinBroadhurstetal.(2005a).Wehave
kept the spectroscopic redshifts fixed as these help to define the
e massscaleofthecluster.ThedifferencebetweenModelsIbandIIb
χ2 = |~βk,i−~βk,j|2 , (6) isagaininthescalinglawadopted.
σ2 /µ +σ2 /µ Inadditiontothe4stronglensingModelsabovewehavealso
k i j>i k,i k,i k,j k,j
XXX constructed2furtherModelsinordertoderiveNSISandNFWpa-
where~βk,ieisthesourcepositionofimageiinsystemk,σk,iis rametersofthetotalclustermassprofileandtofacilitatethecom-
theerrorinthecorresponding imagepositionand µk,i isthelocal parisonofourresultswithearliermethodsusedtomeasurecluster
imagemagnification. masses, and numerical simulations. InModel III wehave fitteda
Theadvantageofχ2 overχ2 isthatforeveryimageposition NSISandanNFWprofiletothetotalmassobtainedwithModels
it isalways possible to calculate a corresponding source position IandII.WithModelIVwecombinethestronglensingconstraints
andsoχ2 canbecalculeatedforallvaluesofthemodelparameters fromModelIIIandtheweaklensingconstraintsfromBroadhurst
makingχ2convergewell. etal.(2005b) andderiveaccurateNSISandNFWparametersfor
Toefindoptimalmodelparameterswehavefirstminimisedχ2 thetotalmassprofileoutto 15’( 2.5Mpc).
∼ ∼
toobtainemodelparametersclosetotheoptimalonestoensurethat Most of the image systems are very well reproduced by the
the identified multiple images can be reproduced by the modeels. Models. Image systems 8, 12, 14, 15, 30, 31 and 32 are located
Theoptimalmodelparameterswerethenfoundbyminimisingχ2 closetocriticallineswheretheimageplane χ2 isdifficulttocal-
properlyintheimageplane. culateduetoilldeterminedimagepositionsfromthemodels.For
theseimagesystemswehavealwayscalculatedtheχ2inthesource
plane.
3.5 Degeneracies
We quantify the quality of fit by the rms distance between
Anymultipleimagesystemcanonlyconstrainthemasscontained anobservedimagepositionandonepredictedbythemodels.For
within the images. This leads to degeneracies in the derived sur- the images systems mentioned above the magnification weighted
facemass profile:the socalledmasssheet degeneracy statesthat sourceseparationwasusedinstead.Thermsdistancebetweenan