Two-dimensional kinematics of SLACS lenses: I. Phase-space analysis of the early-type galaxy SDSS J2321-097 at z=0.1

0.81 MB·English

by  Oliver Czoske

Mon.Not.R.Astron.Soc.000,1–16(2007) Printed2February2008 (MNLATEXstylefilev2.2)
Two-dimensional kinematics of SLACS lenses – I. Phase-space
analysis of the early-type galaxy SDSSJ2321−097 at z ≈ 0.1⋆
Oliver Czoske1†, Matteo Barnabe`1, Le´on V. E. Koopmans1, Tommaso Treu2
8
and Adam S. Bolton3
0
0 1KapteynAstronomicalInstitute,POBox800,9700AVGroningen,theNetherlands
2 2DepartmentofPhysics,UniversityofCalifornia,SantaBarbara,CA93101,USA
3InstituteforAstronomy,UniversityofHawaii,2680WoodlawnDrive,Honolulu,HI96822-1897,USA
n
a
J
0 Accepted2007November27.Received2007November27;inoriginalform2007November6
1

] ABSTRACT
h WepresentthefirstresultsofacombinedVLTVIMOSintegral-fieldunitandHubbleSpace
p
Telescope (HST)/ACS study of the early-type lens galaxy SDSSJ2321−097at z = 0.0819,
-
o extendingkinematicstudiestoalook-backtimeof1Gyr.Thissystem,discoveredintheSloan
r Lens ACS Survey, has been observed as part of a VLT Large Programme with the goal of
st obtainingtwo-dimensionalstellarkinematicsof17early-typegalaxiestoz ≈ 0.35andKeck
a spectroscopy of an additional dozen lens systems. Bayesian modelling of both the surface
[
brightnessdistributionofthelensedsourceandthetwo-dimensionalmeasurementsofvelocity

3 andvelocitydispersionhasallowedus,undertheonlyassumptionsofaxisymmetryandatwo-
v integral stellar distribution function (DF) for the lens galaxy,to dissect this galaxy in three
8 dimensions and break the classical mass–anisotropy, mass-sheet and inclination–oblateness
8 degeneracies.Ourmainresultsarethatthegalaxy(i)hasatotaldensityprofilewelldescribed
8 byasinglepower-lawρ ∝ r−γ′ withγ′ = 2.06+0.03;(ii)isaveryslowrotator(specificstellar
0 angularmomentumparameterλ = 0.075);(ii−i)0.0s6howsonlymildanisotropy(δ ≈ 0.15);and
R
1. (iv)hasadark-mattercontributionof∼30percentinsidetheeffectiveradius.Ourfirstresults
1 fromthislargecombinedimagingandspectroscopiceffortwiththeVLT,KeckandHSTshow
7 thatthestructureofmassiveearly-typegalaxiesbeyondthelocalUniversecannowbestudied
0 ingreatdetailusingthecombinationofstellarkinematicsandgravitationallensing.Extending
: these studies to look-backtimes where evolutionaryeffects become measurable holdsgreat
v
i promisefortheunderstandingofformationandevolutionofearly-typegalaxies.
X
Key words: gravitational lensing — techniques: spectroscopic — galaxies: elliptical and
r
a lenticular,cD—galaxies:kinematicsanddynamics—galaxies:structure.
1 INTRODUCTION relations between their stellar population, dark matter, kinematic
and black hole properties, the origins of which are still not well
Within the hierarchical galaxy formation scenario, early-type
understood (e.g. Djorgovski&Davis 1987; Dressleretal. 1987;
galaxies are assumed to be the end-products of major mergers
Magorrianetal.1998;Boltonetal.2007).
withadditionalaccretionofsmallergalaxies(e.g.Burkert&Naab
2004 for a review). As such the study of their structure, forma-
Considerable observational progress has been made in
tionandevolutionisanessentialstepinunderstandinggalaxyfor-
the last decades in our understanding of the relative contribu-
mation and the standard Λ cold dark matter (ΛCDM) paradigm,
tions of baryonic (mostly stellar), dark matter and black hole
in which the hierarchical galaxy formation model has its foun-
constituents of early-type galaxies through stellar dynamical
dations. Early-type galaxies are observed to follow tight scaling
tracers and X-ray studies (e.g. Fabbiano 1989; Mouldetal.
1990; Saglia,Bertin,&Stiavelli 1992; Bertinetal. 1994;
Franx,vanGorkom,&deZeeuw 1994; Carolloetal. 1995;
⋆ Based on observations made with ESO Telescopes at the La Silla or
Arnaboldietal. 1996; Rixetal. 1997; Matsushitaetal. 1998;
Paranal Observatories under programmeID 075.B-0226 and 177.B-0682
Loewenstein&White 1999; Gerhardetal. 2001; Seljak 2002;
andonobservations madewiththeNASA/ESAHubbleSpaceTelescope,
Borriello,Salucci,&Danese 2003; Romanowskyetal. 2003).
obtainedattheSpaceTelescopeScienceInstitute,whichisoperatedbythe
Morerecently,theSAURONcollaboration(deZeeuwetal.2002)
AssociationofUniversitiesforResearchinAstronomy,Inc.,underNASA
contractNAS5-26555. has extensively studied a large and uniform sample of early-type
† E-mail:[email protected] galaxiesinthelocalUniverse(i.e.z.0.01),combiningSAURON
2 O. Czoskeet al.
integral-field spectroscopy (IFS) on the William Herschel Tele- Table 1. Basic data for SDSSJ2321−097. Thedata are taken
scopewithhigh-resolutionHubbleSpaceTelescope(HST)imaging fromTreuetal.(2006a)andKoopmansetal.(2006).
toobtainathree-dimensionalpictureofthesegalaxiesintermsof
their mass and kinematic structure (Emsellemetal. 2004, 2007; αJ2000 23h21m20s.93
Cappellarietal.2006,2007). δJ2000 −09◦39′10′.′2
Despitemakingagreatleapforward,thecomplexmodelling zl 0.0819
oftheSAURONsystemsandtheunknowndarkmattercontentand zs 0.5342
total mass of these galaxies still require some assumptions to be σap (236±7)kms−1
madeabouttheirgravitationalpotential[e.g.constantstellarmass- σc (245±7)kms−1
to-light ratio (M/L) from stellar population models] and inclina- REinst 1.68arcsec=2.59kpc
tion.Theseissuescanbepartlyovercomeindiscysystemswhere Reff,B (8.47±0.11)kpc
MB −21.72±0.05
theinclinationcanbedeterminedfromthephotometryorpossibly
Reff,V (7.93±0.07)kpc
fromdustlanes.However,systemswithlowerellipticitycanshow MV −22.59±0.05
stronger degeneracies (e.g. Cappellarietal. 2007). Also, near the qiso,2D 0.77
effectiveradiusdarkmatterbecomesamajorcontributortothestel- ϑPA,⋆ 126◦.5
larkinematics(Gerhardetal.2001;Treu&Koopmans2004)andit
isnotyetclearwhatimpactitsneglecthas,eventhoughitisesti- Notes:zl andzs aretheredshifts ofSDSSJ2321−097 andthe
gravitationally lensed background galaxy, respectively. σap is
matedtobenottoosevere(Cappellarietal.2006).
the velocity dispersion measured from the 3-arcsec-diameter
Athigherredshifts,theextractionofdetailedkinematicinfor- SDSS fibre, σc is the derived central velocity dispersion
mation from early-type galaxies becomes increasingly more dif- (Treuetal. 2006b). REinst is the Einstein radius. Reff and M
ficult because of the decrease in apparent size and cosmological are the effective radii and absolute magnitudes determined
surfacebrightnessdimming.However,galaxiesathigherredshifts by fitting de Vaucouleurs profiles to the B- and V-band ACS
aremuchmorelikelytoactasgravitationallensesonbackground images (Treuetal. 2006b). qiso,2D and ϑPA,⋆ are isophotal
galaxies(e.g.Turner,Ostriker,&Gott1984).Theadditionalinfor- axial ratio and position angle of the major axis, respectively
(Koopmansetal.2006).
mation obtained from gravitational lensing on, for example, the
enclosed mass and the density profile has been shown to break
anidealstartingpointfordetailedinvestigationsintotheproperties
some of theclassical degeneracies in stellarkinematics and lens-
ofthelensgalaxies.
ing,namelythemass–anisotropyandthemass-sheetdegeneracies,
In this paper we introduce a follow-up project to SLACS
respectively(e.g.Treu&Koopmans2002a;Koopmansetal.2003;
which aims at combining the lensing information with detailed
Koopmans&Treu2003;Treu&Koopmans2004;Koopmansetal.
two-dimensionalkinematicinformationobtainedwiththeVIMOS
2006;Barnabe`&Koopmans2007,hereafterBK07).
integral-fieldunit(IFU)mountedontheVLT.Thefullsamplecom-
Inordertousegravitationallensingandstellardynamicsina
prises17systems.Inthispaperwedemonstratetheavailabledata
systematic way, the Lenses Structure & Dynamics (LSD) Survey
and the analysis methodology on one system, SDSSJ2321−097.
wasstartedin2001,combiningHSTimagingdataonlenssystems
The lens in this system is an early-type galaxy at z = 0.0819.
with stellar kinematics obtained with Keck (Koopmans&Treu
TheSDSSspectrumshowsadditional[O]andHβemissionfrom
2002,2003;Treu&Koopmans2002a,b,2003,2004).Thisproject
a gravitationally lensed blue galaxy at z = 0.5342 (Boltonetal.
anditsresults,predominantly atz ≈ 0.5−1.0, haveshowntheva- s
2006). In HST imaging (Fig. 1) the lensed source is shown to
lidityofthismethodologyanditseffectivenessathigherredshifts.
forman almost complete Einsteinring of radius 1.68arcsec,cor-
Thesehigh-redshiftsystems,however,arerelativelyrareandhard
responding to an Einstein radius of R = 2.6kpc at the lens
to follow up, which has limited studies to the use of luminosity- Einst
redshift. The projected mass enclosed within the Einstein radius
weightedstellarvelocitydispersions(e.g.Koopmans&Treu2002;
Treu&Koopmans2002b)orlong-slitspectroscopyinafewlarge isM(<REinst)=1.3×1011M⊙.Detailsonthissystemarelistedin
Table1.
apertures along the major axes (e.g. Koopmans&Treu 2003;
InSection2,wedescribetheVIMOSandHST observations
Treu&Koopmans2004).
thatformthebasisofthisanalysis.TheIFUdataarepresentedin
The Sloan Lens ACS Survey (SLACS, Boltonetal. 2005,
Section3, withadescription of the datareduction inSection 3.1
2006) was begun to extend the sample of lens galaxies suit-
and of the extractionof two-dimensional kinematic maps inSec-
able for joint lensing and dynamical studies. SLACS lens can-
tion3.2.InSection3.3wedetermineseveral commonlyusedpa-
didates were selected from the SDSS Luminous Red Galaxy
rameters from the kinematic data alone, in a way that is directly
sample (Eisensteinetal. 2001) and a quiescent subsample [de-
comparabletoearlierstudies,forexamplefromtheSAURONcol-
fined by EW(Hα) < 1.5Å] of the MAIN SDSS galaxy sample
laboration. We apply the joint lensing and dynamics analysis de-
(Straussetal. 2002) by an inspection of their SDSS fibre spec-
veloped by BK07 in Section 4. In Section 5 we present a phys-
troscopy (Boltonetal. 2004). Galaxies whose spectra presented
ical argument that explains why gravitational lensing and stellar
emission linesat ahigher redshift than the redshift of the galaxy
dynamicscanbreakthedegeneracybetweenoblatenessandincli-
itself were observed in an HST snapshot programme to check
nation.Wediscussour resultsandconclude inSection6.Weuse
whether the source of the emission lines was a gravitationally
aconcordance ΛCDMmodel throughout thispaper, described by
lensedbackgroundgalaxy.Todate,SLACShasconfirmedaround
Ω =0.3,Ω =0.7andH =100hkms−1Mpc−1withh=0.7un-
80lenssystems(Boltonetal.,inpreparation).Incontrasttomost M Λ 0
lessstatedotherwise.AttheredshiftofSDSSJ2321−097,1arcsec
earlierlenssurveyswhichdrewtheircandidatesystemsfromcat-
correspondsto1.1h−1kpc=1.5kpc.
alogues of potential sources (e.g. quasi-stellar objects or radio
sources as in the CLASS survey, Browneetal. 2003), SLACS is
alens-selectedsurveyandassuchpreferentiallyfindssystemswith
brightlensgalaxiesbutcomparativelyfaintsources.Itistherefore
Two-dimensionalkinematicsof SLACS lenses– I. 3
Figure1.HST/ACSimagesofSDSSJ2321−097,takenthroughtheF435W (left-handpanel)andF814W (right-handpanel)filters.Theimagesaresingle
exposureswithanintegrationtimeof420s;thepixelscaleis0.05arcsec.Afulldescriptionoftheseobservationsandthedata-reductionprocessisgivenin
Boltonetal.(2006).
2 OBSERVATIONS the telescope was shifted by about 3.5arcsec between the expo-
sures to fill in on dead fibres. The number of observing blocks
In this section, we briefly discuss the VLT VIMOS/IFU spectro-
spent on each lens system was adjusted so as to reach approxi-
scopic and the HST imaging observations, which form the basic
mately equal signal-to-noise ratios (S/N) across the sample. For
datasetsforthesubsequentmodelling.
thesystemdescribedinthispaper,SDSSJ2321−097,fiveobserv-
ingblocks werecarriedout for atotalintegration timeof 8325s.
For the large programme, we switched to the HR-Orange grism
2.1 Integral-fieldspectroscopy whichcoversawavelength rangefrom5250 to7400Åatasimi-
lar spectral resolution of λ/∆λ = 2650 and a dispersion of 0.6Å
We have obtained integral-field spectroscopic observations of
perpixel.Thespectralresolutiontranslatestoarest-framevelocity
17 systems using the IFU of VIMOS (LeFe`vreetal. 2001)
resolution∆vvaryingbetween107and85kms−1,improvingasthe
mounted ontheVLTUT3(Melipal). Theobservations werecon-
redshiftofoursamplelensgalaxiesincreasesfrom0.08to0.35.
ducted in two parts: three systems were observed in a pilot pro-
gramme(ESOprogramme075.B-0226;PI:Koopmans)andthere-
maining14systemsinalargeprogramme(177.B-0682;PI:Koop-
2.2 HSTimaging
mans) with a slightly different observational setup. Observations
were carried out in service mode throughout to ensure uniform High-resolutionimagingwithHST/ACSorHST/WFPC2hasbeen
observing conditions. Inthispaper we report indetail onthe ob- orisbeingobtainedforallconfirmedSLACSlenses(Boltonetal.
servationsofthepilotprogrammeandtheanalysisofonesystem, 2006;Gavazzietal.2007).ThedatausedherecomefromtheACS
SDSSJ2321−097. The observations of the large programme and snapshot survey described by Boltonetal. (2005) and consist of
theanalysisofthefullsamplewillbepresentedinforthcomingpa- single420sexposuresthroughtheF435W andF814W filters,re-
persinthisseries. spectively.Fig.1showstheimagesofSDSSJ2321−097.Thelens
TheIFUheadofVIMOSconsistsofasquarearrayof80×80 modellingrequiresasinputacleanestimateofthebrightnessdis-
microlenses with a sampling of 0.67arcsec per spatial element tribution of the lensed images, so the smooth brightness distri-
(‘spaxel’)attheNasmythfocusofthetelescope.Inhigh-resolution bution of the lens galaxy has to be subtracted from the images.
modeonlythecentral1600lensesareused,resultinginafieldof Boltonetal. (2006) tried a variety of commonly used modelling
view of 27×27arcsec2. The lenses are coupled to optical fibres techniques(e.g.fittingofSersicprofiles),butsettledonthemore
whichtransmitthelighttopseudo-slitmasks,eachofwhichcon- flexible B-spline technique. The galaxy-subtracted image in the
tains400fibresandcorrespondstoaquadrantoftheoriginalfield. F814Wfilterisshowninthetopright-handpanelofFig.7.
Theseeinglimitfortheobservationswassetto0.8arcsec,sothat
eachfibreoftheIFUgivesaspectrumthatisessentiallyindepen-
dentofitsneighbours. 3 IFUDATAANALYSIS
For the pilot programme, we employed the HR-Blue grism
whichcoversawavelengthrangefrom4000to6200Å(theexact 3.1 Datareduction
wavelength range variesslightlyfromquadrant toquadrant). The TheVLTVIMOS/IFUdatawerereducedusingthepackage
spectral resolutionof theHR-Bluegrismisλ/∆λ = 2550, where
whichwasdevelopedwithintheframeworkoftheVIRMOScon-
∆λisthefullwidthathalf-maximum(FWHM)oftheinstrument sortiumandtheVVDSproject.1 hasbeendescribedindetail
profile.ThemeandispersionontheCCDis0.5Åperpixel.Three
exposures of 555s integration time were obtained per observing
block(aone-hoursequenceofscienceandcalibrationexposures); 1 http://cosmos.iasf-milano.inaf.it/pandora/vipgi.html
4 O. Czoskeet al.
Figure2.Left-handpanel:VLTVIMOS/IFUimagereconstructionofSDSSJ2321−097,obtainedbyintegratingalongtheindividualfibrespectrafrom4400
to6100Å.Right-handpanel:mapofthestrengthofthe[O]doubletfromthelensedsource.
byScodeggioetal.(2005)andZanichellietal.(2005);here,were- sentiallymeasurepuresky.identifiesthesefibresinastatisti-
strictourselvestoabriefsummaryofthemainreductionstepsper- calwaybyintegratingthefluxineachfibreoveralargewavelength
formedby. rangeandselectingasskyfibresthosewithafluxwithin±0.1per
ThefourquadrantsoftheIFUaretreatedindependentlyupto centofthemodeoftheresultingdistribution.
thecombinationofallexposuresintothefinaldatacube.Foreach Finally,thefourquadrantsofalltheindividualexposuresare
nightonwhichobservations werecarriedout,fivebiasexposures combinedintoasingledata‘cube’aftercorrectionforthedithering
wereobtainedwhicharemedian-combinedandsubtractedfromthe betweenexposures.Thefinaldataareinrow-stackedform,where
data.Theobservedhalogen-lampilluminatedscreenflatsareused eachrowcorrespondstothespectrumfromonefibre.Afitstable
tolocatethetracesoftheindividual spectraontheCCD,starting extensiongivesthetranslationfromrownumbertothespatialco-
frominteractivelyadjustedfirstguessesofthelayoutofthefibres. ordinateofthefibre.
Thisincludesashiftcorrectionandanidentificationof‘bad’fibres. The left-hand panel of Fig. 2 shows an image of
Thepositionofthepeakofeachspectruminthespatialdirection SDSSJ2321−097,reconstructedfromthereduceddatacubeusing
isthenmodelledasaquadraticpolynomialfunctionoftheposition theprogram 2 by integrating along thewavelength direction
inthedispersiondirection.doesnotperformaCCDpixel-to- from 4400 to 6100Å. The galaxy surface brightness distribution
pixelflat-fieldcorrection(Scodeggioetal.2005). agrees well with that seen in the HST/ACS F435W and F814W
The wavelength calibration is subsequently done using he- snapshotimages,althoughitisnotaspreciseduetothecomplexi-
liumandneonlampobservationstakenimmediatelyafterthesci- tiesinvolvedinthespectralcalibration,inparticularoftherelative
ence exposures. Withabout 20 linesspread between ∼ 4000 and fibretransmissivity.Inthemodellingofthelensgalaxybrightness
∼ 6200Å,thermsdeviationsfromthepolynomial fitareroughly distribution,wethereforeusetheHSTimages.Theimperfectabso-
Gaussian-distributedaroundameanof∼0.075Åandadispersion lutefluxcalibrationoftheIFUdatahasnoeffectonthemeasure-
of∼ 0.02Å.Thevelocityerrorintroducedbyuncertaintiesinthe mentsofthestellarkinematicswhichdonotdependontheoverall
wavelengthcalibrationisthusoftheorderof5kms−1,significantly scalingofeachfibrespectrum.
smallerthanthespectralresolutionandnegligibleinouranalysis. FollowingBolton&Burles(2007),wecanusetheIFUdatato
Theindividualspectraareextractedusingtheoptimalextrac- obtainanimageofthelensedsourcebyisolatingitsredshifted[O]
tionschemeofHorne(1986),thatis,byaveragingthefluxacross emissionlineat5710Å.The[O]doubletismodelledasthesum
thespectral traceof thefibre, weightedby afittoitsspatial pro- of two Gaussians with separation 4.5Å (taking into account the
file.Inordertocorrectforthetransmissivityvariationsfromfibre redshiftofthesource)andrelativestrength3:2.Inregionswithsig-
tofibre(‘spaxel-to-spaxelflat-fielding’),thefluxunderaskyemis- nificantlightfromthelensgalaxythebest-fittingkinematicmodel
sion line ismeasured and scaled relative toa reference fibre. We (Section 3.2) was subtracted prior to the fit; elsewhere a polyno-
use[O]atλ=5577Åforthispurpose. mialfitwassubtractedtoremovearemainingcontinuumlevel.The
Spectrophotometricstandardstarswereobservedduringmost mapofthelinestrengthofthe[O]lineobtainedinthismanneris
nights.Sincethestellarlightonlycoversasmallfractionofthefi- presentedintheright-handpanelpanelofFig.2.Themapclearly
bresintheIFUfieldwhereasthecalibrationisappliedtoallfibres, showsthering-likestructureofthelensedgalaxy(cf.thehigherres-
thespectrophotometriccalibrationonlyaccuratelycorrectstherel- olutiongalaxy-subtractedACSimageinthetopright-handpanelof
ativewavelengthdependenceofthesensitivityfunction(assuming Fig.7)anddemonstratesthatthe[O]lineusedfortheinclusion
thatthisisthesameforallfibres),whiletheabsolutefluxlevelsare ofthissystemintheSLACSsampleindeedoriginatesfromagrav-
determined with a much larger uncertainty due to residuals from itationallylensedbackgroundgalaxy.
thecorrectionofthevaryingfibretransmissivities.
TheVIMOSIFUdoesnothavededicatedskyfibres,butthe
fieldofviewislargeenoughforasufficientnumberoffibrestoes- 2 http://cosmos.iasf-milano.inaf.it/pandora/sadio.html
Two-dimensionalkinematicsof SLACS lenses– I. 5
Figure3.SpectrumofSDSSJ2321−097integratedfromtheVIMOS/IFUdatacubeoveracircularapertureofdiameter3arcsec.Theinsetontheright-hand
sideshowsazoomonthe[O]doubletofthelensedbackgroundgalaxyatzs =0.5342,nexttotheFeCaλ5268lineofthelens.Theinsetontheleft-hand
sideshowsazoomontheweakbutclearlydetected[O]lineofthelensgalaxy.Severalotherspectralfeaturesofthelensaremarked.
ThespectralenergydistributionofSDSSJ2321−097isessen- byalinearfitnestedwithinthenon-linearoptimizationforthekine-
tiallythatofatypicalearly-typegalaxywithstrongmetallines,as maticparametersv andσ .
i v,i
seen inFig. 3which shows the spectrum integrated over acircu- The template spectrum is first resampled to a logarithmic
larapertureof3arcsecdiameter(i.e.theSDSSaperture).Wenote, wavelengthscalewithstep∆lnλ=∆v/c,whereweuseavelocity
however,thatthisspectrumshowsweakbutsignificant[O]emis- stepof∆v=25kms−1.TheconvolutionkernelisaGaussianwith
sion(EW ∼ −4Å)aswellasHδabsorptionfromthelensgalaxy. dispersionσ = σ/∆v,extendingto±8σ .Thegalaxyspectrum
G v G
This might indicate residual star formation or active galactic nu- iscorrectedforthemeanredshiftofthelensgalaxy,z=0.0819in
cleusactivityatalowlevel. thecaseofSDSSJ2321−097,takenfromtheSDSSdatabase.The
model is then resampled on to the wavelength grid of the galaxy
spectrum, a linear wavelength scale λ /(1+z ) in our case.
obs SDSS
3.2 Kinematicanalysis Thisavoidsfurtherresamplingofthedatabeyondwhatwasdonein
thedatareduction.Similarmethodstypicallyresamplebothmodel
We extract two-dimensional maps of systematic velocity v(R)
anddatatothesamelogarithmicwavelengthgridbeforecomparing
and line-of-sight velocity dispersion σ(R) using a direct pixel-
(e.g.vanderMarel1994;Cappellari&Emsellem2004).
fitting method implemented in the statistical language 3
Remainingcosmicrays,emissionlines,andremnantsofsky
(RDevelopmentCoreTeam 2007). Since the analysis is done on
linesaremaskedautomaticallyby3σ-clippingoftheresidualsfrom
spectra from individual IFU fibres, the spectra have relatively
apreliminaryfitwithoutmask.Fig.4showsanexampleofafitto
low signal-to-noise ratio, in particular in the outskirts of the lens
aspectrumfromafibrenearthecentreofSDSSJ2321−097.Error
galaxy where the surface brightness drops rapidly. We therefore
analysisisperformedthroughaMonteCarlotechniquebyadding
restrict the model to a Gaussian line-of-sight velocity distribu-
Gaussiannoisetothebest-fittingmodelatthenoiselevelgivenby
tion (LOSVD) and do not attempt to measure higher order mo-
theresidualsbetweenmodelanddata,andusing1000realizations
ments,suchastheGauss–Hermitecoefficientsh andh commonly
3 4 todeterminethespreadofthebest-fittingparameters.
usedinkinematicstudiesoflow-redshiftearly-typegalaxies(e.g.
The choice of a good template spectrum is important
vanderMarel&Franx1993;Cappellari&Emsellem2004).
for accurate kinematic measurements from galaxy spectra. For
Theobservedgalaxyspectrumd(λ)ismodelled,foreachfi-
i SDSSJ2321−097,westartedbyexperimentingwithasetofeight
breiseparately,bytheconvolutionofahigh-resolutionstellartem-
stellartemplatespectraobtainedwiththeEchelleSpectrographand
platespectrumt(lnλ)withaGaussiankernelg(lnλ)withdisper-
i Imager (ESI) on the Keck II telescope (Koopmans&Treu 2002)
sionσ /c:
v,i appliedtotheglobalspectrumofSDSSJ2321−097.Thesespectra
di(λ)= fi(λ)[t⊗gi](lnλ)+ni(λ). (1) have a velocity resolution of ∆vESI = 22.4kms−1 (ESI manual4)
andcoverarangeofspectraltypesG0toK4,luminosityclassIII.
WeassumeGaussiannoisen(λ)onthespectrum. Sincetheaccu- FittingtotheintegratedspectrumofSDSSJ2321−097itwasfound
racyofthespectrophotometriccalibrationisnotperfectforbothour thatfourtemplatesforstellartypesbetweenG7andK0gavecon-
spectraandthechosentemplatespectra(seebelowandValdesetal. sistentresultsforthevelocitydispersionwithin±10kms−1,which
2004)thereremainsmallrelativetiltsbetweenthecontinuaofthe alsoagreedwiththemeasurementfromtheSDSS.Coolertemplate
galaxyandtemplatespectra.Wemodelthisdifferencebythelinear stars(K2andK4)resultedinasignificantlyhighervelocitydisper-
correctionfunction f(λ)=a+bλwhoseparametersaredetermined sionwhilehottertemplates(G0andG5)gavelowervalues.Onthe
3 http://www.r-project.org/ 4 http://www2.keck.hawaii.edu/inst/esi/echmode.html
6 O. Czoskeet al.
significantlyenhanced inthegalaxy spectrum. Barthetal.(2002)
recommend masking the Mgb line before fitting. Doing this in-
creasesthevelocitydispersiononlybyafewpercentanddoesnot
significantlyimprovethefitoutsidetheMgbregion.
3.3 Resultsfromtheobservedstellarkinematics
Afitofakinematicmodeltotheglobalspectrum(Fig.3)following
theproceduredescribedinSection3.2resultsinavelocitydisper-
sionof σ = 240±8kms−1, which isconsistent withthevalue
tot
σ =236±7kms−1listedintheSDSSdatabase(Boltonetal.
SDSS
2006). Note that the error estimates are not directly comparable
because of differences in the estimation method. We believe our
MonteCarloestimatetobemorereliable,althoughitmightstillun-
derestimatetherealerrorduetotheassumptionofGaussiannoise
inthespectrum(deBruyneetal.2003).
The two-dimensional maps of the velocity and velocity dis-
persionareshowninFig.5.Onlyfibresinwhichthespectrumhas
anS/N perpixelintherest-framewavelengthrangefrom5350Å
to5450Å(aflatpartofthespectrum)ofS/N >8areusedinthese
maps and the subsequent analysis. Spectra with lower S/N were
foundnottoyieldreliablekinematicfits.
Neither the velocity nor the velocity dispersion map shows
significant structure. The velocity map might hint at a pattern of
slightrotation,althoughwithinthenoisethemapisconsistentwith
norotationatall.
Inorder toquantify theamount of rotationinagalaxyfrom
two-dimensionalmapsofline-of-sightstellarvelocityvandveloc-
itydispersionσ,Emsellemetal.(2007)definedastellarangular
v
momentumparameterλ by
R
hR|v|i
λ = , (2)
R
hR v2+σ2i
v
p
Figure4.SpectrumfromasinglefibrenearthecentreofSDSSJ2321−097, whereRistheprojecteddistancefromthegalaxycentre.Wefind
withtheoverlaid best-fitting model(toppanel). Thebottom panelshows λR = 0.075, which in combination with the absolute magnitude
theresidualsofthefit.The[O]doubletfromthesourceat5710Å(zs = MB = −21.72 places SDSSJ2321−097 among the slowly rotat-
0.5342)andtheresidualofthe[O]skylineat5577Åweremaskedauto- ing galaxies at the high-luminosity end of the SAURON sample
matically(indicatedbythepoints). (cf.fig.7ofEmsellemetal.2007).
Following the prescription of Binney (2005) and
Cappellarietal. (2007), we find a low value for the ra-
basisofthevisualappearanceofthefitandtheshapeofthegalaxy
tio between systematic velocity and velocity dispersion of
spectrumitwasdecidedtousetheK0starHR19asthetemplate
(hv2i/hσ2i)1/2 = 0.1, which in combination with the observed
starforSDSSJ2321−097. v
ellipticity of ǫ = 1 − q = 0.23 from isophote fitting
WesubsequentlyexchangedtheESIspectrumofHR19forthe iso,2D iso,2D
places SDSSJ2321−097 well below the curve for an isotropic
corresponding spectrum from the Indo-US library of Coude´ feed
rotating oblate galaxy in an ǫ−(v/σ) diagram. From the tensor
stellarspectra5 (Valdesetal.2004).TheIndo-USlibrarycontains v
virial theorem the global anisotropy parameter (Binney 2005) is
spectrafor1273starscoveringawiderangeofstellarparameters.
found to be δ ∼ 0.15, where no correction for inclination has
The spectra have a resolution of ∼ 1Å FWHM and for the ma-
beenmadeyet.InSection4,wecalculatethesamequantityfrom
jorityofstarscoverawavelengthrangeof3460to9464Å.While
the reconstructed stellar DF and find effectively the same result.
theresolutionoftheIndo-USspectraissomewhatlowerthanthat
Again,SDSSJ2321−097 compares wellwiththehigh-luminosity
oftheESIspectra,thisdisadvantageisoffsetbythelargenumber
galaxiesfromtheSAURONsample(Cappellarietal.2007)inthat
ofspectraavailable,whichwillprovidefreedomofchoiceforthe
itsellipticityrequiresananisotropicvelocitydistributionandisnot
analysisofourfullsample.Wehavecheckedthatthekinematicre-
duetorotation.
sultsobtainedwiththeIndo-USandtheESItemplatesareindeed
Thevelocitydispersion isnearly constant across thefieldas
consistent.
one would expect for a nearly isothermal total mass distribution.
Fig. 4shows an important limitationof any useof observed
Fig.6showsthevelocityandvelocitydispersionasafunctionof
stellarspectraastemplatesinkinematicanalyses,namelythedif-
positionalongthemajorandminoraxes(fibreswithin1arcsecof
feringabundance ratiosbetween starsinthesolar neighbourhood
therespectiveaxiswereused).Themajoraxisvelocitycutshowsa
and early-type galaxies (Barth,Ho,&Sargent 2002). While the
strengthsoftheFe/Caλ5268linesagree,theMgbλ5174lineis hintoftheS-shapeexpectedfromrotationaroundtheminoraxis,
althoughataverylowlevel.Thevelocitydispersionalongthema-
joraxisisperfectlyflat,whereastheminoraxiscutshowsaslight
5 http://www.noao.edu/cflib/ declineoutwards.
Two-dimensionalkinematicsof SLACS lenses– I. 7
Figure5.KinematicmapsofSDSSJ2321−097.Thetopleft-handpanelshowssystematicvelocityrelativetothemeanredshiftofthelensgalaxy;thebottom
left-handpanelshowsthevelocitydispersion.Thepanelsontheright-handsidevisualizethecorresponding1σerrorsonthemeasurements.Spaxelsizeis
0.67arcsecandonlyspectrawithS/N>8wereanalysed.
Toquantifythis,weperformalinearfittothevelocitydisper-
sionprofileasafunctionoftheellipticalradiusµ2 =x2+y2/q2 ,
iso,2D
with q = 0.77 obtained by ellipse fitting to the light distri-
iso,2D
bution.Thelinear slopeisdσ/dµ = (−7.3±4.5)kms−1arcsec−1,
consistentwithbeingfullyisothermal.
4 JOINTSELF-CONSISTENTGRAVITATIONAL
LENSINGANDSTELLARDYNAMICSANALYSIS
Inthissection, wecombine theinformation fromthegravitation-
allylensedimagewiththesurfacebrightnessdistributionfromHST
observationsandtheprojectedvelocitymomentmapsderivedfrom
VLTobservationsofthelensgalaxySDSSJ2321−097inorderto
constrainthetotalmassdensityprofileofthisgalaxy.Theanalysis
iscarriedoutbymakinguseofthealgorithm,presented
in BK07: we refer to that paper for a detailed description of the
method.
4.1 Massmodelandoverviewofthejointanalysis
Figure6.Velocity(toppanels)andvelocitydispersion(bottompanels)as The key idea of a self-consistent joint analysis is to adopt the
afunctionofpositionalongthemajor(left-handpanels)andminor(right-
sametotalgravitationalpotentialΦforboththegravitationallens-
handpanels)axes.Spectrafromfibreswithin1arcsecoftherespectiveaxis
ing and the stellar dynamics modelling of the data. As shown in
wereusedhere.
BK07, these two modelling problems, although different from a
physicalpointofview,canbeexpressedinananalogouswayasa
8 O. Czoskeet al.
Figure7.Bestmodellensimagereconstruction forthesystemSDSSJ2321−097.Fromthetopleft-handtobottomright-handpanel:reconstructedsource
model;HST/ACSdatashowingthelensedimageaftersubtractionofthelensgalaxy;lensedimagereconstruction;residuals.
singlesetofcoupled(regularized)linearequations.Foranygiven bits(asintheclassicalSchwarzschildmethod)butTICs.6 Among
choiceηofthenon-linearpotentialparameters,theequationscan theoutcomes of thejoint analysis presented in thissection isthe
be solved non-iteratively tosimultaneously obtain as thebest so- weightsmapoftheTICsuperposition,whichcanberelatedtothe
lutionfor thechosen potentialmodel theunlensedsourcesurface weightedtwo-integralDF(seeBK07).
brightness distributions and the weights of the elementary stellar As a first-order model and in order to allow a straightfor-
dynamicsbuildingblocks(e.g.orbitsortwo-integralcomponents, ward comparison with the results of the preliminary analysis of
TICs,Schwarzschild 1979;Verolme&deZeeuw 2002).Thislin- Koopmansetal.(2006),weadoptasthetotalmassdensitydistri-
ear optimization scheme is consistently embedded in the frame- bution of the galaxy a power law stratified on axisymmetric ho-
workofBayesianstatistics.Asaconsequence,itispossibletoob- moeoids:
jectivelyassesstheprobabilityofeachmodelbymeansoftheev- ρ
ρ(m)= 0 , 0<γ′<3, (3)
idencemeritfunctionand,therefore,torankdifferentmodels(see mγ′
MacKay 1992, 1999, 2003). In this way, by maximizing the evi- whereρ isadensityscale,γ′willbereferredtoasthe(logarithmic)
0
dence,thesetofnon-linearparametersηwhichcorrespondstothe
slopeofthedensityprofile,and
bestpotentialmodel,giventhedata,canberecovered.
R2 z2 R2 z2
m2= + = + , (4)
a2 c2 a2 a2q2
witha>candc/a≡q.
Whilethe method iscompletely general, itspractical imple-
mentation (inthealgorithm) ismorerestrictedtomake
6 ATICcanbevisualized asanelementary toroidal system,completely
itcomputationallyefficientandappliesspecificallytoaxisymmet-
specifiedbyaparticularchoiceofenergyEandaxialcomponentofthean-
ric potentials, Φ(R,z), and two-integral DFs f = f(E,L). Un-
z gularmomentumLz.TICshavesimple1/Rradialdensitydistributionsand
dertheseassumptions,thedynamicalmodelcanbeconstructedby
analyticunprojectedvelocitymoments,andbysuperposingthemonecan
makinguseofthefastBK07numericalimplementationofthetwo- build f(E,Lz)modelsforarbitraryspheroidalpotentials (cf.Crettonetal.
integralSchwarzschildmethoddeveloped byCrettonetal.(1999) 1999):allthesecharacteristics contributetomakeTICsparticularly valu-
andVerolme&deZeeuw(2002),whosebuildingblocksarenotor- ableand‘inexpensive’buildingblockswhencomparedtoorbits.
Two-dimensionalkinematicsof SLACS lenses– I. 9
Figure8.BestdynamicalmodelforthegalaxySDSSJ2321−097.Firstrow:observedsurfacebrightness distribution, projected line-of-sight velocity and
line-of-sightvelocitydispersion.Secondrow:correspondingreconstructedquantities forthebestmodel.Thirdrow:residuals.Thelow-level ripplesinthe
surfacebrightnessresidualsareduetothediscretenatureoftheTICs.
The (inner) gravitational potential associated with a ho- small value. In addition to the previously mentioned parameters,
moeoidaldensitydistributionρ(m)isgivenbyChandrasekhar’sfor- therearefouradditionalparameterswhichdeterminethegeometry
mula(Chandrasekhar 1969;seealsoe.g.Merritt&Fridman1996 oftheobservedsystem:thepositionangleϑ ,theinclinationiand
PA
andCiotti&Bertin2005).Wefind,forγ′,2, thecoordinatesofthecentreofthelensgalaxywithrespecttothe
skygrid.
+∞ [1/(γ′−2)]m˜2−γ′
Φ(R,z)=−Φ dτ, (5) Thealgorithmisrunonthedatasets(andthecorre-
0Z0 (1+τ) q2+τ spondingcovariancematrices)describedintheprevioussections.
p
whereΦ0 =2πGqa2ρ0and (i) The lensed image surface brightness distribution is defined
R2 z2 on a 100×100 grid (1 pixel = 0.05arcsec) on which the inner-
m˜2 = + . (6) most and outermost low-S/N regions have been masked out, and
a2(1+τ) a2(q2+τ)
thesourceisreconstructedona40×40grid.Toincludetheeffects
Forγ′=2wehave ofseeinginthemodelling,thelensingmatrixisblurredwithablur-
+∞ log(1/m˜) ringoperatorwhichtakesintoaccounttheHST/ACSF814Wpoint
Φ(R,z)=−Φ0Z0 (1+τ) q2+τdτ. (7) spr(eiia)dTfuhneclteionns(gPaSlaFx)yobsutarifnaecdewbriitghhtnessdis(tKrirbiustti1o9n9i3s).given on
p
Therearethreefreenon-linearparametersinthepotentialto a50×50 grid(1pixel = 0.10arcsec),whilethekinematics data
bedeterminedviatheevidencemaximization:Φ (orequivalently, aregivenona9×9grid(1pixel = 0.67arcsec),butmaskingout
0
throughequation[B4]ofBK07,thelensstrengthα ),theslopeγ′ allthepointswithS/N lessthan8.Thedynamicalmodellingem-
0
andtheaxialratioq.OnecanalsoeasilyincludeacoreradiusR ploys100TICs,eachonepopulatedwithN =1×105particles.
s TIC
by modifying the definition of the homoeoidal radius m, but for TheTICsareselectedbyconsidering N = 10elementslogarith-
E
thepurposeofthepresentanalysisitwaskeptfixedtoanegligibly micallysampledintheradiusR ofthecircularorbitwithenergy
c
10 O. Czoskeet al.
Table2.SDSSJ2321−097: non-linear parameters forthe best tothemodel,butthebestreconstructionunambiguouslyexcluded
power-lawmodelρ∝m−γ′(equation3). anyrelevanceofthiscomponent(itslensstrengthisnotsignificant).
Asasanitycheck,wealsoperformedthereconstructionwith
Model:powerlaw Mbest
adifferentset-upforthedynamics,thatis,employingalargernum-
i 67◦.8 ber of TICs (N = 18 and N = 9) or increasing the number
E Lz
pNaornam-lienteearsr αγ0′ 20..046618 ofpopulating particlesto NTIC = 12×105 whilemaintainingthe
same number of TICs. In both cases, the obtained values for the
q 0.739
non-linearparametersdonotdiffersignificantlyfromtheonespre-
logλlen −2.344 sentedinTable4.2.Thereisanimprovementinthetotalevidence
Hyperparameters logλdyn −0.023 mainly because of the slightly better fit to the surface brightness
L distribution.
logλdyn 1.810
E
Elen −78480 Themostremarkableresultofthereconstructionistheloga-
Evidence Edyn −14558 rithmicslopeγ′ ≃2,makingthetotaldensityprofileverycloseto
Etot −93038 isothermal.Thisisingoodagreementwiththepreviousfindingof
γ′ ≃1.9obtainedbyKoopmansetal.(2006)forthesamesystem.
Thefactthat Koopmansetal.(2006)achieved robustresultswith
E =E(R ),andN =5elementslinearlysampledinangularmo- asimplerandnotfullyself-consistentjointlensinganddynamics
c c Lz
mentum,mirroredfornegative L.ThePSFismodelledasGaus- analysis(inwhichtheyonlyusedtheaverageσ fromtheSDSS
z los
sianswith0.10and0.90arcsecFWHMforthetwogrids,respec- fibrespectroscopyasaconstraintforthedynamicsandsolvedthe
tively. sphericalJeansequations)isastrongindicationoftheeffectiveness
ofthecombinedanalysisinpinningdownthecharacteristicsofthe
A curvature regularization (as described in Suyuetal. 2006
totalpotential.
and appendix A of BK07) is adopted for both the gravitational
lensing and the stellar dynamics reconstructions. As discussed in Itshouldbestressedthattheroleofthekinematicconstraints
BK07, the initial guess values of the hyperparameters (which set iscrucialinbreakingtheampledegeneraciesthatwouldarisefrom
thelevel oftheregularization) arechosen tobequitelarge,since areconstructionlimitedtothelensingdata:inparticular,thefairly
the convergence to the maximum is faster when starting from an constant line-of-sight velocity dispersion map unyieldingly rules
overregularizedsystem.Moreover,inordertofurtherspeedupthe outvaluesfortheslopewhicharesignificantlyfarfromisothermal,
reconstruction, the position angle ϑPA and the coordinates of the butwhichwouldturnouttobeperfectlyreasonable(evenbasedon
lens galaxy centre ξc = (xc,yc) are fairly accurately determined thequantitativeevidence)iflensingalonewasconsidered.
bymeansofapreliminaryoptimizationrunandsubsequentlykept
fixedorallowedtovaryonlywithinalimitedrangearoundthede- Thelensgalaxysurfacebrightnessdistributiontogetherwith
terminedvalues. Thepositionangleϑ = 135◦.5obtainedinthis theline-of-sightprojectedvelocitymap(whichshowsaslightin-
PA
wayisveryclosetoboththeobservedvalueforthelightdistribu- dication of rotationof thesystem) provide important information
tionϑ = 126◦.5andthepositionangleϑ = 136◦.2obtainedfor ontheflatteningandtheinclinationofthesystem,althoughthese
⋆ SIE
thesingularisothermalellipsoidlensingmodelinKoopmansetal. quantitiesremainthemostdifficulttodisentangle.Theweak(ofthe
(2006). orderofafewpercent)butdiscerniblepatternintheresidualsof
thesurfacebrightnessreconstruction canbeascribedtotheeffect
ofthediscretenessoftheadoptedTIClibrary,butcouldalsobean
indicationofaslightoffsetintheorientationbetweentheluminous
4.2 Results
matterdistributionandthetotalpotential:whenthereconstruction
Thenon-linearparametersofthebest-fittingmodelobtainedfrom islimitedtolensingalone,theobtainedpositionangleistypically
thecombinedgravitationallensingandstellardynamicsreconstruc- afewdegreeslargerthanϑ foundforthejointreconstruction.
PA
tionarepresentedinTable4.2,togetherwiththebest-recoveredhy-
perparametersandthevaluesoftheevidence.Therecoveredincli- Much information about the best reconstructed dynamical
nationangleisi=67◦.8.Thelensstrength,proportionaltothenor- model is contained in the weighted two-integral DF presented in
malizationconstantofthepotentialΦ (seeappendixBofBK07), Fig.9(refertoappendixDofBK07foraprecisedefinitionofthe
0
isα = 0.468.Thelogarithmicslopeandtheaxialratiooftheto- weightsγ whicharethequantityshowninthisplot).Sinceallthe
0 j
taldensitydistributionofthepower-lawmodelareγ′ =2.061and originalTICshaveequalmassbyconstruction,eachpixelinthein-
q=0.739,respectively. tegralspaceprovidestherelativecontributioninmassofthecorre-
The reconstructed observables corresponding to the best- spondingTIC.TheweightedDFisonlyslightlyasymmetricaround
fittingmodelareshownandcomparedtothedatainFigs7and8 L/L =0,whichisreflectedintheverylowvaluesoftherotation
z max
for,respectively,gravitationallensingandstellardynamics.Fig.7 velocityofthebestmodelsystem(cf.centralpanelofFig.8).The
alsodisplaysthereconstructedsourcemodel,whilethereconstruc- veryweakindicationofcounter-rotationwhichcanbedetectedin
tionoftheweightedtwo-integralDFofthelensgalaxyisgivenin thesamepanelisalsomoreclearlyexhibitedintheweightedDF:
Fig.9. forthehigher R theTICswithnegative L (theredpixels)show
c z
Thepresenceofanexternalshearwasalsoexamined,giving largervaluesthanthecorresponding(mirrored)TICswithpositive
resultsconsistentwithzero.Furthermore,inordertoassessthepos- L,whiletheoppositebehaviourisdisplayedforR .1.Giventhe
z c
sibleeffectofafaintgalaxylocatedat(2′.′55,2′.′10)north-westof lowS/N,however,wedonotplacemuchsignificanceonthispos-
thecentreofthelensgalaxyonthesky(Fig.1),asingularisother- sible counter-rotation, although SAURON has discovered several
malsphere(SIS)centredonthecorrespondingpositionwasadded systemswithsuchabehaviour(Emsellemetal.2004).