Steepened inner density profiles of group galaxies via interactions An N-body analysis

A(MNLTEXstylefilev2.2)

Steepenedinnerdensityprofilesofgroupgalaxiesviainteractions:

AnN-bodyanalysis

BenjaminM.Dobke,LindsayJ.King,MichaelFellhauer

InstituteofAstronomy,UniversityofCambridge,MadingleyRd,CambridgeCB30HA⋆

arXiv:astro-ph/0702741v1 28 Feb 2007Accepted.../Received...

ABSTRACT

WecontinuetoseearangeofvaluesfortheHubbleconstantobtainedfromgravitationallylensedmultipleimagetimedelayswhenassuminganisothermallensdespitearobustvaluefromtheHSTkeyproject(72±8kms−1Mpc−1).Oneexplanationisthatthereisavari-ationinHubbleconstantvaluesduetoafundamentalheterogeneityinlensgalaxiespresentingroups,i.e.centralgalaxieswithahighdarkmattersurfacedensity,andsatellitegalaxieswithapossiblystrippedhalo,lowdarkmattersurfacedensity,andamorecentralconcen-tratedmatterdistribution.Ourgoalistoseeifavarietyofgroupinteractionsbetweenthemostmassivegroupmemberscanresultinsignificantchangesinthegalaxydensityprofilesoverthescaleprobedbystronglensing(󰀁15kpc).Whilestrippingoftheouterpartsofthehalocanbeexpected,theimpactoninnerregionswheretheluminouscomponentisimpor-tantislessclearinthecontextoflensing,thoughstillcrucial,asasteepeneddensityprofilewithinthisinnerregionallowstheselenssystemstobeconsistentwithcurrentHST/WMAPestimatesonH0.Weemploytheparticle-meshcodeSUPERBOXtocarryoutthegroupin-teractionsimulations.Animportantadvantageofusingsuchacodeisthatitimplementsafast,low-storageFFT-algorithmallowingsimulationswithmillionsofparticlesondesk-topmachines.Wesimulateinteractionsbetweengroupmembers,comparingthedensityprofileforthesatellitebeforeandafterinteractionforthemassrangeof1011to1013M⊙.Ourinves-tigationsshowasignificantsteepeningofthedensityprofileintheregionof∼5-20kpc,i.e.thatwhichdominatesstronglensinginlensgalaxies.Thiseffectisindependentoftheinitialmass-to-lightratio.Additionally,thesteepeningintheinnerregionistransientinnature,withconsecutiveinteractionsreturningtheprofiletoanisothermalstatewithinatimeframeof∼0.5-2.0Gyr.ThisfactormayhelpexplainwhylensgalaxiesthatproducelowervaluesofH0(i.e.thosewithpossiblysteeperprofiles)arefarfewerinnumberthanthosewhichagreewithboththeHSTkeyprojectvalueforH0andisothermality,sinceonewouldhavetoobservethelensgalaxyduringthistransientsteepenedphase.

Keywords:Methods:N-bodysimulations–galaxies:structure–galaxies:interactions–galaxies:haloes–Gravitationallensing

1INTRODUCTION

ArangeofvaluesfortheHubbleconstantisobtainedfromdifferentstronggravitationallenssystemswithmeasuredimagetimedelays,assumingisothermalmodelsforthelensinggalax-ies(e.g.Biggsetal.1999;Gil-Merino,Wisotzki&Wambsganss2002).TheestimatesaretypicallylowerthanthevaluesfromtheHSTkeyprojectandWMAPwhichareconsistentwitheachotherat72±8kms−1Mpc−1and72±5kms−1Mpc−1,re-spectively(Freedmanetal.2001;Spergeletal.2003).SincethisHubbleconstantvalueisconsideredrobust,thisvariationraisesquestionsastotherelativeextentofthegalaxyhaloandthelu-minouscomponentsincethederivedvaluesfundamentallyde-⋆

pendontheassumedlenspotentialandmattersurfacedensi-ties.Anumberofdifferentlymotivatedstudieshavefoundthattheinnerregionsoflensgalaxiesexhibitisothermaldensitypro-files(e.g.Rusin,Kochanek&Keeton2003),includingrecentre-sultsfromtheSloanSLACSsurvey(Koopmansetal.2006)which

.02−η

foundη=2.01+0)whenstudying15early-−0.03(1σ;ρtot∝r

typegalaxies.Moreover,studiesinferringtheHubbleconstantfrom10time-delaylensgalaxiesproducedvaluesinagreementwiththeHST/WMAPvalue,whilestillbeingconsistentwithsimu-latedtime-delaysfromN-bodygalaxieswhichwereisothermalinnature(Sahaetal.2006).Evenarelativelysimpleanalysisus-ingtheexistingtimedelaysystemsindicatesaconsistencywithisothermality(Dobke&King2006).Stellardynamicalstudiesalsopointtowardsisothermaldensityprofilesintheinnerregionsof

E-mail:bdobke@;ljk@;madf@;ast.cam.ac.uk

2B.M.Dobkeetal.

galaxies(e.g.Rixetal.1997;Romanowsky&Kochanek1999;Gerhardetal.2001;Treu&Koopmans2002a).

Despitethisapparenttrendtowardsisothermality,westillob-serveanumberofsystemswhereitishardtoreconcileH0val-ueswiththeHSTkeyprojectvaluewhenassuminganisother-malmodel.ModellingofthesystemsPG1115+080,B1600+434,HE2149-2745,andSBS1520+530hasshownthatthesesystemsprefermorecentrallyconcentrateddensityprofilesandhencenon-isothermalmodels(Kochanek2002;Kochanek&Schechter2004).InthespecificcaseofPG1115+080,detailedmodellingdonebycombininglensing,stellarkinematicandmass-to-lightratiocon-straints,inordertobuildatwo-componentmodel,showsamassdensityprofilesignificantlysteeperthanisothermalatη=2.35±0.15,whereρtot∝r−η.However,thederivedvaluefortheHubble

constantcametoH0=59+12−1

Mpc−1,steeperprofileto−7kms

implyingtheneedforanevenagreewiththeHSTkeyprojectvalue(Treu&Koopmans2002b).

Keeton&Zabludoff(2004)haveshownthatlensmodelsoffour-imagesystemsoverestimatetheHubbleparameterbyupto15%whenneglectingtheeffectofthecontributionfromothergroupmembersonthepotentialofthelensgalaxy,withanevenhigherdiscrepancyfortwo-imagesystems.HenceaccountingforenvironmentwouldfurtherreducethederivedvalueofH0forafixedprimarylensmodel,requiringtheslopeoftheprimarylenstobefurthersteepenedtoobtainconsistencywiththeHSTkeyprojectvalue.Additionally,outsidethegroupenvironment,line-ofsightstructurescanprovidecontributionstothelenspotential(Momchevaetal.2006).

Whileisothermalsystemscertainlyappeartobethemoreprevalent,anexplanationofthissub-sampleoflensgalaxiesre-quiringnon-isothermalprofilesisnotclear.Onesuggestionfo-cusesonthepossiblerolethatthegroupmayhaveonthelensgalaxy,specificallytheeffectoftidalstrippingofsatellitegalax-ies(Kochaneketal.2006)asrecentlyobservedinclustersus-inggalaxy-galaxylensing(Limousinetal.2006).Althoughafullcensusoflensgalaxyenvironmentsisnotavailable,anumberofobservations(Tonry&Kochanek1999;Fassnachtetal.2002;Williamsetal.2006)seemtoshowthattypicalearly-typelensgalaxiesshouldbemembersofagroup.InthehalomodeldescribedbyCooray&Sheth(2002),onegalaxytypicallyliesatthecentreofthegrouphalo,whileothergalaxiesaresmallersatellitesorbitinginthathalo.Inthisscenariowehavetwodifferingcasesforgalaxystructure.Inthefirstweseethecentralgalaxieswithahighdarkmattersurfacedensityandtheluminouscomponentmakingonlyasmallfractionoftheoverallmattercontent.Inthesecondcase,weseeasatellitegalaxywithapossiblystrippedhalo,andmorecen-trallycondensedcore.Inessence,itsuggestsafundamentalhetero-geneityinearly-typegalaxiespresentingroups,andwhenappliedtolensgalaxiesthedistinctionbetweenthetwosurfacemassden-sitiescouldgiverisetoadifferenceinthederivedHubbleconstantwereastandardisothermalmodelassumed.

Thisdescriptionoftheheterogeneityingroupsbegsasimplequestion;howmuchcanthegroupmembersactuallyeffecteachotherintermsofchangingtheirmassanddensityprofiles?Insuchgroups,unlikelargeclusters,thereisnotsomuchasingledom-inantcentralgalaxy,butratherapowersharingscenariobetweentwoormoreapproximatelyequivalentgalaxies.Whileinteractionsbetweenmemberswithlargedifferencesintheirrespectivemassesshouldbeexpectedtoproducelargedynamicalandstructuralef-fectsinthelessmassivemember(e.g.ultracompactdwarf(UCD)interactions;seeFellhauer&Kroupa2006),theeffectofinterac-tionsbetweenmoresimilarlymatchedgroupmembersisperhaps

notsoclearwhenconsideringtheeffectinthecontextoflensgalax-ies.Indeed,althoughcertainpaststudies(e.g.Hayashietal.2003;Kazantzidisetal.2004)haveinvestigatedtheeffectofsatellite-halointeractions,thefocushasbeenontheevolutionofsubstruc-tureoroftheveryinnercuspratherthantheeffectoncosmologicalparameterderivationfromlensgalaxies.

Thequestionposedisacomplexone,andtheparameterspaceofgroupinteractionsisvast.Inthispaperouraimistoestablishwhether,overthescaleprobedbystronglensing(󰀁15kpc),signif-icantchangesinthedensityprofileofagalaxyonasatelliteorbitcanbeinducedbyinteractionswithagroupgalaxyatacentrallo-cale.Whilestrippingoftheouterpartsofthesatellite’shalocouldbeeasilyexpected,theimpactoninnerregionswherethelumi-nouscomponentisimportantislessclearinthecontextoflensing,thoughcrucial,asitiswithasteepeneddensityprofilewithinthisinnerregionthatthetimedelaysfortheselenssystemscanagreewithcurrentHST/WMAPestimatesonH0.Inthiswayweascer-tainifanothermechanismisrequiredtoalterdensityprofiles,pos-siblyduringthegroup’sformation.

Theoutlineofthepaperisasfollows;in§2webrieflyintro-ducetheN-bodycodeusedtosimulatetheinteractions,withdetailsoftheexactparametersofthesimulatedobjects.In§3,wegoontopresentresultsanddiscusstheiranalysisintheframeworkoftheproblem.Finally,§4drawssomeconclusionsbaseduponourfindings.

2THEORYANDSIMULATIONSETUP2.1Theparticle-meshcodeSUPERBOX

Ouranalysisemploystheparticle-meshcodeSUPERBOXtocarryoutthesimulations.Thecodeutiliseshighresolutionsub-gridsandanearestgridpoint(NGP)force-calculationschemebasedonthesecondderivativesoftheinputpotential.Animportantadvantageofusingsuchacodeisthefactthatitimplementsafast,low-storageFFT-algorithmgivingthepossibilitytoworkwithmillionsofpar-ticlesondesk-topmachinesorsmallclusters.

Foreachobjectusedinthesimulation,thereare5gridswith3differentresolutions.Forthepurposesofoursimulationsetup,wedefinethesizeoftheinner,middleandoutergridsizes.Theinnergridisthehighest-resolutiongridwhichresolvesthecentreofthegalaxy.Themiddlegridhasanintermediateresolutiontoresolvethegalaxyasawhole,whiletheoutergridisthesizeofthewholesimulationarea(i.e.‘thelocaluniverse’),andhasthelowestresolution.Theinnerandmiddlegridremainfocusedontheobjectthroughoutthesimulation.Eachgridhasapredefinednumberofmesh-points.Inoursimulationsweuse26mesh-points.Doublingthenumberto27showednosignificanteffectontheoverallresultsdiscussedin§3,indicatingahighenoughlevelofresolution.ForafulldescriptionofthecodeandgridstructureseeFellhaueretal.(2000).

2.2SimulationSetup

Wefocusthemainbodyofourinvestigationsonapproximatelyimitatingthedynamicsandparametersoflenssystemswithaknownhistoryoflensmodelswithlowdarkmattersurfacedensi-ties,andhencepossiblesteeperdensityprofiles,e.g.PG1115+080,B1600+434,HE2149-2745,SBS1520+530(Kochanek2002),andassuchareclearcandidatesforpossiblesatellitestripping.Weuse

Steepeneddensityprofilesviagroupinteractions

100

3

80R [10%...(10)...90%] (kpc) 60

40

20

0

0 2 4

6 8Time (Gyrs)

10 12

Figure1.Left:ContourplotofanNFWhaloafterundergoingisolatedintegration.Thecontourshave(logarithmic)magnitudespacingwiththeinnermostandoutermostcontourscorrespondingto16and36magarcsec−2respectively.Theobjectshowsclearlydefinedsymmetricalstructureandisreadytobeusedinasimulationscenario.Right:TheevolutiontodynamicequilibriumasshownbytheLagrangian-radiiofthemassshellsof10%to90%ofparticlesinatypicalsimulation.Theflatteningofthelineswithtimeshowsthatanequilibriumstatehasbeenreached,andthattheobjectisreadytobeusedinaninteractionscenario.

300 200 100y (kpc) 0-100-200-300

-300

Central PotentialSatellite Orbit100kpc radius150kpc radius

-200-100

0x (kpc)

100 200 300

Figure2.Left:Exampleofapathtakeninthex-yplaneduringasimulationrun.Theorbitstartingpositioninthex-yplaneis(0,-300)kpc,withatangentialvelocityVtan=+100kms−1inthex-dimension.Thisparticularrunshowstheinteractionbetweena1011M⊙objectafterundergoingthreepassesofacentral1012M⊙object.Theruntimeis7.3×109years.Alsoshownarethe100kpcand150kpcradialdistancesfromthecentralpotential,beyondwhichthesatellitespends78%and%ofitsorbittimerespectively.Right:Typicalcontourplotforthesame1011M⊙haloafterundergoingtwointeractionswitha1012M⊙centralobject.Notetheformationoftidal-tailstructuresintheouterregions.

thetermsatelliteheretorefertoagalaxyundergoingorbitalinter-actionswithanothergalaxy,whetherthatbeofequivalentorhighermass,andnotinthesenseofe.g.adwarfellipticalsatellite.SomeoftheparametersusedtosetupthemodelgalaxiesweretakeninpartfromImpeyetal.(1998)andKeeton&Zabludoff(2004).Param-eterssuchasmassandscaleareobviouslyoftenmodeldependentandassuchwereonlytakenasapproximateguidevaluesandsanitychecks.Inactuality,variationsofuptoanorderofmagnitudeinthemassesofthemodelobjectshadlittleeffectontheoverallresults;moreimportantweretherelativemassesbetweentheinteractingbodies.

ForthedarkmatterhalocomponentofeachofthegroupgalaxiesweuseaNavarro,Frenk&White(NFW)profile(Navarro,Frenk&White1996)whilefortheluminouscomponentweuseaHernquistprofile(Hernquist1990).Hence,thecombineddensityprofileforeachofthemodelledgalaxieshastheform,

ρtot(r)=ρN(r)+ρH(r)

=

ρ0,Nrs,N

2πr(rs,H+r)3

(1)

WehavetheoptioninSUPERBOXtoeitherdefinethecharac-teristicdensityfortheNFWhalo,ρ0,N,orprovidethetotalmass,MN,inwhichcasethecodewillcalculatethedensityinequation(1)bymeansofthefollowing,ρ0,N=

MN

rs,N+r

󰀃

(2)

Inordertofullydefinetheobjectsusedinthesimulationswemustprovidevaluesforthefollowingparameters;NFWtotalmass(MN)andscalelength(rs,N),theHernquisttotalmass(MH)and

4B.M.Dobkeetal.

Table1.Tableofourinitialmodelparametersforthreeparticularinteractions.ThefirsttworowscorrespondtotheobjectparametersthatwereusedinthesimulationsofFig.3andFig.4.WeincludeNFWandHernquistmasswithcorrespondingscaleandcut-offradii,thecharacteristiccrossingtime(Tcr)andthegridresolutionsusedineachcase.Weusethevirialradiusasanapproximateguideforthecut-offradii.Themiddlegridsizeisthenchosentoencompassthisvalue.Parametersforthe1013M⊙objectweretakeninpartfromMcLaughlin(1999)andVesperinietal.(2003)

1011/10101012/10111012/1011

10/1020/1020/10120/120250/120250/120292.36250.58250.58

1012/10111012/10111013/1012

20/1020/10560/5250/120250/120150/15050/150/100050/300/100050/300/1000

ρi(r)

󰀁

rcut

dφtotal(r′)

r

Steepeneddensityprofilesviagroupinteractions

0-2

-2

Log density (Msolar/pc3)Log density (Msolar/pc3)-4-6-8-10-12-14

-5 0.7

-5 0.7Log density (Msolar/pc3)Initial density profileDensity profile after 2nd interactionDensity profile after 3rd interaction-1

Initial density profileTop solid line: Power-law fit η = 2.01Density profile after 2nd interactionBottom solid line: Power-law fit η = 3.05

-1

Initial density profileTop solid line: Power-law fit η = 2.01Density profile after 3rd interactionBottom solid line: Power-law fit η = 2.04

5

-2

-3-3

-4-4

-1-0.5 0

0.5 1 1.5Log radius (kpc)

2 2.5 3 0.8

0.9 1 1.1Log radius (kpc)

1.2 1.3 0.8

0.9 1 1.1Log radius (kpc)

1.2 1.3

Figure3.Left:Combineddensityprofile(NFW+Hernquist)beforeandaftera2ndand3rdinteractionbetweena1011M⊙galaxyand1012M⊙objectmodelledwithananalyticpotential.Initialx-ypositionandvelocityofthesatellitewere(0,-300)kpcand(+60,0)kms−1respectively.Middle:Enlargedplotofthedensityprofileintheregion󰀁15kpcafterthe2ndinteraction.Dashedlineisthecombineddensityprofilebeforeinteraction,whiledottedlineisthecombineddensityprofileafterinteraction.Topsolidlineshowsapower-lawfitwithη=2.01±0.03i.e.isothermal,whilebottomsolidlineshowsapower-lawfitwithη=3.05±0.07.Theprofilehassteepenedsignificantly.Right:Enlargedplotofthedensityprofileintheregion󰀁15kpcafterthe3rdinteraction,∼2×109yearsafterthe2ndinteraction.Dashedlineisthecombineddensityprofilebeforeinteraction,whiledottedlineisthecombineddensityprofileafterinteraction.Topsolidlinesshowapower-lawfitwithη=2.01±0.03,withthebottomatη=2.04±0.06.Theprofilehasreturnedtoanisothermalstate.

2

Initial density profile

Density profile after 3rd interactionDensity profile after 4th interaction

-1Log density (Msolar/pc3) 0Log density (Msolar/pc3)Initial density profile

Top solid line: Power-law fit η = 2.00Density profile after 3rd interactionBottom solid line: Power-law fit η = 2.70

Log density (Msolar/pc3)-1

Initial density profile

Top solid line: Power-law fit η = 2.00Density profile after 4th interactionBottom solid line: Power-law fit η = 2.01

-2

-4

-2-2

-6

-8

-10

-1-0.5 0

0.5 1 1.5Log radius (kpc)

2 2.5 3

-3 0.8

0.9

1 1.1Log radius (kpc)

1.2 1.3

-3 0.8

0.9

1 1.1Log radius (kpc)

1.2 1.3

Figure4.Left:Combineddensityprofile(NFW+Hernquist)beforeandaftera3rdand4thinteractionbetweena1012M⊙galaxyand1012M⊙objectmodelledwithananalyticpotential.Initialx-ypositionandvelocityofthesatellitewere(0,-300)kpcand(+100,0)kms−1respectively.Middle:Enlargedplotofthedensityprofileintheregion󰀁15kpcafterthe3rdinteraction.Dashedlineisthecombineddensityprofilebeforeinteraction,whiledottedlineisthecombineddensityprofileafterinteraction.Topsolidlineshowsapower-lawfitwithη=2.00±0.02i.e.isothermal,whilebottomsolidlineshowsapower-lawfitwithη=2.70±0.06.TheprofilehassteepenedlesssothaninthecaseshowninFig.3,althoughtheeffectisstillsignificant.Right:Enlargedplotofthedensityprofileintheregion󰀁15kpcafterthe4thinteraction,∼0.5×109yearsafterthe3rdinteraction.Dashedlineisthecombineddensityprofilebeforeinteraction,whiledottedlineisthecombineddensityprofileafterinteraction.Topsolidlinesshowapower-lawfitwithη=2.00±0.02,withthebottomatη=2.01±0.20.Theprofilehasreturnedtoanapproximatelyisothermalstate.

(NFW+Hernquist)forthesimulatedsatellitegalaxy(topmostline),andthedensityprofileafterthe2ndand3rdinteractionsim-mediatelybelow.Themassesoftheinteractingbodieswere1011and1012M⊙forthesatelliteandcentralgalaxyrespectively.Theinitialmass-to-lightratioofbothgalaxieswassetto10:1,whilethestartingpositionwassetat(0,-300)kpc,withaninitialtangen-tialvelocityof+60kms−1inthex-dimension.Forthepurposesofthisstudyweareinterestedinhowtheinteractionsintheouterpartofthehaloeffectboththeluminousanddarkmattercomponentsintheinnerpartofthegalaxy,particularlytheregion󰀁15kpcasthisscaleencompassesthestronglensingregime.InFig.3(mid-dle),weenlargethisregion,justcomparingtheinitialandpost-2ndinteractiondensityprofiles.Here,thedashedlineistheprofilebe-foreinteraction,whilethedottedlineistheprofileafterinteraction.

Thetopsolidlineshowsapower-lawfitwithη=2.01±0.03,i.e.isothermal,whilethebottomsolidlineshowsafitwithη=3.05±0.07.Weclearlyseesignificantsteepeningoccurringafterthe2ndinteractionovertheregionof∼5-20kpc.TherightplotofFig.3focusesoncomparingtheinitialandpost-3rdinteractiondensityprofiles.Thetimeelapsedbetweenthetwoconsecutiveinteractionsis∼2×109years,duringwhichtheslopeoftheprofileremainsapproximatelyconstantatη∼3.0.Wenowseethatafterthe3rdinteraction,theslopehasreturnedoncemoretoanisothermalstateatη=2.04±0.06,andhencethesteepeningobservedafterthe2ndinteractionappearstransientinnature.

WeseeasimilareffecttakingplaceinFig.4,exceptinthiscaseweareinvestigatingtheinteractionsbetweentwogalaxiesofsimilarmass,bothbeing1012M⊙.Weseethatthereductioninthe

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1.4e+041.2e+04

Pot. Energy (Arb. Units) Kin. Energy (Arb. Units)1.0e+048.0e+036.0e+034.0e+032.0e+030.0e+00

0.0e+00-5.0e+03

Pot. Energy (Arb. Units) 0

1

2

3 4Time (Gyrs)

5

6

7

-1.0e+04-1.5e+04-2.0e+04-2.5e+04-3.0e+04-3.5e+04-4.0e+04

-5.0e+03-1.0e+04-1.5e+04-2.0e+04-2.5e+04-3.0e+04-3.5e+04-4.0e+04-4.5e+04

0 1 2

3 4Time (Gyrs)

5 6 7 0 1 2

3 4Time (Gyrs)

5 6 7

Figure5.Left:ThetimeevolutionofkineticenergyoftheboundremnantsoftheinteractingsatelliteofFig.3.Notethethreefluctuationsinenergyresultantfromthethreeinteractionsandthatthisenergychangebecomesprogressivelysmallerwithtimeandhencethesteepeningoftheprofilesbecomeslesssevereaftereachpassage.Middle:ThetimeevolutionofpotentialenergyoftheboundremnantsoftheinteractingsatelliteofFig.3.Againweseethethreeenergyfluctuations,eachbecomeslesssevereaftereachpassage.Thisresultsintheprofilebeingultimatelyunchangeddespiteatransientsteepeningduringthefirstinteractions.Right:Thetimeevolutionofpotentialenergyjustwithintheregionof5-20kpc,forthecaseoftheinteractingsatelliteofFig.3.Weseethepotentialwellincreaseindepthaftertheinteractions,confirmingthesteepeningoftheprofile,whichagainlessensaftereachinteraction.

relativemassnecessitatesagreaternumberofinteractioneventstoinitiatethesteepeningwhencomparedwiththepreviouscase.InthemiddleplotofFig.4thesteepenedprofileafterthe3rdinter-actioncannowbefittedwithaslopeofη=2.70±0.06overtherangeof∼6-20kpc.Althoughlessthaninthe1011vs.1012M⊙case,thisisstillasignificanteffect.Onceagain,afterthenextinteractionweobservetheprofilereturningtoanapproximatelyisothermalstatewithη=2.01±0.20(Fig.4,right).Thistimethetransientsteepeninglastsfor∼0.5×109years.

WenowconsiderthesteepeningeffectseeninFig.3andFig.4intermsofenergytransferduringtheinteractions.DisplayedinFig.5isthetimeevolutionofthekineticandpotentialenergiesoftheboundremnantsfortheinteractionsshowninFig.3.Duringthefly-bysofthesatellitegalaxy,energygetstransformedfromorbitaltointernalenergy.Particlesinthesatellitegainkineticenergyandthosethatarelooselybound,mainlyintheouterparts,getstrippedresultinginthesteepeningoftheouterprofile.Closertowardstheinnerportionofthegalaxytidalshocksaremoreimportantthanstripping(e.g.Readetal.2006;Gnedin,Hernquist&Ostriker1999).Closepericentrepassescangeneratemattercompressionspossiblyresultingintheobservedtransientfluctuationsintheden-sityprofile.Asthemassgetsredistributedandtheobjectreturnsonceagaintoequilibrium,theoriginalslopereturnstoitsapproxi-mateinitialstate.NoticeinFig.5thatthisenergychangebecomesprogressivelysmallerwithtimeandhencethesteepeningoftheprofilesbecomeslessandlesssevereaftereachpassage,resultinginareturntoisothermality.Withgreatertimeweseenochangeintheprofile,confirmingthetransientnatureofthesteepeningseeninFig.3andFig.4.Ultimately,thedensityprofileswillbeapproxi-matelyunchangedfromtheirinitialstate,inagreementwithanum-berofotherstudies(e.g.Kazantzidisetal.2004).IntherighthandplotofFig.5,wedisplaytheevolutionofpotentialenergywithtimeforjustthe5-20kpcregion.Noticehowthepotentialwellin-creasesindepthaftertheinteractions,confirmingthesteepeningoftheprofilethatisseeninFig.3.

ForbothoftheinteractionscenariosdisplayedinFig.3andFig.4,wealsoinvestigatedtheeffectofvaryingtheinitialmass-to-lightratioofthesatellitegalaxy.InadditiontothecasesshownforanM/L=10:1,weperformedsimulationsusingM/L=20:1and5:1.Inallcasestherewasverylittlevariationintheresults,andtheoveralleffectofsteepeningfollowedbyareturntoisothermalitywasobserved.Wealsoexaminedhighermassinteractionsbetween

-1

Initial density profileTop solid line: Power-law fit η = 2.01Density profile after 3rd interactionMiddle solid line: Power-law fit η = 2.70

Density profile after 4th interactionBottom solid line: Power-law fit η = 2.01

Log density (Msolar/pc3)-2

-3 0.8

0.9

1

Log radius (kpc)

1.1 1.2

Figure6.Densityprofilebeforeandaftera3rdand4thinteractionbetweentwo1012M⊙objects.Herewehavereplacedthestaticpotential(asinFig.3andFig.4)withaparticlebasedobject.Asinthestaticpotentialcase,wecontinuetoseethedistinctivesteepeningfollowedbyareturntoanapproximateisothermalstate.

1012and1013M⊙objectsandagainobservedasimilarpatternofsteepeningfollowedbyareturntoisothermality,althoughthelargerofthesetwoobjectsrepresentsamassnormallyonlyasso-ciatedwithcentrallocationsinclustersratherthaningroups.Toconfirmthattheexclusionofdynamicalfrictiondoesnotoverlyef-fectthefundamentalresult,wereplacethecentralstaticpotentialwithaparticlemodelledobject.Weperformasimulationbetweentwo1012M⊙galaxies,i.e.thecaseinwhichdynamicalfrictionshouldbeatitsgreatest.AsshowninFig.6wecontinuetoseethetransientsteepeninginthedensityprofile.

4DISCUSSIONANDCONCLUSIONS

WeappeartorequiresteeperthanisothermaldensityprofilesinordertoreconcilecertainlensgalaxieswithcurrentestimatesoftheHubbleconstantfromHST/WMAP.ThesegalaxiesincludePG1115+080,B1600+434,HE2149-2745,andSBS1520+530.Onepossibleexplanationisthatcertainearly-typelensgalaxiesareoff-setfromtheFundamentalPlaneandassuchearly-typegalaxiesingeneralmightdisplayaheterogeneityintheirstructureandhence

Steepeneddensityprofilesviagroupinteractions

7

massdensityprofiles.Thistheoryproposesthattheheterogene-ityresultsfromgroupinteractionsbetweenthelensgalaxyanditsfellowgroupmembers,resultingincentralgalaxieshavingisother-malprofilesandsatelliteshavingsteeperthanisothermalprofiles.Anumberofstudiesseemtoagreethatthediscrepanciesinde-rivedH0valuescouldbetheresultofstructuraldifferencesinlensgalaxiesasopposedtootherfactors(Treu&Koopmans2002b;Kochaneketal.2006).

Wehaveinvestigatedgroupinteractionsbetweengroupmem-berswithmassesrangingfrom1011to1013Masignificantsteepeningofthedensity⊙.Ourinvestiga-tionsshowprofileinthein-nerregion,pertinenttostronglensing.Thiseffectappearstobeindependentoftheinitialmass-to-lightratio,whilethepreciseini-tialorbitalparametersmadelittledifferencetothefundamentalre-sultprovidedconsecutiveinteractionswiththecentralgalaxytookplace.Themagnitudeofthesteepeningitselfvariedwiththerel-ativemassesoftheinteractingobjects;forcaseswherethemassdifferedbyanorderofmagnitude,asforthecasesininteractionsinvolving1011vs.1012Moriginallyisothermalprofiles⊙and1012vs.1013M(η=2)steepenupto⊙,weobserveη=3.Forthecaseswheremasseswereofsimilarmagnitude,i.e.aninteractionbetweentwo1012Mup⊙groupmembers,weobserveasteepeningoftheprofileslopeto∼η=2.7.Inbothcases,modellingsuchsystemswithisothermalmodelswouldresultinasignificantunder-estimateoftheHubbleconstant.

Westressedin§1thatmostearly-typegalaxiesappeartoresideingroups(Tonry&Kochanek1999;Fassnachtetal.2002;Williamsetal.2006),andasdiscussedbyKochaneketal.(2006),thereisaplay-offinlensingopticaldepthbetweensatellitegalaxiesbeingmorenumerousthanthemostmassivecentralgroupgalax-ies,yethavingalowerlensingcross-section(seealsoOguri2006).Evenforsatellitegalaxies,thesteepeningintheinnerregionseemstransientinnature,withconsecutiveinteractionsoftenreturningtheprofiletoanisothermalstatewithinatimeframeof󰀁2Gyr.Thisfactormayhelpexplainwhylensgalaxiesthatproducelowerval-uesofH0(i.e.thosewithpossiblysteeperprofiles)arefarfewerinnumberthanthosewhichagreewithboththeHSTkeyprojectvalueforH0andisothermality.Inaddition,itisimportanttonotethatthistransientfluctuationinsteepnessisconsistentwithstud-iesshowingthatdensityprofilesoverlongertimescalescanbero-bustdespitestronginteractionsormergers(Kazantzidisetal.2004;Kazantzidis,Zentner&Kravtsov2006).

Weshouldhighlightthatotherfactorsbesidestheenviron-mentdiscussedin§1arethoughttocontributetodiscrepanciesinderivedHubbleconstantvaluesfromcertainsystems.Recently,Saha&Williams(2006)showedinaninvestigationwithmodelsof35galaxylensesthatshape-modellingdegeneracies(e.g.causedbytriaxiality)canalsocontributetochangesinthetimedelaysandhencethederivedHubbleconstantvalue.ThiseffectalonewouldneedtobeverysignificanttofullyaccountforthesystemswiththelowestderivedvaluesofH0.Itseemsprobablethatacombi-nationofeffects,alongwithneglectofphysicalsteepeningduetostripping,canaccountforlowH0values.

Lookingtowardsthefuture,spaceandgroundbasedobser-vatoriessuchasGAIAandLSSTwilluncoverroughlyfiftytimesmoremultiplyimagedquasarsthanwehaveidentifiedtoday.Withagreatersampleoflenssystemstostudy,andbycombininglensmodelswithstellardynamicalconstraints,heterogeneityinearly-typegalaxiesasafunctionofenvironmentandredshiftwillbeex-ploredindetail.Thiswillalsoallowunprecedentedconstraintsontheassemblyanddistributionofdarkmatterongalaxyandgalaxygroupscales.

ACKNOWLEDGMENTS

WethankMarkWilkinsonandChrisFassnachtfordiscussionsandtheanonymousrefereefortheirconstructivecomments.ThisworkwassupportedbyPPARCthroughaPhDstudentship(BMD),bytheRoyalSociety(LJK),andbyPPARC(MF)

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