A Unifying Framework for Self-consistent Gravitati

arXiv:astro-ph/0701372v328Sep2007DraftversionFebruary5,2008PreprinttypesetusingLATEXstyleemulateapjv.10/09/06AUNIFYINGFRAMEWORKFORSELF-CONSISTENTGRAVITATIONALLENSINGANDSTELLARDYNAMICSANALYSESOFEARLY-TYPEGALAXIESMatteoBarnab`eandL´eonV.E.KoopmansKapteynAstronomicalInstitute,UniversityofGroningen,P.O.Box800,9700AVGroningen,TheNetherlandsDraftversionFebruary5,2008ABSTRACTGravitationallensingandstellardynamicsaretwoindependentmethods,basedsolelyongravity,tostudythemassdistributionsofgalaxies.Bothmethodssufferfromdegeneracies,however,thataredifficulttobreak.Inthispaper,wepresentanewframeworkthatself-consistentlyunifiesgravitationallensingandstellardynamics,breakingsomeclassicaldegeneraciesthathavelimitedtheirindividualusage,particularlyinthestudyofhigh-redshiftgalaxies.Foranygivengalaxypotential,themappingofboththeunknownlensedsourcebrightnessdistribu-tionandthestellarphase-spacedistributionfunctionontothephotometricandkinematicobservablescanbecastasasinglesetofcoupledlinearequations,whicharesolvedbymaximizingthelikelihoodpenaltyfunction.TheBayesianevidencepenaltyfunctionsubsequentlyallowsonetofindthebestpotential-modelparametersandtoquantitativelyrankpotential-modelfamiliesorothermodelas-sumptions(e.g.PSF).Wehaveimplementedafastalgorithmthatsolvesforthemaximum-likelihoodpixelizedlensedsourcebrightnessdistributionandthetwo-integralstellarphase-spacedistributionfunctionf(E,Lz),assumingaxisymmetricpotentials.Tomakethemethodpractical,wehavedevisedanewMonteCarloapproachtoSchwarzschild’sorbitalsuperpositionmethod,basedonthesuperpo-sitionoftwo-integral(EandLz)toroidalcomponents,tofindthemaximum-likelihoodtwo-integraldistributionfunctioninamatterofsecondsinanyaxisymmetricpotential.ThenonlinearparametersofthepotentialaresubsequentlyfoundthroughahybridMCMCandSimplexoptimizationoftheevidence.Illustratedbythepower-lawpotentialmodelsofEvans,weshowthattheinclusionofstel-larkinematicconstraintsallowsthecorrectlinearandnonlinearmodelparameterstoberecovered,includingthepotentialstrength,oblatenessandinclination,which,inthecaseofgravitational-lensingconstraintsonly,wouldotherwisebefullydegenerate.Subjectheadings:gravitationallensing—stellardynamics—galaxies:structure—galaxies:ellipticalandlenticular,cD1.INTRODUCTIONUnderstandingtheformationandtheevolutionofearly-typegalaxiesisoneofthemostimportantopenproblemsinpresent-dayastrophysicsandcosmology.WithinthestandardΛCDMparadigm,massiveellipti-calsarepredictedtobeformedviahierarchicalmergingoflowermassgalaxies(Toomre1977;Frenketal.1988;White&Frenk1991;Barnes1992;Coleetal.2000).De-spitethemanytheoreticalandobservationalsuccessesofthisscenario,severalimportantfeaturesofearly-typegalaxiesarestillleftunexplained.Inparticular,theoriginofthe(oftenstrikinglytight)empiricalscal-inglawsthatcorrelatetheglobalpropertiesofellip-ticalsremainunexplained:(i)thefundamentalplane(Djorgovski&Davis1987;Dressleretal.1987),relat-ingeffectiveradius,velocitydispersionandeffectivesur-facebrightness;(ii)theMBH−σ(Magorrianetal.1998;Ferrarese&Merritt2000;Gebhardtetal.2000),relat-ingthemassofthecentralsupermassiveblackholehostedbythegalaxywithitsvelocitydispersion;(iii)thecolor-σ(Bower,Lucey,&Ellis1992)andtheMg2−σ(Guzmanetal.1992;Bender,Burstein,&Faber1993;Bernardietal.2003)relatingthevelocitydispersionwiththestellaragesandthechemicalcompositionofthegalaxy.Eachofthesescalingrelationsrelatestructural,(spectro-)photometric,anddynamical(i.e.stellarve-locitydispersion)quantities.Whereasthefirsttwoaresolelybasedontheobservedstellarcomponent,thelatterisafunctionofthestellaranddarkmattermassdistri-bution.Hence,adetailedstudyoftheinnermassprofileofearly-typegalaxiesatdifferentredshiftsisundoubt-edlynecessarytoproperlyaddressthenumerousissuesrelatedwiththeformationandtheevolutionoftheseob-jectsandtheirscalingrelations,anditwouldalsoconsti-tuteanexcellenttestbedforthevalidityoftheΛCDMscenarioonsmall(i.e.nonlinear)scales.Today,alargenumberofthoroughstellardy-namicandX-raystudieshavebeenconductedtoprobethemassstructureofnearby(z.0.1)early-typegalaxies(Fabbiano1989;Mouldetal.1990;Saglia,Bertin,&Stiavelli1992;Bertinetal.1994;Frank,vanGorkom,&deZeeuw1994;Carolloetal.1995;Arnaboldietal.1996;Rixetal.1997;Matsushitaetal.1998;Loewenstein&White1999;Gerhardetal.2001;Seljak2002;deZeeuwetal.2002;Borriello,Salucci,&Danese2003;Romanowskyetal.2003;Cappellarietal.2006;Gavazzietal.2007),inmost(butnotall)casesfindingevidenceforthepres-enceofadarkmatterhalocomponentandforaflatequivalentrotationcurveintheinnerregions.Whenitcomestodistant(z&0.1)early-typegalaxies,however,onlyrelativelylittleisknown.Thetwomaindiagnostictoolswhichcanbeem-ployedtocarryoutsuchstudies,namelygravitationallensing(Schneideretal.2006)andstellardynamics2M.Barnab`e&L.V.E.Koopmans(Binney&Tremaine1987),bothsufferfromlimita-tionsanddegeneracies.Gravitationallensingprovidesanaccurateandalmostmodelindependentdetermi-nationofthetotalmassprojectedwithintheEin-steinradius,butareliabledeterminationofthemassdensityprofileisoftenpreventedbytheeffectsofthemasssheet(Falco,Gorenstein,&Shapiro1985)an