1996AJ 112. . 105G -2 -1 THE ASTRONOMICALJOURNALVOLUME112,NUMBER1JULY1996 potential withadensityvaryingasp,whichtheyterm potential withacuspandbuildingself-consistentmodels Merritt &Fridman(1996)studiedtheinfluenceofacentral child 1992;Merritt&Fridman1995,1996).Inparticular, consequences forthephasespacestructureofgalaxy that is,theyconserveenergybut havenootherisolatingin- from theorbitlibrary.Theyhaveshownthatinatriaxial density cuspontheorbitalstructurebyfollowingorbitsina which theyterma“weakcusp,” afraction60%-80%(de- inner potentialspike.Theyalso showedthatifpr, tegrals. Thisbehavioriscaused byscatteringoffthesteep a “strongcusp,”fraction80%oftheorbitsareirregular, (Gerhard 1987;Hasan&Norman1990;LeesSchwarzs- 105 Astron.J.112 (1),July1996 0004-6256/96/112(l)/105/9/$6.00 r The massdensitydistributionforgalaxieshasimportant © American Astronomical Society • Provided by theNASA Astrophysics Data System nearly allthegalaxiesinoursample,logarithmicslopeofluminositydensity(S=dlogv/dlogr) We havenon-parametricallydeterminedtheluminositydensityprofilesandtheirlogarithmicslopesfor42 based ontheirsurfacebrightnessprofiles;i.e.,thosewithcuspycoresandwhosesteeppower-law from zero;i.e.,mostellipticalgalaxieshavecusps.Thereareonlytwoforwhichananalyticcore luminosity densityisuniquelydeterminedfromthesurfacebrightnessdatathroughAbelequation.For early-type galaxiesobservedwithHST.Assumingthattheisodensitycontoursarespheroidal,then profiles continueessentiallyunchangedintotheresolutionlimit.Thepeaksslopedistributionoccur the conclusionofLaueretal[AJ,110,2622(1995)]thatearly-typegalaxiescanbedividedintotwotypes measured at0.1"(theinnermostreliablemeasurementwiththeuncorrectedHST)issignificantlydifferent influence ofstochasticorbits.©1996AmericanAstronomicalSociety. elliptical galaxiesareeithernearlyaxisymmetricorsphericalnearthecenter,slowlyevolvedueto the recenttheoreticalworkofMerrittandFridman,theseresultssuggestthatmany(andmaybemost) 3i S=—0.8and-1.9.Morethanhalfofthegalaxieshaveslopessteeper-1.0.Takentogetherwith (S—>0) cannotbeexcluded.Thedistributionoflogarithmicslopesat0.1"appearstobimodal,confirming UCO/Lick Observatories,BoardofStudiesinAstronomyandAstrophysics,UniversityCalifornia,SantaCruz,California95064 THE CENTERSOFEARLY-TYPEGALAXIESWITHHST.III.NON-PARAMETRICRECOVERY Canadian InstituteforTheoreticalAstrophysics,UniversityofToronto,60St.GeorgeStreet,M5S3H8,Canada Kitt PeakNationalObservatory,OpticalAstronomyObservatories,P.O.Box26732,Tucson,Arizona85726 The ObservatoriesoftheCarnegieInstitutionWashington,813SantaBarbaraStreet,Pasadena,California91101 Department ofAstronomy,DennisonBuilding,UniversityMichigan,AnnArbor,Michigan48109 1. INTRODUCTION Institute forAstronomy,UniversityofHawaii,2680WoodlawnDrive,Honolulu,Hawaii96822 Electronic mail:[email protected],[email protected] OF STELLARLUMINOSITYDISTRIBUTIONS Karl GebhardtandDouglasRichstone Received 1995November3;revised1996March8 Edward A.AjharandTodR.Lauer Yong-Ik ByunandJohnKormendy S. M.FaberandCarlGrillmair Scott Tremaine Alan Dressler ABSTRACT pending onenergy)oftheorbitsareirregular.Thisisin box orbits)willbechaotic.Therefore,ifthedensitydistribu- they canbedescribedashaving“analytic”cores. of theorbitsareregular(reviewedindeZeeuw&Franx tion istriaxialandhasdlogp/dr1/2,orsteeper, eral, ifthedensitydistributionhasanonzerologarithmic portant intheaxisymmetriccase sinceorbitsdonotpassnear ritt &Fridman1995).Therole ofacentralcuspislessim- ture andevolutionofthecentral regionsofthegalaxy(Mer- then stochasticityislikelytoplayacriticalroleinthestruc- sharp contrastwithseparabletriaxialpotentials,inwhichall the centerbecausetheyconserve onecomponentofthean- (Taylor seriesexpandable)nearthecenterand,therefore, slope nearthecenter,orbitsthatpasscenter(i.e., 1991). Inallofthesepotentials,thedensityiswellbehaved Merritt andFridman’sworkstronglysuggeststhatingen- © 1996Am.Astron. Soc.105 1996AJ 112. . 105G profile. Theexistenceofacentralmassivedarkobject rected HST.Thesecanbedeprojectedtoobtaintheluminos- ness profilesforearly-typegalaxiesobservedwiththeuncor- important roleofhighresolutionsurfacebrightnessdistribu- practice anynoiseinthedataisamplifiedbyconstruction the massconcentration.Deprojectiontoobtainden- the luminositydensityprobablyprovidesalowerlimitfor ity densitywhichweassumealsorepresentsthemass tions inunderstandingthedynamicsofellipticalgalaxiesand guiar momentum.Merritt&Fridman’sworkemphasizesthe This equationisanAbelintegralwithsolution the surfacebrightnessasafunctionofprojectedradiusis spiral bulges. While thisapproachisreasonableandshouldyieldgood data beforedeprojecting.Themodelhadtheform time honoredprocedureoffittingaparametricmodeltothe construct S=dlogv(r)/d\ogr.InPaperI,wefollowedthe of v(r),andfurtheramplifiedbyaseconddifferentiationto given by symmetry. Ifthegalaxyhasaluminositydensityv(r),then sity isformallysimpleinthecaseofspherical(orspheroidal) (MDO) willonlyincreasethecentralmassconcentration,so The parametricmodelcanthenbeinvertednumericallyus- maximum-likelihood technique(Byunetal.1996,PaperII). This parametricmodelwasfittedtotheobserveddatabya Fig. 3andEq.28ofTremaineetal(1994)(seealsoDehnen is somecauseforconcerninitsapplication. estimates ofviftheparametricmodelfitsdatawell,there ing Eq.2toestimatethestellardensityrunnearcenter. 106 GEBHARDTETAL:EARLY-TYPEGALAXIES.III. projected radius.Analternative andmorefrighteningwayto put thisisthatthe useofEq.3asaparametric model creates havior ofv(r)nearthebreak radius(r~linthe d log/(/?)/*/logÆ=0),despitethefactthatcentralden- The figureandEq.28showthatfor2thecentral not dominatedbythedensitynear r~0.Ratheritisthebe- ies, inthissubsetofthosemodels,overtheopeninterval density cusp,thesurfacebrightness (seeEq.1)nearÆ—0is (-1,0). Theproblemisthatinamodelwithmildcentral sity isinfiniteandthecentralvalueofdlogv{r)ldrvar- surface brightnessisfinite(andtherefore “77-models,” withdensityprofilesoftheform 1993). Thatpaperexaminedavarietyofproperties 77-models), thatdeterminesthe surfacebrightnessnearzero /(Ä)(3) 1{R) Pr,(r) Lauer etal(1995,PaperI)havepresentedsurfacebright- Cast inthisway,thesolutionlooksdeceptivelysimple.In ~(r/r^)^1+' It iseasiesttounderstandthisconcernbycontemplating © American Astronomical Society • Provided by theNASA Astrophysics Data System =/, 1 2 2 _V 1 477- r’-’a+r)^' R Vr-/?' 3 wir dR^-' 1 C™dldR v(r)rdr r a 2^~I b (2) (1) (4) lead toanasymptoticvalueofdlogv{r)ldratleastas problem, wehaverecomputedthedensityanditsderivative potential sensitivityofsomeourearlierconclusionstothis relatively flatinnersurfacebrightnessprofiles.Inviewofthe cant. ThispointisalsoillustratedinFigure8PaperI. inoursample,andthiseffectwillnotbeassignifi- occurs aresmallerthantheHSTresolutionformostof lytic core.Foranyvalueofygreaterthanzero,evenbyan luminosity densityneary=0.Ifthemodelhasyexactly have beendiscussedinthestatisticsliteratureandmanyof Merritt &Fridman1995).VariousmethodstohandleEq.2 reported inPaperIandsomeadditionalgalaxies(asimilar evolution ofthecentergalaxy,andbecause importance ofthedensityslopeindeterminingdynamicsand its derivativeatsmallradiimaybesuspectingalaxieswith arbitrarily smallamount,deprojectingeitherEqs.3or4will a discontinuousmappingfromthesurfacebrightnessto these arenowbeingintroducedtoastronomy(Merritt& analysis wasdoneonasmallersampleofsixgalaxiesby at 0.1"usingnon-parametricmethodsinallofthegalaxies steep as—1.However,theradiiatwhichthisdiscontinuity rithmic slopeofthecentraldensity.Theexistencetwo be twoclassesofobjectsinthissample,basedontheloga- ing splinefittothedata.Wechooselattermethodand Tremblay 1994,Gebhardt&Fischer1995).Inthepresent became publiclyavailable.Allimagesweredeconvolvedus- programs, wereobtainedfromtheHSTarchivewhenthey ments arefromtheuncorrectedHST.The21additionalgal- 45 galaxiesinPaperIand21takenfromtheHST case, twoobviouschoicesareakernelestimatororsmooth- classes continuestoappearprobable. describe itinsomedetailbelow. Measurement ofsurfacebrightnessprofileswasdescribedin PSF closestintimetotheobservationdateforeachgalaxy. composite PSFsusedinPaperI.AsI,wethe ing 80iterationsoftheLucy-Richardsonalgorithm(Lucy axies, originallyobservedaspartofvariousGOandGTO archive (presentedinByunetal1996).Allofthemeasure- those galaxieswhichhaveanAGN,innerdustobscuration, detail inPaperI.Thecentraldensityslopeisnotreportedfor = 0anda2,thenthedeprojectedmodelwillhaveanana- results donotchangeifthosegalaxies areexcludedfromthe initial sample,andwethereforehave42remaininggalaxies of Eq.2requires7(R)atallR. ForR>10"wehaveuseda central regions,butwewereadequatelyabletoexcludethose which areusedinthefollowinganalysis.Includedthis or centralflat-fieldingdefects.Thiswas35%ofthetotal de Vaucouleurs profile.Theeffectiveradius wasei- areas whencalculatingthesurface brightnessprofiles.The sample. sample are12galaxieswhichshowsomedustyareasinthe 1974; Richardson1972)andthesame,highsignal-to-noise This argumentsuggeststhatestimatesofthedensityand We alsorediscussthequestionofwhetherthereappearto The datacomprisethesurfacebrightnessprofilesfor The HSTprofilesextendoutto 10". Formally,thesolution 2. DATA 106 1996AJ 112. . 105G ther takenfromFaberetal(1989),ortheSecondRef- to theouterregionsofHSTdata.Errorsinsurface erence CatalogueofBrightGalaxies(deVaucouleursetal. brightness atlargeradiihaveanegligibleeffectonthecen- problem thathasgottenconsiderableattentionfromstatisti- tral luminositydensityandtheestimatedcentralslopes. approximated basedonthebestfitteddeVaucouleursprofile These useakernelrepresentationforthesurfacebrightness Merritt &Tremblay(1994)useddensityestimationmethods. which useeitherdensityestimationorregressionanalysis. cians. Anumberofpossibletechniquescanbeapplied, distribution ofasymmetricalobjectisanill-conditioned 107 GEBHARDTETAL:EARLY-TYPEGALAXIES.III. penalized likelihood.Inourcase,asmoothcurveisfitto kernel estimators(Scott1992,p.229)oraswedohereusing gression analysis(asinMerritt1993,Gebhardt&Fischer luminosity densityisdirectlyinferred.Here,wewillusere- data toprovideasmoothprobabilitydensity,fromwhichthe but itiscomputationallysimpleandefficient. involve fittingasmoothcurvetothedatapoints,usingeither WFPC ratherthanbyPoissonnoise.Laueretal(1995)and rors ofourellipsefittingmethod,andcalibrationthe regression analysisrequiressmoothingthesurfacebrightness dl/dR, andv(r)isobtainedthroughnumericalintegration 1976). Iftheeffectiveradiuswasnotavailablethenit not affectourresults.Sincesmoothingtechniquesarede- whose accuracyislimitedprimarilybydust,systematicer- estimation directlyprovidestheluminositydensity,whereas of Eq.2.Bothtechniqueshavetheirrelativemerits;density x =logr.Thesecondterminthe aboveequationisthepen- of miscalibratingtheHSTPSFandconcludethattheywill we believethatourchoiceofthespecificsmoothingmethod Kormendy etal(1995)haveseparatelymodelledtheeffects by tabulatingsurfacebrightnessversusprojectedradiusona from theprocessingofrawdatainPaperI.Thechoice will havelittleimpactontheresults. y ¿isoclogIandx=R,astraight linefitisapowerlaw. log-log scale.Foragivenstiffness,X,thebestsmoothing cussed byWabha(1990).Weconstructthesmoothingspline source oferrorinthesurfacebrightnessisnotPoissonnoise, signed toovercomenoiseinthedataandsincedominant where y¿arethedata,datauncertaintiesand statistical estimatorsandfunctionalapproximators,asdis- splines forthispurposeismotivatedbytheirdualqualitiesas surface brightnessprofiles(asafunctionofR)asderived 1995, andMerritt&Fridman1995).Regressiontechniques penalty function. AwisechoiceofXtradesfidelity tothe This isduetoourchoiceof the secondderivativein alty function.Inthelimitofinfinite smoothing(X—>-°o),the spline, g,minimizes spline isforcedtoastraightline (g"—>0).Inourcase,since As notedintheintroduction,invertingprojectedlight The HSTdataprovideaverysmoothprojecteddensity, We havechosentousesmoothingsplinesrepresentthe f=z, -2+k k © American Astronomical Society • Provided by theNASA Astrophysics Data System V x 3. NON-PARAMETRICDEPROJECTION / («-■ 2 (x))dx, (5) 2 2 = the firstterminEq.5.Unfortunately,numberofdegrees make thechoicesoastoachieveanacceptablevalueofxin data againstthesmoothnessofsplinefit,andoneshould of freedominthisproblemisnoteasilydefined.Inthelimit through eachdatapointwithy0.If,ontheotherhand,we 2N—2 unknownsplinecoefficients(thevaluesandsecond X—>0, thereareNdatapoints,splineequationsand Wahba 1979)thattheappropriatechoiceofXinthiscase line tothedata(inourcase,apowerlaw). toward 0,andthenumberofdegreesfreedomtendsto- choose Xverylarge,thesplinesecondderivativesaredriven degrees offreedominthiscaseis~0,andthesplinefitgoes derivatives ofthesplinefitatx¿).Clearly,number estimate theoptimalsmoothing.Onecanshow(Craven& cians advocatetheuseof“predictedmeanerror”to ward A—2,whichisreasonablesincewearefittingastraight minimizes, subjecttoEq.5,thequantityRgivenby where Aisthesplinecoefficientmatrixsuchthatg=Ay proach. ForagivenX,onepointisremovedfromthesample points fromtheestimatedpoints.Agoodexplanationof lying function. of vectora.Equation6isderivedfromthesum estimate thevalueforremovedpoint.Thisisrepeated estimate theoptimalsmoothing.GCVusesajackknifeap- value oftheknowndatauncertainties,and||<2||isnorm mizes thesumofsquareddifferencesactualdata each point,andtheoptimalsmoothingisthatwhichmini- and thesplinefromremainingN-lpointsisusedto suspect, onecanusegeneralizedcrossvalidation(GCV)to squared differencesbetweentheestimatedandactualunder- spline estimate),Iistheidentitymatrix,