Structure of Disk Dominated Galaxies I. BulgeDisk Parameters, Simulations, and Secular Evol

Simulations,andSecularEvolution

LaurenA.MacArthurandSt´ephaneCourteau1

DepartmentofPhysics&Astronomy,UniversityofBritishColumbia,6224Agricultural

Road,Vancouver,BCV6T1Z1

lauren@astro.ubc.ca,courteau@astro.ubc.ca

and

JonA.Holtzman1

DepartmentofAstronomy,NewMexicoStateUniversity,Box30001,Department45000,

LasCruces,NM88003-8001

holtz@astro.nmsu.edu

ABSTRACT

Arobustanalysisofgalaxystructuralparameters,basedonthemodelingofbulgeanddiskbrightnessesintheBVRHbandpasses,ispresentedfor121face-onandmoderatelyinclinedlate-typespirals.Eachsurfacebrightness(SB)profileisdecomposedintoasumofageneralizedS´ersicbulgeandanexponentialdisk.Thereliabilityandlimitationsofourbulge-to-disk(B/D)decompositionsaretestedwithextensivesimulationsofgalaxybrightnessprofiles(1D)andimages(2D).Wehaveusedrepeatobservationstotesttheconsistencyofourdecompositions.

Theaveragesystematicmodelerrorsare<<

∼20%and∼5%forthebulgeanddiskcomponents,respectively.Thefinalsetofgalaxyparametersisstudiedforvari-ationsandcorrelationsinthecontextofprofiletypedifferencesandwavelengthdependences.

GalaxytypesaredividedintothreeclassesaccordingtotheirSBprofileshapes;FreemanType-IandType-II,andathird“Transition”classforgalax-ieswhoseprofileschangefromType-IIintheopticaltoType-Iintheinfrared.Roughly43%,44%,and13%ofTypeI,II,andTransitiongalaxiesrespectively

–2–

compriseoursample.OnlyType-Igalaxies,withtheirfullyexponentialdisks,areadequatelymodeledbyour2-componentdecompositionsandourmainresultsfocusontheseprofiles.WediscusspossibleinterpretationsofFreemanType-IIprofiles.TheS´ersicbulgeshapeparameterfornearbyType-Ilate-typespiralsshowsarangebetweenn=0.1–2but,onaverage,theunderlyingsurfacedensityprofileforthebulgeanddiskofthesegalaxiesisadequatelydescribedbyadouble-exponentialdistribution.Thedistributionofdiskscalelengthsshowsadecreasingtrendwithincreasingwavelength,consistentwithahigherconcentrationofoldstarsordust(orboth)inthecentralregionsrelativetotheouterdisk.Weconfirmacouplingbetweenthebulgeanddiskwithascalelengthratio󰀔re/h󰀕=0.22±0.09,or󰀔hbulge/hdisk󰀕=0.13±0.06forlate-typespirals,inagreementwithrecentN-bodysimulationsofdiskformation.Thisratioincreasesfrom∼0.2forlate-typespiralsto∼0.24forearliertypes.Theseobservationsareconsistentwithbulgesoflate-typespiralgalaxiesbeingmoredeeplyembeddedintheirhostdiskthanearlier-typebulges,asdiscussedbyGraham(2001).Bulgesanddiskscanthuspreserveanearlyconstantre/hbutshowagreatrangeofsurfacebrightnessforanygiveneffectiveradius.Thesimilarscalingrelationforearlyandlate-typespiralssuggestscomparableformationand/orevolutionscenariosfordiskgalaxiesofallHubbletypes.InthespiritofCourteau,deJong,&Broeils(1996)butusingournew,moreextensivedatabase,weinterpretthisresultasfurtherevidenceforregulatedbulgeformationbyredistributionofdiskmaterialtothegalaxycenter,inagreementwithmodelsofsecularevolutionofthedisk.Subjectheadings:galaxies:spiral—galaxies:photometry—galaxies:structure—galaxies:formation—galaxies:simulations

1.Introduction

Stellardensitydistributionsprovideimportantconstraintsforbulgeanddiskformationmodels.Historically,astronomershaveembracedther1/4brightness“law”(deVaucouleurs1948)andexponentialbrightnessprofile2(deVaucouleurs1959a;Freeman1970)tomodelthe

–3–

lightdistributionofthegalaxybulgeanddisk,respectively3.DeparturesfromthestandarddeVaucouleursprofileintheinnerlightdistributionofearly-andlate-typespiralshavehoweverbeendemonstratedinanumberofearlystudies(deVaucouleurs1959;vanHouten1961;Burstein1979),includingtheMilkyWay(Kent,Dame,&Fazio1991).Andredakis&Sanders(1994),deJong(1996a),Courteau,deJong,&Broeils(1996),andCarollo(1999)laterusedsmallsamplesofhigh-qualitysurfacebrightness(SB)profilestoestablishtheexponentialprofileasabettermatchtolate-typediskbulges;thusSBprofilesofmostlate-typespiralsarebestmodeledbyadouble-exponentialfittothebulgeanddisk.Abroaderanalysissuggestsarangeofbulgeshapesfromearly-tolate-typespirals(Andredakisetal.1995;deJong1996a;Courteauetal.1996;Graham2001).MostoftheseanalysesrelyonthemodelingofageneralizedsurfacedensityfunctionsuchasthatproposedbyS´ersic(1968);󰀂󰀓

r

I(r)=I0exp−

󰀃1/n󰀅

.

(2)

r0

whereµ0(I0)isthecentralsurfacebrightness(intensity),r0isascalingradius,andtheexponent1/nisashapeparameterthatdescribestheamountofcurvatureintheprofile.Forn=1or4onerecoversapureexponentialorthedeVaucouleursr1/4profilerespectively.Collectively,theworksabovesuggestthatthebulgeshapeparameterncorrelateswithabsoluteluminosityandhalf-lightradius,suchthatbigger,brightersystemshavelargervaluesofn.ThisresultwasextendedtobrightestclustergalaxiesbyGrahametal.(1996).Courteauetal.(1996)alsodemonstratedatightcorrelationbetweenthebulgeanddiskexponentialscalelengths,forallspiraltypes,withhb/hd=0.1±0.05(whereh=r0andn=1inEq.1).Theexponentialnatureoflate-typegalaxybulgesandthecorrelationbetweenbulgeanddiskscalelengthswasinterpretedbyCourteauetal.(1996)asevidenceforregulatedbulgeformationbyredistributionofdiskmaterialtothegalaxycenterbyabar-likeperturbation.Wewillreturntothisimportantconstraintforsecularevolutionmodelsin§5.3.

–4–

Thisstudyfocusesonthedevelopmentofareliablesetofobservablesandconstraintsforstructureformationmodels.AnimportantgoalistomeasuretherangeoftheS´ersicnparameterforvirializeddisksystems.Theanalysesdescribedabovearereproducedandexpandeduponwiththelargestmulti-bandsurveyofitskindtodateandaclearerunder-standingofmodellimitationsthanpreviouslyattained.Weaimtocharacterizeandquantifytheintrinsicstructuralpropertiesofthebulgeanddiskandtheextentoftheirvariationwithwavelength.ThesecharacterizationsaremadethroughreliablemodelingofbulgeanddiskparametersfromSBprofiledecompositions.Multi-wavelengthinformationalsopro-videsinsightaboutstructuralvariationswithinandamonggalaxiesduetodustandstellarpopulationeffects.Whilesomeoftheseissueshavebeenaddressedbefore,thereremainsanumberofsignificantmeasurementuncertaintiesandtechnicallimitationswhichwenowinvestigatethoroughly.

Thispaperisorganizedasfollows:abriefdescriptionofthedatabaseisgivenin§2andin§3wediscussourB/Ddecompositionalgorithms(1Dand2D)andthesimulationstotestthereliabilityofourtechnique.Forthereadersinterestedmostlyinfinalprofiledecompositionsandresults,asummaryofthesimulationresultsandguidelinesisgivenin§3.4.ActualB/DdecompositionsofgalaxySBprofilesarepresentedin§4,followedbyadiscussionandinterpretationoftheresultsintermsofsecularevolutionmodelsin§5.AdiscussiononthenatureofFreemanType-IIprofilesisalsopresentedin§5.Weconcludewithfuturedirectionsin§6.Twoappendicespresent(A)adiscussionofthefunctionalformfortheS´ersiccoefficientbn,and(B)decompositionresultsforourType-Iprofiles.

2.TheData

Ourstructuralanalysisofgalaxyluminosityprofilesisbasedonthecatalogofmulti-bandimagesoflate-typespiralgalaxiesbyCourteau,Holtzman,&MacArthur(2002;hereafterPaperII).Itconsistsofover1000deepB,V,R,andHimagesof322nearbybrightlate-typespiralgalaxies.Thedatawerecollectedbetween1992and1996atLowellObservatoryandKittPeakNationalObservatory(KPNO).Afulldescriptionofthesampleselection,observations,andreductionsispresentedinPaperII.Asummaryisgivenbelow.

ThegalaxysamplewasselectedfromtheUppsalaGeneralCatalogue(UGC,Nilson1973)withthefollowingcriteria:

•PredominentlylateHubbletypes•ZwickymagnitudemB≤15.5

•BlueGalacticextinctionAB=4×E(B−V)≤0m.5(Burstein&Heiles1984)

–5–

•Inclinationbinscoveringface-on(i≤6◦),intermediate(50◦Thiscatalogisnotcompleteinanysenseofthe(muchabused)term.ThediameterlimitwasconstrainedprimarilybythefieldofviewoftheinfraredcamerasinuseatKPNO(IRIMandCOB)andLowellObservatory(OSIRIS)in1992–1996andtherequirementforblankareasinthefieldofviewforskysubtraction.Additionally,peculiarandinteractinggalaxies(e.g.novisibletidaltails)wereexcludedtoensurethatthesampleconsistedonlyofisolateddiskdominatedgalaxies.Barredgalaxies,asclassifiedintheUGC,werenotexcludedpersebutonlyahandfulwereobserved.Forthepresentanalysis,weuseasub-sampleof121galaxieswithface-onandintermediateinclinationsonly,foratotalof523images4.ThedistributionofHubbletypesinourreducedsampleis:2Sab,26Sb,19Sbc,38Sc,25Scd,11Sd.

AlldistancesarecorrectedtothereferenceframeoftheLocalStandardofRest(Courteau&vandenBergh1999),andweuseH0=70kmsec-1Mpc-1.Thesurveyeffectivedepthis󰀔cz󰀕∼5500kms−1or80Mpc.

2.1.ObservationsandBasicReductions

AllopticalBVRimageswereobtainedfrom1992to1994atLowellObservatorywithaTI800×800chip(scale=0′′.5/pix)onthePerkins72′′telescope.TheinfraredH-bandimageswereacquiredfrom1993to1995atKPNOwiththe2-meterand4-metertelescopesequippedwitheitheraHgCdTe(IRIM)oranInSb(COB)256×256array(1′′.09/pixand0′′.5/pixrespectively),andfrom1995to1996withtheOSIRISimager(1′′.49/pix)mountedontheLowell72′′telescope.Theexposuretimeswere:300sinR,400sinV,1500sinB,andon-targetintegrationof1200sinH.Landolt(1992)standardscoveringawiderangeofairmassesandcolorswereobservedeachnightatLowellObservatory,givingaphotometricaccuracyof∼2%fortheopticalpassbands.UKIRTstandards(Guarnieri,Dixon,&Longmore1991)observedeachnightyieldedH-bandphotometriccalibrationsgoodto∼3%.Starsanddefectswereeditedfromtheimagespriortofurtheranalysis.

Thetypicalseeingfull-widthathalf-maximum(FWHM)atLowellandKPNOwas2′′.0withtypicalstandarddeviationsof∼20%(optical)and∼35%(IR)perimage.These

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measurementswerecomputedasthemeanoftheFWHMsofallnon-saturatedstarsmeasuredautomaticallyoneachimageframe;typically5to40measurementsperframewereused.Wemeasuremeanskylevels(fora5–6dayoldmoon)ofB=21.9±0.8,V=21.2±0.5,R=20.6±0.5,andH=14.1±1.2magarcsec−2.Typicalsystematicerrorsintheskymeasurement,computedfrom4or5skyboxessuitablylocatedbetweenthegalaxyandtheedgeoftheframe,are0.5−1.0%intheopticaland0.005−0.01%intheIR.

Azimuthally-averagedSBprofileswereextractedforallthegalaxiesusingellipsefittingwithafixedcenter.Toensureahomogeneouscomputationofstructuralparametersandcolorgradients,weusetheisophotalmapsfromtheR-bandtodeterminetheSBprofilesinBVH.Eventhoughdusteffectscanstillplayaroleat7000˚A,theR-bandwasadoptedforourisophotaltemplatesasithasthemoststableskyanddeepestprofiles.Weallowedavariablepositionangleandellipticityateachisophote,butacomparisonwithSBprofilesextractedusingconcentricisophotalfitsdemonstratedthatourresultsdonotdependonthefittingtechnique.FurtherinformationaboutprofileextractionandCCDsurfacephotometrycanbefoundinCourteau(1996a)andPaperII.WetraceSBprofilesto∼26magarcsec−2inopticalbandsand∼22magarcsec−2atH-band.Theselevelscorrespondtoasurfacebrightnesserrorof∼0.12magarcsec−2.

2.2.SurfaceBrightnessCorrections

Theobservedsurfacebrightnessofagalaxycanchangewhenviewedatdifferentinclina-tionangles,dependingonthedistributionofagalaxy’sinterstellarmediumanditsopacity.SurfacebrightnessesarealsoaffectedbyGalacticforegroundextinctionandredshiftdim-ming.Weaccountforthelattereffectsbutdeferanytreatmentofinternalextinction,whichvarygreatlyfromauthortoauthor,toPaperII.Ourconclusionsdonotdependontheexactvaluesofthecentralandeffectivesurfacebrightnessesofgalaxies.

WecorrectforGalacticforegroundextinctionusingthereddeningvalues,Aλ,ofSchlegeletal.(1998)andassuminganRV=3.1extinctioncurve(e.g.Cardellietal.1989),

λ

µλc,Gal=µobs−Aλ.

(3)

Wecorrectsurfacebrightnessesforthe(1+z)3cosmologicalredshiftdimming(perunit

frequencyinterval)as

λ

µλ(4)c,z=µobs−7.5log(1+z).

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Thefinalcorrectiontotheobservedsurfacebrightnessesisthus,

λ

µλc=µobs−Aλ−7.5log(1+z).

(5)

ExamplesofthetypesandextentoftheSBprofilesfortypicallate-typespiralgalaxiesin

oursampleareshowninFig.1.ForType-Idisks(Freeman1970),theinnerprofilealwaysliesabovethesurfacebrightnessoftheinwardextrapolationoftheouterdisk,whereasType-IIsystemshaveaportionoftheirbrightnessprofileslyingbelowtheinwarddiskextrapolation.WedefineaTransitioncaseforluminosityprofilesthatchangefromType-IIatopticalwavelengthstoType-Iintheinfrared.ManygalaxiesclassifiedasType-IIshowaweakeningoftheinnerprofiledipatlongerwavelengthsand,inthissensethereisnocleardistinctionbetweentheType-IIandTransitiongalaxies.LikelyinterpretationsforType-IIprofilesarediscussedin§refsubsec:typeII.

3.SimulationsofBulge-to-DiskDecompositions

Inordertomeasuregalaxystructuralparameters,wehavedevelopedtwoindependentalgorithmstodecomposethegalaxy1Dand2Dlightdistributionsintobulgeanddiskcom-ponents.TheseprogramsallowforageneralizedS´ersicbulge,anexponentialdisk,andacentralbarfor2Dimages.Thereareseveralissuesinvolvedwithaccuratedecompositions,particularlywiththemeasurementofbulgeparameters,including;thesensitivityoffinalresultstostartingguesses,effectsofstatisticalandsystematicerrorsinskybrightnessandseeingestimates,choiceoffitbaseline,etc.Weexploretheseingreatdetailbelowusingboth1Dand2Danalysestodeterminetherobustnessofourcodesandthereliabilityofourfinalsolutions.Becauseprojectedsurfacebrightnessprofilescontainfewerdatapointsthanfull2Dimages,wecancreate1Dsimulationsfasterthan2Dmodels.Thusourmostextensivetestsrelyon1Dsimulations,whichareshowntobefullyconsistentwith2Dsimulationswhenconsideringaxisymmetricfeatures.

3.1.1Dand2DAlgorithms

Ourbrightnessprofile(1D)bulge-to-disk(B/D)decompositionalgorithmwasinitiallydevelopedbyBroeils&Courteau(1997)andsubsequentlyimprovedbyLM.Thisprogramreduces1Dprojectedgalaxyluminosityprofilesintobulgeanddiskcomponentssimultane-ouslyusinganon-linearLevenberg-Marquardtleast-squares(NLLS;see§15.5inPressetal.(1992))fittothelogarithmicintensities(i.e.magnitudeunits).RandomSBerrorsare

–8–

accountedforinthe(data−model)minimization,whereassystematicerrorssuchasuncer-taintiesintheskybackgroundanddeterminationoftheimagemeanPSFareaccountedforseparatelyinaseriesofexperimentsdesignedtocalibratetheireffects.Seeingeffectsinourmodelgalaxiesareaccountedforbyconvolvingthetheoreticalbulge-disksurfacebrightnessprofilesandimageswitharadiallysymmetricGaussianPointSpreadFunction(PSF),oftheform󰀎∞

2222

Is(r)=σ−2e−r/2σItotal(x)I0(xr/σ2)e−x/2σxdx(6)

0

whereItotal(x)istheintrinsicsurfacebrightnessprofile,σisthedispersionoftheGaussian

PSFandI0isthezero-ordermodifiedBesselfunctionofthefirstkind(seealsoTrujilloetal.(2001)forastudyoftheMoffatPSF.)

The2DdecompositionprogramisbasedonthesameNLLStechniqueasabovebutusesthefull2Dimageinintensityunitsinsteadofalogarithmicradialsurfacebrightnessprofile.Whilecomputationallymoreintensivethanits1Danalogue,the2Ddecompositionshouldyieldmorephysicallymeaningfulresultssincetheazimuthalinformationislostin1Dprofiles.Byun&Freeman(1995),deJong(1996a),andSimardetal.(2002)havediscussedthemeritsofthe2Dapproach,suchasgreaterabilitytorecovertrueparameters(basedonsimulations),andthepotentialtomodelnon-axisymmetricfeaturessuchasbars,rings,andspiralarms.Theneedfortheimplementationandtestingofarobust2DB/Ddecompositionpackageisthusobvious,butwefindthat1DdecompositionscomparefavorablyforreliabilityandpredictivepowerprovidedhighS/N1Dradialprofilesareused.Notethatneither1Dnor2Ddecompositionsareimpervioustodustextinctioneffects.ExtinctioneffectsarelessenedatH-band,butcanstillbesignificantindiskbulgesandspiralarms.Aproperrecoveryofthetruestellardensityprofilewouldrequireafull3Dradiationtransfertreatment,andsuchananalysisisbeyondthescopeofthiswork.

3.2.Methodology

Afundamentalaspectofprofiledecompositionsisthechoiceoffittingfunctions.Thedisklightismodeledwiththeusualexponentialfunction,

󰀉r

Id(r)=I0exp−

h

󰀌

(8)

–9–

whereµ0≡−2.5logI0andharethediskcentralsurfacebrightness(CSB)andscalelengthrespectively,andristhegalactocentricradiusmeasuredalongthemajoraxis.Inthe2Ddecompositions,thecomputationoftheradiusateachpixelrequirestwoadditionalparam-eters:thepositionangle(PA)ofthediskmajoraxisontheskyandthediskellipticity,ε=1−b/a,whereaandbarethemajorandminoraxesofthediskrespectively).TotestfortheshapeofthebulgeluminosityprofilesweadoptthegeneralizedformulationofS´ersic(Eqs.1&2).

IthasbecomecustomarytoexpressthediskparametersintermsofscalelengthandCSB(handµ0),whilethebulgeparametersareexpressedintermsofeffectiveparameters(reandµe).Weadoptthisformalism,thusparameterswithsubscripterefertothebulge.Eq.1canbere-writtenas:

󰀂󰀈󰀓

r

Ib(r)=Ieexp−bn

re

whereµeistheeffectivesurfacebrightness.

󰀃1/n

−1

󰀋

(11)

ItistrivialtoconvertfromEq.2toEq.11bynotingthat

re=(bn)nr0

µe=µ0+2.5log(e)bn.

Eq.10impliesthat

Γ(2n)=2γ(2n,bn)

(14)

whereΓ(a)isthegammafunctionandγ(a,x)istheincompletegammafunction.Unfor-tunately,Eq.14cannotbesolvedanalyticallyforbn.Variousnumericalapproximations

(12)(13)

–10–

havebeenbeengivenintheliterature(Caonetal.1993;Graham&Prieto1999;Ciotti&Bertin1999;Khosroshahi,Wadadekar,&Kembhavi2000;M¨ollenhoff&Heidt2001).Oneoftenencounterstheapproximationbn≈2n−0.32,validsupposedlyforallvaluesofn(sic).Khosroshahietal.(2000)contendthatthisapproximationisaccuratetoonepartin105,witharangeofvalidityonnunspecified.However,becausethegammafunctiondivergesneartheorigin,mostutilizedapproximationsareinaccurateforvaluesoftheS´ersicexpo-nentn≤1.Differencesbetweennumericalsolutionsforbn(Eq.14)andcommonlyadoptedapproximationscanyieldbrightnessdifferencesgreaterthan0.1magarcsec−2forn󰀂1.AswewishtotestforbulgeswithS´ersicnparametersmallerthan1,wehaveadoptedtheasymptoticexpansionofCiotti&Bertin(1999)toO(n−5)forn>0.36.Forn≤0.36thissolutiondivergesandinsteadweuseapolynomialexpression(4thorder)accuratetoonepartin103.Wecomparedifferentnumericalsolutionsforbn(Fig.23)andpresentouradoptedfunctionalforminAppendixA.

AnillustrationofprofileshapesfordifferentvaluesoftheS´ersicnparameterisshowninFig.2.Thetoppanelshowsprofileswithµe=21magarcsec−2andre=3′′.5forvaluesofnintherange0.21aresteepatsmallradii(≪re),butleveloffasrincreases.Giventhelargedifferencesintheprofileshapesaboveandbelown=1(exponentialcase),onemightexpectdifferentphysicalmechanisms(formation,transport,dynamics,interactions)tobeatworkforsystemswhoselightprofileshaveverydifferentnvalues.Additionally,forthesmallbulgesoflate-typegalaxies,poorseeingcouldconceivablysmeartheimagesuchthatanintrinsicallyn>1bulgecouldbemistakenforann<1structure.

Ofpotentialrelevancetothestudyofgalaxystructureistherelativelightfractioncontributedbythebulgeanddisk.Thisisexpressedintermsofabulge-to-diskluminosityratio,B/D,derivedbyintegratingthebulgeanddiskluminosityprofilestoinfinity.Foraface-onS´ersicprofilethetotalextrapolatedluminosityisgivenby

Lb=

󰀎

∞

Ib(r)2πrdr=

2bn

2πIereenΓ(2n)

0

nb2n

󰀐r

e

Io

󰀃

.(17)

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Eqs.15&16shouldbemultipliedbythefactor(b/a)whenconsideringprojectionsontheplaneofthesky.OnemayuseEq.17inageneralsense,independentofprojection,undertheassumptionthatthebulgeanddiskdensitydistributionshavesimilaraxesratio(nearlytrueforlate-typegalaxies).AweaknessofB/Dratiosforsystematiccomparisonsofgalaxylightprofilesisitsmodeldependenceandthepotentialcovariancesbetweensomeofthemodelparameters.ConsiderFig.2(top)fortherelativelightfractionscontributedbyprofilesofdifferentnvalues,normalizedton=1.Theintegratedbulgelightincreasessteadilyasafunctionofn,forgivenvaluesofreandµe.Thus,theadoptednvalueinabulge-to-diskdecompositionhasastronginfluenceonthecomputedB/Dratio.Additionally,sincelargernprofilescontributelightouttolarger,thecombinationofahighnandalowµe(brightr1/4bulge)couldtakeawaylightfromtheouterdiskandartificiallyboosttheB/Dratio.Adiscussiononnon-parametricstatistics,suchasconcentrationindices(Kent1985;Courteau1996a;Graham2001)whichalleviatemodeldependences,ispresentedinPaperII.

Wemodelthetotalgalaxyluminosityprofileasasumofbulge+diskcomponents:

Itot(r)=Ib(r)+Id(r).

(18)

ProfilesmearingbyatmosphericturbulenceisaccountedforinB/Ddecompositionsbycon-volvingEq.18withaGaussianPSFoftheformofEq.6.

SimilarB/DanalyseshavealsoconsideredadditionaltermsforaGaussianbar(deJong1996a),alensorring(Prietoetal.2001),spiralarms,andstellardiskswithinnerand/oroutertruncations(Kormendy1997;Baggett,Baggett,&Anderson1998).WerestrictourchoiceoffittingfunctionstoaS´ersicbulgeandanon-truncatedexponentialdiskforanumberofreasons.Wefindnoprominentbarsinoursampleandmostourdiskprofilesarefairlylinear(inmagnitudespace).Azimuthalaveragingfor1Dprofilessmoothesoutspiralarmfeatures(toadifferentextentdependingonwhetherthepositionanglewasfixedorallowedtovaryintheprofileextraction.Removalofspiralarmsignaturesfromthelightprofilesorimageswouldrequiremoretimeandeffortthaniswarrantedbyouranalysisatthisstage.)Wedonotconsiderasharpinnerdisktruncationforanumberofreasons:(i)unsharpmaskingtechniquesrevealspiralstructurefromtheinnerdiskintothegalaxycenter(Courteau1992,1996b;Elmegreen,Elmegreen,&Eberwein2001);(ii)usingHSTimagesofinnerdisks,Carollo(1999)alsofindsevidenceforinnerspiralstructureandnuclearstarclustersinthecentersofearly-tointermediate-typespiralgalaxies;(iii)allcomponentsoftheGalaxyhavetheirpeaksurfacebrightnessesinthecenter(e.g.Wyse1999).Thus,atleastsomeevidencesuggeststhatspiraldisksreachinallthewaytothecenteroflate-typesystems.Aloweringofthediskcentralsurfacedensitymayoccurasstarsgetheatedupintoabulgebytheactionofabar-likeinstability.Anexponentialprofilewithacoremay

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thusbeareasonabledescriptionofType-IISBprofiles.Wedonotconsiderthisapproachhere,butpointoutthatresolvedB/Dkinematicsofnearbygalaxieswouldprovideaclearindicationwhetherstellarpopulationshavebeenstronglydepletedand/orsystematicallyscatteredverticallyintoabulge.Someofourgalaxiesshowouterdisktruncation(e.g.Fig.13),butsee§4.3.2.

Thebest-fitparametersofthe(data−model)comparisonarethosewhichminimizethereducedchi-squaremeritfunction,describedinintensityunitsas

χ2ν=

1

σi

󰀇2

(19)

whereNisthenumberofdatapointsused,Misthenumberfreeparameters(i.e.N-M=

ν≡DegreesofFreedom),andσiisthestatisticalintensityerrorateachpixel(2D)orsurfacebrightnesslevel(1D).Fromhereontheνsubscriptwillbeomittedandtheχ2variablereferstoaχ2perdegreeoffreedom(unlessotherwisespecified).

Theglobalχ2ofintensitiesisclearlydominatedbythecontributionfromthedisk,virtuallyirrespectiveofthefittedbulge.Thiseffectwouldbeaccentuatedingalaxieswithprominentfeatures,suchasspiralarms,rings,orlenses,whicharenotaccountedforinourpureexponentialdiskmodels.CasesarefoundwhereB/Ddecompositionswithsignificantlydifferentbulgeexponentnvaluesforagivenprofilehavenearlythesameglobalχ2value(seeFigs.8&9).Thus,inordertorefineourparametersearchforthebest-fitbulgeanddiskmodel,wecomputeaseparate,inner,χ2statisticouttotwicetheradiuswherethebulgeanddiskcontributeequallytothetotalluminosityofthegalaxy(rb=d≡2r(Ib=Id)).Welabelthisstatisticasχ2in(seeGraham2001forasimilarformulation).Forcaseswherethebulgesaresosmallthattheynevertrulydominatethelightprofile(i.e.rb=disundefined),wecomputetheχ2inouttotheradiusatwhichν=1.

3.3.ReliabilityoftheDecompositionResults

Thissectiondescribesextensivetestingofourbulge-to-diskdecompositionprograms.ArtificialSBprofilesandimageswerecreatedwithawiderangeofbulgeprofileshapesandexponentialdisksincludingrealisticnoiseandseeingeffects.Realgalaxiesareclearlymorecomplicatedthanthesumoftwoidealizedmathematicalfunctions,butthesetestsprovideareasonablebaseforaglobalunderstandingofthereliabilityandlimitationsofB/Ddecompositionalgorithms.ThemockcatalogofSBprofilesandimageswillbeusedtoaddressthefollowingquestions:

–13–

•Howreliableandmeaningfularethebulge-to-diskdecompositionsandfittedparame-ters?

•Howcrucialareinitialestimates?Aremodelfitsalwaysconvergingtothelowestχ2minimum?

•Howdoseeingeffectsandskysubtractionerrorsaffectthedecompositions,andcantheybeproperlyaccountedfor?

•Arethesmallbulgesinlate-typediskgalaxiessufficientlyresolvedtopermitareliablesolutionoftheS´ersicnparameterasafreeparameter?Theliteratureaboundswithinvestigationsofprofilefittingalgorithmsbasedonartifi-cialdata,suchasSchombert&Bothun(1987;hereafterSB87)whoperformeddouble-blindexperimentswhereoneoftheauthorscreatedmockluminosityprofilesandtheotherindepen-dentlyfittedthedata.TheSBprofilescombinedadeVaucouleursbulgeandanexponentialdisk.Photonnoise,atalevelmatchingtypicalblueCCDperformances,andasystematic0.5–3.0%erroroftheskybackgroundwereaddedtotheprofiles.SB87foundthatthesimul-taneousfittingofdiskandbulgeusingstandardNLLStechniquescouldreproducetheinputparameterstowithin10–20%incaseswheregalaxyprofilescanbedecomposedperfectlyasthesumofabulgeanddisk(whichfailsforType-IIprofiles.)SB87claimthataskyestimateuncertaintyofupto3%doesnotaffecttheirderivedparameters,butwefindthatskyerrorsassmallas1%canhaveasignificanteffectontheshapeoftheouterdiskprofileandthederivedbulgeanddiskparameters(see§3.3.5below).SB87didnotconsiderotherfittingfunctionsbutrecognizedthatbulgesmaynotbeadequatelydescribedbythedeVau-couleursr1/4function.Andredakis&Sanders(1994)laterexaminedtheinadequacyofther1/4functionalformforthebulge(1D)profile,andfirstestablishedthedouble-exponentialnatureoflate-typespirals.

2DB/Ddecompositiontechniques,whichexploitthefullgalaxyimage,werealsode-velopedandtestedinsimilarfashioninthemid-nineties(Byun&Freeman1995;deJong1996a).DeJongperformedextensivetestswithmockgalaxiesmodeledaspureexponen-tialbulgesanddisks,exploringtheeffectsoferrorsinthemeasuredobservablesincludingtheseeingFWHM,skybackgroundlevel,minorovermajoraxisratio,b/a,andpositionangle,PA.Theseobservableswereusedasfixedinputparameterstothefittingroutine,anddeJongcalibratedtheeffectofmeasurementerroronthedeterminedparametersbydecomposingtheartificialgalaxiesusingerroneousvaluesforeachobservable.Heconcludedthat:errorsinµ0arepredominantlycausedbyskysubtractionerrorsandcanbeaslargeas0.1magarcsec−2;errorsinhcanreach10%andaredominatedbyskybackgroundandellipticitymeasurementerrors;bulgeparametererrors,oforder20%,arecontrolledbythe

–14–

B/Dsizeandbrightnessratios.Brightbulgesaremostaffectedbyseeingerrors,andfainterbulgescanalsobeaffectedbyskybackgrounderrors.

OurowninvestigationreachessimilarconclusionsandfurtherextendsdeJong’ssim-ulations.Wetestfortherobustnessofthefittingprocedureandaccuracyofthederivedparameterswithvariousvaluesofthefitinitialestimates,seeingFWHM,skyvalue,andtheirerrors,and–unlikedeJong(1996a)–wemodelthebulgewithageneralizedS´ersicprofile.ThesesimulationswereinitiatedbyBroeils&Courteau(1997)butareextendedhereinmuchgreaterdetail,especiallywithrespecttothedeterminationofthebulgeshapeparametern.

3.3.1.SimulatedProfilesandImages

OurtestsusealargesetofartificialSBprofilesandimageswhichspanawiderangeofthebulge,disk,andseeingparameters.Themathematicalformsofthebulgeanddiskcomponentsarethosediscussedin§3.1.NoisewasaddedtothemodelprofilesandimagesfromaGaussiandistributionwithdeviationrepresentativeofthestandardbrightnesserrorsofourluminosityprofilesatagivensurfacebrightnesslevel(seeCourteau1996a,Fig.9;PaperII).

OnehundredSBprofilesandfourtyimageswithrealisticnoisewerecreatedforeachbulge,disk,andseeingcombination.Mostofthesimulatedprofilesandimageshadthesamediskparameters,

µ0=20magarcsec−2,

h=12′′,

whicharerepresentativeofatypicalgalaxyinoursample6.Forexponential(n=1)bulges,thestructuralparameterswereselectedfrom

µe=16,17,...,22magarcsec−2,

re=0.1,0.2,...,3′′.0

correspondingtoB/Dratiosrangingfrom0to5andB/Tfrom0to0.8(seeEq.17).

Allprofilesandimageswereconvolvedwithaseeingdiskof

–15–

FWHM=1.0,1.5,...,3′′.0.

Ourmodelsspanthefullrangeofparameterstypicallyfoundinlate-typebulges(e.g.de

Jong(1996b);Courteau(1996a);Broeils&Courteau(1997))andtheseeingvaluesmatchtheexpectedrangeatLowellObservatoryandKPNO(see§2).

WealsoexploredthefollowingrangeofS´ersicnvalues

n=0.2,0.4,...,4.0

forthesetofcombinationswithre=0.8,1.5,and2′′.5,µe=18,20,and22magarcsec−2andforseeingFWHMsof1.5,2.0and2′′.5.HerethecorrespondingB/Dratiosrangefromabout0to1,andB/Tfrom0to0.5.Wealsosimulatedthefullrangeofre,FWHM,andµe=18,20,22,and24magarcsec−2forn=0.2andn=4.0foratestregardingtheinitialestimates(see§4.2).

Atotalofabout223,000artificialSBprofilesand16,200imageswerecreatedandmodeled.Themockprofileswereallsampledat0′′.5/pixeltomatchthedata.The1Dand2Ddecompositionsoftheartificialprofilesandimagesweredeemedsatis-factoryiftheymetthefollowing(ratherliberal)criteria:•solutionfoundwithin100iterations•0•0<µe(fit)<30magarcsec−2•0.05•0<µ0(fit)<30magarcsec−2

Foreachsetofparameters,werequirethatatleasttwo-thirdsofthe100(40)simulatedprofile(image)decompositionspassthesecriteriatobeincludedintheanalysis.The2Dtestswerenotdevelopedasfullyduetoprohibitivecomputingtimes.Thusourtestsrelymoreheavilyonthe1Dtechnique,butwehaveconfirmedthattheresultsfrombothtechniquescorroborateeachother.

–16–

3.3.2.DiskInitialEstimates

NLLSalgorithmsrequireinitialestimatesasinputparameters,andwemustverifywhetherourfinalsolutionsaresensitivetoourinitialguesses.Differentinitialestimatesmayyielddifferentsolutionswithcomparableχ2valuesespeciallyifthetopologyoftheχ2distributionisnon-trivialorshallow(e.g.Schombert&Bothun1987;deJong1996a).Oursimulationsconfirmtherobustnessofouralgorithmstoawiderangeofinitialdiskparameterestimates.Theresultsareslightlymoresensitiveforlargevaluesofn,butingeneralthediskparameterswereperfectlyrecoveredindependentoftheinitialguesses.Onemuststillcautionthatifthebulgeisfitwiththewrongn,thefitteddiskparameterswilldifferfromtheirintrinsicvalues,eveniftheinitialestimatesweregood.Fig.3showstherelativefiterrorforh,∆h,where

hfit(mean)−hmodel

∆h≡

–17–

Thetestsforthen=1casedemonstratedthatbulgeparameterinitialestimatesarenotimportantinboththe1Dand2Ddecompositions,aslongasre󰀁0.3∗FWHM.Belowthislimitinitialestimatesaremoreimportant;errorsonrecanexceed50%anderrorsofupto∆µe±0.3magarcsec−2canoccur.Inthen=0.2casetheparameterrecoveryislargelyindependentoftheseeingFWHM,asexpectedforprofilesthatareflatinthecenter.Givenincorrectinitialguesses,recoveredparametersforprofileswithre󰀁1′′.0canstillbetrusted,butforsmallerbulgesthealgorithmistrappedinalocalminimumandtheoutputvalueisnearlythesameastheinput,e.g.a±25(50)%inputerroryieldsa±25(50)%outputerror.Forn=4.0profiles,theparameterrecoveryisstronglydependentontheseeingFWHMsuchthatdecompositionresultsforprofileswithre󰀂0.7∗FWHMcannotbetrusted.Hereagaininputandoutputerrorsareapproximatelyequal.Moreover,evenwithcorrectinitialestimates,themodelparametersarenotperfectlyrecoveredforprofileswithre󰀂1′′.0.Wehaveinterpolatedtheseresultsfordifferentvaluesofnanddefineaparameterspaceforwhichoursolutionsarenotaffectedbythechoiceofinitialestimates:

re󰀁(0.3)1/n∗FWHM

and

re󰀁

󰀍

−0.75n+1.150.2n+0.2

forn≤1.0forn≥1.0

Thecorrespondingresultsforthe2Ddecompositionalgorithmcloselymatchthosefromthe1Dtests.

3.3.4.SeeingEffects

TheeffectofanuncertaintyintheseeingFWHMmeasurementonthemodelparameters(n,re,µe,h,andµ0)mustbeaccountedforinoursimulations.Wefiteachmodelusingnotonlythenominalseeingvalue,butwealsovariedtheseeingFWHMbytypicalseeingmeasurementerrors(1σ;∼15−20%foroptical,∼35%forinfrared;seePaperII).Figs.4&5showtheeffectofanincorrectseeingestimateonthefittedreforthe1Dand2Dalgorithmsrespectively.Plottedaretherelativefiterroronre,∆re,where

∆re≡

re,fit(mean)−re,model

–18–

bulgeparametersandforallvaluesoftheseeingFWHMtested.However,evenmoderateseeinguncertaintiescanseverelyaffectbulgeparametersdependingonthesizeofthebulgerelativetotheseeingFWHM.Iftheseeingwidthisunder(over)-estimatedreissystematicallyover(under)-estimated,worseningforsmallerandfainterbulgesandlargerseeingvalues.Similartrendsareseenforµe.OurtestsshowthatthefiterrorscanbesignificantlylargeriftheseeingFWHMisover-estimatedthanifitisunder-estimated.Asaroughruleofthumb,forre≃FWHMandaseeingmeasurementuncertaintyatthe35%(15%)level,thebulgerecanbetrustedtowithin10–25%(0–10%),andµetowithin±0.1–0.4(0–0.2)magarcsec−2,thelowerendoftherangeapplyingtothebrightestbulgesandincreasingtowardstheupperendforthefainterbulges.Forre≃FWHM+1theerrorsimprovetowithin0–15%(0–10%)forre,and±0.0–0.2(0.0–0.05)magarcsec−2forµe.

Thereisnoappreciableeffectduetoseeingonthediskparameters(lessthan1%)exceptfortheworstcaseofaFWHMof3′′.0anda35%seeingover-estimate.Inallothercases,thediskparametersarevirtuallyunaffectedbyseeing,asthesizeofthediskismuchlargerthantheseeingprofile.However,itisofparamountimportancetouseaccurateseeingestimatesandrealisticseeingerrorsinordertosamplethetruerangeofbulgeparametersinB/Ddecompositions.

3.3.5.SkyUncertaintyEffects

Wenowtestfortheeffectsofanimproperskysubtractiononthedecompositions.ThetremendoussensitivityofB/Ddecompositionandscalelengthdeterminationstoskyerrorshasbeenhighlightedbefore(Courteau1992;deJong1996a).Hereweaimtoprovideafirmquantitativeassessmentofsucherrors.Were-modelthesamesimulatedprofilesasintheprevioussectionbutusingskyvaluesthatare±1%ofthenominalskylevel(typicalerrorintheopticalpassbands),andusingatypicalopticalskybrightnessof21magarcsec−27.Sincebulgebrightnessesaretypicallygreaterthantheskylevel,atleastatopticalwavelengths,onemightexpectbulgeparameterstobesomewhatinsensitivetoskysubtractionerrors.However,theouterdiskisverysensitivetoskysubtractionerrorsandamodifieddiskultimatelyaffectsbulgestructureduetotheircoupling.Quantitatively,ourtestsshowthatiftheskyisover-orunder-estimatedby1%,theerroronthediskscalelength,∆h,willbeoforder5–15%andthediskCSB,∆µ0,willbe±0.1–0.25magarcsec−2.Thesedispersionsholdforthefullrangeofbulgebrightnessesexceptthetwofaintestbulgeswhichareoneand

–19–

twomagnitudesfainterthanthesky;thesehaderrorsinexcessof50%.Theerrorrangesarecontrolledbytherelativesizesofthebulgeanddisksuchthatthediskparametererrorsincreaseslightlyfromsmallertolargerbulges.Thisissimplybecausealargerbulgeweakenstheimportanceofthediskinthecentralparts,thusgivingmoreweighttothesky-sensitiveouterdisk.

Theerrorsonthebulgeparametersarenegligibleforbulgeswith(µsky−µe)>1magarcsec−2fortheentirerangeofreandseeingFWHMs,butincreaseupto∆re≥15%and|∆µe|≥0.1magarcsec−2(increasingasthebulgegetssmallerandasseeingconditionsdegrade)forbulgeswith(µsky−µe)<1magarcsec−2.Inotherwords,ifthebulgeeffectivesurfacebrightnessislessthanonemagnitudegreaterthantheskybrightness,thebulgeparameterswillbestronglyaffectedbyskysubtractionerrors.Thiseffectisoftenneglectedinstudiesofbulge/diskstructure.

Thebulgeanddiskparametersaremostaffectedforthecaseofanunder-subtractedsky.Thisislargelyduetoamagnitudethresholdof26.5magarcsec−2inourdecompositionalgorithm.Thedataaretoonoisybelowthisvalue(PaperII)andweexcludethemfromthefits.Thisthresholdprovidessomeprotectionagainstover-subtractedskiesinthemea-surementofthediskscalelength.Similartestswereperformedwithour2Ddecompositionalgorithmwhichconfirm,onceagain,theresultsabove.

3.3.6.S´ersicnTests

Anumberofrecentstudieshavedescribedthevariationofbulgeshapesasafunction

ofHubbletype(Andredakisetal.1995;Moriondo,Giovanardi,&Hunt1998;Khosroshahietal.2000;M¨ollenhoff&Heidt2001),goingsofarassuggestingthatprecisevaluesofn(i.e.±0.1)couldbedetermined(Graham2001).Toourknowledge,nostudytodatehastestedthereliabilityoftherecoveryoftheS´ersicnparameter.InordertotestthesensitivityofthedecompositiontothefullrangeofbulgeprofileshapesweusemockluminosityprofileswithvaluesoftheS´ersicnparameterrangingfromn=0.2ton=4.0.Thesuiteofprofilesusedallcombinationsofre=0.8,1.5,and2′′.5,µe=18,20,and22magarcsec−2andseeingFWHMsof1.5,2.0,and2′′.5.Theprofilefitsusedinitialestimatesofn=0.4,1,2,and4,andcorrectinitialestimatesforre,µe,andthediskparameters.TheseeingFWHMwasfixedtothecorrectmodelvalue.Theresultsforthen=1initialestimatearepresentedinFig.6,whereweplottheaveragerelativefiterroronn(forthe100profileswiththesamesimulatedparameters)where

n(mean)−nmodel

∆n≡fit

–20–

versusthemodeln(thedashedlineat∆n=0indicatesaperfectrecoveryofthemodelnparameter.)Eachpanelshowsoneparticularcombinationofµeandre,andthepanelsarearrangedsuchthattheB/Dratio,foragivenvalueofn,decreasesfromtoptobottomandrighttoleft.ThedifferentseeingFWHMvaluesarerepresentedbythreepointtypes:circles,triangles,andsquaresforseeingvaluesof1′′.5,2′′.0,and2′′.5respectively.Fig.6reinforcesthatthebulgesofevennearbylate-typespiralsaresmallandnotsampledathighenoughspatialresolutiontoyieldastable,robustsolutionfornasafloatingparameter.Giventhecorrectvalueofnasaninitialestimate(alongwiththecorrectinitialestimatesfortheotherfourparameters),thealgorithmnormallyfindsthecorrectvalueofthemodeln,butanydeparturefromthemodelvalueevenbyasmallamount,yieldssignificantlydifferentsolutionsforn,orthefitmaysimplyfail(asindicatedbytheverticallinesinthefigures).Formostoftheparametercombinations,anoffsetof∼50%intheinitialestimateofnyieldsa∼50%erroronitsdeterminedvalue.

Similartestsusingthe2Dalgorithmshowaslightlymorerobustrecoveryofthemodelnparameterbasedonincorrectinitialestimates,buttherecoveryefficiencyisstillpoorandresultsbasedonafloatinginitialestimateofnarequestionable.Weareherefacedwithanunder-determinedoptimizationwithtoofewindependentdatapointsfortoomanymodelparameters(atleastthreeforthebulge).Thestrongcovariancesbetweenn,µe,andre(σn,µe,σn,re)preventauniquedeterminationofnwiththisNLLScode.Weactuallyfindthebest-fitnbygridsearch,holdingnasafixedparameter,solvingforarangeofvalues,andusingtheχ2inasdefinedin§3.1todeterminethebestfit.Furthersimulationsshowedthistechniquetobefullyreliableforthebulgesconsideredhere.

Itisdifficulttoestimatetheerroronn.Eitherweuseagridsearchandnisfixed,orniskeptasafloatingparameterandvarieswidelygivenwronginitialestimates.Basedonatestwithfloatingnbutcorrectinitialestimatesforbulgeparameters,typicalseeingandskyerrorsmodifynbynomorethan20%.

Basedon2DB/DdecompositionsofsimulatedimagesandJHKspirals,M¨ollenhoff&Heidt(2001)estimatethatrecoveryerrorsforparameters(Id,h,Ie,re,andn)arelessthan15%,comparabletoourtestedforvariableestimatesoftheskylevelandseeingwidth,thoughoftheirtechniqueispresented.

imagesof40brightallthestandardfitfindings.Theyalsonocleardescription

–21–

3.4.

SummaryoftheSimulations

Thetestsperformedin§3.3,foridealizedgalaxies,allowustodefineasetofguidelinesforthereliabilityandlimitationsofour1D/2Ddecompositions:

•Initialestimatesforbulgeanddiskparametersareunimportantprovidedthat

re󰀁(0.3)1/n∗FWHM

and

re󰀁

󰀍

−0.75n+1.150.2n+0.2

forn≤1.0forn≥1.0

•Seeingerrorsmustbeaccountedforinallbulgeparameterstudies.Forre≃FWHMandaseeingmeasurementuncertaintyatthe35%(15%)level,thebulgerecanbetrustedtowithin10–25%(0–10%),andµetowithin±0.1–0.4(0–0.2)magarcsec−2.Forre≃FWHM+1theerrorsimprovetowithin0–15%(0–10%)forre,and±0.0–0.2(0.0–0.05)magarcsec−2forµe.Thereisnoappreciableeffectduetoseeingonthediskparameters(lessthan1%).

•Skysubtractionerrorsdominatediskparametererrors(∼5–15%)andarenon-negligible(upto25%)forbulgeswhoseeffectivesurfacebrightnessesarelessthanonemagnitudebrighterthantheskybrightness(i.e.for{µsky−µe}󰀂1).

•Thesamplingoflate-typenearbybulgesmaynotbehighenoughtoconstraintheS´ersicnexponentuniquelyasafreeparameter.Iterativemodelfittingschemesshouldbetestedforthis.Ourapproachusesagridsearch.

•Typicalseeingandskyerrorsmodifynbynomorethan20%.

•The2Ddecompositiontechniquedoesnotprovideasignificantimprovementoverthe1Dmethodfortherecoveryofaxisymmetricstructuralparameterstowarranttheextracomputationaleffort.

Armedwiththesebasicguidelineswecannowturnourattentiontorealdatadecom-positionsusingthe1Dtechnique.

–22–

4.

Bulge-to-DiskDecompositions

4.1.

Outline

Ourstudyofstructuralpropertiesandthevariationofgalaxianparametersasafunctionofwavelengthusesthemulti-band(BVRH)datasetoflate-typespiralgalaxiesofCourteau,Holtzman,&MacArthur(§2;PaperII).MostgalaxieshaveatleastonesetofBVRHimages,andweusemultipleobservationsfor54galaxiestoestimatesystematicerrors.OtherB/Ddecompositionanalyseshaveusedlargersamples(e.g.Baggettetal.(1998))butlackthecrucialmulti-wavelengthinformation.

Weaimtodevelopastableandversatileprescriptiontocharacterizestructuralevolutionofthebulgesanddisksofgalaxies.However,justasanymorphologicaldescriptionofgalax-ies(e.g.Hubbletypes)dependsonthewaveband,intrinsicstructuralparametersarealsoexpectedtovarywithwavelengthduetostellarpopulationanddustextinctioneffects.Thus,multi-wavelengthinformationisrequiredforanyaccuratedescriptionofgalaxianstructuralparameters.

Physicaldifferencesintheshapeandsizeofbulgesamonggalaxiesarealsoexpecteddependingonhowtheywereformed.Formationbyaccretionprocesses(e.g.major/minormergers)canaccountforsteeplyrisingdeVaucouleurslightprofilesinthecentralpartsofgalaxies(e.g.vanAlbada1982),whilesecularevolutionwouldyieldexponentialdistribu-tions,withorwithoutacore,ofthecentrallight.Theformationofsmallbulgesisindeedlargelyattributedtosecularprocessesandredistributionofdiskmaterial(see§5.3).ThepresentstudyisanaturalextensionofdeJong’s(1996a)structuralanalysisof86face-onspiralswithBVRIHKimaging,andGraham’s(2001)re-investigationofdeJong’sdata.DeJong’s1Dand2DB/Ddecompositionsestablishedsignificantparametricvariationsatdifferentwavelengths.Giventheintrinsiclimitationsofthedatamodeling(i.e.over-determinationoftheparameterspace),hisB/DfitsalsousedafixedS´ersicnparameter(see§3.3.6),butlimitedtovaluesofn=1,2,and4bulges.DeJong’sanalysis,andthatofCourteauetal.(1996)whoperformed1Dprofiledecompositionsfor290r-bandluminosityprofiles,supportedthenotionofexponentialbulgesanddisksandatightcorrelationofB/Dscaleparametersinlate-typespirals.EvidenceforthiscorrelationwaschallengedbyGraham&Prieto(1999)butlatervalidatedbyGraham(2001)whore-modeleddeJong’sthesissamplewitha1DB/Ddecompositiontechnique8.HisresultssupportarangeintheS´ersicshapeparameterfromlarge(n≃2−3)tosmall(n󰀁0.5)valuesforearly-tolate-

–23–

typespirals.AwareoftheinadequacyofbasicB/Ddecompositionsinfittingthebulgeshapeparameterduetopoordataresolutionandstrongcovarianceswithotherbulgeparameters(deJong1996b;Broeils&Courteau1997,see§3.3.6),wewerecompeledtorevisitthisissuewithourownwell-testedtechniqueandamoreextensivedatabase.

OurapproachinvolvesB/Ddecompositionswithfixednvaluesthatsamplethefullparameterspaceofspiralbulges,fromn=0.1,0.2,...,4.0.Thefitsolutionsarefilteredoutonbasisofrelativeχ2andcriteriabasedonoursimulations(§3.4).WeareconcernedbelowwiththederivationofrobustB/Dparametersforeachgalaxyprofile.WecompareourresultswithGraham(2001)andothers,andtestforanyB/Dparametercorrelationsin§5.

Thefollowingisbasedexclusivelyonresultsfrom1DB/Ddecompositions.Thefactthatwedonotmodelnon-axisymmetricshapes(bars,rings,ovaldistortions)lessenstheneedformorecomputationallyintensive2DB/Ddecompositions,asoursimulationsshowednoimprovementsusingthe2Doverthe1Ddecompositionmethodforaxisymmetricstucture.

4.2.B/DInitialEstimates

Inordertodeterminetherangeofbestfittedbulgeanddiskparameters,weneedtoassisttheminimizationprograminfindingthelowestpossible(data−model)χ2.Fromanalysisofourmockimagesandprofiles,wehavefoundthatanyreasonableinitialestimatesforthediskparametersyieldsarobustsolution.Webaseourinitialestimatesforthediskparametershandµ0onthe“markingthedisk”technique,wherethelinearportionofaluminosityprofileis“marked”andtheselectedrangeisfitusingstandardleastsquarestechniquestodetermineitsslope.Clearly,theresultingfitsareverysensitivetotheadoptedbaseline.Wetestedvariouschoicesforthefitstartandendpointsforourgalaxyprofilesincluding:fullprofilefit,startingpointsof0.2rmaxand0.4rmaxouttormax,andafixedbaselineshiftedalongthelengthoftheprofileandtracking8differentlocations.Additionally,wealsotestedthe“momentsmethod”ofWillick(1999).Thediscrepanciesbetweenthedifferentfitsarelarge;(∼10%onaverageandupto∼100%fortheworstcases),butwefoundthatthe0.2rmaxtormaxbaselineyieldedthemostreliablefits(asjudgedbyeye).TheinnerboundaryischosentoexcludethemajorcontributionofaputativebulgeorType-IIdipandrmaxistheradiusatwhichthesurfacebrightnesserrorhassystematicallyreachedvaluesgreaterthan0.12magarcsec−2(beyondwhichthedatabecometoonoisytobetrusted).Thefitsusingthe0.2rmaxtormaxbaselineprovidedfitsthatweremorethanadequateasinitialestimatesforthediskparametersinthedecompositions.

–24–

Flexibilityinthechoiceofbulgeinitialparametersis,however,onlyaffordedoutsideacertainrangeofbulgesizesrelativetotheseeingdisk.Moreover,observedgalaxypro-filesshowsignificantlymorevarietythantheidealizedprofilesfromwhichtheseconclusionsweredrawn(e.g.wedidnotmodelType-IIgalaxies,orthepresenceofstrongspiralfea-tures).Accordingly,weexplorethreedifferentsetsofinitialbulgeparameterestimatestoprotectagainstlocalminimaintheparameterspace.Initialbulgeeffectiveparametersweredeterminedfrom:

•Subtractionofthediskfit(basedonthe“markingthedisk”technique)fromtheoriginalprofileleavingonlythebulgelight.reisthencomputednon-parametricallyfromthedatabysummingupthelightuptotheradiuswhichencloseshalfthetotallightofthebulge.Thusµe=µ(re).

•re=0.15handµe=µ0,wherehandµ0aredeterminedfromthe“markingthedisk”technique.

•re=(bn/log(e))∗0.15handµe=(bn−bn=1)+µ0.

Thesecondsetofinitialestimateswasmotivatedbythebulge/diskstructuralcorrelationfoundbyCourteauetal.(1996)Weaddedthethirdsetofinitialestimateswhichattempttoscalereandµemoreappropriatelytothedifferentvaluesofn.Nospecificsetofinitialestimatesworkedbetterforallcases,thoughthe3rdmethodmaybetheleastattractive.Itfailedtoprovidereliablesolutions(i.e.thefitfailedortheχ2valueswerelarge)inmostcases,butinafewcasesitalsoyieldedtheonlyviablesolution.

4.2.1.SeeingandSkyTreatment

“Bulges”oflate-typespiralsaresmallandtheirluminosityprofilescanbeseverelyaffectedbyatmosphericblur.Inprinciple,iftheblurringfromtheatmosphere(seeing)canbemeasuredaccurately,itcanalsobecorrectedbyFourierdeconvolution.Inpractice,however,deconvolutionamplifiesnoise,andtheseeingFWHMissubjecttomeasurementerrors.In§3weusedextensivesimulationswithawiderangeofinputparametersandvariousvaluesofntoderiveaspaceofrecoverableparametersunderspecificseeingconditions,accountingforthetypicalmeasurementerrorsofourdata.SeeingisaccountedforbyconvolvingthemodelprofileswithaGaussianPSF(Eq.6)whosedispersionismeasuredfromfieldstars.Inordertoaccountforseeingmeasurementerrors,eachprofileismodeledwiththreedifferentvaluesoftheseeingFWHM:thenominalmeasuredvalueand±15%ofthatvalue.Ameanseeing

–25–

uncertaintyof±15%wasusedratherthantheindividualerrorspermeasurementasthesefluctuategreatlydue,inlargepart,tothedifferentnumberofstarsineachmeasurement.Skysubtractionerrors,oforder∼0.5–1.0%intheopticaland∼0.02%intheH-band,werealsoexaminedcarefully(§3.3.5).TheskybrightnessmeasurementerrorisaccountedforinB/Ddecompositionsbyusingthreedifferentskylevels:thenominalmeasuredvalueand±0.5%(optical)or±0.01%(H-band),ofthatvalue.

Eachprofileisthusreduced3timesforeachdifferentcombinationofreandµeinitialestimates,times3seeingFWHMvalues,times3skyvalues,andtimes40differentfixedvaluesofn,foratotalof1080decompositionsperprofile.

4.3.DataFiltering

The1080decompositionsforeachprofilearefirstvettedonthebasisofstructuralcriteriadeterminedfromoursimulations(§3.3).Adecompositionisdeemedacceptableifitmeetsthefollowingcriteria:

•re󰀁(0.3)1/n∗FWHMandre󰀁•B/D<5

•h<15kpc;re<50kpc•re/h<1

Thefirstconstraintisderivedfromoursimulationsandeffectivelyeliminatessmallbulgeswhosesizesarecomparableto,orsmallerthan,theseeingdisk.Theremainingconstraintsarebasedonphysicalconsiderationsandhelpeliminatesolutionswithsmallχ2valuesbutunrealisticparametersforlate-typegalaxies.Note,however,thatthesephysicalconstraintsarerathergenerousanddonotcontributeanysubjectivebias.

Thesuccessfuldecompositionsarethenrankedonthebasisoftwoindicators:(a)a

2

globalχ2,χ2gl,computedforthefullSBprofilefromr=0tormax;and(b)aninnerχ,χ2in,whichincludesonlythecentralregionsofthegalaxyfromr=0totwicetheradius,rb=d,wheretheintensitiesofthefittedbulgeanddiskareequal(see§3.2).χ2inwasadopted

󰀍

−0.75n+1.15

0.2n+0.2

forn≤1.0forn>1.0

–26–

toincreasethesensitivityofthegoodness-of-fitindicatortothebulgearea9.Theradiusrb=disclearlyafunctionofthebulgeshapeandmaychangefromsmalltolargen(seee.g.Fig.2).Thusweuseaχ2perdegreeoffreedomtoremoveanydependenceofthenormalχ2toachangingre.Becauseofthepresenceofspiralarmsandothernon-axisymmetricfeatureswhichwedonotattempttomodel,thereducedχ2isalways,inprinciple,greaterthanunity.However,someofoursolutionsmayhaveχ2valueslessthanunityindicativeofanover-determinedsystem(correlatedparameters),orover-estimatederrors.

Wefirstrankthesolutionsaccordingtotheirχ2glandpreserveonlythebetterhalf.Thereducedsetisthenrankedaccordingtoχ2invaluesandthebottomhalfofthedistributionisdiscarded.Thisprocessisiteratedatleasttwice,oruntilwereach50orfewersolutions.Solutionswithχ2glgreaterthan50inthisfinalsubsetarediscarded.

2

Ideally,theminimaforthedistributionsofχ2glandχinvaluesshouldagreetoacommonvalueofn,butdifferencesmayexist.Wesearchthefinal≤50solutionsforacommonsolution,startingattheminimaofeachχ2distribution.Ifthenvaluescorrespondingtothetwoχ2minimadonotagree,thenvaluesforthenextsmallestχ2valuesarecompared(withthelowerχ2valuesandwitheachother),andthisprocessisiterateduptothreetimesuntilamatchisfound.Ifthisprocessdidnotconverge,i.e.thereisnotrueminimumin

2

the((χ2gl,χin)-nspace),afinalsolutionischosencorrespondingtotheminimumvalueof

222

(χ2gl/min(χgl)+χin/min(χin)).

22

Figs.7&8showexamplesofthedistributionsof(χ2gl,χin)versusnwhereχglobal′≡

222222

χ2gl/min(χgl,filt)andχinner′≡χin/min(χin,filt),wheremin(χfilt)istheminimumχvaluefromthesetof(≤50)filteredsolutions.Notethattheseminimadonotnecessarilycor-respondtothelowestvalueoftherespectivedistributionsfromall1080solutions,asthe

2

initialabsoluteminimamayhavebeenfilteredout(i.e.apoorcombinationofχ2gl,χinforagivensolution).Thus,thenormalizedχ2’smaybelessthanone(asiseasilyseenintheleftmostplotofFig.8).Intheseplots,theleftpanelsshowtheχ2distributionsforall1080decompositions,whiletherightpanelsdisplayonlythe≤50solutionsremainingaftertheiterativefilteringschemedescribedabove.

Fig.7highlightsthesensitivityofourtechniquefortwoV-bandobservationsofUGC929takenunderdifferentseeing/skyconditions.Theleftfiguresshowafairlywell-behavedsolutionfavoringn=0.6andthefiguresontherightplotshowarathermessysolutionfavoringn=0.8.Theseeingconditionswereworseandtheskywasmuchbrighterfortheobservationshownontherightwhichcouldexplainthenoisydistributionsofboththeχ2gl

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andχ2in.

2

Fig.8showstwodifferentbehaviorsofχ2glforprofileswithverywell-behavedχin.TheplotontheleftforourUGC784B-bandprofileillustratestheneedforanadditional,morediscriminatingstatisticforthebulgeregion.Decompositionsbasedsolelyontheχ2glgoodness-of-fitindicatormayresultinfits,liketheoneshownontherightsideofFig.9(dashed-dottedblueline).However,Fig.8a)clearlyshowsthatthefitwithn=0.6isafarsuperiormatchtothebulgeshape,asindicatedbytheχ2inbehaviour.

Thefinalstepofourfilteringprocedureentailsavisualinspectionofthefinaldecom-positions.Thecriteriaforuserexaminationincludeinformationfrommultipleexposuresandmulti-bandreductionsforagivengalaxy.Profilesand/orsolutionswiththefollowingpathologieswereeliminatedfromthefinalsample:•diskprofilesthataretooshortforproperfitting

•noobvious,extended,underlyingexponentialstructureforthedisk(occurspredomi-nantlyinType-IIprofiles)•unphysicallylargefittedbulge

•unrealisticdiskfitforType-IIprofiles.ThefitistippedbelowthetruedisktoaccountfortheType-IIdipnearthebulge-disktransitionregionleadingtoscalelengthsthatarebiasedhigh(e.g.seeFig.12forUGC12527forexamplesof“bad”fitswhichwereeliminatedfromthefinalsample)

•largedeviationsbetweensolutionsformultipleobservationsofagivengalaxy.Notsurprisingly,mostoftheeliminatedprofilesareType-IIsystems.WecautionthateventheType-IIprofiledecompositionsthatsurvivedthefullsortingprocessmaynotprovidetheidealdescriptionoftheircomplexsurfacebrightnessdistributions.ClearytheseType-IIprofilescannotbeproperlymodeledwithjustaS´ersicbulgeandexponentialdisk.Outof523images/profiles,atotalof341passedouracceptancecriteria.

4.3.1.PreferredSkyandSeeing

HistogramsofthepreferredseeingFWHMandskyoffsetsforalldecompositionsinallfourbandsareshowninFig.10.Typically,alowerskybrightnesslevelispreferredbyouralgorithm.Insomecases,thiscouldbeexplainedbyanover-estimatedskylevel,butitmay

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alsobedueinparttoprofileswithtruncatedouterdisksasinFig.13.OurprogramprefersaslightlylargerseeingFWHMthanmeasured.Thiscouldbetheresultofanunder-estimatedFWHM,orperhapsdifferencesbetweentheidealizedGaussianmodelandtherealseeingPSF.Solutionswithvariablesky/seeingestimateswereretainedinthefinalsolutionsetforassessmentofparametererrors.

4.3.2.Effectofrmax

Ofsignificancetothefitresultsisthemaximumradiususedinthedecompositions.Wehaveusedthefullobservedprofileouttoradiiwherethesurfacebrightnesserrorsreachedabove0.12magarcsec−2.Thereisnoabsolutedefinitiontotheedge,rmax,ofadiskandadifferentselectioncouldyielddifferentresults.Totestthesensitivityofourparameterdeterminationstothechosenvalueofrmax,were-decomposedtheprofilesasdescribedabove,butwithafitbaselineextendingonlyto0.75×rmax.Acomparisonoftheresultsfromthetwotechniques,priortoeyeballfiltering,showsgoodagreementandwechosetokeepthelargerbaselinetoavoiddiscardinggooddata.

4.4.DecompositionExamples

Thereisnoroomforafulldisplayofourcatalogoffinaldecompositions,butafewexamplesareshowninFigs.11–14.Thefullcatalogofdecompositionplotsisavailableuponrequestfromtheauthors.

Inthesefigures,thesolidblackcirclesarethedatapoints,theblackdotsshowtheskyerrorenvelope(fromthemeasuredskyerror),thedashedanddashed-dottedlinesshowthediskandbulgefitsrespectively,andthesolidlineisthetotal(bulge+disk)fit.Thefitsareallseeing-convolvedusingthebestselectedseeingvalues.Thebottompanelshowsthefitresidualswhere∆µ(r)representsthe(data−model).Fig.11showsanexampleofthequintessentialType-Iprofileatallwavelengths.Fig.12showsaType-II/TransitiongalaxywhoseType-IIsignaturesignificantlyweakensfromtheopticaltotheinfrared.Fig.13showsaType-Iprofilewithanoutertruncateddisk.Suchdecompositionswillpresumablyfavoranunder-subtractedskyinattempttoaligntheinnerandouterpartsofthedisk.Hereisanexamplewhereourprocedurewithanover/under-estimationoftheskyandaninfiniteexponentialdiskmodelmaynotbeadequatesincetheouterdisktruncationappearsreal(asdetectedinallfourbands).WealsoshowanexampleofanearlybulgelesssysteminFig.14.Oursampleisdividedinto52Type-I,53Type-IIand16transitionsystems,ofwhich18

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truncatedand7bulgelessdisksareidentified10.

4.5.DistributionoftheS´ersicnparameter

Fig.15showshistogramsoftheS´ersicnparameterforallthefinalfits(left)andgood

fitsonly(right)afteruser-examinationasdescribedabove.Thedistributionofnhasadefiniterange,implyingthatnotalllate-typebulgesarebestdescribedbyanexponentialprofile,butthemeanvalueisveryclosetoone.ThisresultagreeswithGraham(2001)andrecentN-bodysimulationsofgalaxyevolution(§5.3).

4.5.1.FloatingS´ersicn

In§3.3.6weshowedthatresolutionlimitationspreventedstablefittingoftheS´ersic

shapeparameternasafreeparameter.Toillustratetheeffectafloatingncanhaveonfittedparameterswere-decomposedallofourgalaxyprofilesleavingnasafreeparameter(e.g.akintoGraham2001).TheresultsareshowninFig.16forthreedifferentinitialguessesforn(0.2,1.0,and4.0).

Thehistogramsoftheresultingdistributionsofnrevealastrongbiastowardstheadoptedinitialestimate.All3distributionsshowalargepeakatn=0.1,indicativeofpoorbulgefits.Thehistogramforthen=1.0initialestimatelookssomewhatsimilartoourownconstrainedsolution(Fig.15),butthisissomewhatfortuitousgiventheclosely-exponentialnatureofspiralbulges.Notealsothenon-GaussiantailinFig.15isnotreproducedinFig.16forthen=1initialestimatecase.Fig.17showsacomparisonoftheχ2valuesfromthefloatedversusfixednsolutions.(Whennosuitablefitwasfound,allparametersweresetto0asindicatedbythepointslyingontheaxes;notethelargenumberoffitfailuresinthe

2

floatedncase.)Notethelargediscrepanciesinχ2inandχglbetweenthetwomethods.Thus,whilethefinaldistributionsforthen=1initialestimateandourconstrainednprocedurelooksimilar,significantdifferencesmayexistbetweenindividualdecompositions.

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4.6.

ErrorofaSingleMeasurement

Animportantfeatureofanydecompositiontechniqueisthestabilityofthefinalresultsforrepeatobservationsofagivensystem.Oursamplehas50profilesforwhichmultiple(twotofour)observationsexist,allowingforadirectmeasureofthereliabilityofourdecompo-sitions.Table1givesthemeanandmeanstandarddeviationofthefivemodelparametersfromrepeatobservationswith,

x)2

(23)

N

wherexisthefitparameter,nisthenumberofobservationsforagivenprofile,andNisthenumberofprofileswithrepeatobservations.TheaverageerrorsfromrepeatobservationsofType-Iprofilesare±14%forn,±0.2magarcsec−2forµe,±13%forre,±0.05magarcsec−2forµ0,and±3%forh.Clearly,determinationsofdiskparametersaremuchmorestablethanforbulges.Errortermsquotedbelowcorrespondtothe1-σdeviation,unlessotherwisenoted.

4.7.ComparisonwithOtherAuthors

TheoverlapbetweenoursampleanddeJong’sthesiscatalog(deJong&vanderKruit1994)amountstoonly3galaxies.DirectcomparisonofourSBprofilesshowsexcellentzero-pointandoverallshapeagreement(PaperII);however,ourB/Ddecompositionsdiffersomewhat,asshowninTable2(notethatdeJongusesfixedn=1).AlsoshowninthattablearedecompositionparametersforthesamegalaxiesbyGraham(2001)(samedataasdeJong,butusingarangeofn).Wefindscalelengthdifferencesatthe10%levelwithdeJongandGraham,consistentwith,orslightlybetterthan,typicalvariationsbetweendifferentauthors(Knapen&vanderKruit1991).AcomparabledispersionismeasuredbetweenthescalelengthsofGrahamanddeJongbasedon82R-bandprofiledecompositions.Graham’sscalelengthsare,onaverage,smallerforsmallgalaxiesandlargerforbiggalaxies(apparentsize)thandeJong’s.WefindsystematicallylargerdiskscalelengthsthandeJong(basedononly7comparisons.)Weverifiedthatskyunder/over-estimatescannotaccountforanydifferencewithdeJong.DeJong’salgorithmgivesmoreweighttotheouterpartofthedisk,possiblyexplainingtheshorterdiskscalelengths.ForprofileswithoutertruncateddisksorType-IIdecrements,greaterweightintheouterpartsfavorstheouterdiskcurvatureandthussteeperdiskfits.

BulgeparametersbetweenusandGrahammatchreasonablywellforthefirsttwogalax-iesbutdiffersubstantiallyforUGC3140.Itishoweverdifficulttoestablishtrendsbased

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onjust3comparisons.Wecan,instead,broadlycompareourrespectivedistributionsofS´ersicnwithmorphologicaltype.ThisisdoneinFig.18forcomparisonwithFig.10ofGraham(2001).Thegeneralfeaturesaresimilar,butwefindawiderrangeofnvaluesforthelater-typespossiblyduetothelargernumberofScd/Sdgalaxiesinoursample.AnotherfavorablecomparisonofbulgeparameterswithGrahamisshowninFig.20(see§5.2.1).Wealsofindanoverlapoftwogalaxies,NGC3512andNGC7782,withthesampleofBaggettetal.(1998).AswithdeJongandGraham,diskparametersagreewithin10%.BulgeparametersfromBaggettetal.aremissingforNGC3512,andthoselistedforNGC7782(bothVband)differquitesubstantiallyfromours.Theseauthorsfindµe=10.88andre=0.2foradeVaucouleursbulgeandwehaveµe=20.1andre=3.2forn=1(bestfit)orµe=25.2andre=83.8forn=4(verybadfitwithhighreducedχ2).Notethatseeingestimateswerecomparable.Surprisingly,theirµeisnearly10magnitudesbrighterthantheirµ0=20.3(wealsofindµ0=20.3)!Wefindthispathologyinanumberoftheirbulgedecompositions(seee.g.theirFig.2)wherethemodelsoftenovershootthedataatthecenter.

Weconcludethissectionbynotingthatdiskscalelengthsbetweenusandotherauthorsdifferatthe10%level.OurbulgeparametersarealsoqualitativelyconsistentwiththoseofGraham.

5.Discussion

Simulationsofgalaxyprofilesandimages(§3)andcarefulB/Ddecompositions(§4)haveledtoafinalsetofstructuralparametersforlate-typespiralgalaxies(Table4inAp-pendixB).Thesedatacannowbeexaminedforintrinsicstructuralvariationsandsensitivitytodustandstellarpopulationeffects.Theoutlineofthissectionisasfollows:First,wever-ifyin§5.1thatoursolutionsarenotaffectedbyprojectioneffects.Wethendiscussin§5.2B/Dparametervariationsbothinthecontextofprofiletypedifferencesandwavelengthde-pendence.InlightofexistinglimitationsinourmodelingofType-IIprofiles,ourconclusionswillbebasedmostlyonpropertiesderivedfromType-Iprofiles.Thesewillenableustoexaminetheviabilityofsecularevolutionmodelsfordiskgalaxies(see§5.3).

5.1.InclinationDependence

Inordertotestforprojectioneffects,weplotthedistributionsofµeandre,aswellasdiskµ0andhasafunctionofellipticity,ε=1−b/a,inFig.19.Thesurfacebrightnesses

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areonlycorrectedforGalacticextinctionandcosmologicaldimming(asin§2);thustheµ0andµevaluesshouldbeconsideredasupperlimits(i.e.effectivebrightnessesaretoolow).Notrendswithellipticityareseen,includingtheS´ersicnparameterandratioofdiskscalelengths(notplotted).Furthermore,TypesI,IIandTransitionarenotconfinedtoanyparticularinclinationrangeshowingthattheType-IIphenomenonisnotanaccentuatedfeatureduetoline-of-sightextinction(e.g.Type-IIgalaxiesarenotpreferentiallyinclinedwiththeplaneofthesky).

5.2.Bulge/DiskParameters

Table3showstherangeoffittedparametersatBVRHwavelengthsforallgalaxyprofiletypes(Type-I,II,andTransition).ThenumberofType-IIandTransitionsystemsincludedinthistable(e.g.only4decompositionsforTransitiongalaxiesintheB-band)isdrasticallyreducedfromouroriginaldistributionasmanyofthemdidnotpassourvaliditycriteria(§4.3).NotethattheparametersforTransitionprofilesatH-bandbroadlymatchthoseofType-I’satthatwavelength.

TheS´ersicshapeparameternforType-Igalaxiesisnearunity,withintheerrors,forallwavelengths.Thus,weadvocatethatthenatural,intrinsicdistributionoftheS´ersicnparameterforlate-typespiralshasameannear1.0(withσn≃0.4;seeFig.15).Byallaccounts,bulgesoflate-typespiralsarewell-approximated,onaverage,byapureexponential(luminosity/mass)densitydistribution.

ThedistributionsofS´ersicµeandrearebroad,indicativeoftherangeofbulgetypesinoursample.Effectiveradiiaretypicallylessthan1kpc.ThoseofType-IIprofilesareevensmallerandseeminglybetterdeterminedthanType-I’sbutthisispredominentlyanartifactofourlimited2-componentmodeling.ExaminationofType-IIprofilefitsshowsthatthemodeldiskistypicallyshallower(thanthe“true”disk),asthefitaccountsforthefainterbulge/disktransitiondip,andbulgeeffectiveradiiarenaturallyconfinedtoasmallerrange.Thedistributionsofdiskscalelengthsandtheirratiosshowacleardecreasingtrendasafunctionofwavelength(asnotedbydeJong(1996b)).Thisstatisticallysignificanteffect,detectedforallprofiletypes,canbeexplainedeitherbyahighconcentrationofolderstarsand/ordustinthecentralregionsofthedisk.Absorptionbydustalonecanaccountforthescalelengthratiosthatwemeasure(seee.g.Evans(1994),Fig.5).Evans’modelsdonotconsiderscatteringbutforthenearlyface-ongalaxiesconsideredhere,itseffectsarenegligible(Byunetal.1994;deJong(1996c)).ThecolorgradientanalysisofdeJong(1996c)usingstellarpopulationanddustextinctionmodelssuggests,however,thatdustand

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metallicityplayaminorrolebutthatageisbethedominantfactor.Preliminaryanalysisofourphotometricdatawiththelateststellarevolutionaryanddustmodels(MacArthuretal.,inprep.;hereafterPaperIII)suggestsacombinationofeffects.Theinterpretationofcolorgradientsisnon-trivialandmayultimatelyrequireafullspectroscopicinvestigationtoconvincinglydisentangletheeffectsofage,dust,andmetallicity.

5.2.1.B/DScaleRatios

InFig.20,weplotrevs.hforourType-Idecompositions(solidsymbols)andthoseofGraham(2001)fordeJong’sBRKType-ISBprofiles(opensymbols).ThisfigureprovidesthebasisforareneweddiscussionofthesuggestionbyCourteauetal.(1996)ofstructuralcouplingbetweenthebulgeanddiskoflate-typegalaxies.Thelargedispersionsintheλreandhλ(seeTable3)nearlycancelouttoyieldtighterre/hcorrelations.ForType-Iprofileswefind󰀔re/h󰀕≃0.22±0.09atallwavelengths,correspondingto󰀔hbulge/hdisk󰀕=0.13±0.06forn=1.ThisresultisalsoborneoutintheH-bandTransitionprofiles(seeTable3).Forcomparison,Courteauetal.(1996)found󰀔hbulge/hdisk󰀕∼0.10±0.05(or󰀔re/h󰀕=0.15±0.08)11.ThelatterresultisalsoinagreementwithGraham(2001)whofinds󰀔re/h󰀕=0.2(noquoteddispersion,butitissomewhatlargerthanoursjudgingfromFigs.20and21)forearlyandlate-typespiralsintheK-band.Thisisconsistentwithascenariowherebulgesoflate-typespiralgalaxiesaremoredeeplyembeddedintheirhostdisk,thanearlier-typebulges.Insuchan“iceberg”scenario(e.g.Graham2001),bulgesanddiskscanpreserveanearlyconstantre/hbutshowagreatrangeofµeforanygivenre.InFig.21,weshowre/hasafunctionofmorphologicaltypefromour(solidsymbols)andGraham’s(opensymbols)decompositions.AmildtrendwithHubbletypeisseenwith󰀔re/h󰀕=0.20−0.013(T−5)(1σ=0.09),rangingfrom󰀔re/h󰀕∼0.20forlate-typespiralsto󰀔re/h󰀕∼0.24forearliertypes.ComparisonofourandGraham’sdecompositionparametersinTable2showsthatlargedeviationsmayexist,thusonlyourdatapointswereincludedinthefitof󰀔re/h󰀕vsTabove.Datafordifferentbandsscatterevenlyaboutthemeanline.Moredataatearlierandlatertypeswouldbeneededtofirmupthistrend.Itisnonethelessremarkablethatearlyandlate-typesystemsaredescribedbyverysimilarscalingrelations,thussuggestingcomparableformationand/orevolutionscenarios.

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5.3.

TestofSecularEvolutioninLate-TypeSpirals

Thisworkhasprovidedconfirmationoftwoimportantstructuralsignaturesofspiralgalaxieswhichmustbeaddressedbymodelsofstructureformation:

•Theunderlyingsurfacebrightnessdistributionoflate-typespiralshasarangefortheS´ersicnparameterfrom0.1–2,butisbestdescribed,onaverage,byadouble-exponentialmodelofbulgeanddisk,suchasfoundinType-Iprofilegalaxies.

•Bulgesanddisksoflate-typespiralsarecoupled,with󰀔re/h󰀕=0.22±0.09,or󰀔hbulge/hdisk󰀕=0.13±0.06,atallwavelengths.AmildtrendwithHubbletypeisalsodetectedwitharange󰀔re/h󰀕∼0.20–0.24,fromlatetoearly-typespirals.

Thefirstresultdescribesthelarge-scaleappearanceofbulges.AnalysesofHSTimageshaveshownthatasignificantfractionofbulgenucleihavepower-lawprofiles(r<500pc;e.g.Phillipsetal.1996;Balcells2001)andhostacentralcompactsource(Carollo1999).Theextentofthesenuclearsources(<0′′.3forcz<2500kms−1)issmallerthanourimages’pixelsizeandsmoothedoutbyseeing.Wethusignoretheireffectsonthebulgelightprofileinthisanalysis,butcautionthatourbulgeparametersaretobeconsideredupperlimitsifasignificantnuclearcomponentispresent.

AnaturalinterpretationofthenearconstancyofB/Dsizeratiosinlate-typespiralsisthattheirbulgesformedviasecularevolutionofthedisk.Thisscenarioispossibleifdisksarebar-unstable,whichcanbetriggeredbytheglobaldynamicalinstabilityofarotationallysup-porteddiskorinducedbyinteractionswithasatelliteandifsignificantangularmomentumtransportisfeasible(e.g.Martinet1995;Combes2000;seethecollectionofpapersinCarollo,Ferguson&Wyse1999forcomprehensivereviews).Forbar-unstabledisks,inparticulartoverticaldeformations,theinnerdiskmaterialisheatedupto1–2kpcabovetheplaneintoa“bulge”viaresonantscatteringofthestellarorbitsbythebar-forminginstability.Thisinturn,catalyzesfunnelingofdiskmaterialintothecentralregionsandgeneratesoutwardtransportofdiskmaterialintheouterparts.Gasflowsmustalsobeinvokedtoexplainthehigherspatialdensitiesofbulgescomparedtotheinnerdisk.Suchamodelisexpectedtoproducecorrelatedscalelengthsandcolorsbetweenthediskanditscentralregions,asob-served(e.g.Terndrupetal.1994;Peletier&Balcells1996;Courteau1996b).A“bulge-like”componentwithanearlyexponentialprofileisexpectedfromnon-axisymmetricdisturbancesthatinduceinwardradialflowofdiskmaterial(Pfenniger&Friedli1991;Zhang&Wyse2000,andreferencestherein).Thelongerthedisk-barheatinginteraction,thegreatertheextentofthediskexponentialprofile(Valenzuela&Klypin2002).Theevolvingexponentialattractorisanempiricalresultwellestablishedinsimulations,butitlacks,atpresent,atheoreticalexplanation(Pfenniger1999).

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Althoughabarcangrowspontaneously(<∼20Myr)fromsmallscalefluctuationsintheinnerdisk,anexternalfiniteperturbationcancatalyseitsgrowth.However,collisionlessmergersseemunsuitedtogrowingtheexponentialbulgesofpresent-daylate-typespirals,thoughtheymaycontributetotheincreaseinS´ersicnparameterseentowardearliertypesinproportiontotheaccretedsatellitemass(Barnes1988,Aguerrietal.2001).ThespontaneousortriggeredformationofbarsalsosuggeststhattheHubbletypeofgalaxiescanchangewellaftertheformationofthedisk(Pfenniger1999).Allofourbulgesaresmallerthanadiskscalelengthandcouldbecreatedbypurelybar-relatedprocesses.Instead,accretionofgalaxysatellitesisrequiredtomakebiggerbulges,eitherbeforeorafterformationofthehostdisk.

Secularevolutionmodelsofstellarandgaseousdisks,especiallythroughcosmologically-motivatedthree-dimensionalN-bodysimulations,haveseensignificantdevelopmentsinthelastdecade.Forexample,thecolddarkmatter(CDM)hierarchicalhydrodynamicalsimula-tionsbyS´aizetal.(2001)andScannapieco&Tissera(inprep.)showthatsecularprocessescanoccurnaturallyduringtheformationofspiraldisksandplayanimportantroleintheregulationofstarformationandthedeterminationofthedynamicalandstructuralproper-tiesofthesesystems.Onaverage,thesimulateddisksystemsareshowntobecharacterizedbyadoubleexponentialprofilewhichnaturallyemergeswithinthehierarchicalclusteringscenario.Theseresultsarebasedonastellarformationprocessimplementedinsuchawaythatitsucceedsinformingcompactbulgesthatstabilizedisk-likestructureallowingthecon-servationofanimportantfractionoftheirangularmomentumduringtheviolentphasesoftheirassembly.Fig.22showsthedistributionoffinalS´ersicnparametersforrelaxedpresent-daylate-typedisksbyScannapieco&Tissera(inprep.);theirmodelsreproduceourresults(Fig.15)verynicely.Thedouble-exponentialstructureofbulgeanddiskmaynotalwaysbethefinalrelaxedstateofanobject,butwhenevern∼1,theB/Dscaleratio󰀔hbulge/hdisk󰀕takesitsnominalvalueof0.15.Thesemodelsdonothavebulgeswithn<0.7,possiblyduetolimitedresolutionand/orexcessiveangularmomentumtransferthatsupernovafeedbackcouldhelpprevent.

SimulationsbyPfenniger(2002;privatecomm.)ofself-gravitatingdisksformingbarswhichmaylaterdissolveintoabulge-likecomponentalsoshowanearlyuniversalratiore/h,inagreementwithobservedvalues,whichisrelatedtothestellardynamicsofthebarredsystem(i.e.relativepositionoftheverticaltohorizontalresonances).Thebarlengthisrelatedtotheinitialrisingpartoftherotationcurve(yieldingascale),andthecorotationofbarsisproportionaltotheirlength.Thecorotationfixesthepositionsoftheotherresonances,whichinturnfixthemaximumextensionofbulgesmadefromresonantheating,asindeedtheverticalresonancesarestrongonlywithinthebar.Thismechanismwouldthussettheupperlimitfortheallowedrangeinre/h.

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TheN-bodysimulationsofAguerrietal.(2001),whichconsiderthegrowthofgalacticbulgesbymergers,alsosuggestthatthefinalB/Dscaleratio󰀔re/h󰀕doesnotscalewiththeB/Dluminosityratio.Theseauthorsshowthatthediskscalelengthhcanincreasefrom15%(lowmassretrogradesatellite)to65%(highmassdirectsatellite)while󰀔re/h󰀕woulddecreasefrom0.21to0.14,or33%,inthemostextremecase.Onecanthusinfer󰀔re/h󰀕=0.17±0.03,independentofB/Dluminosityratio,ingoodagreementwithourfindings.TheirsimulationsarehoweverlimitedtoasmallrangeofinitialreandamorecompleteinvestigationwithabroadrangeofreandhvaluesisneededtoestablishthefundamentalnatureoftheB/Dscaleratio.

5.4.Type-IIProfiles

Theabovescenariosforsecularevolutionnaturallyproducethedouble-exponentialchar-acterofthebulgeanddiskradialluminosityprofilesforlate-typesystems.However,overhalfoursampleof121late-typespiralgalaxiesshowstrongdeviationsfromthissimpletwo-componentdescription.Otherauthors(Kormendy1977;Baggettetal.1998)havecon-sideredinnerdisktruncation(plusdeVaucouleursbulges)asanalternativetomodelingType-IIlightprofiles,with

Idisk(r)=I◦exp{−[r/r◦+(rhole/r)n]}

(24)

whererholeisthetruncationradiusandn∼3.Asdiscussedin§3.2,wedonotconsiderthisapproachatthepresent,butitspotentialmeritsshouldnotbeoverlooked.

N-bodysimulations(e.g.Norman,Sellwood,&Hasan1996;Valenzuela&Klypin2002)reproduceType-IIsurfacedensityprofilesasaresultoftheredistributionofcentralstarsintoaringbyabar-likeperturbation.Approachingthecenterswherethecomponentcalledbulgeandthecomponentcalledexponentialdiskoverlap,onecannottell,inthesesimulations,ifastarorparticlebelongstowhichcomponent.Galaxycentersmayrecurrentlymovefromabarredtoanunbarredphaseandundergocontinuingbulgebuildingasthebarsdissolve12(Normanetal.1996).Thus,thepaucityofbarredgalaxiesinoursampledoesnotprecludebar-inducedeffectsasapossibleexplanationforType-IIprofiles(e.g.Gadotti&dosAnjos(2001)).Pre-existingbarsmaysimplyhavedissolved.Forexample,outof8barred-classifiedgalaxiesinoursample,6haveType-IIprofilesthuslendingsomecredencetothebar-lensscenario.Ontheotherhand,themoststronglybarredgalaxiesintheShellflow

–37–

sampleof∼300brightlate-typegalaxies(Courteauetal.2000)havemostlyType-Iprofiles,indistinguishableinshapeandglobalpropertiesfromtheprofilesofunbarredType-Igalaxies(Courteauetal.,inprep.).¿Fromaninhomogeneoussampleof167spiralgalaxies,Baggettetal.(1996)findonlyaweaktendencyforbarredgalaxiestohaveahigheroccurenceofType-IIprofiles.ThelinkbetweenType-IIprofilesandbarredgalaxiesisthusunsecuredatpresent.

Type-IIprofilesmayalsobeexplainedbyextinctioneffectsinthedisk.Increasedopacitytowardsthecentraldiskcancauseadepressionintheluminosityprofile,especiallyatshorterwavelengths.RealisticType-IIprofiles(inshapeandcolors)havebeenproducedwithexponentialdistributionsofstarsanddustandvariablelayeringparameters(Evans1994).Ifdustextinctioncausestheinnerdiskprofiledip,Transitiongalaxieswouldjustbeacaseoflesserdustcontent,whereasbonafideType-IIsystemsremainopticallythick,evenatH-band.Usingfar-infrared(FIR)toB-bandfluxratios,andradiationtransfermodelsforthedust(Gordonetal.2001),wehavetestedfortheoriginofType-IIsignatureasbeingduetoextinction.TheFIR/Bfluxratioshouldbehigherforthedustiersystems.Unfortunately,ourmeasuredtotalFIR/Bfluxratiosarestatisticallyidentical(withlargescatter)forType-I,Type-II,andTransitiongalaxies(PaperIII),thusthwartinganyclearinterpretation.TheIRAS60and100µmfluxeshavetoolowresolutionandtoolargeerrorstoseparatetheinnerdiskdustemissionfromthewholegalaxy.

Ifstellarpopulationeffectsarerelevant(Prietoetal.1992),age/metallicitygradientsshouldbedetectedatthebulge/disktransitioninTransitionsystems.Wewillfurtherinvestigatethedustand/orstellarpopulationoriginoftheType-IIdipinPaperIII.

6.SummaryandConcludingRemarks

ThisstudyhasfocusedonthedevelopmentofrigorousB/Ddecompositiontechniquesusinganew,comprehensive,multi-bandsurveyoflate-typespiralgalaxies.WeexaminethreetypesofSBprofiles,FreemanType-IandType-II,andathird“Transition”classforgalaxieswhoseprofileschangefromType-IIintheopticaltoType-Iintheinfrared.ThisdistinctionisimportantsinceType-IIandTransitionprofilescannotbeadequatelymodeledbyasimpletwo-componentmodelofthebulgeanddisk.Thus,ourmainresultsarebasedonType-Iprofiles.

Basedonextensivesimulations,carefultreatmentofskyandseeingmeasurementerrors,andrepeatobservationsweareconfidentthatsystematicerrorsare󰀂20%forthebulgecomponents,includingtheS´ersicshapeparameter,and󰀂5%fordiskcomponents.

–38–

Themainconclusionsfromoursimulationsandfinalprofiledecompositionsareasfol-lows:

•SimulationstodeterminetherangeofacceptablesolutionsforanyB/Ddecompositionprogramarecrucial.Thereliabilityofbulgemodelparametersislimitedbytherelativesizeofthebulgeandseeingdisk,seeingerrors,theintrinsicbulgeshape,skybrightnessanderrors.Diskparametersarefairlyrobusttosystematicerrors,withtheexceptionofimproperbulgeshapesandskyerrorswhichcanhavedramaticeffectsonbothmodeleddiskandbulgecomponents.

•TheS´ersicbulgeshapeparameterfornearbylate-typegalaxiesshowsarangebetweenn=0.1−2,but,onaverage,theirunderlyingsurfacebrightnessdistributionisbestdescribedbyadouble-exponentialmodelofbulgeanddisk.

•Diskscalelengthsdecreaseatlongerwavelengths,indicativeofahigherconcentrationofolderstarsand/ordustinthecentralregionsrelativetotheouterdisk.

•WeconfirmandreinforcetheresultofCourteauetal.(1996)ofastructuralcouplingbetweenthebulgeanddiskoflate-typespirals.Wefind󰀔re/h󰀕=0.22±0.09,or󰀔hbulge/hdisk󰀕=0.13±0.06,independentofwavelength.AmildtrendwithHubbletypeisobservedwith󰀔re/h󰀕=0.20−0.013(T−5)(1σ=0.09),rangingfrom󰀔re/h󰀕∼0.20forlate-typespiralsto󰀔re/h󰀕∼0.24forearliertypes.Theseresultsareconsistentwithscenariosofbulgeformationinwhichbulgesoflate-typespiralgalaxiesaremoredeeplyembeddedintheirhostdiskthanearlier-typebulges.Underthis“iceberg”scenario,bulgesanddiskscanthuspreserveanearlyconstantre/hbutshowagreatrangeofµeforanygivenre.Theobservedscaleratioisconsistentwithnumericalsimulationsofself-gravitatingdisksandprobablyrelatedtothestellardynamicsofanactualorpre-existingbarredsystem.

•TheinnerbrightnessprofilesignaturesofType-IIgalaxiesarelikelyexplainedbyacombinationofdustextinctionandstellarpopulationeffectsandperhapslinkedtotheoccurenceofabar,butnodecisiveconclusioncanbederivedatpresent.

–39–

WearegratefultoMarcBalcells,EricBell,RoelofdeJong,andDanielPfennigerfortheircommentsonearlierversionsofthismanuscript.CeciliaScannapiecoandPatriciaTisseraarealsothankedforsharingtheirmaterial(Fig.22)inadvanceofpublication.AlisterGrahamkindlyprovidedtablesofhisprofiledecompositionsofdeJong’sSBprofilesforcomparisonwithouranddeJong’ssimilarresults.Wealsowishtothanktheanonymousrefereeforsuggestionsandcommentsthathelpedimprovedthepresentationandcontentofthepaper.ThisresearchhasmadeuseoftheNASA/IPACextragalacticdatabase(NED)whichisoperatedbybytheJetPropulsionLaboratory,CaliforniaInstituteofTechnology,undercontractwiththeNationalAeronauticsandSpaceAdministration.LMandSCacknowledgefinancialsupportfromtheNationalScienceandEngineeringCouncilofCanada.

A.FunctionalformfortheS´ersicbnparameter

Eq.14cannotbesolvedinexplicitclosedformforbn.Manyofthenumericalandana-lyticalsolutionsfoundintheliteratureagreewellforn>1butdiffersignificantlyforsmaller

valuesofn.Fig.23showsacomparisonofthetwomostcommonlyusedapproximations(short-andlong-dashedcurves)withtheexactsolutionforbn,computedtoanumericalprecisionofonepartin107foralln≤10(seealsoFig.2inGraham1999).

AswewishtotestforspiralbulgeswithS´ersicn’sassmallas0.1,wehaveadoptedaformalismthatisvalidforalln.Tomaintaincomputationalsimplicity,andensureasuitablyaccuratesolutionwefounditpracticaltodividethecurveintotwosegments.Foralln>0.36weusetheasymptoticexpansionofCiotti&Bertin(1999)uptoO(n−5)(theirEq.18),

bn∼2n−

1

405n

+

46

1148175n3

−

2194697

–40–

B.

DecompositionResultsfortheTypeIProfiles

Table4givesrelevantphotometricinformationand1DB/DdecompositionresultsforthefinalsetofTypeIgalaxyprofiles.DecompositionresultsforType-IIandTransitiongalaxiesareavailablefromtheauthorsuponrequest,withthecautionthatparametersfortheseprofiletypesshouldbeinterpretedwithcare.Theentriesarearrangedasfollows:

Column(1):(UGCnumber)(observationnumber)(passband)foreachprofile;Column(2):Ellipticity,ε≡(1−b/a).Thefinalellipticity(andpositionangle)estimatescorrespondtoanaverageofthosevaluesfromthefivecontourssurroundingthebestisophotalfitintheouterdisk,asdeterminedbyeye.Thisestimateisclearlysensitivetothepresenceofspiralarms.Thetypicalinclinationerroris∼3deg,independentofellipticity;

Column(3):Skybrightnessinmagarcsec−2,measuredfrom4skyboxeslocatedbetweenthedetectoredgesandafairdistanceawayfromthegalaxy.Typicalrmsskyerrors,computedfromthedeviationsofthemeanskycountsinthoseskyboxes,are∼0.5−1.0%intheopticaland0.05%intheIR.Thesubscriptsindicatetheskyoffsetpreferredbyourselectionprocessasdescribedin§4.2.1and§4.3(andseeFig.10),where“+”and“−”indicate0.5%foropticaland0.01%forH-bandover-andunder-subtractedskiesrespectively.Nosubscriptindicatesthatthemeasuredskywaspreferred;

Column(4):SeeingFWHMvalues,computedastheaverageoftheFWHMsofallnon-saturatedstarsmeasuredautomaticallyoneachimageframe;typically10to30measure-mentsperimagewereusedforeachFWHMestimate.Theaccuracyoftheseeingestimateperimageisroughly20%fortheopticalbandsand30%fortheH-band.Thesubscriptsindicatetheseeingoffsetpreferredbyourselectionprocessasdescribedin§4.2.1and§4.3,(andseeFig.10)where“+”and“−”indicate15%over-andunder-estimatedseeingFWHMrespectively.NosubscriptindicatesthatthemeasuredseeingFWHMwaspreferred;Theupperandlowerboundariesintheremainingcolumnscorrespondtothemaximumandminimumvaluesofthe≤50(outof1080total)solutionsleftafterfiltering(see§4.3);

Column(5):BestfitS´ersicnbulgeshapeparameter;

Column(6):Bulgeeffectivesurfacebrightness,µe,inmagarcsec−2,correctedforGalacticextinctionandcosmologicalredshiftdimming,asdescribedin§2.2;

Column(7):Bulgeeffectiveradius,re,inarcseconds;

Column(8):Bulgeeffectiveradius,re,inkpc.ConvertedtoaphysicalscaleusingtheLocalStandardofRestvelocity,VLG(seePaperII);

–41–

Column(9):Exponentialdiskcentralsurfacebrightness,µ0inmagarcsec−2,correctedforGalacticextinctionandcosmologicalredshiftdimmingasdescribedin§2.2;

Column(10):Exponentialdiskscalelengthh,inarcseconds;

Column(11):Exponentialdiskscalelengthh,inkpc.ConvertedtoaphysicalscaleusingtheLocalStandardofRestvelocity,VLG(seePaperII);

Column(12):Bulge-to-diskluminosityratio,B/D,calculatedusingEq.17in§3.1.

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Fig.1.—ExamplesofType-I(left),Type-II(middle),and“Transition”(right)SBprofiles.Bluecircles,greensquares,redtriangles,andpurpleasterisksareforB,V,R,andH-bandrespectively.ThesolidblacklinesplottedontheB-bandandH-bandprofilesarefitstotheouterexponentialdiskprofile.

–48–

Fig.2.—S´ersicnprofilesfordifferentvaluesofn.Thetoppanelshowsprofileswithµe=21magarcsec−2andre=3′′.5forvaluesofnintherange0.2–49–

Fig.3.—EffectoffittinganincorrectS´ersicnbulgeonthediskscalelengthh.Eachpanelplotstheaveragerelativefittedherrors(∆h≡(hfit(mean)-hmodel)/hmodel)withsolidsymbolsandconnectedbysolidlinesasafunctionofthemodelnforabulgewithre=2′′.5andµe=20magarcsec−2.Redcircles,greentriangles,andbluesquarescorrespondtoseeingvaluesof1.5,2.0,and2′′.5respectively.Thethreepanelsareformodelnvaluesof0.2,1.0,and4.0fromlefttoright.

–50–

Fig.4.—Effectonthefittedrevalueofanincorrectseeingvalueinthe1Ddecomposition.Thecolumnplotsarebasedondifferentvaluesforthefractionalseeingerrorusedinthefit,where∆FWHM≡(FWHMused−FWHMmodel)/FWHMmodel.EachrowisforadifferentvalueofthemodelFWHM,1′′.5(top)and2′′.5(bottom).Eachpanelshowstheaveragerelativeerroronre,∆re(Eq.21),versusthemodelre.Thesevencurvesarefordifferentvaluesofµe:16(darkpurple),17(blue),18(red),19(green),20(magenta),21(cyan),and22(orange)magarcsec−2.

–51–

Fig.5.—Effectonthefittedrevalueofanincorrectseeingvalueinthe2Ddecomposition(comparewithFig.4.)Thesolidsymbolsconnectedbysolidlinesindicatetheaverage(ofthe40imagedecompositionsforeachparameterandinitialestimatecombination)relativeerroronre,∆re(Eq.21).Bluecirclesandredtrianglesareforµevaluesof18and22magarcsec−2respectively.∆FWHMisasdefinedinFig.4

–52–

Fig.6.—Differencebetweenmodeledandrecoveredvaluesofnforarangeofartificialprofilesfromn=0.2−4.TheS´ersicexponentnisafreefitparameterandtheinitialestimateissetton=1.Eachpanelshowstheaveragerelativefittednerrors(∆n≡(nfit(mean)-nmodel)/nmodel)withsolidsymbolsandconnectedbysolidlinesversusthemodelnforthe9combinationsofre=0.8,1.5,2′′.5,andµe=18,20,22magarcsec−2.Redcircles,greentriangles,andbluesquarescorrespondtoseeingvaluesof1.5,2.0,and2′′.5respectively.ThepanelsareorderedsuchthattheB/Dratio,foragivennvalue,decreasesfromtoptobottomandrighttoleftpanels.

–53–

2

Fig.7.—Examplesofχ2inner′(openbluetriangles)andχglobal′(filledredsquares)versusS´ersicndistributionsforthe1080decompositionsoftwodifferentV-bandobservationsofthesamegalaxy(UGC929).Inthetwosetsofplots(a)andb)),theleftpaneldisplaysall1080pointsandtherightpanelshowsonlythe(≤50)pointsremainingafteriterativefiltering.Seta)showsareasonablywell-behavedsolutionfavoringn=0.6whilesetb)showsarathernoisysolutionfavoringn=0.8.

–54–

2

Fig.8.—Examplesofχ2inner′andχglobal′distributionsforasolutionwithawell-behaved

2

χ2inner′,butaflatχglobal′distribution(UGC784B-band),plota),andforaverywell-behavedsolutioninbothχ2distributions(UGC929B-band),plotb).

–55–

Fig.9.—Comparisonofdifferentbulgefitsforthesameprofile(UGC784B-band).Theplotontherighthasabulgefit(dashed-dottedblueline)whichislikelyunphysical.Itsχ2gl,however,islowerthanthatofthedecompositionontheleftplot,whosebulgefitlooksmore

2

realistic.Withoutadoptingtheχ2instatistic,theplotontherightisfavored.Usingtheχininadditiontotheχ2glasadiscriminator,theplotontheleftisfavored.(SeeleftplotofFig.8forthecorrespondingχ2vs.ndistributions.)Symbols,colorsandline-typesareasdefinedinFig.11.

–56–

Fig.10.—HistogramsofskyandseeingFWHMoffsetspreferredinouranalysisforallprofilessurvivingthefinalcut,separatedintothefourdifferentbands.NotethattheH-bandskyerrorismorethananorderofmagnitudesmallerthanintheoptical(asintheactualmeasurements).

–57–

Fig.11.—DecompositionresultsforaTypeIgalaxy(UGC9908).Intheupperpanelsofeachplot,thedatapointsandmeasuredskyerrorenvelopesareshownwithsolidblackcirclesanddotsrespectively.Thebluedashed-dottedandgreendashedlinesshowthebulgeanddiskfitsrespectively,andthesolidredlineisthetotal(bulge+disk)fit.Thefitsareallseeing-convolvedusingthebestselectedseeingvalues.Thebottompanelsshowthefitresidualswhere∆µ(r)≡data(r)−model(r).

–58–

Fig.12.—DecompositionresultsforaType-II/Transitiongalaxy(UGC12527).

–59–

Fig.13.—Decompositionresultsforagalaxywithatruncateddisk(UGC927).Notethatskyerrorscouldnotaccountforthetruncation.

–60–

Fig.14.—Decompositionresultsforagalaxywitha“bulgeless”disk(UGC10757).

–61–

Fig.15.—HistogramsofS´ersicnparameterfor“final”solutions(left),andthereducedsetofsolutionsafterfurthervisualexamination(right).Seetextfordetails.

–62–

Fig.16.—HistogramsofS´ersicnparameterfittingnasafreeparameterinthedecompo-sitions.Resultsusingthreedifferentvaluesfortheinitialestimateofnareshown:n=0.2(left),n=1.0(middle),n=4.0(right).Notethedifferenty-axisscalesineachoftheplots.Theselectioncriteriaforthefitsisasdescribedinthetext.

–63–

Fig.17.—χ2comparisonoffloatednversusfixednsolutions.Thepointtypesandcolorsareasfollows:B-band(triangles),V-band(squares),R-band(pentagons),H-band(asterisks),Type-I(blue),Type-II(red),andTransition(green).Notethedifferentaxisscalesforχ2in.

–64–

Fig.18.—S´ersicnversusmorphologicaltypeindex.Bluecircles,redtriangles,andgreensquaresindicateType-I,Type-II,andTransitiongalaxiesrespectively.

–65–

Fig.19.—Bulgeanddiskparametersversusellipticity(1−b/a).Thepointtypesareasfollows:Type-I(bluetriangles),Type-II(redsquares),Transition(greencircles).

–66–

λ

Fig.20.—reversushλforourcurrentType-Idata(solidsymbols)andthedecompositionsofGraham(2001)ofdeJong&vanderKruit(1994)’sdata(opensymbols).BluecirclesareB-band,greenpentagons(ourdataonly)areV-band,redtrianglesareR-band,andmagentasquaresareH-band(us)andK-band(Graham(2001)).Thedashedlineshavea

λ

slope󰀔re/h󰀕=0.22forlate-typespirals.Notethatthelargedispersionsinthereandhλ(Table3)counteracttoyieldsignificantre/hcorrelations.Theleftplotisinapparentunits(arcsec)andtherightplotshowsthephysicalscaleinkpc.Thediscretenatureofourdata

λ

intherightplotisduetothelimitedprecisionoftheremeasurement(onedecimal).

–67–

Fig.21.—Distributionofre/hwithHubbletypesforourType-IgalaxiesandthoseofGraham(2001).SymbolsandcolorsareasinFig.20.Thedashedlinedescribesthefit󰀔re/h󰀕=0.20−0.013(T−5)with1σ=0.09errors(dottedlines)basedonourdataonly.

–68–

543210

012Sersic n

34

Fig.22.—DistributionoftheS´ersicnparameterfromcosmologicalsimulationsbyScanna-pieco&Tissera(inprep.).

–69–

Fig.23.—Differencebetweentheexactnumericalvalueforbnandseveralcommonlyadoptedapproximations.Theshort(red)andlong(green)dashedlinesarethetwomostcommonlyusedapproximationsfoundintheliterature.ThesolidbluelineshowsCiotti&Bertin’sasympoticexpansionandthedottedpurplelinedepictsouradoptedextensionatn≤0.36.

–70–

Table1.Tableofmeanvaluesandmeanrmsdeviationsforrepeatobservations.

n

µe

0.130.070.160.14

21.8820.1620.0217.52

0.190.270.170.26

re1.130.810.770.85

0.080.110.110.14

µ020.9019.9619.5917.33

0.030.030.040.10

h

N

1.000.980.880.95

TotalII

BRH

5.354.063.763.490.180.080.070.17

32242

0.660.200.650.140.650.111.180.40

Total

20.1221.4120.1918.14

0.260.270.330.41

0.670.780.710.60

0.120.110.370.09

19.6220.6919.6017.27

0.050.050.030.04

3.735.004.512.58

0.140.140.060.17

10

–71–

Table2.ComparisonofB/Ddecompositionparameters.

UGC

band

µ0

h

µe

re

n

463

deJong

463463

usGraham

3080

usdeJong

30803140

deJong

3140

usdeJong

3140

BBH

20.7720.5916.80

13.014.212.0

20.5120.1516.73

1.61.11.9

0.40.41.0

B21.9917.219.880.21.0

HBB

18.2821.0620.52

18.213.712.9

19.2122.1620.75

2.74.82.8

0.92.91.1

H17.2212.017.284.11.9

–72–

Table3.Meanandstandarddeviation,µ(σ),forbulgeanddiskparametersfordifferenttypesandbandpasses.Ndenotesthenumberofdatapoints.Multipleobservationscount

asindependententries.

Param

Type

B

V

R

H

–73–

Table4.DecompositionResultsforTypeIProfiles

FWHM

UGCobsband(1-b/a)(mag/⊔⊓)(′′)

′′

Profile(1)

εSky(3)

n(5)

µe(mag/⊔⊓

(6)

′′

)

(2)(4)

re(′′)(7)re(kpc)(8)

µ0(mag/⊔⊓

(9)

′′

)

h(′′)(10)h(kpc)(11)

B/D(12)

–74–Table4—Continued

FWHM

UGCobsband(1-b/a)(mag/⊔⊓)(′′)

′′

Profile(1)

εSky(3)

n(5)

µe(mag/⊔⊓

(6)

′′

)

(2)(4)

re(′′)(7)re(kpc)(8)

µ0(mag/⊔⊓

(9)

′′

)

h(′′)(10)h(kpc)(11)

B/D(12)

–75–Table4—Continued

FWHM

UGCobsband(1-b/a)(mag/⊔⊓)(′′)

′′

Profile(1)

εSky(3)

n(5)

µe(mag/⊔⊓

(6)

′′

)

(2)(4)

re(′′)(7)re(kpc)(8)

µ0(mag/⊔⊓

(9)

′′

h

)

(′′)(10)

h(kpc)(11)

B/D(12)

–76–Table4—Continued

FWHM

UGCobsband(1-b/a)(mag/⊔⊓)(′′)

′′

Profile(1)

εSky(3)

n(5)

µe(mag/⊔⊓

(6)

′′

)

(2)(4)

re(′′)(7)re(kpc)(8)

µ0(mag/⊔⊓

(9)

′′

h

)

(′′)(10)

h(kpc)(11)

B/D(12)

–77–

Table4—Continued

FWHM

UGCobsband(1-b/a)(mag/⊔⊓)(′′)

′′

Profile(1)

εSkyn(5)

µe(mag/⊔⊓

(6)

′′

)

(2)(3)(4)

re(′′)(7)re(kpc)(8)

µ0(mag/⊔⊓

(9)

′′

h

)

(′′)(10)

h(kpc)(11)

B/D(12)