The Opacity of Spiral Galaxy Disks VIII Structure of the Cold ISM

TheOpacityofSpiralGalaxyDisksVIII:

StructureoftheColdISM

B.W.Holwerda1,B.Draine2,K.D.Gordon3,R.A.Gonz´alez4,D.Calzetti5,M.Thornley

6

,B.Buckalew7,RonaldJ.Allen1andP.C.vanderKruit8

holwerda@stsci.eduABSTRACT

Thequantityofdustinaspiraldiskcanbeestimatedusingthedust’stypicalemissionortheextinctionofaknownsource.Inthispaper,wecomparetwotechniques,onebasedonemissionandoneonabsorption,appliedonsectionsoffourteendiskgalaxies.Thetwomeasurementsreflect,respectivelytheaverageandapparentopticaldepthofadisksection.Hence,theydependdifferentlyontheaveragenumberandopticaldepthofISMstructuresinthedisk.

ThesmallscalegeometryofthecoldISMiscriticalforaccuratemodelsoftheoverallenergybudgetofspiraldisks.ISMgeometry,relativecontributionsofdifferentstellarpopulationsanddustemissivityareallfreeparametersingalaxySpectralEnergyDistribution(SED)models;theyarealsosometimesdegenerate,dependingonwavelengthcoverage.OuraimistoconstraintypicalISMgeometry.TheapparentopticaldepthmeasurementcomesfromthenumberofdistantgalaxiesseeninHSTimagesthroughtheforegrounddisk,calibratedwiththe

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“SyntheticFieldMethod”(SFM).WediscusswhatcanbelearnedfromtheSFMmeasurementaloneregardingISMgeometry.

WemeasuretheIRfluxinimagesfromtheSpitzerInfraredNearbyGalaxySurveyinthesamesectionofthediskthatwascoveredbyHST.AphysicalmodelofthedustisfittotheSEDtoestimatethedustsurfacedensity,meantemper-ature,andbrightnessinthesedisksections.Thesurfacedensityissubsequentlyconvertedintotheaverageopticaldepthestimate.

ThetwomeasurementsgenerallyagreeandtheSEDinordermodelfindsamostlycolddust(T<25K.).Theratiosbetweenthemeasuredaverageandapparentopticaldepthsofthedisksectionsimplyopticallythin(τc=0.4)cloudsinthesedisks.Opticallythickdisks,arelikelytohavemorethanasinglecloudalongtheline-of-sight.

Subjectheadings:(ISM:)dust,extinction,ISM:structure,galaxies:ISM,galax-ies:spiral,infrared:galaxies,infrared:ISM

1.Introduction

Thedustcontentofaspiralgalaxydiskcanbemappedeitherbythecharacteristicdustemissioninthefar-infrared(FIR)andsub-mmregimes,orbyusingtheattenuationofknownbackgroundsources.Bothtechniqueshaveseenrecentsignificantimprovementsinaccuracyandsensitivity,withcomplementaryresultssheddinglightonthedustyinterstellarmediuminspiraldisks.

Theemissionfromtheinterstellardustinthedisksofspiralgalaxieshasbeencharac-terizedwithincreasingaccuracybyseveralinfraredspacemissions(IRAS,ISOand,recently,Spitzer),aswellasthesub-mmobservationsofSCUBAontheJCMT.Theimprovementsinspatialresolutionandwavelengthcoveragehaveledtosignificantinsightintothetem-peraturecomponentsofthedustinspiraldisks,andintotherelationbetweendustycloudsandstar-formation.TheFIRemissionfromspiralgalaxieshasrevealedthatthedustcanbedescribedbytwodominantthermalcomponents:warm(25KFIRandsub-mmobservationsofgalaxiesfindindications(e.g.,Trewhellaetal.2000;Altonetal.1998)ordirectevidenceofcolddustdisksextendingbeyondthestellardisk

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(Nelsonetal.1998;Daviesetal.1999;Popescu&Tuffs2003).Thestudiesofedge-onspi-ralsbyRadovichetal.(2001)andXilourisetal.(1999)quantifiedtheradialprofileasascalelengthofthedustthatis40%largerthanthatofthestarlight.ThecontributionofthecoldISM(T<25K)totheoverallemissionofspiraldiskshasbeendifficulttoconstrainbe-causeofthedegeneracybetweendusttemperatureandmass.Hence,thecoldISM’srelationwithHIandtheirrelativedistributionremainunknown.

Spitzerobservations,mostlyfromtheSpitzerInfraredNearbyGalaxySurvey(SINGS,Kennicuttetal.2003),havealreadycontributedgreatlytotheunderstandingofspiraldisks.Therelationsbetweenthetracersofcolddust(70and160micronemission)andstar-formation,bothobscured(24micron)andunobscured(UVandHαemission)havealreadybeenstudiedwiththismulti-wavelengthsurveyinseveralcanonicalgalaxiesandtheirsubstructure:thestarburstM51(Calzettietal.2005;Thornleyetal.2006),thegranddesignspiralM81(Gordonetal.2004;P´erez-Gonz´alezetal.2006),theringsofNGC7331(Reganetal.2004)andM31(Gordonetal.2006),thesuperwindinM82(Engelbrachtetal.2006),andthedwarfNGC55(Engelbrachtetal.2004).Daleetal.(2005,2007)discusstheSEDofallSINGSgalaxiesoverallavailablewavelengths.Draineetal.(2007)findampleev-idencefordustinalltheSINGSgalaxies,withthegas-to-dustratiorelatedtothemetallicity.Theyfindnoevidenceforverycold(T<10K)dust,however.

Thesestudiesfindampleevidenceofcolddustthroughouttheopticaldisksofspi-ralsbut,interestingly,alsooutsidetheminvariousplaces:ontheedgeoftheopticaldisk(Thornleyetal.2006;Gordonetal.2006),outsideM82’ssuperwind(Engelbrachtetal.2006)andextendingbeyondthestellardisk(Hinzetal.2006).

Paralleltotheseinvestigationsofdustemissionhasgoneanobservationalefforttoquan-tifytheabsorptionbydustinspiraldisksusingknownbackgroundsources.White&Keel(1992)proposedusingoccultinggalaxypairsforthispurpose.Nearbyoccultinggalaxypairswereinitiallyinvestigatedwithground-baseddata,bothimages(Andredakis&vanderKruit1992;Berlindetal.1997;Domingueetal.1999;Whiteetal.2000)andspectra(Domingueetal.2000).Subsequently,withtheHubbleSpaceTelescope(HST),amoredetailedpictureofdustinthesenearbydisksemerged(Keel&White2001a,b;Elmegreenetal.2001).Theresultsofthesestudiesarethatextinctionisgray1whenmeasuredoverdisksectionsgreaterthan100pc,butresemblestheGalacticextinctionlawatsmallerscales–thosethatcanonlyberesolvedwithHST.Armsarefoundtobemoreopaquethanthegeneraldisk,and

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someevidencesuggeststhatthedustdiskisafractal,similartotheHIdisk.

Gonz´alezetal.(1998)investigatedtheuseofthecalibratednumberofdistantgalaxiesseenthoughtheforegrounddiskinHSTimages.Thecalibratedcountsofdistantgalaxieshavebeenexploredfurtherinthepreviouspapersinthisseries(Gonz´alezetal.1998,2003;Holwerdaetal.2005a,b,c,d,e,2006).Boththeoccultinggalaxytechniqueandcountsofdistantobjectsyieldverysimilaropacitiesfordisksandspiralarms(Holwerdaetal.2005b).Inrecentyears,modelshavebeendevelopedtoexplaintheSpectralEnergyDistri-bution(SED)ofedge-onspiralsspanningwavelengthsrangingfromtheUVtotheFIR.(e.g.,Popescuetal.2000;Misiriotisetal.2001;Popescuetal.2000;Misiriotisetal.2001;Tuffsetal.2004;Boissieretal.2004;Dasyraetal.2005;Tuffsetal.2004;Dasyraetal.2005;Calzettietal.2005;Dopitaetal.2006b,a;Draine&Li2007;Draineetal.2007).Threescenarioshavebeenproposedtoexplainthediscrepancybetweentheapparentab-sorptioninUVandopticalwavelengthsandtheemissionofdustintheFIRandsub-mmregimes:

1.Ayoungstellarpopulationisembeddedinthedenseplaneofthedisk.ThisisproposedbyPopescuetal.(2000)andcorroboratedbyDriveretal.(2007).TheembeddedyoungstarspumptheFIRemissionradiatedbythedustplane.2.Astronglyclumpeddustymedium.Theclumpingwouldleadtounderestimatethedustmassfromopticalextinctioninedge-onsystems(Bianchietal.2000b;Witt&Gordon2000;Misiriotis&Bianchi2002);thedustmasswouldalsobeunderestimatedbyaUVtoFIRSED(Bianchietal.2000a).3.Adifferentemissivityofthecolddustgrains–higherthancanonical–intheFIRandsub-mm.AchangeofemissivityhasbeenproposedfordenserISMregions(Altonetal.2004;Dasyraetal.2005)or,alternatively,forthelowerdensityregionsofthedisk(Bendoetal.2006).Inallthreeofthesescenarios,theclumpinessofthedustyISMisanimportantfactor.WhileinsomemodelsthelargescalestructureofthedustyISMhasbeensomewhatconstrained(e.g.,Xilourisetal.1999;Sethetal.2005;Bianchi2007;Kamphuisetal.2007),thesmall-scalegeometry(“clumpiness”)ofthecoldISMremainsunknown.Therefore,anestimateoftheprevalentdustycloudsizeforspiraldiskswouldprovideaconstraintfortheSEDmodelsofspiraldisks.

GiventhatSEDandextinctiontechniquesaresensitivetodifferentcharacteristicsofthedustycloudsinthespiraldisk,acomparisonbetweentheopticaldepthderivedfrom

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thesetwomethodshasthepotentialtofigureoutthestructureofthedustyISM.Here,wecomparetheI-bandopticaldepthsforasectionofthespiraldisk,onederivedfromaSEDmodeloftheSpitzerfluxes(“average”),andonedeterminedfromthenumberofdistantgalaxiesfoundinanHSTimage(“apparent”).Theterm“average”referstowhattheopticaldepthproportionaltothedustmass,uniformlydistributedoverthedisksection.Theterm“apparent”meanstheeffectiveopticaldepthofthedisksectionforabackgrounduniformlightsource(originaldefinitionsfromNatta&Panagia1984).Theterm“opacity”isusedthroughoutourpreviouspapersfortheapparentopticaldepthmeasuredoverasectionofthediskforitswholeheight.

Thispaperisstructuredasfollows:§2discussestheSpitzerandHSTdataused.In§3,thetwodifferentmethodstoderiveopticaldepthsarepresented.Wediscusstherelationbetweendustgeometryandgalaxycountsin§4.In§5,wepresentthederivedopticaldepths;§6presentsasimplegeometricmodeltointerprettheresults,and§7listsourconclusionsandfuturework.

2.Data

Thedataforthispapercomefromtwoarchives,theHSTarchiveandthefourthdatarelease(SINGSteam2006)oftheSpitzerInfraredNearbyGalaxySurvey(SINGS,Kennicuttetal.2003)2.ThereisanoverlapoffourteengalaxiesbetweentheSINGSsampleandthatofHolwerdaetal.(2005b).Twoofthese,NGC3621andNGC5194,havetwoWFPC2exposuresanalyzedinHolwerdaetal.(2005b).TheHST/WFPC2datareductionisdescribedinHolwerdaetal.(2005a).ThereasoningbehindtheHSTsampleselectionfromthearchiveisexplainedin§3.1.

TheoverlapbetweenHSTandSpitzerdataisillustratedinFigure1,withtheWFPC2footprintprojectedonthe24micronSpitzerimages.OnlythesolidanglecoveredbytheWFchipsisusedforfurtheranalysis(thePCchipisexcluded).

TheInfraredArrayCamera(IRAC)mosaicismadewiththecustomSINGSditherscript,byM.Regan,thatcombinesthescanimagesintoasinglemosaicusingthe“drizzle”algo-rithm(FourthDataReleaseNotes,SINGSteam2006).TheMultibandImagingPhotometerforSpitzer(MIPS)dataproductsarecalibrated,sky–subtractedmosaicsinallthreebands,reducedasdescribedinGordonetal.(2005);Bendoetal.(2006);SINGSteam(2006).Thebasicinstrumentparameters,pixelscale,andadoptedPSFFWHMforthesevenmainSpitzer

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imagingmodesaresummarizedinTable1.

3.Analysis

Twoparallelestimatesoftheopticaldepthofdisksareusedinthispaper:first,theapparentopticaldepthofthespiraldisksisdeterminedfromthenumberofdistantgalaxiesidentifiedintheHST/WFPC2images,calibratedwiththe“SyntheticFieldMethod”(SFM).Secondly,theopticaldepthofthesamesectionofthespiraldisksisderivedfromthedustsurfacedensity,whichisaresultoftheSEDmodelfittotheSpitzerfluxesusingthemodelfromDraine&Li(2007).

3.1.Galaxycounts:“SyntheticFieldMethod”

Inprinciple,thenumberofdistantgalaxiesseenthroughaspiraldiskisafunctionofthedustextinction,aswellasthecrowdingandconfusionbytheforegrounddisk.InitialapplicationsofthenumberofdistantgalaxiesasanextinctiontracerwereontheMagellanicClouds(Shapley1951;Wesselink1961;Hodge1974;MacGillivray1975),buttheylackedaccuracy.The“SyntheticFieldMethod”(SFM)wasdevelopedbyGonz´alezetal.(1998)tocorrectanextinctionmeasurementbasedonthenumberofdistantgalaxiesinanHSTimage,fortheeffectsofcrowdingandconfusionbytheforegroundspiraldisk.

The“SyntheticFieldMethod”followsaseriesofsteps.First,thenumberofdistantgalaxiesinanHSTsciencefieldisdetermined.Second,aseriesofsimulated(“synthetic”)fieldsaremade.Ineachofthesefields,atypicalbackground(e.g.,theHubbleDeepField)isfirstdimmedbyagrayscreenandaddedtothesciencefield.Third,theaddeddistantobjectsareidentifiedinthesesyntheticfields.Thefourthstepistomeasuretherelationshipbetweenthenumberoftheseidentifiedsyntheticdistantgalaxiesandbackgrounddimming.Fromthisrelationandtheoriginalnumberofactualdistantgalaxiesfoundinthesciencefield,anaverageopacitycanbefound.Itisimportanttoremakethesyntheticfieldsforeachsciencefieldbecausethecrowdingandconfusionissuesareuniqueineachcase.

Anadditionaluncertaintyintheresultingaverageextinctionmeasurementisthecos-micvarianceintheintrinsicnumberofdistantgalaxiesbehindtheforegrounddisk.Theuncertaintyduetocosmicvariancecanbeestimatedfromthetwo-pointcorrelationfunctionofdistantgalaxiesandfoldedintothePoissonianerror.ThecosmicvarianceuncertaintyisofthesameorderasthePoissonstatisticalerrorforsmallnumbers(Foracompletedis-cussionontheuncertaintiesoftheSFM,seeHolwerdaetal.2005a).Therefore,single-field

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SFMmeasurementsremainuncertain,butameaningfulconclusioncanbedrawnfromacombinedsetofsciencefields.

WehaveappliedthismethodsuccessfullyonarchivalWFPC2data.Holwerdaetal.(2005b)presenttheaverageradialopacityprofileofspiraldisksandtheeffectofspiralarms.Thespiralarmsaremoreopaqueandshowastrongradialdependence,whilethemoretransparentdiskshowsaflatprofile.Holwerdaetal.(2005c)compareHIradialprofilestotheopacityones,andconcludethatnogoodrelationbetweendiskopacityandHIsurfacedensityradialprofilescanbefound.However,Holwerdaetal.(2005c)findthatthesub–mmprofilefromMeijerinketal.(2005)generallyagreeswiththeiropacitymeasurementsofM51.Holwerdaetal.(2005e)comparetherelationbetweensurfacebrightnessanddiskopacity;thisrelationisstronginthespiralarms,butweakintherestofthedisk.

Gonz´alezetal.(2003)predicted,basedonsimulateddata,thattheoptimumdistancefortheapplicationoftheSFMwithcurrentHSTinstrumentsisapproximatelythatofVirgo.Theidentificationofbackgroundgalaxiessuffersincloserdisks,asthestellardiskbecomesmoreresolved,compoundingconfusion.Thisoptimumdistance,combinedwiththeavailabilityofdeepHST/WFPC2imagesfromtheCepheidExtragalacticDistanceScaleKeyProject,resultedinthesamplepresentedinHolwerdaetal.(2005b).Holwerdaetal.(2005d)confirmedtheresultsfromGonz´alezetal.(2003)usingthissample,withforegrounddisksspanningdistancesbetween3.5and35Mpc.AselectioneffectoftheKeyProjectisthatthemajorityoftheHSTsciencefieldsisconcentratedonspiralarmsandexcludethecentersofthegalaxies.

Here,wepresentaverageextinctionvaluesforthewholeWFPC2field–of–view,mini-mizingtheuncertaintiestotheextentpossible.Inourinitialpapers,wedidnotapplyaninclinationcorrectiontotheopticaldepthsbecausethecorrectiondependsstronglyonthedustgeometry(seethediscussioninHolwerdaetal.2005b).However,inthispaperweassumeasimpledustmodelin§6.Theappropriateinclinationcorrection–×cos(i)–hasbeenappliedtothepointsinFigures4and5,andinTable4.Theuncertaintiesinthetablesandfiguresreflectthe1-sigmaconfidencelevelsproducedbythecombinationofthePoissonerrorandthecosmicvarianceofbackgroundgalaxies.

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3.2.

SEDopticaldepthestimate

TheaveragediskopticaldepthisderivedfromtheSpitzerobservations.First,thesurfacebrightnesseswithintheWFchips’footprintaremeasured3(seeFigure1).Second,theseareconvertedintoadustsurfacedensityusinganSEDmodel.Third,thissurfacedensityistranslatedintoanI-bandopticaldepth.

AlltheIRACandMIPSdataareconvolvedtothepoorestresolutionofthe160micronobservations(seeTable1),andthepixelscaleissetto9”.Thisisdonewiththegauss,wcsmapandgeotrantasks,underIRAF.Subsequently,theoverallfluxismeasuredintheWFPC2fieldofview(Figure1).BecausetheL-shapedapertureisahighlyunusualone,theaperturecorrectionremainsuncertain,butnotnegligible,sincetheFWHMat160micron

(40′′.0)isoftheorderoftheaperturediameter(3×1′.3×1′

.3inanL-shape).PublishedaperturecorrectionsfortheIRACinstruments(Horaetal.2004)overestimatethecorrectionforextendedobjects(Jarrett2005).HereweusetheaperturecorrectionsforextendedsourcesfromJarrett(2005)forIRACfluxes,4andfromMuzerolleetal.(2005)fortheMIPSfluxes5(seealsoTable1).Table2givestheaveragesurfacebrightnessesforthesevenSpitzerchannelsinthefield-of-viewofthethreeWFchipsoftheWFPC2array.Theuncertaintiesarederivedfromthevarianceinthesky.Generally,thesurfacebrightnessesagreewiththeresultspresentedbyDaleetal.(2005)fortheentiredisks.

Thesecondstepistoconvertthesesurfacebrightnessestoadustsurfacedensity.Ini-tially,wefittedonlytheMIPSfluxeswithtwoblackbodiesandderivedsurfacedensitiesfromthese(Holwerdaetal.2006).However,amorerigoroustreatmentoftheIRfluxescanbedonewithaSEDmodel,suchastheonepresentedinLi&Draine(2001).Thismodelusesthephysicsofgrainheatingandreradiation,andamodeldistributionofgrainsizesandtypes.TheupdatedversionfromDraine&Li(2007)hasbeenfittothedata,andtheresultsareshowninFigure2.Thederivedstellaranddustsurfacebrightnesses,dustsurfacedensity,andmeantemperaturearepresentedinTable3.Dustsurfacedensitiesarebetween0.1and1.4×106M⊙kpc−2,withmeantemperaturesbetween14.6and17.8K.

Themeandusttemperaturesareobtainedfromthemeanradiationscaling,U

¯,6intheDraine&Li(2007)model(T2intheirequation18).Themodelusesadistribution

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oftemperaturesandgrainsizes.Thereforethemeantemperatureisanindicationofthethermalequilibriumpointofthebulkofthedust.Mostofthedustiscold(T<25K).Theseresultsareanobviousimprovementoverasimplesingle-temperaturefit,butthemodelparametersinTable3arestillnotfullyconstrained;disagreementbetweendataandmodelatthePAHpeakat8microncouldbeaneffectofmetallicityorthepresenceofabrightHIIregion.The70and160µmfluxeshintatcolderormoredustinthedisk(Figure2).Therearethreecaveatstothefits:(1)alackofsub–mmdata,(2)thesinglecolorISRFusedand(3)averagingoverdifferenttypesofemissionregions,i.e.,HIIregionsandthegeneraldisk.

Additionalsub-mmdataofcomparablequality,neededtobetterconstrainthemodelandespeciallythecolddustemission,willnotbeavailableuntilSCUBA2startsoperationsontheJCMTandthelaunchofHerschel.7Atpresent,comparable-qualitysub-mmmapsareavailableonlyforNGC5194(Meijerinketal.2005)andNGC7331(Reganetal.2004).Draineetal.(2007)discussSEDmodelswithandwithoutsub-mmdata.

Thesecondcaveatinthederivationofdustmassistheassumptionofaconstantcolorfortheinterstellarradiationfield(ISRF)illuminatingtheemittingdust.Inreality,dustgrainsdeeperinaduststructurewillencounterradiationfieldthatisnotonlydimmedbutalsoreddenedand,hence,willcontributelessfluxtotheFIRemission.Locally,theISRFwillalsodependontheageofthenearbystellarpopulation.DuetothereddeningoftheISRFdeeperinthecloud,densecloudscouldcontainmoredustmassintheircentersthaninferredfromjusttheFIRemission.Additionalsub-mmobservationswillhelpresolvethisuncertaintyintheSEDopticaldepth.Draineetal.(2007)discussfitsofthemodeltothetotalfluxesoftheSINGSgalaxieswithandwithoutadditionalsub-mmdata.TheyfindthattheFIRestimateunderestimatesof1.5timesthedustmassmore(5outof17cases,notablyNGC3627and7331ofoursample)thanoverestimatesit(onlyM51ofthe17).

ThethirdcaveatisthatthemodelvaluesareanaverageovermanydifferenttypesofISMregions,eachwithadifferentheatingmechanism,duststructureandcomposition(e.g.,photo-dissociationregions,cirrus,andstar-formingregionsinspiralarms).TheDraine&Li(2007)model’sassumptionsholdbetterforsomeregionsthanforothers,butweusetheresultsas“typical”forthesedisks.TherelativecontributionofPAHemissiontotheSEDisafunctionofISMgeometryaswellasirradiationandcomposition(e.g.,Silvaetal.1998;Piovanetal.2006).Togetherwithabetter-constrainedFIR/sub-mmSEDonecouldconstrainISMgeometrysolelyfromtherelativecontributionstotheSED.Draineetal.

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(2007)discusstheapplicationoftheDraine&Li(2007)modeltowholedisksoftheSINGSgalaxies.

Thedustsurfacedensityistranslatedintoanaverageopticaldepthwiththeabsorptioncrosssectionperunitdustmass,κabs(λ),andgrainalbedofromDraine(2003)fortheJohnsonI-band(865.5˚A.):τm=κabs×(1−Albedo)×Mdust/area.TheseopticaldepthvaluesarepresentedinTable4.

4.CloudsizeandtheSFM

Itcontinuestobedifficulttoconstraindustycloudgeometryfrommodelsofeitherextinctionoremission.Inthissection,wereviewwhatcanbelearned,solelyfromtheSFMmeasurements,aboutthegeometryoftheextinctingmedium.

InHolwerdaetal.(2005b),twoindicationsthatthedustdiskisclumpyareidentified:(1)theaveragecolorofthedistantgalaxiesisindependentofthediskopacityimpliedbytheirnumber,and(2)themeasurementofdiskopacityisindependentofinclination.Thelackofarelationbetweentheaverageopacityofthediskandtheaveragecolorofthedetectedgalaxiescanbeexplainedbytwoscenarios:(1)someofthebackgroundgalaxiesarecompletelyblockedbylargeclouds,andsomearenot.Thecolormeasurementisdoneonbackgroundobjectsthatdonotsufferfromextinctionand,hence,reddening.Alternatively,(2)allbackgroundgalaxiesaredimmedbycloudssmallerthantheprojecteddistantgalaxies.Consequently,someofthedistantgalaxiesaredimmedenoughtodropbelowthedetectionthreshold.Anydetectedgalaxy’scoloris,however,measuredfrommostlyunreddenedflux.8However,wenotethatarelationbetweenthereddeningandderivedextinctionfromthedistantgalaxiesisdifficulttodetectbecause(1)thespreadincolorsofdistantgalaxiesissubstantialand(2)colorismeasuredfromthedetectedgalaxies–automaticallytheleastdimmed–whereasopacityismeasuredformthemissinggalaxies.

Theinclinationeffectontheapparentopticaldepthfromthenumberofdistantgalaxiesisminimal(seeHolwerdaetal.2005b,,§5.1andFigure3).Assumingathinlayerofopticallythickclouds,theapparentopticaldepthofthedisk,measuredfromthenumberofdistantgalaxies,isdominatedbytheapparentfillingfactorofclouds.Theprojectedfillingfactor

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doesnotchangemuchwithinclination:aflatcloudcovering40%ofacertaindisksectionstillcovers40%oftheinclinedsection.Onlywhentheheightofthecloudbecomesimportant–whentheinclinationisclosertoedge-on–,doestheapparentfillingfactorchange.Thisexplanationforthelackofaninclinationeffectintheopacityprofilesdoesnotdependonthesizeoftheclouds.Itcouldbeasingle,large,cloudormanysmallonesintheplaneofthedisk.However,theopticaldepthvaluesinHolwerdaetal.(2005b)arefromdifferentsectionsofthedisks–althoughgenerallycenteredonaspiralarm–andtheeffectofsmallinclinationdifferencescouldwellhavebeenmaskedbycomparingdifferentregionsinthedisks.InHolwerdaetal.(2005b)wedidnotapplyaninclinationcorrectionbecauseitdependsontheassumeddustgeometry.Inthispaperwedoassumeadustgeometryandhencemakeaninclinationcorrection(§6).

ThesimulationsintheSFMassumeagrayscreen,anuniformunclumpeddustlayerwithopacityequalintheVandIbands.TheSFMopacitymeasurementsinthispaperarebasedonsuchsimulations.InHolwerda(2005),weranaseriesofsimulationsonNGC13659tocharacterizetheeffectofaveragecloudcross-sectiononthenumberofdistantgalaxiesobservablethroughadisk.

Figure3showstheeffectofcloudsize,expressedinpixels,onthesimulatedrelationbetweenaverageopacityandnumberofdistantobjects.Ineachsimulation,wefixacloudsizeandvarytheirnumbertoincreasediskopacity.Anensembleofunresolvedcloudsiseffectivelythegrayscreen.ForcloudsresolvedwithHST,morethan2pixels10,therelationbetweenopacity(cloudfillingfactor)andnumberofdistantobjectsbecomesmuchshallower.Thesamenumberofdistantgalaxiesobservedwouldthenimplyamuchhigheropacityofthedisk.BecausetheSFM(calibratedwithagrayscreen)generallyagreeswellwithmeasurementsfromoverlappinggalaxies(Holwerdaetal.2005b),itseemsunlikelythatthedisk’sopacityispredominantlyduetolarge,resolvedclouds.Apixelof0′′.05atthedistanceofNGC1365(18Mpc)is4pcinlinearsize.ItisthereforeimpliedthatthestructureoftheISMresponsibleforthediskopacitymeasuredwiththeSFMvariesinopticaldepthonscalesof∼10pcorless.

FromtheSFMmeasurementsalone,thecloudgeometryisimpossibletodetermine.Onlywhenadditionalinformationisused–e.g.,thegeneralagreementwiththeoccultinggalaxytechnique–itfavorssmall(unresolved)scalesfortheclouds.Therefore,toconstraincloudgeometry,informationfromtwodifferenttechniquesneedstobecombined.

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5.

Opticaldepths

Table4presentstheopticaldepthestimatesfromtheSyntheticFieldMethodandtheSEDmodel(Draine2003;Draine&Li2007),fortheWFPC2field-of-view.Theopticaldepthsrangebetween0.1and3.5magnitudesintheI-band.Themeasurementsarefordifferentpartsofdifferentspiraldisks(Figure1),explaininginparttherangeinvalues.TheopticaldepthestimatespresentedheremayappearhighfortheJohnsonI-band,comparedwithotherextinctionestimates(e.g.,thosefrominclinationeffectsorreddening),buttheseare(1)fortheentireheightofthedisk,and(2)generallycenteredonaspiralarm.TypicalextinctionvaluesintheI-bandareseveraltenthsofamagnitudeforadustscreeninfrontofthestellarspiraldisk(e.g.,Meyeretal.2006).TwoofthederivedSFMopacitiesarenegative,possiblytheeffectofanoverdensityofdistantgalaxiesbehindthetargetgalaxy.ThediscrepancyinNGC5194-1maybeduetomisidentificationofbackgroundgalaxies,astheyaredifficulttoidentifyinthisfield.

Figures4and5showthevaluesofdiskopacitybybothmethods,overthesamesectionofthedisk.Bothmethodsgenerallyagreewithintheuncertaintiesofthemeasurements.TheagreementisbetterthanourinitialestimatefromablackbodyfittotheMIPSfluxesinHolwerdaetal.(2006).Thegeneralagreementandthemeantemperatureofthedust(Table3)implythatmostofadisk’sopacityisduetothecolddustinthedisk.

6.Modelofcloudgeometry

TherelationbetweentheapparentandaverageI-bandopticaldepthmeasurements–thefirstfromthenumberofdistantgalaxiesinHSTimagesandthesecondderivedfromtheSpitzerSED–couldrevealthenatureoftheprevalentstructureintheISM.

Theaprioriassumptionsarethat(1)allduststructureistransparenttotheFIRemissionfromwhichthedustsurfacedensityisestimatedintheSEDmodel,and(2)theentirevolumeofthecloudemitsintheFIR.WeadoptModelCfromNatta&Panagia(1984),inwhicharandomlydistributedseriesofclumpscoversthearea.Theseauthorsdefinetwoopticaldepths:(1)thetypicalopticaldepth(τm),i.e.,theaverageopticaldepththatisproportionaltothedustmass;and(2)theapparentopticaldepth(˜τ),ortheopticaldepthifauniformlayerwouldcoverthearea.

Ourtwomeasurementsofopticaldepth–SEDandSFM–correspondtothesetwoopticaldepths,averageandapparent.AnopticaldepthbasedontheSEDdependsonthedustmasswithinthearea,andhencecorrespondstoτm.TheSFMopticaldepthistheapparentoptical

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depthaveragedoverthearea,andhenceτ˜.Thesetwoopticaldepthsneednotbethesame,andtheirrelationisanindicationofhowclumpedthemediumis.

Letusassumeanumberofsmallduststructureswithaheight(h),anaveragegraincross-section(σ),andagrainemissivity(Q).Thegrainnumberdensityinthecloudsisdenotedbynd,andtheaveragenumberofcloudsinaline-of-sightisn.Weassumeallclumpshavethesameopticaldepthτc:

τc=hndσQ.

Theaverageopticaldepth(τm)isthen:

τm=n×τc,

(2)(1)

andtheapparentopticaldepthcanbederivedifoneassumesaGaussiandistributionofthenumberofcloudsalongthepossiblelines-of-sightandsumthecontributionsofallclouds(seeequations15and17-19inNatta&Panagia1984):

τ˜=n×(1−e−τc),

withasymptoticvaluesforbothopticallythinandthickclouds:

τc→0,⇒τm→0,τ˜→τm,

(4)(3)

τc→∞,⇒τm→∞,τ˜→n.

(5)

Wenotethattheinferreddustmassinthedisk(Mdust)isproportionaltothenumberof

clouds(n)andthecloudopticaldepth(τc).Theratiooftheapparentovertheaverageopticaldepthis:

τ˜

τc

,

(6)

leavingonlytheopticaldepthoftheclouds(τc),andhencethegraindensity(nd)andthe

cloudsize(h)asthevariables.OurfittotherelationbetweenSFMandSEDopticaldepthestimatesonlyhasτcasthevariable(Figure4).

–14–

Wecannowuseequation6toderiveτcfromafittotheopticaldepthsfromSED(τm),andfromthenumberofdistantgalaxies(˜τ).Wewanttoanswerthreequestions.Arethecloudsinthediskstypicallyopticallythinorthick?Ifalldisksareequal,whatistheimpliedcloudopticaldepth?Howmanycloudslietypicallyalongagivenline-of-sight?

6.1.Opticallythickorthinclouds?

Opticallythinclouds(τc<<1)resultinaratioofopticaldepthsclosetounity:

τ˜

τc

≈

1−(1−τc)

τm

=

1−e−τc

–15–

σ=0.03µm2

andQ=3

󰀁β

=1.5×10−3withβ=2(Hildebrand1983)11.Thevalueof0.4forτacloud160

cimpliesheight,h,of∼60pc!Muchlargercloudscouldberesolvedinextinctionmapsofthesedisksbasedonstellarreddening.InthecaseofNGC3627,NGC5194,NGC6946andNGC7331,thereisaclearspiralarminthereddeningmap(Meyeretal.2006;Holwerdaetal.2007);theotherreddeningmapsaresmoothordonotextendouttocoverthewholeWFPC2pointing.

Amorelikelyscenarioisthatthereisaninverserelationbetweentheclouddensityndandscaleh.Sucharelationcanbeseeningiantmolecularclouds(GMC)ofourownGalaxy(e.g,Solomonetal.1987).Inthiscase,thesingleopticallythincloudcanbereplacedbysmalleropticallythickones.WenotethatthelargestGMCsareoftheorderof60pc.Thevalueof60pccloudsappearsincontradictiontotheimpliedsizeintheSFMcali-bration(∼10pc).However,thetypicalcloudsizecanbe60pcandstillthediskopacitycanchangeoversmallerscales,ifseveralpartiallyoverlappingcloudsareseeninprojection.Theinverserelationbetweenscaleanddensitywouldalsohelpmakethetwoscalescompatible.Ourdataareconsistentwithopticallythinclouds(τc=0.4)andtheiraverageopacityvalueimpliesatypicalcloudsizethatisunresolvedwithSpitzerinourgalaxies.

6.3.CloudNumbers

Theabovefittotherelationbetweenτ˜andτmindicatesthat,onaverage,morethanonecloudisneededalongtheline-of-sightinmostdisks(¯n=2.6).However,theassumptionwasthatτchadasinglevalueforallthedisks.In§6.1,wearguedthattheratiobetweenτ˜andτmimpliedtheτcisopticallythin.Itlogicallyfollowsthatopticallythickdisksmusthavemorethanasinglecloudalongtheline-of-sight.

Figure5illustratestheeffectofnumberofcloudsalongtheline-of-sight(n)ontherelationbetweenτ˜andτm,whenτcisfreelyincreasedfrom0toτm/n(thelineshavebeendrawnaccordingtoequations2and3).Wenotethatalldisksareconsistentwithmanycloudsalongtheline-of-sight–includingtheopticallythinones,alsobecausetheylieintheopticallythincloud(˜τ/τm∼1,τc<1)regime.

–16–

6.4.

PotentialImprovements

Therearemanyrefinementstobemadetothesimplemodelpresentedhere.Someimprovementsforfuturecomparisonsbetweenthesetwomeasurementsofopticaldepthare:(1)adistributionofcloudsizesforboththeSFMcalibration,aswellasinthemodelexplainingtheratiobetweenSFMandSEDopticaldepths.AmodeldistributioncanbetakenfromobservationsofGMCsinourownGalaxyandnearbyones(Heyeretal.2001;Rosolowsky2005).Thecross-sectiondistributioncouldbeusedinSFMmeasurementsinthefuture.Across-sectiondistributioncanonlybeapplied,iftheforegrounddiskisatasingle,fixeddistance;onlyonecountsthoughasingleforegroundgalaxyareused.Therearethreeface-onspiralswithenoughsolidangleinHSTimagingaswellasadditionalSpitzerdata:M51,M81andM101.(2)TheSEDmodelcanbemuchbetterconstrainedwithadditionalsub-mmobservations,theopportunitiesforwhichwillexpanddramaticallyinthenearfuture(SCUBA2ontheJCMTandHerschelsatellite).(3)Theeffectsofgranddesignspiralarmsandgalacticradiuscouldbeidentifiedinasingledisk;thecomparisonSFMandSEDisnotmadefordifferentsectionsofthediskscombined.(4)AfutureSEDmodelcantakeintoaccountthereddeningoftheinterstellarradiationfield,asitpenetratestheISM.ThiswouldrequireacomprehensivetreatmentoftheISMstructureinadditiontoitstemperature,compositionandirradiation.

7.Conclusions

Toconstrainmodelsofthespiraldisk’senergybudgetwithtypicalvaluesforthesizeofdustycloudsintheISM,wecomparetwotechniquestoextracttheaverageandapparentopticaldepthsofasectionofspiraldisk.FromthecomparisonbetweenSFMandSEDresults,weconcludethefollowing:

1.TheSFM’scalibrationaloneimpliesprojectedcloudscalespredominantlyunresolvedwithHST(oftheorderof10pcinNGC1365,see§4andFigure3).2.Thedustresponsibleforthedisk’sopacityispredominantlycold(T<25K,Table3).3.Theaverageandapparentopticaldepthsofthesedisksections,measuredfromSEDandSFMrespectively,generallyagree(Figure4and5).Thisimpliesgenerallyopticallythinclouds(τc<1,§6.1).4.Thefittotheratiobetweenapparentandaverageopticaldepthmeasurements,τ˜/τm,indicatesacloudopticaldepth,τc,of0.4,morethanasinglecloudalongtheline-of-sight,andacloudsizeof∼60pc.Ifseveralpartiallyoverlappingcloudsareseenin

–17–

projectionthroughthedisk,thedisk’sopacitywillchangeoversmallerscales,consistentwithconclusion1.

6.Opticallythickdisksappeartohavemorethanasinglecloudalongtheline-of-sight(Figure5)andopticallythindisksmayhaveseveralcloudsaswell.Futureworkusingcountsofdistantgalaxiesthroughaforegrounddiskcouldbeusedtofindcoldduststructuresatlargergalacticradii,providedasufficientlylargesolidanglehasbeenimagedwithHST/ACS’ssuperbresolution.12Notably,theACSdataonM51,M81andM101areverypromisingforsuchananalysis.13Spitzerobservationsofthesenearbydisksarealsoavailable,makingasimilarcomparisonbetweenSEDandapparentopticaldepthpossibleforportionsofthesedisks.Thetypicalcloudscaleforspiralarmsordisksectionsorasafunctionofgalacticradiuscouldthenbefound.TheSCUBA-2instrumenthasrecentlybeeninstalledontheJamesClerkMaxwellTelescope.AprojectwithSCUBA-2tomaptheSINGSgalaxiesintwosub-mmbandswillimprovefutureSEDmodellingofthesespiraldiskssignificantlyovertheSEDmodelspresentedhere.

ThisworkisbasedinpartonarchivaldataobtainedwiththeSpitzerSpaceTelescope,whichisoperatedbyJPL,CalTech,underacontractwithNASA.ThisworkisalsobasedonobservationswiththeNASA/ESAHubbleSpaceTelescope,obtainedattheSTScI,whichisoperatedbytheAssociationofUniversitiesforResearchinAstronomy(AURA),Inc.,underNASAcontractNAS5-26555.

TheauthorswouldliketothankT.Jarrett,formakinghisaperturecorrectionsofextendedsourcesavailabletousatanearlystage,andErikRosolowsky,forusefuldiscussionsonLocalGroupcloudsizes.WewouldliketothankMaartenBaesfordiscussionofthemotivation.WewouldalsoliketothankGeorgeBendo,ErikHollenbackandKristenKeenerfortheircommentsonearlierdraftsofthispaper.

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Fig.1.—ThefootprintsoftheWPFC2cameraonboardHST,onthe24micronimagesfromtheMIPSdetectoronboardSpitzer.MostoftheHSTimagesdonotincludethecenter,andarepointedonaspiralarm.NGC3621andNGC5195(M51)havetwoseparateWFPC2fieldsassociatedwiththem.ThePCchip,i.e,thesmallchipinthenookofthe“L”ofthethreeWFchips,isnotusedfortheSFManalysis,noraspartofourSpitzeraperture.

–25–

Fig.2.—TheSpectralEnergyDistributioninthe7Spitzerbands(IRAC/MIPS),foreachoftheWFPC2aperturesinFigure1.ThebestfittingmodelfromDraine&Li(2007)isshown.TherelevantparametersforeachfitareinTable3.

–26–

Fig.3.—DifferentsyntheticfieldswithdifferentdustdiskmodelsfromHolwerda(2005).AsetofsyntheticfieldsismadeusingdimmedHDFimages.Thisdimmingcanbeasmoothuniformgrayscreen(triangles),orsomedistributionofopaqueclouds(curves).TherelationbetweenthenumberofdistantgalaxiesfromtheHDFthatcanstillberetrievedandtheaverageopacityofthedimmingdependsontheassumedmodel.Large,resolvedcloudsblockfewerbackgroundobjectsgiventhesamefillingfactor.Asaresult,oneahigheropticaldepthisimpliedbytheintersectionofthecurveandnumberofobservedbackgroundgalaxies.ThedistancetoNGC1365is18Mpc,soapixelof0′′.05isequivalentto4pcinlinearsize.Thescalesintheabovesimulationscorrespond,therefore,tocloudswithcross-sectionswitharadiusof2,4,8,100and400pc,respectively

–27–

Fig.4.—ThemeanandapparentopticaldepthintheI-band,fromSED(τm)andSFM(˜τ).Thebestfitwithequation6totheratiosofthesevaluesisalsoshown(τc=0.4).Theaveragevaluefornumberofcloudsalongtheline-of-sight,n,isthen2.6.IfthenegativeSFMvaluesareincludedinthefit,thecloudopticaldepthrisestoτc=0.6andtheaveragenbecomes1.9.

–28–

Fig.5.—Modelvaluesofτmandτ˜,fromequations2and3,forfixedvaluesoftheaveragenumberofcloudsalongtheline-of-sight(n=0.5,1,2,3,4).Thecloudopticaldepthτcislefttovaryproportionallytoτm(τm=n×τc,equation2).Hencemaximumcloudopticaldepthsare6,3,1.5,1and0.75respectively.Ifopticallythincloudsaremadeupofopticallythinclouds,itseemsnecessarythatmorethanonecloudwillliealongtheline-of-sight.

–29–

Table1.SINGSdatadescription.IRACandMIPSpixelscales,PSFFWHM,and

aperturecorrections(thefactorbywhichthefluxesaremultiplied).

Instrument(band)

pixelscale

1

PSF(FWHM)

2

aperturecorrection

3

12

PixelscalesweresetbytheSINGSteam.

FWHMvaluesfortheIRACareconserva-tiveestimates.ActualFWHMvaluesarebetterthan2′′.5.TheIRACvaluesaretheinitialresultsfromT.Jarrett(privatecommunication),buttheydonotdiffersubstantiallyfromthefinalresults.

3

–30–

Table2.SpitzerchannelsurfacebrightnessintheWFPC2aperture(3WFs×1.3′×1.3′).

Galaxy

3.6µmMJy/sr

4.5µmMJy/sr

5.8µmMJy/sr

8µmMJy/sr

24µmMJy/sr

70µmMJy/sr

160µmMJy/sr

Table3.Modeloutput:stellaranddustsurfacebrightness,dustsurfacedensity,dustmeantemperature(T2inDraine&Li2007),fitquality,theparametersofstellar

irradiation–minimumandmaximumofthedistribution,meanirradiativefield,fractionof

dustexposedtomorethanUmin.Modelscanbefoundathttp://www.astro.princeton.edu/∼draine/dust/irem.html

Galaxyname

Lstar/area(L⊙/kpc2)×108

Ldust/area(L⊙/kpc2)×108

Mdust/area(M⊙/kpc2)×106

T(K)

χ2

Umin

Umax×106

¯U

γ×10−3

1e7

NGC2841NGC3198NGC3621-1NGC3627NGC4536NGC4725NGC5194-2NGC7331

1.80.60.73.70.50.51.80.6

0.30.20.52.10.10.11.70.2

0.50.30.41.10.10.21.40.2

14.615.016.217.015.714.616.015.4

4.153.732.452.112.323.011.832.15

0.700.501.001.000.800.500.800.70

10101.010101.01.010

0.80.61.01.40.80.50.90.7

1.07.52.624.52.70.011.74.5

U0.70

1e6

U0.50

1e6

U1.00

1e7

U1.00

1e7

U0.80

1e6

U0.50

1e6

U0.80

1e6

U0.70

MW3.1MW3.1

50

MW3.1

50

MW3.1

30

MW3.1

30

MW3.1

30

MW3.1

10

MW3.1

3020

model

name

–31–

Table4.Theapparent(˜τ)andaverage(τm)opticaldepthsinI-bandmeasuredinthe

′′

WFPC2field(3WFs1.3×1.3),uncorrectedandcorrectedforinclination.

Galaxy

τ˜

(SFM)

τ˜×cos(i)

τm(SED)

τm×cos(i)

τ˜/τm