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Astronomy&Astrophysicsmanuscriptno.m33-fourier (cid:13)c ESO2012 January13,2012 ⋆ Dust and gas power-spectrum in M33 (HERM33ES) F.Combes1,M.Boquien2,C.Kramer3,E.M.Xilouris4,F.Bertoldi5,J.Braine6,C.Buchbender3,D.Calzetti7,P. Gratier8,F.Israel9,B.Koribalski10,S.Lord11,G.Quintana-Lacaci3,M.Relan˜o12,M.Ro¨llig13,G.Stacey14,F.S. Tabatabaei15,R.P.J.Tilanus16,F.vanderTak17,P.vanderWerf9,andS.Verley12 1 Observatoire de Paris, LERMA (CNRS:UMR8112), 61 Av. de l’Observatoire, F-75014, Paris, France e-mail: [email protected] 2 Laboratoired’AstrophysiquedeMarseille,UMR6110CNRS,38rueF.Joliot-Curie,13388,Marseille,France 3 InstitutoRadioastronomiaMilimetrica,Av.DivinaPastora7,NucleoCentral,E-18012Granada,Spain 2 4 InstituteofAstronomyandAstrophysics,NationalObservatoryofAthens,P.Penteli,15236Athens,Greece 1 5 ArgelanderInstitutfu¨rAstronomie.AufdemHu¨gel71,D-53121Bonn,Germany 0 6 Laboratoired’AstrophysiquedeBordeaux,Universite´ Bordeaux1,ObservatoiredeBordeaux,OASU,UMR5804,CNRS/INSU, 2 B.P.89,FloiracF-33270,France n 7 UniversityofMassachusetts,DepartmentofAstronomy,LGRT-B619E,Amherst,MA01003,USA a 8 IRAM-InstitutdeRadioAstronomieMillime´trique,300RuedelaPiscine,38406-St.Martind‘He`res,France J 9 LeidenObservatory,LeidenUniversity,POBox9513,NL2300RALeiden,TheNetherlands 2 10 ATNF,CSIRO,POBox76,Epping,NSW1710,Australia 1 11 IPAC,MS100-22CaliforniaInstituteofTechnology,Pasadena,CA91125,USA 12 Dept.F´ısicaTeo´ricaydelCosmos,UniversidaddeGranada,Spain 13 KOSMA,I.PhysikalischesInstitut,Universita¨tzuKo¨ln,Zu¨lpicherStraße77,D-50937Ko¨ln,Germany ] O 14 DepartmentofAstronomy,CornellUniversity,Ithaca,NY14853,USA 15 MaxPlanckInstitutfu¨rAstronomie,Ko¨nigstuhl17,D-69117Heidelberg,Germany C 16 JAC,660NorthA’ohokuPlace,UniversityPark,Hilo,HI96720,USA h. 17 SRONNetherlandsInstituteforSpaceResearch,Landleven12,9747ADGroningen,TheNetherlands p Received16October2011/Accepted10January2012 - o r ABSTRACT t s a Powerspectraofde-projectedimagesoflate-typegalaxiesingasand/ordustemissionareveryusefuldiagnosticsofthedynamics [ andstabilityoftheirinterstellarmedium.Previousstudieshaveshownthatthepowerspectracanbeapproximatedastwopower-laws, ashallowoneatlargescale(largerthan500pc)andasteeperoneatsmallscale,withthebreakbetweenthetwocorrespondingtothe 1 line-of-sightthicknessofthegalaxydisk.Thebreakseparatesthe3Dbehaviouroftheinterstellarmediumatsmallscale,controlled v bystarformationandfeedback,fromthe2Dbehaviouratlargescale,drivenbydensitywavesinthedisk.Thebreakbetweenthese 8 tworegimesdependsonthethicknessoftheplanewhichisdeterminedbythenaturalself-gravitatingscaleoftheinterstellarmedium. 5 WepresentathoroughanalysisofthepowerspectraofthedustandgasemissionatseveralwavelengthsinthenearbygalaxyM33. 5 Inparticular,weusetherecentlyobtainedimagesatfivewavelengthsbyPACSandSPIREonboardHerschel.Thelargedynamical 2 range(2-3dexinscale)ofmostimagesallowsustodetermineclearlythechangeinslopesfrom-1.5to-4,withsomevariationswith 1. wavelength.Thebreakscaleisincreasingwithwavelength,from100pcat24and100µmto350pcat500µm,suggestingthatthe 0 cooldustliesinathickerdiskthanthewarmdust,maybeduetostarformationmoreconfinedtotheplane.Theslopeatsmallscale 2 tendstobesteeperatlongerwavelength,meaningthatthewarmerdustismoreconcentratedinclumps.Numericalsimulationsofan 1 isolatedlate-typegalaxy,richingasandwithnobulge,likeM33,arecarriedout,inordertobetterinterprettheseobservedresults. : Varyingthestarformationandfeedbackparameters,itispossibletoobtainarangeofpower-spectra,withtwopower-lawslopesand v breaks,whichnicelybracketthedata.Thesmall-scalepower-lawisindeedreflectingthe3Dbehaviourofthegaslayer,steepening i stronglywhilethefeedbacksmoothesthestructures,byincreasingthegasturbulence.M33appearstocorrespondtoafiducialmodel X withanSFRof∼0.7M /yr,with10%supernovaeenergycoupledtothegaskinematics. ⊙ r a Keywords.galaxies:individual:M33–galaxies:spiral–galaxies:infrared–galaxies:starformation 1. Introduction ture,orapower-lawpower-spectrum.TostudytheatomicISM, Crovisier&Dickey(1983)andGreen(1993)studiedthepower- Quantifyingthemulti-scalestructureoftheinterstellarmedium spectrum as a function of inverse scale of the 21cm emission, (ISM)ingalaxiesisadifficultenterprise,butrewardingsinceit and found a power-law of slope between -2 and -3, indepen- betrays the underlyingphysicalphenomena,controllingits dy- dent of the distance and the velocity of the HI. This power- namics.IthasbeenknownforalongtimethattheISMreveals law slope was observed to be steeper from -3 to -4, when the no preferentialscale and is well representedby a fractal struc- broadestvelocitieswereaveraged,thusincludingthewarmgas, withmoreturbulence(Dickeyetal.2001).Goldman(2000)and Sendoffprintrequeststo:F.Combes ⋆ Herschel is an ESA space observatory with science instruments Stanimirovicetal.(2000)extendedtheISMstudiesintheMilky providedbyEuropean–ledPrincipalInvestigatorconsortiaandwithim- Way to the Small Magellanic Cloud (SMC), where a power- portantparticipationfromNASA. law of slope -3.1 to -3.4 was found for the dust and gas, and 1 F.Combesetal.:Dustpower-spectruminM33 interpreted as 3D turbulence, although the index is shallower of 3D-turbulence,with a flattening at large scales, and a break than the -3.7 Kolmogorov spectrum (Stanimirovic & Lazarian corresponding to several hundred parsecs, typical of the plane 2001).A breakin scale hasbeensearchedforin orderto iden- thickness. This is true for flocculent and density-wave galax- tifytheenergyinjectionmechanism,butnonehasbeenfoundat ies,althoughthespiralarmwavedominatesatverylargescales. smallscale(i.e.smallerthan500pc).Insteadaflatteningtowards In M33, Elmegreen etal.(2003b) analysed optical and Hα im- larger scales has been measured by Elmegreen etal.(2001) in ages and found a very shallow power-law of slope -0.7, which theatomicgasoftheLargeMagellanicCloud(LMC).Theyin- theyattributetoflatteningduetocontaminationbystrongpoint terpret this flattening as a change of dimensionality, from 3D sources(foregroundstars). For the face-ongalaxyNGC 628, a at small scale to 2D at large scale, where the dynamics of the shallowslopeof-1.5isfound,withnobreakinspiteofthevery galaxy plane is dominating,and identify the scale of the break high spatial resolution obtained with HST (scale below 10 pc, asthethicknessoftheplane.Thismethodtofindthescaleheight Elmegreen etal.2006). Odekon (2008) however finds a transi- ofdiskshasbeendevelopedbyPadoanetal.(2001),whofound tion scale of about 300 pc between a steep spectrum at small ∼ 180 pc in the LMC. Recently, Block etal.(2010) have anal- scale and flattening at large scale, in the stellar distribution of ysed thedustmapsof theLMC fromSpitzer,andalso confirm M33.Thisscaledoesnotchangemuchwiththeageofthestel- thisbreakscale at 100-200pc. Fromhighresolutionnumerical lar population considered, however the slope is steeper for the simulationsfittedtoalate-typegalaxyliketheLMC,Bournaud bluest and youngest stars. Sanchez etal.(2010) investigate all etal.(2010) were able to retrieve the characteristics of the ob- the young components in M33, including young stars, HII re- served dust power-spectrum. The source of turbulence could gions and molecular clouds, and also find a clear transition re- thenbeduetolarge-scaledifferentialrotation,coldgasaccretion gion,butintherange0.5-1kpc.Thefactthatdifferentresultsare orgalaxyinteractions(Lazarianetal.2001,Elmegreen&Scalo obtainedforthesamegalaxiescallsfornewstudiestoclarifythe 2004).Khaliletal.(2006),froma 2DwaveletanalysisofHI in situation. ourown Galaxy,find a power-lawslope of -3 everywhereover The galaxy M33 has been observed with Herschel PACS thesecondquadrant,anddonotdetectthevariationswithveloc- andSPIREatallwavelengths,bytheKeyProjectHERM33ES, ity slices predicted by Lazarian & Pogosyan (2004). However and the photometric maps have been presented in Kramer they discover an anisotropy linked to spiral arms. The fractal etal.(2010),Boquienetal.(2010)andVerleyetal.(2010).These approach was widely developed for molecular clouds, which providehighsignal-to-noisemapswithalargedynamicalrange aremoreself-gravitatingthantheHIgas(Falgaroneetal.1991, of scales ofthe dustemission at manywavelength,allowingto Pfenniger&Combes1994,Stutzkietal.1998). probewarmandcooldust.Thelargeandsmallscale structures Uptonow,thebreakscaleintheHIdistributionhasnotbeen of the gas and dust are studied through Fourier analysis. We identified unambiguously in external galaxies other than the present in Section 2 the differentmaps used, and how they are LMC,althoughithasbeensearchedfor.Duttaetal.(2008)stud- processed,anddeprojectedtoaface-onorientation.Theresults iedtheface-ongalaxyNGC628,andfoundanHIpowerspec- will be comparedto numericalmodels, which technique is de- trumwithasinglepower-lawslopeof-1.6,overthescales0.8to scribedinSection3.Resultsarepresentedforbothobservations 8kpc,theirspatialresolutionbeinginsufficienttoreachthescale andmodelsinSection4,anddiscussedinSection5.Weadopta height of the gaseous plane. In DDO210, Begum etal.(2006) distanceof840kpcforM33(Freedmanetal.1991). founda power-lawslope of -2.75overthe scales 80 to 500pc. ThissteepslopeiscompatiblewithwhatisfoundintheMWand LMC/SMC at small scales, driven by 3D-turbulence. In other dwarfsalso(Duttaetal.2009b)asingleslopeisfound.Theface- ongalaxyNGC1058isthefirstoneforwhichthetransitionfrom 3D to 2D turbulencehas been claimed (Dutta etal.2009a):the slopechangesfrom-2.5over0.6to1.5kpc,to-1over1.5-10kpc scales. The scale heightis thenestimated at490pc, which isa relativelylargevaluefortheHIlayer.IntheSMC,whileasteep power-law is found consistently by several studies, no break scale has been found, and no transition to the 2D-turbulence (Roychowdhuryetal.2010).It is suggested thatdwarf galaxies havegasdisksconsiderablythickerthannormalspiralgalaxies, whichsupportpreviousstudiesfromthethicknessofdustlanes Fig.1.Rotationcurveforourlate-typeScdmodel,comparedto in edge-on galaxies (e.g. Dalcanton etal.2004). The transition theobservedpoints(symbolsanderrorbars)compiledforM33 between thick and thin gas disks was found at a mass corre- byCorbelli&Salucci(2007).Inadditiontothecircularvelocity spondingtoacircularvelocityof120 kms−1.Forgalaxieswith ofthemodel,thecharacteristicfrequenciesΩ,Ω−κ/2,etc.are lowinclinationontheskyplane,thethicknessofgasdiskscanbe plotted. estimatedthroughtheratioofgasvelocitydispersiontorotation (Dalcanton&Stilp 2010).ThecaseofM33isexactlyinterme- diate,havingarotationalvelocityof120 kms−1.Boththickand thindiskscouldbeexpected. 2. Observationaldata Anothertooltoexplorethescaleheightofdiskswithhigher spatial resolution, apart from studying edge-on galaxies (e.g. To study the interstellar medium structure, we use all possible Bianchi & Xilouris 2011), is to study young components re- tracers,beginningbygasanddustemission (HI, CO lines, and lated to the gas, as young stars just formed out of the ISM. far-infrared photometry), and also star formation from the gas Elmegreenetal.(2003a)studiedthepower-spectrumofstarfor- (Hα,ultra-violetcontinuum). mationmapsinnearbygalaxies,andfoundthesamebehaviour Forthedustemission,weuseHerschelobservationstakenin asfortheLMCinHI:asteeppower-lawatsmallscales,typical the context of the HERM33ES open time key project (Kramer 2 F.Combesetal.:Dustpower-spectruminM33 Table1.Characteristicsofthedifferentmapsused Map Resolution Npix pixelsize (′′) (pc) (′′) (pc) Galex-FUV 4.3” 17 2560 1.5” 6 Galex-NUV 5.3” 21 2560 1.5” 6 Hα 6” 24 1920 2” 8 MIPS-24µm 6” 24 2560 1.5” 6 MIPS-70µm 18” 72 850 4.5” 18 PACS-100µm 7” 28 2268 1.7” 7 PACS-160µm 11” 44 1344 2.85” 12 SPIRE-250µm 18” 72 800 6” 24 SPIRE-350µm 26” 104 480 10” 41 SPIRE-500µm 36” 144 342 14” 57 CO(2-1) 12” 48 1470 2” 8 Fig.3.PowerspectrumoftheSpitzer-MIPS70µmmap,andthe HI-21cm 12” 48 1200 4” 16 PACS-100µmdustemissionmapofM33.Theverticaldashline atrightrepresentsthe spatialresolutionof the observations,on Adistanceof840kpchasbeenadoptedforM33.1”=4pc the major axis. Since the de-projectionto a face-ongalaxyim- Resolutionscorrespondtothemajoraxisdirection, theyare1/cos(56◦)=1.79largerintheotherdirection plies a factor 1/cos(56◦) = 1.79largerbeam on the minoraxis, Npixisthenumberofpixelsineachdimensionofthemap we have also plotted a second dashedvertical line at left, indi- catingthe geometricmeanof the resolutiononboth minorand major axis. Statistical error bars are given in this first figure, and then omitted later for clarity. They are significant only at the largest scales (small k), where the statistical errors are the largest. gasanddustemissiontracedirectlythecollisionalmaterial,with nointerveningandperturbingemission(likegalacticstars). Fig.2. 2D power-spectrum for the PACS 100µm image. The colorscaleisinarbitraryunits. etal.2010) in combination with Spitzer MIPS data (Verley etal.2007),spanningwavelengthsfrom24µmto500µm. The Herschel PACS data covering the 100µm and 160µm bands, have been described in Boquien etal.(2010, Fig.4.Same asFigure3, forthe powerspectrumofthe PACS- 2011). The SPIRE observations obtained at 250, 350 and 160µmandSPIRE-250µmdustmap. 500µm are displayed in Kramer etal.(2010) and Verley etal.(2010). We also use ground-based Hα observations presented in Hoopes etal.(2001) in order to trace the current (∼ 10 Myr) starformationandGALEXdata(Thilkeretal.2005,GildePaz 3. Numericalmodel etal.2007) to trace the recent (∼ 100 Myr) star formation. To trace the atomic gas, we use the HI 12′′resolution map pub- In order to comparewith what is expectedfrom the dynamical lished in Gratier etal.(2010), and for the molecular gas, their instabilities of the interstellar medium, we have run idealised partialCO(2-1)12′′resolutionmap(Gratieretal.2010). simulations,fittedforalate-typegalaxywithoutbulgeandrich All useful characteristics of the images are summarized in ingaslikeM33. Table1.Allmapshavebeendeprojected,withtheadoptedincli- nationof56◦ andpositionangleof22.5◦,beforeFourieranaly- 3.1.Technique sis. Thedataanalysedinthepresentworkaremoreappropriate The numericalcode is based on the TREE/SPH version devel- thanearlierdata(Elmegreenetal.2003b,Odekon2008,Sanchez opedbySemelin&Combes(2005)anddiMatteoetal.(2007), etal.2010)tostudythepowerspectraandthestructureofthein- with star formationandfeedback.The TREE gravitationalpart terstellarmedium,sincetheyreferdirectlytothegasanddustin of the code deals with two collisionless components: the stars M33,andtheyhavethedynamicalrange(from40pcto20kpc) andthedarkmatter,whichdifferonlybytheirinitialconditions. andspatialresolutiontorevealthethicknessofthegaslayer.The It deals also with the gas component,whose hydrodynamicsis 3 F.Combesetal.:Dustpower-spectruminM33 feedbackisnotwellknown,anddependsonthecouplingofthe supernovae and the gas (whether the young stars have already left their birth clouds when they explode, etc..). Some energy and supernovaexplosionsare also due to the old populationof stars(e.g.Dopita1985). 3.2.Initialconditions The initial stellar componentof the galaxy is selected to be of typeScd,similartotheM33galaxy. The stellar and gaseous disks are represented initially by Miyamoto-Nagai functions, with characteristic height parame- tersof170pcand70pcrespectively.Thestellardiskhasamass of0.5×1010M ,withacharacteristicradiusof2kpc,whilethe Fig.5.SameasFigure3,forthepowerspectrumoftheSPIRE- ⊙ gasdiskhasamassof1.5×109M ,witharadius2.5kpc.Both 350µmand500µmcooldustmap. ⊙ areembeddedinasphericalhaloofdarkmatter,representedby a Plummer sphere of mass 4 × 1010 M within 18 kpc, with a ⊙ treated with the SPH (Smooth Particle Hydrodynamics) algo- characteristicradiusof9.5kpc. rithm,usingadaptiveresolution.Thesofteningparameter,orthe Theinitialstabilityofthegasandstellardisksiscontrolled spatialresolutionofgravity,istakentobe30pc,andtheaverage by the Toomre parameter Q, ratio of the radial component of size ofa gaseousparticle(theSPH kernel,orspatialresolution the velocity dispersion to the critical one σcrit = 3.36GΣ/κ, Σ ofthehydrodynamics)isalsoaround30pc,althoughitcandrop being the disk surface density of the component(gas or stars), down to 3 pc in overdense regions. The number of particles is and κ the epicyclic frequency.We select equal Toomre param- 1.2million,with0.6milliongasparticles,0.4milliondarkmat- eters for gaseous and stellar disks. The azimuthalvelocity dis- terparticles,and0.2million”stars”atthebeginningofthesim- persions verify initially the epicyclic ratio σθ/σr = κ/2Ω, and ulations.Thegasparticlesareassumedtofollowanisothermal thez-componentsarefixedbytheisothermalequilibriumofthe equationofstate,withpressureforcescorrespondingtoasound disks,intheverticaldirection. velocity of 6 kms−1, in the fiducial model. Shocks are treated Thegasisisothermal,butitsequivalentvelocitydispersion, withaconventionalformofartificialviscosity,withparameters governingthepressure,maybevariedbetween6and12 kms−1. givenindiMatteoetal.(2007).Allphysicalquantities(density Therecipesforstarformation:efficiency,densitythreshold,and andforces)arecomputedaveragingoveranumberof50neigh- feedback,are also varied;theyhavea significantimpactonthe bours. The equationsof motion are integratedusing a leapfrog interstellargasstructure. algorithmwithafixedtimestepof0.3Myr. The initial conditions of the runs described here are given Star Formation is taken into account in most models, with inTable2.Therotationcurvecorrespondingtooneoftheruns an assumed Schmidt law, of exponent n=1.5, i.e. for each gas isplottedincomparisontothedatapointsinFigure1.Allruns particleofmass M , andvolumicdensityρ computedover havesimilarrotationcurves.Giventheuncertainties,allthero- gas gas itsneighbours: tationcurvesarecompatiblewiththedata. dM gas =C ρ1/2 Mgasdt ∗ gas Table2.Initialconditionsofthesimulations whereinmostruns,thestarformationconstantC iscalibrated ∗ Run Q V C threshold ǫ suchthatthegasconsumptiontime-scaleisoftheorderof2Gyr ∗ s ∗ =Q (km/s) at.cm−3 (feedback) (then C = C ). C is increased in some of the runs, to check g ∗ 0 ∗ its influence on the gas scale height. In most simulations, the run1 2. 6. C 2×10−6 10−4 0 densitythresholdforstar formationis set verylow,and isthen run2 2. 6. C 2×10−6 10−3 0 increasedinafewruns.Starformationisthentakenintoaccount run3 2. 6. C 2×10−6 1. 0 foreach gasparticle, usingthe hybridparticlesscheme (e.g.di run4 2. 6. C 2×10−6 10−1 0 Matteoetal.2007),beforesufficientstellarmassisaccumulated run5 1.5 6. C x20 2×10−3 10−1 0 anda star particlecanbe created.Continuousmasslossis also run6 1. 6. C0x20 4 10−1 considered, with about 40% of the stellar mass loss by young run7 1. 6. 0. – – run8 1. 12. 0. – – massivestarsafter5Gyr(Jungwiertetal.2001). run9 1. 12. C 2×10−6 10−1 The energy reinjected by supernovae into the interstellar 0 mediumisassumedtobeakinematicfeedbackonneighboring C0issuchthatthegasdepletiontimeis2Gyr gas particles: each neighbourof a SPH particle having formed δmofstars isgivena radialkickin velocityawayfromthe su- pernovaformed.Theenergyavailableiscomputedfromδm,as- sumingaScaloIMF(about0.5%ofthestarsformedhaveamass 4. Results larger than 8 M and explode as supernovae), the energy of a ⊙ supernova E = 1051 erg is distributed to the 50 neighbours 4.1.Power-spectrumofthedata SN accordingtotheirweight(theSPHkernel),withanefficiencyǫ, The power spectra of the 2D projected maps were analysed whichhasbeenvariedfrom10−4to1,with10%beingthefidu- throughthe2DFouriertransform: cial value (see Table 2). For the fiducial feedback efficiency, a particlemightgetakickof∼1 kms−1ateachtime-step,forthe extreme feedback, it can reach 5 kms−1. The efficiency of the F∗(kx,ky)=Z Z F(x,y)e−i(kxx+kyy)dxdy x y 4 F.Combesetal.:Dustpower-spectruminM33 where F(x,y) is the intensity of each pixel, and k and k are influencetheresults.Wehavetruncatedthepowerspectraatthe x y thewavenumber,conjugatetoxandy,varyingastheinverseof Nyquistsampling,inthefigures. scales.Thefull2Dpowerspectrumisgivenby Lookingat Tables 1 and 3, it is oviousthat the break scale isaffectedbythespatialresolutionofthemaps.Fivemapshave P(k ,k )=(Re[F∗])2+(Im[F∗])2 a ratio between break and resolution scales lower or equal to x y 3 (NUV, the 3 SPIRE, and CO), for the others, the ratio is 3.8 An example of 2D-power spectrum is given for the PACS- (FUV), 12.5 (Hα), 4.2 (MIPS-24µm), 3.9 (MIPS-70µm), 3.9 100µm image, in Figure 2. The one-dimensional power- (PACS-100µm),3.6(PACS-160µm)and3.3(HI).Forthose,the spectrum P(k) can be calculated through azimuthal averaging, determinationofthe breakappearssignificant.Asfortheslope inthe(k ,k )plane,usingk2 = k2+k2.Thepowerspectra P(k) determination, the large-scale ones are not affected, and their x y x y are plotted for the various maps in Figures 3, 4, 5, 6, 7 and 8. variation is meaningful: the LS-slopes are flatter for the star- Two slopeshavebeenfittedatsmall(ss) andlargescales(LS), formationtracersthanforthedustandgas,andtheyaregrowing and the differentvalues can be found in Table 3, together with steeperwithwavelength. thebreakscaleseparatingthetworegimes.Inmostofthecases, thepowerspectrashowclearlytwodifferentregimes,withdiffer- entslopes,andsmallandlarge-scales,forscaleslargerthanthe spatial resolution.A knee is apparentbetween the two regimes (ssandLS),andindicatesthebreakscale.Wehaveusedaleast- square-fit program to determine the slopes of the two power- laws, and left open the position of the separation between the twostraightlines.Varyingthevalueoftheseparation,weselect forthebreakthevaluewhichminimisesthesumoftheχ2ofthe twolinefits(ssandLS).Ingeneraltheminimumisquiteclear. There is one ambiguous case, however, which is the HI map, inFigure8.Thetwoslopesareclose,andthedeterminationless trustworthy.ExamplesofthedeprojectedimagesoftheHerschel dustemission are displayedin Figures9 and 10. In additionto the power spectra, the azimuthal large-scale structure develops spiral arms, that are quantified and studied in the AppendixA. Fig.6.SameasFigure3,forthepowerspectrumoftheHαand The determinationof the break scale is obviously more secure MIPS-24µmmaps. when it is found significantly larger than the resolution scale of the observations. We have tested the effect of spatial reso- lution by smoothing the PACS-100µm map in Appendix B. In theobservedpowerspectra(e.g.Figures3-7),thelimitofthein- strumentalresolutionisclearlyseen,throughthesuddenchange fromastraightlinetoadroppingcurve,atlargek.Wehaveindi- catedwithtwoverticaldashedlines,thevalueoftheresolution scaleonthemajoraxis,andalsothegeometricmeanofthereso- lutiononbothminorandmajoraxis,sincethede-projectiontoa face-ongalaxyimpliesafactor1/cos(56◦)=1.79largerbeamon theminoraxis.WeconcludefromAppendixBthatbreakscales atleasttwicelargerthanthebeamscalearerobustwithrespect toresolutioneffect. Wehavealsotestedtherobustnessofourdeterminationsby computingthepowerspectrausingonlyonesideofthegalaxy. Theeasiestwayto computethat, withthesame algorithm,was to remove in the map one half of the galaxy, say the southern part, and complete the map by extending the northern part by Fig.7.SameasFigure3,forthepowerspectrumoftheGALEX point symmetry through the center. For all maps, the different NUVandFUVmaps. slopesandbreakscales forthe twosidesofthe galaxyare also displayedinTable3:thelastcolumnsdisplaytheextremevalues foundfortheslopesandbreaks.Thisproceduremaximizesthe ItisinterestingtonotethattheHαmap,tracinggasionized error bars, especially in the LS-slopes, since the Norhern part by the UV photons from massive stars and, indirectly, current ofM33isquitedifferentfromtheSouthernpart.Theslopesare star formation, has power spectrum slopes significantly flatter determinedwithanerrorbaroftheorderof±0.2ingeneral,and thantheotherpowerspectraoftheISMcomponents.Thismight thebreakscalesareevenmorerobust,within15%.Wehavealso beexpected,sincestarformationgivesmorepowertothesmall checkedthe effect of smoothingon the maps. As expected,the scales. Its break scale, related to the plane thickness is higher. power-spectrumatlarge-scaleisunchanged,andonlythepower The GALEX FUV and NUV, tracing the recent, but past, star attheextremeofthesmall-scalesofthespatialresolutiondrops. formation,haveslopesmorecomparabletotheotherISMslopes The break scale also is enlarged by the resolution, when it is however,andhavesmallerbreakscales,implyingthinnerlayers. tooclose,andwecanseeforinstancethatthebreakat70µmis ThebreakscaleinHαislargerthanintheUVbecauseHαtraces largerthanat24and100µm(Table3)onlybecauseofthecoarser theionisedgas,andonlyindirectlytheyoungstars.Theionised resolution. The fact that some maps have a pixel size smaller layer may be thicker, affected by bubbles, filaments, fountain than the Nyquist size (half of the nominalresolution) does not effectsassociatedtostarformation. 5 F.Combesetal.:Dustpower-spectruminM33 has less small-scale structure, since individual cloud cores are invisible. 4.2.Power-spectrumofthemodel Thevariousrunsdevelopdifferentgasstructureandclumpiness, ascanbeseeninFigures11to16,duetobothinitialconditions that are selected more or less stable, or with different sub-grid “temperature”ofthegas,andtothestarformationrateandfeed- back.Somegalaxies,whicharelaunchedwithinitialconditions amongthemoststable,canendupwithamoreperturbedmor- phologyandlargerturbulencethanothersstartedmoreunstable, duetoalargersupernovaefeedback. Fig.8.SameasFigure3,forthepowerspectrumoftheHIand We computed the Fourier analysis in the same way on the CO(2-1)maps. gasmapsobtainedinthenumericalsimulations.Thederivedre- sultsareconvergingtowardsthesamevaluesatthemiddleofthe runs, and we plotall of them for the finalsnapshotsat T= 667 Table3.Power-lawslopesandbreaksforobservations Myr for the sake of simplicity (cf Figures 17, 18 and 19). For comparison,we also plotthepower-spectrumofsomeoldstel- Map slope slope break Extremevalues larcomponents,whichhavealmostonlyonepower-law(Figure LS ss (pc) LS ss (pc) 20).Thesecorrespondtorun2andrun7atT=667Myr.There FUV -1.2 -3.4 44 -0.9/-1.2 -3.4/-3.9 40-45 isstructureuntilabout300pc,andnosmall-scalecomponent,as NUV -1.2 -3.6 56 -1.0/-1.2 -3.0/-3.6 50-56 expectedforacollisionlesscomponent. Hα -.77 -2.4 300 -0.7/-1.0 -2.4/-2.7 250-300 ThefirstresultwhichisobviousinTable4isthatthepredic- 24µm -.9 -2.6 93 -0.9/-1.0 -2.6/-2.7 93-105 tions of the modelsbracketthe valuesobtainedin the observa- 70µm -1.5 -3.5 236 -1.4/-1.5 -3.4/-3.8 230-270 tions,bothintheslopesatlargeandsmallscales,andinthebreak 100µm -1.5 -3.4 93 -1.4/-1.5 -3.2/-3.6 93-105 scales.Thosecanbecomparedtotheactualthicknessofthegas 160µm -1.5 -3.6 165 -1.5/-1.6 -3.6/-4.0 140-165 layer, in the simulations. This is done in Table 4, showing that 250µm -1.6 -4.7 208 -1.4/-1.6 -4.3/-4.7 200-210 thebreakscale doesnotfollowcloselythethickness.Although 350µm -1.3 -4.0 315 -1.3/-1.5 -4.0/-4.4 290-320 bothshouldbe of thesame orderwhen onlythe self-gravityof 500µm -1.1 -4.4 390 -1.1/-1.4 -4.2/-4.6 320-390 CO(2-1) -1.5 -3.7 103 -1.3/-1.5 -3.6/-3.8 100-115 the gas is taken into account, their close relation is blurred by HI -2.4 -3.2 110 -2.0/-2.4 -3.2/-3.6 100-120 otherfactors,andinparticularthefeedbackefficiency.Whenthe feedback is too extreme, the small scale structure of the gas is LS=LargeScale(>500pc),ss=smallscale(<500pc) smoothedout,eventowardslargerscalethantheverticalthick- ness.Sotherangeofbreakscalesiswiderthanthatofpossible Table4.Power-lawslopesandbreaksforsimulations planeheights. The main parametersthat control the turbulence and there- Run slope slope break Comments H forethepower-spectrumofthegasstructurearethestarforma- z LS ss (pc) (pc) tion rate, the amountof feedback,but also the assumed equiv- alent temperature of the gas, or sound speed, directly involved run1 -0.8 -3.6 75 lowfeedback 80 run2 -0.9 -3.4 75 ′′ 80 in the pressureforces.Increasingthe turbulenceby supernovae run3 -2.3 -4.9 650 extremefeedback 217 feedbacksuppressesefficientlythesmall-scalestructure(power- run4 -1.3 -3.8 63 fiducial 120 lawslopesof∼-5.),andthickenssubstantiallythegaslayer,and run5 -2.1 -5.0 300 moreSF(C*) 142 thecorrespondingbreakscale.Withoutstarformationandfeed- run6 -1.1 -4.0 430 higherthreshold 103 back,itisonlypossibletohavegasmorphologiescomparableto run7 -0.8 -2.8 75 noSF 64 theobservedonesifaminimumsoundspeedof12 kms−1isin- run8 -0.9 -2.9 201 noSF,Vs=12 kms−1 110 jected.Figures11to16showthelargevarietyofgasstructures, run9 -1.1 -5.2 250 SF,Vs=12 kms−1 108 according to the variationsof those parameters. It appears that LS=LargeScale,ss=smallscale ourfiducialmodel(run4)istheonewhichcorrespondsthebest ThegasscaleheightHz=2.35(<z2 >−<z>2)1/2 to theM33galaxy,bothforits morphology,andforthetypical power-lawslopesandbreakscale.Ithasastarformationrateof ∼0.7M /yr,andafeedbacksuchthat10%ofsupernovaeenergy ⊙ iscoupledtothekinematicsofthegas.Verleyetal.(2009)found The dustemissionsatvariouswavelengthshavequitecom- forM33astarformationrateof0.5M /yr. patiblepowerspectra.Theirdifferentbreakscalesreflectthein- ⊙ creasingbeamsizewithwavelength.Asforthegas,themolecu- larmediumappearsthinnerthantheatomicone,althoughspatial 5. Discussionandconclusions resolutionmightinfluenceitsbreakscale. Independent of the spatial resolutions, the slope of the WehavepresentedtheFourieranalysisofdustandgasemission power-spectraatsmallscalearesteeperatlargewavelengthsfor maps of M33 at severalwavelengths,and comparedthem with the dust emission. Between 250 and 500µm, the SPIRE maps the HαandUV maps, tracersof recentstar formation.We find have slopes between -4.3 and -4.7, while the MIPS and PACS thatallthepowerspectracanbedecomposedintotworegimes, mapsbetween24and100µmhaveslopescomprisedwithin-2.7 a large-scale(>500pc)power-law,of−1.5 < slope < −1.0, LS and-3.8.Thisissignificant,andtendstoshowthatthecooldust and a steeper small-scale one, with −4.0 < slope < −3.0. ss 6 F.Combesetal.:Dustpower-spectruminM33 Fig.9.Deprojectedmapsofthedustat100(left)and160µm(right).Themajoraxishasbeenrotatedtobevertical.Thecolorscale isinarbitraryunits. Fig.10.Deprojectedmapsofthedustat250(left)and500µm(right).Themajoraxishasbeenrotatedtobevertical.Thecolorscale isinarbitraryunits. Fig.11. Face-on projection of the run2 model, at T=667 Myr. Fig.12. Face-on projection of the run3 model, at T=667 Myr. Leftarethestars,andRight,thegas.Thelinearscalesonboth Leftarethestars,andRight,thegas.Thelinearscalesonboth axesareinkpc,andthecolorscaleisinarbitraryunits. axesareinkpc,andthecolorscaleisinarbitraryunits. Thelatterslopescorrespondquitewelltowhatisfoundforthe lowerpower-spectra,indicatingmorepoweratsmallscales.The MilkyWay (Dickeyetal.2001)orthe SmallMagellanicCloud break scale is of the order of 100-150 pc for the dust and the (Stanimirovicetal.2000).TheHαstarformationtracerhasshal- gas,increasingatlargewavelength,probablyinfluencedbythe 7 F.Combesetal.:Dustpower-spectruminM33 Fig.13. Face-on projection of the run4 model, at T=667 Myr. Leftarethestars,andRight,thegas. Fig.17.Powerspectrumofthegasinmodelsrun1andrun2.The verticaldashlinerepresentsthesofteninglengthofthegravity. Fig.14. Face-on projection of the run7 model, at T=667 Myr. Leftarethestars,andRight,thegas. Fig.18.Powerspectrumofthegasinmodelsrun3andrun4. Fig.15. Face-on projection of the run8 model, at T=667 Myr. Leftarethestars,andRight,thegas. Fig.19.Powerspectrumofthegasinmodelsrun7andrun8. Fig.16. Face-on projection of the run9 model, at T=667 Myr. Leftarethestars,andRight,thegas. 24µmmapisalsorelatedtotheyoungstarformation,thereare nosuchlargebubbles:onlythepointsourcesareprominentin- sidethebubble,ascanbeseeninFigure3ofVerleyetal.(2010). progressivelackofspatialresolution.TheHαbreakscaleissig- ThisexplainsthedifferentbreakscalesbetweenHαandMIPS- nificantly higher (300 pc), implying a thicker layer for ionized 24µmmaps. gas, while the UV layers appears thinner (65 pc). This might It is also possible that the large thickness suggested by the be understood considering that UV originates from stars that Hαanalysisisrelatedtothediffuseionizedgas(DIG)observed are concentrated in the disk whereas the ionised gas could be inseveraledge-ongalaxiestofollowaratherthicklayerofabout present quite far from the plane of the disk: giant HII regions 1kpcheight(see areviewbyDettmar1992).Thisextra-planar havetypicallyroundshapes,andarenotasconfinedtothedisk. ionizedgasisthoughttobeduetostarformation,itispresentin Tabatabaei etal.(2007) have already shown with wavelets that about40% of 74 edge-ongalaxies(Rossa & Dettmar 2003). It theHαmapisdominatedbygiantHIIregions,whicharebubble- canbewidelydiffuse,ortaketheshapeoffilaments,plumesand like with sizes from 100 to 500 pc. By comparison the diffuse bubbles (Rand, 1997). These extra-planar extensions could be component is weaker, which flattens the Fourier ss-slope, and due to UV ionizingphotonsescaping the star formingregions, enlargesthebreakscale.AlthoughthewarmdustintheMIPS- supernovaefeedback,fountaineffects,chimneys,andbepartof 8 F.Combesetal.:Dustpower-spectruminM33 terminethethicknessofthegasplane,atanyorientationonthe plane of the sky, and the actual physical state of the gas, in its star formationphase.Notehoweverthatthecurrentmodelsare notdevoidofshortcomings:theyonlypartlyallowtodistinguish betweenthe12differenttracersofthegasphaseobservedhere, asradiativetransport,dustproperties,etc.arenottakenintoac- count. Comparison of power spectrum analysis between galaxies shouldallowadeterminationoftheirrelativestarformationrate and feedback. This will need however high spatial resolution maps,suchastheonesthatwillbeprovidedwithALMA. ThegasplaneinM33appearsrelativelythickandturbulent, withrespecttoourclosestfiducialmodel,whichcouldbeacon- sequenceofitsrecentheatinginparticularthroughtheM31in- Fig.20. Power spectrum of the stellar component in run2 and teraction(McConnachieetal.2009). run7. The vertical dash line represents the softening length of thegravity. Acknowledgements. Wewarmlythanktheanonymousrefereeforhis/hercon- structive comments and suggestions. We have made use of the NASA/IPAC ExtragalacticDatabase(NED). anactivedisk-halointeraction,withexchangeofmatter(Howk &Savage2000). Thecomparisonwithsimulatedmodelstellsusthatthe ob- References servationofabreakinthepower-spectrumofobservedgasdis- Begum,A.,Chengalur,J.N.,Bhardwaj,S.:2006,MNRAS372,L33 tributionisnotalwaysrelatedtothe2D/3Dtransition,andindi- BianchiS.,XilourisE.M.:2011,A&A531,L11 catesthethicknessofthegaseousdiskonlywhenthelatterisnot BlockD.L.,Puerari,I.,Elmegreen,B.G.,Bournaud,F.:2010ApJ718,L1 heated by a strong star formationrate and a strong supernovae BoquienE.,Calzetti,D.,Kramer,C.etal.:2010,A&A518,L70 BoquienE.,Calzetti,D.,CombesF.etal.:2011AJ142,111 feedback. For a quiescent galaxy, the break should still be an Bournaud,F.,Combes,F.,Jog,C.J.,Puerari,I.:2005,A&A438,507 indicationofthegaslayerthickness. Bournaud F., Elmegreen, B. G., Teyssier, R., Block, D. L.,Puerari, I.: 2010, Block etal.(2010) found in the LMC that the small scale MNRAS409,1088 slopesofthepowerspectraweresteeperatlongerwavelengths, BraineJ.,GratierP.,KramerC.etal.:2010,A&A518,L69 BronfmanL,CohenR.S.,AlvarezHetal.,1988,ApJ324,248 i.e. the cool dust was smoother than the hot emission. We find CorbelliE.,SalucciP.:2007,MNRAS311,441 a similar result. The slope of the SPIRE mapsare around-4.5, CrovisierJ.,DickeyJ.:1983A&A122,282 whilethewarmerdustat24-100µmhasanaverageslopeof-3.2. DalcantonJ.J.,StilpA.M.:2010,ApJ721,547 This observation can be interpreted in terms of star formation DalcantonJ.J.,YoachimP.,BernsteinR.:2004,ApJ608,189 occuringintheclumpymedium.Oncethestarsareformed,they DameT.M.,ThaddeusP.,1994,ApJ436,L173 DettmarR.J.:1992,Fund.CosmicPhysics,15,143 heatthedustaround,andthisexplainswhytheemissionatshort- DiMatteoP.,,CombesF.,MelchiorA.-L.,SemelinB.:2007,A&A468,61 est wavelength (warmer dust) is more clumpy. Also the break Dickey, J. M., McClure-Griffiths, N. M., Stanimirovic, S., Gaensler, B. M., scale, and thusthe thicknessof the plane,is largerforthe cold Green,A.J.:2001ApJ561,264 gas.Thiscouldbedueinadditiontothefactthatthecoldcom- DopitaM.A.:1985ApJ295,L5 DuttaP.,Begum,A.,Bharadwaj,S.,Chengalur,J.N.:2008MNRAS384,L34 ponent is much more extended in radius (Kramer etal.2010), DuttaP.,Begum,A.,Bharadwaj,S.,Chengalur,J.N.:2009aMNRAS397,L60 anditiswellknownthatthegaslayerisflaringintheouterparts DuttaP.,Begum,A.,Bharadwaj,S.,Chengalur,J.N.:2009bMNRAS398,887 ofgalaxies.Asforthegasemission,themoleculargasappears ElmegreenB.G.,Kim,S.,Staveley-Smith,L.:2001ApJ548,749 moreclumpy,butonlyslightlymoresothantheatomicgas.And ElmegreenB.G.,Leitner,S.N.,Elmegreen,D.M.,Cuillandre,J-C.:2003bApJ the break occursat aboutthe same scale. Since it is likely that 593,333 ElmegreenB.G.,ElmegreenD.M.,Leitner,S.N.:2003a,ApJ590,271 in the relatively quiescent M33 galaxy, the break indicates the ElmegreenB.G.,ScaloJ.:2004,ARAA42,211 thickness of the gas layer, this situation is not alike the Milky ElmegreenB.G.,Elmegreen,D.M.,Chandar,R.,Whitmore,B.,Regan,M.:2006 Way,wheretheCO componentis significantlythinnerthanthe ApJ644,879 HI(Bronfmanetal.1988).AthickCOlayer,3timesaswideas FalgaroneE.,Phillips,T.G.,WalkerC.K.:1991,ApJ378,186 Freedman,W.L.,Wilson,C.D.,Madore,B.F.1991,ApJ,372,455 the dense moleculargas layer, and comparableto the HI layer, GildePaz,A.,Boissier,S.,Madore,B.F.,etal.:2007,ApJS,173,185 has been observed(Dame & Thaddeus1994), but it involvesa GoldmanI.:2000ApJ541,701 small mass. M33 being a galaxy of intermediate mass, could Gratier,P.,Braine,J.,Rodriguez-Fernandez,N.J.,etal.2010,A&A,522,A3 haveindeedathickergaslayerthantheMilkyWay(Dalcanton GreenD.A.:1993,MNRAS262,327 etal.2004). HoopesC.G.,WalterbosR.A.M.,BothunG.D.:2001,ApJ559,878 HowkJ.C.,SavageB.D.:2000,AJ119,644 We also showed that numerical models predict gas mor- JungwiertB.,CombesF.,PalousJ.:2001,A&A36,85 phologiesandsmallscale power-lawswhichbracketthe obser- Khalil, A.,Joncas,G.,Nekka,F.,Kestener, P.,Arneodo,A.:2006,ApJS165, vationaldata.Itispossibletoobtainacolderoramoreturbulent 512 interstellarmedium,byvaryingthestarformationfeedbackand Kramer,C.Buchbender,C.,Xilouris,E.M.etal.:2010,A&A518,L67 Lazarian,A.,Pogosyan,D.:2004,ApJ616,943 the sound speed of the gas. The Fourier analysis of the mod- Lazarian,A.,Pogosyan,D.,Vzquez-Semadeni,E.,Pichardo,B.:2001ApJ555, elsrevealsthatthebreakscale isa goodindicatorof theactual 130 thicknessofthegaslayer,althoughitvarieswithinawiderrange McConnachieA.W.,Irwin,M.J.,Ibata,R.A.etal.2009,Nature461,66 whenextremefeedbackisused,andsmoothesoutthesmallscale OdekonM.C.:2008ApJ681,1248 PadoanP.,Kim,S.,Goodman,A.,Staveley-Smith,L.:2001ApJ555,L33 gasstructures.The power-lawslopesobtainedin the modelsat PfennigerD.,CombesF.:1994,A&A285,94 largeandsmallscalereflectquiteaccuratelythegasmorphology Rand,R.J.:1997,ApJ474,129 andturbulence.Fourieranalysesarethereforeagoodtooltode- RossaJ.,DettmarR.J.:2003,A&A406,493 9 F.Combesetal.:Dustpower-spectruminM33 Roychowdhury, S., Chengalur, J. N., Begum, A., Karachentsev, I. D.: 2010, MNRAS404,L60 SanchezN.,Anez,N.,Alfaro,E.J.,CroneOdekon,M.:2010ApJ720,541 SemelinB.,CombesF.:2005,A&A441,55 StanimirovicS.,LazarianA.:2001,ApJ551,L53 Stanimirovic S.,Staveley-Smith, L.,vanderHulst,J.M.etal.2000:MNRAS 315,791 Stutzki,J.,Bensch,F.,Heithausen,A.,Ossenkopf,V.,Zielinsky,M.:1998,A&A 336,697 TabatabaeiF.S.,BeckR.,KrauseM.etal.:2007,A&A466,509 Thilker,D.A.,Hoopes,C.G.,Bianchi,L.etal.:2005,ApJ619,L67 Verley,S.,Hunt,L.K.,Corbelli,E.,Giovanardi,C.:2007,A&A,476,1161 VerleyS.,Corbelli,E.,Giovanardi,C.,Hunt,L.K.:2009,A&A493,453 VerleyS.,Relano,M.,Kramer,C.etal.:2010,A&A518,L68 Fig.B.1.PowerspectrumofthePACS100µmmap,atdegraded AppendixA: Fourieranalysisofthedensity resolutionsof11and18′′. Theverticaldashlineatrightrepre- Until now, we have essentially analysed the small scale struc- sents the smoothed spatial resolution, on the major axis. Since tureoftheinterstellarmediuminvarioustracers,butignoredthe thede-projectiontoaface-ongalaxyimpliesafactor1/cos(56◦) azimuthal dependence. The different tracers also reveal a con- =1.79largerbeamontheminoraxis,wehavealsoplottedasec- trasted spiral structure, that is interesting to present, at least in onddashedverticallineatleft,indicatingthegeometricmeanof tworepresentativedeprojectedmaps,theHerschel100µmimage theresolutiononbothminorandmajoraxis. (dustandgasimagesshowsimilarfeatures),andtheHαimage (starformation,similartoUVimages). WehavecomputedtheFouriertransformofthecorrespond- ingsurfacedensities,decomposedas: µ(r,φ)=µ (r)+ a (r)cos(mφ−φ (r)) 0 m m X m where the normalisedstrength of the Fourier componentm isA (r) = a (r)/µ (r).Thus,A representsthenormalizedam- m m 0 2 plitudeforam = 2distortionlikeabarorspiralarm,atagiven radius,andA representsthesameforthedisklopsidedness.The 1 quantityφ denotesthepositionangleorthephaseoftheFourier m componentm.Suchadensityanalysishasbeenmadeforexam- plebyBournaudetal.(2005),tostudyasymmetriesandlopsid- ednessingalaxies,andcorrelateitwithm=2features. Fig.B.2.SameasFigureB.1forthedegradedresolutionsof26 FigureA.1showsaplotoftheamplitudeAm(r)andthephase and36′′. φ (r) versus radius r for M33, at 100µm and Figure A.2, the m sameforHα.Bothshowaprominentspiralfeatureatradii∼3-4 kpc, with a phase decreasing slightly with radius, in the direct sense. The m = 2 distortions show accompanying m = 4 har- monics. The lopsidednessalso increases with radius, while the thecorrespondingphaseisfairlyconstant. These plots show that the spiral structure in M33 is mildly contrasted,inbothgasandSFRtracers.Indeed,thenormalised m = 2 amplitude is between 0.4 and 0.8. There is also a sig- nificant asymmetry and lopsidedness, as testified by the strong m=1amplitudecomparablewiththem=2.Thisexplainswhy the slopes at large-scale were more different between the two halvesofthegalaxythanthesmall-scaleones.Thislargeasym- metrycouldbeduetothecurrentinteractionbetweenM33and M31(McConnachieetal.2009). AppendixB: Theeffectofresolution We have explored the effect of spatial resolution in the deter- mination of the various slopes and break scales, in smoothing the PACS-100µm map at variousresolutions, from the original 7′′to11,18,26and36′′,correspondingto44,72,104and144 pconthemajoraxis.TheresultingfitsareshowninFigureB.1 andB.2,tobecomparedwithFigure3.Thebreakscalesarere- spectively97,111,135,195pc,insteadof93pcfortheoriginal 7′′resolution.Thedeterminationispossibleonlyforthetwofirst cases. 10

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Astronomy&Astrophysicsmanuscriptno.m33-fourier (cid:13)c ESO2012 January13,2012 ⋆ Dust and gas power-spectrum in M33 (HERM33ES) F.Combes1,M.Boquien2,C.Kramer3,E.M.Xilouris4,F.Bertoldi5,J.Braine6,C.Buchbender3,D.Calzetti7,P. Gratier8,F.Israel9,B.Koribalski10,S.Lord11,G.Quintana-Lacaci3,M.Relan˜o12,M.Ro¨llig13,G.Stacey14,F.S. Tabatabaei15,R.P.J.Tilanus16,F.vanderTak17,P.vanderWerf9,andS.Verley12 1 Observatoire de Paris, LERMA (CNRS:UMR8112), 61 Av. de l’Observatoire, F-75014, Paris, France e-mail: [email protected] 2 Laboratoired’AstrophysiquedeMarseille,UMR6110CNRS,38rueF.Joliot-Curie,13388,Marseille,France 3 InstitutoRadioastronomiaMilimetrica,Av.DivinaPastora7,NucleoCentral,E-18012Granada,Spain 2 4 InstituteofAstronomyandAstrophysics,NationalObservatoryofAthens,P.Penteli,15236Athens,Greece 1 5 ArgelanderInstitutfu¨rAstronomie.AufdemHu¨gel71,D-53121Bonn,Germany 0 6 Laboratoired’AstrophysiquedeBordeaux,Universite´ Bordeaux1,ObservatoiredeBordeaux,OASU,UMR5804,CNRS/INSU, 2 B.P.89,FloiracF-33270,France n 7 UniversityofMassachusetts,DepartmentofAstronomy,LGRT-B619E,Amherst,MA01003,USA a 8 IRAM-InstitutdeRadioAstronomieMillime´trique,300RuedelaPiscine,38406-St.Martind‘He`res,France J 9 LeidenObservatory,LeidenUniversity,POBox9513,NL2300RALeiden,TheNetherlands 2 10 ATNF,CSIRO,POBox76,Epping,NSW1710,Australia 1 11 IPAC,MS100-22CaliforniaInstituteofTechnology,Pasadena,CA91125,USA 12 Dept.F´ısicaTeo´ricaydelCosmos,UniversidaddeGranada,Spain 13 KOSMA,I.PhysikalischesInstitut,Universita¨tzuKo¨ln,Zu¨lpicherStraße77,D-50937Ko¨ln,Germany ] O 14 DepartmentofAstronomy,CornellUniversity,Ithaca,NY14853,USA 15 MaxPlanckInstitutfu¨rAstronomie,Ko¨nigstuhl17,D-69117Heidelberg,Germany C 16 JAC,660NorthA’ohokuPlace,UniversityPark,Hilo,HI96720,USA h. 17 SRONNetherlandsInstituteforSpaceResearch,Landleven12,9747ADGroningen,TheNetherlands p Received16October2011/Accepted10January2012 - o r ABSTRACT t s a Powerspectraofde-projectedimagesoflate-typegalaxiesingasand/ordustemissionareveryusefuldiagnosticsofthedynamics [ andstabilityoftheirinterstellarmedium.Previousstudieshaveshownthatthepowerspectracanbeapproximatedastwopower-laws, ashallowoneatlargescale(largerthan500pc)andasteeperoneatsmallscale,withthebreakbetweenthetwocorrespondingtothe 1 line-of-sightthicknessofthegalaxydisk.Thebreakseparatesthe3Dbehaviouroftheinterstellarmediumatsmallscale,controlled v bystarformationandfeedback,fromthe2Dbehaviouratlargescale,drivenbydensitywavesinthedisk.Thebreakbetweenthese 8 tworegimesdependsonthethicknessoftheplanewhichisdeterminedbythenaturalself-gravitatingscaleoftheinterstellarmedium. 5 WepresentathoroughanalysisofthepowerspectraofthedustandgasemissionatseveralwavelengthsinthenearbygalaxyM33. 5 Inparticular,weusetherecentlyobtainedimagesatfivewavelengthsbyPACSandSPIREonboardHerschel.Thelargedynamical 2 range(2-3dexinscale)ofmostimagesallowsustodetermineclearlythechangeinslopesfrom-1.5to-4,withsomevariationswith 1. wavelength.Thebreakscaleisincreasingwithwavelength,from100pcat24and100µmto350pcat500µm,suggestingthatthe 0 cooldustliesinathickerdiskthanthewarmdust,maybeduetostarformationmoreconfinedtotheplane.Theslopeatsmallscale 2 tendstobesteeperatlongerwavelength,meaningthatthewarmerdustismoreconcentratedinclumps.Numericalsimulationsofan 1 isolatedlate-typegalaxy,richingasandwithnobulge,likeM33,arecarriedout,inordertobetterinterprettheseobservedresults. : Varyingthestarformationandfeedbackparameters,itispossibletoobtainarangeofpower-spectra,withtwopower-lawslopesand v breaks,whichnicelybracketthedata.Thesmall-scalepower-lawisindeedreflectingthe3Dbehaviourofthegaslayer,steepening i stronglywhilethefeedbacksmoothesthestructures,byincreasingthegasturbulence.M33appearstocorrespondtoafiducialmodel X withanSFRof∼0.7M /yr,with10%supernovaeenergycoupledtothegaskinematics. ⊙ r a Keywords.galaxies:individual:M33–galaxies:spiral–galaxies:infrared–galaxies:starformation 1. Introduction ture,orapower-lawpower-spectrum.TostudytheatomicISM, Crovisier&Dickey(1983)andGreen(1993)studiedthepower- Quantifyingthemulti-scalestructureoftheinterstellarmedium spectrum as a function of inverse scale of the 21cm emission, (ISM)ingalaxiesisadifficultenterprise,butrewardingsinceit and found a power-law of slope between -2 and -3, indepen- betrays the underlyingphysicalphenomena,controllingits dy- dent of the distance and the velocity of the HI. This power- namics.IthasbeenknownforalongtimethattheISMreveals law slope was observed to be steeper from -3 to -4, when the no preferentialscale and is well representedby a fractal struc- broadestvelocitieswereaveraged,thusincludingthewarmgas, withmoreturbulence(Dickeyetal.2001).Goldman(2000)and Sendoffprintrequeststo:F.Combes ⋆ Herschel is an ESA space observatory with science instruments Stanimirovicetal.(2000)extendedtheISMstudiesintheMilky providedbyEuropean–ledPrincipalInvestigatorconsortiaandwithim- Way to the Small Magellanic Cloud (SMC), where a power- portantparticipationfromNASA. law of slope -3.1 to -3.4 was found for the dust and gas, and 1 F.Combesetal.:Dustpower-spectruminM33 interpreted as 3D turbulence, although the index is shallower of 3D-turbulence,with a flattening at large scales, and a break than the -3.7 Kolmogorov spectrum (Stanimirovic & Lazarian corresponding to several hundred parsecs, typical of the plane 2001).A breakin scale hasbeensearchedforin orderto iden- thickness. This is true for flocculent and density-wave galax- tifytheenergyinjectionmechanism,butnonehasbeenfoundat ies,althoughthespiralarmwavedominatesatverylargescales. smallscale(i.e.smallerthan500pc).Insteadaflatteningtowards In M33, Elmegreen etal.(2003b) analysed optical and Hα im- larger scales has been measured by Elmegreen etal.(2001) in ages and found a very shallow power-law of slope -0.7, which theatomicgasoftheLargeMagellanicCloud(LMC).Theyin- theyattributetoflatteningduetocontaminationbystrongpoint terpret this flattening as a change of dimensionality, from 3D sources(foregroundstars). For the face-ongalaxyNGC 628, a at small scale to 2D at large scale, where the dynamics of the shallowslopeof-1.5isfound,withnobreakinspiteofthevery galaxy plane is dominating,and identify the scale of the break high spatial resolution obtained with HST (scale below 10 pc, asthethicknessoftheplane.Thismethodtofindthescaleheight Elmegreen etal.2006). Odekon (2008) however finds a transi- ofdiskshasbeendevelopedbyPadoanetal.(2001),whofound tion scale of about 300 pc between a steep spectrum at small ∼ 180 pc in the LMC. Recently, Block etal.(2010) have anal- scale and flattening at large scale, in the stellar distribution of ysed thedustmapsof theLMC fromSpitzer,andalso confirm M33.Thisscaledoesnotchangemuchwiththeageofthestel- thisbreakscale at 100-200pc. Fromhighresolutionnumerical lar population considered, however the slope is steeper for the simulationsfittedtoalate-typegalaxyliketheLMC,Bournaud bluest and youngest stars. Sanchez etal.(2010) investigate all etal.(2010) were able to retrieve the characteristics of the ob- the young components in M33, including young stars, HII re- served dust power-spectrum. The source of turbulence could gions and molecular clouds, and also find a clear transition re- thenbeduetolarge-scaledifferentialrotation,coldgasaccretion gion,butintherange0.5-1kpc.Thefactthatdifferentresultsare orgalaxyinteractions(Lazarianetal.2001,Elmegreen&Scalo obtainedforthesamegalaxiescallsfornewstudiestoclarifythe 2004).Khaliletal.(2006),froma 2DwaveletanalysisofHI in situation. ourown Galaxy,find a power-lawslope of -3 everywhereover The galaxy M33 has been observed with Herschel PACS thesecondquadrant,anddonotdetectthevariationswithveloc- andSPIREatallwavelengths,bytheKeyProjectHERM33ES, ity slices predicted by Lazarian & Pogosyan (2004). However and the photometric maps have been presented in Kramer they discover an anisotropy linked to spiral arms. The fractal etal.(2010),Boquienetal.(2010)andVerleyetal.(2010).These approach was widely developed for molecular clouds, which providehighsignal-to-noisemapswithalargedynamicalrange aremoreself-gravitatingthantheHIgas(Falgaroneetal.1991, of scales ofthe dustemission at manywavelength,allowingto Pfenniger&Combes1994,Stutzkietal.1998). probewarmandcooldust.Thelargeandsmallscale structures Uptonow,thebreakscaleintheHIdistributionhasnotbeen of the gas and dust are studied through Fourier analysis. We identified unambiguously in external galaxies other than the present in Section 2 the differentmaps used, and how they are LMC,althoughithasbeensearchedfor.Duttaetal.(2008)stud- processed,anddeprojectedtoaface-onorientation.Theresults iedtheface-ongalaxyNGC628,andfoundanHIpowerspec- will be comparedto numericalmodels, which technique is de- trumwithasinglepower-lawslopeof-1.6,overthescales0.8to scribedinSection3.Resultsarepresentedforbothobservations 8kpc,theirspatialresolutionbeinginsufficienttoreachthescale andmodelsinSection4,anddiscussedinSection5.Weadopta height of the gaseous plane. In DDO210, Begum etal.(2006) distanceof840kpcforM33(Freedmanetal.1991). founda power-lawslope of -2.75overthe scales 80 to 500pc. ThissteepslopeiscompatiblewithwhatisfoundintheMWand LMC/SMC at small scales, driven by 3D-turbulence. In other dwarfsalso(Duttaetal.2009b)asingleslopeisfound.Theface- ongalaxyNGC1058isthefirstoneforwhichthetransitionfrom 3D to 2D turbulencehas been claimed (Dutta etal.2009a):the slopechangesfrom-2.5over0.6to1.5kpc,to-1over1.5-10kpc scales. The scale heightis thenestimated at490pc, which isa relativelylargevaluefortheHIlayer.IntheSMC,whileasteep power-law is found consistently by several studies, no break scale has been found, and no transition to the 2D-turbulence (Roychowdhuryetal.2010).It is suggested thatdwarf galaxies havegasdisksconsiderablythickerthannormalspiralgalaxies, whichsupportpreviousstudiesfromthethicknessofdustlanes Fig.1.Rotationcurveforourlate-typeScdmodel,comparedto in edge-on galaxies (e.g. Dalcanton etal.2004). The transition theobservedpoints(symbolsanderrorbars)compiledforM33 between thick and thin gas disks was found at a mass corre- byCorbelli&Salucci(2007).Inadditiontothecircularvelocity spondingtoacircularvelocityof120 kms−1.Forgalaxieswith ofthemodel,thecharacteristicfrequenciesΩ,Ω−κ/2,etc.are lowinclinationontheskyplane,thethicknessofgasdiskscanbe plotted. estimatedthroughtheratioofgasvelocitydispersiontorotation (Dalcanton&Stilp 2010).ThecaseofM33isexactlyinterme- diate,havingarotationalvelocityof120 kms−1.Boththickand thindiskscouldbeexpected. 2. Observationaldata Anothertooltoexplorethescaleheightofdiskswithhigher spatial resolution, apart from studying edge-on galaxies (e.g. To study the interstellar medium structure, we use all possible Bianchi & Xilouris 2011), is to study young components re- tracers,beginningbygasanddustemission (HI, CO lines, and lated to the gas, as young stars just formed out of the ISM. far-infrared photometry), and also star formation from the gas Elmegreenetal.(2003a)studiedthepower-spectrumofstarfor- (Hα,ultra-violetcontinuum). mationmapsinnearbygalaxies,andfoundthesamebehaviour Forthedustemission,weuseHerschelobservationstakenin asfortheLMCinHI:asteeppower-lawatsmallscales,typical the context of the HERM33ES open time key project (Kramer 2 F.Combesetal.:Dustpower-spectruminM33 Table1.Characteristicsofthedifferentmapsused Map Resolution Npix pixelsize (′′) (pc) (′′) (pc) Galex-FUV 4.3” 17 2560 1.5” 6 Galex-NUV 5.3” 21 2560 1.5” 6 Hα 6” 24 1920 2” 8 MIPS-24µm 6” 24 2560 1.5” 6 MIPS-70µm 18” 72 850 4.5” 18 PACS-100µm 7” 28 2268 1.7” 7 PACS-160µm 11” 44 1344 2.85” 12 SPIRE-250µm 18” 72 800 6” 24 SPIRE-350µm 26” 104 480 10” 41 SPIRE-500µm 36” 144 342 14” 57 CO(2-1) 12” 48 1470 2” 8 Fig.3.PowerspectrumoftheSpitzer-MIPS70µmmap,andthe HI-21cm 12” 48 1200 4” 16 PACS-100µmdustemissionmapofM33.Theverticaldashline atrightrepresentsthe spatialresolutionof the observations,on Adistanceof840kpchasbeenadoptedforM33.1”=4pc the major axis. Since the de-projectionto a face-ongalaxyim- Resolutionscorrespondtothemajoraxisdirection, theyare1/cos(56◦)=1.79largerintheotherdirection plies a factor 1/cos(56◦) = 1.79largerbeam on the minoraxis, Npixisthenumberofpixelsineachdimensionofthemap we have also plotted a second dashedvertical line at left, indi- catingthe geometricmeanof the resolutiononboth minorand major axis. Statistical error bars are given in this first figure, and then omitted later for clarity. They are significant only at the largest scales (small k), where the statistical errors are the largest. gasanddustemissiontracedirectlythecollisionalmaterial,with nointerveningandperturbingemission(likegalacticstars). Fig.2. 2D power-spectrum for the PACS 100µm image. The colorscaleisinarbitraryunits. etal.2010) in combination with Spitzer MIPS data (Verley etal.2007),spanningwavelengthsfrom24µmto500µm. The Herschel PACS data covering the 100µm and 160µm bands, have been described in Boquien etal.(2010, Fig.4.Same asFigure3, forthe powerspectrumofthe PACS- 2011). The SPIRE observations obtained at 250, 350 and 160µmandSPIRE-250µmdustmap. 500µm are displayed in Kramer etal.(2010) and Verley etal.(2010). We also use ground-based Hα observations presented in Hoopes etal.(2001) in order to trace the current (∼ 10 Myr) starformationandGALEXdata(Thilkeretal.2005,GildePaz 3. Numericalmodel etal.2007) to trace the recent (∼ 100 Myr) star formation. To trace the atomic gas, we use the HI 12′′resolution map pub- In order to comparewith what is expectedfrom the dynamical lished in Gratier etal.(2010), and for the molecular gas, their instabilities of the interstellar medium, we have run idealised partialCO(2-1)12′′resolutionmap(Gratieretal.2010). simulations,fittedforalate-typegalaxywithoutbulgeandrich All useful characteristics of the images are summarized in ingaslikeM33. Table1.Allmapshavebeendeprojected,withtheadoptedincli- nationof56◦ andpositionangleof22.5◦,beforeFourieranaly- 3.1.Technique sis. Thedataanalysedinthepresentworkaremoreappropriate The numericalcode is based on the TREE/SPH version devel- thanearlierdata(Elmegreenetal.2003b,Odekon2008,Sanchez opedbySemelin&Combes(2005)anddiMatteoetal.(2007), etal.2010)tostudythepowerspectraandthestructureofthein- with star formationandfeedback.The TREE gravitationalpart terstellarmedium,sincetheyreferdirectlytothegasanddustin of the code deals with two collisionless components: the stars M33,andtheyhavethedynamicalrange(from40pcto20kpc) andthedarkmatter,whichdifferonlybytheirinitialconditions. andspatialresolutiontorevealthethicknessofthegaslayer.The It deals also with the gas component,whose hydrodynamicsis 3 F.Combesetal.:Dustpower-spectruminM33 feedbackisnotwellknown,anddependsonthecouplingofthe supernovae and the gas (whether the young stars have already left their birth clouds when they explode, etc..). Some energy and supernovaexplosionsare also due to the old populationof stars(e.g.Dopita1985). 3.2.Initialconditions The initial stellar componentof the galaxy is selected to be of typeScd,similartotheM33galaxy. The stellar and gaseous disks are represented initially by Miyamoto-Nagai functions, with characteristic height parame- tersof170pcand70pcrespectively.Thestellardiskhasamass of0.5×1010M ,withacharacteristicradiusof2kpc,whilethe Fig.5.SameasFigure3,forthepowerspectrumoftheSPIRE- ⊙ gasdiskhasamassof1.5×109M ,witharadius2.5kpc.Both 350µmand500µmcooldustmap. ⊙ areembeddedinasphericalhaloofdarkmatter,representedby a Plummer sphere of mass 4 × 1010 M within 18 kpc, with a ⊙ treated with the SPH (Smooth Particle Hydrodynamics) algo- characteristicradiusof9.5kpc. rithm,usingadaptiveresolution.Thesofteningparameter,orthe Theinitialstabilityofthegasandstellardisksiscontrolled spatialresolutionofgravity,istakentobe30pc,andtheaverage by the Toomre parameter Q, ratio of the radial component of size ofa gaseousparticle(theSPH kernel,orspatialresolution the velocity dispersion to the critical one σcrit = 3.36GΣ/κ, Σ ofthehydrodynamics)isalsoaround30pc,althoughitcandrop being the disk surface density of the component(gas or stars), down to 3 pc in overdense regions. The number of particles is and κ the epicyclic frequency.We select equal Toomre param- 1.2million,with0.6milliongasparticles,0.4milliondarkmat- eters for gaseous and stellar disks. The azimuthalvelocity dis- terparticles,and0.2million”stars”atthebeginningofthesim- persions verify initially the epicyclic ratio σθ/σr = κ/2Ω, and ulations.Thegasparticlesareassumedtofollowanisothermal thez-componentsarefixedbytheisothermalequilibriumofthe equationofstate,withpressureforcescorrespondingtoasound disks,intheverticaldirection. velocity of 6 kms−1, in the fiducial model. Shocks are treated Thegasisisothermal,butitsequivalentvelocitydispersion, withaconventionalformofartificialviscosity,withparameters governingthepressure,maybevariedbetween6and12 kms−1. givenindiMatteoetal.(2007).Allphysicalquantities(density Therecipesforstarformation:efficiency,densitythreshold,and andforces)arecomputedaveragingoveranumberof50neigh- feedback,are also varied;theyhavea significantimpactonthe bours. The equationsof motion are integratedusing a leapfrog interstellargasstructure. algorithmwithafixedtimestepof0.3Myr. The initial conditions of the runs described here are given Star Formation is taken into account in most models, with inTable2.Therotationcurvecorrespondingtooneoftheruns an assumed Schmidt law, of exponent n=1.5, i.e. for each gas isplottedincomparisontothedatapointsinFigure1.Allruns particleofmass M , andvolumicdensityρ computedover havesimilarrotationcurves.Giventheuncertainties,allthero- gas gas itsneighbours: tationcurvesarecompatiblewiththedata. dM gas =C ρ1/2 Mgasdt ∗ gas Table2.Initialconditionsofthesimulations whereinmostruns,thestarformationconstantC iscalibrated ∗ Run Q V C threshold ǫ suchthatthegasconsumptiontime-scaleisoftheorderof2Gyr ∗ s ∗ =Q (km/s) at.cm−3 (feedback) (then C = C ). C is increased in some of the runs, to check g ∗ 0 ∗ its influence on the gas scale height. In most simulations, the run1 2. 6. C 2×10−6 10−4 0 densitythresholdforstar formationis set verylow,and isthen run2 2. 6. C 2×10−6 10−3 0 increasedinafewruns.Starformationisthentakenintoaccount run3 2. 6. C 2×10−6 1. 0 foreach gasparticle, usingthe hybridparticlesscheme (e.g.di run4 2. 6. C 2×10−6 10−1 0 Matteoetal.2007),beforesufficientstellarmassisaccumulated run5 1.5 6. C x20 2×10−3 10−1 0 anda star particlecanbe created.Continuousmasslossis also run6 1. 6. C0x20 4 10−1 considered, with about 40% of the stellar mass loss by young run7 1. 6. 0. – – run8 1. 12. 0. – – massivestarsafter5Gyr(Jungwiertetal.2001). run9 1. 12. C 2×10−6 10−1 The energy reinjected by supernovae into the interstellar 0 mediumisassumedtobeakinematicfeedbackonneighboring C0issuchthatthegasdepletiontimeis2Gyr gas particles: each neighbourof a SPH particle having formed δmofstars isgivena radialkickin velocityawayfromthe su- pernovaformed.Theenergyavailableiscomputedfromδm,as- sumingaScaloIMF(about0.5%ofthestarsformedhaveamass 4. Results larger than 8 M and explode as supernovae), the energy of a ⊙ supernova E = 1051 erg is distributed to the 50 neighbours 4.1.Power-spectrumofthedata SN accordingtotheirweight(theSPHkernel),withanefficiencyǫ, The power spectra of the 2D projected maps were analysed whichhasbeenvariedfrom10−4to1,with10%beingthefidu- throughthe2DFouriertransform: cial value (see Table 2). For the fiducial feedback efficiency, a particlemightgetakickof∼1 kms−1ateachtime-step,forthe extreme feedback, it can reach 5 kms−1. The efficiency of the F∗(kx,ky)=Z Z F(x,y)e−i(kxx+kyy)dxdy x y 4 F.Combesetal.:Dustpower-spectruminM33 where F(x,y) is the intensity of each pixel, and k and k are influencetheresults.Wehavetruncatedthepowerspectraatthe x y thewavenumber,conjugatetoxandy,varyingastheinverseof Nyquistsampling,inthefigures. scales.Thefull2Dpowerspectrumisgivenby Lookingat Tables 1 and 3, it is oviousthat the break scale isaffectedbythespatialresolutionofthemaps.Fivemapshave P(k ,k )=(Re[F∗])2+(Im[F∗])2 a ratio between break and resolution scales lower or equal to x y 3 (NUV, the 3 SPIRE, and CO), for the others, the ratio is 3.8 An example of 2D-power spectrum is given for the PACS- (FUV), 12.5 (Hα), 4.2 (MIPS-24µm), 3.9 (MIPS-70µm), 3.9 100µm image, in Figure 2. The one-dimensional power- (PACS-100µm),3.6(PACS-160µm)and3.3(HI).Forthose,the spectrum P(k) can be calculated through azimuthal averaging, determinationofthe breakappearssignificant.Asfortheslope inthe(k ,k )plane,usingk2 = k2+k2.Thepowerspectra P(k) determination, the large-scale ones are not affected, and their x y x y are plotted for the various maps in Figures 3, 4, 5, 6, 7 and 8. variation is meaningful: the LS-slopes are flatter for the star- Two slopeshavebeenfittedatsmall(ss) andlargescales(LS), formationtracersthanforthedustandgas,andtheyaregrowing and the differentvalues can be found in Table 3, together with steeperwithwavelength. thebreakscaleseparatingthetworegimes.Inmostofthecases, thepowerspectrashowclearlytwodifferentregimes,withdiffer- entslopes,andsmallandlarge-scales,forscaleslargerthanthe spatial resolution.A knee is apparentbetween the two regimes (ssandLS),andindicatesthebreakscale.Wehaveusedaleast- square-fit program to determine the slopes of the two power- laws, and left open the position of the separation between the twostraightlines.Varyingthevalueoftheseparation,weselect forthebreakthevaluewhichminimisesthesumoftheχ2ofthe twolinefits(ssandLS).Ingeneraltheminimumisquiteclear. There is one ambiguous case, however, which is the HI map, inFigure8.Thetwoslopesareclose,andthedeterminationless trustworthy.ExamplesofthedeprojectedimagesoftheHerschel dustemission are displayedin Figures9 and 10. In additionto the power spectra, the azimuthal large-scale structure develops spiral arms, that are quantified and studied in the AppendixA. Fig.6.SameasFigure3,forthepowerspectrumoftheHαand The determinationof the break scale is obviously more secure MIPS-24µmmaps. when it is found significantly larger than the resolution scale of the observations. We have tested the effect of spatial reso- lution by smoothing the PACS-100µm map in Appendix B. In theobservedpowerspectra(e.g.Figures3-7),thelimitofthein- strumentalresolutionisclearlyseen,throughthesuddenchange fromastraightlinetoadroppingcurve,atlargek.Wehaveindi- catedwithtwoverticaldashedlines,thevalueoftheresolution scaleonthemajoraxis,andalsothegeometricmeanofthereso- lutiononbothminorandmajoraxis,sincethede-projectiontoa face-ongalaxyimpliesafactor1/cos(56◦)=1.79largerbeamon theminoraxis.WeconcludefromAppendixBthatbreakscales atleasttwicelargerthanthebeamscalearerobustwithrespect toresolutioneffect. Wehavealsotestedtherobustnessofourdeterminationsby computingthepowerspectrausingonlyonesideofthegalaxy. Theeasiestwayto computethat, withthesame algorithm,was to remove in the map one half of the galaxy, say the southern part, and complete the map by extending the northern part by Fig.7.SameasFigure3,forthepowerspectrumoftheGALEX point symmetry through the center. For all maps, the different NUVandFUVmaps. slopesandbreakscales forthe twosidesofthe galaxyare also displayedinTable3:thelastcolumnsdisplaytheextremevalues foundfortheslopesandbreaks.Thisproceduremaximizesthe ItisinterestingtonotethattheHαmap,tracinggasionized error bars, especially in the LS-slopes, since the Norhern part by the UV photons from massive stars and, indirectly, current ofM33isquitedifferentfromtheSouthernpart.Theslopesare star formation, has power spectrum slopes significantly flatter determinedwithanerrorbaroftheorderof±0.2ingeneral,and thantheotherpowerspectraoftheISMcomponents.Thismight thebreakscalesareevenmorerobust,within15%.Wehavealso beexpected,sincestarformationgivesmorepowertothesmall checkedthe effect of smoothingon the maps. As expected,the scales. Its break scale, related to the plane thickness is higher. power-spectrumatlarge-scaleisunchanged,andonlythepower The GALEX FUV and NUV, tracing the recent, but past, star attheextremeofthesmall-scalesofthespatialresolutiondrops. formation,haveslopesmorecomparabletotheotherISMslopes The break scale also is enlarged by the resolution, when it is however,andhavesmallerbreakscales,implyingthinnerlayers. tooclose,andwecanseeforinstancethatthebreakat70µmis ThebreakscaleinHαislargerthanintheUVbecauseHαtraces largerthanat24and100µm(Table3)onlybecauseofthecoarser theionisedgas,andonlyindirectlytheyoungstars.Theionised resolution. The fact that some maps have a pixel size smaller layer may be thicker, affected by bubbles, filaments, fountain than the Nyquist size (half of the nominalresolution) does not effectsassociatedtostarformation. 5 F.Combesetal.:Dustpower-spectruminM33 has less small-scale structure, since individual cloud cores are invisible. 4.2.Power-spectrumofthemodel Thevariousrunsdevelopdifferentgasstructureandclumpiness, ascanbeseeninFigures11to16,duetobothinitialconditions that are selected more or less stable, or with different sub-grid “temperature”ofthegas,andtothestarformationrateandfeed- back.Somegalaxies,whicharelaunchedwithinitialconditions amongthemoststable,canendupwithamoreperturbedmor- phologyandlargerturbulencethanothersstartedmoreunstable, duetoalargersupernovaefeedback. Fig.8.SameasFigure3,forthepowerspectrumoftheHIand We computed the Fourier analysis in the same way on the CO(2-1)maps. gasmapsobtainedinthenumericalsimulations.Thederivedre- sultsareconvergingtowardsthesamevaluesatthemiddleofthe runs, and we plotall of them for the finalsnapshotsat T= 667 Table3.Power-lawslopesandbreaksforobservations Myr for the sake of simplicity (cf Figures 17, 18 and 19). For comparison,we also plotthepower-spectrumofsomeoldstel- Map slope slope break Extremevalues larcomponents,whichhavealmostonlyonepower-law(Figure LS ss (pc) LS ss (pc) 20).Thesecorrespondtorun2andrun7atT=667Myr.There FUV -1.2 -3.4 44 -0.9/-1.2 -3.4/-3.9 40-45 isstructureuntilabout300pc,andnosmall-scalecomponent,as NUV -1.2 -3.6 56 -1.0/-1.2 -3.0/-3.6 50-56 expectedforacollisionlesscomponent. Hα -.77 -2.4 300 -0.7/-1.0 -2.4/-2.7 250-300 ThefirstresultwhichisobviousinTable4isthatthepredic- 24µm -.9 -2.6 93 -0.9/-1.0 -2.6/-2.7 93-105 tions of the modelsbracketthe valuesobtainedin the observa- 70µm -1.5 -3.5 236 -1.4/-1.5 -3.4/-3.8 230-270 tions,bothintheslopesatlargeandsmallscales,andinthebreak 100µm -1.5 -3.4 93 -1.4/-1.5 -3.2/-3.6 93-105 scales.Thosecanbecomparedtotheactualthicknessofthegas 160µm -1.5 -3.6 165 -1.5/-1.6 -3.6/-4.0 140-165 layer, in the simulations. This is done in Table 4, showing that 250µm -1.6 -4.7 208 -1.4/-1.6 -4.3/-4.7 200-210 thebreakscale doesnotfollowcloselythethickness.Although 350µm -1.3 -4.0 315 -1.3/-1.5 -4.0/-4.4 290-320 bothshouldbe of thesame orderwhen onlythe self-gravityof 500µm -1.1 -4.4 390 -1.1/-1.4 -4.2/-4.6 320-390 CO(2-1) -1.5 -3.7 103 -1.3/-1.5 -3.6/-3.8 100-115 the gas is taken into account, their close relation is blurred by HI -2.4 -3.2 110 -2.0/-2.4 -3.2/-3.6 100-120 otherfactors,andinparticularthefeedbackefficiency.Whenthe feedback is too extreme, the small scale structure of the gas is LS=LargeScale(>500pc),ss=smallscale(<500pc) smoothedout,eventowardslargerscalethantheverticalthick- ness.Sotherangeofbreakscalesiswiderthanthatofpossible Table4.Power-lawslopesandbreaksforsimulations planeheights. The main parametersthat control the turbulence and there- Run slope slope break Comments H forethepower-spectrumofthegasstructurearethestarforma- z LS ss (pc) (pc) tion rate, the amountof feedback,but also the assumed equiv- alent temperature of the gas, or sound speed, directly involved run1 -0.8 -3.6 75 lowfeedback 80 run2 -0.9 -3.4 75 ′′ 80 in the pressureforces.Increasingthe turbulenceby supernovae run3 -2.3 -4.9 650 extremefeedback 217 feedbacksuppressesefficientlythesmall-scalestructure(power- run4 -1.3 -3.8 63 fiducial 120 lawslopesof∼-5.),andthickenssubstantiallythegaslayer,and run5 -2.1 -5.0 300 moreSF(C*) 142 thecorrespondingbreakscale.Withoutstarformationandfeed- run6 -1.1 -4.0 430 higherthreshold 103 back,itisonlypossibletohavegasmorphologiescomparableto run7 -0.8 -2.8 75 noSF 64 theobservedonesifaminimumsoundspeedof12 kms−1isin- run8 -0.9 -2.9 201 noSF,Vs=12 kms−1 110 jected.Figures11to16showthelargevarietyofgasstructures, run9 -1.1 -5.2 250 SF,Vs=12 kms−1 108 according to the variationsof those parameters. It appears that LS=LargeScale,ss=smallscale ourfiducialmodel(run4)istheonewhichcorrespondsthebest ThegasscaleheightHz=2.35(<z2 >−<z>2)1/2 to theM33galaxy,bothforits morphology,andforthetypical power-lawslopesandbreakscale.Ithasastarformationrateof ∼0.7M /yr,andafeedbacksuchthat10%ofsupernovaeenergy ⊙ iscoupledtothekinematicsofthegas.Verleyetal.(2009)found The dustemissionsatvariouswavelengthshavequitecom- forM33astarformationrateof0.5M /yr. patiblepowerspectra.Theirdifferentbreakscalesreflectthein- ⊙ creasingbeamsizewithwavelength.Asforthegas,themolecu- larmediumappearsthinnerthantheatomicone,althoughspatial 5. Discussionandconclusions resolutionmightinfluenceitsbreakscale. Independent of the spatial resolutions, the slope of the WehavepresentedtheFourieranalysisofdustandgasemission power-spectraatsmallscalearesteeperatlargewavelengthsfor maps of M33 at severalwavelengths,and comparedthem with the dust emission. Between 250 and 500µm, the SPIRE maps the HαandUV maps, tracersof recentstar formation.We find have slopes between -4.3 and -4.7, while the MIPS and PACS thatallthepowerspectracanbedecomposedintotworegimes, mapsbetween24and100µmhaveslopescomprisedwithin-2.7 a large-scale(>500pc)power-law,of−1.5 < slope < −1.0, LS and-3.8.Thisissignificant,andtendstoshowthatthecooldust and a steeper small-scale one, with −4.0 < slope < −3.0. ss 6 F.Combesetal.:Dustpower-spectruminM33 Fig.9.Deprojectedmapsofthedustat100(left)and160µm(right).Themajoraxishasbeenrotatedtobevertical.Thecolorscale isinarbitraryunits. Fig.10.Deprojectedmapsofthedustat250(left)and500µm(right).Themajoraxishasbeenrotatedtobevertical.Thecolorscale isinarbitraryunits. Fig.11. Face-on projection of the run2 model, at T=667 Myr. Fig.12. Face-on projection of the run3 model, at T=667 Myr. Leftarethestars,andRight,thegas.Thelinearscalesonboth Leftarethestars,andRight,thegas.Thelinearscalesonboth axesareinkpc,andthecolorscaleisinarbitraryunits. axesareinkpc,andthecolorscaleisinarbitraryunits. Thelatterslopescorrespondquitewelltowhatisfoundforthe lowerpower-spectra,indicatingmorepoweratsmallscales.The MilkyWay (Dickeyetal.2001)orthe SmallMagellanicCloud break scale is of the order of 100-150 pc for the dust and the (Stanimirovicetal.2000).TheHαstarformationtracerhasshal- gas,increasingatlargewavelength,probablyinfluencedbythe 7 F.Combesetal.:Dustpower-spectruminM33 Fig.13. Face-on projection of the run4 model, at T=667 Myr. Leftarethestars,andRight,thegas. Fig.17.Powerspectrumofthegasinmodelsrun1andrun2.The verticaldashlinerepresentsthesofteninglengthofthegravity. Fig.14. Face-on projection of the run7 model, at T=667 Myr. Leftarethestars,andRight,thegas. Fig.18.Powerspectrumofthegasinmodelsrun3andrun4. Fig.15. Face-on projection of the run8 model, at T=667 Myr. Leftarethestars,andRight,thegas. Fig.19.Powerspectrumofthegasinmodelsrun7andrun8. Fig.16. Face-on projection of the run9 model, at T=667 Myr. Leftarethestars,andRight,thegas. 24µmmapisalsorelatedtotheyoungstarformation,thereare nosuchlargebubbles:onlythepointsourcesareprominentin- sidethebubble,ascanbeseeninFigure3ofVerleyetal.(2010). progressivelackofspatialresolution.TheHαbreakscaleissig- ThisexplainsthedifferentbreakscalesbetweenHαandMIPS- nificantly higher (300 pc), implying a thicker layer for ionized 24µmmaps. gas, while the UV layers appears thinner (65 pc). This might It is also possible that the large thickness suggested by the be understood considering that UV originates from stars that Hαanalysisisrelatedtothediffuseionizedgas(DIG)observed are concentrated in the disk whereas the ionised gas could be inseveraledge-ongalaxiestofollowaratherthicklayerofabout present quite far from the plane of the disk: giant HII regions 1kpcheight(see areviewbyDettmar1992).Thisextra-planar havetypicallyroundshapes,andarenotasconfinedtothedisk. ionizedgasisthoughttobeduetostarformation,itispresentin Tabatabaei etal.(2007) have already shown with wavelets that about40% of 74 edge-ongalaxies(Rossa & Dettmar 2003). It theHαmapisdominatedbygiantHIIregions,whicharebubble- canbewidelydiffuse,ortaketheshapeoffilaments,plumesand like with sizes from 100 to 500 pc. By comparison the diffuse bubbles (Rand, 1997). These extra-planar extensions could be component is weaker, which flattens the Fourier ss-slope, and due to UV ionizingphotonsescaping the star formingregions, enlargesthebreakscale.AlthoughthewarmdustintheMIPS- supernovaefeedback,fountaineffects,chimneys,andbepartof 8 F.Combesetal.:Dustpower-spectruminM33 terminethethicknessofthegasplane,atanyorientationonthe plane of the sky, and the actual physical state of the gas, in its star formationphase.Notehoweverthatthecurrentmodelsare notdevoidofshortcomings:theyonlypartlyallowtodistinguish betweenthe12differenttracersofthegasphaseobservedhere, asradiativetransport,dustproperties,etc.arenottakenintoac- count. Comparison of power spectrum analysis between galaxies shouldallowadeterminationoftheirrelativestarformationrate and feedback. This will need however high spatial resolution maps,suchastheonesthatwillbeprovidedwithALMA. ThegasplaneinM33appearsrelativelythickandturbulent, withrespecttoourclosestfiducialmodel,whichcouldbeacon- sequenceofitsrecentheatinginparticularthroughtheM31in- Fig.20. Power spectrum of the stellar component in run2 and teraction(McConnachieetal.2009). run7. The vertical dash line represents the softening length of thegravity. Acknowledgements. Wewarmlythanktheanonymousrefereeforhis/hercon- structive comments and suggestions. We have made use of the NASA/IPAC ExtragalacticDatabase(NED). anactivedisk-halointeraction,withexchangeofmatter(Howk &Savage2000). Thecomparisonwithsimulatedmodelstellsusthatthe ob- References servationofabreakinthepower-spectrumofobservedgasdis- Begum,A.,Chengalur,J.N.,Bhardwaj,S.:2006,MNRAS372,L33 tributionisnotalwaysrelatedtothe2D/3Dtransition,andindi- BianchiS.,XilourisE.M.:2011,A&A531,L11 catesthethicknessofthegaseousdiskonlywhenthelatterisnot BlockD.L.,Puerari,I.,Elmegreen,B.G.,Bournaud,F.:2010ApJ718,L1 heated by a strong star formationrate and a strong supernovae BoquienE.,Calzetti,D.,Kramer,C.etal.:2010,A&A518,L70 BoquienE.,Calzetti,D.,CombesF.etal.:2011AJ142,111 feedback. For a quiescent galaxy, the break should still be an Bournaud,F.,Combes,F.,Jog,C.J.,Puerari,I.:2005,A&A438,507 indicationofthegaslayerthickness. Bournaud F., Elmegreen, B. G., Teyssier, R., Block, D. L.,Puerari, I.: 2010, Block etal.(2010) found in the LMC that the small scale MNRAS409,1088 slopesofthepowerspectraweresteeperatlongerwavelengths, BraineJ.,GratierP.,KramerC.etal.:2010,A&A518,L69 BronfmanL,CohenR.S.,AlvarezHetal.,1988,ApJ324,248 i.e. the cool dust was smoother than the hot emission. We find CorbelliE.,SalucciP.:2007,MNRAS311,441 a similar result. The slope of the SPIRE mapsare around-4.5, CrovisierJ.,DickeyJ.:1983A&A122,282 whilethewarmerdustat24-100µmhasanaverageslopeof-3.2. DalcantonJ.J.,StilpA.M.:2010,ApJ721,547 This observation can be interpreted in terms of star formation DalcantonJ.J.,YoachimP.,BernsteinR.:2004,ApJ608,189 occuringintheclumpymedium.Oncethestarsareformed,they DameT.M.,ThaddeusP.,1994,ApJ436,L173 DettmarR.J.:1992,Fund.CosmicPhysics,15,143 heatthedustaround,andthisexplainswhytheemissionatshort- DiMatteoP.,,CombesF.,MelchiorA.-L.,SemelinB.:2007,A&A468,61 est wavelength (warmer dust) is more clumpy. Also the break Dickey, J. M., McClure-Griffiths, N. M., Stanimirovic, S., Gaensler, B. M., scale, and thusthe thicknessof the plane,is largerforthe cold Green,A.J.:2001ApJ561,264 gas.Thiscouldbedueinadditiontothefactthatthecoldcom- DopitaM.A.:1985ApJ295,L5 DuttaP.,Begum,A.,Bharadwaj,S.,Chengalur,J.N.:2008MNRAS384,L34 ponent is much more extended in radius (Kramer etal.2010), DuttaP.,Begum,A.,Bharadwaj,S.,Chengalur,J.N.:2009aMNRAS397,L60 anditiswellknownthatthegaslayerisflaringintheouterparts DuttaP.,Begum,A.,Bharadwaj,S.,Chengalur,J.N.:2009bMNRAS398,887 ofgalaxies.Asforthegasemission,themoleculargasappears ElmegreenB.G.,Kim,S.,Staveley-Smith,L.:2001ApJ548,749 moreclumpy,butonlyslightlymoresothantheatomicgas.And ElmegreenB.G.,Leitner,S.N.,Elmegreen,D.M.,Cuillandre,J-C.:2003bApJ the break occursat aboutthe same scale. Since it is likely that 593,333 ElmegreenB.G.,ElmegreenD.M.,Leitner,S.N.:2003a,ApJ590,271 in the relatively quiescent M33 galaxy, the break indicates the ElmegreenB.G.,ScaloJ.:2004,ARAA42,211 thickness of the gas layer, this situation is not alike the Milky ElmegreenB.G.,Elmegreen,D.M.,Chandar,R.,Whitmore,B.,Regan,M.:2006 Way,wheretheCO componentis significantlythinnerthanthe ApJ644,879 HI(Bronfmanetal.1988).AthickCOlayer,3timesaswideas FalgaroneE.,Phillips,T.G.,WalkerC.K.:1991,ApJ378,186 Freedman,W.L.,Wilson,C.D.,Madore,B.F.1991,ApJ,372,455 the dense moleculargas layer, and comparableto the HI layer, GildePaz,A.,Boissier,S.,Madore,B.F.,etal.:2007,ApJS,173,185 has been observed(Dame & Thaddeus1994), but it involvesa GoldmanI.:2000ApJ541,701 small mass. M33 being a galaxy of intermediate mass, could Gratier,P.,Braine,J.,Rodriguez-Fernandez,N.J.,etal.2010,A&A,522,A3 haveindeedathickergaslayerthantheMilkyWay(Dalcanton GreenD.A.:1993,MNRAS262,327 etal.2004). HoopesC.G.,WalterbosR.A.M.,BothunG.D.:2001,ApJ559,878 HowkJ.C.,SavageB.D.:2000,AJ119,644 We also showed that numerical models predict gas mor- JungwiertB.,CombesF.,PalousJ.:2001,A&A36,85 phologiesandsmallscale power-lawswhichbracketthe obser- Khalil, A.,Joncas,G.,Nekka,F.,Kestener, P.,Arneodo,A.:2006,ApJS165, vationaldata.Itispossibletoobtainacolderoramoreturbulent 512 interstellarmedium,byvaryingthestarformationfeedbackand Kramer,C.Buchbender,C.,Xilouris,E.M.etal.:2010,A&A518,L67 Lazarian,A.,Pogosyan,D.:2004,ApJ616,943 the sound speed of the gas. The Fourier analysis of the mod- Lazarian,A.,Pogosyan,D.,Vzquez-Semadeni,E.,Pichardo,B.:2001ApJ555, elsrevealsthatthebreakscale isa goodindicatorof theactual 130 thicknessofthegaslayer,althoughitvarieswithinawiderrange McConnachieA.W.,Irwin,M.J.,Ibata,R.A.etal.2009,Nature461,66 whenextremefeedbackisused,andsmoothesoutthesmallscale OdekonM.C.:2008ApJ681,1248 PadoanP.,Kim,S.,Goodman,A.,Staveley-Smith,L.:2001ApJ555,L33 gasstructures.The power-lawslopesobtainedin the modelsat PfennigerD.,CombesF.:1994,A&A285,94 largeandsmallscalereflectquiteaccuratelythegasmorphology Rand,R.J.:1997,ApJ474,129 andturbulence.Fourieranalysesarethereforeagoodtooltode- RossaJ.,DettmarR.J.:2003,A&A406,493 9 F.Combesetal.:Dustpower-spectruminM33 Roychowdhury, S., Chengalur, J. N., Begum, A., Karachentsev, I. D.: 2010, MNRAS404,L60 SanchezN.,Anez,N.,Alfaro,E.J.,CroneOdekon,M.:2010ApJ720,541 SemelinB.,CombesF.:2005,A&A441,55 StanimirovicS.,LazarianA.:2001,ApJ551,L53 Stanimirovic S.,Staveley-Smith, L.,vanderHulst,J.M.etal.2000:MNRAS 315,791 Stutzki,J.,Bensch,F.,Heithausen,A.,Ossenkopf,V.,Zielinsky,M.:1998,A&A 336,697 TabatabaeiF.S.,BeckR.,KrauseM.etal.:2007,A&A466,509 Thilker,D.A.,Hoopes,C.G.,Bianchi,L.etal.:2005,ApJ619,L67 Verley,S.,Hunt,L.K.,Corbelli,E.,Giovanardi,C.:2007,A&A,476,1161 VerleyS.,Corbelli,E.,Giovanardi,C.,Hunt,L.K.:2009,A&A493,453 VerleyS.,Relano,M.,Kramer,C.etal.:2010,A&A518,L68 Fig.B.1.PowerspectrumofthePACS100µmmap,atdegraded AppendixA: Fourieranalysisofthedensity resolutionsof11and18′′. Theverticaldashlineatrightrepre- Until now, we have essentially analysed the small scale struc- sents the smoothed spatial resolution, on the major axis. Since tureoftheinterstellarmediuminvarioustracers,butignoredthe thede-projectiontoaface-ongalaxyimpliesafactor1/cos(56◦) azimuthal dependence. The different tracers also reveal a con- =1.79largerbeamontheminoraxis,wehavealsoplottedasec- trasted spiral structure, that is interesting to present, at least in onddashedverticallineatleft,indicatingthegeometricmeanof tworepresentativedeprojectedmaps,theHerschel100µmimage theresolutiononbothminorandmajoraxis. (dustandgasimagesshowsimilarfeatures),andtheHαimage (starformation,similartoUVimages). WehavecomputedtheFouriertransformofthecorrespond- ingsurfacedensities,decomposedas: µ(r,φ)=µ (r)+ a (r)cos(mφ−φ (r)) 0 m m X m where the normalisedstrength of the Fourier componentm isA (r) = a (r)/µ (r).Thus,A representsthenormalizedam- m m 0 2 plitudeforam = 2distortionlikeabarorspiralarm,atagiven radius,andA representsthesameforthedisklopsidedness.The 1 quantityφ denotesthepositionangleorthephaseoftheFourier m componentm.Suchadensityanalysishasbeenmadeforexam- plebyBournaudetal.(2005),tostudyasymmetriesandlopsid- ednessingalaxies,andcorrelateitwithm=2features. Fig.B.2.SameasFigureB.1forthedegradedresolutionsof26 FigureA.1showsaplotoftheamplitudeAm(r)andthephase and36′′. φ (r) versus radius r for M33, at 100µm and Figure A.2, the m sameforHα.Bothshowaprominentspiralfeatureatradii∼3-4 kpc, with a phase decreasing slightly with radius, in the direct sense. The m = 2 distortions show accompanying m = 4 har- monics. The lopsidednessalso increases with radius, while the thecorrespondingphaseisfairlyconstant. These plots show that the spiral structure in M33 is mildly contrasted,inbothgasandSFRtracers.Indeed,thenormalised m = 2 amplitude is between 0.4 and 0.8. There is also a sig- nificant asymmetry and lopsidedness, as testified by the strong m=1amplitudecomparablewiththem=2.Thisexplainswhy the slopes at large-scale were more different between the two halvesofthegalaxythanthesmall-scaleones.Thislargeasym- metrycouldbeduetothecurrentinteractionbetweenM33and M31(McConnachieetal.2009). AppendixB: Theeffectofresolution We have explored the effect of spatial resolution in the deter- mination of the various slopes and break scales, in smoothing the PACS-100µm map at variousresolutions, from the original 7′′to11,18,26and36′′,correspondingto44,72,104and144 pconthemajoraxis.TheresultingfitsareshowninFigureB.1 andB.2,tobecomparedwithFigure3.Thebreakscalesarere- spectively97,111,135,195pc,insteadof93pcfortheoriginal 7′′resolution.Thedeterminationispossibleonlyforthetwofirst cases. 10

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