Dust and gas power-spectrum in M33 (HERM33ES)

0.93 MB·English

by  F. Combes

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).
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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