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Astronomy & Astrophysics manuscript no. f-r5 c ESO 2008 (cid:13) February 5, 2008 A multi–scale study of infrared and radio emission from Scd galaxy M33 7 0 0 F. S. Tabatabaei1, R. Beck1, M. Krause1, E. M. Berkhuijsen1, R. Gehrz2, K. D. Gordon3, J. L. 2 Hinz3, R. Humphreys2, K. McQuinn2, E. Polomski2, G. H. Rieke3, C. E. Woodward2 n a J 1 Max-Planck Institutfu¨r Radioastronomie, Aufdem Hu¨gel 69, 53121 Bonn, Germany 2 School of Physicsand Astronomy,University of Minnesota, Minneapolis, MN 55455 1 3 Steward Observatory,University of Arizona, 933 North Cherry Avenue,Tucson, AZ 85721 3 1 Preprint online version: February 5, 2008 v 7 ABSTRACT 9 8 Aims. We investigate the energy sources of the infrared(IR) emission and their relation to the radio continuum 1 emission at various spatial scales within theScd galaxy M33. 0 Methods. We use the data at the Spitzer wavelengths of 24, 70, and 160µm, as well as recent radio continuum 7 mapsat3.6cmand20cmobservedwiththe100–mEffelsbergtelescopeandVLA,respectively.Weusethewavelet 0 transform of these maps to a) separate the diffuse emission components from compact sources, b) compare the / h emission at different wavelengths, and c) study the radio–IR correlation at various spatial scales. An Hα map p serves as a tracer of thestar forming regions and as an indicator of thethermal radio emission. o- Results. The bright HII regions affect the wavelet spectra causing dominant small scales or decreasing trends r towards the larger scales. The dominant scale of the 70µm emission is larger than that of the 24µm emission, st while the 160µm emission shows a smooth wavelet spectrum. The radio and Hα maps are well correlated with a all 3 MIPS maps, although their correlations with the 160µm map are weaker. After subtracting the bright HII v: regions, the24 and 70µm mapsshow weakercorrelations with the20cm map thanwith the3.6cm map at most i scales. We also finda strong correlation between the 3.6cm and Hαemission at all scales. X Conclusions. Comparing the results with and without the bright HII regions, we conclude that the IR emission r is influenced by young, massive stars increasingly with decreasing wavelength from 160 to 24µm. The radio–IR a correlations indicate that the warm dust–thermal radio correlation is stronger than the cold dust–nonthermal radio correlation at scales smaller than 4kpc. A perfect 3.6cm–Hα correlation implies that extinction has no significant effect on Hα emitting structures. Keywords.Methods:dataanalysis–ISM:dust–ISM:HIIregions–Galaxies:individual:M33–Infrared:galaxies – Radio continuum:galaxies 1. Introduction emission, as a component of the far–infrared(FIR) emission,mightnotbe directly linkedto the young One of the most important discoveriesof the IRAS stellar population. For the nearby galaxy M31, missionwasthe correlationbetweenthe IRand ra- Hoernes et al. (1998), using IRAS data, found a dio continuum luminosities for a sample of galax- good correlation between the emission of thermal– ies (e.g., Helou et al. 1985; de Jong et al. 1985; radio/warm dust and nonthermal–radio/cold dust Gavazzi et al. 1986). Beck & Golla (1988) showed with slightly different slopes. They explained the thatthiscorrelationalsoholdswithinseveralspiral lattercorrelationbyacouplingofthemagneticfield galaxiesincludingM31andM33.Theradio–IRcor- to the gas mixed with the cold dust. Such a cou- relation was explained by Helou et al. (1985) and pling was also considered by Niklas & Beck (1997) de Jong et al.(1985)asadirectandlinearrelation- as the origin of the global radio–FIR correlation. ship between star formation and IR emission. On Hinz et al. (2004) found similar behavior in M33. theotherhand,therewasconcernthatthecolddust 2 Tabatabaei et al.: A multi–scale studyof IRand radio emission from M33 Table 1. Images of M33 used in this study 1994), it is seen nearly face on. The central position of M33 given by de Vaucouleurs & Leach Wavelength Resolution Telescope (1981) is RA(2000)=1h33m51.0s and 20cm 51′′ VLA 1 DEC(2000)=30◦39′37.0′′. The position angle of 20cm 540′′ Effelsberg 2 the major axis is PA 23◦ (Deul & van der Hulst 3.6cm 84′′ Effelsberg1 ∼ 1987). 160µm 40′′ Spitzer3 70µm 18′′ Spitzer3 With the Multiband Imaging Photometer 24µm 6′′ Spitzer3 Spitzer (MIPS, Rieke et al. 2004) data at 24, 70, 6570˚A(Hα) 2′′ (pixelsize) KPNO4 and160µm,itis possible tointerpretthe morphol- 1 Tabatabaei et al. (in prep.) ogy of M33. Hinz et al. (2004) compared the first 2 Fletcher, (2001) epoch data of MIPS with Hα images and radio 3 Hinzet al. (2004) and this paper continuum data at 6cm, using a Fourier filtering 4 Hoopes & Walterbos (2000) technique to distinguish different emission components. We now have multiple epochs of MIPS data that provide a significantly higher–quality image The possibility that the relationship between of M33, for example in the suppressing of stripes the radio and IR emission might vary within alongthe directionofscancausedby slowresponse galaxies (e.g., Gordon et al. 2004) due to the from the far infrared arrays. In addition, we use a range of individual conditions makes it necessary wavelet analysis technique that Frick et al. (2001) to repeat these kinds of studies with improved have shown to be more robust against noise than data in the most nearby galaxies. Furthermore, Fourier filtering. Furthermore, the wavelet tech- it is uncertain what component of a galaxy nique provides more precise and easier analysis of provides the energy that is absorbed and re– the scale distribution of emission energy,especially radiated in the IR (Kennicutt 1998). For exam- atsmaller scalesof the emitting structures.We ap- ple, from the close correspondence between hy- ply a 2D–wavelet transformation to separate the drogen recombination line emission and IR mor- diffuse emission components from compact sources phologies, Devereux et al. (1994); Devereux et al. inMIPSmid–andfar–infrared(hereafterIR),radio (1996,1997)andJones et al.(2002)arguedthatthe (at 3.6 and 20cm), and Hα images (Table 1). FIRispoweredpredominantlybyyoungO/Bstars. The classical correlation between IR and ra- Deul (1989), Walterbos & Greenawalt(1996), and dio emission has been at the whole galaxy level. Hirashita et al. (2003) argued that about half of However, such a correlation can be misleading the IR emission or more is due to dust heated by a when a bright, extended central region or an ex- diffuse interstellar radiation field that is not dom- tended disk exists in the galactic image. This inated by any particular type of star or star clus- technique gives little information in the case of ter. Sauvage & Thuan (1992) suggested that the an anticorrelation on a specific scale (Frick et al. relative role of young stars compared with the dif- 2001). The ‘wavelet cross–correlation’ introduced fuse interstellar radiation field increases with later byFrick et al.(2001)calculatesthecorrelationco- galaxy type. efficient as a function of the scale of the emitting Another problem that hinders a better under- regions. Hughes et al. (2006) applied this method standing of the radio–IR correlation is the sepa- to study the radio–FIR correlation in the Large ration of thermal and nonthermal components of Magellanic Cloud (LMC). the radio continuum emission using a constant nonthermal spectral index (the standard method). Hippelein et al. (2003) found evidence for a lo- Althoughthisassumptionleadstoaproperestima- cal radio–FIR correlation in the star forming re- tionforglobalstudies,it cannotproducearealistic gions of M33. In this paper, we investigate this image of the nonthermal distribution in highly re- correlation not only for the star forming regions solved and local studies within a galaxy, because but also for other structures within M33 at spa- it is unlikely that the nonthermal spectral index tial scales between 0.4 and 4 kpc using the wavelet remains constant across a galaxy (Fletcher et al. cross–correlation. Instead of the standard method 2004). to separate the thermal and nonthermal compo- M33 (NGC598) is the nearest late–type spiral nents of the radio continuum emission, we use the galaxy, at a distance of 840kpc (Freedman et al. radio continuum data both at high and low fre- 1991) and is ideal to compare the morpholo- quency (3.6 and 20cm, respectively). We compare gies of different components of such a galaxy. ourradioimagestotheHαimage(whichistakenas With an inclination of i=56◦ (Regan & Vogel atracerofthethermalfree–freeemission)todistin- Tabatabaei et al.: A multi–scale studyof IRand radio emission from M33 3 Fig.1. The 3.6cmradiomapofM33 (left panel)observedwith the 100–mradiotelescope inEffelsberg, and the combined 20cm radio map (right panel) from the VLA and Effelsberg observations. The half power beam widths (HPBWs) of 84′′ and 51′′, respectively, are shown in the lower left corners. guishthecomponentsoftheradiocontinuumemis- (18 nights) between August 2005 and March 2006. sion. The reduction was done in the NOD2 data reduc- We study the structural characteristics of the tion system (Haslam 1974). The r.m.s. noise after MIPS IR images using the 2D-wavelet transfor- combination (Emerson & Graeve 1988) of 36 cov- mation in Sect. 3. The distribution of the emis- erages is 220 µJy/beam. We also observed M33 sion energy at both IR and radio regimes and the at 20cm w∼ith the B–band VLA2 D-array during 5 dominant spatial scale at each wavelength are dis- nights in November 2005 and January 2006 (from cussed in Sect. 4. We show how the different MIPS 06to08-11-05,13-11-05,and06-01-06).Thereduc- images are correlated at different scales and how tion, calibration, and mozaicing of 12 fields were the radio–IR correlation varies with components accomplished using the standard AIPS programs. of the galaxy (e.g. gas clouds, spiral arms, ex- The VLA interferometric data will miss much of tended central region and extended diffuse emis- the extended emission of the galaxy. For example, sion) in Sect. 5. In Sec. 6, we use the Hα map to Viallefond et al.(1986a)showedthattheWSRTin- probe the energy sources of IR and radio emission. terferometric map at 1.4 GHz accounted for only Results and discussion are presentedin Sect. 7. An about16%ofthetotalemission.Therefore,wecom- overview of the preliminary results was presented bined the VLA map with the new Effelsberg 20cm in Tabatabaei et al. (2005). map (Fletcher, 2001) to recover the emission from extendedstructuresinM33.The r.m.s.noiseofthe 2. Observations and data reduction finalmapis 180µJy/beam.Detaileddescriptions ∼ of the observationsand data reduction of the radio Table 1 summarizes the data used in this work. data will be given in Tabatabaei et al. (in prep.). The 3.6cm radio observations were made with the Both radio maps are shown in Fig. 1. 100-m Effelsberg telescope1 during several periods 2 The VLA is a facility of the National Radio 1 The 100–m telescope at Effelsberg is operated by Astronomy Observatory. The NRAO is operated by theMax-Planck-Institut fu¨r Radioastronomie (MPIfR) Associated Universities, Inc., under contract with the on behalf on theMax–Planck–Gesellscahft. National Science Foundation. 4 Tabatabaei et al.: A multi–scale studyof IRand radio emission from M33 Fig.2. Top: the Spitzer MIPS images of M33 at 24µm (left panel) and 70µm (right panel). Bottom: the Spitzer MIPS image at 160µm (left panel) and the Hα map (right panel) from KPNO Hoopes & Walterbos (2000). The resolutions are given in Table 1. M33 was mapped in the IR by MIPS offsetsandcoveringthe fullextentofM33.Theba- (Rieke et al. 2004) four times on 29/30 December sic data reduction was performed using the MIPS 2003, 3 February 2005, 5 September 2005, and instrument team Data Analysis Tool versions 3.02- 9/10 January 2006. Each observation consisted of 3.04 (Gordon et al. 2005). At 24µm extra steps medium-rate scan maps with 1/2 array cross-scan were carried out to improve the images includ- Tabatabaei et al.: A multi–scale studyof IRand radio emission from M33 5 Fig.3.MIPSmapsofM33at40′′resolution(firstcolumn)andtheirwaveletdecompositionsfor4different scales:80′′ (secondcolumn), 383′′ (third column), 626′′ (forth column), and 1024′′ (fifth column). At the distance of 840 kpc, 1′′ is equivalent to 4pc. The maps at 24, 70, and 160µm are shown from top to bottom. Before the decomposition, huge HII regions like NGC604 were subtracted from the original images(Sect.4).MapsareshowninRA–DECcoordinatesystemandcenteredatthecenterofthegalaxy. The field size is 34′ 41′. × ing readout offset correction, array averaged back- the 70µm array were changed after the first M33 ground subtraction (using a low order polynomial map was made and before the second. The 160µm fit to each leg, with the region including M33 ex- image used consisted of a combination of all four cluded from this fit), and exclusion of the first five maps. The combination of multiple maps of M33 images in each scan leg due to boost frame tran- taken with different scan mirror angles results in a sients.At 70 and160µm, the extraprocessingstep significantsuppressionofresidualinstrumentalsig- was a pixel dependent background subtraction for natures (seen mainly as low level streaking along eachmap(usingaloworderpolynomialfit,withthe the scanmirror direction). The images used in this region including M33 excluded from this fit). The workhaveexposuretimesof 100,120,and36sec- ∼ background subtraction should not have removed onds/pixel for 24, 70, and 160µm, respectively. real M33 emission as the scan legs are nearly par- The Hα observations by Hoopes & Walterbos allel to the minor axis resulting in the background (2000) were carried out on the 0.6 meter Burrell– regions being far above and below M33. Schmidt telescope at the Kitt Peak National Observatory,providinga68′ 68′fieldofview.The The24µmimageusedconsistedofjustthe9/10 × MIPS and Hα images are shown in Fig. 2. January 2006 observations as the depth reached in a single mapping was sufficient for this work. To obtain a proper comparison with the 3.6cm The 70µm image used was a combination of the radio map, all images of M33 were convolved to last three observations as the 29/30 December the angular resolution of 84′′. For higher angular 2003 70µm map suffered from significant instru- resolution studies, maps at 18′′, 40′′, and 51′′ were mentalresiduals.Theseinstrumentalresidualswere also made. The PSF (point spread function) of the much reduced when the operating parameters of MIPS data is not Gaussian, in contrast to that of 6 Tabatabaei et al.: A multi–scale studyof IRand radio emission from M33 theradiodata.Thus,convolutionsoftheMIPSim- ages were made using custom kernels created using Fast Fourier Transforms (FFTs) to account for thedetailedstructureoftheMIPSPSFs.Detailsof the kernel creation can be found in Gordon et al. (2007, in prep.). After convolution, the maps were normalized in grid size, rotation and reference co- ordinates. Then, they were cut to a common field of view (34′ 41′ in RA and DEC, respectively). × In the following sections, we discuss the results with and without bright compact sources at each resolution to investigate how these sources affect the energy distribution ateachwavelengthandthe correlationsbetweenwavelengths.After convolving toeachresolution,thebrightsources,commontoall Fig.4. The wavelet spectra of the 24 and 70 µm images,weresubtractedusingGaussianfitsinclud- images before (or.) and after (s.s.) subtraction of ing baselevels (‘Ozmapax’ program in the NOD2 the same sources from the two images at 18′′ res- datareductionsystem).Forinstance,the giantHII olution. The data points correspond to the scales complexes NGC604 and NGC595 were subtracted 34,50, 73,107,155,226,329,479, 697,1015′′. The atallresolutions.We also subtractedstrongsteep– spectra are shown in arbitrary units. spectrum backgroundradio sourcesfrom the 20cm image. Detailed descriptions of the source subtrac- in M33, we use the ‘Pet–Hat’ wavelet that was in- tion at each resolution are given in Sect. 4. troduced by Frick et al. (2001) and applied there to NGC6946. It is defined in Fourier space by the 3. Wavelet analysis of IR emission formula: Waveletanalysisisbasedonaspatial–scaledecom- ψ(k)= cos2(π2log22kπ) π ≤k≤4π (2) (cid:26)0 π >k or k>4π, position using the convolution of the data with a family of self–similar basic functions that depend where k is the wavevectorand k= k. on the scale and location of the structure. Like the | | We decomposed the Spitzer images into 10 dif- Fouriertransformation,the wavelettransformation ferent scales a to compare the morphologies at 24, includes oscillatory functions; however, in the lat- 70,and160µmateachscale3.Fig.3showstheorig- ter case these functions rapidly decay towards in- inal maps and the decomposed maps at 4 scales. finity. As a result, Frick et al. (2001) show that The original 24 and 70µm maps were smoothed the wavelet method is more resistant to noise and to the MIPS 160µm PSF with FWHM (full width the smootherspectra allowbetter determinationof half maximum)=40′′ before decomposition. The the true frequency structure. The cross–correlation maps at scale a=80′′ (0.3kpc) show the smallest of wavelet spectra lets us compare the structures detectable emitting structures. Most of the mor- of different images systematically as a function of phological differences among the MIPS images are scale. found at this scale. At scale a=383′′ (1.5kpc), the The wavelet coefficients of a 2D continuous prominent structures are spiral arms and the cen- wavelet transform are given by: ter of the galaxy. The central extended region is 1 +∞ x′ x more pronounced at scale a=628′′ (2.5kpc). The W(a,x)= f(x′)ψ∗( − )dx, (1) emission emerges from a diffuse structure at scale aκ Z−∞ a a=1024′′ (4kpc), and it is not possible to distin- where ψ(x) is the analysing wavelet, x = (x,y), guish the arm–structure anymore. This structure f(x)isatwo–dimensionalfunction(animage),and can be identified as an underlying diffuse disk with a and κ are the scale and normalization parame- ageneralradialdecreaseinintensity.Thesimilarity ters, respectively, (the ∗ symbol denotes the com- ofthe4kpcmapsatdifferentwavelengthsindicates plex conjugate). The above transformation decom- posesanimageinto‘maps’ofdifferentscale.Ineach 3 To have both physically meaningful results and a map, structures with the chosen scale are promi- sufficiently large number of independent points, the nentastheyhavehighercoefficientsthanthosewith scale a varies between a minimum of about twice the smallerorlargerscales.Toobtainagoodseparation resolution and a maximum of about half of the image ofscalesandtofindthescaleofdominantstructures size, a<1100′′, for all images studied in this paper. Tabatabaei et al.: A multi–scale studyof IRand radio emission from M33 7 Table 2. Source subtraction thresholds S(λ) and number of subtracted sources at different spatial resolutions. Resolution Subtracted sources S(24µm) S(70µm) S(160µm) S(20cm) S(3.6cm) arcsec # µJy/arcsec2 µJy/arcsec2 µJy/arcsec2 µJy/beam µJy/beam 18 40 460 2650 – – – 51 15 115 1570 2865 7190 – 84 11 85 900 2340 8260 5780 that the large scale diffuse emission has the same minimum energy inthe originalspectrum at24µm structure at different mid– and far–infrared wave- is larger(by a factorof3)thanthatat70µm. This lengths. indicates that either star forming regions provide Atthesmallestscale,theemissionemergesfrom more energy for the 24µm emission, or the large– spot–like features aligned along filaments with the scale diffuse emission is stronger at 70µm than width ofthe scale.At 24µmthe spots,correspond- 24µm. ingtoHIIregions,containmostoftheenergyatthis scale (see Sect. 6). Diffuse filaments are more pro- At the next angular resolution, 51′′, the nounced at 160µm . As shown in the next section, HPBW (half power beam width) of the 20cm ra- the fraction of the energy at this scale at 160µm is dio map, 15 bright sources (HII regions) visible at lessthanthatat24µm.The situationinthe 70µm all maps were subtracted from the 20cm map and mapisinbetweenthe24and160µmmaps.Itseems the smoothed 24, 70, and 160µm IR maps. In ad- that the star forming regions provide most of the dition to these sources,9 backgroundradio sources energy of the 24 and 70µm emission, if the spots plus the supernova remnants SN1, SN2, and SN3 correspond to these regions, as is discussed in the (Viallefond et al. 1986b) were subtracted from the following sections. 20cm radio map. The waveletspectra of the MIPS and20cmradiomapsatthis resolutionareplotted 4. Spectral characteristics of IR and radio in Fig. 5. The Hα spectrum is also given for com- maps parison. The 24 and 70µm maps show a smoothed version of their distributions in Fig. 4 (the linear In this section, we demonstrate how the wavelet smoothing factor is 3 between Figs. 4 and 5). ∼ spectracanbe usedtoinvestigatethe scalingprop- However, the effect of the sources can still be seen erties of the emission. This spectrum is defined as by comparing the left panel with the source sub- the energy in the wavelet coefficients of scale a tracted spectra in the right panel. (Frick et al. 2001): +∞ +∞ The 160µm map is hardly influenced by the M(a)= W(a,x)2dx. (3) smoothing, as its original resolution is 40′′ (the Z−∞ Z−∞ | | smoothing factor is 1.3). The 160µm spectrum ∼ After smoothing the 24µm map to the MIPS is more similar to the 70µm spectrum than to 70µm PSF with FWHM=18′′, about 40 bright the other spectra. It seems that the compact sources(mostly correspondingto HII regions)were sourceshavelesseffectontheenergydistributionat subtracted from both maps (equivalent to remov- 160µm, because the smallest scale is not the dom- ingsourceswithfluxeshigherthanthelowerlimits, inant scale. Hence, there is no important change in S(λ), given in Table 2.). Fig. 4 shows the wavelet the spectrum after the source subtraction. The en- spectra, M(a), of the 24µm and 70µm maps be- ergy shows an increase at the scale of complexes of foreandaftersourcesubtraction.Clearly,thesmall dustandgasclouds( 250′′or1kpc),thenasecond ∼ scales are the dominant scales before source sub- increase in transition to the large–scale structures traction. The scale at which the wavelet energy is or diffuse dust emission. maximum is 70′′ (280pc) at 24µm and 110′′ ∼ ∼ (440pc) at 70µm. After source subtraction, the The spectrum of the 20cm radio image is also spectra become flatter. The larger size of the dom- dominated by bright sources. There is a maximum inant scale at 70µm caused by these sources (by a at the scale a=140′′, then a decrease towards the factorof 1.6atthisresolution)indicatesaslightly larger scales with a slope of -0.9. However, a flat ∼ more extended distribution of dust grains emitting spectrum remains for the scales less than the size at 70µm than 24µm in the vicinity of star form- of the central extended region ( 600′′) after the ∼ ingregions.Moreover,the ratioofthe maximumto source subtraction. 8 Tabatabaei et al.: A multi–scale studyof IRand radio emission from M33 Fig.5. The wavelet spectra of the 20cm radio and IR images at 51′′ resolution before (left) and after (right) subtractionofthe same sources.For comparison,the waveletspectrum of the Hα emissionis also plotted. The data points correspondto the scales 112, 143,183,234, 299,383,490, 626,800,1024′′. The spectra are shown in arbitrary units. Fig.6. The wavelet spectra of the 3.6cm radio and IR images at 84′′ resolution before (left) and after (right)subtractionofthe same sources.Forcomparison,the spectrumofthe Hα emissionis alsoplotted. The data points correspond to the scales 210, 250, 298, 356, 424, 505, 602, 718, 856, 1020′′. The spectra are shown in arbitrary units. The spectra of all the maps at the resolution of region, 600′′ (2.5kpc). All three spectra become the 3.6cm data (HPBW=84′′)4 are shown in Fig. steeper∼between a 600′′ and a 850′′, which ∼ ∼ 6 (left panel). The spectra after subtracting the 11 means that the wavelet energy from structures brightestHII regionsareplottedin the rightpanel. with scales of about half of the size of the galaxy Again, because of these sources, the small scales ( 900′′or3.7kpc)isnotassignificantasthatfrom ∼ aredominantat3.6cm.Asshownincomparingthe smaller structures. It seems that a minimum in left and right panels, they also are seen in all the the wavelet spectra of the radio and Hα emission, infrared bands, strongly at 24µm and much more as was also shown in NGC6946 (see Fig. 7 in weaklyat160µm.Here,thesmoothingeffectisseen Frick et al. 2001), is a characteristic of this scale. in all 3 MIPS wavelet spectra. However, while this decrease disappears in the Before source subtraction, the 20 and 3.6cm 3.6cm and Hα spectra after source subtraction, it spectra are similar to each other and to the Hα remains in the 20cm spectrum. This implies that spectrum up to the scale of the central extended besides the bright HII regions there are sources of nonthermal emission within the spiral arms and 4 GaussianPSFswithFWHM=51′′ and84′′ arecon- central extended region which emitt significantly sidered to convolve the IR maps to the angular resolutions of the20 and 3.6cm radio maps, respectively. Tabatabaei et al.: A multi–scale studyof IRand radio emission from M33 9 Table 3. Fractions of the wavelet energy provided by the 11 brightest HII regions at different scales (Eq. 4). λ ∆(210′′) ∆(505′′) ∆(1020′′) 24µm 0.94 0.81 0.35 70µm 0.80 0.57 0.14 160µm 0.52 0.33 0.11 3.6cm 0.86 0.69 0.16 at 20cm. To estimate how much of the wavelet energy M(a)isprovidedbythesubtractedsources,wecon- sider the following definition: M (a) M (a) ∆(a) or. − s.s. , (4) ≡ M (a) or. where M (a) and M (a) represent the wavelet or. s.s. energy before and after source subtraction. Table 3 shows the fraction of energy produced by the 11 HIIregionsinIRandat3.6cmfor3differentscales (at 84′′ resolution). The corresponding fractions at 20cmarenotshowninthis table,assomenonther- malradiosourceswerealsosubtractedatthiswave- length. The emission at 24µm has the largest energy fraction ∆(a) at all scales. The smallest ∆(a) occurs at 160µm. This indicates that HII regions play a more important role in providing energy for dust emission at 24µm than at 160µm. Weestimatetheuncertaintiesofthewaveleten- ergies of each map by making a noise map with the distribution and amplitude of the real noise in the corresponding observed map. A linear combination of uniform and Gaussian distributions sim- ulates the real noise in each observed image. The wavelet spectra of the derived noise maps were ob- Fig.7.Thecross–correlationbetween24and70µm tained using Eq. 3. The resulting wavelet energies (top), 24 and 160µm (middle), and 70 and 160µm of the noise maps are taken as the uncertainties of (bottom) images at 51′′ resolution before (or.) and the waveletenergiesofthe observedmapsatdiffer- after (s.s.) source subtraction. ent scales. The estimated values are at least three orders of magnitude less than the wavelet energies of the observed maps. Therefore, the results from wherethesubscriptsrefertotwoimagesofthesame ourwaveletspectrumanalysisarenotsubstantially sizeandlinearresolution.Thevalueofrw variesbe- affected by the noise in the maps. tween -1 (total anticorrelation) and +1 (total correlation). Plotting r against scale shows how well w structures at different scales are correlated in in- 5. Wavelet cross–correlations tensity and location. The error is estimated by the A useful method to compare images at different degree of correlation and the number of indepen- wavelengths is the wavelet cross-correlation. The dent points, n: waveletcross-correlationcoefficientatscaleaisde- 1 r2 fined as: ∆r (a)= − w, (6) w p√n 2 rw(a)= R R W[M11(a(a,)xM) 2W(a2∗)(]a1/,2x)dx, (5) where,n=2.13−(La)2, andL is the size ofthe maps. 10 Tabatabaei et al.: A multi–scale studyof IRand radio emission from M33 First, we examine the cross–correlations between the 3 MIPS maps of M33 (Fig.7). There are significant correlations between each pair of the MIPS maps at different scales within 0.4<a<4kpc, as r > 0.75 (Frick et al. 2001). w These scales include gas clouds, spiral arms, and the central extended region. Comparing the plots, the 24–160µm correlation is weaker than the two other correlations,especially at scales smaller than the spiral arms (a<400′′). This is due to significantly different contributions ofHII regionsin providing energy for dust emission at 24 and 160µm (see Table 3), as the 24–160µm correlation coeffi- cientsincreaseaftersubtractingthesesources.This indicatesthatHIIregionsinfluencethecorrelations at scales larger than their sizes (the width of the huge HII regionNGC604 is about 100′′ or 0.4kpc). Second,westudythecross–correlationsbetween the infrared and the radio continuum emission at 3.6cm. As shown in Fig. 8, the correlations between 3.6cm radio and the three infrared emission are strong at all scales. At scales smaller than the width of the spiral arms (1.6kpc), the correlations are higher between 3.6cm and 24µm emission (and also 70µm) than between 3.6cm and 160µm emission. The source subtraction reduces the 3.6cm–24µm and 3.6cm–70µm correlationsat scales smaller than the spiral arms. This indicates that young massive stars are the most common andimportantenergysourcesofthe 3.6cm,24and 70µm emission at these scales. Third,thecross–correlationswiththeradiocon- tinuum emission at 20cm are presented in Fig. 9. Before the source subtraction, the correlations be- tween20cmradioandthethreeIRbandsarestrong at scales smaller than the width of the spiralarms. After the source subtraction, the coefficients of the IR–20cmcorrelationsdecreasedramaticallysothat Fig.8. The cross–correlationbetween 3.6cm radio theybecomelessthan0.75atsmallscales.Therea- and IR before and after subtraction of the same son is possibly that besides HII regions there are sources at 84′′ resolution. otherunresolvedsourcesofthe20cmradioemission like supernova remnants that are stronger than at 3.6cmbutdonotemitsignificantlyinthe infrared. 6. Comparison with Hα Studies within our Galaxy have demonstrated that the ratio of far-infrared to radio continuum emis- Ifwe takethe Hαemissionasatracerofstarform- sion for supernova remnants is much smaller than ing regions that both heat the warm dust and ion- that for HII regions (Fuerst et al. 1987). ize the gas giving the free–free emission, we expect Comparing Fig. 8 with Fig. 9, a drop in cor- acorrelationbetweenHαemission,IRandthermal relation coefficient at the scale of 800′′ (3.2kpc) radioemission.Thecross–correlationsinvolvingHα is more prominent in the MIPS correlations with andIRimagesareshowninFig.10.Thesignificant 20cm than with 3.6cm. This is due to the strong correlationof Hα with the 24 and 70µm images at minimum in the 20cm spectrum at this scale (Fig. the smallest scale confirms that most of the com- 5) that persists even after the source subtraction pact structures in the MIPS decomposed maps at (incontrastto thesituationinthe3.6cmspectrum 24and70µm(Fig.3)correspondtotheHIIregions. shown in Fig. 6). The 160µm emission is also correlated with Hα