CARMA Survey Toward Infrared-bright Nearby Galaxies (STING): Molecular Gas Star Formation Law in NGC4254
Nurur Rahman, Alberto D. Bolatto, Tony Wong, Adam K. Leroy, Fabian Walter, Erik Rosolowsky, Andrew A. West, Frank Bigiel, Juergen Ott, Rui Xue, Rodrigo Herrera-Camus, Katherine Jameson, Leo Blitz, Stuart N. Vogel
aa r X i v : . [ a s t r o - ph . C O ] J a n Draft version October 3, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
CARMA SURVEY TOWARD INFRARED-BRIGHT NEARBY GALAXIES (STING): MOLECULAR GAS STARFORMATION LAW IN NGC 4254
Nurur Rahman , Alberto D. Bolatto , Tony Wong , Adam K. Leroy , Fabian Walter , Erik Rosolowsky ,Andrew A. West , Frank Bigiel , J¨urgen Ott , Rui Xue , Rodrigo Herrera-Camus , Katherine Jameson , LeoBlitz , Stuart N. Vogel Draft version October 3, 2018
ABSTRACTThis study explores the effects of different assumptions and systematics on the determination of thelocal, spatially resolved star formation law. Using four star formation rate (SFR) tracers (H α withazimuthally averaged extinction correction, mid-infrared 24 µ m, combined H α and mid-infrared 24 µ m, and combined far-ultraviolet and mid-infrared 24 µ m), several fitting procedures, and differentsampling strategies we probe the relation between SFR and molecular gas at various spatial resolutions(500 pc and larger) and surface densities (Σ H ≈ −
245 M ⊙ pc − ) within the central ∼ . f DE , thus determined is a strong inversefunction of the size of the filtering kernel. We find that in the high surface brightness regions ofNGC 4254 the form of the molecular gas star formation law is robustly determined and approximatelylinear ( ∼ . − .
1) and independent of the assumed fraction of diffuse emission and the SFR traceremployed. When the low surface brightness regions are included, the slope of the star formationlaw depends primarily on the assumed fraction of diffuse emission. In such case, results range fromlinear when the fraction of diffuse emission in the SFR tracer is f DE .
30% (or when diffuse emissionis removed in both the star formation and the molecular gas tracer), to super-linear ( ∼ .
4) when f DE & µ m SFR tracer by itself shows the tightest correlationwith the molecular gas surface density, whereas the H α corrected for extinction using an azimuthally-averaged correction shows the highest dispersion. We find that for R < . the local star formationefficiency is constant and similar to that observed in other large spirals, with a molecular gas depletiontime τ dep ∼ Subject headings: galaxies: general — galaxies: individual(NGC 4254) — galaxies: spiral — galaxies:ISM — ISM:molecules — stars:formation INTRODUCTION
The formation and evolution of galaxies is driven bythe complex processes of star formation (SF) that oc-cur inside them. Some galaxies produce stars at verylow rates . . ⊙ yr − , some do at modest rates ∼ ⊙ yr − , while others host ongoing starbursts with SFR, ∼ − ⊙ yr − . The processes responsible for con-verting gas into stars in various galactic environments arestill poorly understood. Observations find that the SFRand the gas content in galaxies are related by,Σ SFR = A Σ
Ngas , (1) Department of Astronomy, University of Maryland, CollegePark, MD 20742, USA; [email protected] Department of Astronomy, University of Illinois, Urbana-Champaign, IL 61801, USA National Radio Astronomy Observatory, Charlottesville, VA, USA Max-Planck-Institute fur Astronomie, Konigstuhl 17, 69117,Heidelberg, Germany I. K. Barber School of the Arts & Science, University ofBritish-Columbia, Kelowna, BC V1V1V7, Canada Department of Astronomy, Boston University, Boston, MA02215, USA Department of Astronomy, University of California, Berke-ley, CA 94720, USA National Radio Astronomy Observatory, Socorro, NM 87801,USA where Σ
SFR and Σ gas are the star formation rate surfacedensity and the gas (atomic and molecular) surface den-sity, respectively; and A is the normalization constantrepresenting the efficiency of the processes (Schmidt1959, 1963; Sanduleak 1969; Hartwick 1971; Kennicutt1989). For disk averaged surface densities, both normalstar-forming and starburst galaxies follow this relation-ship with a power law index of N ∼ . − . I , CO, andSFR are useful for characterizing global disk properties,understanding the mechanisms behind the SF law re-quires resolved measurements. Only recently has it be-come possible to probe the form of the gas-SF relation-ship on kpc and sub-kpc scales, through the availabilityof high-resolution interferometric H I and single-dish COobservations and of a suite of multi-wavelength SFR trac-ers. Studies of the local SF law on nearby galaxies pro-vide substantial evidence that the molecular gas is well-correlated with the SFR tracers, whereas the atomic gasshows little or no correlation with SF activity (e.g., Wong& Blitz 2002; Bigiel et al.. 2008). This is a natural con-sequence of stars forming out of giant molecular clouds Rahman et al.(GMCs), as we observe in the local universe. Moreover, ithas long been known that the spatial distribution of COemission follows closely that of the stellar light and H α (Young & Scoville 1982; Scoville & Young 1983; Solomonet al. 1983; Lord & Young 1990; Tacconi & Young 1990;Boselli et al. 1995).Spatially resolved SF law studies frequently reach dis-similar conclusions on the value of the exponent in Equa-tion 1 when relating molecular gas to SFR. Hereafter wewill express the exponent as N mol to represent the molec-ular gas SF law. Wong & Blitz (2002) used azimuthallyaveraged radial profiles for gas and SFR in a sample ofseven molecule-rich spiral galaxies, finding that the bestfit power law index for the molecular gas and SFR densityradial profiles is N mol ∼ . − .
4, very dependent on theextinction correction applied to their SFR tracer (H α ).Boissier et al. (2003) used CO observations along the ma-jor axes of sixteen disk galaxies with spatial resolution of ∼ − mol ∼ − . mol ∼ . α as theSFR tracer. In the most recent comprehensive study,Bigiel et al. (2008) analyzed a sample of 18 normal diskand irregular galaxies using a combination of GALEX far-ultraviolet (FUV, 1350-1750 ˚A) emission and
Spitzer µ m to trace SFR, and CO 2 − ∼
750 pc. They found a best fitslope N mol ≈ . H and Σ SFR .This spread in the value of the power law index is ob-served by in-depth case studies of just one galaxy. An ex-ample is M 51, where Kennicutt et al. (2007) studied therelation between gas and SFR on ∼ . − . II regions to find N mol ∼ . − . mol ∼ .
64 using bolometric (combiningultraviolet and infrared luminosity) SFR tracer at 400pc resolution. By comparison, Schuster et al. (2007)used the λ = 20 cm radio-continuum as SFR tracer andCO J = 2 − mol .
1, changing with galactocentricdistance. Similarly, Blanc et al. (2009) studied the cen-tral ∼ α dataat 170 pc resolution, finding a slightly sub-linear rela-tionship (N mol ∼ . ± . mol ∼ . − .
8) depending on thechoice of SFR tracers, data sampling and fitting tech-niques. For the same galaxy but with a different SFRtracer Heyer et al. (2004) found N mol ∼ . conversion, or the range of spatial scalesprobed. It is important to keep in mind that the choiceof SFR tracers and spatial scales means that differentstudies effectively sample different time scales, thus theSF history of any particular galaxy potentially plays animportant role in determining the result of the measure-ment. It is also possible that these differences correspondto a spectrum of physical SF mechanisms present in awide range of environments: in that case, the local SFlaw would not be universal. It is, therefore, vital to un-derstand the impact of systematics on the measurementof the parameters of the local SF law. Whether the lo-cal SF law is linear or non-linear has implications for thedominant SF mechanisms as well as for modeling efforts.The objective of this paper is to explore the moleculargas SF law in the galaxy NGC 4254 (M 99), at ∼
500 pcand ∼ conversion factor, the extinction correc-tion, and various other assumptions pertinent to localSF law studies. These issues will be addressed in thefuture using other sample galaxies of the STING survey.The organization of the paper is as follows. In § §
3. Section § §
5. The mainfindings of our study are summarized in §
6. A brief in-troduction to NGC 4254, details on the construction ofvarious data products such as surface density maps, thediscussion on the treatment of DE, and the details ofthe sampling and the regression analysis can be found inseveral sections of the Appendix. DATA
The target of this study, NGC 4254, is an almost face-on ( i ∼ ◦ ) SA(s)c spiral located at a kinematic distanceof 16.6 Mpc (Prescott et al. 2007). It lies in the outskirtsof Virgo cluster, ∼ . ′′ corresponds to ≈
80 pc. Theoptical radius of this galaxy is R ≈ . J = 1 − survey, andsingle-dish CO J = 2 − GALEX
NearbyGalaxy Survey (NGS; Gil de Paz et al. 2007), and H α and24 µ m images from the Spitzer
Infrared Nearby GalaxiesSurvey (SINGS ; Kennicutt et al. 2003) archive.Here we describe the multi-wavelength data. The ba-sic information for the data set is provided in Table 1.For various other properties of NGC 4254 the reader isreferred to Table 1 of Kantharia et al. (2008). CARMA STING Data
The interferometric CO J = 1 − δ > − ◦ ), moderately inclined ( i < ◦ )galaxies within 45 Mpc culled from the IRAS
RevisedBright Galaxy Survey (RBGS; Sanders et al. 2003).These galaxies have been carefully selected to have uni-form coverage in stellar mass, SF activities, and morpho-logical types. The survey is complementary to BIMASONG (Helfer et al. 2003) but targeted to have betterdisk coverage, sensitivity and resolution. The details ofthe CARMA STING survey will be published in a forth-coming paper (Bolatto et al., in preparation).Observations with the CARMA interferometer wereconducted in the D array configuration during June 2008for a total of 8.5 hours. Passband and phase calibrationwere performed using 3C273, and 3C274 was used as asecondary phase calibrator to assess the quality of thephase transfer and coherence. The absolute flux scalefor the interferometer was determined by observing Mars.At λ = 2 . ′′ (54 ′′ )whichdefines their effective field of view. The observationswere obtained using a Nyquist-sampled 19-pointing mo-saic pattern that provides an effective field of view of100 ′′ in diameter. The synthesized beam produced usingnatural weighting has 4.3 ′′ FWHM, which is the resolu-tion of the resulting map.In our frequency setup, the receivers were tuned to theCO J = 1 − λ = 2 . − with a intrinsic velocity resolution of 2.6 kms − . The maps were produced with 10 km sec − veloc-ity resolution. The sensitivity of the interferometric mapbefore combination with the single-dish data is σ ≈ ∼ bolatto/STING/ http://ssc.spitzer.caltech.edu/legacy/singshistory.html mJy Beam − in 10 km sec − wide channels (see Fig.1).The high angular resolution measurements obtainedwith interferometers filter out the large spatial scales ofthe source, giving rise to the “missing flux” problem.As a result, surface densities for extended sources de-rived from interferometric data alone may be underes-timated. To overcome this limitation it is necessary tomerge single-dish observation of the object with the in-terferometric data.The single dish IRAM CO J = 2 − λ = 1 . ∼ . ′′ beam and extendbeyond the region mapped by STING. The sensitivity ofthe single dish map is σ ≈
27 mK in 2.6 km sec − widechannels. We have used a gain factor of T / S ∼ .
14 K / Jyappropriate for the IRAM 30m single dish telescope andconverted the units of the CO J = 2 − − .Comparison of the enclosed fluxes shows that (assum-ing thermalized optically-thick CO emission, see below)the STING map recovers most of the single-dish flux inits inner 60 ′′ , progressively losing flux beyond that point(see Fig. 1). We converted the CO J = 2 − J = 1 − mb ) is approxi-mately the same for the CO J = 1 − J = 2 − J = 1 − ′′ , theglobal CO flux for NGC 4254 estimated in the FCRAOCO J = 1 − ± − (Younget al. 1995). Our estimated flux using the IRAMCO J = 2 − ∼ − out toR , showing that IRAM reproduces the FCRAO mea-surement within ∼ − − UV, Optical and Mid-infrared Data
NGC 4254 has been observed in the near-ultraviolet(NUV, 1750-2750 ˚A) by
GALEX . To convert the mapfrom NUV to FUV we use a morphology dependent colorcorrection (FUV − NUV) ≈ .
46, following Gil de Pazet al. (2007). The morphology parameter T = 5 ofNGC 4254 is obtained from the Third Reference Cat-alog (RC3; de Vaucouleurs et al. 1991). To correctthe FUV map for line-of-sight Galactic extinction we useA
FUV ∼ .
24 E(B − V) (Weyder et al. 2007), where the Rahman et al.
Table 1
Basic Information of the Data SetTelescope Wavelength Pixel FWHM Sensitivity (1 σ ) Sensitivity UnitGALEX 0.2271 µ m 1.5 5.6 3 . ± .
00 10 − erg sec − cm − KPNO 0.6563 µ m 0.3 1.5 (6 . ± . × − − erg sec − cm − Spitzer 24 µ m 1.5 6.0 3 . ± .
82 10 − erg sec − cm − CARMA 2.6 mm 1.0 4.3 0.22 Jy Beam − km sec − IRAM 1.3 mm 2.0 12.5 1.00 Jy Beam − km sec − Σ SFR – 3.0 6.0 0.10 M ⊙ Gyr − pc − Σ H2 – 3.0 6.0 3.70 M ⊙ pc − Note . —
The pixel resolution and the FWHM of FUV, H α , and MIR 24 µ m maps are in unit of arcsec. The limitingsensitivities of the CARMA CO J = 1 − J = 2 − σ = [ σ ch √ N] ∆v, where σ ch is the rms noise in avelocity channel map, N is the number of channel, and ∆v is the velocity resolution. The velocity resolutions for the CARMAand the IRAM observations are ∆v = 10 and 2.6 km sec − , respectively. In this table only the surface densities are inclinationcorrected. Galactic reddening E(B − V) ≈ .
04 is estimated fromSchlegel et al. (1998). The FUV map is converted toAB magnitudes using the following formula (Gil de Paz,private communication),m AB = − . / sec] + 18 .
82 (2)The SINGS project public data archive provides cal-ibrated and stellar continuum subtracted H α image ofthis galaxy. Comparing the R-band and H α images weidentify foreground stars which are then masked, partic-ularly those within the optical diameter. The resultingimage is then corrected for [N II ] λλ α filter,using the factors obtained for this galaxy by Prescott etal. (2007).The SINGS archive also provides the mid-infrared(MIR) 24 µ m image, which s a scan map taken with theMIPS instrument on board the Spitzer
Space Telescope(Rieke et al. 2004). The MIPS data were processed usingthe MIPS Instrument Team Data Analysis Tool (Gor-don et al. 2005). No stellar masking was necessary forthe MIR map of NGC 4254. Basic information of theseimages are given in Table 1.
Data Products
The spatial resolution of our study is limited by thepoint spread function (PSF) of the MIR data, which hasa FWHM of 6 ′′ . This angular scale corresponds to aphysical length of ∼
480 pc in the disk of NGC 4254. Weshould note that, although we approximate it as a Gaus-sian, the mid-infrared PSF is complex. It has prominentfirst and second Airy rings, with the second ring stretch-ing out to ∼ ′′ . Nevertheless, approximately 85% ofthe total source flux is contained within the central peakwith FWHM of 6 ′′ (Engelbracht et al. 2007).The higher resolution H α and CO images wereGaussian-convolved to have the same resolution and sam-pling as the MIR image. In both cases, the convolutionand regridding used the AIPS package . No convolutionwas necessary for the FUV image, since it has a resolu-tion similar to the MIR (see Table 1). For our high-resolution analysis we regrid the images to 3 ′′ pixels toNyquist-sample the PSF. The Astronomical Image Processing System (AIPS) has beendeveloped by the National Radio Astronomy Observatory (NRAO)
Figure 2 shows the FUV, H α , and 24 µ m images ofNGC 4254 used to construct the SFR maps in logarith-mic color scale. The black contours correspond to the 3and 12 Jy Beam − levels of the CO J = 1 − ∼
100 K) traced by the 24 µ m emis-sion and the cold molecular gas traced by CO J = 1 − J = 1 − i ∼ o . In this study we constructfour different SFR surface density maps from combin-ing FUV and 24 µ m (Σ FUV+24 µ m), extinction correctedH α (Σ H α ), observed H α and 24 µ m (Σ H α +24 µ m), and 24 µ m (Σ µ m) following various prescriptions in the litera-ture. All four of SFR surface density maps are expressedin units of M ⊙ Gyr − pc − . The limiting 1 σ sensitiv-ity in surface density varies from map to map where theH α + 24 µ m map has the highest rms sensitivity ( ∼ . ⊙ Gyr − pc − ) among all four SFR tracer maps. Weadopt this value as the limiting sensitivity for all Σ SFR maps.The inclination corrected limiting surface densities cor-responding to the sensitivity limits (Table 1) of theCARMA interferometer and the IRAM single dish mapsare ∼ . ⊙ pc − and ∼ . ⊙ pc − , respectively. Asmentioned in section 2.1, we apply a multiplicative factorof 4 to the sensitivity limit of the IRAM single dish mapto derive the limiting surface density. We combine theinterferometric and single-dish maps to create the molec-ular gas surface density map (Σ H ). This combined mapfinally convolved with a Gaussian beam to the obtain thespatial resolution of 6 ′′ . The typical 1 σ sensitivity of thisΣ H map is ∼ ⊙ pc − (inclination corrected). Westudy the molecular gas-SFR surface density relation forΣ H ∼ −
245 M ⊙ pc − . Sampling and fitting considerations
Since one of the goals of this study is to explore how thefunctional form of the SF law depends on the treatmentof the data, we analyze the images using 1) pixel anal-olecular Gas Star Formation Law in NGC 4254 5ysis, incorporating all the data above a signal-to-noisecut, 2) aperture analysis, where we average over circu-lar apertures selecting bright regions, and 3) azimuthallyaveraged annuli, with a width of 500 pc.In local SF law studies, especially for normal star-forming galaxies such as NGC 4254, the dynamic rangeprobed by the molecular gas is rather small. The ob-served dispersion in the SFR tracer, on the other hand, isusually quite large depending on the selection of the SFRtracer. Due to this characteristic of the gas-SFR surfacedensity relation, the determination of the functional formof this relation depends critically on statistical method-ologies and fitting procedures. We describe the samplingand fitting strategies in detail in the appendix sectionsD and E. DIFFUSE EXTENDED EMISSION
Several components contribute to the total emissionin galaxies. Images contain emission from backgroundsor foregrounds, which are not physically related to thegalaxy. Besides the emission of the localized SF, theyalso contain diffuse components that are extended overthe entire disk and not necessarily associated with SFactivity, which we discuss in the following section. Sincethe calibration of SFR tracers is frequently performed instar-forming regions, it may be important to remove thecontribution from diffuse components to the brightnessdistribution before interpreting it in terms of a SFR.The CO distribution of a galaxy can also contain dif-fuse emission not necessarily associated with the indi-vidual star-forming regions. A collection of unresolvedsmall molecular clouds, in particular Taurus-like cloudsin the Milky Way with masses M ∼ M ⊙ , will fallbelow our detection threshold as individual entities butwould contribute to the diffuse extended emission. Itshould be removed if those clouds do not host massiveSF contributing to the SFR tracers. Diffuse Emission in the SFR Tracers
The DE is ubiquitous in the UV, H α and 24 µ m mapsand it spreads out across the disk over a few tens tohundreds of parsecs from the clustered OB associationand resolved H II regions. The origin of this emission anactive area of research over the past four decades (seeMonnet 1971; Reynolds et al. 1971; Haffner et al. 2009and references therein for diffuse emission from H II re-gions; see Witt 1968; Hayakawa & Yoshioka 1969; Meureret al. 1995; Pellerin et al. 2007 and references therein fordiffuse UV emission; see Dale et al. 2007; Draine et al.2007; Verley et al. 2007, 2009 for diffuse 24 µ m emission).The extended H α emission originates in diffuse ( n e ∼ . − ) warm (T ∼ ∼ ∼ ∼ −
15% to the total H II emission. For ex-ternal galaxies, however, observational evidences suggestthat the DIG may contribute a substantial fraction ( ∼ II regions have beenused to argue that early type OB stars in H II regions arethe sources of this diffuse emission. Leakage of ionizingphotons from porous H II regions has been invoked toexplain the widespread distribution of this component.This requires the interstellar medium (ISM) to have lowFUV extinction along certain lines-of-sight, allowing alarge mean free path for these photons likely through in-terconnecting ionized bubbles (Tielens 2005, Seon 2009).On the other hand, observational studies suggest that theDIG may not be entirely associated with the early-typemassive OB stellar clusters in H II regions. A popula-tion of late-type field OB stars (Patel & Wilson 1995a,b;Hoopes et al. 2001) or supernovae shocks may also pro-vide energy to the DIG (Collins & Rand 2001; Rand et al.2007). The relative contribution of each of these sourcesto the DIG energy balance is uncertain. Recent numer-ical simulations, however, suggest that the contributionfrom H II regions to the energy budget of the DIG couldbe ∼
30% or less (Seon 2009).Because mostly non-ionizing photons contribute to it,the diffuse UV continuum emission has an origin differentfrom that of the DIG. In starburst galaxies, Meurer etal. (1995) found that about 80% of the UV flux at 2200˚A is produced outside clustered OB associations and ithas an extended character. Popescu et al. (2005) sug-gested UV light scattered by dust as the possible originof the diffuse UV emission. Tremonti et al. (2001) andChander et al. (2003, 2005), however, have noted thatfor starburst galaxies the spectral UV lines from clustersare different from those in the inter-cluster environment.Their studies show that the UV stellar signature in clus-ters is dominated by O-type stars, while the inter-clusterenvironment is dominated by less massive B-type stars.Late type OB field stars were also suggested by Hoopeset al. (2001) as the origin of the diffuse UV emission innormal spirals. In a recent study, Pellerin et al. (2007)find that ∼ ∼
100 Myr)and less massive ( ∼ − ⊙ ) than O-type stars (age ∼
10 Myr, mass ≥
20 M ⊙ ), with the latter types mostlyresiding in clustered associations.The 24 µ m continuum emission also has a diffuse com-ponent associated with it. In galaxy disks, 24 µ m dustemission is frequently found near discrete H II regions(Helou et al. 2004). This extended 24 µ m emission is dueto small dust grains out of equilibrium with the radia-tion field, for which single-photon events produce large Rahman et al. Table 2
Diffuse Fraction at Various Filter ScalesNumber Filter Width Filter Width Diffuse Fraction(arcsec) (kpc) f FUV f DIG f MIR
I 75 6.03 0.72 0.55 0.68II 90 7.23 0.69 0.50 0.62III 105 8.44 0.66 0.45 0.56IV 120 9.64 0.62 0.40 0.50V 135 10.85 0.59 0.35 0.44VI 150 12.06 0.55 0.30 0.38VII 165 13.27 0.52 0.25 0.34VIII 180 14.47 0.48 0.21 0.30IX 195 15.68 0.45 0.18 0.25X 210 16.88 0.42 0.15 0.22XI 225 18.10 0.39 0.11 0.19
Note . — Diffuse fractions as a function of galacto-centric radius that area obtained from different SFR maps are shown in Fig. 14 in the appendixF. temperature excursions (Desert et al. 1990). In additionto this localized emission, 24 µ m sources are surroundedby a diffuse component associated with overall galaxyprofile and internal structure such as spiral arms (Helouet al. 2004; Presscott et al. 2007; Verley et al. 2007,2009). The old stellar population is thought to be respon-sible for such component, which comprises ∼ − ∼ −
80% of the integrated emission in the extendeddisk (Draine et al.2007; Verley et al. 2009).Understanding the nature and sources of DE is of greatimportance in studies of SF. Kuno et al. (1995) and Fer-guson et al. (1996) discussed the role of the diffuse com-ponent when deriving the SFR based on H α emission. Anassessment of the magnitude of DE contribution is neces-sary in order to use the SFR tracers derived from FUV,H α , and MIR 24 µ m dust emission. Thus, it is interest-ing to study the contribution of the DE in these tracersand its consequences on spatially resolved molecular gasSF law studies. The contribution from DE is most im-portant in the low surface brightness regime, where it canflatten the power law index of the SF law if unaccountedfor. Diffuse Emission in the Molecular Gas
In the Milky Way most CO emission is associated withGMCs, which have a top heavy mass function (most ofthe mass is in the most massive GMCs; Solomon et al.1987). Similar “top heavy” GMC mass functions areobserved in most Local Group galaxies with the excep-tion of M33 (Engargiola et al. 2003). Even in the MilkyWay, however, there exists a population of CO-emittingmolecular clouds that are considerably more diffuse andhave lower masses and column densities than GMCs thathost massive SF. Examples are the high latitude clouds(Magnani et al. 1985), with typical column densities ofN H2 ∼ cm − and very low SF activity.We do not know whether the GMC mass functionin NGC 4254 is top heavy or bottom heavy. Even ifit is top heavy, at 500 pc resolution our observations(1 σ Σ H ∼ . ⊙ pc − ) will be sensitive only to GMCswith masses M & M ⊙ as distinct entities. Local-ized, lower mass GMCs will be blended together and willappear as a blurred diffuse emission background in thegalaxy disk. If these lower mass GMCs host no massiveSF contributing to the selected SFR tracer they shouldnot be included in the determination of the SF law, oth- erwise their inclusion will artificially steepen the powerlaw index of the SF law. Removing Diffuse Emission
We will discuss first the treatment of the DE in theSFR tracers. The procedure for removing the DE in themolecular gas map is very similar, and is discussed in § II regions and the DIG in the nebularemission map. For example, forbidden line ratio of[S II ] λλ α (Walterbos & Bruan 1994),the equivalent width of H α line (Veilleux et al. 1995),surface brightness cut (Ferguson et al. 1996), unsharpmasking (Hoopes et al. 1996), H II region luminosityfunction (Thilker et al. 2002). Although the forbiddenline ratios are powerful probes to separate the compo-nents unambiguously, in absence of such information two-dimensional (2D) spatial filtering techniques such as un-sharp masking is the most robust method to obtain in-formation about DE.We use unsharp masking to model and remove the DE,taking advantage of its extended nature. Several authorshave used unsharp masking to separate DE and discreteH II regions in Local Group galaxies (Hoopes et al. 1996;Greenawalt et al. 1998; Thilker et al. 2005). Our ap-proach, slightly different than these studies, is simple,easy to implement, and avoids an ad hoc surface bright-ness cut used in other studies. The process involves cre-ating a smoothed or blurred image produced by a 2Dmoving boxcar kernel (middle panels, Fig. 3) and thenthe subtraction of the smoothed map from the originalimage (see Appendix F for a more detailed explanation).It is possible to use other kernels (Gaussian, Hanning,etc), but boxcar is the computationally simplest. Theresulting final image (lower panels, Fig. 3) has a reducedcontribution from the background as well as the DE,since most of it is contained in the spatially smoothedmap. Indeed deep H α images of the distribution of DIGin the local group members, such as M 33 and M 81,show it to be quite smooth (Greenawalt et al. 1998).Since we are using multi-wavelength SFR tracers, it isimportant to understand the nature and brightness of theDE as observational studies suggest that these propertiesolecular Gas Star Formation Law in NGC 4254 7depend on the wavelength studied. For example, studiesof the fraction of DE at 24 µ m in the disk of M 33 find f MIR ∼ − f DIG ∼
60% at the center and decreasing toalmost zero towards the outer disk. It is generally near40% across the disk (Thilker et al 2005). The diffusefraction in FUV shows a remarkably flat profile f FUV ∼ .
65 (Thilker et al. 2005).The crucial aspect of unsharp masking is the choice ofthe size of the median filter kernel. The filtering kernelsize affects the fraction of the total emission of the orig-inal map contained in the smoothed image, f DE , withthe larger fractions in the smooth or diffuse componentcorresponding to the smaller filtering kernel sizes. In thesubsequent analysis we will use f DE to refer the diffusefraction in general. To refer to the diffuse fraction in theFUV, H α , and 24 µ m we use the notation f FUV , f DIG ,and f MIR , respectively.The details of the unsharp masking process can befound in the appendix F. In NGC 4254 we explore anumber of filter sizes in each SFR tracer, carrying outour analysis for each case (see Table 2). The diffuse frac-tion as a function of filter scale is shown in the panel (D)in Fig. 14. At a given filter scale the diffuse fractionis different for different SFR tracers. For convenience ofpresentation, therefore, we use f DIG as the reference DEsince it is widely known in the astronomical community.We will refer the DE as the dominant, significant, sub-dominant, and negligible part of the total disk emissionin the H α map for f DIG & ∼ ∼ . RESULTS AND DISCUSSION
We begin our analysis presenting azimuthally averagedradial distributions of surface densities to demonstratetheir spatial variations. Fig. 4 shows the derived surfacedensities of the SFR tracers as well as CO gas maps at f DE = 0. Each panel shows the 1 σ dispersion in theradial distributions out to 0 . , where R ∼ . ′′ ( ∼ . ∼ . The SF Law using Pixels and Apertures
In this section we present our results for the case whenDE is subtracted only from the SFR tracer maps. Amore general case which addresses DE in both the SFRand CO maps will be presented in the following section.The Nyquist sampling rate at the fixed 2 . σ cut resultsin ∼
800 approximately independent pixels for both gasand SFR surface density maps at the dominant diffusefractions. The number of pixels increases to ∼
950 when the contribution of DE is sub-dominant or negligible, be-cause fewer pixels fall below the signal-to-noise cut afterDE subtraction.We show our results for the Σ
SFR - Σ H relation in Fig.6. The figure shows the gas-SFR surface density relationfor pixel sampling at various f DE . The gray scale rep-resent the two dimensional histogram of the frequencyof points, and the contours are placed at 90%, 75%,50%, and 25% of the maximum frequency. The diago-nal dotted lines represent lines of constant SF efficiency( ǫ ), or constant molecular gas exhaustion timescale ( τ dep )with values of ǫ = 1%, 10%, and 100% corresponding toexhaustion times of τ dep = 10, 1, and 0.1 Gyr respec-tively. The filled circle in each panel represents the diskaveraged surface densities measured within R beforeunsharp masking. Within the range of diffuse fractionsprobed the Σ µ m - Σ H relation shows the tightest cor-relation whereas the Σ H α - Σ H relation shows the largestscatter. We compute the linear Pearson correlation coef-ficient (r) for these two relations in the range of exploreddiffuse fractions, finding r ∼ . − . ∼ . − .
55 for the latter. The observed dispersion is σ i ∼ . µ m - Σ H and ∼ . H α -Σ H .Note that the scatter in the SF law is substantiallylower when no DE is subtracted from the total emissionof the SFR tracers. Furthermore, since the DE is propor-tionally more important in fainter regions, its subtractionincreases scatter in the gas-SFR relation mostly at lowsurface densities. For the same reason, removing the DEsteepens the SF law.The results of aperture sampling are shown in Fig. 7for a 105 ′′ unsharp masking kernel. The distributions ofpoints are overlaid on the contours obtained from thepixel analysis. By construction the apertures samplemostly the high density regions, and the overall agree-ment in these regions is excellent between the pixel andthe aperture analysis. The lack of the low surface bright-ness tail along the vertical axes, however, has importantimplications for the slope of the SF law, as we discussnext.We show the measurements from different bivariate re-gression methods in Fig. 8. The figure highlights theΣ H α +24 µ m - Σ H relation for pixel sampling at 105 ′′ fil-ter scale. The scale corresponds to a case of significantto dominant diffuse fraction, f DIG =0.45 and f MIR =0.56.The figure shows that the FITEXY method yields theshallowest slope. For this gas-SFR surface density rela-tion the power-law index is in the range N mol ∼ . − . σ i ∼ . − . mol ∼ . − .
2) and smallerscatter ( σ i ∼ . − . σ mol , fit . The mean ( ¯N mol , fit ) obtained Rahman et al.from averaging the results of the three fits for each f DE is shown with filled circles. The panels in this figure il-lustrate how the measurement depends not only on thechosen SFR tracer, but also on the type of analysis andon the treatment of the DE. The linearity of the func-tional form of the molecular SF, in particular, dependson the amount of DE assigned to either axis. It is impor-tant to notice, however, that this slope change is drivenby the lowest surface brightness regions of the disk. Inthe high surface-brightness regions sampled in the aper-ture analysis the choice of f DE is unimportant, and aunique slope is consistent with the data for any (rea-sonable) amount of DE. In these regions the SF law isapproximately linear, although its precise value dependson the SFR tracer.We observe a direct relation between the slope of theSF law and the magnitude of the DE subtracted inthe pixel analysis (left panels, Fig. 9). For a domi-nant diffuse fraction ( f DE & −
60% of the total diskemission), all the resolved SF law relations in the pixelanalysis show systematically the steepest power-law in-dices, ¯N mol , fit ∼ . − .
7. For sub-dominant to neg-ligible diffuse fraction ( f DE .
30% of the total diskemission), however, the slope clearly becomes shallower,¯N mol , fit ∼ − .
2. Thus higher f DE corresponds to asteeper power-law index. This is only observed in thepixel analysis, which contains the low surface brightnessregions. Furthermore, the scatter in the results yieldedby the different fitting algorithms is also a monotonicallyincreasing function of the amount of DE subtracted. Thismethodological scatter is driven by the corresponding in-crease in the scatter of the low surface brightness pixels,which have a very broad distribution for large f DE .For the aperture analysis the fitted power indices aresystematically shallower than for pixel analysis, and ro-bust to the choice of f DE and the aperture size (rightpanels in Fig. 9). For a dominant diffuse fraction( f DE & mol , fit ∼
1. For small diffuse frac-tions ( f DE . mol , fit ∼ . −
1. Furthermore, theH α corrected for azimuthally averaged extinction tendsto consistently have the steeper slopes (and the highestmethodological scatter). Although with this data sam-pling the slope still flattens monotonically with the re-duction of the amount of subtracted DE, the dependenceon f DE is very weak and the variation in ¯N mol , fit is withinthe scatter of the different fitting methods.The power law index is slightly steeper for the larger,1 kpc diameter apertures at the dominant and signifi-cant diffuse fractions where the fitting procedures divergemore from one another. This is likely due to a combi-nation of the fact that the larger apertures encompasssome area of low surface density material, and to the re-duction in the number of data points by a factor of ∼ f DE . . ′′ , 105 ′′ and 180 ′′ . The row representedby dash in each table show the parameter for f DE =0, i.e., when the filter size is the same of the entire map.The filter widths are chosen to show the representativecases of the dominant, significant, sub-dominant, andnegligible diffuse fractions. At a given filter scale theestimates from the OLS bisector, FITEXY, and LINEX-ERR methods are shown by the top, middle, and, bottomrow, respectively. The quoted error in each parametercomes from bootstrap sampling of 1000 realizations ofdata points. The SF Law in Annuli
Many of the early resolved studies of the relation be-tween gas and star formation in galaxies analyzed thedata using azimuthal averages (e.g., Wong & Blitz 2002).Following the procedure similar to that discussed in ap-pendix D to select the common regions from the Σ H andΣ SFR maps, we also explore the SF law for azimuthally-averaged radial profiles (Fig. 10). Sampled in this man-ner, the functional form of the SF law in NGC 4254 islinear ( ¯N mol , fit ∼
1) for f DE = 0, and approximately lin-ear ( ¯N mol , fit ∼ − .
2) in the range of diffuse fractionsstudied. The linear form stems from the fact that theazimuthal averages are dominated by the high surfacebrightness regions, and there is no “extended tail” of lowΣ
SFR points steepening the fit to the distribution. Sincethe data have low dispersion all fitting methods yieldconsistent results. Table 7 shows the fitted parametersderived from the OLS bisector method in all SFR tracersat the diffuse fractions highlighted in Fig. 10.
Dispersion in the Relations
The intrinsic dispersion ( σ i ; see appendix E) in the gas-SFR surface density relations at various diffuse fractionsis shown in Fig. 11. The figure shows the mean (¯ σ i , fit )and the scatter (“dispersion of dispersion”) obtained bythe three regression methods. The SFR obtained from24 µ m displays the tightest correlation with the molec-ular gas, among all tracers ( σ i ∼ . − . µ m. Bright H α emissionwill only happen when the HII is older and the parentcloud is at least partially cleared (see also Helou et al.2004; Relano & Kennicutt 2009). And, 2) by its nature,this SFR tracer does not need to be corrected by extinc-tion. The spatial correspondence between the 24 µ m andCO maps is striking (Fig. 2).The extinction-corrected Σ H α , on the other hand,shows the largest scatter ( σ i ∼ . − . II ] λλ α mapis also another potential (likely minor) contributor to thescatter, since it may well vary with the position.Due to its large dispersion, the results for Σ H α fromthe different regression methods differ substantially fromone another ( σ mol , fit ∼ . α + 24 µ m tracer, which applies the same un-derlying extinction correction locally, yields a tightercorrelation (¯ σ i , fit ∼ . − . mol , fit ∼ − . µ m yieldsvery similar results to H α + 24 µ m. The observed scat-ter also becomes somewhat smaller for larger apertures(dashed-dot line in the right panels of Fig. 11), partic-ularly for H α which clearly benefits from averaging overlarger regions. Diffuse CO Emission
So far we have only considered the effect of diffuseemission, possibly unrelated to recent massive SF (onthe vertical axis of the Σ
SFR - Σ H plots). Should webe also concerned about analogous effects in the hori-zontal axis ( § µ m andH α + 24 µ m using pairs of lines to illustrate the method-ological dispersion. The thin solid and dashed lines showthe dependence of N mol on f DE when both axes are sub-ject to unsharp masking, at two different values of thesignal-to-noise threshold for including points. To serveas comparison, the thick solid lines represents the casewhen only the SFR tracer maps have undergone un-sharp masking. In most of our analysis we have onlyincluded points where the gas surface density map isΣ H ≥ . σ ∼ . ⊙ pc − (dashed lines in Fig. 12).To explore the effects of this threshold on the analysiswe also plot the results for Σ H ≥ σ ∼ . ⊙ pc − (thin black lines). The threshold for the SFR maps iskept at 2 . σ .This figure shows that our attempt at removing a dif-fuse molecular component in NGC 4254 has only verymild impact on the results of the analysis, and only forthe pixel analysis. The results derived from the aper-tures in the high surface brightness regions are essen-tially unchanged. Interestingly, the consistency betweenthe different fitting methods is better than in the casewhere only the DE in SFR is removed. This is likely be-cause errors in f DE subtraction smear the data along themain relation, rather than only in the vertical directionin gas-SFR surface density relation.Lowering the threshold after unsharp masking the COproduces somewhat flatter slopes at higher f DE , whileincreasing the dispersion of the results. For example, theslope is approximately unity below f MIR ∼ . H cutoff value of 7.4 M ⊙ pc − , while for a threshold of 10M ⊙ pc − it would be unity only below f MIR ∼ .
25. Thepower-law index remains unchanged at the extremes of f DE . The results for other two tracers are qualitativelysimilar to those presented in Fig. 12.The dispersion in the gas-SFR surface density relationsystematically goes down when both variables are subjectto unsharp masking. We find ∼ −
20% reduction inthe scatter depending on the SFR tracer. The fittingmethods tend to converge with one another because ofthe reduction in the scatter in the range f DE . .
3, whichis clearly evident in Fig. 12.
Goodness of Fit
How well does the power law functional form representthe SFR and molecular gas surface density relationship analyzed in this study? A measure of the goodness-of-fit of a model is to derive the χ statistic based on theleast square method (Deming 1943). The best fit linesprovided by the FITEXY estimator have reduced- χ ∼ ∼ . − .
5. This is achieved byiteratively adjusting the error along the Y-axis, σ = σ + σ , where σ i is the intrinsic scatter in the gas-SFRsurface density relation and σ m is the measurement error.A graphical alternative to evaluate the goodness-of-fit is to test the normality of residuals. The residualsare the deviations of observational data from the best fitline. We perform this test for the fitted lines producedby all three estimators. At large f DE the distributionsof residuals for various SFR tracers are approximatelyconsistent with a normal distribution, and they becomemore so when f DE decreases. Our analyses suggest thatthe observed relation between the molecular gas and SFRsurface densities in NGC 4254 is consistent, at least tofirst order, with the power law form. Star Formation Efficiency
The star formation efficiency (SFE) is a convenient,physically motivated way to parametrize the relationshipbetween molecular gas and SFR. The SFE has been de-fined in various ways in literature. For example, it isdefined as the ratio of the produced stellar mass to thetotal gas mass. This definition is more commonly seenin the Galactic studies (Myers et al. 1986) but also usedin galaxy modeling (Vazquez-Semadeni et al. 2007). Forextra-galactic studies, the molecular gas SFE is usuallydefined as (Young & Scoville 1991; McKee & Ostriker2007), SFE = ǫ = Σ SFR Σ H = A [Σ H ] N mol − . (3)The inverse of the SFE is considered as the gas deple-tion timescale, τ dep = SFE − . This parameter is used todiscern between the starburst and normal star-forminggalaxies (Rowand & Young 1999). For starburst galaxiesthe typical depletion time is hundreds of Myr whereasnormal star-forming galaxies have depletion timescale of ∼ τ dep ) as a function of diffuse fraction. We derive the SFEfinding the ratio of SFR to molecular gas for each pixelor aperture in the map, and plot the average with errorbars computed from the standard deviation. The SFE0 Rahman et al. Table 3
Disk Averaged ParametersParam. DE FUV + 24 µ m H α H α + 24 µ m 24 µ m UnitSFR f DE =0 4 . ± . . ± . . ± . . ± . ⊙ yr − Σ SFR f DE =0 13 . ± . . ± . . ± . ± ⊙ Gyr − pc − τ dep max. f DE . ± . . ± . . ± . . ± . τ dep max. f DE . ± . . ± . . ± . . ± . τ dep f DE =0 1 . ± . . ± . . ± . . ± . Note . — The parameters are estimated within R from the maps with zero subtraction of DE, i.e., maps with f DE =0. The τ dep and τ dep are estimates at maximum f DE when 1) only the SFR tracer map, and 2) the SFR tracer and gas maps bothare subject to unsharp masking. The total molecular gas mass within R is, M H , tot = (5 . ± . × M ⊙ . The disk averagegas surface density is, Σ H = (46 ±
3) M ⊙ pc − . is robustly determined and independent of f DE when f DE . . f DE observed in the Milky Way and LocalGroup galaxies (e.g., Thilker et al. 2002), and also inrecent spectroscopic determinations in the central regionof M 51 (Blanc et al. 2009). At higher f DE the SFEchanges (becomes lower) by up to a factor of 1 . − f DE (within 40%) when both the SFR and thegas map have a diffuse component removed (gray points),reflecting the fact that the SF law is linear in that case.At a given f DE , the global SFEs derived from our fourtracers are approximately consistent with one another,although the SFE obtained from H α is only marginallyso (see Table 3). The SFE in NGC 4254 is essentiallyindependent of radius up to R ∼ . ≈ . the SFR derived from the extinction-corrected H α image does not agree with the other SFR tracers, likelybecause of a problem with the extinction correction.Figure 13 also shows that from sub-dominant to negli-gible f DE the molecular SFE in NGC 4254 is fairly typ-ical of large spirals (see Table 3). The disk averaged τ dep ≈ . ± . H α +24 µ m map at f DE = 0 is in good agreement with the τ dep ≈ . J = 1 − Systematics Affecting the Local SF Law
Effect of the Non-detections
In this study we analyze regions that have values overthe adopted thresholds in both the Σ
SFR and Σ H maps.To check for the effect of not including pixels that aredetected in one axis but not the other, we include everypixel out to R . This results in about 20 additionalpoints, all with measurable SFR but no CO detectionand thus having only upper limits for their gas surfacedensities. We find that these points closely follow theoriginal distribution of points detected in both Σ SFR andΣ H at the limiting end of gas surface density. Thus,there are no new data trends hidden in the limits. Theycomprise only 3% of the total number of points obtainedwith the data selection criteria as mentioned in appendixD. The impact of these points on the determination ofthe functional form of the SF law is negligible. Variations in the Data Selection Cuts
It is a common practice to adopt one specific sensitivitylimit in analyzing the gas-SFR surface density relation(for example, 3 σ in Kennicutt et al. 2007; 2.5 σ in Bigielet al. 2008; 2 σ in Verley et al. 2010). The reason foradopting these sensitivity cuts is to ensure the reliabilityof the data. The choice of sensitivity limit, however,may have an impact on the determination of the localSF law, particularly given the limited dynamic range ofthe data. To explore the effect of this choice we haveanalyzed the gas-SFR surface density relation at severalthresholds above 2 σ sensitivity. We find that the choiceof limit has a measurable effect on the slope of the SF lawfor the pixel analysis, such that lower thresholds steepenthe slope by as much as 30% −
40% (depending on theSFR tracer considered), with a simultaneous increase inthe dispersion of the low surface brightness points. Thedetermination of the slope for the aperture sampling andthe azimuthally averaged radial profile are, on the otherhand, robust to the choice of the sensitivity limit.
Sensitivity to the Error Maps
The results presented in this section are computed fora set of measurement error maps of SFR and gas surfacedensities. These maps are constructed under certain as-sumptions of the observational uncertainties. However,measurement uncertainties in flux calibration, continuumsubtraction, and other parameters are propagated intothe error maps. Variations in the assumptions made toinclude their contributions lead to changes in the errormaps, which directly influence the regression analysis.For several sets of error maps with varying assumptionsabout the measurement uncertainties we find up to 40%variations in the slope measurements provided by the bi-variate regression methods. COMPARISON WITH PREVIOUS STUDIES
In this section, we compare our results with recentstudies of the spatially resolved SF law in nearby galaxiesby Kennicutt et al. (2007), Bigiel et al. (2008), Blanc etal. (2009), and Verley et al. (2010). While making com-parison it should be borne in mind that, for a given SFRtracer and at a given kernel size, the table provides fittingresults from three different bivariate regression methods.Our main results presented in various panels in Fig. 9,on the other hand, show the mean ( ¯N mol , fit ) and the dis-persion ( σ mol , fit ) of these three measurements.olecular Gas Star Formation Law in NGC 4254 11Kennicutt et al. (2007) obtain a super-linear power law(N mol ∼ . ± .
03) and an observed scatter of σ i ∼ . α + 24 µ m as the SFRtracer, and apertures 520 pc in diameter centered on armand inter-arm star-forming regions of M 51. They find asomewhat shallower power-law index in larger apertures(1850 pc in diameter). The authors subtract the diffusecomponent contribution using measurements in rectan-gular regions of the image, and employ FITEXY for thefitting. Although our aperture placement method is notstrictly identical since we place apertures on the highsurface density regions of NGC 4254, we find an approx-imately linear power law ( ¯N mol , fit ± σ mol , fit ∼ . ± . σ i , fit ∼ . − . µ m tracer, sim-ilar methodology in data analysis, OLS bisector fitting,and the same conversion factor, X CO . The authors findan approximately linear form (N mol ∼ . ± .
07) for theresolved molecular gas SF law in a sample of star-formingdisk galaxies at 700 pc resolution. They measure a typi-cal molecular gas depletion timescale ∼ ∼ . f DE = 0 studied in this paper. Comparing our results ofpixel analysis at f DE = 0, we find a very similar powerlaw index ( ¯N mol , fit ± σ mol , fit ∼ . ± .
03) in NGC 4254(see panel a of Fig. 9). The molecular gas depletiontimescale for this galaxy is τ dep ∼ . σ i ∼ . ∼ H α . Theyfind a slightly sub-linear (N mol ∼ . ± .
05) functionalform of the SF law with an intrinsic scatter σ i ∼ . II ] /H α ratio,finding f DE ≈ mol , fit ± σ mol , fit ∼ . ± .
2) at a simi-lar DIG fraction in NGC 4254 (see panel d of Fig. 9).The authors use a Monte Carlo method for fitting theSF law, which treats the intrinsic scatter in the relationas a free parameter. This method allows the inclusion ofnon-detections in both the Σ H and Σ SFR maps, fittingthe data in linear space. The advantage of this approachis that it is free from the systematics involved in per-forming linear regressions over incomplete data sets inlogarithmic space. A drawback of this method is that itdoes not treat the data symmetrically, as it employs Σ H as the independent variable in the fits. Bivariate statis-tical methods such as the ones in our study do not easily allow for the inclusion of upper limits. Nonetheless, theypermit a robust parametrization of the data with a sym-metric treatment of both axes, assuming that there is agood understanding of the data uncertainties.Verley et al. (2010) study the local SF law in M 33using azimuthally-averaged radial profiles at a resolutionof 240 pc, and using apertures sampling at various spa-tial scales (180 − α , extinction-corrected FUV, and a combination of ob-served FUV and total infrared luminosities. From aper-ture analysis they find that the molecular gas SF law isalways super-linear and steeper (N mol ∼ . − .
8) in H α than in the other SFR tracers (N mol ∼ . − . mol ∼ . − .
3) by all three tracers.The results of our study are qualitatively similar to thoseof Verley et al. (2010). For example, our study also sug-gests that the power-law index is systematically steeperin H α . We also obtain a shallower SF law when usingazimuthally-averaged radial profiles. At coarser spatialresolutions, Verley et al. find a slight steepening of theSF law in all SFR tracers. However, we find little changein the power-law index over the explored range of spatialscales. SUMMARY AND CONCLUSIONS
We study the spatially resolved molecular gas SF lawin NGC 4254 within the central ∼ . α with azimuthally-averaged extinction correction, 24 µ m, combined H α and24 µ m, and combined FUV and 24 µ m. We utilize vari-ous fitting procedures (the OLS bisector, FITEXY, andLINEXERR; described in appendix E) to constrain theparameters of the local SF law. We explore the effectsof error weighting and signal-to-noise cuts on the results.We employ three different sampling strategies (pixel-by-pixel, aperture, and azimuthal averages) to probe thegas-SFR surface density relation at various spatial reso-lutions (500 pc and 1 kpc) and surface densities.We investigate the effect of diffuse emission on our abil-ity to measure the local SF law. Diffuse emission is anubiquitous component of the maps in FUV, H α , and 24 µ m, which may not be associated with star formationand comprises an unknown but perhaps significant frac-tion of the total disk emission (in the Milky Way, thediffuse emission in H α related to the DIG constitutes10 −
15% of the total emission; Reynolds 1991). Simi-larly, there may be an analogous component of DE in themolecular gas axis of the SF law, comprised of unresolvedclouds that are too small to host massive star formation.The contribution from DE is most important in the lowsurface brightness regime. To extract the DE from themaps we use spatial filtering (unsharp masking).We study the gas-SFR surface density relation for themolecular gas surface density range ∼ −
245 M ⊙ pc − .This range is typical of normal star-forming galaxies butsignificantly smaller than starburst galaxies. The lower2 Rahman et al.limit of molecular gas surface density is consistent withrecent studies which suggest that the atomic to molecularphase-transition occurs in the ISM at surface densitiesΣ gas ∼
10 M ⊙ pc − (Wong & Blitz 2002; Kennicutt etal. 2007; Leroy et al. 2008)Without additional data (for example, optical spec-troscopy) we cannot establish the fraction of DE cor-responding to the different SFR tracers in NGC 4254.Therefore, we take f DE as an independent parameter andwe explore the molecular gas-SFR surface density rela-tion for varying diffuse fraction. We find that in the highsurface brightness regions sampled by our aperture anal-ysis (and dominating our azimuthal averages) the valueof f DE has little or no impact on the power-law index ofthe SF law, which is approximately linear in NGC 4254for all the SFR tracers considered ( ¯N mol , fit ∼ . − . f DE . . mol , fit ∼ . − . f DE . . f DE & .
5) all tracers yield a super-linear SFlaw ( ¯N mol , fit & . f DE is H α with a radially-dependent extinction correction.Since this is the SFR tracer that yields that largest scat-ter (¯ σ i , fit ∼ . − . f DE ( ¯N mol , fit ∼ − . σ i ) of the SF lawvaries with the choice of the SFR tracer (Fig. 11). In par-ticular, the 24 µ m emission shows the tightest correlationwith the molecular gas surface density (¯ σ i , fit ∼ . − . µ m emission is closelycorrelated with embedded SF, still associated with theparent molecular material. This suggests that 24 µ m isa very good tracer of SFR on timescales similar to thelifetimes of GMCs over spatial scales of several hundredparsecs. The combined H α + 24 µ m tracer, which essen-tially applies a local extinction correction to H α , yieldsthe second tightest correlation (¯ σ i , fit ∼ . − . R ∼ . (Fig. 13). The disk averaged depletiontimescale of the molecular gas ( ∼ α , which yields a somewhat longer molecular gasdepletion time. Like the power-law index, the SFE isindependent of f DE when the DE is sub-dominant. Re-moving a diffuse component from both the SFR and themolecular gas yields a SFE that is independent of f DE .Although the presence of diffuse emission not associ-ated with star formation in the tracer used to determinethe SFR or the molecular surface density should be aconcern, at least in NGC 4254 the SF law can be de-termined in a precise and robust manner with no exactknowledge of the diffuse fraction in two cases: 1) in thehigh surface brightness regions independent of the valueof f DE , and 2) throughout the disk if f DE < α images; Richard Rand for valuable sug-gestions regarding extended DE. The authors thankthe teams of SINGS and GALEX NGS for makingtheir outstanding data set available. This research hasmade use of the NASA/IPAC Extragalactic Database(NED) which is operated by the Jet Propulsion Lab-oratory, California Institute of Technology, under con-tract with the National Aeronautics and Space Admin-istration. We acknowledge the usage of the Hyper-Ledadatabase (http://leda.univ-lyon1.fr). We have made useof NASA’s Astrophysics Data System NASA/ADS. Sup-port for CARMA construction was derived from theGordon and Betty Moore Foundation, the Eileen andKenneth Norris Foundation, the Caltech Associates, thestates of California, Illinois, and Maryland, and the Na-tional Science Foundation. Funding for ongoing CARMAdevelopment and operations are supported by the Na-tional Science Foundation (NSF) and the CARMA part-ner universities. This research is supported in part bygrant nsf-ast0838178. Facilities: GALEX , KPNO,
Spitzer , CARMA, IRAM.
REFERENCESAkritas, M. G., & Barshedy, M. A., 1996, ApJ, 470, 706Bigiel, F., Leroy, A., Walter, F., Brinks E., de Blok, W. J. G.,Madore, B., & Thornley, M. D., 2008, AJ, 136, 2846Blanc, G. A., Heiderman, A., Gebhardt, K., Evans, N. J., &Adams, J. 2009, ApJ, 704. 842Blitz, L., Fukui, Y., Kawamura, A., Leroy, A. K., Mizuno, N., &Rosolosky, E. 2007, in ProtoStars and Planets V, eds. B.Reipurth, D. Jewitt, & K. Keil (Tucson: Univ. Arizone Press),p. 81Bolatto, A. D., Leroy, A. K., Rosolosky, E., Walter, F. & Blitz, L.2008, ApJ, 686, 948Boissier, S., Prantzos, N., Boselli, A., & Gavazzi, G. 2003,MNRAS, 346, 1215Boissier, S., et al. 2007, ApJS, 173, 524Boselli, A., & Gavazzi, G., Lequeux, J., Buat, V., Casoli, F.,Dickey, F., & Donas, J. 2003, A&A, 300, L13 olecular Gas Star Formation Law in NGC 4254 13
Bournaud, F., Combes, F., Jog, C. J., & Puerari, I. 2005, A&A,438, 507Calzetti, D., Sheth, K., Churchwell, E., Jackson, J. 2009, in TheEvolving ISM in the Milky Way & Nearby Galaxies, Ed. K.Sheth, A. Noriega-Crespo, J. Ingalls, and R. Paladini,Published online athttp://ssc.spitzer.caltech.edu/mtgs/ismevol/Calzetti, D., et al. 2007, ApJ, 666, 870Chander, R., Leitherer, C., Tremonti, C., & Calzetti, D. 2003,ApJ, 586, 939Chander, R., Leitherer, C., Tremonti, C., Calzetti, D., Aloisi, A.,Meurer, G. R., & de Mello, D. 2005, ApJ, 628, 210Chemin, L., et al. 2006, MNRAS, 366, 812Collins, J. A., & Rand, R. 2001, ApJ, 551, 57Dale, D., et al. 2007, ApJ, 655, 863Dame, T. M., Hartmann, D., & Thaddeus, P. 2001, ApJ, 547, 792de Vaucouleurs, G., de Vaucouleurs, A., Corwin, H. G., Buta, R.,Petural, G., & Fouque, P. 1991, Third Reference Catalog ofBright Galaxies, (Austin: University of Texas Press), RC3catalogDeming, W. E. 1943, Statistical Adjustment of Data (New York:Wiley)Desert, F.-X., Boulanger, F., & Puget, J. L. 1990, A&A, 237, 215Draine, B. T., et al. 2007, ApJ, 663, 866Duc, P.-A., & Bournaud, F. 2008, ApJ, 673, 787Egusa, F., Sofue, Y., & Nakanishi, H. 2004, PASJ, 56, 45LEngargiola, G., Plambeck, R. L., Rosolowsky, E., & Blitz, L.,2003, ApJS, 149, 434Engelbracht, C. W., et al. 2007, PASP, 119, 994Feigelson, E. D., & Babu, G. J. 1992, ApJ, 397, 55Ferguson, A. M. N., Wyse, R. F. G., Gallagher III, J. S., &Hunter, D. A. 1996, AJ, 111, 2265Gardan, E., Braine, J., Schuster, K. F., Brouillet, N., & Sievers,A. 2007, A&A, 473, 91Gil de Paz, A., et al. 2007, ApJS, 173, 185Gordon, K. D., et al. 2005, PASP, 117, 503Greenawalt, B. E.,, Walterbos R. A. M., Thilker, D., & Hoopes,C. G. 1998, ApJ, 506, 135Guhathakurta, P., van Gorkom, J. H., Kotanyi, C. G., &Balkowski, C. 1988, AJ, 96, 851Haffner, L. M., Dettmar, R.-J., Beckman, J. E., Wood, K., Slavin,J.-D., Giammanco, C., Madsen, G. J., Zurita, A., & Reynolds,R. J., 2009, RvMP, 81, 969Hartwick, F. D. A. 1971, ApJ, 163, 431Helfer, T. T., Thronley, M. D., Regan, M. W., Wong, T., Sheth,K., Vogel, S. N., Blitz, L., & Bock, D. C.-J. 2003, ApJS, 145,259Helou, G., et al. 2004, ApJS, 154, 253Heyer, M. H., Corbelli, E., Schneider, S. E., & Young, J. S. 2004,ApJ, 602, 723Hoopes, C. G., Walterbos R. A. M., & Greenawalt, B. E. 1996,AJ, 112, 1429Hoopes, C. G., & Walterbos R. A. M. 2000, ApJ, 541, 597Hoopes, C. G., Walterbos R. A. M., & Bothun, G. D. 2001, ApJ,559, 878Hunter, D. A., & Gallagher, J. S. 1990, ApJ, 362, 480Hunter, D. A., & Gallagher, J. S. 1992, ApJ, 391, L9Hunter, S. D. et al. 1997, ApJ, 481, 205Hayakawa, S., Yamashita, K., & Yoshioka, S. 1969, AP&SS, 5, 493Isobe, T., Feigelson, E. D., Akritas, M. G., & Babu, G. J. 1990,ApJ, 364, 104Iye, M., et al. 1982, ApJ, 256, 103Kantharia, N. G., Rao, A. P., & Sirothia, S. K. 2008, MNRAS,383, 173Kelly, B. C. 2007, ApJ, 665, 1489Kennicutt, R. C., Jr. 1989, ApJ, 344, 685Kennicutt, R. C., Bresolin, F., Bomans, D. J., & Bothun, G. D.1995, AJ, 109, 594Kennicutt, R. C., Jr. 1998a, ApJ, 498, 541Kennicutt, R. C., Jr. 1998b, ARA&A, 36, 189Kennicutt, R. C., Jr, et al. 2003, PASP, 115, 928Kennicutt, R. C., Jr, et al. 2007, ApJ, 671, 333Koopmann, R. A., Kenney, J. D. P., & Young, J. 2001, ApJSS,135, 125Koopmann, R. A., & Kenney, J. D. P. 2004, ApJ, 613, 866Komugi, S., Sofue, Y., Nakanishi, H., Onodera, S., & Egusa, F.2005, PASJ, 57, 733 Krumholz, M. R., & Tan, J. C. 2007, ApJ, 654, 304Kuno, N., Nakai, N., Handa, T., & Sofue, Y. 1995, PASJ, 47, 745Leroy, A. K., Walter, F., Brinks, E., Bigiel, F., de Blok W. J. G.,Madore, B., & Thornley, M. D. 2008, AJ, 136, 2782Leroy, A. K., Walter, F., Bigiel, F., Usero, A., Weiss, A., Brinks,E., de Blok W. J. G., Kennicutt, R. C., Schuster, K.-F.,Kramer, C., Wiesemeyer, H. W., & Roussel, H. 2009, AJ, 137,4670Lord, S. D., & Young, J. S. 1990, ApJ, 356, 135Madore, B. F. 1977, MNRAS, 178, 1Magnani, L., Blitz, L., & Mundy, L. 1985, ApJ, 295, 402Maoz, D., Filippenko, A. V., Ho, L. C., Rix, H.-W., Bahcall, J.N., Schneider, D. P., & Macchetto, F. D. 1995, ApJ, 440, 91Martin, C. L., & Kennicutt, R. C. 2001, ApJ, 555, 301McKee, C. F., & Ostriker, E. C. 2007, ARA&A, 45, 565Meurer, G. R., Heckman, T. M., Leitherer, C., Kinney, A.,Robert, C., Garnet, D. R. 1995, AJ, 110, 2665Monnet, G. 1971, A&A, 12, 379Morrissey, P., et al. 2007, APJSS, 173, 682Nakanishi, H., Kuno, N., Sofue, Y., Sato, N., Nakai, N., Shioya,Y., Tosaki, T., Onodera, S., Sorai, K., Egusa, F., & Hirota, A.2006, ApJ, 651, 804Patel, K., & Wilson, C. D. 1995a, ApJ, 451, 607—-. 1995b, ApJ, 453, 162Pellerin, A., Meyer, M., Harris, J., & Calzetti, D. 2007, Apj, 658,L87Polk, K., Knapp, G., Stark, A., Wilson, R. 1988, ApJ, 332, 432Popescu, C. C. et al. 2005, ApJ, 619, L75Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B.P. 1992, Numerical Recipes (2d ed.; Cambridge: CambridgeUniv. Press)Prescott, M. K., et al. 2007, ApJ, 668, 182Phookun, B., Vogel, S. N., & Mundy, L. E. 1993, ApJ, 418, 113Quirk, W. J. 1972, ApJ, 176, L9Rand, R. J., Kulkarni, S., & Hester, J. J. 1990, 352, L1Rand, R. J., Wood, K., & Benjamin, R. A. 2007, in The EvolvingISM in the Milky Way and Nearby Galaxies, The FourthSpitzer Science Center Conference, ed. K. Sheth, A.Noriega-Crespo, J. Ingalls, and R. Paladini, published online:http://ssc.spitzer.caltech.edu/mtgs/ismevol/Reynolds, R. J., Roesler, F. L., Scherb, F., & Boldt, E. 1971, inThe Gum Nebula and Related Problems, eds. S. P. Maran, J.C.Brandt, and T. P. Stecher (NASASP-332), p. 169Reynolds, R. J. 1991, in The Disk-Halo Connection in Galaxies,IAU Symposium No. 144, ed. H. Bloemen (Kluwer, Dordecht),p. 67Reynolds, R. J. 1993, in Back to the Galaxy, AIP Conf. Proc. No.278, eds. S. S. Holt and F. Verter (AIP, New York), p. 156Rieke, G. H., et al. 2004, ApJS, 154, 25Rosolowsky, E., Engargiola, G., Plambeck, R., & Blitz, L. 2003,ApJ, 599, 258Rownd, B. K., & Young, J. S. 1999, AJ, 118, 670Sanders, D. B., Mazzarella, J. M., Kim, D.-C., Surace, J. A., &Soifer, B. T. 2003, AJ, 126, 1607Sanduleak, N. 1969, AJ, 74, 47Sault, R. J., Teuben, P. J., & Wright, M. C. H., in ARetrospective View of Miriad , Astronomical Data AnalysisSoftware and Systems IV, ed. R.A. Shaw, H.E. Payne andJ.J.E. Hayes. PASP Conf Series 77, 433 (1995).Schmidt, M. 1959, ApJ, 129, 243Schmidt, M. 1963, ApJ, 137, 758Schuster, K. F., et al. 2004, A&A, 423, 1171Schuster, K. F., Kramer, C., Hitschfeld, M., Garcia-Burillo, S., &Mookerjea, B. 2007, A&A, 461, 143Scoville, N. Z., & Young, J. S. 1983, ApJ, 265, 148Seon, K.-I. 2009, ApJ, 703, 1159Sofue, Y., Koda, J., Nakanishi, H., & Hidaka, M. 2003, PASJ, 55,75Solomon, P. M., Barrett, J. M., Sanders, D. B., & de Zafra, R.1983, ApJ, 266, L103Solomon, P. M., Rivolo, A. R., Barrett, J., & Yahil, A. 1987, ApJ,319, 730Stanimirovi´c, S., Staveley-Smith, L., Dickey, J. M., Sault, R. J., &Snowden, S. L. 1999, MNRAS, 302, 417Strong, A. W. & Mattox, J. R. 1996, A&A, 308, L21Tacconi, L. J., & Young, J. S. 1986, ApJ, 308, 600Tacconi, L. J., & Young, J. S. 1990, ApJ, 352, 595
R.A. Offset (") D e c . O ff s e t ( " ) −120−90−60−300306090120−90−60−300306090 Figure 1.
The contours of velocity integrated CARMA CO J = 1 − J = 2 − − km sec − .The peak flux of CARMA map is 26 Jy Beam − km sec − .Tielens, A. G. G. M. 2005, in The Physics and Chemistry of theInterstellar Medium, Cambridge University PressThilker, D. A, Walterbos, R. A. M., Braun, R., & Hoopes, C. G.2002, ApJ, 124, 3118Thilker, D. A. et al. 2005, ApJ, 619, L67Thilker, D. A. et al. 2007, APJSS, 173, 572Tremoni, C. A., Calzetti, D., Leitherer, C., & Heckman, T. M.2001, ApJ, 555, 322Vazquez-Semadeni, E., Gomez, G. C., Jappsen, A. K.,Ballesteros-Parades, J., Gonzalez, R. F., & Klessen R. S. 2007,ApJ, 657, 870Verley, S., Hunt, L. K., Corbelli, E., & Giovanardi, C., 2007,A&A, 476, 1161Verley, S., Corbelli, E., Giovanardi, C., & Hunt, L. K., 2009,A&A, 493, 453Verley, S., Corbelli, E., Giovanardi, C., & Hunt, L. K., 2010,A&A, 510, 64 Vila-Costas, M. B. & Edmunds, M. G. 1992, MNRAS, 259, 121Vollmer, B., Huchtmeier, W., & van Driel, 2005, A&A, 439, 921Walterbos, R. A. M. & Braun, R. 1994, ApJ, 431, 156Wang, J., Heckman, T. M., & Lehnert, M. D. 1997, ApJ, 491, 114Wilson, C., et al. 2009, ApJ, 693, 1736Wong, T., & Blitz, L. 2002, ApJ, 569, 157Wyder, T. K., et al. 2007, ApJS, 173, 293Wyder, T. K., et al. 2009, ApJ, 696, 1834Witt, A. N. 1968, ApJ, 152, 59Young, J. S., & Scoville, N. Z. 1982, ApJ, 258, 467Zaritsky, D., Kennicutt, Robert, R. C., Jr., & Huchra, J. P. 1994,ApJ, 420, 87 olecular Gas Star Formation Law in NGC 4254 15 FUV
R.A. Offset (") D e c . O ff s e t ( " ) −100−50050100−100−50050100 H α R.A. Offset (") −100−50050100 µ m R.A. Offset (") −100−50050100
Figure 2.
Multi-wavelength maps of NGC 4254. The panels show FUV, H α , and MIR 24 µ m emission maps from left to right. Theimages are shown in logarithmic scale. The black contours represent the layout of CO J = 1 − − km sec − . The figure shows a striking similarity between the distributions of hotdust traced by the MIR map and the cold molecular gas traced by CO J = 1 − Figure 3.
Two-dimensional unsharp masking. The top panels show the original FUV, H α and MIR 24 µ m emission maps from left toright. The middle panels show the corresponding smoothed images where the smoothing is performed by a 2D box kernel with the kernelsize of 105 ′′ . The smoothed images are subtracted from the corresponding un-smoothed original images to produce unsharp masked maps(bottom panels). Figure 4.
Azimuthally averaged radial profiles of surface density maps shown at 1 σ up and below the mean trend at each radial point.Left Panel: The radial distribution of molecular gas surface density is shown for the most sensitive CARMA interferometer CO J = 1 − FUV+24 µ m, Σ H α , Σ H α +24 µ m, and Σ µ m mapsare shown by long dashed line, thick hatched region, thin hatched region, and short dashed line, respectively. Figure 5.
The Σ H α +24 µ m map with the 6 ′′ (500 pc) diameter apertures overlaid. The total number of such apertures is 102. olecular Gas Star Formation Law in NGC 4254 17 Figure 6.
Pixel-by-pixel analysis all four SFR tracers. The figure shows the molecular gas-SFR surface density relations at two represen-tative filter scales, 105 ′′ (filled contours) and 180 ′′ (black unfilled contours) as well as the case of no filtering, i.e., when f DE =0 (coloredunfilled contours). At any smoothing scale the amount of DE varies in the maps as shown at the bottom right in each panel (see also Table2). The gray scale represent the two dimensional histogram of the frequency of points, and the contours are placed at 90%, 75%, 50%,and 25% of the maximum frequency. The diagonal dotted lines represent constant SFE ( ǫ ) where ǫ ∼ . The correlationstrength varies among the maps. For a given SFR map, it depends on the degree of unsharp masking. The Pearson correlation coefficientsat 105 ′′ and 180 ′′ are r ∼ . − .
55 (weak correlation) and ∼ . − . Figure 7.
Aperture analysis for all SFR tracers for the selected apertures as shown in Fig. 5. The faint lines correspond to the contoursof pixel distributions at 105 ′′ filter scale (shown as filled contours in Fig. 6). The diagonal dotted lines present constant efficiencies as inFig. 6. olecular Gas Star Formation Law in NGC 4254 19 Figure 8.
Measurements provided by bivariate regression methods for Σ H α +24 µ m - Σ H relation using pixel-by-pixel sampling. TheSFR tracer is subject to unsharp making with a smoothing scale of 105 ′′ which results in f DIG =0.45 and f MIR =0.56. The gray scalerepresent the two dimensional histogram of the frequency of points, and the contours are placed at 90%, 75%, 50%, and 25% of themaximum frequency. For this particular distribution of points, the FITEXY provides the shallowest slope (N mol ∼ . mol ∼ . mol ∼ . σ i ∼ . ∼ . −
114 M ⊙ Gyr − pc − and ∼ −
245 M ⊙ pc − , respectively. Figure 9.
Dependence of the power law index (N mol ) of the molecular gas SF law on various diffuse fractions. The figure summarizesresults for both pixel (left panels) and aperture (right panels) samplings. Results from 6 ′′ ∼
500 pc (points) and 12 ′′ ∼ mol , fit ) and dispersion ( σ mol , fit ) from threemeasurements are shown, respectively, by gray filled circles and vertical lines. The black points connected by dashed lines in panels aand b (e and f) represent the locus of ¯N mol , fit at f FUV ( f DIG ). The horizontal dotted lines represent N mol = 1 and N mol = 1 .
5. Thevertical dotted lines demarcate the regions where the DE is assumed to be (from left to right) the dominant, significant, sub-dominant, andnegligible component of the total disk emission. Each point along the horizontal axis has 15% uncertainty (see appendix F for details). olecular Gas Star Formation Law in NGC 4254 21
Figure 10.
Azimuthally averaged radial profile analysis all four SFR tracers. Figure shows the radial relations at the dominant (filledcircle), significant (open square) and negligible ( f DE = 0) diffuse fractions. The faint lines correspond to the contours of pixel distributionsat 105 ′′ filter scale (shown as filled contours in Fig. 6). The diagonal dotted lines represent constant efficiencies as in Fig. 6. The bin sizeis 500 pc. The absence of an extended tail along the vertical axis at the low surface density regions, irrespective of the subtraction of theamount of diffuse fractions, indicates that azimuthally averaged radial profile systematically sample the high SFR surface density regionscompared to pixel sampling. See Table 7 for the derived fitted parameters for all SFR tracer maps at these three diffuse fractions. Figure 11.
The intrinsic scatter ( σ i ) in the gas-SFR surface density relations at various diffuse fractions for pixel (left panels) andaperture (right panels) samplings. Results from 6 ′′ ∼
500 pc (points) and 12 ′′ ∼ σ i , fit ; “mean dispersion”)by filled circle and thedispersion in the observed scatter (“dispersion of dispersion”) from linear regression methods by vertical solid lines. Figure highlights thatthe estimate of intrinsic scatter depends on the choice of the SFR tracer as well as on the treatment of DE. For example, the scatter issystematically larger in the Σ H - Σ H α relation (methodological dispersion, ¯ σ i , fit ∼ . − . olecular Gas Star Formation Law in NGC 4254 23 Figure 12.
Dependence of the power law index (N mol ) of molecular gas SF law when both SFR tracer and CO J = 1 − H unsharp-masked map is placed at 1 σ (3.7 M ⊙ pc − ) and 2.5 σ (9.2 M ⊙ pc − ). The thick solid line illustrates the casewhen only the SFR tracers are subject to masking. Presentation style is similar to Fig. 9. Figure 13.
Disk averaged SFE (Gyr − ) and the depletion timescale, τ dep (Gyr), at various diffuse fractions. The SFE and the τ dep areshown on the left and right vertical axis, respectively. Results for both (6 ′′ by points and 12 ′′ by dashed-dot line) aperture samplings areshown in the right panels. Parameters estimated from applying unsharp masking to the SFR tracer only, and to both the SFR and the gasmaps, are shown in black and gray respectively. For a given SFR tracer, the subtraction of DE introduces ∼
50% (Σ H α ) up to a factor 3variations in the global SFE. At a given f DE , the global SFE derived from four tracers are approximately consistent with one another (seeTable 3), although H α yields a lower depletion time. The dashed lines added by black points shown in Figs. 9 and 11 have omitted forclarity of presentation. The vertical lines have similar meanings as in Fig. 9. olecular Gas Star Formation Law in NGC 4254 25 Table 4
Fitted Parameters from Pixel AnalysisFilt. f DIG
FUV+24 µ m H α H α +24 µ m 24 µ m( ′′ ) Log A N mol σ i Log A N mol σ i Log A N mol σ i Log A N mol σ i − . ± .
11 1 . ± .
06 0.50 − . ± .
13 1 . ± .
07 0.54 − . ± .
11 1 . ± .
06 0.49 − . ± .
10 1 . ± .
05 0.4275 0.55 − . ± .
14 1 . ± .
07 0.37 − . ± .
20 1 . ± .
11 0.47 − . ± .
17 1 . ± .
09 0.40 − . ± .
13 1 . ± .
07 0.33 − . ± .
16 1 . ± .
09 0.36 − . ± .
21 1 . ± .
11 0.47 − . ± .
18 1 . ± .
10 0.41 − . ± .
15 1 . ± .
08 0.31 − . ± .
10 1 . ± .
05 0.40 − . ± .
10 1 . ± .
05 0.50 − . ± .
09 1 . ± .
05 0.41 − . ± .
08 1 . ± .
04 0.33105 0.45 − . ± .
13 1 . ± .
07 0.30 − . ± .
16 1 . ± .
09 0.42 − . ± .
15 1 . ± .
08 0.32 − . ± .
12 1 . ± .
06 0.23 − . ± .
13 1 . ± .
07 0.30 − . ± .
18 1 . ± .
10 0.42 − . ± .
14 1 . ± .
08 0.33 − . ± .
11 1 . ± .
06 0.23 − . ± .
05 1 . ± .
03 0.24 − . ± .
07 1 . ± .
04 0.36 − . ± .
05 1 . ± .
05 0.25 − . ± .
04 0 . ± .
02 0.20180 0.21 − . ± .
09 1 . ± .
05 0.15 − . ± .
13 1 . ± .
07 0.30 − . ± .
09 1 . ± .
05 0.17 − . ± .
07 1 . ± .
04 0.10 − . ± .
08 1 . ± .
04 0.15 − . ± .
12 1 . ± .
07 0.30 − . ± .
08 1 . ± .
04 0.17 − . ± .
07 1 . ± .
04 0.10 − . ± .
04 0 . ± .
02 0.18 − . ± .
06 1 . ± .
03 0.33 − . ± .
04 1 . ± .
02 0.21 − . ± .
04 0 . ± .
02 0.16– 0.0 − . ± .
06 0 . ± .
03 0.10 − . ± .
11 1 . ± .
06 0.27 − . ± .
07 0 . ± .
04 0.13 − . ± .
05 0 . ± .
03 0.10 − . ± .
06 0 . ± .
03 0.10 − . ± .
11 1 . ± .
06 0.26 − . ± .
07 1 . ± .
04 0.13 − . ± .
05 0 . ± .
03 0.10
Note . — The normalization constant, power-law index, and intrinsic scatter of the molecular gas SF law are derived from unsharp maskedSFR tracers (i.e., unsharp masking along the vertical axis). The filter scales from the top correspond to the dominant ( f DIG & f DIG ∼ − f DIG ∼ − f DIG =0. Ata given filter scale, the three rows under FUV+24 µ m provide the measurements of the OLS bisector, FITEXY, and LINEXERR fitting method,respectively. Typical uncertainty in the intrinsic scatter is few percent. Table 5
Fitted Parameters from 6 ′′ Aperture AnalysisFilt. f DIG
FUV+24 µ m H α H α +24 µ m 24 µ m( ′′ ) Log A N mol σ i Log A N mol σ i Log A N mol σ i Log A N mol σ i − . ± .
17 1 . ± .
09 0.28 − . ± .
22 1 . ± .
11 0.37 − . ± .
17 1 . ± .
08 0.30 − . ± .
16 1 . ± .
08 0.2575 0.55 − . ± .
23 0 . ± .
12 0.25 − . ± .
37 0 . ± .
20 033 − . ± .
23 0 . ± .
12 0.20 − . ± .
23 0 . ± .
12 0.20 − . ± .
26 0 . ± .
14 0.23 − . ± .
35 0 . ± .
19 0.33 − . ± .
28 0 . ± .
15 0.30 − . ± .
24 0 . ± .
13 0.20 − . ± .
14 1 . ± .
07 0.22 − . ± .
18 1 . ± .
09 0.32 − . ± .
14 1 . ± .
07 0.23 − . ± .
13 1 . ± .
07 0.20105 0.45 − . ± .
16 0 . ± .
08 0.16 − . ± .
30 0 . ± .
16 0.27 − . ± .
21 0 . ± .
11 0.18 − . ± .
18 0 . ± .
10 0.16 − . ± .
20 0 . ± .
11 0.17 − . ± .
30 0 . ± .
16 0.28 − . ± .
22 0 . ± .
12 0.19 − . ± .
19 0 . ± .
10 0.14 − . ± .
12 0 . ± .
06 0.17 − . ± .
14 1 . ± .
07 0.24 − . ± .
11 0 . ± .
06 0.17 − . ± .
10 0 . ± .
05 0.15180 0.21 +0 . ± .
16 0 . ± .
08 0.12 − . ± .
20 0 . ± .
11 0.22 +0 . ± .
15 0 . ± .
08 0.12 +0 . ± .
15 0 . ± .
08 0.10 − . ± .
16 0 . ± .
08 0.11 − . ± .
23 0 . ± .
12 0.20 − . ± .
17 0 . ± .
09 0.12 − . ± .
14 0 . ± .
07 0.10 − . ± .
10 0 . ± .
05 0.14 − . ± .
15 1 . ± .
08 0.23 − . ± .
11 0 . ± .
06 0.16 +0 . ± .
09 0 . ± .
05 0.13– 0.0 +0 . ± .
12 0 . ± .
07 0.10 − . ± .
18 0 . ± .
09 0.20 +0 . ± .
16 0 . ± .
08 0.10 +0 . ± .
13 0 . ± .
06 0.10+0 . ± .
14 0 . ± .
07 0.10 − . ± .
22 0 . ± .
12 0.20 − . ± .
15 0 . ± .
08 0.10 +0 . ± .
12 0 . ± .
06 0.10
Note . — The normalization constant, power-law index, and intrinsic scatter of the molecular gas SF law are derived from unsharp maskedSFR tracers (i.e., unsharp masking along the vertical axis). The filter scales from the top correspond to the dominant ( f DIG & f DIG ∼ − f DIG ∼ − f DIG =0. Ata given filter scale, the three rows under FUV+24 µ m provide the measurements of the OLS bisector, FITEXY, and LINEXERR fitting method,respectively. Typical uncertainty in the intrinsic scatter is few percent. Table 6
Fitted Parameters from 12 ′′ Aperture AnalysisFilt. f DIG
FUV+24 µ m H α H α +24 µ m 24 µ m( ′′ ) Log A N mol σ i Log A N mol σ i Log A N mol σ i Log A N mol σ i − . ± .
30 1 . ± .
15 0.23 − . ± .
33 1 . ± .
17 0.26 − . ± .
33 1 . ± .
16 0.26 − . ± .
30 1 . ± .
15 0.2575 0.55 − . ± .
44 0 . ± .
23 0.20 − . ± .
67 0 . ± .
36 0.20 − . ± .
52 0 . ± .
28 0.20 − . ± .
50 0 . ± .
26 0.20 − . ± .
50 0 . ± .
26 0.20 − . ± .
56 0 . ± .
29 0.25 +0 . ± .
56 0 . ± .
30 0.24 − . ± .
49 0 . ± .
26 0.20 − . ± .
27 1 . ± .
13 0.18 − . ± .
35 1 . ± .
18 0.25 − . ± .
26 1 . ± .
13 0.20 − . ± .
25 1 . ± .
12 0.21105 0.45 − . ± .
84 0 . ± .
42 0.15 − . ± .
85 0 . ± .
43 0.15 +0 . ± .
73 0 . ± .
37 0.10 +0 . ± .
42 0 . ± .
21 0.10 − . ± .
39 0 . ± .
21 0.15 − . ± .
56 0 . ± .
29 0.23 − . ± .
44 0 . ± .
23 0.18 − . ± .
37 0 . ± .
20 0.14 − . ± .
21 1 . ± .
11 0.12 − . ± .
25 1 . ± .
13 0.20 − . ± .
21 1 . ± .
11 0.15 − . ± .
21 1 . ± .
10 0.12180 0.21 − . ± .
22 0 . ± .
12 0.10 − . ± .
44 0 . ± .
23 0.12 − . ± .
27 0 . ± .
14 0.10 − . ± .
25 0 . ± .
14 0.10 − . ± .
29 0 . ± .
15 0.10 − . ± .
42 0 . ± .
22 0.17 − . ± .
32 0 . ± .
17 0.11 +0 . ± .
27 0 . ± .
14 0.10 − . ± .
18 0 . ± .
09 0.10 − . ± .
26 1 . ± .
13 0.18 − . ± .
20 1 . ± .
10 0.12 − . ± .
18 0 . ± .
01 0.10– 0.0 − . ± .
31 0 . ± .
17 0.10 − . ± .
32 0 . ± .
17 0.13 − . ± .
30 0 . ± .
16 0.10 − . ± .
15 0 . ± .
08 0.10+0 . ± .
25 0 . ± .
13 0.10 − . ± .
39 0 . ± .
20 0.15 − . ± .
28 0 . ± .
15 0.10 +0 . ± .
24 0 . ± .
13 0.10
Note . — The normalization constant, power-law index, and intrinsic scatter of the molecular gas SF law are derived from unsharp maskedSFR tracers (i.e., unsharp masking along the vertical axis). The filter scales from the top correspond to the dominant ( f DIG & f DIG ∼ − f DIG ∼ − f DIG =0. Ata given filter scale, the three rows under FUV+24 µ m provide the measurements of the OLS bisector, FITEXY, and LINEXERR fitting method,respectively. Typical uncertainty in the intrinsic scatter is few percent. Table 7
Fitted Parameters from Azimuthally Averaged Radial Profile AnalysisFilt. f DIG
FUV+24 µ m H α H α +24 µ m 24 µ m( ′′ ) Log A N mol Log A N mol
Log A N mol
Log A N mol
75 0.55 − . ± .
11 1 . ± . − . ± .
15 1 . ± . − . ± .
13 1 . ± . − . ± .
11 1 . ± . − . ± .
08 1 . ± . − . ± .
12 1 . ± . − . ± .
10 1 . ± . − . ± .
07 1 . ± . − . ± .
04 1 . ± . − . ± .
08 1 . ± . − . ± .
05 1 . ± . − . ± .
03 1 . ± .
02– 0.0 − . ± .
03 0 . ± . − . ± .
06 1 . ± . − . ± .
04 1 . ± . − . ± .
02 0 . ± . − . ± .
03 0 . ± . − . ± .
06 1 . ± . − . ± .
04 1 . ± . − . ± .
02 0 . ± . − . ± .
06 1 . ± . − . ± .
08 1 . ± . − . ± .
06 1 . ± . − . ± .
05 1 . ± . − . ± .
10 1 . ± . − . ± .
08 1 . ± . − . ± .
08 1 . ± . − . ± .
10 1 . ± .
04– 0.0 − . ± .
08 1 . ± . − . ± .
08 1 . ± . − . ± .
06 1 . ± . − . ± .
07 1 . ± . Note . — Top: parameters in the top four rows are derived from unsharp masked SFR tracer maps. Bottom: parameters in the bottom four rowsare derived when both molecular gas and SFR maps are subject to unsharp masking. The intrinsic scatter σ i is 0 in radial profile analysis and hencethe parameters are obtained from the OLS bisector method. The filter scales from the top correspond to the dominant ( f DIG & f DIG ∼ − f DIG ∼ − f DIG =0. olecular Gas Star Formation Law in NGC 4254 27
APPENDIX A. NGC 4254
The optical structure of NGC 4254 shows a one-armed appearance (m = 1 mode), unlike most grand-design spiralswhere symmetric modes are prominent. This unusual morphology has made it the target of several observational andnumerical studies (Phookun et al. 1993; Chemin et al. 2006; Sofue et al. 2003). Various explanations for its asymmetryhave been put forth including the superposition of spiral modes induced by global gravitational instability (Iye etal. 1982), the asymmetric accretion of gas onto the disk (Phookun et al. 1993; Bournaud et al. 2005), ram pressurestripping (Sofue et al. 2003; Kantharia et al. 2008), a close high-speed encounter plus ram pressure (Vollmer et al.2005), a high-speed encounter only (Duc & Bournaud 2008), and harassment while entering Virgo (Haynes et al. 2007).On the other hand, detailed photometric studies of NGC 4254 show that it is photometrically similar to other Sc typespirals, and it has no close companion (Phookun et al. 1993).NGC 4254 has a flat rotation curve with v rot ∼
150 km sec − up to 200 ′′ ( ∼
16 kpc) from its center (Guhathakurtaet al. 1988). Apart from its morphological peculiarity, it does not show any anomaly in its properties. For example,the distribution of SF through out the disk based on H α emission has been classified as normal (Koopman & Kenney2004). Its molecular gas fraction in the disk is similar to field galaxies, suggesting that it is entering Virgo for the firsttime and external agents (such as ram pressure) have not yet been effective at striping its gas, despite its morphology(Nakasihsi et al. 2006). NGC 4254 is a metal rich galaxy with a median 12+log[O / H] ∼ . B. SFR SURFACE DENSITY MAPS
We construct four different SFR maps from combining FUV and MIR maps (Leroy et al. 2008), optical H α emissionmap (Kennicutt et al. 2007; Prescott et al. 2007), optical H α emission and MIR maps (Calzetti et al. 2007), andMIR map (Calzetti et al. 2007). The details of the construction of these SFR tracers can be found in the references.These different SFR tracers probe different time scales and hence the SF history of any particular galaxy. Forexample, the H α emission traces gas ionized by massive (M >
10 M ⊙ ) stars over a timescale of <
20 Myr. TheFUV luminosity corresponds relatively older ( <
100 Myr), less massive (M & ⊙ ) stellar populations. The MIR24 µ m emission mostly traces re-processed radiation of newborn (few Myr) OB stellar associations embedded insidethe parent molecular clouds. Although star clusters break from their parent clouds in less than 1 Myr, they remainassociated with it for a much longer time scale, t ∼ −
30 Myr. Thus this is the time scale associated with the MIR24 µ m emission as a tracer. However, if there is significant heating of the small dust grains by late B and A stars, thetime scales associated with 24 µ m emission may be as long as ∼
100 Myr. For composite SFR tracer such as FUV +24 µ m or H α + 24 µ m while the MIR emission traces the dust-obscured fraction of the SF, and the un-obscured SFcan be traced either by FUV continuum or optical nebular emission.Each pixel in the FUV, H α , and 24 µ m emission maps has some uncertainties coming from observation and dataprocessing. These errors propagate to the uncertainties in the surface density maps. For example, uncertainties in theflux calibration, the stellar continuum subtraction, and the background subtraction contribute to the total error budgetof H α . We assume 10% uncertainty for each of these components and construct H α error map σ H α by adding theseterms quadratically. A flat 10% uncertainty in stellar continuum subtraction for the entire galaxy disk is probablyunrealistic because stellar emission dominates in the galaxy center. Therefore, it will have higher contribution at thehigh surface brightness regions. We take this limit on ad hoc basis. The assumed fraction in the continuum subtractionmay a lower limit since it varies along the galaxy disk contributing as much as 30% to I H α flux uncertainty (Koopmanet al. 2001). Finally, there is a considerable (calibration) uncertainty in H α to SFR conversion (Calzetti et al. 2007).We take all these factors into consideration when constructing the SFR error map ( σ H α ) from the H α image. TheFUV and mid-IR error maps ( σ FUV and σ MIR ) are constructed in a similar manner but without the contribution fromthe stellar continuum. The error in the composite SFR maps (FUV + 24 µ m or H α + 24 µ m) are constructed bycombining the appropriate terms from the respective images. C. MOLECULAR GAS SURFACE DENSITY MAP
A conversion factor (X CO ) is frequently used to determine the distribution of molecular hydrogen (H ) from theCO J = 1 − CO = 2 . × to be consistent with several other current studies(Wong & Blitz 2002; Komugi et al. 2005; Gardan et al. 2007; Bigiel et al. 2008; Leroy et al. 2008). The choice of thecalibration factor linearly scales the estimated gas densities. It is worthwhile asking how accurate is the assumption ofa single conversion factor. Although the CO-to-H appears to be approximately constant for resolved GMCs (Bolattoet al. 2008), on the hundreds of parsecs scales sampled in this study it may vary across the disk, especially in thelow density low metallicity regions (e.g., Garcia-Burillo et al. 1993). We adopt a constant X CO for simplicity andconsistency with most recent studies (Wong & Blitz 2002; Komugi et al. 2005; Gardan et al. 2007; Bigiel et al. 2008;Leroy et al. 2008). Furthermore, we expect the effects of a varying conversion factor to be most important in theouter portions of the disk, which are beyond the central region sampled by this study. The H surface densities aremultiplied by a factor of 1.40 to account for the mass contribution of helium and are expressed in units of M ⊙ pc − .To obtain the integrated (interferometer + single) CO map, first, the CARMA cube was de-convolved using theimplementation of the CLEAN algorithm in the MIRIAD task mossdi and the velocity channels (2.6 km sec − ) in theIRAM cube has been re-binned to have the same velocity resolution (10 km sec − ) of the CARMA cube. The MIRIAD8 Rahman et al.task immerge was then used to combine the CARMA and the IRAM cube in the image plane (e.g., Stanimirovi´c etal. 1999). The integrated intensity map was created by convolving each plane of the original data-cube with a 15 ′′ Gaussian to degrade its resolution, selecting regions in each velocity plane where the signal is larger than 2.5 σ , andusing these regions as masks on the original cube to compute an integrated intensity. The corresponding error mapwas computed by considering the 1 σ rms value of 22 mJy Beam − km sec − for each plane and multiplying by thesquare root of the number of planes that make up each integrated intensity point. Although the RMS in each plane isapproximately constant within the field-of-view of the mosaic, the noise in the integrated intensity map is a functionof position because of the change in line width of the CO emission. D. DATA SAMPLING
Existing studies of spatially resolved SF law in galaxies use three different data sampling methodologies. These are:azimuthally averaged radial profile (Kennicutt 1989; Martin & Kennicutt 2001; Wong & Blitz 2002; Boissier et al. 2003;Heyer et al. 2004; Schuster et al. 2007; Bigiel et al. 2008); aperture analysis encompassing the star forming regions andcentering on H α and MIR 24 µ m emission peaks (Kennicutt et al. 2007; Blanc et al. 2009); and finally, pixel-by-pixelanalysis (Bigiel et al. 2008; Leroy et al. 2008). To understand the methodological impact on the determination of thelocal SF law we have incorporated all three methods in our study since each of these methodologies has strengths andweaknesses.Pixel analysis probes the gas-SFR density relation at the smallest spatial resolution. Aperture averages, on theother hand, provide information relevant to ensemble averages of representative regions. Azimuthally averaged radialprofiles, in general, provide estimates over much larger scales. This method of sampling produces overall radial trendssuppressing local variations. Radial profiles, by construction, are less susceptible to angular resolution. Although thefirst sampling probes the smallest spatial scales in the map, it is heavily dominated by its low surface brightness regions.The latter two samplings, by construction, probe only the high surface brightness regions where our signal-to-noise isbest and DE in both the SFR and the molecular gas tracers is likely to be a minor contaminant.In our pixel analysis, we consider regions of the galaxy where both Σ SFR and Σ H are higher than 2 . σ . All imagesare resolution-matched to 6 ′′ , and re-sampled at the Nyquist rate (3 ′′ × ′′ pixels). For the aperture analysis we laydown a total of 102 circular apertures 6 ′′ (500 pc) in diameter, in regions of high surface density, mostly following thespiral arms (see Fig. 5). These apertures are approximately independent, with only a small fraction (5%) of overlap.We also consider 34 circular apertures 12 ′′ (1 kpc) in diameter with a similar overlapping fraction. It should be stressedhere that sampling the same region in all four SFR surface density maps simultaneously determines the total numberof apertures. The aperture centers for both small and large radii were chosen independently. For any Σ SFR map all theapertures combined cover approximately ≈
45% of the total area covered by all the pixels above 2 . σ limit. Because ofthe nature of the selection method the aperture samplings contain ∼
65% of the total emission when compared pixelsampling. E. STATISTICAL METHODOLOGY AND FITTING PROCEDURES
The SF law, as expressed by Eq. 1, is a power law relationship between the SFR surface density and the gas surfacedensity. The outcome of any regression analysis, with the object of finding A and N, depends on the treatment ofthe data and its measurement errors, and on the intrinsic scatter of the observables. The intrinsic scatter reflects thevariations of local physical properties of star-forming regions (for example, evolutionary stages, stellar populations,metallicity, obscuration, etc). The measurement errors, on the other hand, depend entirely upon the observationsand the subsequent data reduction. The factors that contribute to the measurement errors include flux calibration,continuum subtraction, and background subtraction.The linear regression methods can be divided into two broad classes depending on whether the intrinsic scatter orthe measurement error dominate. When seeking a linear relationship between two variables in a data set where bothof them have small measurement errors but substantial yet unknown intrinsic scatter, the ordinary least square (OLS)bisector method provides one the best solutions, (Isobe et al. 1990; Feigelson & Babu 1992).When both the dependent and the independent variables are subject to measurement errors and intrinsic scatterof comparable magnitude, the regression analysis becomes more complex. Several bivariate regression methods havebeen developed to deal with astronomical problems but each method has its own advantages and disadvantages (seeFeigelson & Babu 1992 for a detailed account). The most widely used bivariate regression analysis is based on theleast squares technique (FITEXY; Press et al.1992 and references therein). Akritas & Bershady (1996) extendedthe OLS bisector method incorporating measurement error and the intrinsic scatter. This estimator is known asthe bivariate correlated error and scatter (BCES) estimator. Kelly (2007) developed a bivariate estimator based onBayesian statistics (LINEXERR).The FITEXY and LINEXERR estimators differ in the underlying assumption of the nature of the true relationshipbetween independent and dependent variables. Both the BCES and LINEXERR estimators assume that the datapoints are scattered around the true linear relationship. The FITEXY, on the other hand, assumes that there is nointrinsic scatter and the true points lie exactly on a straight line, providing a solution for data showing a perfectcorrelation. While the BCES and LINEXERR estimators incorporate the correlated measurement error, FITEXYdoes not account for it. Correlated error arises when the dependent and the independent variables both are subjectedto the same uncertainty. For example, surface densities are defined as the ratio of the total mass to the de-projectedarea of the disk. An uncertainty in the inclination measurement leads to a correlated measurement error in the surfaceolecular Gas Star Formation Law in NGC 4254 29density of gas and SFR. The covariance term broadens the actual distribution of data points and thus provides a flatterrelationship if unaccounted for (Akritas & Bershady 1996).In this study we have used the OLS bisector, FITEXY, and LINEXERR estimators. We note here that while theOLS bisector method will guard our analysis against possible flaw in constructing measurement error maps, the othertwo methods will be required for a complete analysis of data. Our experience shows that the BCES bisector method ishighly sensitive to the measurement error resulting in unstable slope estimates compared to the three other methodsmentioned above. We do not use this estimator in our analysis.We construct surface density measurement error maps as mentioned in section §
2. The covariant term for eachpixel is calculated from the error map. The intrinsic scatter is directly provided by the LINEXERR estimator. Forthe OLS bisector and FITEXY estimators we estimate it using the best fit line. To include scatter in the FITEXYestimator we iteratively adjust the error along the Y-axis iteratively until we achieve a reduced chi-squared ∼
1. Forthis estimator, the total dispersion along the y-axis is σ = σ + σ . The adjustment is made for σ i which is a measureof the intrinsic dispersion in the gas-SFR surface density relation, while σ m is the measurement uncertainty obtainedfrom error propagation. The formal error in each parameter quoted in this paper comes from the bootstrap samplingof 1000 realizations of the data points. F. UNSHARP MASKING
We use unsharp masking to model and remove the local variations of DE from the maps. The choice of the size ofthe median filter kernel plays a vital role in selecting the amount of diffuse component in the total disk emission. Asmentioned earlier, we explore a number of filter sizes in each SFR tracer, carrying out our analysis for each case (seeTable 2). We should stress here that our intention is to use the simplest model for a reliable estimate of the diffusefraction in the disk of NGC 4254 which can be used with a reasonable confidence to explore the main goal of thisstudy.For a filter size of 75 ′′ the fraction of the total emission contained in the smooth H α map is f DIG ∼ α is likely overestimated for kernel sizes below 75 ′′ (corresponding to physical scales ∼ ′′ or a physical length of ∼
13 kpc. With this scale we obtain f DIG ∼
15% which is approximately the f DIG observed in the Galaxy (Reynolds 1991). Similar kernel sizes are alsoexplored in the FUV and 24 µ m maps. As expected, the diffuse fractions in the FUV, H α , and 24 µ m emission mapsvary substantially from one another. Within our range of kernel sizes, f FUV ∼ − f MIR ∼ − f FUV , f DIG , and f MIR respectively. In each panel, we show our results for six differentsmoothing filters.It is evident from the figure that the f DE profiles increase radially outward. In spite of their distinct origins, all threediffuse fractions show this remarkably similar trend. Beyond ∼
10 kpc these profiles flatten out. For smaller smoothingscales, f MIR varies as much as a factor of four whereas f DIG varies up to a factor three. The f FUV shows a factor oftwo radial variation. For longer smoothing scales the variation is about a factor of two for all f DE . For f DIG , a similartrend was observed in the Sculptor group spiral NGC 7793 (Ferguson et al. 1996). Panel D shows the average fractionof diffuse emission, ¯f DE , as a function of smoothing scale, summarizing the results of the first three panels. The filledcircles with roman numerals on the color coded line represent the radial average of the corresponding lines in panelsA, B, and C. It is interesting to note that NGC 4254 is not unusual compared to the Local Group galaxies in termsof its multi-wavelength diffuse components. As expected, the FUV map has the highest diffuse fraction compared tothe MIR 24 µ m and H α maps. The f FUV has slowest variation ( ∼ − f DIG ,on the other hand, falls sharply with larger smoothing scales. The amount of diffuse fraction in the 24 µ m map isintermediate between those in the H α and the FUV images.At every filter scale, we combine the DE subtracted FUV, H α , and 24 µ m emission maps to construct the desiredΣ SFR maps. Finally, a small global value is subtracted from the images. As long as most of the pixels sample thebackground, sky background subtraction can be considered an extreme case of unsharp masking where the medianfilter size is the same as the entire image. In this case, DE estimated from all three maps is negligibly small and thiswhat we call as the zero diffuse fraction ( f DE = 0) in the remaining sections. For each Σ SFR map as discussed inAppendix B, we use the Σ H map derived from the combination of single-dish and interferometer data.To remove the diffuse extended component from the CO distribution we employ the same techniques we have appliedto the SFR tracers. To simplify the range of parameter space to explore we use filtering kernels of the same size inboth the Σ SFR and the Σ H maps. Although this is not necessarily correct, we take it as an illustration of the effectsof removing a diffuse contribution in both axes. F.1.
Image PSF and Unresolved H II Regions
To determine the fraction of DE in the disk of NGC 4254 it is extremely important to verify that the DE that wedetect is a distinct source of emission in various SFR tracer maps, and that, it is not simply the emission spreads outfrom the star-forming regions. If, for example, the extended tail of the PSF contains a large fraction of the source flux,light from star-forming H II regions might scatter over a considerable area. This will bias the estimate of DE.0 Rahman et al. Figure 14.
Azimuthally averaged radial profiles of diffuse fractions. Panels A, B, and C show respectively, the f FUV , f DIG , and f MIR .In each panel, lines with Roman numerals correspond to the DE maps constructed from six different filters as given in Table 2. Panel Dshows the average diffuse fraction (¯f DE ) as a function of filter scale. The filled circles with roman numerals in panel D represent the radialaverages of the corresponding lines in panels A, B, and C. The figure shows clearly that f DE increases radially outward with a trend that isindependent of wavelength or filter scale. The profiles flatten out at the edge of the disk (beyond ∼
10 kpc). For smaller smoothing scalesboth f DIG and f MIR vary as much as a factor of three along the disk. For longer smoothing scales the variation is about a factor of two forall f DE . The FUV map has the highest diffuse fraction compared to the other two maps (see Table 2). Measurements for only six kernelsizes are shown for the ease of demonstration. The horizontal axes in panels A, B, and C span the optical radius, R ∼ To understand the nature of the bias we need to have clear picture about the shapes of the PSFs in the SFR tracerimages. The PSF of these images show distinctive characteristics. For example, the PSF of I
MIR image has a complexpattern showing first and second Airy rings with radially extending artifacts. The linear scale of the second Airy ringfrom the center of the PSF is ∼ ′′ . While approximately 85% of the total source flux is contained within the centralpeak of the PSF which has a Gaussian shape with FWHM of 6 ′′ , more than 99% flux is contained with an aperture ofdiameter 40 ′′ (see Table 1 of Engelbracht et al. 2007). The PSF of GALEX
NUV channel varies along the field-of-viewfrom a symmetric 2D Gaussian profile to extended structure further from the center. The FWHM of the PSF containsmore than 80% of the source flux and 95% of the total flux is contained within an aperture size of 40 ′′ in diameter(see Fig. 12 of Morrisset et al. 2007). The FWHM of I H α map is fairly consistent with a 2D symmetric Gaussian. Forthis map, we find two isolated field stars far from the source and construct their light profiles. We find that 100% ofthe stellar light is confined within a region of radius 15 ′′ .The smallest filter scale used in our study is 75 ′′ . By construction, the median filter will remove any sub-structureof the map whose linear dimension is equal to or smaller than half of the filter scale, i.e. 38 ′′ in this case. For largerfilter scales sub-structures of even more larger dimension will be removed. This implies that for MIR 24 µ m map, at75 ′′ filter scale, the estimate of DE would have ∼
15% of the total emission that would come from the star-formingregions and not from the truly diffuse component un-associated with the region. For