Molecular Gas and Star Formation in Nearby Disk Galaxies

Molecular Gas and Star Formation in Nearby Disk Galaxies

Adam K. Leroy11affiliation: National Radio Astronomy Observtory, 520 Edgemont Road, Charlottesville, VA 22903, USA , Fabian Walter22affiliation: Max Planck Institute für Astronomie, Königstuhl 17, 69117, Heidelberg, Germany , Karin Sandstrom22affiliation: Max Planck Institute für Astronomie, Königstuhl 17, 69117, Heidelberg, Germany , Andreas Schruba33affiliation: California Institute for Technology, 1200 E California Blvd, Pasadena, CA 91125 , Juan-Carlos Munoz-Mateos11affiliation: National Radio Astronomy Observtory, 520 Edgemont Road, Charlottesville, VA 22903, USA , Frank Bigiel44affiliation: Theoretische Astrophysik, Albert-Ueberle-Str. 2, 69120, Heidelberg, Germany , Alberto Bolatto 66affiliation: Department of Astronomy, University of Maryland, College Park, MD, USA , Elias Brinks77affiliation: Centre for Astrophysics Research, University of Hertfordshire, Hatfield AL10 9AB, United Kingdom , W.J.G. de Blok88affiliation: ASTRON, Netherlands Foundation for Radio Astronomy, Postbus 2, 7990 AA Dwingeloo, The Netherlands 99affiliation: Astrophysics, Cosmology and Gravity Centre, Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa , Sharon Meidt22affiliation: Max Planck Institute für Astronomie, Königstuhl 17, 69117, Heidelberg, Germany , Hans-Walter Rix22affiliation: Max Planck Institute für Astronomie, Königstuhl 17, 69117, Heidelberg, Germany , Erik Rosolowsky1010affiliation: University of British Columbia, Okanagan Campus, Kelowna, BC Canada , Eva Schinnerer22affiliation: Max Planck Institute für Astronomie, Königstuhl 17, 69117, Heidelberg, Germany , Karl-Friedrich Schuster1111affiliation: IRAM, 300 rue de la Piscine, 38406 St. Martin d’Hères, France , Antonio Usero55affiliation: Observatorio Astronómico Nacional, C/ Alfonso XII, 3, 28014, Madrid, Spain ,
Abstract

We compare molecular gas traced by CO(2-1) maps from the HERACLES survey, with tracers of the recent star formation rate (SFR) across 30 nearby disk galaxies. We demonstrate a first-order linear correspondence between and but also find important second-order systematic variations in the apparent molecular gas depletion time, . At the 1 kpc common resolution of HERACLES, CO emission correlates closely with many tracers of the recent SFR. Weighting each line of sight equally, using a fixed equivalent to the Milky Way value, our data yield a molecular gas depletion time,  Gyr with 0.3 dex scatter, in very good agreement with recent literature data. We apply a forward-modeling approach to constrain the power-law index, , that relates the SFR surface density and the molecular gas surface density, . We find for our full data set with some scatter from galaxy to galaxy. This also agrees with recent work, but we caution that a power law treatment oversimplifies the topic given that we observe correlations between and other local and global quantities. The strongest of these are a decreased in low-mass, low-metallicity galaxies and a correlation of the kpc-scale with dust-to-gas ratio, D/G. These correlations can be explained by a CO-to-H conversion factor () that depends on dust shielding, and thus D/G, in the theoretically expected way. This is not a unique interpretation, but external evidence of conversion factor variations makes this the most conservative explanation of the strongest observed trends. After applying a D/G-dependent , some weak correlations between and local conditions persist. In particular, we observe lower and enhanced CO excitation associated with nuclear gas concentrations in a subset of our targets. These appear to reflect real enhancements in the rate of star formation per unit gas and although the distribution of does not appear bimodal in galaxy centers, does appear multivalued at fixed , supporting the the idea of “disk” and “starburst” modes driven by other environmental parameters.

Subject headings:
galaxies: evolution — galaxies: ISM — radio lines: galaxies — stars: formation
slugcomment: Accepted to the Astronomical Journal

1. Introduction

The relationship between gas and star formation in galaxies plays a key role in many areas of astrophysics. Its (non)evolution over cosmic time informs our understanding of galaxy evolution at high redshift (Daddi et al., 2010; Tacconi et al., 2010; Genzel et al., 2010). The small-scale efficiency of star formation is a key input to galaxy simulations and scaling relations measured for whole galaxies provide important benchmarks for the output of these simulations. Measurements of gas and star formation at large scales give context for studies focusing on parts of the Milky Way (e.g., Lada et al., 2010; Heiderman et al., 2010) and the nearest galaxies (e.g., Schruba et al., 2010; Chen et al., 2010). Ultimately, a quantitative understanding of the gas-stars cycle is needed to understand galaxy evolution, with implications for the galaxy luminosity function, the galaxy color-magnitude diagram, the structure of stellar disks, and chemical enrichment among other key topics.

Recent multiwavelength surveys make it possible to estimate the surface densities of gas and recent star formation in dozens of nearby galaxies. This has lead to several studies of the relationship between gas and stars. Many of these focus on a single galaxy (e.g., Heyer et al., 2004; Kennicutt et al., 2007; Blanc et al., 2009; Verley et al., 2010; Rahman et al., 2011) or a small sample (e.g., Wilson et al., 2009; Warren et al., 2010). Restricted by the availability of complete molecular gas maps, studies of large sets of galaxies (e.g., Young et al., 1996; Kennicutt, 1998b; Rownd & Young, 1999; Murgia et al., 2002; Leroy et al., 2005; Saintonge et al., 2011) mostly use integrated measurements or a few low-resolution pointings per galaxy.

From 2007-2010, the HERA CO-Line Extragalactic Survey (HERACLES, first maps in Leroy et al., 2009) used the IRAM 30-m telescope111IRAM is supported by CNRS/INSU (France), the MPG (Germany) and the IGN (Spain). to construct maps of CO emission from 48 nearby galaxies. The common spatial resolution of the survey is  kpc, sufficient to place many resolution elements across a typical disk galaxy. Because the targets overlap surveys by Spitzer (mostly SINGS and LVL, Kennicutt et al., 2003a; Dale et al., 2009) and GALEX (mostly the NGS, Gil de Paz et al., 2007), a wide variety of multiwavelength data are available for most targets. In this paper, we take advantage of these data to compare tracers of molecular gas and recent star formation at  kpc resolution across a large sample of 30 galaxies.

This paper builds on work by Leroy et al. (2008, hereafter L08) and Bigiel et al. (2008, hereafter B08). They combined the first HERACLES maps with data from The H i Nearby Galaxies Survey (THINGS Walter et al., 2008), SINGS, and the GALEX NGS data to compare H i, CO, and tracers of recent star formation in a sample of nearby galaxies. In the disks of large spiral galaxies, they found little or no dependence of the star formation rate per unit molecular gas on environment. The fraction of interstellar gas in the molecular phase, on the other hand, varies strongly within and among galaxies, exhibiting correlations with interstellar pressure, stellar surface density, and total gas surface density among other quantities (Wong & Blitz, 2002; Blitz & Rosolowsky, 2006, L08). They advocated a scenario for star formation in disk galaxies in which star formation in isolated giant molecular clouds is a fairly universal process while the formation of these clouds out of the atomic gas reservoir depends sensitively on environment (see also, Wong, 2009; Ostriker et al., 2010).

The full HERACLES sample spans a wider range of masses, morphologies, metallicities, and star formation rates (SFRs) than the spirals studied in L08 and B08. Schruba et al. (2011) and Bigiel et al. (2011) used these to extend the findings of L08 and B08. Using stacking techniques, Schruba et al. (2011) demonstrated that correlations between star formation tracers and CO emission extend into the regime where atomic gas dominates the ISM, . This provides the strongest evidence yet that star formation in disk galaxies can be separated into star formation from molecular gas and the balance between atomic and molecular gas, a hypothesis that has a long history (e.g., Young & Scoville, 1991, and references therein). Bigiel et al. (2011) demonstrated that a fixed ratio of CO to recent star formation rate remains a reasonable description of the ensemble of 30 galaxies.

In this paper we expand on L08, B08, Bigiel et al. (2011), and Schruba et al. (2011) and examine the general relationship between molecular gas and SFR in nearby disk galaxies. We divide the analysis into two main parts. In Section 3, we consider the scaling relation linking to , the molecular analog to the “Kennicutt-Schmidt law” or “star formation law.” We show the distribution of data in - parameter space using different weightings (Section 3.1) and examine how this distribution changes with different approaches to physical parameter — varying the choice of SFR tracer, the CO transition studied, the processing of SFR maps, or the adopted conversion factor (Section 3.2). Using an expanded version of the Monte Carlo modeling approach of Blanc et al. (2009), we carry out power law fits to our data, avoiding some of the systematic biases present in previous work (Section 3.3). Finally, we compare our results to a wide collection of literature data, demonstrating an emerging consensus with regard to the region of parameter space occupied by - data, if not the interpretation (Section 3.4). We conclude that in the disks of normal, massive star-forming galaxies, to first order the relationship between and can be described by a single depletion time with a factor of two scatter. This expands and reinforces the results of L08, B08, and Schruba et al. (2011).

In Section 4, we show important second-order deviations from this simple picture, which can easily be missed by comparing only and . We find systematic variations in the molecular gas depletion time, as a function of global galaxy properties (Section 4.1) and local conditions (Section 4.2). This analysis includes the first resolved comparison of to the local dust-to-gas ratio, which is expected to play a key role in setting the CO-to-H conversion factor, and we discuss the possibility that drives some of the observed variations. We explicitly consider the central regions of our targets (Section 4.4) and show strong evidence for lower , i.e., more efficient star formation, in galaxy centers compared to galaxy disks - a phenomenon that we discuss in context of recent proposals for “disk” and “starburst” modes of star formation (Daddi et al., 2010; Genzel et al., 2010). In our best-resolved targets we examine how the scatter in depends on spatial scale (Section 4.3). The relationship appears much shallower than one would expect for uncorrelated averaging in a disk. This suggests either a high degree of large-scale synchronization in the star formation process or, more likely, widespread systematic, but subtle, variations in due to still-undiagnosed drivers.

Thus our conclusions may be abstracted to: molecular gas and star formation exhibit a first order one-to-one scaling but we observe important second order variations about this scaling. These include likely conversion factor effects, efficient nuclear starbursts, and weak systematic variations in that emerge considering scale-dependent scatter or global galaxy properties. The remainder of this section presents a brief background, Section 2 describes our data and physical parameter estimation, Sections 3 and 4 motivate these conclusions, and Section 5 synthesizes these results and identifies several key future directions.

1.1. Background

Following Schmidt (1959, 1963), astronomers have studied the relationship between gas and the SFR for more than 50 years. Most recent work follows Kennicutt (1989, 1998b) and compares the surface densities of SFR, , and neutral (H i+H) gas mass, . Recent work focuses heavily on the power law relationship between these surface densities (the “Schmidt-Kennicutt law” or the “star formation law”):

(1)

An alternative approach treats the ratio of gas and SFR as the quantity of interest (Young et al., 1986, 1996, L08). This ratio can be phrased as a gas depletion time () or its inverse, a star formation efficiency (). These share the same physical meaning, which is the SFR per unit gas. Both convolve a timescale with a true efficiency, for example the lifetime of a molecular cloud with the fraction of gas converted to stars over this time. We focus exclusively on molecular gas in this paper and phrase this ratio as the molecular gas depletion time,

(2)

which is the time for star formation to consume the current molecular gas supply.

The state of the field is roughly the following. Kennicutt (1998b) demonstrated a tight, non-linear () scaling between galaxy-averaged and spanning from normal disk galaxies to starbursts. Including H i improved the agreement between disks and starbursts, but most of the dynamic range and the nonlinear slope was driven by the contrast between the disks and merger-induced starbursts, especially the local ultraluminous infrared galaxy (ULIRG) population. The adoption of a single conversion factor for all systems also had significant impact; adopting the “ULIRG” conversion factor suggested by Downes & Solomon (1998) for the Kennicutt (1998b) starburst data drives the implied slope to . Subsequent studies resolved galaxy disks — often as radial profiles — and usually revealed distinct relationships between , , and with shallower indices for H than H i (Wong & Blitz, 2002; Heyer et al., 2004; Kennicutt et al., 2007; Schruba et al., 2011, B08, L08). This suggests that the immediate link is between SFR and H. A more aggressive conclusion, motivated by the steep, relatively weak relation between and is that star formation in galaxy disks may be broken into two parts: (1) the formation of stars in molecular clouds and (2) the balance between H and H i (Wong & Blitz, 2002; Blitz & Rosolowsky, 2006, L08, B08).

The fraction of dense molecular gas also appears to be a key parameter. Gao & Solomon (2004) found a roughly fixed ratio of SFR to HCN emission, a dense gas tracer, extending from spiral galaxies to starbursts. Over the same range, the ratio of SFR to total H, traced by CO emission, varies significantly (though the two relate roughly linearly in their normal galaxy sample). Galactic studies also highlight the impact of density on the SFR on cloud scales (Wu et al., 2005; Heiderman et al., 2010; Lada et al., 2010). It remains unclear how the dense gas fraction varies inside galaxy disks, but merger-induced starbursts do show high HCN-to-CO ratios (Gao & Solomon, 2004; García-Burillo et al., 2012).

SFR tracers and CO can both be observed at high redshift. Genzel et al. (2010) demonstrated broad consistency between local H-SFR relations and those at . One key difference at high- is the existence of disk galaxies with only slightly lower than those in local disk galaxies but H surface density, , as high as that found in starbursts in the local universe (Daddi et al., 2010; Tacconi et al., 2010). Merger-driven starbursts with similar can have much lower , suggesting the relevance of another parameter to set . Density and the dynamical timescale are both good candidates (Daddi et al., 2010; Genzel et al., 2010).

Meanwhile, investigations of the Milky Way and the nearest galaxies have attempted to connect observed scaling relations to the properties of individual star-forming regions. These are able to recover the galaxy-scale relations at large scale but find enormous scatter in the ratios of SFR tracers to molecular gas on small scales (Schruba et al., 2010; Chen et al., 2010; Onodera et al., 2010). Detailed studies of Milky Way and LMC clouds suggest the time-evolution of individual star-forming regions as a likely source of this scatter (Murray, 2010; Kawamura et al., 2009), with the volume density of individual clouds a key parameter (Heiderman et al., 2010; Lada et al., 2010).

For additional background we refer the reader to the recent review by Kennicutt & Evans (2012).

2. Data

Galaxy res. Multi-
[Mpc] [kpc] [] [] [] [kpc] scale
(1) (2) (3) (4) (5) (6) (7) (8)
NGC 0337 19.3KKfootnotemark: 1.24aaToo distant to convolve to 1 kpc resolution. Included in analysis at native resolution. 51 90 1.5 10.6
NGC 0628 7.2KKfootnotemark: 0.46 7 20 4.9 10.4
NGC 0925 9.1KKfootnotemark: 0.59 66 287 5.4 14.3
NGC 2403 3.2WWfootnotemark: 0.21 63 124 7.9 7.4
NGC 2841 14.1KKfootnotemark: 0.91 74 153 3.5 14.2
NGC 2903 8.9WWfootnotemark: 0.57 65 204 5.9 15.2
NGC 2976 3.6KKfootnotemark: 0.23 65 335 3.6 3.8
NGC 3049 19.2KKfootnotemark: 1.24aaToo distant to convolve to 1 kpc resolution. Included in analysis at native resolution. 58 28 1.0 2.7
NGC 3184 11.8KKfootnotemark: 0.76 16 179 3.7
NGC 3198 14.1KKfootnotemark: 0.91 72 215 3.2 13.0
NGC 3351 9.3KKfootnotemark: 0.60 41 192 3.6 10.6
NGC 3521 11.2KKfootnotemark: 0.72 73 340 4.2 12.9
NGC 3627 9.4KKfootnotemark: 0.61 62 173 5.1 13.8
NGC 3938 17.9KKfootnotemark: 1.15aaToo distant to convolve to 1 kpc resolution. Included in analysis at native resolution. 14 15 1.8 6.3
NGC 4214 2.9WWfootnotemark: 0.19 44 65 3.4 2.9
NGC 4254 14.4KKfootnotemark: 0.93 32 55 2.5 14.6
NGC 4321 14.3KKfootnotemark: 0.92 30 153 3.0 12.5
NGC 4536 14.5KKfootnotemark: 0.94 59 299 3.5 14.9
NGC 4559 7.0KKfootnotemark: 0.45 65 328 5.2 10.7
NGC 4569 9.86KKfootnotemark: 0.64 66 23 4.6 26.5
NGC 4579 16.4KKfootnotemark: 1.06aaToo distant to convolve to 1 kpc resolution. Included in analysis at native resolution. 39 100 2.5 15.0
NGC 4625 9.3KKfootnotemark: 0.60 47 330 0.7 1.9
NGC 4725 11.9KKfootnotemark: 0.77 54 36 4.9 13.2
NGC 4736 4.7KKfootnotemark: 0.30 41 296 3.9 5.3
NGC 5055 7.9KKfootnotemark: 0.51 59 102 5.9 17.3
NGC 5194 7.9WWfootnotemark: 0.52 20 172 3.9 9.0
NGC 5457 6.7KKfootnotemark: 0.43 18 39 12.0 25.8
NGC 5713 21.4KKfootnotemark: 1.38aaToo distant to convolve to 1 kpc resolution. Included in analysis at native resolution. 48 11 1.2 9.5
NGC 6946 6.8KKfootnotemark: 0.44 33 243 5.7 9.8
NGC 7331 14.5KKfootnotemark: 0.94 76 168 4.6 19.5

Note. – Sample used in this paper. Columns give (1) galaxy name; (2) adopted distance in Mpc; (3) FWHM spatial resolution of HERACLES data at that distance, in kiloparsecs; (4) adopted inclination and (5) position angle in degrees; adopted radius of the the B-band 25 magnitude isophote, used to normalize the radius in (6) arcminutes and (7) kiloparsecs. Most analysis in this paper considers data inside . Column (8) indicates if the galaxy is close and large enough for the multiscale analysis in Section 4.3. K,WK,Wfootnotetext: Distance adopted from K: Kennicutt et al. (2011) or W: Walter et al. (2008).

Table 1Sample

2.1. Data Sets

We use HERACLES CO(2-1) maps to infer the distribution of H and GALEX far-ultraviolet (FUV), Spitzer infrared (IR), and literature H data to trace recent star formation. We supplement these with H i data used to mask the CO, derive kinematics, and measure the dust- to-gas ratio and with near-IR data used to estimate the stellar surface density, .

HERACLES CO: The HERA CO Line Extragalactic Survey (HERACLES) used the Heterodyne Receiver Array (HERA, Schuster et al., 2004) on the IRAM 30m telescope to map CO(2-1) emission from nearby galaxies, of which we use 30 in this paper (see Section 2.3). HERACLES combines an IRAM Large Program and several single-semester projects that spanned from 2007 to 2010. Leroy et al. (2009) presented the first maps (see also Schuster et al., 2007). The additional data here were observed and reduced in a similar manner. The largest change is a revised estimate of the main beam efficiency, lowering observed intensities by . This propagates to a revised CO (2-1)/(1-0) line ratio estimate, so our estimates of are largely unaffected compared to B08 and L08. The HERACLES cubes cover out to radii of with angular resolution and typical sensitivity  mK per 5 km s channel.

We integrate each cube along the velocity axis to produce maps of the integrated intensity along each line of sight. We wish to avoid including unnecessary noise in this integral and so restrict the velocity range over which we integrate to be as small as possible while still containing the CO line, i.e., we “mask” the cubes. To be included in the mask a pixel must meet one of two conditions: 1) lie within  km s of the local mean H i velocity (derived from THINGS, Walter et al., 2008, supplemented by new and archival H i) or 2) lie in part of the spectrum near either two consecutive channels with SNR above or three consecutive channels with SNR above . Condition (2) corresponds to traditional radio masking (e.g., Helfer et al., 2003; Walter et al., 2008). Condition (1) is less conventional, but important to our analysis. Integrating over the H i line, which is detected throughout our targets, guarantees that we have an integrated intensity measurement along each line of sight, even lines of sight that lack bright CO emission (see Schruba et al., 2011, for detailed discussion of this approach). This avoids a traditional weakness of masking, that nondetections are difficult to deal with quantitatively. We calculate maps of the statistical uncertainty in the integrated CO intensity from the combination of the mask and estimates of the noise derived from signal-free regions. The result is an integrated intensity and associated uncertainty for each line of sight in the HERACLES mask.

SINGS and LVL IR: We use maps of IR emission from 3.6–160m from the Spitzer Infrared Nearby Galaxies Survey (SINGS, Kennicutt et al., 2003a) and the Local Volume Legacy survey (LVL, Dale et al., 2009). We describe the processing of these maps in Leroy et al. (2012, hereafter L12).

SINGS, LVL, and Literature H: Both SINGS and LVL published continuum-subtracted H images for most of our sample. We supplement these with literature maps, particularly from the GoldMine and Palomar-Las Campanas surveys. L12 describe our approach to these maps (masking, N ii correction, flux scaling, background subtraction) and list the source of the H data for each galaxy.

GALEX UV: For 24 galaxies, we use NUV and FUV maps from the Nearby Galaxy Survey (NGS, Gil de Paz et al., 2007). For one galaxy, we use a map from the Medium Imaging Survey (MIS) and we take maps for five targets from the All-sky Imaging Survey (AIS). L12 describe our processing.

BIMA+12-m and NRO 45-m CO (1-0) Maps: A subset of our targets have also been observed by the BIMA SONG (Helfer et al., 2003) or the Nobeyama CO Atlas of Nearby Spiral Galaxies (Kuno et al., 2007). Where these data are available, we apply our HERACLES masks to these maps and measure CO (1-0) intensity. We only use BIMA SONG maps that include short-spacing data from the Kitt Peak 12-m.

THINGS and Supplemental Hi: We assemble H i maps for all targets, which we use to mask the CO, estimate the dust-to-gas ratio, explore 24m cirrus corrections, and derive approximate rotation curves. These come from THINGS (Walter et al., 2008) and a collection of new and archival VLA data (programs AL731 and AL735). These supplemental H i are C+D configuration maps with resolutions . We reduced and imaged these in a standard way using the CASA package.

2.2. Physical Parameter Estimates

Following standard practice in this field, we estimate physical parameters from observables. Despite the intrinsic uncertainty involved in this process, these estimates play a fundamental role in enhancing our understanding of the physics of galaxy and star formation, as demonstrated from the earliest works in this subfield (Young & Knezek, 1989; Kennicutt, 1989). We adopt an approach largely oriented to physical quantities, but discuss the impact of our assumptions throughout.

CO Intensity to H: We convert CO (2-1) intensity to H mass via

(3)

where is the CO(2-1)-to-CO(1-0) line ratio and is the CO(1-0)-to-H conversion factor. By default, we adopt a Galactic conversion factor,  M pc  equivalent to (Strong & Mattox, 1996; Dame et al., 2001) and a line ratio of . This line ratio is slightly lower than the derived by Leroy et al. (2009), reflecting the revised efficiency used in the reduction. The appendix motivates this value using integrated flux ratios and follow-up spectroscopy of HERACLES targets. Equation 3 and all “” in this paper include a factor of 1.36 to account for helium. Because we consider only molecular gas, any results that we derive using a fixed can be straightforwardly restated in terms of CO intensity.

We adopt this “Galactic” to facilitate clean comparison to previous work, but improved estimates exist for HERACLES. Sandstrom et al. (2012) solved directly for the CO-to-H conversion factor across the HERACLES sample using dust as an independent tracer of the gas mass. They find a somewhat lower average . We quote this as a CO (1-0) conversion factor, though Sandstrom et al. (2012) directly solve for the CO (2-1) conversion factor. They find a CO(2-1) conversion factor of  M pc , compared to our “Galactic” CO(2-1) conversion factor  M pc . Sandstrom et al. (2012) find  dex point-to-point scatter, of which may be intrinsic with the remainder solution uncertainties. Because Sandstrom et al. (2012) solve directly for a CO (2-1) conversion factor using the same HERACLES data employed in this paper, these values should be borne in mind when reading our results. Our results remain pinned to a Galactic CO(1-0) conversion factor of  M pc  that may be too high, on average. As a result, a systematic bias of in appears plausible with factor of two variations in the conversion factor point-to-point.

In addition to a fixed conversion , we consider the effects of variations in due to decreased dust shielding at low metallicity and variations in the linewidth, optical depth, and temperature of CO in galaxy centers. Our “variable” builds on the work of Sandstrom et al. (2012), who compare H i, CO(2-1), and in 22 HERACLES galaxies and Wolfire et al. (2010), who consider the effects of dust shielding on the “CO-dark” layer of molecular clouds, where most H is H. The prescription combines three terms

(4)

Here  M pc  is our fiducial CO (2-1) conversion factor in the disk of a galaxy at solar metallicity (Equation 3). The term represents a correction to the H mass to reflect the H in a CO-dark layer not directly traced by CO emission. We calculate this factor following Wolfire et al. (2010), assuming that all GMCs share a fixed surface density, and adopting a linear scaling between the dust-to-gas ratio and metallicity (see also Glover & Mac Low, 2011). In this case

(5)

Here is normalized to our adopted “Galactic” value of 0.01, with the normalization constructed to yield for .  M pc. The appendix presents this calculation in detail.

We consider two cases:  M pc (“”) and  M pc (“”). “” reflects a typical surface density that is often assumed and observed for extragalactic GMCs (e.g., Bolatto et al., 2008; Narayanan et al., 2012). Over the range of that we consider, this prescription reasonably resembles the shallow power law dependences of on metallicity calculated from simulations by Feldmann et al. (2012) and Narayanan et al. (2012), who both suggest . “ M pc” yields a steeper dependence of on metallicity over our range of interest but we will see in Sections 4.1 and 4.2 that it offers a simple way to account for most of the dependence of on dust-to-gas ratios in our observations. Low surface density GMCs or significant contribution of “translucent” ( mag) gas to the overall CO emission are supported by observations of the Milky Way Heyer et al. (2009) and Liszt et al. (2010), LMC observations by Hughes et al. (2010); Wong et al. (2011), and M31 (Schruba et al. in prep.) but may not be appropriate for more actively star-forming systems (Hughes et al., submitted). We return to this issue in Section 4.

We calculate using derived for fixed . Given observations of , , , and a prescription for , one can simultaneously solve for and . The solution is often multivalued and unstable, though not intractable. However, after experimentation and comparison with the self-consistent Herschel-based results of Sandstrom et al. (2012), we found that the process does not clearly improve our estimates. In the interests of clarity and simplicity, we work with only calculated using fixed throughout the paper. This simplification biases our estimate high by (“”) and (“”). With improved estimates, we expect that the self-consistent treatment will become necessary.

The third term, , accounts for depressed values of in the centers of galaxies. Sandstrom et al. (2012) find such depressions in the centers of many systems (see also Israel, 2009a, b). These likely reflect the same line-broadening and temperature effects that drive the commonly invoked “ULIRG conversion factor” (Downes & Solomon, 1998), though the depression observed by Sandstrom et al. (2012) have lower magnitude than the factor of 5 depression found by Downes & Solomon (1998). Sandstrom et al. (2012) could not identify a unique observational driver for these depressions, though they correlate well with stellar surface density. Instead, they appear to be present with varying magnitudes in the centers of most systems with bright central CO emission. Following their recommendation, we apply this correction where in systems that have such central CO concentrations. Whenever available, we adopt directly from Sandstrom et al. (2012), taking the factor by which the central falls below the mean for the disk of that galaxy. For systems with central CO concentrations but not in the sample of Sandstrom et al. (2012), we apply a factor of two depression, again following their recommendations. The appendix presents additional details.

SFR from H, UV, and 24m Emission: L12 combined UV, H, and IR emission to estimate the recent star formation rate surface density, , at 1 kpc resolution (the limiting common physical resolution of the HERACLES survey) for our sample. We adopt their estimates and refer the reader to that work for detailed discussion. Briefly, our baseline estimate of combines H and infrared emission at 24m via

(7)

where and refer to the line-integrated H intensity and intensity at m.

The H emission captures direct emission from H ii regions powered by massive young stars while the 24m emission accounts for recent star formation obscured by dust. Before estimating , we correct our 24m maps for the effects of heating of dust by a weak, pervasive radiation field (i.e., a “cirrus”) with magnitude derived from modeling the infrared spectral energy distribution. The cirrus removed corresponds to the expected emission from the local dust mass illuminated by a quiescent radiation field, typically times the Solar neighborhood interstellar radiation field (see L12 for details). We derive the appropriate weighting for the combination of H and 24m emission based on comparing our processed H and 24m maps to literature estimates of H extinction. The resulting linear combination resembles that of Kennicutt et al. (2007) but places slightly more weight on the 24m term. For comparison, we also estimate from combining FUV and 24m emission and taking H alone while assuming a typical 1 magnitude of extinction.

L12 estimate a substantial uncertainty in the absolute calibration of “hybrid” UV+IR or H+IR tracers, with magnitude . In addition to this overall uncertainty in the calibration, they derive a point-to-point uncertainty in of  dex based on intercomparison of different estimates.

Dust Properties: In order to measure dust properties, we convolve the Spitzer 24, 70, and 160m data and the CO and H i maps to the resolution of the Spitzer 160m data. At this resolution, we build radial profiles of each band and then fit the dust models of Draine & Li (2007) to these profiles. These fits, presented in L12, provide us with radial estimates of the dust-to-gas ratio, , and are used to help account for “cirrus” contamination when estimating . Note that the resolution of the 160m data used to measure these dust properties is significantly coarser than the 1 kpc resolution used for the rest of our data. Where possible, we have compared our Spitzer-based dust masses to masses estimated using the improved SED coverage offered by Herschel (e.g., Aniano et al., 2012); above  M pc the median offset between the Herschel and Spitzer based dust masses is only ; however the dust masses derived for individual rings using only Spitzer do scatter by  dex (a factor of two) compared to Herschel-based dust masses and show weak systematic trends with the sense that Spitzer underestimates the mass of cooler ( K) dust in the outskirts of galaxies (both consistent with the analysis of Draine et al., 2007). We expect that once Herschel images become available, they will significantly improve the accuracy of dust-based portion of this analysis.

Stellar Mass: To estimate the stellar mass for whole galaxies, we draw 3.6m fluxes from Dale et al. (2007, 2009), convert to a luminosity using our adopted distance, and apply a fixed m mass-to-light ratio. Based on comparison to Zibetti et al. (2009), we use

(8)

which is lower than L08. This value is uncertain by .

We estimate the stellar surface density, , for each kpc-sized element from the contaminant-corrected 3.6m maps of Meidt et al. (2012). Starting from a reprocessing of the SINGS data (as part of the S4G survey Sheth et al., 2010), they used independent component analysis to remove contamination by young stars and hot dust from the overall maps. These contaminants make a minor contribution to the overall 3.6m flux but may be important locally. We convert the contaminant-corrected 3.6m maps to estimates using Equation 8.

Rotation Velocities: Following L08 and Boissier et al. (2003) we work with a simple two-parameter fit to the rotation curve of each galaxy

(9)

with and free parameters. We derive these from fits to the rotation curves of de Blok et al. (2008) wherever they are available. Where these are not available, we carry out our own tilted ring fits to the combined H i and CO first moment maps. We use these fits to calculate the orbital time for each line of sight.

2.3. Sample and Galaxy Properties

We present measurements for galaxies meeting the following criteria: 1) a HERACLES CO map containing a clear CO detection (S/N over a significant area and multiple channels), 2) Spitzer data at 24m, and 3) inclination . The first condition excludes low mass galaxies without CO detections (these are discussed in Schruba et al., 2012). The second removes a few targets with saturated or incomplete Spitzer coverage. The third removes a handful of edge-on galaxies. We are left with the disk galaxies listed in Table 1.

For each target, Table 1 gives the distance, physical resolution of the HERACLES maps at that distance, inclination, position angle, and optical radius. The Table notes note the subset of galaxies that that are close and large enough for us to carry out the multi-resolution analysis in Section 4.3. We adopt distances from Kennicutt et al. (2011) where possible and from Walter et al. (2008) elsewhere. We take orientations from Walter et al. (2008) and from LEDA (Prugniel & Heraudeau, 1998) and NED elsewhere.

Table 2 reports integrated and disk-average properties for our sample. We report our integrated stellar mass estimate, galaxy morphology, metallicity and dust-to-gas ratio at , average gas mass and star formation rate surface density inside , our parameterized rotation curve fit, and the orbital time at . We take metallicities from Moustakas et al. (2010), averaging their PT05 and KK04 strong-line calibrations. They argue that these two calibrations bracket the true metallicity and that the relative ordering of metallicities is robust (see also Kewley & Ellison, 2008), but the uncertainty in the absolute value is considerable. For cases where Moustakas et al. (2010) do not present a metallicity, we draw one from the recent compilations by Marble et al. (2010) and Calzetti et al. (2010).

Galaxy log() Morphology
log [M] T-Type [12+log[O/H]] [M pc] [ ] [km s] [kpc] [ yr]
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
NGC 0337 9.9 6.7 8.51 0.004 21 14 130 2.9 2.4
NGC 0628 10.0 5.2 8.69 0.012 15 4.0 200 0.8 1.3
NGC 0925 9.7 7.0 8.52 0.004 7.5 1.3 140 6.9 4.4
NGC 2403 9.6 6.0 8.57 0.009 10 3.3 120 0.95 1.6
NGC 2841 10.7 3.0 8.88 0.037 4.6 1.4 310 2.3 1.2
NGC 2903 10.4 4.0 8.90ccMetallicity from compilation of Calzetti et al. (2010) and Marble et al. (2010) or Kennicutt et al. (2003b) (NGC 5457). 0.012 12 5.7 210 2.4 2.0
NGC 2976 9.0 5.2 8.67 0.008 7.6 4.4 88 1.1 1.4
NGC 3049 9.5 2.5 8.82 0.005 8.0 10 180 3.0 1.5
NGC 3184 10.2 6.0 8.83 0.018 14 2.8 200 2.5 1.8
NGC 3198 10.0 5.2 8.62 0.012 8.4 2.3 150 3.0 2.6
NGC 3351 10.1 3.1 8.90 0.018 8.5 5.2 200 1.1 1.2
NGC 3521 10.7 4.0 8.70 0.012 22 7.8 229 1.5 1.5
NGC 3627 10.5 3.1 8.67 0.016 13 7.7 190 1.1 1.8
NGC 3938 10.3 5.1 8.71ccMetallicity from compilation of Calzetti et al. (2010) and Marble et al. (2010) or Kennicutt et al. (2003b) (NGC 5457). 0.018 22 7.9 140 0.73 1.6
NGC 4214 8.7 9.8 8.36ccMetallicity from compilation of Calzetti et al. (2010) and Marble et al. (2010) or Kennicutt et al. (2003b) (NGC 5457). 0.0038 9.2 8.4 350 11 2.0
NGC 4254 10.5 5.2 8.79 0.01 47 18 170 1.4 1.6
NGC 4321 10.6 4.1 8.84 0.012 30 9.0 229 1.8 1.4
NGC 4536 10.2 4.3 8.61 0.005 13 6.8 180 0.7 2.0
NGC 4559 9.5 6.0 8.55 0.005 10 1.8 100 2.1 2.9
NGC 4569 10.2 2.4 8.88ccMetallicity from compilation of Calzetti et al. (2010) and Marble et al. (2010) or Kennicutt et al. (2003b) (NGC 5457). 0.017 8.5 1.9 220 3.2 1.8
NGC 4579 10.7 2.8 8.93ccMetallicity from compilation of Calzetti et al. (2010) and Marble et al. (2010) or Kennicutt et al. (2003b) (NGC 5457). 0.021 13 3.8 270 1.7 1.2
NGC 4625 8.9 8.8 8.70 0.011 8.5 6.6 27 0.53 2.3
NGC 4725 10.5 2.2 8.73 0.03 4.5 0.75 220 1.1 1.9
NGC 4736 10.2 2.4 8.66 0.008 17 10 170 0.25 0.77
NGC 5055 10.5 4.0 8.77 0.02 18 4.1 200 0.71 1.7
NGC 5194 10.5 4.0 8.87 0.02 53 20 210 0.58 1.0
NGC 5457 10.4 6.0 8.46ccMetallicity from compilation of Calzetti et al. (2010) and Marble et al. (2010) or Kennicutt et al. (2003b) (NGC 5457). 0.013 10 2.4 210 1.2 2.7
NGC 5713 10.3 4.0 8.64 0.006 54 37
NGC 6946 10.5 5.9 8.73 0.007 37 21 190 1.2 1.5
NGC 7331 10.8 3.9 8.68 0.01 16 4.4 260 1.9 1.9

Note. – Properties of sample galaxies. Columns give (1) galaxy name; (2) integrated stellar mass of whole galaxies based on 3.6m flux of Dale et al. (2007, 2009); (3) morphology; (4) “characteristic” metallicity at 0.4  from Moustakas et al. (2010), averaging their PT05 and KK04 calibrations; (5) dust-to-gas ratio at 0.4  based on our modeling of Spitzer data; (6) average H i+H surface density inside ; (7) average star formation rate surface density inside ; parameters for simple rotation curve fit, (8) and (9) ; and (10) orbital time at based on the rotation curve.

Table 2Sample Properties

2.4. Methodology

We sample our targets at 1 kpc resolution. This is fine enough to isolate many key physical conditions in the interstellar medium (ISM): metallicity, coarse kinematics, gas and stellar surface density. At the same time, we expect to average several star forming regions in each element (e.g., Schruba et al., 2010), with  M and  M formed over the last  Myr in each element. This minimizes concerns about evolution of individual regions, sampling the IMF, and drift of stars or leakage of ionizing photons from their parent region.

We convolve each map to have a symmetric gaussian beam with FWHM 1 kpc. For the Spitzer 24m maps we first convert from the MIPS PSF to a gaussian beam using a kernel kindly provided by K. Gordon, then we convolve to  kpc. This exercise effectively places our targets at a common distance but does not account for foreshortening along the minor axis. Five galaxies are too distant to convolve to 1 kpc. We mark these in Table 1 and include them in our analysis at their native resolution.

We sample each map to generate a set of intensity measurements. The sampling points are distributed on a hexagonal grid with points spaced by 0.5 kpc, one half-resolution element. At each sampling point we measure CO(2-1) intensity, Hi intensity, a suite of star formation rate tracers (described in L12), dust properties, and . We use these to estimate physical conditions as described above and in L12, taking into account the inclination of the galaxy.

We also identify a sample of galaxies to study the effects of physical resolution. Nine galaxies, marked in Table 1, have both the proximity and extent to allow us to test the effect of physical resolution on our results. We convolve these to a succession of physical resolutions from 0.6 to 2.4 kpc for further analysis (Section 4.3).

We treat regions with  M yr kpc or as upper limits and consider only points with — the HERACLES maps contain signal outside this radius (Schruba et al., 2011) but mostly not significant emission over individual lines-of-sight. In total we have lines of sight with at least one significant measurement, of which have CO upper limits and have SF upper limits. Points for which neither measurement is significant are not considered in the analysis. Nyquist sampling the maps in a hexagonal pattern leads to oversampling by a factor of , so that this corresponds to independent measurements. The maximum () upper limit on is  M pc, the median upper limit is  M pc.

2.5. Literature Data

We compare our results to recent measurements of SFR and molecular gas. These employ a variety of sampling schemes and SFR tracers. We adjust each to match our adopted CO-to-H conversion factor and IMF. Contrasting our approach with these data illuminates the impact of methodology and allows us to explore whether diverse observations yield consistent results under matched assumptions.

Kennicutt (1998b) presented disk-averaged measurements for 57 normal spiral galaxies and 15 starburst galaxies. He used literature CO with a fixed to estimate . To estimate , he used H in disk galaxies and IR emission in starbursts.

Calzetti et al. (2010) estimated disk-averaged for a large set of nearby galaxies. We cross-index these with integrated CO fluxes from Young et al. (1995), Helfer et al. (2003), and Leroy et al. (2009) to derive assuming that CO emission covers the same area as H. From the combination of these data we have disk-average and estimates for galaxies.

Saintonge et al. (2012), following Saintonge et al. (2011), present the COLDGASS survey, which obtained integrated molecular gas mass and SFRs for galaxies with  M, with secure CO detections. This large survey represents the best sample of integrated galaxy measurements to date. To convert to surface densities, we take the area of the star-forming disk in these galaxies to be . Saintonge et al. (2012) derive their SFRs from SED modeling that yields results close to what one would obtain converting the UV+IR luminosity directly to a SFR. This yields higher SFRs than our approach for matched measurements. Comparing galaxies with matched stellar mass or molecular gas content, we find the offset to be  dex, a factor of . This agrees well with what one would expect accounting for our subtraction of an IR cirrus with magnitude and our 24m coefficient, which is lower than what one would adopt to match a bolometric TIR SFR indicator (see L12 for calculations and discussion).

Leroy et al. (2005) combined new data with measurements by Young et al. (1995), Elfhag et al. (1996), Taylor et al. (1998), Böker et al. (2003), and Murgia et al. (2002) to compare and for individual pointings in a wide sample of nearby galaxies. They estimate from the 20cm radio continuum (Condon, 1992). These low-resolution pointings typically cover several kpc, a larger area than our resolution elements but less than an average over a whole galaxy disk.

Wong & Blitz (2002), Schuster et al. (2007), and Crosthwaite & Turner (2007) presented radial profiles of and for several nearby galaxies. Wong & Blitz (2002) targeted 7 nearby spirals, using H to calculate . Schuster et al. (2007) targeted M51 and derived from 20-cm radio continuum to estimate . We only present the Wong & Blitz (2002) and Schuster et al. (2007) profiles down to  M pc, below which we consider them somewhat unreliable. Crosthwaite & Turner (2007) targeted NGC 6946 and used IR emission to estimate .

Kennicutt et al. (2007), Blanc et al. (2009), Rahman et al. (2011), and Rahman et al. (2012) targeted small regions, similar to B08 and the work presented here. Kennicutt et al. (2007) focused on luminous regions in M51, mainly in the spirals arms. They infer from a combination of H and 24 µm emission. Rahman et al. (2011) explored a range of methodologies. We focus on their most robust measurements, drawn from bright regions in NGC 4254 with from a combination of NUV and 24 µm emission. Rahman et al. (2012) extended this work to consider the full set of CARMA STING galaxies, using only the 24m emission with a nonlinear calibration to infer . Blanc et al. (2009) studied the central  kpc of M51, deriving from H spectroscopy corrected using the Balmer decrement.

3. - Scaling Relations: First Order Constancy of

We estimate and for points in nearby galaxies. In this section we analyze these data in the context of a traditional “star formation law” scaling relation (§1.1). We show the data distribution in - parameter space (§ 3.1) and examine how this depends on methodology (§3.2). Using a Monte Carlo technique based on that of Blanc et al. (2009), we consider the best fit power-law to the ensemble data and individual galaxies (§3.3). We compare our results to a broad sample of literature data (§3.4).

3.1. Combined Measurement

Figure 1.— Star formation rate surface density, , estimated from H+24m emission, as a function of molecular gas surface density, , derived from CO(2-1) emission for 30 nearby disk galaxies. The top left panel shows individual points (dark gray points show upper limits) with the running median and standard deviation indicated by red points and error bars. The red points with error bars from the first panel appear in all four panels to allow easy comparison. Dotted lines indicated fixed H depletion times; the number indicates in yr. The top right panel shows the density of the data in the top left panel. In the bottom panels we vary the weighting used to derive data density. The bottom left panel gives equal weight to each galaxy. The bottom right panel gives equal weight to each galaxy and each radial bin.
Tracer Median Scatter
[Gyr] [ dex]
Weighting as equal each …
line-of-sight
… fixed 2.2 0.28
2.6 0.26
3.1 0.28
galaxy
… fixed 1.3 0.32
… … only  M 1.7 0.21
… … only  M 0.4 0.29
1.8 0.20
… … only  M 2.0 0.13
… … only  M 1.1 0.26
2.4 0.26
… … only  M 2.1 0.31
… … only  M 2.7 0.21
Tracing with …
(weighting lines-of-sight equally)
H+24m
best estimate 2.2 0.28
… no cirrus 2.0 0.22
… double cirrus 3.0 0.37
FUV+24m
best estimate 2.2 0.27
… no cirrus 1.9 0.21
… double cirrus 3.2 0.39
H +  mag 2.1 0.30

Note. – Median molecular gas depletion time, scatter, and correlation between and in our sample. Line-of-sight averages treat each kpc-resolution line of sight as equal. Galaxy averages refer to inside for each galaxy. Unless otherwise noted, we calculate using fixed and from Hm. Quoted error bars on report scatter, uncertainties on the rank correlation arise from randomly repairing data.

Table 3 at 1 kpc Resolution

Figure 1 compares , estimated from H+24m, and at 1 kpc resolution for our whole sample. Individual kpc resolution lines of sight appear as gray points and the red points show the median and standard deviation after binning the data by . In the top right panel and bottom row, blue contours show data density adopting different weightings. The top right panel gives identical weight to each line of sight, treating each kpc as equal regardless of location. The bottom left panel gives equal weight to each galaxy and so weights measurements from small galaxies with little area more than measurements from large galaxies. The bottom right panel treats each radial ring in each galaxy equally, and so gives more weight to the central parts of galaxies than their outer regions. Dashed lines here and throughout this paper indicate fixed and a horizontal line indicates the limit of our measurements. In the top left panel, dark points show measurements where one quantity is an upper limit. Table 3 summarizes key values from the plots in this section.

The top rows of Figure 1 and Table 3 show the good correspondence between and that we have previously found in the HERACLES sample (B08,L08, Schruba et al., 2011; Bigiel et al., 2011). Our dynamic range at 1 kpc resolution spans from to  M yr kpc and from a few to  M pc. Across this range, and correlate well, exhibiting a Spearman rank correlation coefficient for most tracers and weightings. This quantifies the tight, one-to-one relationship visible by eye in the top row.

The median weighting each line of sight equally is  Gyr with a scatter of dex, a factor of two. The absolute value of the median , i.e., the scale of the - and -axes in Figure 1, depends on the calibration of our SFR tracer and CO-to-H conversion factor. Each remains uncertain at the level and we suggest that an overall uncertainty of 60% on the absolute value of represents a realistic, if somewhat conservative, value. Our and estimates can be compared internally with much better accuracy than this (L12, Sandstrom et al., 2012), so we suggest that this uncertainty be viewed as an overall scaling of our results.

The bottom row in Figure 1 begins to reveal the deviations from a simple one-to-one scaling that will be the subject of Section 4. Weighting all galaxies equally (bottom left panel) reveals a significant population of low , high , low data. This drives the median depletion time for the sample from  Gyr, weighting by line-of-sight, to  Gyr, weighting by galaxy. In Section 4.1 we show that these low apparent originate from low-mass, low-metallicity systems (see also Schruba et al., 2011; Krumholz et al., 2011; Schruba et al., 2012). Because of their small size, these systems do not contribute many data compared to large, metal-rich spirals. Therefore, they only weakly influence the overall data distribution seen in the top row. We examine as a function of host galaxy properties and local conditions in Sections 4.1 and 4.2. In the appendix we present relations for individual galaxies (see also Table 2), allowing the reader to see how Figure 1 emerges from the superposition of individual systems (see also Section 3.3).

Weighting radial rings equally (bottom right panel) highlights these same low -low galaxies and brings out an additional low population at higher . These point emerge because the radial weighting emphasizes points in the central parts of galaxies relative to their outskirts. We show in Section 4.4 that the central regions of many of our targets exhibit enhanced efficiency compare to their disks. As will small galaxies, these central regions contribute only a tiny fraction of the area in our survey and thus exert little impact on the plots in the top row.

Figure 1 thus illustrates our main conclusions: a first order simple linear correlation between and and real second order variations. It also illustrates the limitation of considering only - parameter space to elicit these second-order variations. Metallicity, dust-to-gas ratio, and position in a galaxy all play key roles but are not encoded in this plot, leading to double-valued at fixed in some regimes. We explore these systematic variations in and motivate our explanations throughout the rest of the paper.

3.2. Relationship for Different SFR and Molecular Gas Tracers

Figure 1 shows our best-estimate and computed from fixed . Many approaches exist to estimate each quantity (see references in Leroy et al., 2011, L12) and the recent literature includes many claims about the effect of physical parameter estimation on the relation between and . In this section, we explore the effects of varying our approach to estimate and .

3.2.1 Choice of SFR Tracer

Figure 2.— Star formation rate surface density, , estimated using different tracers as a function of molecular gas surface density, . Light gray points show individual measurements, dark gray points indicate upper limits, contours show data density, and red points show the running median and standard deviation using each tracer. For comparison, black-and-white points show the bins from the top right panel of Figure 1. We show estimated from: (top left) H assuming 1 magnitude of extinction, (top right) a combination of FUV and 24m emission (L12), (bottom left) H combined with 24m emission with no IR cirrus correction applied to the 24m, (bottom right) H combined with 24m with double our IR cirrus correction applied to the 24m.

Figure 2 and the lower part of Table 3 report the results of varying our approach to trace the SFR. We show estimated from only H, with a fixed, typical  mag (top left), along with results combining FUV, instead of H, with 24m emission (top right). We also show the results of varying the approach to the IR cirrus. Our best-estimate combines H or FUV with 24m after correcting the 24m emission for contamination by an IR cirrus following L12. We illustrate the impact of this correction by plotting results for two limiting cases of IR cirrus correction: no cirrus subtraction (bottom left) and removing double our best cirrus estimate (bottom right), which we consider a maximum reasonable correction. Data density contours in Figure 2 weight each point equally and the large black points indicate the original binned results from Figure 1.

The top left panel of Figure 2 and Table 3 show that the basic relationship between and persists even when we derive from H alone. The median and scatter using only H resemble what we find for our best estimate and the correlation between H and CO appears only moderately weaker than for the hybrid SFR tracer. It also appears moderately flatter than relations that incorporate IR emission as we underestimate extinction in the central parts of galaxies (§3.3). Inasmuch as H represents an unambiguous tracer of recent star formation, the top left panel in Figure 2 demonstrates that subtle biases in the treatment of IR emission, e.g., 24m emission tracing the ISM rather than recent star formation, do not drive our results.

The top right panel shows traced by FUV+24m emission. The distribution agrees well with what we found using H+24m, as do the median and scatter in . The agreement of FUV+24m and H with our best estimate H+24m occur partially because we have designed our SFR tracers to yield self-consistent results (L12). However, that procedure considered only the overall normalization and did not require the detailed agreement we see comparing Figures 1 and 2.

In the bottom row, we vary our approach to the infrared cirrus. By default, we correct the 24m map for infrared cirrus following L12. The bottom left panel shows the results of applying no cirrus subtraction, while in the bottom right panel we double our cirrus subtraction. Turning off the cirrus subtraction yields median shorter than our best estimate with notably lower scatter and our strongest observed correlation. The tighter correlation reflects the fact that the relationship between 24m and CO emission is the strongest in the data (see also Schruba et al., 2011). tracers that more heavily emphasize 24m exhibit the strongest correlation with traced by CO.

Doubling the cirrus subtraction leads to an longer , larger scatter, and a mildly weaker correlation between and . This partially reflects uncertainty in the cirrus calculation, which relies on model fits to observed data. It also reflects the deemphasis of 24m emission, which exhibits a very tight correspondence to CO, in favor of H, which still exhibits a good correspondence but with more scatter. The fraction of data that have upper only limits for also increases, so that extending this analysis to lower surface density will require improved data and methodology.

Thus we observe subtle variations in the relation between and depending on the exact treatment of 24m emission, including up to a variation in across the full plausible range of cirrus treatments. However, our main results of a simple correspondence between and hold even when we omit IR data from the analysis. Note that this conclusion relies on the assumption that H emission traces recent star formation. If a substantial fraction of H emission arises from sources other than recent star formation or if the mean free path of an ionizing photon regularly exceeds one of our kpc-sized resolution elements, then this general agreement may break down (Rahman et al., 2011; Liu et al., 2011, see discussion in L12). These more exotic situations aside, overall Figure 2 and Table 3 show good qualitative and quantitative agreement among different approaches to estimate . We will find the same when fitting the data in Section 3.3.

Throughout the rest of the paper, we adopt H+24m, corrected for the effects of a 24m cirrus, as our single, best estimate of . L12 justify this choice and we refer to that paper for more discussion.

3.2.2 Choice of CO Line

Figure 3.— Star formation rate surface density, , as a function of molecular gas surface density, , here estimated from literature CO(1-0) data (Section 2). Gray points show individual measurements, contours indicate data density, and red points show the running median and standard deviation using each tracer. For comparison, large black-and-white circles show the same bins from Figure 1.

HERACLES consists of maps of CO (2-1) emission, which we use to estimate the distribution of H. CO (1-0) has been more commonly used to trace the distribution of H 222 We emphasize that Sandstrom et al. (2012) demonstrate the ability of CO (2-1) to robustly trace molecular gas in our sample (Section 2.2). CO (1-0) maps exist for a subset of our targets (Helfer et al., 2003; Kuno et al., 2007). Though these do not have the same overall quality as the HERACLES maps, we use them to assess the impact of our choice of molecular gas tracer. Figure 3 plots as a function of estimated from literature CO(1-0) data. We allow repeats, so that if Helfer et al. (2003) and Kuno et al. (2007) each mapped a galaxy we include each data set in Figure 3.

Overall, results for CO (2-1) and CO (1-0) agree fairly well. The CO (1-0) data tend to yield higher . This is exclusively a product of the Kuno et al. data. In the overlap of our sample and the Kuno et al. data, the Kuno et al. data yield median  Gyr. Our data yield median  Gyr for the same points. However, for the overlap with BIMA SONG (Helfer et al., 2003), the BIMA SONG CO (1-0) data give median  Gyr. Over the same points, HERACLES implies  Gyr. The disagreement between our CO (2-1) data and CO (1-0) data thus appears no larger than the disagreement among published CO (1-0) data sets.

3.2.3 CO-to-H Conversion Factor

Figure 4.— Star formation rate surface density, , as a function of molecular gas surface density, , here estimated from CO (2-1) data combined with our “” conversion factor , which reflects variations in the dust-to-gas ratio and the central depressions in found by Sandstrom et al. (2012). Panels and annotations are as Figure 1.

In Figures 1 - 3 we adopt a fixed CO-to-H conversion, . This assumption may be too coarse given the wide range of metallicities, dust-to-gas ratios, and central CO concentrations in our targets. Figure 4 shows and after the application of our “” conversion factor (Section 2.2), which attempts to account for the presence of CO-poor envelopes of molecular clouds and central CO depressions. This conversion factor assumes that all CO emission originates from clouds with surface densities of  M pc with PDR structure like that described in Wolfire et al. (2010). The resulting dependence of on metallicity approximates the current consensus (Wolfire et al., 2010; Glover & Mac Low, 2011; Leroy et al., 2011; Feldmann et al., 2012; Narayanan et al., 2012), but note that this remains highly uncertain because of limited observational constraints. We also report results for the “” conversion factor in Table 3. This conversion factor makes the more aggressive assumption that a large amount of molecular emission emerges from weakly shielded parts of clouds, rendering very sensitive to the dust-to-gas ratio.

Applying the “ to the data in Figure 1 yields Figure 4. The top rows, which show the bulk distribution of the data, appear qualitatively similar in the two plots, though Table 3 and close inspection of the plots do indicate that the normalization of the - relation changes between the two plots. The median weighting each line of sight equally rises from  Gyr to  Gyr with the application of the variable conversion factor.

The most dramatic contrast between Figures 1 and 4 appears in the bottom rows. Many of the low (high ) data in Figure 1 arise from small galaxies with low dust-to-gas ratios. With the application of a variable conversion factor, our estimate of in these galaxies moves to higher values while remains constant. The result, visible in the bottom rows of Figure 4, is that data from these low-mass, low metallicity galaxies now overlap the other points, forming a (more) continuous single - trend. Table 3 reports that the scatter among galaxies drops from  dex to  dex with the application of the “” conversion factor, with the systematic difference in between high and low mass galaxies reduced from a factor of to a factor of .

Thus, as we will see in Section 4, application of a dust-to-gas ratio-dependent conversion factor CO-to-H conversion factor does affect the derived - relation, with the sense of moving many points with low apparent into closer agreement with the distribution defined by large galaxies. This scenario of a rapidly varying conversion factor and a weakly varying has been discussed in the context of the Small Magellanic Cloud by Bolatto et al. (2011) and in a theoretical context by Krumholz et al. (2011). They interpret weak variations of but strong variations of as evidence that the requisite preconditions for star formation more closely resemble those for H formation than those required for a high CO abundance.

In §4, we will show that the even more aggressive “” case may in fact explain most apparent galaxy-wide variations in , but note that while the ”” conversion factor may offer an explanation for our observed trends, it also requires that a substantial fraction of the integrated CO emission from galaxies arise from relatively low regions.

The other portion of the “” conversion factor, the depressions in the central parts of galaxies, affect too few data to be prominent in Figure 4. However, one can see many individual low points at moderate on close inspection. We return to this point in §4.4.

3.3. Power Law Index

Figure 5.— Results of power-law fitting for (left) our combined data set and (right) individual galaxies. Left: Goodness of fit as a function of power law index, , for fits to our entire ensemble of data where  M pc. The fitting uses a modified version of the Monte Carlo technique outlined by Blanc et al. (2009). From top to bottom, we show results for fixed using our best H+24m and FUV+24m estimates; H+24m with no cirrus removed or double the best-estimate cirrus removed; and H with an assumed fixed 1 magnitude of extinction. We also show results for our “” and “” conversion factors. Vertical lines indicate where reaches its minimum value, bold for the value in that panel, dotted for other panels. Each case shows a broad minimum in with a best-fit value near . Right: Best fit power-law index, , for individual galaxies using H+24m and a fixed CO-to-H conversion factor. The distribution of best-fit for individual galaxies peaks near with most galaxies in the range but outliers up to .
Tracer
(M yr kpc) (dex)
H+24m
no cirrus removed
double cirrus removed
FUV+24m
H with mag
and H+24m
and H+24m

Note. – Results of Monte Carlo fitting to Equation 10 for different combinations of and tracers. Column (1) reports the tracer used; in the top five rows we vary the tracer while adopting fixed . The last two rows adopt our best estimate, H+24m, and vary the adopted conversion factor. Columns (2)–(4) report the best-fit coefficient at  M pc, the power law index, and the intrinsic scatter. We quote uncertainties from the Monte Carlo simulations described in the appendix.

Table 4Results of Monte Carlo Fitting to Equation 10

Studies of the star formation-gas connection in galaxies have treated the relationship as a power law and focused on the index of this power law. While this single parameter undoubtedly makes for easy shorthand, the fixation on this parameter obscures environmental factors other than . Recent observations offer good evidence that relates to and in fundamentally different ways (L08, B08 Wong & Blitz, 2002; Schruba et al., 2011) and that is even a multivalued function of (Daddi et al., 2010; Genzel et al., 2010; Schruba et al., 2011; Saintonge et al., 2011, 2012, and this paper).

Despite the shortcomings of this approach, we consider the best-fit index in our data as a useful, or at least expected, point of comparison to previous studies. We derive best-fit relations for our ensemble of measurements and individual galaxies. We fit a relation with three parameters: a normalization, , power-law index, , and intrinsic, log-normally distributed scatter with RMS magnitude . Then

(10)

with data intrinsically scattered by . We derive the best-fit , , and using a Monte Carlo approach based on the work of Blanc et al. (2009). This resembles Hess diagram fitting used for optical color-magnitude diagrams. It includes observational uncertainties, upper limits, and intrinsic scatter in the relation. This approach also avoids important biases that can easily arise fitting scaling relations to noisy, bivariate data. We illustrate these biases, which affect many commonly adopted approaches, in the appendix (see also Blanc et al., 2009) and note that they can easily shift the derived index by a few tenths for realistic data distributions.

Following Blanc et al. (2009) we grid our data, deriving a two-dimensional image of data density in regularly-spaced cells in - space. Unlike Blanc et al. (2009) we work in logarithmic space. This gives us a better ability to resolve the distribution of our data, but forces a coarser approach to upper limits. We treat upper limits by essentially creating an “upper limit row” along the axis. In detail, we adopt the following approach:

  1. We exclude all data with  M pc. This gives us a data set with a well-defined -axis.

  2. We generate Monte Carlo data sets for a wide range of , , and in the following way. We take our observed to represent the true physical distribution. We draw 100,000 data points from this distribution (allowing repeats) for each combination of , , and . We derive for each of these points. We then apply the expected uncertainty to (the statistical uncertainty from HERACLES) and (0.15 dex). We grid these data in - space, using cells  dex wide in both dimensions. We treat this grid as the expected probability distribution function for those underlying parameters , , and .

  3. We grid our observed data in - using the same grid on which we derived probability distribution functions. We create different grids for each set of estimates.

  4. We compare our gridded data to the Monte Carlo realization for each , , combination and calculate a goodness-of-fit estimate, which we here loosely refer to as . After re-normalizing the Monte Carlo grid to have the same amount of data as the observed grid, we calculate:

    (11)

    where the sum runs across all cells, , refers to the observed number of data in the grid cell, and refers to the expected number of data in that cell given , , and and our observational uncertainties. The goodness-of-fit statistic is thus analogous to calculated for the case of Poisson noise in each cell. Points with only upper limits on (where  M yr kpc) are included in the calculation. These have an associated value but all upper limits are treated as having the same .

We apply this method to our ensemble of data, repeating the exercise for each SFR tracer discussed in Section 3.2 and for our fixed, “”, and “” conversion factors. We also fit each galaxy on its own333Due to the lower density of points, we use 0.2-dex cell sizes and require only points to populate the theoretical distribution.. Figure 5 and Table 4 report our fits to the combined data set. Figure 5 plots the approximate reduced as a function of power law index, marginalizing over and . We observe clear minima in the range for all SFR tracers. The appendix presents a Monte Carlo treatment that considers a number of effects: robustness to removal of individual data or galaxies, statistical noise, calibration (gain) uncertainties for each data set, and exact choice of fitting methodology. We quote uncertainties derived from this Monte Carlo treatment in Table 4.

The fits in Table 4 suggest a power law with , intrinsic scatter of a factor of (0.3 dex), and  Gyr at  M pc. The slope remains consistent with a linear relation between H and star formation (B08, Bigiel et al., 2011) or with the weakly super-linear slope of Genzel et al. (2010) or Daddi et al. (2010), though note that our 1-kpc scale does not precisely match their observations. The mild difference between the best-fit coefficient, , and the median reported in Table 3 reflect the inadequacy of the power law to capture the full distribution of the data.

The choice of SFR tracer affects the fit, but offers more of a refinement than a qualitative shift in these conclusions. The sense of the shifts resemble those seen in Section 3.2. Replacing FUV for H as the unobscured tracer has minimal effect. Using only H to estimate SFR yields a slightly shallower slope. Based on the observed H-to-IR ratio, extinction increases with increasing (see plots in Prescott et al., 2007, and L12). By assuming a fixed we would expect to underestimate at the high end and overestimate it at low end, somewhat “tilting” the relationship to shallower slope. If we do not remove any cirrus from the data, working only with the measured 24m emission, the scatter in the relation diminishes to less that  dex. This underscores the point that it is the tight observed correlation between CO and 24m emission that drives much of the recent work on this topic (see more discussion in Rahman et al., 2011; Schruba et al., 2011; Liu et al., 2011) so that SFR tracers that emphasize 24m data tend to yield the tightest relations. Conversely, increasing the cirrus removed leads to a somewhat longer overall with larger intrinsic scatter. Adjusting the conversion factor exerts only a mild impact on the fit because largest corrections apply to small galaxies and often to low apparent regions. These significant variations to a small subset of the data do not drive substantial variations in the fit.

The simple nearly linear scaling given by our fits could result from the superposition of a varied set of distinct relations for individual galaxies (see Schruba et al., 2011). In the right hand panel of Figure 5 we show that indeed the best-fit index for individual galaxies exhibits significant scatter. We find a median but best-fit values span . We report best-fit indices for individual galaxies in the appendix and stress two general conclusions here. First, we see variation in index from galaxy to galaxy, but the range is still , consistent with the idea that to first order the molecular gas supply regulates the star formation distribution and in sharp contrast to the steep indices relating to atomic gas Bigiel et al. (2008); Schruba et al. (2011). Second, the fact that these galaxy-to-galaxy variations wash out into Figure 1 implies that while may correlate with within an individual galaxy, simply knowing at 1 kpc resolution with no other knowledge of local conditions or host galaxy does not allow one to predict better than simply adopting a median . That is, in the absence of knowledge of other conditions, is not a good predictor of the molecular gas depletion time.

3.4. Comparison to Literature Data

Figure 6.—  versus  for our results and literature measurements. The type of literature measurement varies with panel: the top left shows galaxy averages, the top right shows radial profiles, the lower left shows sparse, coarse () sampling of galaxy disks, and the lower right shows fine sampling (). In all four panels, red points show a running median and standard deviation (error bars) for our kpc-scale measurements (Figure 1) using H+24m and CO (2-1). Orange circles in the top left panel show integrated measurements for our data. Literature measurements are described in Section 2; references: K98 — Kennicutt (1998a), C10 + lit. — Calzetti et al. (2010) combined with literature CO, S12 — Saintonge et al. (2012) with offset to bring SFR estimates into agreement (Section 2.2), W02 — Wong & Blitz (2002), S07 — Schuster et al. (2007), C07 — Crosthwaite & Turner (2007), M02 — Murgia et al. (2002), L05 — Leroy et al. (2005), K07 — Kennicutt et al. (2007), B09 — Blanc et al. (2009), R11 — Rahman et al. (2011), R12 — Rahman et al. (2012).
Galaxy Median Scatter
[Gyr] [dex]
This study …
… weighting by measurement 2.2 0.28
… weighting by galaxy 1.3 0.32
Median of literature …
… weighting by measurement 2.7aaDominated by Rahman et al. (2012). Without Rahman et al. (2012) median is 2.1 Gyr. 0.36
… weighting by study 2.0 0.23
Disk Averages
Saintonge et al. (2012) 0.7 0.37
… offset SFRbbSFR estimate offset to match our estimates (Section 2.2). 1.1 0.37
Calzetti et al. (2010) + literature COccCO from Young et al. (1995) and Helfer et al. (2003). 1.5 0.38
Kennicutt (1998b)
… disks 1.1 0.46
… starbursts 0.4 0.39
Radial Profiles
Schuster et al. (2007) 2.0 0.12ddStudy considered a singe galaxy. Others combine multiple galaxies.
Crosthwaite & Turner (2007) 4.4 0.06ddStudy considered a singe galaxy. Others combine multiple galaxies.
Wong & Blitz (2002) 2.0 0.36
Low Resolution
Leroy et al. (2005) 2.1 0.33
Murgia et al. (2002) 2.8 0.41
High Resolution
Rahman et al. (2012) 2.9 0.37
Rahman et al. (2011) 1.6 0.15ddStudy considered a singe galaxy. Others combine multiple galaxies.
Blanc et al. (2009) 3.2 0.64ddStudy considered a singe galaxy. Others combine multiple galaxies.
Kennicutt et al. (2007) 2.2 0.37ddStudy considered a singe galaxy. Others combine multiple galaxies.

Note. – Average molecular gas depletion time, in Gyr, for matched assumptions —  cm (K km s), a Kroupa IMF, and including helium in the gas estimate. The left column gives the study, with the list broken down by sampling approach and the right column reports the median molecular gas depletion time in that study. Error bars report the scatter, in dex, for each study.

Table 5Comparison to Literature

Many studies have assessed the relationship between gas and star formation in nearby galaxies (Section 1). Figure 6 and Table 5 compare our measurements to a compilation of these studies (see Section 2.5 and Bigiel et al., 2011). We adjust each set of measurements to share our adopted CO-to-H conversion factor and stellar initial mass function.

Table 5 gives by study. These span from 0.4 for the Kennicutt (1998b) starbursts to 4.4 Gyr for the study of NGC 6946 by Crosthwaite & Turner (2007). Considering all measurements equally, the median literature is 2.7 Gyr, which drops to 2.1 Gyr if we exclude the large data set of Rahman et al. (2012), which otherwise dominates the statistics. Treating each study as a single independent measurement, the median is  Gyr. These are in good agreement with the estimates of this study (Table 3)  Gyr weighting all measurements equally. The scatter among individual literature data is  dex and from study to study the scatter is dex.

Figure 6 shows a more detailed comparison between our measurements and individual literature data. We separate the literature studies according to the scale sampled. The top left panel shows measurements where one point corresponds to one galaxy. The top right panel shows data from studies that measure azimuthally averaged and in a series of concentric tilted rings. The bottom left panel shows data for individual pointings with comparatively poor angular resolution, . The bottom right panel shows studies that obtain high-angular-resolution sampling of each target. In each panel we plot the running median and standard deviation for our data, binned by , as red points.

The final three panels of Figure 6 demonstrate excellent agreement between our data and previous studies that resolve the disks of galaxies (see also Bigiel et al., 2011). This agreement may not be surprising given that our study shares targets with many of these literature studies, which also heavily overlap one another. Nonetheless, we show here that repeated measurements of the distribution of and in the nearest star-forming spiral galaxies mostly cover the same part of parameter space regardless of exact methodology. Uncertainty in interpretation and fitting techniques have clouded this basic agreement in where the data lie. Given our basic approach to physical parameter estimation, there appears to be overall agreement for a typical  Gyr in local disk galaxies.

The first panel of the Figure 6 looks qualitatively different from the other three. This panel shows galaxy-integrated measurements, so that one point is one galaxy. These tend to scatter from overlapping our data up to lower at low and comparatively high . This same effect appears in Table 5 as low values of for studies that focus on measurements of whole galaxies. Our synthesis of literature CO and SFR measurements yields  Gyr while for the Kennicutt (1998a) disk galaxies the median  Gyr. Treating our own sample as a set of integrated measurements we find a similar value,  Gyr.

This disk-integrated is significantly shorter than the that we measure treating each point equally. We noted this effect in Section 3.1. It arises because weighting each galaxy equally emphasizes small, low-mass, low SFR galaxies relative to large, massive galaxies. These low mass galaxies have less physical area than large disks, so that they do not affect the ensemble of measurements much. However these small galaxies do exhibit short apparent and when given equal weight they drive the median down by a factor of . We explore this and other systematic variations in in the second part of this paper.

4. Systematic Second-Order Variations in : Global Correlations, Efficient Galaxy Centers, and Correlated Scatter

In Section 3 we demonstrate that our ensemble of data can be described to first order by a roughly linear relation between and with a slope corresponding to a typical depletion time  Gyr with a factor of two scatter from line of sight to line of sight. However, we also show that the apparent uniformity of results at least partially from the emphasis that our approach places on the disks of large, star-forming galaxies. These contribute most of the area in our sample. When we apply weightings that emphasize small galaxies or the inner parts of galaxies, we observe departures from this simple picture.

In this section, we explore these variations. We examine the dependence of disk-average on integrated galaxy properties (§4.1) and the dependence of on local physical conditions (§4.2). We highlight apparent variations in <