THE RELATIONSHIP BETWEEN MOLECULAR GAS, H i, AND STAR FORMATION IN THE LOW-MASS, LOW-METALLICITY MAGELLANIC CLOUDS

The far-IR images come from the Herschel Inventory of the Agents of Galaxy Evolution in the Magellanic Clouds key project (HERITAGE; Meixner et al. 2013). HERITAGE mapped both the LMC and SMC at 100, 160, 250, 350, and 500 μm with the Spectral and Photometric Imaging Receiver (SPIRE), and the Photodetector Array Camera and Spectrometer (PACS) instruments. Information on the details of the data calibration, reduction, and uncertainty can be found in Meixner et al. (2013).

For this work, we apply further background subtraction. First, we remove the foreground Milky Way cirrus emission. Following Gordon et al. (2014) (based on Bot et al. 2004), we estimate the foreground cirrus emission by using the relationship between IR dust emission and H i from Desert et al. (1990) and scaling the integrated H i intensity map over the velocities of the Milky Way emission in the direction of the LMC by applying the conversion factors 1.073, 1.848, 1.202, and 0.620 (MJy sr⁻¹/10²⁻ cm⁻²) for 100 μm, 160 μm, 250 μm, and 350 μm images, respectively. The median estimated cirrus emission was 5.7, 9.9, 6.4, and 3.3 MJy sr⁻¹ for the 100, 160, 250, and 350 μm images.

Second, we set the images to comparable zero-points: the outskirts of the PACS images were set to the COBE and IRAS data emission levels while the outskirts of the SPIRE images were set to zero due to the lack of similar large-scale coverage at the longer wavelengths. After subtracting the cirrus emission, we chose six regions in the outskirts of the LMC with no emission in the Herschel or H i images, fit a plane to the median values of the regions, and subtract the plane. The cirrus subtraction and background subtraction had primarily minor effects on the images, with the final image values being lower by 7%, 9%, 2%, and 2% on average for the 100, 160, 250, and 350 μm images in regions with a signal-to-noise ratio (S/N) > 3.

The neutral atomic gas data come from 21 cm line observations of H i. We use the LMC H i map from Kim et al. (2003) and the SMC H i map from Stanimirović et al. (1999), both combine Australian Telescope Compact Array (ATCA) and Parkes 64 m radio telescope data. The interferometric ATCA data set the map resolution at 1' (r ∼⃒ 20 pc in the SMC and r ∼⃒ 15 pc in the LMC), but the data are sensitive to all size scales due to the combination of interferometric and single-dish data.

The observed brightness temperature of the 21 cm line emission is converted to H i column density (${N}_{{\rm{H}}{\rm{I}}}$) assuming optically thin emission using

$$\begin{eqnarray*}&&{N}_{{\rm{H}}{\rm{I}}}=1.823\times {10}^{18}\displaystyle \frac{{{\rm{cm}}}^{-2}}{{{\rm{K}}{\rm{km}}{\rm{s}}}^{-1}}\int {T}_{B}(v)\;{dv}.\end{eqnarray*}$$

We find rms column densities of 8.0 × 10¹⁹ cm⁻² in the LMC map and 5.0 × 10¹⁹ cm⁻² in the SMC map. We convert column density to surface mass density (${{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}$) using

$$\begin{eqnarray*}&&{{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}=1.4\mathrm{cos}i\left(8.0\times {10}^{-21}\displaystyle \frac{\;{M}_{\odot }\;{{\rm{pc}}}^{-1}}{{{\rm{cm}}}^{-2}}\right){N}_{{\rm{H}}{\rm{I}}},\end{eqnarray*}$$

where the factor of 1.4 accounts for He and i is the inclination angle.

While the assumption of optically thin H i emission is likely appropriate throughout much of the galaxies, there are regions with optically thick emission, which would cause ${N}_{{\rm{H}}{\rm{I}}}$ to be underestimated. While a statistical correction for H i optical depth in the SMC exists (Stanimirović et al. 1999), none exists for the LMC. Additionally, nearby surveys of H i (i.e., THINGS; Walter et al. 2008) make no optical depth corrections. We chose not to make any optical depth corrections to the H i maps as the statistical corrections in the SMC are generally small (increases the total H i mass by 10%; Stanimirović et al. 1999) and an accurate optical depth correction would require assuming a spin temperature.

We use integrated ¹²CO (1–0) intensity maps from the 4 m NANTEN radio telescope (half power beam width of 26 at 115 GHz) for the LMC (Fukui et al. 2008) and SMC (Mizuno et al. 2001). The LMC and SMC velocity integrated maps have typical 3σ noise of ∼⃒1.2 K km s⁻¹ and ∼⃒0.45 K km s⁻¹, respectively. For the LMC, there is also the higher resolution and sensitivity Magellanic Mopra Assessment (MAGMA) Survey, which used the 22 m Mopra telescope of the Australia Telescope National Facility to follow-up the NANTEN survey with 40'' angular resolution and 1σ sensitivity of 0.2 K km s⁻¹ (Wong et al. 2011). However, the MAGMA survey is not complete as they only mapped regions with detected CO in the NANTEN map. Because the CO maps are only used to identify molecular regions and do not affect the final resolution of our molecular gas maps, we use the higher coverage NANTEN maps in our molecular gas mapping process and then, in the LMC, we compare the final dust-based molecular gas maps to the higher resolution MAGMA data.

We combine images of ${\rm{H}}\alpha$ and 24 μm dust emission to trace recent star formation. For the LMC we use the calibrated, continuum-subtracted Hα map from the Southern Hα Sky Survey Atlas (Gaustad et al. 2001) at 08 resolution. We correct the ${\rm{H}}\alpha$ maps for the line-of-sight Milky Way extinction using A_V (LMC) = 0.2 mag and A_V (SMC) = 0.1 mag (Schlafly & Finkbeiner 2011). We found background emission outside the LMC, on the order of 10% of the total flux observed in the main part of the galaxy, likely from the diffuse Milky Way Hα emission. We apply additional background subtraction by removing a polynomial fit to the regions outside the galaxy. In the SMC, we use the continuum-subtracted Hα map from the Magellanic Cloud Emission Line Survey (MCELS; Smith & MCELS Team 1999) at 23 resolution. For both the SMC and LMC we use the Multiband Imaging Photometer 24 μm map from the Spitzer Survey "Surveying the Agents of Galaxy Evolution" (SAGE; Meixner et al. 2006; Gordon et al. 2011).

To convert observational measurements to surface mass density (Σ), we need both the distance to the galaxy and inclination angle (i). For the LMC, we use an inclination angle of i = 35°, which is the approximate intermediate value of the three fits to stellar proper motions and line-of-sight velocity measurements in van der Marel & Kallivayalil (2014), which range from i = 262 ± 59 to i = 396 ± 45, and is consistent with their previous work that found i = 347 ± 62 (van der Marel & Cioni 2001). We assume that the inclination of the stellar disk is comparable to the gas disk given the disk-like morphology of the LMC. While Kim et al. (1998) fit an inclination angle to the H i kinematics, they found it was unreliable and much higher than the morphological fit (i = 22° ± 6°). Ultimately, Kim et al. (1998) adopted the inclination angle found from the stellar dynamics.

The inclination of the SMC is poorly constrained due to its irregular morphology. Recent work by Scowcroft et al. (2016) shows that assuming a disk with an inclination angle inaccurately represents the detailed morphology of the SMC. However, comparing the SMC to the LMC and studies of other galaxies requires knowing the mass surface densities and adopting the simple model of an inclined disk. Bolatto et al. (2011) adopted i = 40° ± 20° based on the analysis of the H i rotation curve by Stanimirović et al. (2004). The recent estimate of the SMC inclination based on three-dimensional structure traced by cepheid variable stars found i = 74° ± 9° (Haschke et al. 2012), which is consistent with the previous studies using cepheids (Caldwell & Coulson 1986; Groenewegen 2000). While cepheids, as old stars, may not trace the gaseous disk, a new analysis of the H i rotation also indicates a higher possible inclination of i ≈ 60°–70° (P. Teuben 2016, private communication). A higher inclination angle scales the surface mass densities to lower values. We adopt i = 70° for the inclination of the SMC and compare to the previous results in Bolatto et al. (2011) to determine how the higher inclination angle affects the results.

We combine the dust emission with a self-consistently estimated gas-to-dust ratio to estimate the total amount of gas. By removing the atomic gas, we are left with an estimate of the amount of molecular gas. The benefit of this method, particularly at low metallicity, is its ability to trace ${{\rm{H}}}_{2}$ where CO has photo-dissociated. This method is based on the previous work by Israel (1997) and Leroy et al. (2011) in both Magellanic Clouds, Dame et al. (2001) in the Milky Way, Bernard et al. (2008) in the LMC, and Leroy et al. (2007, 2009) and Bolatto et al. (2011) in the SMC, all of which have demonstrated that dust is a reliable tracer of the molecular gas. In Figure 1, we show that the optical depth of the dust correlates well with ${N}_{{\rm{H}}{\rm{I}}}$, which represents the majority of the gas, and the 1σ scatter in the distribution is comparable to the uncertainty of τ₁₆₀ of ∼⃒1 × 10⁻⁵, suggesting there is an intrinsically tight relationship. The variation in the relationship between ${N}_{{\rm{H}}{\rm{I}}}$ and τ₁₆₀ that is observed in nearby clouds (see below) could contribute to the observed scatter. The dust is also well correlated with the molecular gas traced by ¹²CO, which is shown for the SMC in Figure 4 in Lee et al. (2015a) using the HERITAGE and MAGMA data. We summarize the specific steps in our methodology, which closely follow the methodology by Leroy et al. (2009) for the SMC, but with improvements allowed by the increased IR coverage and resolution from Herschel.

Zoom In Zoom Out Reset image size

Following Leroy et al. (2009) and Bolatto et al. (2011), we model the dust emission in order to obtain the optical depth of the dust emission at 160 μm (τ₁₆₀). We use the results from two different dust emission fitting techniques for the LMC, one presented in this paper and another from Gordon et al. (2014), both based on the assumption of modified blackbody emission, ${S}_{\nu }\propto {\nu }^{\beta }{B}_{\nu }({T}_{d})$. We describe the fitting techniques in more detail in Appendix A. For the SMC, we only produce one molecular gas map using the modeling results from Gordon et al. (2014) since Bolatto et al. (2011) produced a molecular gas map using a fixed β simple modified blackbody model and a similar methodology. The Gordon et al. (2014) dust modeling may produce a more accurate measure of T_d since it allows β to vary while reducing the amount of degeneracy between T_d and β (Dupac et al. 2003; Shetty et al. 2009) by accounting for the correlated errors between the Herschel bands.

While the dust temperature along the line-of sight throughout the Magellanic Clouds likely has a distribution of temperatures (Bernard et al. 2008; Galliano et al. 2011; Galametz et al. 2013), the assumption of a single dust temperature on the small spatial scales we cover (∼⃒20 pc) is reasonable since temperature mixing is restricted. Leroy et al. (2011) ran both simple modified blackbody fits and more complex dust models from Draine & Li (2007) to find τ₁₆₀ using the Spitzer data for the LMC and SMC, and found that both produced similar results. A future follow-up study of Gordon et al. (2014) will run more complex dust modeling of the HERITAGE Herschel data.

This study focuses on using dust emission as a means to estimate the amount of molecular gas, which does not require a measurement of the dust mass. By only using τ₁₆₀ we avoid making any assumptions about the conversion to dust mass, which would introduce a further layer of uncertainty. We define our effective gas-to-dust (${\delta }_{{\rm{GDR}}}$) ratio in terms of τ₁₆₀,

$$\begin{eqnarray*}&&{\delta }_{{\rm{GDR}}}={{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}/{\tau }_{160},\end{eqnarray*}$$

such that any proportionality constant between the IR intensity and τ₁₆₀ will be incorporated into ${\delta }_{{\rm{GDR}}}$ and not affect our final results.

We expect, in principle, that the relationship between ${N}_{{\rm{H}}{\rm{I}}}$ and τ₁₆₀ should go through the origin, but our measurements show indications of an offset (see Figure 1). We regionally fit and then remove the offset and find that the relationship has a positive and roughly constant offset in ${N}_{{\rm{H}}{\rm{I}}}$ in both the LMC (${N}_{{\rm{H}}{\rm{I}}}\sim 4\times {10}^{20}$ cm⁻²) and SMC (${N}_{{\rm{H}}{\rm{I}}}\sim 1.5\times {10}^{21}$ cm⁻²). A similar offset was observed by Leroy et al. (2011), Bolatto et al. (2011), and Roman-Duval et al. (2014). As opposed to Bolatto et al. (2011), we remove the offset to avoid overestimates when creating maps of the gas-to-dust ratios, which would result in higher estimates of the total amount of gas. This offset could be due to a layer of H i gas with little to no dust, it could be due to the issues with background subtraction with the Herschel images (particularly in the LMC where the HERITAGE maps to not extend much past the main part of the galaxy), or some combination of the two effects. Another possibility is that the relationship between ${N}_{{\rm{H}}{\rm{I}}}$ and τ₁₆₀ is nonlinear and the slope (gas-to-dust ratio) decreases at low ${N}_{{\rm{H}}{\rm{I}}}$, which we explore as part of the systematic uncertainty estimation (see Sections 3.1.1 and 4.1.2). Determining the true nature of the offset is beyond the scope of this work, but warrants further investigation. We subtract the offset in ${N}_{{\rm{H}}{\rm{I}}}$ from the H i map and use the offset-subtracted map for the rest of the analysis. For further discussion on the offset subtraction see Appendix B.

Steps to produce molecular gas map:

• 1.  
  Model the dust emission in the Herschel images to obtain τ₁₆₀ (see Appendix A for more details).
• 2.  
  Fit the H i offset in the ${N}_{{\rm{H}}{\rm{I}}}$ versus τ₁₆₀ distribution regionally (see Appendix B for more details).
• 3.  
  Produce first iteration map of the spatially varying effective gas-to-dust ratio (${\delta }_{{\rm{GDR}}}$) at 500 pc scales determined from the diffuse regions (${{\rm{\Sigma }}}_{{\rm{gas}}}={{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}$)

  • a.  
    Compute ${\delta }_{{\rm{GDR}}}$ for each pixel.
  • b.  
    Mask all pixels that likely have molecular gas: all regions within 2' of bright CO emission (I_CO > 3σ).
  • c.  
    Use averaging of nearest neighbors to iteratively fill in the masked (molecular) regions in the map.
  • d.  
    Convolve map with symmetric Gaussian with FWHM = 500 pc.
• 4.  
  Estimate ${{\rm{\Sigma }}}_{{\rm{mol}}}$ using the first iteration of the smoothed effective ${\delta }_{{\rm{GDR}}}$:

  $$\begin{eqnarray*}&&{{\rm{\Sigma }}}_{{\rm{mol}}}=({\delta }_{{\rm{GDR}}}{{\rm{\Sigma }}}_{\mathrm{dust}})-{{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}.\end{eqnarray*}$$
• 5.  
  Produce second iteration of map of spatially varying ${\delta }_{{\rm{GDR}}}$ smoothed to 500 pc. Same as step 4 with the modification that both regions within 2' of bright CO emission (I_CO > 3σ) and points that have estimated ${{\rm{\Sigma }}}_{{\rm{mol}}}\gt 0.5{{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}$ are masked.
• 6.  
  Produce final map of ${{\rm{\Sigma }}}_{{\rm{mol}}}$ map using the second iteration of the smoothed ${\delta }_{{\rm{GDR}}}$ map.

The final steps in producing the molecular gas maps remove unphysical artifacts. First, we remove small regions of estimated ${{\rm{H}}}_{2}$ that are likely spurious by masking pixels that have positive molecular gas in less than 50% of the pixels surrounding them within a 4' × 4' box (12 × 12 pixels in the modified blackbody map from this work and 4 × 4 pixels in the maps from Gordon et al. (2014); ∼⃒60 × 60 pc in the LMC and ∼⃒70 × 70 pc in the SMC). Generally, this removes emission smaller than ∼⃒2' (r ∼⃒ 30 pc in the LMC and r ∼⃒ 35 pc in the SMC)–two times the beam size of the lower resolution H i data—and regions of negative values (from underestimated total gas). Second, we median-filter the map over 3 pixels (∼⃒1' in the LMC map from this work) to smooth out the ${{\rm{\Sigma }}}_{{\rm{mol}}}$ map and remove spikes that are unphysical and below the resolution of the H i map, largely due to the residual striping from the HERITAGE PACS images (Meixner et al. 2013).

There are a few caveats to this methodology that can potentially bias our molecular gas estimate. In addition to tracing the molecular gas (including any "CO-faint" component), our methodology may also trace optically thick and/or cold H i gas that emits disproportionately to the optically thin H i. Stanimirović et al. (1999) takes a statistical approach and estimates the optical depth correction in the SMC based on column density using the absorption line measurements from Dickey et al. (2000) and finds the correction only changes the total H i mass by ∼⃒10%. Lee et al. (2015b) takes a similar approach to estimate an optical depth correction in the Milky Way and finds that the correction only increases the mass of H i in the Perseus molecular cloud by ∼⃒10%. Braun (2012) attempted to measure the H i optical depth from the flattening of the line profile in M31, M33, and the LMC, and found non-negligible optical depth corrections for high column densities ($22\lt \mathrm{log}\ {{\rm{N}}}_{{\rm{H}}{\rm{I}}}\lt 23$) in compact (∼⃒100 pc) regions, which increases the total H i mass by ∼⃒30%. The Braun (2012) estimate relies on the assumption of Gaussian line profiles to look for flattening of the H i line due to optical depth, which is a difficult measurement in the low S/N data. McKee et al. (2015) find ∼⃒30% to be the appropriate H i optical depth correction for the solar neighborhood based on the average correction factors found using absorption line measurements in the plane of the Milky Way. Fukui et al. (2015), on the other hand, find more extreme opacity correction factors, as high as a factor of ∼⃒2 in the plane of the Milky Way using a relationship between ${N}_{{\rm{H}}{\rm{I}}}$ and the optical depth at 353 GHz from Planck. The possible H i opacity corrections coming from a variety of methods and data show that the factors are uncertain.

The manner in which the optical depth correction will affect our molecular gas estimates is complex. It can increase the H i column density in the regions used to estimate the gas-to-dust ratio, leading to an increase in the total gas estimated in the molecular regions, and/or in the molecular regions, resulting in a decrease in the amount of molecular gas. We choose to use the H i statistical opacity corrections from Stanimirović et al. (1999) and Lee et al. (2015b) to explore how correcting for optical depth effects our methodology in Section 4.1.3.

We note that, in Perseus where the structure of the molecular cloud is resolved, Lee et al. (2015b) compares their map of H i with the statistical optical depth correction to their inferred "CO-faint" gas, observing that the structures are not spatially coincident (see Figure 4 and Lee et al. 2015b). This suggests that the "CO-faint" gas cannot be explained by optically thick H i alone. Additionally, Lee et al. (2015b) comment that their opacity corrected H i map does not show the the sharp peaks seen in maps from Braun (2012).

Our methodology also relies on the assumption that the gas-to-dust ratio in the diffuse, atomic gas is the same in the molecular regions; we only measure the relationship between gas and dust in the atomic phase. There is observational evidence that the gas-to-dust ratio may vary from the diffuse to the dense gas in the Magellanic Clouds (Bot et al. 2004; Roman-Duval et al. 2014). In the Milky Way, Planck results show an factor of 2 increase in the far-IR dust optical depth per unit column density (τ₂₅₀/N_H) from the diffuse to the dense gas (Planck Collaboration et al. 2011), which would could indicate a lower gas-to-dust ratio in the dense gas. Both optically thick H i and a decrease in the gas-to-dust ratio from the diffuse to the dense gas would mimic the effect of molecular gas and would result in our methodology overestimating the amount of molecular gas. We explore how these factors could affect our measurement of ${{\rm{H}}}_{2}$ in our systematic uncertainty estimate.

3.1.1. Map Sensitivity and Uncertainty

We use a Monte Carlo method to estimate the uncertainty in our molecular gas maps and determine the sensitivity levels. For the maps produced with the dust fitting from this work, we select three sub-regions (shown in Figure 2) with different levels of molecular gas (high, moderate, and low). We add normally distributed noise with an amplitude equal to the uncertainty to each of the Herschel bands and fit T_d for a fixed β for each sub-region and then calculate τ₁₆₀. For the dust modeling results from Gordon et al. (2014), we add normally distributed noise to the τ₁₆₀ maps with an amplitude equal to the uncertainty estimates from Gordon et al. (2014). Finally, we add noise to the H i map and create new ${{\rm{\Sigma }}}_{{\rm{mol}}}$ maps. The process is repeated 100 times for each of the different maps. We use the distribution of ${{\rm{\Sigma }}}_{{\rm{mol}}}$ for each pixel from the Monte Carlo realizations to estimate a realistic uncertainty. The sensitivity of the maps is estimated by finding the lowest ${{\rm{\Sigma }}}_{{\rm{mol}}}$ that is consistently recovered at ≧̸2σ.

Zoom In Zoom Out Reset image size

We know that the systematic uncertainty from the methodology will dominate the uncertainty in our molecular gas maps (Leroy et al. 2009; Bolatto et al. 2011). To estimate the level of systematic uncertainty, we see how changes to various aspect of the mapping methodology affect the estimated total molecular mass ${M}_{{\rm{mol}}}$ (which includes the factor of 1.4 to account for He). We explore the effects of different assumptions in the dust modeling and determination of the gas-to-dust ratio, and produces map that:

• 1.  
  change the value of β in our dust modeling and re-run the fitting with β = 1.5 and β = 2.0;
• 2.  
  do not remove an H i offset, which explores the idea that the relationship between ${N}_{{\rm{H}}{\rm{I}}}$ and τ₁₆₀ may not be linear at low column densities;
• 3.  
  apply a single gas-to-dust ratio using the high and low values from Roman-Duval et al. (2014) (as opposed to using the map of ${\delta }_{{\rm{GDR}}}$);
• 4.  
  scale the ${\delta }_{{\rm{GDR}}}$ map down by a factor of 2 in the molecular regions to account for a possible change in the gas-to-dust ratio from the diffuse to the dense gas, where we define the dense gas as regions in the map that are likely to have molecular gas (step 5 in Section 3.1);
• 5.  
  apply a single gas-to-dust ratio for the diffuse gas and a lower value for the dense gas using the values from Roman-Duval et al. (2014), where the dense gas value is applied to regions with bright CO emission (as in Roman-Duval et al. 2014).

For the versions of the maps where we use gas-to-dust ratios found in Roman-Duval et al. (2014), we use the maps of ${{\rm{\Sigma }}}_{\mathrm{dust}}$ in place of τ₁₆₀. We use the range in ${M}_{{\rm{mol}}}$ values to estimate the amount of systematic uncertainty in our molecular gas estimate.

3.1.2. Estimating H₂ from CO

For the purposes of this work, we want to compare the amount of ${{\rm{H}}}_{2}$ traced by detected, bright ¹²CO emission to the molecular gas traced by the dust emission. To convert the CO intensity (I_CO) into column density of mass, we use the following equations:

$$\begin{eqnarray}&&N({{\rm{H}}}_{2})={X}_{{\rm{CO}}}\;{I}_{{\rm{CO}}}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{M}_{{\rm{mol}}}={\alpha }_{{\rm{CO}}}\;{L}_{{\rm{CO}}},\end{eqnarray} \tag{ 2 }$$

where proportionality constants appropriate for Galactic gas are X_CO = 2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ and ${\alpha }_{{\rm{CO}}}=4.3\;\;{M}_{\odot }$ (K km s⁻¹ pc²)⁻¹, I_CO is the integrated intensity of the ¹²CO $J=1\to 0$ transition (in K km s⁻¹), and L_CO is the luminosity of the same transition (in K km s⁻¹ pc²). On small spatial scales and in CO-bright regions, using the Galactic values is a good approximation (Bolatto et al. 2008).

We use ${\rm{H}}\alpha$, locally corrected for extinction using 24 μm emission, to trace the star formation rate surface density (Σ_SFR). Following Bolatto et al. (2011), we use the star formation rate calibration by Calzetti et al. (2007) to convert ${\rm{H}}\alpha$ and 24 μm luminosities:

$$\begin{eqnarray}{\rm{SFR}}(\;{M}_{\odot }\;{\mathrm{yr}}^{-1}) & = & 5.3\times {10}^{-42}[L({\rm{H}}\alpha )\\ & & +\;(0.031\pm 0.006)L(24\;\mu {\rm{m}})],\end{eqnarray} \tag{ 3 }$$

where luminosities are in erg s⁻¹ and L(24 μm) is expressed as νL(ν). The average contribution from 24 μm to the total star formation rate is ∼⃒20% in the LMC and ∼⃒10% in the SMC. A significant fraction (∼⃒40%) of the ${\rm{H}}\alpha$ emission in both the LMC and SMC is diffuse. We include all of the ${\rm{H}}\alpha$ emission in this analysis since Pellegrini et al. (2012) showed that all of the ionizing photons could have originated from H ii regions from massive stars (see Appendix C for further discussion). The rms background value of the star formation rate map is 1 × 10⁻⁴ $\;{M}_{\odot }\;{{\rm{yr}}}^{-1}\;{{\rm{kpc}}}^{-2}$ in the LMC and 4 × 10⁻⁴ $\;{M}_{\odot }\;{{\rm{yr}}}^{-1}\;{{\rm{kpc}}}^{-2}$ in the SMC.

This conversion to star formation rate assumes an underlying broken power-law Kroupa initial mass function (IMF) and was calibrated against Paschen-α emission for individual star-forming regions. Ideally, ${\rm{H}}\alpha$ and 24 μm emission would only be used for size scales that fully sample the IMF and sustain star formation for >10 Myr; for smaller scales, pre-main sequence stars are more appropriate and a better indicator of the current star formation rate. Hony et al. (2015) found that the star formation rate from pre-main sequence stars matches that from ${\rm{H}}\alpha$ at scales of ∼⃒150 pc in the N66 region in the SMC. Our highest resolution of ∼⃒20 pc resolves H ii regions, and the mapping of the star formation rate on these scales is questionable. Nonetheless, we apply the star formation rate conversion even to our highest resolution data to allow us to compare to other studies and investigate the relationships in terms of a physical quantity, although it is important to keep these limitations in mind when interpreting the results.

To produce the lower resolution molecular gas maps, we first convolve the maps from the Herschel beam to a Gaussian with FWHM of 30'' for the β = 1.8 map (appropriate for the 350 μm image resolution) and 40'' for the BEMBB map (appropriate for the 500 μm image resolution) using the kernels from Aniano et al. (2011). We then produce the range of lower resolution maps (from 20 pc to ∼⃒1 kpc) by convolving the images of ${{\rm{\Sigma }}}_{{\rm{SFR}}}$, ${{\rm{\Sigma }}}_{{\rm{mol}}}$, and ${{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}$ with a Gaussian kernel with FWHM $=\quad \sqrt{({r}^{2}-{r}_{0}^{2})}$, where r is the desired resolution and r₀ is the starting resolution of the image. The images are then resampled to have approximately independent pixels (one pixel per resolution element). To mitigate edge effects from the convolution, we remove the outer two pixels (two beams) for all resolution images of the LMC. In the SMC, we remove two outer pixels for r ≦̸ 600 pc and remove one pixel from the edges for r ≧̸ 700 pc due to the small size of the images.

We find molecular gas fractions that are comparable to the Milky Way in the LMC (17%), but much lower in the SMC (3%). These molecular gas fractions come from our new estimates of the total molecular gas mass: we find a total molecular gas mass (including He) in the LMC of ${M}_{\mathrm{LMC}}^{\mathrm{mol}}={6.3}_{-3.2}^{+6.3}\times {10}^{7}$ $\;{M}_{\odot }$ and ${M}_{\mathrm{SMC}}^{\mathrm{mol}}={1.3}_{-0.65}^{+1.3}\times {10}^{7}$ $\;{M}_{\odot }$ in the SMC. These values are the sums (with no cuts) of our fiducial molecular gas maps that use the broken emissivity modified blackbody (BEMBB) dust modeling results from Gordon et al. (2014) with a spatially varying ${\delta }_{{\rm{GDR}}}$ and include a factor of 2 systematic uncertainty. Table 1 shows the results from our exploration of varying the map making methodology to estimate the systematic uncertainty combined with estimates of the molecular gas mass from the literature. In Table 2 we list the integrated properties of both galaxies. The molecular gas maps are sensitive to ${{\rm{\Sigma }}}_{{\rm{mol}}}\sim 15$ $\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$ (∼⃒7 × 10²⁰ cm⁻²) based on the Monte Carlo estimates, which is comparable to the sensitivity of the SMC map from Bolatto et al. (2011). Our molecular gas fraction in the SMC is lower than previous estimates (Leroy et al. 2007; Bolatto et al. 2011), but is consistent with the factor of ∼⃒2 estimate of systematic uncertainty for all of the estimates (see Appendix D for further discussion).

Table 1.  Total Molecular Gas Mass Estimates for the LMC and SMC

	Data	Dust Fitting a	Method	${M}_{{\rm{mol}}}$ [107 $\;{M}_{\odot }$] b
LMC
1	Herschel 100–350 μm	MBB, β = 1.8	${\delta }_{{\rm{GDR}}}$ map c	9.9
2	Herschel 100–500 μm	BEMBB, 0.8 < β < 2.5	${\delta }_{{\rm{GDR}}}$ map c	6.3
3	Herschel 100–350 μm	MBB, β = 1.5	${\delta }_{{\rm{GDR}}}$ map c	6.8
4	Herschel 100–350 μm	MBB, β = 2.0	${\delta }_{{\rm{GDR}}}$ map c	10.1
5	Herschel 100–500 μm	BEMBB, 0.8 < β < 2.5	${\delta }_{{\rm{GDR}}}$ map c , no H i offset	13.4
6	Herschel 100–500 μm	BEMBB, 0.8 < β < 2.5	GDR = 540 d	3.9
7	Herschel 100–350 μm	MBB, β = 1.8	${\delta }_{{\rm{GDR}}}$ map c , ${\delta }_{{\rm{GDR,dense}}}=0.5{\delta }_{{\rm{GDR,map}}}$	4.5
8	Herschel 100–500 μm	BEMBB, 0.8 < β < 2.5	${\delta }_{{\rm{GDR}}}$ map c , ${\delta }_{{\rm{GDR,dense}}}=0.5{\delta }_{{\rm{GDR,map}}}$	4.0
SMC
9	Herschel 100–500 μm	BEMBB, 0.8 < β < 2.5	${\delta }_{{\rm{GDR}}}$ map c	2.0

Notes.

^a MBB = modified blackbody, BEMBB = broken emissivity modified blackbody. ^b Assuming d_LMC = 50 kpc and d_SMC = 62 kpc. ^c Map of spatially varying ${\delta }_{{\rm{GDR}}}$, see Section 3.1. ^d Does not include factor of 1.36 contribution from helium.

Download table as:  ASCIITypeset image

Table 2.  Global Properties

Property	LMC	SMC
${M}_{{\rm{mol}}}^{{\rm{dust}}}$	${6.3}_{-3.2}^{+6.3}\times {10}^{7}$ $\;{M}_{\odot }$	${2.0}_{-1.0}^{+2.0}\times {10}^{7}$ $\;{M}_{\odot }$
LCO	7 × 106 K km s−1 a	1.7 × 105 K km s−1 b
${M}_{{\rm{H}}{\rm{I}}}$	4.8 × 108 $\;{M}_{\odot }$ c	3.8 × 108 $\;{M}_{\odot }$ d
${M}_{* }$	2 × 109 $\;{M}_{\odot }$ e	3 × 108 $\;{M}_{\odot }$ f
SFR d	0.20 $\;{M}_{\odot }$ yr−1	0.033 $\;{M}_{\odot }$ yr−1
${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$	${0.37}_{-0.19}^{+0.37}$ Gyr	${0.61}_{-0.31}^{+0.61}$ Gyr

Notes.

^a Fukui et al. (2008), no sensitivity cuts. ^b Mizuno et al. (2001), no sensitivity cuts. ^c Staveley-Smith et al. (2003). ^d Stanimirović et al. (1999). ^e Skibba et al. (2012). ^f Assuming A_V (LMC) = 0.2 mag and A_V (SMC) = 0.1 mag.

Download table as:  ASCIITypeset image

The fiducial molecular gas maps (β = 1.8, broken emissivity modified blackbody (BEMBB) dust modeling) were produced using maps of the effective dust-to-gas ratio (${\delta }_{{\rm{GDR}}}$) that had average values for ${N}_{{\rm{H}}{\rm{I}}}/{\tau }_{160}$ of 1.8 ± 0.6 × 10²⁵ cm⁻² (LMC β = 1.8), 1.3 ± 0.3 × 10²⁵ cm⁻² (LMC BEMBB), and 4.8 ± 0.9 × 10²⁵ cm⁻² (SMC BEMBB). In the Milky Way, Planck Collaboration et al. (2014) found ${N}_{{\rm{H}}{\rm{I}}}/{\tau }_{160}=1.1\times {10}^{25}$ cm⁻² in the diffuse ISM. Our ${N}_{{\rm{H}}{\rm{I}}}/{\tau }_{160}$ values are on average a factor of ∼⃒1.5 (LMC) and ∼⃒4.4 (SMC) times higher than the diffuse Milky Way ISM, which is consistent with the expectation that the gas-to-dust ratio should increase with decreasing metallicity.

Given the total NANTEN CO luminosities of $L{(\mathrm{CO})}_{\mathrm{LMC}}=7\times {10}^{6}$ K km s⁻¹ pc² and $L{(\mathrm{CO})}_{\mathrm{SMC}}=1.7\quad \times {10}^{5}$ K km s⁻¹ pc² (using no sensitivity cuts), we find ${\alpha }_{\mathrm{CO}}^{\mathrm{LMC}}={10}_{-6}^{+9}$ and ${\alpha }_{\mathrm{CO}}^{\mathrm{SMC}}={76}_{-38}^{+77}$, where all units for α_CO are given in $\;{M}_{\odot }$ (K km s⁻¹ pc²)⁻¹. Compared to the Milky Way value of α_CO = 4.3 (Bolatto et al. 2013), the conversion factor for the LMC is ∼⃒2 times higher and the SMC is ∼⃒17 times higher. Our α_CO values are comparable to the dust-based α_CO found by Leroy et al. (2011) of 6.6 (K km s⁻¹ pc²)⁻¹ and 53–85 (K km s⁻¹ pc²)⁻¹ for the LMC and SMC, respectively.

4.1.1. Structure of the Molecular Gas

One of the most striking results is the similarity of the structure of the molecular gas traced by dust to that traced by CO throughout the entire LMC (see Figures 2 and 4) and SMC (see Figure 3). Since our methodology only indirectly uses the CO map as a mask (see Section 3.1) the similarity is confirmation that our methodology traces the structure of the gas. Figure 4 shows that both dust modeling techniques produce maps with similar structure, although the BEMBB map tends to predict systematically lower amounts of ${{\rm{H}}}_{2}$.

Zoom In Zoom Out Reset image size

The details of the structures traced by CO are different from the dust-based molecular gas map. All of the regions shown in Figure 4 show molecular gas traced by dust, but not by CO at the 3σ level. This is likely a layer of self-shielded ${{\rm{H}}}_{2}$ where CO has mostly dissociated, as expected from models (Wolfire et al. 2010; Glover & Mac Low 2011). The same is generally true for the SMC, but having only the lower resolution full coverage NANTEN ¹²CO map (r = 26) makes detailed comparison of the structure difficult. Conversely, Region 2 in Figure 4 shows a molecular gas cloud traced by CO and not by the dust-based method. As discussed in Leroy et al. (2009), one possible explanation is that the dust is cold and faintly emitting in the far-IR, below the sensitivity of the HERITAGE Herschel images. The peak in the CO emission of this cloud is detected from 250–500 μm, but only weakly detected at 160 μm and marginally detected (∼⃒3σ) at 100 μm, consistent with the interpretation of cold dust. There are a few other detections of CO without a dust-based molecular gas counterpart, although the cloud in Region 2 is the clearest example with the strongest CO emission.

Zoom In Zoom Out Reset image size

4.1.2. Systematic Uncertainty

4.1.3. Estimating the Effect of the Optical Depth of H i

4.1.4. Comparison to Previous Work

Understanding whether or not metallicity and galaxy mass affect the conversion of gas into stars is important for understanding galaxy evolution throughout cosmic time. The relationship between molecular gas and star formation rate has been studied extensively in nearby, high-metallicity, star-forming galaxies. With the dust-based molecular gas estimates of the nearby Magellanic Clouds, we are in a unique position to probe how the relationship between the molecular gas and star formation rate behave as a function of metallicity and the size scale considered. Figure 5 shows the relationships for the LMC and SMC using the new dust-based molecular map at the highest resolution of 20 pc, 200 pc (the scale where multiple star-forming regions are being averaged), and 1 kpc (comparable to the ¹²CO surveys of nearby galaxies).

Zoom In Zoom Out Reset image size

We compare the relationship between the molecular gas and the star formation rate in the SMC and LMC to that for the HERACLES sample of nearby galaxies by Leroy et al. (2013b). The HERACLES sample resolves the galaxies and compares the gas and star formation at a resolution of ∼⃒1 kpc. Figure 6 shows that the LMC and SMC data (convolved to a comparable resolution of 1 kpc) lie within the scatter in the data for high-metallicity, star-forming galaxies, although above the main cluster of data points for a given molecular gas surface density.

Zoom In Zoom Out Reset image size

4.2.1. Molecular Gas Depletion Time

A convenient way to quantify the relationship between molecular gas and star formation is in terms of the amount of time it would take to deplete the current reservoir of gas given the current rate of star formation, the molecular gas depletion time:

$$\begin{eqnarray}&&{\tau }_{{\rm{dep}}}^{{\rm{mol}}}={{\rm{\Sigma }}}_{{\rm{mol}}}/{{\rm{\Sigma }}}_{{\rm{SFR}}}.\end{eqnarray} \tag{ 4 }$$

The data for the LMC and SMC appear consistent with a well-defined depletion time. We find average molecular gas depletion times at 1 kpc scales of ∼⃒0.4 Gyr in the LMC and ∼⃒0.6 Gyr in the SMC. Weighting the average of ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ by the molecular gas mass and star formation rate does not significantly affect the averages at 1 kpc scales; at 200 pc scales, weighting of the average typically changes the value of ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ by ∼⃒20%. The main exception is for the star formation rate weighted ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ average in the LMC, which is shorter by ∼⃒50% and likely due to the significant contribution of 30 Doradus at these scales. The range of possible molecular gas depletion times given the factor of up to ∼⃒2 systematic uncertainty in the molecular gas estimate is ∼⃒0.2–1.2 Gyr. This is shorter than the molecular gas depletion time found for the SMC by Bolatto et al. (2011) of ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}\sim 1.6$ Gyr at 1 kpc resolution, but within the factor of 2 systematic uncertainty on both estimates. The molecular gas depletion time found in the Magellanic Clouds is lower than the average value of ∼⃒2 Gyr for nearby normal disk galaxies at comparable ∼⃒1 kpc size scales, but within the range of observed values for the STING sample (Rahman et al. 2012) and the larger HERACLES sample (Bigiel et al. 2008, 2011; Leroy et al. 2013b).

Figure 7 shows that the median ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ is ∼⃒2–3 Gyr at the highest resolution of 20 pc. The molecular gas depletion time changes with resolution because the peaks in the molecular gas are physically separated from the peaks in the star formation rate at scales where the star-forming regions are spatially resolved. The tendency of low star formation rates at the peaks in the molecular gas, and low to no molecular gas at the peaks in the star formation rate (${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ is only defined for regions with ${{\rm{\Sigma }}}_{{\rm{mol}}}$) biases ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ at high resolutions toward longer times. A scale of 200 pc is typically large enough to include both the recent star formation and the molecular gas and sample star-forming regions at a range of evolution stages (Schruba et al. 2011). While the molecular gas depletion time gets closer to the integrated ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ value at a scale of 200 pc, the median ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ reaches the integrated value at ∼⃒500 pc in the LMC and SMC.

Zoom In Zoom Out Reset image size

The lower metallicities of the SMC and LMC and the lack of a metallicity bias in our dust-based molecular gas estimate allow us to investigate whether there is any trend in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ with metallicity. Figure 9 shows that there is no clear trend in the average molecular gas depletion times when comparing the LMC and SMC to the HERACLES sample of galaxies. Leroy et al. (2013b) also saw no trend with metallicity as long as they allowed for a variable CO-to-${{\rm{H}}}_{2}$ conversion factor. We also compare our measurements to the integrated ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ using a metallicity-dependent CO-to-${{\rm{H}}}_{2}$ conversion factor for the Herschel Dwarf Galaxy Survey (DGS; Cormier et al. 2014). Over the range of metallicities studied, the main cause of variations in the molecular gas depletion time does not appear to be metallicity.

We use the Spearman's rank correlation coefficient to quantitatively gauge how well the gas correlates with star formation rate at different size scales. Spearman's rank correlation coefficient (r_s ) measures the degree to which two quantities monotonically increase (r_s > 0) or decrease (r_s < 0). We computed the 3σ confidence intervals using the Fisher z-transformation, which is appropriate for bivariate normal distributions. Figure 8 shows the rank correlation coefficient as a function of resolution for the relationship between star formation rate and molecular gas and atomic gas.

Zoom In Zoom Out Reset image size

The change in the rank correlation coefficient with resolution is similar for both the LMC and SMC. As expected for atomic-dominated galaxies, the correlation of ${{\rm{\Sigma }}}_{{\rm{gas}}}$ versus ${{\rm{\Sigma }}}_{{\rm{SFR}}}$ follows that of ${{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}$ and ${{\rm{\Sigma }}}_{{\rm{SFR}}}$, therefore we only show ${{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}$ versus ${{\rm{\Sigma }}}_{{\rm{SFR}}}$ in Figure 8. The correlation between ${{\rm{\Sigma }}}_{{\rm{H}}{\rm{I}}}$ versus ${{\rm{\Sigma }}}_{{\rm{SFR}}}$ in both the LMC and SMC is high (r_s ∼⃒ 0.6–0.7) at the smallest size scale of 20 pc and remains high across all size scales. The correlation of H i with the star formation rate, even at small spatial scales, is due to the extended nature of both components combined with the general trend that regions with more total gas have more star formation and more molecular gas.

The ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ versus ${{\rm{\Sigma }}}_{{\rm{SFR}}}$ distribution reaches the maximum correlation coefficient of r_s ∼⃒ 0.9 at a size scales ∼⃒200 pc, past which it is better correlated than the relationship with H i. While the H i is correlated with the star formation rate tracer, we see that molecular gas is best correlated with recent star formation in the LMC and SMC at size scales ⪆200 pc. The 200 pc scale indicates the average size scale where both molecular gas and the star formation rate tracer, ${\rm{H}}\alpha$, are found together and enough independent star-forming regions at different evolutionary stages (i.e., different ratios of ${\rm{H}}\alpha$ to molecular gas) are averaged together. While the correlation peaks at 200 pc, the average molecular gas depletion time decreases until it reaches the integrated value at a size scale of ∼⃒500–700 pc in the LMC and SMC. The molecular gas and star formation rate tracer have a strong positive correlation, stronger than that with H i, supporting the physical connection between molecular gas and recent massive star formation.

The range of possible molecular gas depletion times for the LMC and SMC at 1 kpc scales given the systematic uncertainty in our estimate of the molecular gas of ∼⃒0.2–1.2 Gyr falls below the average ∼⃒2 Gyr found for nearby normal disk galaxies. This is consistent with the previous work by Bolatto et al. (2011) that found ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}=1.6\;{\rm{Gyr}}$ at 1 kpc scales in the SMC using similar dust-based molecular gas estimates, with the value being higher due to a higher estimate of the molecular gas. The shorter molecular gas depletion times do not appear to be directly due the lower metallicities as there is no trend in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ with metallicity (see Figure 9).

Zoom In Zoom Out Reset image size

The other remaining environmental factors, besides metallicity, that could affect the ratio of the amount of molecular gas to the amount of current star formation are the lower galaxy masses of the Magellanic Clouds and the interaction between the LMC, SMC, and Milky Way (Besla et al. 2012). Lower mass galaxies tend to have lower dark matter and stellar densities, making them more susceptible to stochastic bursts of star formation. Both the star formation histories of the SMC and LMC (Harris &#38; Zaritsky 2004, 2009) indicate that there have been recent bursts in star formation in both galaxies. A burst in star formation over a short period of time could lead to a depletion of the molecular gas reservoir combined with higher star formation rates that together can produce low ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ values.

The molecular gas depletion time in M33 is ∼⃒0.5 Gyr when the diffuse ${\rm{H}}\alpha$ emission is included (Schruba et al. 2010), which is comparable to our measurements of the Magellanic Clouds. If the diffuse ionized gas (DIG) is removed from the ${\rm{H}}\alpha$ emission, then the molecular gas depletion time increases to ∼⃒1 Gyr. This highlights the importance of understanding the connection between the DIG and recent massive star formation as it represents a significant fraction of the ${\rm{H}}\alpha$ emission and changes ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$. Rahman et al. (2011) found a similar increase in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ by a factor of ∼⃒2 when the DIG was removed in the disk galaxy NGC 4254. If the diffuse ionized component is excluded in the star formation rate determination, the ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ in M33, LMC, and SMC is ∼⃒1 Gyr.

Like the Magellanic Clouds, M33 is low-mass, atomic-dominated, and has likely interacted with M31 within the past 0.5–2 Gyr (Davidge &#38; Puzia 2011). The LMC, SMC, and M33 show evidence for bursts in the star formation history within the last Gyr and the most recent epochs show lower star formation rates, which suggest that the star-forming gas reservoir has been depleted. The observed shorter depletion times appear to be caused by catching these galaxies after a period of higher star formation rate and does not necessarily indicate that these low-mass, low-metallicity galaxies are forming stars differently from normal disk galaxies.

Saintonge et al. (2011) also found that for the volume-limited COLD GASS survey, lower stellar mass galaxies (∼⃒10¹⁰ $\;{M}_{\odot }$) had shorter depletion times of ∼⃒0.5 Gyr. While consistent with the integrated depletion times in the LMC and SMC, the data are not completely comparable since a value for the CO-to-${{\rm{H}}}_{2}$ conversion factor has to be assumed and single-dish CO observations from the COLD GASS survey will mainly detect the central regions of the galaxies. Saintonge et al. (2011) conjecture that the shorter depletion time is due to the tendency for smaller galaxies to have more "bursty" star formation. Similarly, Cormier et al. (2014) suggest that the observed short molecular gas depletion depletion times for their DGS sample of dwarf galaxies are due to recent bursts in star formation. Kauffmann et al. (2003) found that low redshift galaxies with stellar mass <3 × 10¹⁰ $\;{M}_{\odot }$ in the Sloan Digital Sky Survey have younger stellar populations and that the star formation histories are correlated with the stellar surface density, also indicative of recent bursts in star formation as seen in the LMC, SMC, and M33. In Figure 10, we show the average ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ time as a function of the average stellar surface density (${{\rm{\Sigma }}}_{* }$) for the LMC, SMC, and the HERACLES sample of galaxies and see that all of the low molecular depletion times are found at low ${{\rm{\Sigma }}}_{* }$. The fact that low-mass galaxies are more susceptible to stochastic star formation can produce bursts in star formation (Hopkins et al. 2014) and lead to shorter molecular gas depletion times.

Zoom In Zoom Out Reset image size

The scatter in the ${{\rm{\Sigma }}}_{{\rm{mol}}}\mbox{--}{{\rm{\Sigma }}}_{{\rm{SFR}}}$ relationship, which we quantify in terms of the scatter in $\mathrm{log}{\tau }_{{\rm{dep}}}^{{\rm{mol}}}$, can be produced by both physical mechanisms and the imperfect nature of the observable tracers of the physical quantities. The previous observational work that focused on the scatter in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$, or the "break down" of the ${{\rm{\Sigma }}}_{{\rm{mol}}}\mbox{--}{{\rm{\Sigma }}}_{{\rm{SFR}}}$ relationship, by Schruba et al. (2010), Verley et al. (2010), and Onodera et al. (2010), studied the ${{\rm{\Sigma }}}_{{\rm{mol}}}-{{\rm{\Sigma }}}_{{\rm{SFR}}}$ relationship in M33 over ⪆100 pc size scales. Schruba et al. (2010) compared ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ found for apertures centered on CO peaks to apertures centered on ${\rm{H}}\alpha$ peaks for various aperture sizes from 75 to 1200 pc and found that the ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ values differed for CO and ${\rm{H}}\alpha$ peaks for ⪅300 pc size scales. There are a number of possible causes of the difference between the CO and ${\rm{H}}\alpha$ molecular gas depletions times: a difference in evolutionary stage of the star-forming region, drift of the young stars from their parent cloud, actual variation in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$, differences in how the observables map to physics quantities, and noise in the maps. Schruba et al. (2010) identify the evolution of individual star-forming regions as the likely cause for the variations.

At high resolution (scales of ∼⃒20–50 pc), the star formation and molecular gas are resolved into discrete regions that span a range evolutionary stages (e.g., Kawamura et al. 2009; Fukui &#38; Kawamura 2010) and have different ratios of molecular gas to star formation rate tracers. Averaging over larger size scales samples regions at a range of evolutionary stages resulting in a "time-averaged" ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$. The change in the scatter in the molecular gas depletion time (σ) with resolution informs us about whether the star-forming regions are spatially correlated due to synchronization of star formation by a large-scale process. We see this effect in Figure 5; the scatter in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ decreases as the size scales are increased.

Theoretical studies can be used to explore which mechanism produces the scatter in the ${{\rm{\Sigma }}}_{{\rm{mol}}}$–${{\rm{\Sigma }}}_{{\rm{SFR}}}$ relationship. We compare our results in the SMC and LMC to the hydrodynamical simulations of galaxies by Feldmann et al. (2011) and to the analytical model by Kruijssen &#38; Longmore (2014). Both provide predictions of the amount of scatter in the ${{\rm{\Sigma }}}_{{\rm{mol}}}$–${{\rm{\Sigma }}}_{{\rm{SFR}}}$ relationship for small size scales (50 pc for predictions from Kruijssen &#38; Longmore 2014, and 300 pc for the simulations from Feldmann et al. 2011) and how the scatter changes with size scale.

The simulations by Feldmann et al. (2011) show that the time-averaging of the star formation rate (or, our inability to measure the instantaneous star formation rate) combined with ${{\rm{\Sigma }}}_{{\rm{mol}}}$ estimates that are instantaneous alone can generate most of the scatter observed in the ${{\rm{\Sigma }}}_{{\rm{SFR}}}$–${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ relation. If we possessed a perfect, instantaneous tracer of the star formation rate, then we would expect to see high star formation rates while there is still a large amount of molecular gas. As the molecular gas is depleted and destroyed by the previous episode of star formation, both the molecular gas and star formation rate would decrease. Instead, we observe the tracers of the star formation rate (namely ${\rm{H}}\alpha$) peak when the molecular gas is partially or mostly dissipated because the tracers show the average star formation rate over up to ∼⃒10 Myr (Kennicutt &#38; Evans 2012). The time evolution of star-forming regions alone does not cause the observed offset between the observed star formation rate and molecular gas, rather the time-averaging of the star formation rate combined with the time evolution of star-forming regions produces different ratios of molecular gas to star formation rate and scatter in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$. Hony et al. (2015) show evidence of this effect in the N66 region of the SMC where the star formation rate from ${\rm{H}}\alpha$ disagrees with that from pre-main sequence stars at small (∼⃒6–150 pc) size scales.

Kruijssen &#38; Longmore (2014) quantify how the scatter in the ${{\rm{\Sigma }}}_{{\rm{gas}}}\mbox{--}{{\rm{\Sigma }}}_{{\rm{SFR}}}$ relationship should change with the size scale due to the incomplete statistical sampling of independent star-forming regions, including the effect of the different timescales associated with the gas and star formation tracers discussed by Feldmann et al. (2011), and add the additional scatter associated with incomplete sampling of star formation rate tracers from the initial stellar mass function (IMF), and the spatial drift between gas and stars. The model requires having an estimate of the lifetime of GMCs (t_gas), the timescale for the star formation rate tracer (t_stars), the time where both the gas and star formation rate overlap (t_over), the typical separation between independent star-forming regions (λ), the flux ratio between peaks in the overlap phase and in isolation for the gas and star formation (β₁, β₂), the scatter due to the time evolution of gas and star formation flux (${\sigma }_{{\rm{evol,1g}}}$, ${\sigma }_{{\rm{evol,1s}}}$), the scatter due to the mass spectrum (σ_MF), and the observational error (σ_obs). The predictions for the scale dependence of the scatter in the gas depletion time from Kruijssen &#38; Longmore (2014) agree with the predictions by Feldmann et al. (2011) at size scales >300 pc where the two are directly comparable. Kruijssen &#38; Longmore (2014) find that the scatter varies from ∼⃒0.9 dex at 50 pc scales to ∼⃒0.2 dex at 1 kpc. The trend in the prediction of scatter (valid for their fiducial parameter values for disks and dwarfs) is consistent with the observation from the HERACLES galaxies (Leroy et al. 2013b) and M33 (Schruba et al. 2010).

For the LMC and SMC predictions of the scatter from Kruijssen &#38; Longmore (2014), we use estimates of the input parameters based on observational data when possible. For both the LMC and SMC, we set t_stars = 6 Myr based on the lifetime of ${\rm{H}}\alpha$, make the assumption that β₁, β₂ = 1, and set λ = 150 pc, the typical Toomre length (for ${{\rm{\Sigma }}}_{{\rm{gas}}}\sim 10\;\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$ and Ω ∼⃒ 0.03 Myr⁻¹). In the LMC, we use the results of Kawamura et al. (2009) to set t_gas = 26 Myr and σ_MF = 0.4 dex (the mean logarithmic scatter of the Class I GMC mass). For the upper limit in the LMC, we take t_over = 0 Myr and ${\sigma }_{{\rm{evol,1g}}}$, ${\sigma }_{{\rm{evol,1s}}}=0.3$ dex, based on a linear time evolution to or from zero. For the lower limit in the LMC, we adopt t_over = 3 Myr (the supernova timescale) and ${\sigma }_{{\rm{evol,1g}}}$, ${\sigma }_{{\rm{evol,1s}}}=0.15$ dex; while it can vary from 0–0.3 dex, half the amount of scatter as linear evolution is a reasonable lower limit since the parameters must be >0 due to the existence of molecular clouds without massive stars and H ii regions without molecular clouds. We note that Kruijssen &#38; Longmore (2014) assume that the galactic star formation rate is roughly constant over the entire lifetime of the GMCs (∼⃒30 Myr), which stands in contrast to the multiple bursts over the past ∼⃒50 Myr identified in the star formation history of the both LMC (Harris &#38; Zaritsky 2009) and SMC (Harris &#38; Zaritsky 2004).

In the LMC, where the morphology is more clearly a disk and the metallicity is not much lower than solar metallicity, we observe scatter at the level of ∼⃒0.45 dex at ∼⃒100 pc and ∼⃒0.18 dex at ∼⃒ kpc scales. The Feldmann et al. (2011) simulations show that the behavior of the scatter in $\mathrm{log}{\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ with an averaging size scale from ∼⃒100–1000 pc for the solar metallicity and radiation field are remarkably similar to the observations for the LMC. The simulations from Feldmann et al. (2011) predict scatter of ∼⃒0.4–0.6 dex at ∼⃒100 pc scales and ∼⃒0.1–0.3 dex at ∼⃒ kpc scales for their fiducial solar metallicity simulations (across the range of their parameter exploration). The Kruijssen &#38; Longmore (2014) model produces a range in the predicted scatter in $\mathrm{log}{\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ in the LMC of 0.46–0.51 dex at 100 pc scales and 0.19–0.23 dex at 1 kpc scales, which are comparable to the results from the Feldmann et al. (2011) and close to the observed values for the LMC (see Figure 11). The dominant source of scatter at large (>100 pc) size scales for the lower limit predictions (closest to the observations) from the Kruijssen &#38; Longmore (2014) model comes from the Poisson statistics of the number of times each evolutionary phase of star formation is sampled, which is determined primarily by the timescale of the star formation rate tracer, the lifetime of GMCs, and the separation between star-forming regions. The similarity between our observations and both the Feldmann et al. (2011) simulation and Kruijssen &#38; Longmore (2014) model at large (>100 pc) size scales, where both are comparable and individual star-forming regions are unresolved, supports the interpretation that the scatter in the ${{\rm{\Sigma }}}_{{\rm{mol}}}-{{\rm{\Sigma }}}_{{\rm{SFR}}}$ relationship can be largely attributed to star formation rate tracers that time-average the "true" or instantaneous star formation rate.

Zoom In Zoom Out Reset image size

5.2.1. Scatter in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ as a Function of Size Scale

As a means to quantify the behavior of scatter with different size scales, Feldmann et al. (2011) fit a power law to the relationship between size scale and the scatter in $\mathrm{log}{\tau }_{{\rm{dep}}}^{{\rm{mol}}}$. Leroy et al. (2013b) used a subset of nearby HERACLES galaxies to study the scatter in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ at linear resolutions of 0.6–2.4 kpc, which we can compare to our results in the Magellanic Clouds spanning linear resolutions of 0.02–1 kpc. Following Feldmann et al. (2011) and Leroy et al. (2013b), we quantify the scale dependence of the scatter in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ in the LMC and SMC by

$$\begin{eqnarray*}&&\sigma (l)={\sigma }_{100}{\left(\displaystyle \frac{l}{100{\rm{pc}}}\right)}^{-\gamma }\end{eqnarray*}$$

where l is the spatial resolution, σ₁₀₀ is the scatter in $\mathrm{log}({\tau }_{{\rm{dep}}}^{{\rm{mol}}})$ at 100 pc resolution, and the power-law index γ measures the rate that the scatter changes with resolution (γ = 1 for uncorrelated star formation in a disk). ¹⁵ We fit only resolutions greater than 100 pc, since below that resolution $\mathrm{log}({\tau }_{{\rm{dep}}}^{{\rm{mol}}})$ will be biased by negative and zero values. Figure 11 shows how the scatter in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ changes with resolution, including the best fit power-law functions with γ = 0.43 for the LMC and γ = 0.24 for in the SMC. Leroy et al. (2013b) find a best fit γ for the scatter in $\mathrm{log}({\tau }_{{\rm{dep}}}^{{\rm{mol}}})$ in the range of 0–0.8 with an average of γ = 0.5 (shown by the thick dashed line in Figure 11).

If a galaxy has a fixed ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ and star formation proceeds randomly and independently in separate regions within the resolution element, then behaves like Poisson noise and $\sigma \propto \sqrt{{N}^{-1}}$, where N is the number of star-forming regions. For a region of size l, $N\propto {l}^{2}$ so that $\sigma \propto {l}^{-1}$ or a power-law scaling of γ = 1. Both Feldmann et al. (2011) and Kruijssen &#38; Longmore (2014) find that the scatter in $\mathrm{log}({\tau }_{{\rm{dep}}}^{{\rm{mol}}})$ scales with a rough power-law scaling with an index of γ = 0.5 at larger (⪆200 pc) scales. Feldmann et al. (2011) expect this shallow scaling as a result of star formation occurring in a 2D disk galaxy. However, Kruijssen &#38; Longmore (2014) find similar shallower slopes with uncorrelated, independent star-forming regions due to the contribution to the scatter from the time evolution of the star-forming regions and from the underlying distribution of GMC masses. The model from Kruijssen &#38; Longmore (2014) shows that the scatter due to Poisson noise dominates at large size scales that sample multiple star-forming regions (>100 pc). At small scales, the Poisson noise disappears due to the fact only one star-forming region will be sampled and the scatter from the time evolution of the star-forming regions and from the underlying distribution of GMC masses drives the variation in the ratio of star formation rate to molecular gas and ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$.

Figure 11 shows the data for the LMC and SMC, the HERACLES galaxies, and the corresponding Kruijssen &#38; Longmore (2014) model predictions. For the SMC model predictions, we set the upper limit to the upper limit values for the LMC and the lower limit the same as for the LMC but with t_gas = 10 Myr (approximate free-fall time of a GMC) and σ_MF = 0.2 dex (60% of the scatter in GMC masses in the LMC). We see general agreement between the trend in the observed relationship and the model predictions, however, the LMC data fall below the predicted lower limit (see Section 5.2) at large size scales (>200 pc). The most uncertain parameter in Kruijssen &#38; Longmore (2014), due to lack of observational constraints, is the scatter due to the time evolution of the gas flux and star formation rate flux (${\sigma }_{{\rm{evol,1g}}}$, ${\sigma }_{{\rm{evol,1s}}}$). Decreasing σ_evol further from 0.15 dex to 0.1 dex brings the model predictions much closer to the LMC observations at scales >100 pc.

If we apply the interpretation of Feldmann et al. (2011), the shallower decline in the amount of scatter with increasing averaging scale seen in the LMC (and SMC) could be caused by increased spatial correlation between individual star-forming regions. Correlation of star-forming regions, both spatially and temporally, would cause individual star-forming regions to be at similar evolutionary phases throughout large parts of the galaxy and could explain the need for a lower amount of scatter from the time evolution for the gas flux and star formation rate. Large-scale spatial correlation in star formation requires a physical mechanism to synchronize star formation, such as bursts of star formation throughout large parts of the galaxies driven by tidal interactions or ram pressure. The star formation histories of both the LMC and SMC indicated that there have been recent bursts of star formation throughout large parts of the galaxies, likely due to interaction between the galaxies and the Milky Way, and is possibly driving the shorter molecular gas depletion times. The lower amount of scatter in $\mathrm{log}{\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ at larger size scales observed in the LMC (and more tenuously in the SMC) could also be due to large-scale synchronization of star formation.

The Magellanic Clouds provide ideal laboratories to test models of star formation given their low metallicity. Although it is higher metallicity, the geometry of the LMC is better understood than the more irregular SMC. There are few nearby low-mass, low-metallicity systems and measuring their molecular gas content is challenging as they are often weakly emitting in CO and, when CO is observed, it is unclear what CO-to-${{\rm{H}}}_{2}$ conversion factor should be applied. The dust-based molecular gas estimates for the LMC and SMC allow us to test metallicity-dependent models of star formation at high resolution. We examine the model predictions from Ostriker et al. (2010, hereafter OML10) and Krumholz (2013, hereafter KMT+), a recent update of the Krumholz et al. (2009) model modified for atomic-dominated regions. Both models take the total gas surface density (${{\rm{\Sigma }}}_{{\rm{gas}}}$) and metallicity (Z') as input parameters and predict the fraction of molecular gas, and from that the star formation rate.

The OML10 model determines the star formation rate based on a balance between vertical gravity in the disk and the pressure of the diffuse ISM, which is controlled by star formation feedback. OML10 relates thermal pressure to ${{\rm{\Sigma }}}_{{\rm{SFR}}}$, whereas Ostriker &#38; Shetty (2011) relate turbulence to ${{\rm{\Sigma }}}_{{\rm{SFR}}}$. While the star-forming gas in the OML10 model is not strictly molecular gas, but rather bound clouds, we identify our estimate of ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ with the model parameter Σ_gbc, the surface density of gas in gravitationally bound clouds (note that in our methodology both H₂ and any optically thick H i are effectively indistinguishable). This ignores the fact that in very dense regions a significant fraction of the molecular gas could be not self-gravitating, a concern that is probably important in starburst environments but unlikely to matter in the Magellanic Clouds. The reverse concern, that H i may make a significant contribution to the cloud bounding mass, is likely a more significant consideration in these sources, although its magnitude is difficult to evaluate. The KMT+ model is based on the assumption that the fraction of molecular gas is mainly determined by the balance between the dissociating UV radiation field and the shielding of the gas. The KMT+ model adds to Krumholz et al. (2009) the condition that in a region with a low star formation rate, hence a low UV field, the threshold density of the cold neutral medium is no longer set by two-phase equilibrium between the cold and warm neutral medium, but rather by hydrostatic equilibrium.

Both the KMT+ and OML10 models use the mid-plane pressure, which requires an estimate of the density of stars and dark matter in the disk to determine the gravitational pressure. We estimate the stellar surface density by applying the mass-to-light conversion from Leroy et al. (2008) to the 3.6 μm Spitzer SAGE images of the LMC and SMC. The Σ_* is then converted to volume density by assuming a stellar disk thickness of 600 pc for the LMC (van der Marel et al. 2002) and 2 kpc for the SMC (following Bolatto et al. 2011). For the LMC, we use the dark matter density profile from Alves &#38; Nelson (2000), for the SMC we use the profile from Bekki &#38; Stanimirović (2009) to estimate the dark matter density as a function of radius from the centers of the galaxies. We find that the combined stellar and dark matter densities have ranges of 0.6–0.1 $\;{M}_{\odot }$ pc⁻³, with the higher values concentrated in the stellar bar and an ∼⃒10% dark matter contribution in the LMC, and 0.006–0.1 $\;{M}_{\odot }$ pc⁻³ with an ∼⃒20% dark matter contribution in the SMC.

We adopt most of the fiducial model parameter values as described in OML10 and KMT+. The exception is the depletion time in gravitationally bound clouds, ${t}_{{\rm{SF,gbc}}}$, for which OML10 uses 2 Gyr based on the average observed value in nearby galaxies. This value was also applied in the SMC results in Bolatto et al. (2011) since the observed ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ was not much lower than 2 Gyr. In this study we find a wider range of depletion timescales, and a measurable change in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ as a function of spatial scale (Figure 7). It is important to note that ${t}_{{\rm{SF,gbc}}}$ is an input parameter of the OML10 model, obtained from observations rather than theory, and its main impact is to change the relation between Σ_gbc and Σ_SFR since ${{\rm{\Sigma }}}_{\mathrm{gbc}}={t}_{{\rm{SF,gbc}}}{{\rm{\Sigma }}}_{\mathrm{SFR}}$

The self-regulation in the model operates to make Σ_SFR insensitive to the choice of ${t}_{{\rm{SF,gbc}}}$ over a wide range of total gas surface densities. The results we show in Figure 12 are computed for ${t}_{{\rm{SF,gbc}}}=0.5\;{\rm{Gyr}}$, which corresponds approximately to the value of ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ we observe at large spatial scales. The main effect of changing ${t}_{{\rm{SF,gbc}}}$ from 0.5 to 2 Gyr is to slightly lower the predicted star formation rate, particularly at high surface densities (${{\rm{\Sigma }}}_{{\rm{gas}}}\gtrsim 50$ $\;{M}_{\odot }\;{{\rm{pc}}}^{-2}$). The robustness of Σ_SFR to the choice of ${t}_{{\rm{SF,gbc}}}$ in turn means that Σ_gbc depends significantly on the value of the depletion timescale. Since we identify Σ_gbc with ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$, the consequence is that the comparison of our measurements of ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ with the model predictions are extremely dependent on the assumed ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$, and on the constancy of ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ with ${{\rm{\Sigma }}}_{{\rm{gas}}}$. In other words, in the context of the model they are very uncertain. Adopting the approximate value observed at large spatial scales, ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}\approx 0.5\;{\rm{Gyr}}$, results in a predicted Σ_gbc very similar to the observed ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 12 shows the model predictions for the ${{\rm{\Sigma }}}_{{\rm{SFR}}}$ for the LMC and SMC. Since both models require averaging over the different gas phases, we only compare the predictions to the data at r ∼⃒ 200 pc and r ∼⃒ 1 kpc. While the models can predict the molecular-to-atomic ratio (or, diffuse to gravitationally bound for OML10), the models self-regulate in diffuse gas, which is the dominant regime in the LMC and SMC, and predict similar star formation rates. Both KMT+ and OML10 models predict the general trend observed in the relationship between ${{\rm{\Sigma }}}_{{\rm{gas}}}$ and ${{\rm{\Sigma }}}_{{\rm{SFR}}}$.

The models do not predict the amount of scatter seen in the higher resolution 200 pc data. This is not surprising since both OML10 and KMT+ predict a time-averaged star formation rate and do not recover variations in the star formation rate based on the details on the star formation rate tracer combined with differences in the evolutionary stages of individual star-forming regions. Assuming that the physical interpretation from Feldmann et al. (2011) is correct, which is supported by the predictions of the amount of scatter in ${\tau }_{{\rm{dep}}}^{{\rm{mol}}}$ at ∼⃒100 pc and 1 kpc scales matching our observations, then an important, possibly the dominant, source of scatter is the time-averaging of the star formation rate (over as little as 10 Myr, Kennicutt &#38; Evans 2012) inherent in using ${\rm{H}}\alpha$ and 24 μm as star formation rate tracers. We think that it is likely that the under-prediction of the amount of scatter in the OML10/KMT+ model predictions at 200 pc is due to the fact that the star formation models do not include the time-averaging effect of the star formation rate tracer (${\rm{H}}\alpha$). The scatter present in the OML10/KMT+ model predictions come only from the spatial variation in the stellar and dark matter density, which affects the pressure and therefore the predicted amount of star-forming gas. When averaging over larger (∼⃒ kpc) scales, the difference between the scatter in the data and the scatter in the predictions decreases and the two are comparable. The model predictions are most appropriate at large scales where many independent star-forming regions are averaged over to account for the fact that star formation is treated as a time-averaged process.

The main differences between the star formation rate from OML10 versus KMT+ appear at high ${{\rm{\Sigma }}}_{{\rm{gas}}}$ and low metallicity, where the predictions diverge. All of the KMT+ predictions, independent of metallicity, converge at high ${{\rm{\Sigma }}}_{{\rm{gas}}}$ because once the ISM transitions to ${{\rm{H}}}_{2}$-rich, which happens past a column density based on the amount of shielding, then the star formation will not behave any differently from high-metallicity galaxies. The OML10 model will tend to continue to predict a lower ${{\rm{\Sigma }}}_{{\rm{SFR}}}$ at high ${{\rm{\Sigma }}}_{{\rm{gas}}}$ because the lower metallicity increases the thermal pressure and reduces the star formation at all surface densities. At the same metallicity and at high ${{\rm{\Sigma }}}_{{\rm{gas}}}$, OML10 will predict lower ${{\rm{\Sigma }}}_{{\rm{SFR}}}$ than KMT+.