What Sets the Massive Star Formation Rates and Efficiencies of Giant Molecular Clouds?

We use the Magellanic Mopra Assesment (MAGMA) DR3 (Wong et al. 2011, T. Wong et al. 2017, in preparation) CO intensity map to determine molecular masses using ${M}_{\mathrm{mol}}={\alpha }_{\mathrm{CO}}{L}_{\mathrm{CO}}$, where ${L}_{\mathrm{CO}}$ is the CO luminosity and ${\alpha }_{\mathrm{CO}}=8.6$ (K km s⁻¹ pc²)⁻¹ is the proportionality constant appropriate for the LMC (Bolatto et al. 2013). In addition, we use the dust-based molecular hydrogen map of Jameson et al. (2016, hereafter J16). The J16 map combines far-infrared dust emission (modeled with a single-temperature blackbody modified by a broken power-law emissivity; Gordon et al. 2014) and atomic hydrogen maps to estimate the ${{\rm{H}}}_{2}$ distribution. By doing so, J16 attempted to circumvent the biases of ¹²CO(1–0), which is known to probe a limited range in volume densities because of critical density, depletion, opacity, and photochemical effects. Furthermore, at the reduced metallicity of the LMC, a significant part of ${{\rm{H}}}_{2}$ may be in a "CO-dark" phase (Madden et al. 1997; Leroy et al. 2011). We test the dependency of our results on the choice of molecular gas tracer (shown in Figure 1) by performing our analysis on both the MAGMA and J16 maps.

Zoom In Zoom Out Reset image size

We decompose the molecular cloud maps using the dendrogram technique (Rosolowsky et al. 2008). Dendrograms need three user-defined inputs in order to identify regions while minimizing contamination by spurious artifacts. First, there must be a lower threshold below which data is to be excluded; this is set to the 2σ sensitivity of our CO maps, ∼10 ${M}_{\odot }$ pc⁻² for MAGMA (Wong et al. 2011) and ∼15 ${M}_{\odot }$ pc⁻² for the dust-based ${{\rm{H}}}_{2}$ map (J16). Second, there must be a minimum surface density contrast for a structure to be considered a separate entity; this is set to 1σ. Third, there is a minimum area that each structure must subtend, which set to twice the beam area of the MAGMA (45'' FWHM) or J16 (1') maps. Uncertainties in the molecular masses from MAGMA include the intrinsic noise of the data and the systematic uncertainty of the CO-to-H₂ conversion factor (accurate to within a factor of ∼2; Bolatto et al. 2013). We estimate the uncertainty introduced by the intrinsic noise of the data by adding random normally distributed noise with amplitude 1σ to the data; we find that the systematic uncertainty in the CO-to-H₂ conversion factor always dominates. Thus, the total uncertainty on the molecular masses is estimated at ±0.3 dex; a similar uncertainty was derived by J16 for the dust-based ${{\rm{H}}}_{2}$ map.

The total mass contained in the MAGMA clouds is 2.4 × 10⁷ ${M}_{\odot }$, which is a factor of 2 higher compared to the value of Wong et al. (2011). This is because we use a higher ${\alpha }_{\mathrm{CO}}$ factor of 8.6 (K km s⁻¹ pc²)⁻¹ versus the 4.3 (K km s⁻¹ pc²)⁻¹ used by Wong et al. (2011). The total mass found in the J16 map is 4.0 × 10⁷ ${M}_{\odot }$, consistent with values reported by Jameson et al. (2016).

We provide cloud properties such as median mass (${\tilde{M}}_{\mathrm{cloud}}$), radius (${\tilde{R}}_{\mathrm{cloud}}$), mean density ($\tilde{\langle \rho \rangle }$), and freefall time ($\tilde{{\tau }_{\mathrm{ff}}}$) in Table 1. The GMC freefall time equals ${\tau }_{\mathrm{ff}}=\sqrt{3\pi /32G\rho }$, and $\rho ={M}_{\mathrm{cloud}}$/(4/3πR³) is its mean density. Among cloud types, Type 3 clouds are larger and more massive compared to Type 1 and Type 2 clouds (Kawamura et al. 2009), while mean densities and freefall times do not differ significantly. Comparing MAGMA and J16 clouds, one can see that J16 clouds are, on average, larger and more massive and have lower internal densities (and higher freefall times). This can also be discerned by examining Figure 1: despite the lower sensitivity of the J16 versus MAGMA map (see above), the J16 clouds appear more extended, suggesting that the GMCs in the LMC are surrounded by extensive diffuse "CO-dark" envelopes (Leroy et al. 2011; Bolatto et al. 2013) that constitute roughly 50% of the molecular mass in the LMC.

Table 1.  Cloud and Star Formation Properties

	MAGMA				J16				MW
	Type 1	Type 2	Type 3	All	Type 1	Type 2	Type 3	All
${\tilde{M}}_{\mathrm{cloud}}$ (105 ${M}_{\odot }$)	0.27	0.33	0.84	0.40	0.73	1.04	1.74	1.28	4.8
${\tilde{R}}_{\mathrm{cloud}}$ (pc)	19.9	20.6	28.3	21.5	34.2	36.7	44.6	39.0	43
$\tilde{\langle \rho \rangle }$ (cm−3)	35.2	38.2	30.1	36.4	19.7	21.1	13.8	20.1	59.5
$\tilde{{\tau }_{\mathrm{ff}}}$ (Myr)	8.7	8.3	8.8	8.5	11.6	11.2	12.2	11.5	6.7
${\tilde{v}}_{\mathrm{CO}}$ (km s−1)	1.4	1.3	1.9	1.5	⋯	⋯	⋯	⋯	4.8
${p}_{\mathrm{MYSO}}$	0.10	0.43	0.66	0.42		0.47	0.69	0.52
${\tilde{\mathrm{SFR}}}_{\mathrm{MYSO}}$ (103 ${M}_{\odot }$ Myr−1)	0.3	0.4	1.1	0.7	⋯	0.3	1.5	0.7	⋯
${\tilde{\mathrm{SFR}}}_{{\rm{H}}\alpha }$ (103 ${M}_{\odot }$ Myr−1)	⋯	0.3	1.3	0.8	⋯	0.4	2.3	0.9	1.1b
${\alpha }_{\mathrm{SFR},\mathrm{MYSO}}$	⋯	⋯	0.35 ± 0.07	0.30 ± 0.04	⋯	0.51 ± 0.08	0.57 ± 0.06	⋯	⋯
${\alpha }_{\mathrm{SFR},{\rm{H}}\alpha }$	⋯	⋯	⋯	0.26 ± 0.13	⋯	⋯	0.47 ± 0.15	0.55 ± 0.12	0.65
${\alpha }_{\mathrm{SFR},{\rm{H}}\alpha }$ a	⋯	⋯	⋯	0.28 ± 0.15	⋯	⋯	0.53 ± 0.17	0.56 ± 0.15	0.65
${\alpha }_{\epsilon ^{\prime} ,\mathrm{MYSO}}$	⋯	−0.60 ± 0.10	−0.39 ± 0.12	−0.43 ± 0.06	⋯	−0.41 ± 0.16	⋯	−0.21 ± 0.09	⋯
${\alpha }_{\epsilon ^{\prime} ,{\rm{H}}\alpha }$	⋯	−0.61 ± 0.22	−0.61 ± 0.18	−0.50 ± 0.13	⋯	−0.42 ± 0.29	⋯	⋯	−0.35
${\alpha }_{\epsilon ^{\prime} ,{\rm{H}}\alpha }$ a	⋯	−0.59 ± 0.27	−0.44 ± 0.21	−0.38 ± 0.15	⋯	−0.37 ± 0.25	⋯	⋯	−0.35

Notes. Cloud and star formation properties for the LMC clouds (both MAGMA and J16) and the Milky Way (MW). Listed are the median cloud mass (${\tilde{M}}_{\mathrm{cloud}}$), cloud radius (${\tilde{R}}_{\mathrm{cloud}}$), cloud density ($\tilde{\langle \rho \rangle }$), cloud freefall time ($\langle {\tau }_{\mathrm{ff}}\rangle$), velocity dispersion (${\tilde{v}}_{\mathrm{CO}}$), fraction of clouds that contain MYSOs (${p}_{\mathrm{MYSO}}$), median ${\mathrm{SFR}}_{\mathrm{MYSO}}$ (${\tilde{\mathrm{SFR}}}_{\mathrm{MYSO}}$), median ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ (${\tilde{\mathrm{SFR}}}_{{\rm{H}}\alpha }$), and various slope indices α between SFR and $\epsilon ^{\prime}$ with ${M}_{\mathrm{cloud}}$.

^aThese values refer to the fits by excluding the 30 Doradus and SMR region (see text). ^bThis number is derived from free–free flux as opposed to Hα + 24 μm.

Download table as:  ASCIITypeset image

The most direct measure of SFR can be obtained through counting YSOs, given by SFR = N(YSOs) × $\langle {M}_{\star }\rangle$/${t}_{\star }$, where the average mass is $\langle {M}_{\star }\rangle \approx 0.5$ for a fully sampled initial mass function (Kroupa 2001) and ${t}_{\star }$ is the age of the YSO population. This technique has been applied to the nearest ensemble of molecular clouds (Evans et al. 2009; Heiderman et al. 2010; Lada et al. 2010). However, counting low-mass YSOs becomes cumbersome at large distances because of completeness and/or crowding. This can be overcome by considering only the most luminous sources: MYSOs. Ochsendorf et al. (2016) presented an MYSO catalog that builds on the results of galaxy-wide searches of MYSOs in the LMC (Whitney et al. 2008; Gruendl & Chu 2009; Seale et al. 2014). This catalog has been estimated to be complete for Stage 1 MYSOs of mass M > 8 ${M}_{\odot }$ (Figure 1). We utilize this catalog to obtain a census of massive star formation by counting the number of MYSOs in each cloud identified in the dendrogram decomposition (Section 2.1). We assume that the observed luminosity is dominated by a single massive source (see below) and use the Robitaille et al. (2006) models to obtain a source mass (see Robitaille 2008 for a complete discussion of this conversion). Subsequently, we multiply the obtained source mass with an IMF (Kroupa 2001) to account for stars below our completeness limit. For ${t}_{\star }$, we choose 0.5 Myr, which is the most recent value obtained for the observationally derived "Class 1" low-mass sources (which largely overlap with the theoretically based "Stage 1" sources; Heiderman & Evans 2015; Heyer et al. 2016) in the Gould's Belt (Dunham et al. 2015). It is not clear whether this value applies to massive stars; however, it should represent a reasonable first-order estimate of ${t}_{\star }$. Our completeness limit of M > 8 ${M}_{\odot }$ for MYSOs then translates to a lower limit of ${\mathrm{SFR}}_{\mathrm{MYSO}}\sim 100\,{M}_{\odot }$ Myr⁻¹.

In many cases, our MYSO sources will break into small clusters at higher resolution (Vaidya et al. 2009; Stephens et al. 2017). To estimate the uncertainty in our ${\mathrm{SFR}}_{\mathrm{MYSO}}$ measurement, we consider a case in which we were to observe the Orion Trapezium cluster at the distance of the LMC, where the reprocessed IR luminosity would appear as a compact source in our IR maps. Using the stellar atmosphere models of Vacca et al. (1996), we estimate that the main ionizing source, ${\theta }^{1}$ Ori C, contains ≃50% of the total luminosity of the Trapezium (O'dell et al. 1993; Simón-Díaz et al. 2003). Thus, our measured IR luminosity would overestimate the luminosity of the most massive source by ∼0.3 dex, which translates to an overestimation of the stellar mass by a factor of ∼0.1 dex (Mottram et al. 2011). We adopt this as our systematic uncertainty in ${\mathrm{SFR}}_{\mathrm{MYSO}}$.

Counting Stage 1 MYSOs (Section 2.2) provides a census of the young, embedded phase of massive star formation. In contrast, the widely used SFR diagnostic Hα traces a more evolved population of massive stars of 3–10 Myr (Kennicutt & Evans 2012). Given its sensitivity to dust attenuation, we correct the Hα emission (from the Southern H-Alpha Sky Survey Atlas (SHASSA); Gaustad et al. 2001) for extinction using 24 μm emission (from Spitzer's Surveying the Agents of a Galaxy's Evolution; Meixner et al. 2013). Emission at 24 μm traces star formation activity up to 100 Myr, probably less on small scales if associated with Hα. In this case, young massive stars will dominate ionization and local dust heating (for a discussion, see Vutisalchavakul & Evans 2013).

We convolve the 24 μm map to the resolution of our Hα map (08) and use the Calzetti et al. (2007) relationship to transform Hα and 24 μm luminosity, L(Hα) and L(24 μm), to an SFR:

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

Koepferl et al. (2016) showed that the 24 μm SFR tracer alone can significantly underestimate the SFR. However, in the LMC, 24 μm typically contributes only ≲20% to the measured SFR (see also J16). We again use dendrograms (Rosolowsky et al. 2008) to characterize the diffuse emission across the face of the LMC and automate the identification of star formation regions (Figure 1). The Hα observations show that the diffuse background in the LMC has values ranging between 10 and 200 R (Rayleigh; 1 R = 10⁶/4π photons s⁻¹ cm⁻² sr⁻¹), with a typical value of ∼100 R. We thus use 300 R as a lower threshold for our dendrogram, 150 R as a minimum surface brightness contrast, and a minimum area of 5 times the beam size of SHASSA (∼48''). We choose this rather large minimum area because the SHASSA map shows artifacts of imperfect subtraction of bright (foreground) stars, which are typically several pixels in size. These parameters set the detection limit of ${\mathrm{SFR}}_{{\rm{H}}\alpha }$: using Equation (1), a star formation region of five SHASSA beam areas filled with a surface brightness of 300 R (i.e., the adopted lower threshold) contains ${\mathrm{SFR}}_{{\rm{H}}\alpha }\sim 5{M}_{\odot }$ Myr⁻¹. Note that only stars with spectral type B2 or earlier can maintain an H ii region, which intrinsically sets a lower limit to ${\mathrm{SFR}}_{{\rm{H}}\alpha }$. With an ionizing flux of ∼10^47.5 photons s⁻¹ (Schaerer & de Koter 1997) and assuming that all ionizing photons get absorbed, we estimate the expected Hα luminosity (assuming case B recombination; Osterbrock & Ferland 2006), and thereby ${\mathrm{SFR}}_{{\rm{H}}\alpha }$, through Equation (1). We find that an H ii region powered by a single B2V star should exhibit ${\mathrm{SFR}}_{{\rm{H}}\alpha }\sim 2.5{M}_{\odot }$ Myr⁻¹, comparable to our sensitivity limit.

By summing the Hα luminosities of all star-forming regions, we recover ∼50% of the total Hα luminosity of the entire LMC. This is consistent with an escape fraction of ionizing photon luminosity from H ii regions of ∼0.5 observed by Pellegrini et al. (2012). Consequently, we correct for the effects of ionizing photon leakage by multiplying our ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ measurements by a factor of 2.

We note that Equation (1) assumes a fully sampled IMF, and averaging over large enough spatial scales so that each phase of the star-forming process, i.e., from the deeply embedded phase to fully exposed clusters, is adequately probed (Krumholz et al. 2014). At small scales (Kruijssen & Longmore 2014) or at a low SFR (da Silva et al. 2012), these conditions are likely to be violated, which will introduce scatter and a potential bias in the derived ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ values because of stochastic sampling of the high-end tail of the IMF. These effects become important below cluster masses of ∼3000 ${M}_{\odot }$ (Cerviño et al. 2003). To estimate the stellar masses associated with our observed ${\mathrm{SFR}}_{{\rm{H}}\alpha }$, we relate the Hα luminosity to the ionizing photon rate ${Q}_{0}$ through L(Hα) = $\tfrac{{\alpha }_{{\rm{H}}\alpha }^{\mathrm{eff}}}{{\alpha }_{{\rm{B}}}}$ ${E}_{{\rm{H}}\alpha }$ ≈ 1.37 × 10⁻¹²${Q}_{0}$, where ${\alpha }_{{\rm{H}}\alpha }^{\mathrm{eff}}$ is the effective recombination coefficient at Hα, ${\alpha }_{{\rm{B}}}$ is the case B recombination coefficient, and ${E}_{{\rm{H}}\alpha }$ is the energy of an Hα photon (Osterbrock & Ferland 2006). From this, we estimate the stellar mass through ${M}_{\star }$ = 6.3 × 10⁻⁴⁶${Q}_{0}$, where the constant on the right hand represents the IMF averaged photon rate divided by the IMF averaged stellar mass (in units of s⁻¹ M_⊙; Murray 2011). It follows that the median stellar mass of our Hα-emitting star-forming regions is ∼10³ ${M}_{\odot }$, implying that a significant portion of our SFR measurements will be affected by stochasticity.

We use the Stochastically Lighting Up Galaxies (SLUG) tool (Krumholz et al. 2014) to estimate the bias and scatter (defined in da Silva et al. 2012) associated with stochasticity for our ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ measurements. We use the default SLUG library (da Silva et al. 2012) at solar metallicity and a 500 Myr continuous star formation scale. Note that the metallicity of the LMC is roughly half solar (Russell & Dopita 1992). In this regard, metallicity induces variations in the conversion between Hα luminosity and SFR on the order of ±0.1 dex (Calzetti et al. 2007). In addition, on the scale of individual H ii regions, a <10 Myr star formation timescale may be more appropriate. However, assuming a constant SFR, the Hα luminosity per unit SFR reaches a steady state after only a few Myr (as stars arriving on the zero-age main sequence are balanced by those that evolve toward supernovae; Krumholz & Tan 2007; Kennicutt & Evans 2012). Nonetheless, on the scale of individual star-forming regions, the SFR is likely not continuous but will proceed in discreet bursts (Faesi et al. 2014). In this regard, the ubiquity of "two-stage" starbursts (Walborn & Parker 1992) in the LMC (Ochsendorf et al. 2016) shows that massive star formation typically clusters over multiple generations on timescales up to at least ∼10 Myr. Depending on the amount of subclusters formed, approximating the star formation history as continuous may be a reasonable first-order estimate. Finally, uncertainties in our flux measurements are estimated at ±0.05 dex.

The total scatter in our ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ measurements can be estimated by adding in quadrature the uncertainties from stochasticity (evaluated at log [${\mathrm{SFR}}_{{\rm{H}}\alpha }$ (M_⊙ Myr⁻¹)] = 2.8, the median of Type 2 and Type 3 clouds), metallicity, and photometry. We do not consider the bias induced by stochasticity, which is generally small at log [${\mathrm{SFR}}_{{\rm{H}}\alpha }$ (M_⊙ Myr⁻¹)] = 2.8 compared to the estimated total scatter of our measurements, ±0.4 dex (a similar scatter is found by comparing the ${\mathrm{SFR}}_{\mathrm{MYSO}}$ and ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ tracers; see Figure 2 and Section 3).

Zoom In Zoom Out Reset image size

We aim to derive the SFR associated with each individual GMC found in our cloud decompositions (Section 2.1). For the MYSOs, we simply add all MYSOs found within the footprint of a GMC, from which we calculate ${\mathrm{SFR}}_{\mathrm{MYSO}}$ (Section 2.2). For Hα, we cross-match each structure defined by our Hα dendrogram (Section 2.3) with our GMC dendrogram and define a match when there is a physical overlap between both structures. We then calculate ${\mathrm{SFR}}_{{\rm{H}}\alpha }$, taking into account the local 24 μm emission (Section 2.3). If there is more than one emission structure per GMC, we sum the single components. Likewise, if there are more GMCs found within a given emission structure, we add the single cloud components. We perform the above-described routine for the MAGMA and J16 molecular maps and ultimately obtain a list of GMCs (with ${M}_{\mathrm{cloud}}$) with associated ${\mathrm{SFR}}_{\mathrm{MYSO}}$ and ${\mathrm{SFR}}_{{\rm{H}}\alpha }$.

Kawamura et al. (2009) classified the GMCs in the LMC as Type 1 (GMCs with no massive star formation), Type 2 (GMCs with associated H ii regions), and Type 3 (GMCs with associated H ii regions and optical stellar clusters). This classification was based on GMCs detected in the NANTEN survey (at resolution 26; Fukui et al. 2008). As in Ochsendorf et al. (2016), we match our GMC lists with the Kawamura et al. (2009) catalog. For MAGMA, we consider all GMCs detected within the footprint of a NANTEN GMC as being of the same type. The J16 clouds are typically more extended than the NANTEN clouds of Kawamura et al. (2009), and, in some cases, individual NANTEN clouds of different types are joined in a single structure. In these cases, we label the cloud as the highest type (i.e., most evolved) found within the individual NANTEN components.

Star formation efficiencies are often defined differently in extragalactic and Galactic studies, which stems from the (in)capability of resolving individual stars, clusters, and clouds in specific environments. Given that the LMC represents a special case in this sense (i.e., both extragalactic and resolved in individual star-forming regions), we consider both cases, as this could be useful for intercomparisons across extragalactic and Galactic studies. We follow the notation of Kennicutt & Evans (2012) and write $\epsilon ^{\prime} =\mathrm{SFR}/{M}_{\mathrm{cloud}}$, which is often used in extragalactic studies. In addition, we are particularly interested in the star formation efficiency per freefall time, ${\epsilon }_{\mathrm{ff}}$ = ${\tau }_{\mathrm{ff}}$/${t}_{\mathrm{dep}}$ = ${\tau }_{\mathrm{ff}}$ × $\epsilon ^{\prime}$ (where ${t}_{\mathrm{dep}}$ is the depletion timescale).

We provide the results of our cloud decomposition and SFR measurements in four machine-readable tables in the Appendix: (1) MAGMA clouds and MYSOs, (2) MAGMA clouds and Ha + 24 μm, (3) J16 clouds and MYSOs, and (4) J16 clouds and Ha + 24 μm.

Figure 2 plots ${\mathrm{SFR}}_{\mathrm{MYSO}}$ versus ${\mathrm{SFR}}_{{\rm{H}}\alpha }$; a similar comparison has been made for small samples in the Milky Way (Chomiuk & Povich 2011; Lee et al. 2016). Here, we extend these studies by sampling massive star-forming regions on a galaxy-wide scale. Assuming ${t}_{\star }$ = 0.5 Myr (Section 2.2), we find good agreement between ${\mathrm{SFR}}_{\mathrm{MYSO}}$ and ${\mathrm{SFR}}_{{\rm{H}}\alpha }$, with a mean ratio of 1.4 and 1.0 for the MAGMA and J16 decompositions, respectively. Nonetheless, there is considerable scatter between both SFR tracers, i.e., ∼0.6 and ∼0.4 dex for the MAGMA and J16 clouds, respectively. This is only slightly larger than what is expected from stochastic sampling of the IMF alone (Section 2.3), although the exact extent of this scatter is known to vary with SFR (da Silva et al. 2012; Krumholz et al. 2014). If stochasticity dominates the scatter in Figure 2, this would in turn imply that ${\mathrm{SFR}}_{\mathrm{MYSO}}$ is a robust, unbiased tracer of SFR on individual cloud scales: the largest uncertainty remains ${t}_{\star }$ (Kennicutt & Evans 2012).

A substantial fraction of clouds are only observed in a single SFR tracer (Figure 2), which may be expected given that we are sampling star formation regions in different evolutionary states (Kawamura et al. 2009; Kruijssen & Longmore 2014). Star formation regions captured in the earliest stages of evolution may only be detected through MYSOs, while regions with fully exposed clusters may no longer have MYSOs, as the molecular gas reservoir is disrupted. In this regard, simulations show that stellar feedback quickly clears cavities around young star clusters (≳1 Myr) depending on the initial properties of the cloud (Dale et al. 2012, 2014). By contrast, observations may imply longer timescales (≳5–10 Myr) for clusters to clear their immediate surroundings of CO-emitting gas (Leisawitz et al. 1989; Kawamura et al. 2009). The sensitivity of surveys targeting the molecular gas also plays a role here, which may cause MYSOs to fall outside CO cloud footprints (Wong et al. 2011). Consistent with the above, we find a ratio of ${\mathrm{SFR}}_{\mathrm{MYSO}}$/${\mathrm{SFR}}_{{\rm{H}}\alpha }\sim 3$ for Type 2 clouds (both MAGMA and J16 clouds), while this ratio drops below unity for Type 3 clouds. However, the mean ratio of ${\mathrm{SFR}}_{\mathrm{MYSO}}$/${\mathrm{SFR}}_{{\rm{H}}\alpha }$ over the entire LMC shows that, averaged over the lifetime of star-forming regions, both SFR tracers are consistent with one another.

Figure 3(a) plots SFR versus ${M}_{\mathrm{cloud}}$ for both tracers (${\mathrm{SFR}}_{\mathrm{MYSO}}$ and ${\mathrm{SFR}}_{{\rm{H}}\alpha }$) and both cloud decompositions (MAGMA and J16). We recover systematically higher ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ and ${\mathrm{SFR}}_{\mathrm{MYSO}}$ along the Type 1–Type 3 evolutionary sequence (Kawamura et al. 2009). We emphasize this result in a whisker plot (Figure 4) and provide absolute numbers in Table 1. In addition, the fraction of clouds containing MYSOs, ${p}_{\mathrm{MYSO}}$, increases steadily between Type 1, Type 2, and Type 3 clouds. Thus, it becomes clear that, in the presence of stellar clusters (i.e., Type 2 and Type 3 clouds), ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ increases, which appears to go hand-in-hand with an increased SFR of younger generations of massive stars as reflected by ${\mathrm{SFR}}_{\mathrm{MYSO}}$ (see also Ochsendorf et al. 2016).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We fit a linear model log[SFR] = log[${M}_{\mathrm{cloud}}$]α + β to the data points of Figure 3(a). We perform a linear least-squares fit on three different subsets: Type 2, Type 3, and the entire set of clouds. To estimate the uncertainties on our fit, we approximate the errors on ${M}_{\mathrm{cloud}}$, ${\mathrm{SFR}}_{\mathrm{MYSO}}$, and ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ as being normally distributed with amplitude 0.3, 0.1, and 0.4 dex, respectively (Sections 2.1–2.3). We subsequently add random noise to our data and perform the fit 100 times. Table 1 presents the mean and 1σ uncertainty on the derived slope α in case a statistically significant correlation is found (i.e., Kendall rank correlation p-value <0.05; Figure 3). In all cases, we find that ${\mathrm{SFR}}_{\mathrm{MYSO}}$ and ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ depend on ${M}_{\mathrm{cloud}}$ sublinearly for both the MAGMA and J16 clouds.

The 30 Doradus region contains the largest and most massive H ii region in the Local Group (harboring ∼2400 OB stars; Parker 1993) and is often considered the nearest super star cluster that is reminiscent of starburst environments (Walborn 1991). Conversely, the South Molecular Ridge (SMR) is a GMC that contains about 30% of the total molecular mass of the LMC, yet exhibits a remarkably low SFR (Indebetouw et al. 2008). Given the extremities in GMC and star formation behavior of both these systems, we illustrate how our results are affected by these systems. Table 1 lists fit results for ${\alpha }_{\mathrm{SFR},{\rm{H}}\alpha }$ and ${\alpha }_{\epsilon ^{\prime} ,{\rm{H}}\alpha }$ (see below) by excluding 30 Doradus and the SMR. We note that ${\alpha }_{\mathrm{SFR},\mathrm{MYSO}}$ and ${\alpha }_{\epsilon ^{\prime} ,\mathrm{MYSO}}$ are not significantly affected in this way, as 30 Doradus and the SMR do not appear as outliers in the ${\mathrm{SFR}}_{\mathrm{MYSO}}$ tracer.

We compare our results with the Galactic sample of Lee et al. (2016) in Figure 3(a) and Table 1. We find good agreement between both galaxies. For example, the star-forming regions in the LMC and Milky Way occupy the same region in "M_cloud–SFR" space. The Milky Way does have a larger dynamic range on both axes, which may explain some of the offset in median SFR between the MAGMA clouds and the Milky Way (a factor of ∼1.5; Table 1). However, the derived slopes of ${\alpha }_{\mathrm{SFR},\mathrm{MYSO}}$ and ${\alpha }_{\mathrm{SFR},{\rm{H}}\alpha }$ all agree within ∼1σ for the J16 clouds and the Milky Way, showing that a sublinear relation between SFR and total cloud mass persists in both the LMC and the Milky Way (Lee et al. 2016; Vutisalchavakul et al. 2016).

Figure 3(b) plots the corresponding $\epsilon ^{\prime}$ of individual star-forming complexes. Naturally, a higher ${\mathrm{SFR}}_{\mathrm{MYSO}}$ and ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ leads to increased values of ${\epsilon }_{\mathrm{MYSO}}^{\prime }$ and ${\epsilon }_{{\rm{H}}\alpha }^{\prime }$ along the Type 1–Type 3 evolutionary sequence. However, the most striking result from Figure 3(b) is the decline of ${\epsilon }_{{\rm{H}}\alpha }^{\prime }$ and ${\epsilon }_{\mathrm{MYSO}}^{\prime }$ with ${M}_{\mathrm{cloud}}$, which holds for the MAGMA and J16 clouds and is readily recognized in the Galactic GMCs. Thus, larger GMCs have higher depletion times (${\tau }_{\mathrm{dep}}$ = 1/$\epsilon ^{\prime}$). We defer a further discussion of the implications of this observation to Section 4.

From Larson's laws (Larson 1981), it is known that more massive clouds have lower $\tilde{\langle \rho \rangle }$, which may impact the star formation efficiency of GMCs. While we see a slight tendency for $\tilde{\langle \rho \rangle }$ to decrease along Type 1–Type 3 clouds (Table 1), this difference is not significant given the uncertainties on cloud mass (∼0.3 dex; Section 2.1). Alternatively, we may explore the structure of individual clouds by plotting the ratio ${M}_{\mathrm{cloud},\mathrm{MAGMA}}/{M}_{\mathrm{cloud},{\rm{J}}16}$, since the ${M}_{\mathrm{cloud},{\rm{J}}16}$ values are more likely to include diffuse gas (Section 2.1). Here, we only take into account mass above 15 ${M}_{\odot }$ pc⁻², i.e., the sensitivity limit for the dust-based ${{\rm{H}}}_{2}$ map (J16). While the predicted errors are large, Figure 5 shows that this ratio declines as a function of ${M}_{\mathrm{cloud},{\rm{J}}16}$. This implies that larger clouds have larger diffuse envelopes and grow beyond ∼10⁵${M}_{\odot }$ by adding "CO-dark" gas. The decline of $\epsilon ^{\prime}$ may therefore be caused by an increasing "contamination" of GMCs by the addition of diffuse gas at higher ${M}_{\mathrm{cloud}}$.

Zoom In Zoom Out Reset image size

3.3.1. Observations of ${\epsilon }_{{\rm{ff}}}$

In Figure 6(a), we plot ${\epsilon }_{\mathrm{ff}}$ ($=\epsilon ^{\prime}$ × ${\tau }_{\mathrm{ff}}$) versus ${M}_{\mathrm{cloud}}$ for both the LMC and the Milky Way (Lee et al. 2016). While the large scatter in ${\epsilon }_{\mathrm{ff}}$ on individual star formation region scales has been noted before (Mooney & Solomon 1988; Lee et al. 2016; Vutisalchavakul et al. 2016), the LMC data provide insight into the time evolution of ${\epsilon }_{\mathrm{ff}}$. Indeed, a large part of the scatter in ${\epsilon }_{\mathrm{ff}}$ stems from the systematic offset of ${\epsilon }_{\mathrm{ff}}$ between Type 1, Type 2, and Type 3 clouds (Table 3), most notably for ${\epsilon }_{\mathrm{ff},{\rm{H}}\alpha }$. As in Section 3.2, we fit the different subsets of cloud types, which in all cases results in a negative slope ${\alpha }_{{\epsilon }_{\mathrm{ff}}}$ (Table 3).

Zoom In Zoom Out Reset image size

The decline of ${\epsilon }_{\mathrm{ff}}$ with ${M}_{\mathrm{cloud}}$ (similar to that observed for $\epsilon ^{\prime} ;$ Figure 3) is also present in the Galactic samples of Lee et al. (2016), Murray (2011), and Vutisalchavakul et al. (2016). Murray (2011) argued that this trend is largely explained through an observational selection effect introduced by a lower luminosity limit (shaded areas in Figure 3) and because ${M}_{\mathrm{cloud}}$ appears on both axes. However, while an uncertainty in ${M}_{\mathrm{cloud}}$ would scatter sources along a line with slope −1, this cannot reconcile a systematic decline of ${\epsilon }_{\mathrm{ff}}$ with ${M}_{\mathrm{cloud}}$. Instead, we show that negative slopes derived for $\epsilon ^{\prime}$ and ${\epsilon }_{\mathrm{ff}}$ with ${M}_{\mathrm{cloud}}$ originate from uncorrelated or sublinear-correlated SFRs with ${M}_{\mathrm{cloud}}$ (Section 3.2 and Table 1). Combining the results of the LMC and the Milky Way, it becomes apparent that these relations are inherent to the star-forming properties of individual GMCs when total cloud masses are measured through ¹²CO(1–0), dust, or ¹³CO(1–0).

Disruption of GMCs through stellar feedback may contribute to the observed scatter in ${\epsilon }_{\mathrm{ff}}$. In other words, Type 3 clouds may have lost a significant amount of mass through destructive feedback from their associated clusters, thereby systematically altering ${\epsilon }_{\mathrm{ff}}$ (for a discussion, see Krumholz et al. 2014). The average age of massive star clusters powering ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ is ∼4 Myr (Murray 2011). Observations (Ochsendorf et al. 2015) and simulations (Dale et al. 2014) of feedback on GMCs reveal typical photo-evaporation rates of ∼10⁻² ${M}_{\odot }$ yr⁻¹, implying that only ∼5% of a 6 × 10⁵ ${M}_{\odot }$ GMC (average Type 3 cloud mass; Kawamura et al. 2009) is lost in ∼4 Myr. In fact, Type 3 clouds are slightly more massive than their Type 2 counterparts (Kawamura et al. 2009). Thus, it seems unlikely that GMC disruption alone can be responsible for the systematic increase in ${\epsilon }_{\mathrm{ff},\mathrm{MYSO}}$ and ${\epsilon }_{\mathrm{ff},{\rm{H}}\alpha }$ between Type 1, Type 2, and Type 3 clouds (Table 3). The timescale involved in GMC disruption may potentially be estimated through the uncertainty principle proposed by Kruijssen & Longmore (2014); however, this method does not distinguish between GMC displacement/redistribution of star formation (e.g., H ii region expansion) and actual GMC disruption.

The large scatter in ${\epsilon }_{\mathrm{ff}}$ may instead be driven by an increasing SFR (i.e., accelerating star formation) per freefall time ${\tau }_{\mathrm{ff}}$ in more evolved GMCs. Similar patterns have been observed in nearby clusters (Palla & Stahler 2000) and M17 (Povich et al. 2016) and have been explored theoretically (Zamora-Avilés et al. 2012). While it is clear that the SFR increases within Type 2 and Type 3 clouds (Section 3.2), there is no significant change of ${\tau }_{\mathrm{ff}}$ between cloud types (Table 1), driving ${\epsilon }_{\mathrm{ff}}$ upward. Note, however, that ${\tau }_{\mathrm{ff}}$ is calculated from a single average density. If the dense gas forming massive stars constitutes a small fraction of the total GMC mass, this will not be reflected in our estimation of ${\tau }_{\mathrm{ff}}$. The LMC and Milky Way data may therefore imply that the massive star-forming rates and efficiencies of GMCs are not regulated by specific, small volumes of GMCs instead of their global properties (Evans et al. 2014; Ochsendorf et al. 2016; Vutisalchavakul et al. 2016).

Finally, we note that $\langle {\epsilon }_{\mathrm{ff},\mathrm{MYSO}}\rangle$ and $\langle {\epsilon }_{\mathrm{ff},{\rm{H}}\alpha }\rangle$ for the LMC clouds (0.10–0.25; Table 3) are significantly higher than in Galactic clouds as measured by free–free flux (∼0.03; Lee et al. 2016). However, given the similarity of ${\epsilon }_{\mathrm{ff}}$ of GMCs at mass 4.0 < log(${M}_{\mathrm{cloud}}$) < 5.5, this can largely be explained by a dearth of GMCs in the LMC at high ${M}_{\mathrm{cloud}}$, which tends to have lower $\langle {\epsilon }_{\mathrm{ff}}\rangle$.

3.3.2. Predictions of ${\epsilon }_{{\rm{ff}}}$ with Analytical Star Formation Models

In this section, we compare the observations of the LMC and the Milky Way with analytical star formation models from KM05, PN11, and HC11. In addition, we consider the HC11 model extended to include magnetic fields by FK12. For a detailed explanation and comparison between these models, we refer the reader to FK12 and Padoan et al. (2014).

The aforementioned models rely on the probability distribution function of gas density, often assumed to be log-normal in turbulent media (Vazquez-Semadeni 1994; Padoan et al. 1997; Burkhart & Lazarian 2012). The width of the density PDF is determined by ${\sigma }_{{\rm{p}}}$ = ln(1 + b²${ \mathcal M }$), where b is the turbulence forcing parameter (b = 0.33, 0.4, and 1 for purely solenoidal, mixed, or purely compressive forcing, respectively; FK12) and ${ \mathcal M }$ is the internal Mach number (${ \mathcal M }$ = ${v}_{\mathrm{CO}}$/${c}_{{\rm{s}}},$ where ${c}_{{\rm{s}}}$ is the isothermal speed of sound). The SFR per freefall time, ${\epsilon }_{\mathrm{ff}}$, is then determined by estimating the gas mass above a given density threshold, ${x}_{\mathrm{crit}}$. The exact definition of ${x}_{\mathrm{crit}}$ differs between individual models (Table 2) but is dependent on the virial parameter ${\alpha }_{\mathrm{vir}}$ = $5{v}_{\mathrm{CO}}{R}_{\mathrm{cloud}}$/${{GM}}_{\mathrm{cloud}}$ (where ${R}_{\mathrm{cloud}}$ and ${v}_{\mathrm{CO}}$ are the radius and CO 1D line width, respectively), ${ \mathcal M }$, b, and the plasma β = $2{ \mathcal M }$/${{ \mathcal M }}_{{\rm{A}}}$, where ${{ \mathcal M }}_{{\rm{A}}}$ is the Alfvénic Mach number relating to the strength of the magnetic field.

Table 2.  Analytical Star Formation Models, Parameters, and Observables

Model	${x}_{\mathrm{crit}}$	${\epsilon }_{\mathrm{ff}}$
KM05	(${\pi }^{2}$/45)${\phi }_{{\rm{x}}}^{2}{\alpha }_{\mathrm{vir}}$ ${{ \mathcal M }}^{2}$	ξ $\{1+\mathrm{erf}[{({\sigma }_{{\rm{p}}}^{2}-2\mathrm{ln}{x}_{\mathrm{crit}})/(8{\sigma }_{{\rm{p}}}^{2})}^{1/2}]\}$
PN11	(0.067)${\theta }^{-2}{\alpha }_{\mathrm{vir}}$ ${{ \mathcal M }}^{2}f(\beta )$	ξ $\{1+\mathrm{erf}[{({\sigma }_{{\rm{p}}}^{2}-2\mathrm{ln}{x}_{\mathrm{crit}})/(8{\sigma }_{{\rm{p}}}^{2})}^{1/2}]\}$ ${e}^{1/2{x}_{\mathrm{crit}}}$
HC11	(${\pi }^{2}$/5)${y}_{\mathrm{cut}}^{-2}{\alpha }_{\mathrm{vir}}$ ${{ \mathcal M }}^{-2}$	ξ $\{1+\mathrm{erf}[{({\sigma }_{{\rm{p}}}^{2}-\mathrm{ln}{x}_{\mathrm{crit}})/(2{\sigma }_{{\rm{p}}}^{2})}^{1/2}]\}$ ${e}^{3{\sigma }_{{\rm{p}}}^{2}/8}$
FK12	(${\pi }^{2}$/5)${y}_{\mathrm{cut}}^{-2}{\alpha }_{\mathrm{vir}}$ ${{ \mathcal M }}^{-2}{(1+{\beta }^{-1})}^{-1}$	ξ $\{1+\mathrm{erf}[{({\sigma }_{{\rm{p}}}^{2}-\mathrm{ln}{x}_{\mathrm{crit}})/(2{\sigma }_{{\rm{p}}}^{2})}^{1/2}]\}$ ${e}^{3{\sigma }_{{\rm{p}}}^{2}/8}$
Parameter	Value	References
${\phi }_{{\rm{x}}}$	1.19	KM05
${y}_{\mathrm{cut}}$	0.1	HC11
θ	0.35	PN11
b	0.33, 0.4, 1.0	FK12
β	0.04, 0.20, 1.8, 3.6	FK12
Observable	Uncertainty (dex)	References
${v}_{\mathrm{CO}}$	0.08	Wong et al. (2011)
${c}_{{\rm{s}}}$	0.20	Roman-Duval et al. (2010)
${R}_{\mathrm{cloud}}$	0.11	Wong et al. (2011)
${M}_{\mathrm{cloud}}$	0.30	Bolatto et al. (2013)

Note. The table gives the analytical descriptions of the star formation models (formulations similar to those in FK12 and Padoan et al. 2014), the parameters used in the model realizations, and the observables with their estimated uncertainties. The function f(β) is given in PN11. The turbulent forcing parameter b and plasma β are chosen to be consistent with those used in FK12. The uncertainties on the observables are assumed to be normally distributed.

Download table as:  ASCIITypeset image

We measure the second velocity moment (T. Wong et al. 2017, in preparation) within our cloud footprints (Section 2.1) to determine ${v}_{\mathrm{CO}}$. We then perform model realizations by propagating the uncertainties of all of the observables—${v}_{\mathrm{CO}}$, ${c}_{{\rm{s}}}$, ${R}_{\mathrm{cloud}}$, and ${M}_{\mathrm{cloud}}$—while randomly choosing between specified values of the turbulent forcing parameter b and the plasma β (Table 2). For ${c}_{{\rm{s}}}$, we choose a distribution with a mean value of 0.2 km s⁻¹ (for a gas temperature of 15 K) with a 1σ uncertainty of ±0.2 dex to account for differences in UV field or cosmic-ray environments. This spread is based on the CO excitation temperatures of Galactic GMCs (Roman-Duval et al. 2010). As the excitation temperature may differ from the actual gas temperature at low densities due to subthermal excitation, we only use the spread of excitation temperatures derived in their sample to estimate the uncertainty on the cloud temperatures and thereby ${c}_{{\rm{s}}}$ (Table 2).

To facilitate direct comparisons between observations and theory, we must choose for each model the unique parameter that prescribes the criterion for gravitational collapse (i.e., the model "fudge factor"; FK12). We adopt the author-preferred values, i.e., ${\phi }_{{\rm{x}}}$ = 1.12 (KM05), θ = 0.35 (PN11), and ${y}_{\mathrm{cut}}=0.1$ (HC11 and FK12). In addition, the normalization constant ξ, which relates to the core efficiency and characteristic timescale over which gas becomes gravitationally unstable, is chosen such that the median value of the model realization matches that of the LMC sample. The results are shown in Figure 6(b).

The most striking discrepancy between the model and the observations is the derived slope ${\alpha }_{{\epsilon }_{\mathrm{ff}}}$. None of the analytical star formation models are able to reproduce the decline of ${\epsilon }_{\mathrm{ff}}$ with ${M}_{\mathrm{cloud}}$ that is seen in the observations of the LMC and the Milky Way. The increase of ${\epsilon }_{\mathrm{ff}}$ with ${M}_{\mathrm{cloud}}$ in the models likely results from the GMC size–linewidth relation (Larson 1981; Bolatto et al. 2008; Heyer et al. 2009; Roman-Duval et al. 2010; Wong et al. 2011). In this case, larger clouds will inevitably have higher ${ \mathcal M }$, where theory predicts higher ${\epsilon }_{\mathrm{ff}}$ as stronger and denser compression leads to higher SFRs. We have verified that the discrepancy between the model and the observations is independent of any choice/combination of fudge factor, normalization constant, and level of observational uncertainty.

Milky Way GMCs have larger ${v}_{\mathrm{CO}}$ and ${\alpha }_{\mathrm{vir}}$, which tend to decrease ${\epsilon }_{\mathrm{ff}}$, but this is counteracted by a simultaneous increase in ${ \mathcal M }$ elevating ${\epsilon }_{\mathrm{ff}}$. Consequently, at the same ξ and model fudge factors, the LMC and Milky Way clouds populate a continuous parameter space in Figure 6(b) in the PN11, HC11, and FK12 models. However, in the KM05 model, the LMC and Milky Way clouds significantly differ in ${\epsilon }_{\mathrm{ff}}$, since the KM05 model is relatively insensitive to an increase in Mach number (KM05; FK12).

None of the models predict significant changes in ${\epsilon }_{\mathrm{ff}}$ between cloud types, which is likely because ${v}_{\mathrm{CO}}$ is independent of SFR (Table 1), as turbulent driving and dissipation scale together (Kim et al. 2011). Nonetheless, the observed offset between cloud types can be reproduced if the cloud types experience different turbulent forcing, i.e., solenoidal, mixed, or compressive (Figure 6). Indeed, Type 3 clouds contain nearby stellar clusters and may encounter a higher rate of (nearby) stellar feedback, which may inject substantial compressive forcing locally (i.e., higher b). Nonetheless, any cloud compression must be acting on a relatively small mass fraction, since $\tilde{{\tau }_{\mathrm{ff}}}$ does not differ significantly between Type 1, Type 2, and Type 3 clouds.

The model realizations reveal a large scatter similar to that of the observations when all observational uncertainties are propagated (see Table 3), which was not performed by Lee et al. (2016). However, the model-predicted scatter is largely driven by lower-mass clouds (Figure 6), which is not reflected in the observations. This indicates that a different physical mechanism drives the scatter in the observations, which is not captured in the analytical star formation models (see Section 4).

Table 3.  Star Formation Efficiencies per Freefall Time: Observations vs. Theory

	Observations
	MAGMA				J16				MW
	Type 1	Type 2	Type 3	All	Type 1	Type 2	Type 3	All
$\langle {\epsilon }_{\mathrm{ff},\mathrm{MYSO}}\rangle$	0.08	0.12	0.12	0.12	⋯	0.04	0.06	0.05	⋯
$\langle {\epsilon }_{\mathrm{ff},{\rm{H}}\alpha }\rangle$	⋯	0.09	0.37	0.25	⋯	0.07	0.12	0.10	0.03
$\langle {\epsilon }_{\mathrm{ff},{\rm{H}}\alpha }\rangle$ a	⋯	0.09	0.24	0.17	⋯	0.07	0.12	0.10	0.03
${\alpha }_{{\epsilon }_{\mathrm{ff},\mathrm{MYSO}}}$	⋯	−0.54 ± 0.07	−0.31 ± 0.07	−0.37 ± 0.05	⋯	−0.30 ± 0.10	⋯	⋯
${\alpha }_{{\epsilon }_{\mathrm{ff},{\rm{H}}\alpha }}$	⋯	−0.56 ± 0.25	−0.53 ± 0.21	−0.38 ± 0.14	⋯	−0.30 ± 0.29	⋯	⋯	−0.25
${\alpha }_{{\epsilon }_{\mathrm{ff},{\rm{H}}\alpha }}$ a	⋯	−0.59 ± 0.27	−0.39 ± 0.20	−0.30 ± 0.15	⋯	−0.28 ± 0.31	⋯	⋯	−0.25
${\sigma }_{\mathrm{log}{\epsilon }_{\mathrm{ff},\mathrm{MYSO}}}$	⋯	0.38	0.37	0.41	⋯	⋯	0.35	⋯	⋯
${\sigma }_{\mathrm{log}{\epsilon }_{\mathrm{ff},{\rm{H}}\alpha }}$	⋯	0.36	0.57	0.58	⋯	0.47	⋯	⋯	0.91
${\sigma }_{\mathrm{log}{\epsilon }_{\mathrm{ff},{\rm{H}}\alpha }}$ a	⋯	0.36	0.44	0.49	⋯	0.47	⋯	⋯	0.91
	Analytical Models
	MAGMA				Turbulent Forcing Parameters				MWb
	Type 1	Type 2	Type 3	All	b = 0.33	b = 0.40	b = 1.00	All
$\langle {\epsilon }_{\mathrm{ff},\mathrm{KM}05}\rangle$	0.06	0.07	0.07	0.07	0.04	0.05	0.14	0.07	0.04
$\langle {\epsilon }_{\mathrm{ff},\mathrm{PN}11}\rangle$	0.05	0.05	0.06	0.05	0.02	0.04	0.16	0.05	0.12
$\langle {\epsilon }_{\mathrm{ff},\mathrm{HC}11}\rangle$	0.05	0.05	0.09	0.06	0.03	0.05	0.14	0.06	0.27
$\langle {\epsilon }_{\mathrm{ff},\mathrm{FK}12}\rangle$	0.05	0.05	0.07	0.05	0.05	0.05	0.07	0.08	0.23
${\alpha }_{{\epsilon }_{\mathrm{ff},\mathrm{KM}05}}$	0.34 ± 0.18	0.29 ± 0.06	0.29 ± 0.10	0.25 ± 0.07	0.27 ± 0.11	0.29 ± 0.13	0.22 ± 0.12	0.26 ± 0.06	0.21
${\alpha }_{{\epsilon }_{\mathrm{ff},\mathrm{PN}11}}$	0.40 ± 0.19	0.43 ± 0.11	0.37 ± 0.10	0.41 ± 0.04	0.61 ± 0.11	0.49 ± 0.09	0.18 ± 0.05	0.42 ± 0.08	0.30
${\alpha }_{{\epsilon }_{\mathrm{ff},\mathrm{HC}11}}$	0.52 ± 0.25	0.47 ± 0.09	0.34 ± 0.10	0.40 ± 0.05	0.58 ± 0.08	0.49 ± 0.11	0.38 ± 0.08	0.44 ± 0.06	0.15
${\alpha }_{{\epsilon }_{\mathrm{ff},\mathrm{FK}12}}$	0.51 ± 0.23	0.40 ± 0.08	0.30 ± 0.09	0.43 ± 0.05	0.36 ± 0.10	0.38 ± 0.12	0.27 ± 0.09	0.32 ± 0.05	0.15
${\sigma }_{\mathrm{log}{\epsilon }_{\mathrm{ff},\mathrm{KM}05}}$	0.61	0.55	0.49	0.55	0.59	0.52	0.33	0.56	0.28
${\sigma }_{\mathrm{log}{\epsilon }_{\mathrm{ff},\mathrm{PN}11}}$	0.47	0.45	0.42	0.47	0.38	0.30	0.09	0.48	0.10
${\sigma }_{\mathrm{log}{\epsilon }_{\mathrm{ff},\mathrm{HC}11}}$	0.74	0.64	0.52	0.65	0.64	0.59	0.48	0.65	0.19
${\sigma }_{\mathrm{log}{\epsilon }_{\mathrm{ff},\mathrm{FK}12}}$	0.63	0.65	0.56	0.62	0.62	0.54	0.33	0.51	0.17

Notes. Listed are the average per freefall time measured with MYSOs ($\langle {\epsilon }_{\mathrm{ff},\mathrm{MYSO}}\rangle$) and Hα ($\langle {\epsilon }_{\mathrm{ff},{\rm{H}}\alpha }\rangle$); slope indices ${\alpha }_{\mathrm{ff}}$, and the scatter around the fitted regression (${\alpha }_{{\epsilon }_{\mathrm{ff}}}:$ in dex). The same parameters are listed for the KM05, PN11, HC11, and FK12 turbulence-regulated star formation models (see text).

^aThese values refer to the fits by excluding the 30 Doradus and SMR region (see text). ^bObservational uncertainties for several of the values in Table 2 are not given in Lee et al. (2016) and are thus not propagated. In addition, for consistency between the LMC and the Galaxy, these values are calculated using the normalization constant ξ of the LMC and b = 0.4 (see text and Figure 6).

Download table as:  ASCIITypeset image

In the LMC, the SFR of GMCs increases over multiple generations (i.e., ${\mathrm{SFR}}_{\mathrm{MYSO}}$ and ${\mathrm{SFR}}_{{\rm{H}}\alpha }$) across the evolutionary sequence of GMCs proposed by Kawamura et al. (2009). Whereas ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ is known to be affected by various systematics creating scatter and potential bias (Section 2.3), direct counting of MYSOs is not affected in this sense. The relative contribution of ${\mathrm{SFR}}_{\mathrm{MYSO}}$ and ${\mathrm{SFR}}_{{\rm{H}}\alpha }$ varies between cloud types, but both tracers show good agreement for the entire ensemble of GMCs (Table 1), albeit with the scatter that is expected for stochastic sampling of the high-end tail of the IMF at these SFRs (Table 2; da Silva et al. 2012; Krumholz et al. 2014). Comparison of SFRs from indirect star formation tracers with MYSOs counting on large scales remains complicated in the Milky Way because of confusion and distance ambiguities. Such studies are currently limited to the Magellanic system but will soon be possible in nearby galaxies with the upcoming James Webb Space Telescope.

The increase of SFR between cloud types drives the observed scatter of $\epsilon ^{\prime}$ and ${\epsilon }_{\mathrm{ff}}$ with ${M}_{\mathrm{cloud}}$, and it is very likely that the observed scatter in the Milky Way originates from the same physical mechanism (Mooney & Solomon 1988; Mead et al. 1990; Murray & Rahman 2010; Lee et al. 2016; Vutisalchavakul et al. 2016). In addition, we have determined that both $\epsilon ^{\prime}$ and ${\epsilon }_{\mathrm{ff}}$ decline with ${M}_{\mathrm{cloud}}$ (Figures 3(b) and 6(a)), as SFRs are not linearly correlated with total ${M}_{\mathrm{cloud}}$ on individual cloud scales in the LMC and the Galaxy (Figure 3(a) and Table 1). Massive clouds appear to have larger diffuse (i.e., non-star-forming) envelopes, which potentially affects the global star formation potential of GMCs (Figure 5). One implication of this result is a higher molecular depletion time ${\tau }_{\mathrm{dep}}$ with increasing GMC mass. This may be related to trends observed recently on kpc and galactic scales by Leroy et al. (2013) and Bigiel et al. (2016), who found higher ${\tau }_{\mathrm{dep}}$ in regions of high surface density.

We have tested the turbulence-regulated analytical star formation models of KM05, PN11, HC11, and FK12 and shown that they are unable to reproduce the decline of ${\epsilon }_{\mathrm{ff}}$ with increasing ${M}_{\mathrm{cloud}}$ (Figure 6). This discrepancy likely originates from the GMC size–linewidth relation, resulting in higher Mach numbers and higher ${\epsilon }_{\mathrm{ff}}$ in all considered models (Section 3.3.2). In this regard, several authors have questioned the existence of the size–linewidth relation (Vázquez-Semadeni et al. 1997; Ballesteros-Paredes & Mac Low 2002; Ballesteros-Paredes et al. 2012) because of the limited dynamic range of the observations. These challenges were in turn challenged by Lombardi et al. (2010) using near-infrared extinction mapping, a technique that can probe large dynamic ranges. Regardless of the method used, an intrinsic column density threshold defining molecular clouds may introduce a bias, as this will affect clouds of varying sizes differently. Further research to disentangle the effects of GMC definition and its impact on derived star formation properties is necessary in this sense.

In relation to the above, our results emphasize the importance of a consistent definition of GMCs. Comparison of SFR, $\epsilon ^{\prime}$, and ${\epsilon }_{\mathrm{ff}}$ between two LMC cloud sets (MAGMA and J16) reveals significant differences in the absolute values of these quantities. Unfortunately, in many cases, the exact definition of GMCs will be dictated by the sensitivity, resolution, and cloud decomposition of the observations, as well as the tracer of molecular material that is used. Once resolved maps of nearby galaxies become readily available (e.g., this work; Bigiel et al. 2016), this may complicate the comparison of SFRs and efficiencies of nearby clouds (Evans et al. 2009; Heiderman et al. 2010; Lada et al. 2010) with those in galaxies at different metallicities and/or surface densities (Hughes et al. 2013). Nonetheless, the main results of this work (i.e., the systematic offset between cloud types and the decline of $\epsilon ^{\prime}$ and ${\epsilon }_{\mathrm{ff}}$ with cloud mass) are unchanged regardless of the cloud map used.

While the absolute scatter of the observations can be reproduced with the star formation models, the model-predicted scatter is largely driven by lower-mass clouds (Figure 6), which is not in agreement with the observations. Instead, we have shown that the increasing SFR as a function of time can perfectly drive the scatter (Sections 3.2 and 3.3). Our results may therefore confirm that SFR is a time-variable quantity that appears to increase as soon as stellar clusters emerge and start to disrupt the parent GMCs (Murray 2011; Lee et al. 2016).

While the KM05, PN11, HC11, and FK12 models naturally explain the observed core mass functions at masses of ≲10 ${M}_{\odot }$ (André et al. 2010), suggesting that the large-scale turbulent properties of GMCs are important in low-to-intermediate-mass star formation (Padoan et al. 2014), it has been unclear if this extends to the formation of massive stars (Offner et al. 2014). Here, we have shown that these analytical models, idealizing global turbulence levels, cloud densities, and assuming a stationary SFR, do not reproduce observations from modern large data sets tracing massive star formation on galaxy-wide scales (this work; Lee et al. 2016). The discrepancy may originate from the assumption of a stationary SFR in the models but may also reflect the fact that global turbulent properties are irrelevant to the formation of massive stars.

The large scatter in SFR and star formation efficiency at individual cloud scales arises when total GMC masses are considered (this work; Mooney & Solomon 1988; Mead et al. 1990; Murray & Rahman 2010; Lee et al. 2016; Vutisalchavakul et al. 2016). In contrast, SFR shows a roughly linear relation with dense gas with significantly less scatter (Wu et al. 2005; Heiderman et al. 2010; Lada et al. 2010; Evans et al. 2014; Vutisalchavakul et al. 2016). This illustrates the inability of ¹²CO(1–0), ¹³CO(1–0), or dust-based measures of molecular gas to probe the actual star-forming component of the parent GMC (e.g., Gao & Solomon 2004). The observed scatter in star formation efficiency then reflects a varying fraction of dense gas in GMCs (Krumholz et al. 2012; Lada et al. 2012).

Following the above, the observed decline in star formation efficiency and ${\epsilon }_{\mathrm{ff}}$ with ${M}_{\mathrm{cloud}}$ then simply reflects that the amount of gas participating in the formation of massive stars does not scale with total cloud mass. This is substantiated by the observation that GMCs mostly grow by adding diffuse ("CO-dark") gas (Figure 5). Indeed, massive star formation appears to occupy only a small volume of its parent GMC (Ochsendorf et al. 2016), a configuration that is readily recognized in nearby massive star-forming regions such as Orion (Bally et al. 1987) and M17 (Povich et al. 2007). In this regard, it has been questioned whether the Kennicutt–Schmidt law (Schmidt 1959; Kennicutt 1998) is the result of an actual underlying star formation law or simply a means of counting clouds (Lada et al. 2012). In other words, by averaging over kpc scales, one mainly traces gas that may be irrelevant to the process of massive star formation.

It is clear that the dense gas fraction is crucial to the massive star-forming properties of GMCs. However, the physical mechanisms regulating the dense gas fraction are less clear. In this sense, the increase in SFR between Type 1, Type 2, and Type 3 clouds and the decline of $\epsilon ^{\prime}$ and ${\epsilon }_{\mathrm{ff}}$ with increasing cloud mass (Figures 3, 4, and 6) may show that stellar feedback injected locally is of key importance in shaping GMCs by controlling the dense gas fraction and thereby the massive SFR. In other words, the first effect of massive star feedback is "positive," increasing the local SFR (Elmegreen & Lada 1977). Only later does feedback seem to be negative (Hopkins et al. 2011). Understanding this transition will be important for a thorough understanding of the massive star-forming process.