THE STAR FORMATION RATE AND GAS SURFACE DENSITY RELATION IN THE MILKY WAY: IMPLICATIONS FOR EXTRAGALACTIC STUDIES

We derive cloud masses (M_gas,cloud) and mean surface densities (Σ_gas,cloud) from the extinction maps, which were produced from a combination of Two Micron All Sky Survey (2MASS) and Spitzer data, ranging from 1.25 μm to 24 μm. In this wavelength range, the spectral energy distributions (SEDs) of sources classified as stars provide measurements of the visual extinction (A_V) along lines of sight through the clouds (Evans et al. 2007; T. Huard et al. 2010, in preparation). Line-of-sight extinctions were determined by fitting the SEDs, adopting the Weingartner & Draine (2001) extinction law with R_V = A_V/E(B − V) = 5.5. Extinction maps were constructed by convolving these line-of-sight measures with uniformly spaced Gaussian beams. The c2d team observed "off-cloud fields" for four of the molecular clouds: Chamaeleon, Perseus, Ophiuchus, and Lupus. The line-of-sight extinction measurements from these off-cloud fields suggested A_V calibration offsets of 1–2 mag; therefore, in constructing the maps for these four clouds, Evans et al. (2007) subtracted these calibration offsets. Since no off-cloud field had been observed for Serpens, they used a weighted mean of the A_V calibration offsets to correct the calibration of their Serpens extinction maps. No off-cloud fields were observed for the GB survey. Further analysis of the fitting of the line-of-sight extinctions demonstrates that the inferred calibration offsets strongly depend on which wavebands had detections (T. Huard et al. 2010, in preparation). For example, sources with only near-infrared (2MASS) detections may suggest no calibration offset, while sources with only mid-infrared (Spitzer) detections show greater calibration offsets, perhaps as high as 2–3 mag. This finding suggests that the Weingartner & Draine (2001) extinction law does not accurately characterize the reddening through the full range 1.25–24 μm spectral range. For if it did, the inferred A_V calibration offsets should be independent of the detected wavebands. For this reason, the extinction maps delivered by the GB survey make use of the cataloged line-of-sight extinctions with no correction for potential calibration offsets, and, for consistency, they suggest that the previously adopted A_V calibration "offsets" of 1–2 mag be added to the c2d extinction maps (T. Huard et al. 2010, in preparation). After revising accordingly the extinctions in the clouds mapped by c2d, we find that the gas masses, and thus the cloud surface densities, are ∼20%–30% greater than those previously published by Evans et al. (2009). The extinction maps used in this study probe to higher A_V (up to 40 mag) than some previous studies (e.g., Pineda et al. 2008; Lombardi et al. 2008, 2010) due to the inclusion of both 2MASS and mid-IR Spitzer data.

In order to compute the M_gas,cloud, we chose extinction maps with 270'' beams for all clouds. We base this choice on the best resolution map available for Ophiuchus, which is limited in resolution due to a large extended region of high extinction with relatively few background stars detected. M_gas,cloud and Σ_gas,cloud were calculated by summing up extinction map measurements and converting to the column density using the relation N_H/A_V = (1.086C_ext(V))⁻¹ = 1.37 × 10²¹ cm⁻² mag⁻¹ (Draine 2003) for a Weingartner & Draine (2001) R_V = 5.5 extinction law, where C_ext(V) = 6.715 × 10⁻²² cm²/H from the online tables,⁵ using Equations (5) and (6), respectively. The uncertainties in M_gas,cloud are computed from maps of extinction uncertainty, which account for the statistical photometric uncertainties, but not systematic uncertainties in using the extinction law calibration.

We compute M_gas,cloud by summing up all pixels $(\ssty\sum A_{V})$ above A_V = 2 in all clouds except for Serpens and Ophiuchus which are covered by the c2d survey completely down to A_V = 6 and 3, respectively. M_gas,cloud is then

$$\begin{eqnarray} M_{\rm gas, cloud} &=& \mu \,m_{\rm H}\bigl (1.086C_{\rm ext}(V)\bigr)^{-1} \times \sum A_{V} \times {A_{\rm pixel}} \nonumber \\ &\approx& 1.58\times 10^{-36} \times \sum \ \biggl (\frac{A_{V}}{\rm mag}\biggr) \times \biggl (\frac{A_{\rm pixel}}{\rm cm^{2}}\biggr) \, (M_{\odot }),\nonumber\\ \end{eqnarray} \tag{ 5 }$$

where the mean molecular weight (μ) is 1.37, the total number of hydrogen atoms is N(H) ≡ N(H i) + 2N(H₂), and we assume a standard molecular cloud composition of 63% hydrogen, 36% helium, and 1% dust, m_H is the mass of hydrogen in grams, the area of a pixel in square cm (A_pixel) in the extinction map is (π/180/3600)² D(cm)²R('')², where R('') is the pixel size in arcseconds, and A_cloud is the area of the cloud measured in square pc. We divide M_gas,cloud by the area to obtain Σ_gas,cloud for each cloud:

$$\begin{eqnarray} \Sigma _{\rm gas,cloud} &=& \left(\frac{M_{\rm gas, cloud}}{M_{\odot }} \right) \times \left(\frac{A_{\rm cloud}}{\rm pc^{2}}\right)^{-1}\ (M_{\odot } \;\rm pc^{-2}) \\ \Sigma _{\rm gas,cloud} &=& 15 \biggl (\frac{A_{V}}{\rm mag}\biggr) (M_{\odot } \;\rm pc^{-2}). \end{eqnarray} \tag{ 6 }$$

Measured cloud properties for c2d and GB clouds within a contour of A_V>2 or A_V completeness limit are shown in Table 1.

We estimate the SFR from the total number of YSOs (N_YSO,tot) contained in an area where A_V>2, as described in Section 2.1. We assume a mean YSO mass (〈M_YSO〉) of 0.5 ± 0.1 M_☉, where the mean estimated error in mass is derived from the mass distribution of YSOs in Cha II from Spezzi et al. (2008). The mean YSO mass is consistent with IMF studies by Chabrier (2003), Kroupa (2002), and Ninkovic & Trajkovska (2006). We also assume a period for star formation (t_Class II) of 2 ± 1 Myr, based on the estimate of the elapsed time between formation and the end of the Class II phase (Evans et al. 2009). This SFR assumes that star formation has been continuous over a period greater than t_Class II. All clouds have Class III objects, indicating that star formation has continued for longer than t_Class II. The SFR measured in this way could be underestimated or overestimated in any particular cloud, but over an ensemble of 20 clouds, it should be the most reliable SFR indicator available because no extrapolation from the massive star tail of the IMF is needed. We base our error estimates by choosing the largest error from either the systematic error, combined in quadrature from mean YSO mass and period of star formation, or the Poisson error from YSO number counts.

$$\begin{eqnarray} \Sigma _{\rm SFR} &=& N_{\rm YSO, tot}\times \biggl (\frac{\langle M_{\rm YSO}\rangle }{M_{\odot }}\biggr) \times \biggl (\frac{t_{\rm Class\,II}}{\rm Myr} \biggr)^{-1} \nonumber \\ &&\times\biggl (\frac{A_{\rm cloud}}{\rm kpc^{2}} \biggr)^{-1} (M_{\odot } \;\rm yr^{-1} \;\rm kpc^{-2}). \end{eqnarray} \tag{ 8 }$$

Table 1 lists values for clouds within a contour of A_V>2 for all c2d and GB clouds. We show our estimated Σ_gas,cloud and Σ_SFR for the c2d and GB clouds in Figure 1. Σ_gas,cloud ranges from ∼50 to 140 M_☉ pc⁻², and Σ_SFR ranges from ∼0.4 to 3.4 M_☉ kpc⁻² yr⁻¹. We use these units for convenience in comparing to the extragalactic relations.

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We compare the observations to the predicted values for Σ_SFR using Σ_gas,cloud that we calculate for the c2d and GB clouds. We plot these extragalactic relations in Figure 1 and will include them in all the following SFR–gas relation figures. The solid lines represent the regime where they were fitted to data and the dashed lines are extrapolated relations spanning the range of Σ_gas. The blue line is from disk-averaged or global SFR measurements based on Hα emission and the total (H i + CO) gas surface densities in a sample of normal spirals and starburst galaxies from Kennicutt (1998b). The red line is from Bigiel et al. (2008), who made sub-kpc resolution measurements in a sample of spiral and dwarf galaxies using SFRs based on a combination of Spitzer 24 μm and Galaxy Evolution Explorer (GALEX) UV data and use CO measurements to obtain a relation for H₂ gas surface density. Both of these studies trace either obscured (24 μm) or unobscured (Hα and UV) massive star formation and are blind to regions of low-mass star formation that we are measuring in this work. We also compare to the theoretical total (H i + CO) gas and SFR relation of Krumholz et al. (2009; orange solid line). This prediction takes into account three factors: the conversion of atomic to molecular gas, metallicity, and clumping of the gas. For our comparisons, we choose galactic solar metallicity and a clumping factor of 1, which corresponds to clumping on 100 pc scales. We include data points for the Taurus molecular cloud, including YSO counts from Rebull et al. (2010), Σ_gas from a 2MASS extinction map (Pineda et al. 2010), and the total ¹³CO and ¹²CO gas mass from Goldsmith et al. (2008).

If we take the average Σ_gas,cloud defined as the total M_gas,cloud divided by the total area (A_cloud) in pc², we would find that the average molecular cloud in this study has a surface density of 91.5 M_☉ pc⁻² and a Σ_SFR of 1.2 M_☉ kpc⁻² yr⁻¹. Taking this average Σ_gas,cloud and calculating what the extragalactic relations would predict for the average cloud SFR surface density, we would get 0.13, 0.07, and 0.03 M_☉ kpc⁻² yr⁻¹ for Kennicutt (1998b), Bigiel et al. (2008), and Krumholz et al. (2009), respectively. The observed values exceed the observed extragalactic Σ_SFR predictions by factors of ∼9–17 and the theoretical prediction by a factor of ∼40. While the star formation surface density, Σ_SFR of 1.2 M_☉ kpc⁻² yr⁻¹, seem high, the clouds fill only a small fraction of the local square kpc. From Table 1, the total SFR is 781 M_☉ Myr⁻¹. If we remove the IC5146 clouds, which are more distant than 0.5 kpc, the SFR within 0.5 kpc is 748 M_☉ Myr⁻¹ or 7.5×10⁻⁴ M_☉ yr⁻¹. Extrapolated to the Galaxy with a star-forming radius of 10 kpc, this would amount to 0.3 M_☉ yr⁻¹, less than the rate estimated for the entire galaxy of 0.68–1.45 M_☉ yr⁻¹ (Robitaille & Whitney 2010). This local, low-mass star formation mode thus could account for a substantial, but not dominant, amount of star formation in our Galaxy.

The last section gave us estimates over the whole molecular cloud including all YSOs in each cloud. Early work surveying large areas of clouds (e.g., Lada 1992) suggested that star formation is concentrated in regions within molecular clouds in regions of high densities (n ∼ 10⁴ cm⁻³). The c2d and GB studies of many whole clouds have clearly established that star formation is not spread uniformly over clouds, but is concentrated in regions at high extinction. Furthermore, the youngest YSOs and dense cores (Enoch et al. 2007) are the most highly concentrated at high A_V (Evans et al. 2009; Bressert et al. 2010). Older YSOs can leave their original formation region or even disperse the gas and dust. Taking the average velocity dispersion of a core to be 1 km s⁻¹, a 2 Myr old YSO could travel ∼2 pc, roughly the average radius of a cloud in this study. We therefore apply a conservative approach and only estimate the SFRs using the youngest Class I or Flat SED YSOs (see Greene et al. (1994) for the definition of classes) that have not yet migrated from their birthplace. To classify YSOs as Class I or Flat SED, we use the extinction corrected spectral index from Evans et al. (2009) for the c2d clouds and the uncorrected spectral index for the GB clouds (Table 1). These two classes of YSOs have timescales of 0.55 ± 0.28 and 0.36 ± 0.18 Myr, respectively (L. Allen et al. 2010, in preparation).

In order to measure Σ_SFR and Σ_gas for the youngest YSOs, we divide the clouds into equally spaced contour levels of A_V or Σ_gas,con and measure the SFR, mass (M_con), and area (A_con) enclosed in that contour level. The contour intervals start from the extinction map completeness limits (Section 2.1) and are spaced such that they are wider than our map beam size of 270'' as shown in Figure 2. We compute the gas surface density (Σ_gas,con) in the same way as in Equation (6), but this time using only the mass (M_gas,con) and area (A_con) enclosed in the A_V contour region:

$$\begin{equation} \Sigma _{\rm gas,con} = \left(\frac{M_{\rm gas, con}}{M_{\odot }} \right) \times \left(\frac{A_{\rm con}}{{\rm pc}^{2}}\right)^{-1}\;(M_{\odot } \;{\rm pc}^{-2}). \end{equation} \tag{ 9 }$$

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If there are no YSOs found in the contour region, we compute an upper limit to the SFR by assuming that there is one YSO in that region. The upper limits are denoted by the asterisks in Table 2. We estimate the uncertainties in the both the SFR and Σ_SFR by choosing the largest error: either the systematic or Poisson error from YSO counts.

2.3.1. MISidentified YSOs from SED FITS (MISFITS)

After removing the MISFITS from our Class I and Flat SED YSO sample, we show the number, M_gas,con, Σ_gas,con, SFRs, and Σ_SFR in Table 2 for all contour levels in each cloud or separate cloud component (see Section 2). In Figure 3, we show the Σ_gas,con and Σ_SFR densities for both Class I and Flat SED sources (green and magenta stars) and upper limits for each class (green and magenta inverted triangles) that we measured in contour regions described in Section 2.3, with extragalactic observational relations overplotted. A wider range in both Σ_gas (∼45–560 M_☉ pc⁻²) and Σ_SFR (∼0.03–95 M_☉ kpc⁻² yr⁻¹) are found for contour regions compared to the total cloud measurements. We note that the points for Sco and Cep (Kirk et al. 2009) clouds are obtained by co-adding the separate cloud regions using the same contour intervals. Since these points sample regions with non-uniform A_V, they only provide an estimate of Σ_SFR and Σ_gas. These points lie at Σ_SFR < 6 M_☉ kpc⁻² yr⁻¹ and high Σ_gas>330 M_☉ pc⁻².

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We compare our YSO contour results to extragalactic relations and find that most points lie well above the extragalactic relations. Excluding upper limits, the mean values of Σ_SFR and Σ_gas,con are of 9.7 M_☉ kpc⁻² yr⁻¹ and 225 M_☉ pc⁻², respectively. Evaluated at this mean gas surface density, the extragalactic relations underpredict Σ_SFR by factors of ∼21–54. The mean YSO contour lies above the Krumholz et al. (2009) extragalactic SFR–gas relation prediction by ∼2 orders of magnitude. We explore the differences between the Galactic and extragalactic SFR–gas relations in Section 3.

Since extinction maps are direct probes of Σ_gas, they provide the best measure of the total gas and are optimal for use in determining the Σ_gas of molecular clouds. However, A_V maps are not easily obtainable in extragalactic studies, which instead employ CO maps, particularly ¹²CO J = 1–0, to determine Σ_gas of molecular hydrogen. Since the molecular hydrogen (H₂) rotational transitions require high temperatures not found in the bulk of molecular clouds, other tracers of dense gas are used to estimate the amount of H₂. The next most abundant molecule with easily observable excitation properties in a molecular cloud is ¹²CO J = 1–0. In this study, we want to explore how well CO traces A_V as a function of M_gas or Σ_gas. We can directly test this in two galactic clouds, Perseus and Ophiuchus, which both have ¹²CO J = 1–0 and ¹³CO J = 1–0 maps from the Five College Radio Astronomy Observatory (FCRAO) COordinated Molecular Probe Line Extinction Thermal Emission (COMPLETE) Survey of star-forming regions (Ridge et al. 2006).

In order to directly compare the CO maps from the COMPLETE survey to the A_V maps in this study, we interpolate the CO data onto the A_V map grid with a pixel size of 45''. We integrate the publicly available CO data cubes over the velocity range from 0 to 15 km s⁻¹ to create moment zero maps of integrated intensity defined as I_CO(x, y) ≡ ∫T_mb(x, y, z)dV K km s⁻¹, where T_mb ≡ T*_A/η_mb is the main beam brightness temperature defined as the antenna temperature (T*_A) divided by the main beam efficiency (η_mb) of 0.45 and 0.49 for ¹²CO J = 1–0 and ¹³CO J = 1–0, respectively, from Pineda et al. (2008). Corresponding rms noise maps (σ_T(x,y)) were constructed by calculating the standard deviation of intensity values within each spectroscopic channel where no signal is detected. In order to determine the gas surface density of H₂ ($\Sigma _{\rm H_{2}}$), we must first compute the column density of H₂. The column density of H₂ is estimated from the ¹²CO map by using a CO-to-H₂ conversion factor (X_CO), which is defined as the ratio of H₂ column density to the integrated intensity ($X_{\rm CO} \equiv N_{\rm H_{2}}/I_{\rm CO}$). Similarly for the ¹³CO map, the column density is derived from the ¹²CO and ¹³CO maps, assuming local thermodynamic equilibrium (LTE) and an abundance ratio of H₂ to ¹³CO. To compare $\Sigma _{\rm H_{2}}$ from ¹²CO and ¹³CO to the Σ_gas from A_V, we use only regions in the CO maps that have emission lines with positive integrated intensities and line peaks that are greater than five times the rms noise. Our masses from extinction measurements used for this comparison are therefore slightly lower than those in Tables 1 and 2 by ∼5% (see Tables 4 and 5).

Table 4. A_V, ¹²CO, and ¹³CO Masses and Σ_gas for Per and Oph Clouds

Cloud	Mcloud,gas	$M_{{\rm cloud},^{12}{\rm CO}}$	$M_{{\rm cloud},^{13}{\rm CO}}$	Σcloud,gas	$\Sigma _{{\rm cloud},^{12}{\rm CO}}$	$\Sigma _{{\rm cloud},^{13}{\rm CO}}$
	(M☉)	(M☉)	(M☉)	(M☉ pc−2)	(M☉ pc−2)	(M☉ pc−2)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
Per	5997 ± 3387	9657 ± 2416	1073 ± 110	82 ± 33	132 ± 33	15 ± 2
Oph	2270 ± 1533	2596 ± 659	348 ± 39	77 ± 42	88 ± 22	12 ± 1
Data from literature
Taurusa	27207	16052		108	64

Notes. (1) Cloud name; (2) mass from A_V map (M_☉) where there is positive ¹²CO and ¹³CO emission; (3) ¹²CO mass (M_☉); (4) ¹³CO mass (M_☉); (5) surface gas density from A_V map (M_☉ pc⁻²); (6) ¹²CO surface gas density (M_☉ pc⁻²); (7) ¹³CO surface gas density (M_☉ pc⁻²). ^aCombined ¹² CO and ¹³CO mass from Goldsmith et al. (2008) and A_V mass from Pineda et al. (2010).

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Table 5. A_V, ¹²CO, and ¹³CO Masses and Σ_gas for Per and Oph Clouds in A_V Contours

Cloud	Contour	Mcon,gas	$M_{{\rm con}, ^{12}{\rm CO}}$	$M_{{\rm con}, ^{13}{\rm CO}}$	Σcon,gas	$\Sigma _{{\rm con},^{12}{\rm CO}}$	$\Sigma _{{\rm con},^{13}{\rm CO}}$
	Levels	(M☉)	(M☉)	(M☉)	(M☉ pc−2)	(M☉ pc−2)	(M☉ pc−2)
	(mag)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
Per	6.5	3140 ± 2100	6050 ± 1500	518 ± 54	60.0 ± 31	116 ± 29	9.90 ± 1.0
	11.0	1770 ± 880	2560 ± 640	349 ± 35	115 ± 33	166 ± 42	22.7 ± 2.3
	15.5	653 ± 300	654 ± 160	120 ± 12	172 ± 40	173 ± 43	31.7 ± 3.2
	20.0	370 ± 160	344 ± 86	76.2 ± 7.6	248 ± 47	231 ± 58	51.1 ± 5.1
	24.5	49.5 ± 23	49.4 ± 12	8.91 ± 0.89	225 ± 53	225 ± 56	40.5 ± 4.1
	30.0	6.06 ± 2.8	3.35 ± 0.84	0.671 ± 0.067	339 ± 81	187 ± 47	37.5 ± 3.8
Oph	10.5	1580 ± 1200	1980 ± 510	183 ± 24	59.4 ± 40	74.4 ± 19	6.88 ± 0.89
	18.0	349 ± 160	328 ± 82	71.1 ± 7.3	188 ± 46	176 ± 44	38.2 ± 3.9
	25.5	161 ± 73	136 ± 34	42.6 ± 4.3	282 ± 58	239 ± 60	74.7 ± 7.5
	33.0	112 ± 51	97.1 ± 24	32.2 ± 3.2	331 ± 72	287 ± 72	95.3 ± 9.6
	41.0	70.6 ± 32	52.9 ± 13	18.6 ± 1.9	406 ± 90	304 ± 76	107 ± 11

Notes. (1) Cloud name; (2) A_V contour level in mag at which mass measurement was made. The contour levels start at A_V = 2 or the cloud completeness limit and increase in even intervals to the listed contour level; (3) A_V mass (M_☉) where there is positive ¹²CO and ¹³CO emission; (4) ¹²CO mass (M_☉); (5) ¹³CO mass (M_☉); (6) A_V surface gas density (M_☉ pc⁻²); (7) ¹²CO surface gas density (M_☉ pc⁻²); (8) ¹³CO surface gas density (M_☉ pc⁻²).

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The X_CO factor for ¹²CO has been derived using a variety of methods such as gamma-ray emission caused by the collision of cosmic rays with hydrogen (Bloemen et al. 1986), virial mass methods (Solomon et al. 1987; Blitz et al. 2007), maps of dust emission from IRAS and assuming a constant dust-to-gas ratio (Frerking et al. 1982), extinction maps from optical star counts (Duvert et al. 1986; Bachiller & Cernicharo 1986; Langer et al. 1989) and 2MASS data (Lombardi et al. 2006; Pineda et al. 2008), and theoretically by the assumption that giant molecular clouds are in gravitational equilibrium (Dickman et al. 1986; Heyer et al. 2001). All these studies find a range of X_CO of (0.9–4.8) × 10²⁰ cm⁻² K⁻¹ km⁻¹ s, but they were almost all restricted to regions with A_V < 6 mag. Studies of extragalactic SFR–gas relations chose values close to the average galactic X_CO measurements in the literature: 2.0 × 10²⁰ cm⁻² K⁻¹ km⁻¹ (Bigiel et al. 2008) or 2.8 × 10²⁰ cm⁻² K⁻¹ km⁻¹ s from Bloemen et al. (1986; Kennicutt 1998b; Kennicutt et al. 2007; Blanc et al. 2009). Since the goal of this study is to compare to extragalactic measurements, we choose a X_CO of (2.8 ± 0.7) × 10²⁰ cm⁻² K⁻¹ km⁻¹ s from Bloemen et al. (1986) to be consistent with the study of Kennicutt (1998b).

We compute the column density of H₂ from ¹²CO measurements using the equation

$$\begin{equation} N_{\rm H_{2}(^{12}\rm CO)} = X_{^{12}\rm CO} \times I_{^{12} \rm CO} \ (\rm cm^{-2}). \end{equation} \tag{ 10 }$$

This can be rewritten in terms of gas surface density:

$$\begin{eqnarray*} \Sigma _{\rm H_{2}(^{12}\rm CO)} &=& \bigl (2 m_{\rm H}\times N_{\rm H_{2}(^{12}\rm CO)} \times A_{\rm pixel} \bigr) / A_{\rm cloud}\nonumber \\ &\approx & 10^{-33}\times m_{\rm H} \times \biggl (\frac{N_{\rm H_{2}(^{12}\rm CO)}}{\rm cm^{-2}}\biggr)\times \biggl (\frac{A_{\rm pixel}}{\rm cm^{2}}\biggr) \nonumber \\ &&\times\biggl (\frac{A_{\rm cloud}}{{\rm pc}^{2}}\biggr)^{-1} \;(M_{\odot } \;{\rm pc}^{-2}), \end{eqnarray*}$$

where we take the total number of hydrogen atoms is N(H) ≡ 2N(H₂). We use this factor of two instead of the mean molecular weight of H₂ ($\mu _{\rm H_{2}}$ = 2.8, derived from cosmic abundances of 71% hydrogen, 27% helium, and 2% metals; e.g., Kauffmann et al. 2008) to be consistent with the extragalactic studies of Kennicutt (1998b) and Bigiel et al. (2008). This factor of two does not account for helium, which is an additional factor of ∼1.36 (Hildebrand 1983). The errors in our gas surface density measurements include both the error from the rms intensity maps and the error in the CO-to-H₂ conversion factor from Bloemen et al. (1986).

The top two panels of Figure 4 show the ¹²CO integrated intensity versus the visual extinction derived from the 2MASS and Spitzer data for both Per (left) and Oph (right). We overplot the conversion factor derived from the gamma-ray study of Bloemen et al. (1986; dashed line). ¹²CO is seen to correlate with A_V out to A_V∼ 7–10, where ¹²CO starts to saturate and the distribution flattens out to higher A_V. A large difference in the amount of ¹²CO integrated intensity produced relative to that predicted by $X_{^{12}\rm CO}$ between the Per and Oph molecular clouds is seen. This may be due to higher opacity at higher A_V in Oph relative to Per. Extinction values around 10–20 mag are found to be essentially invisible to ¹²CO. This figure demonstrates the nonlinear, non-monotonic behavior of CO emission with A_V.

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¹²CO, however, is not the most reliable tracer of star-forming gas because of high opacity and varying ¹²CO-to-H₂ abundance due to photodissociation or depletion on to dust grains. Studies of molecular clouds (Carpenter et al. 1995; Heyer et al. 1996; Goldsmith et al. 2008) show that ¹²CO contains a significant diffuse component in the low column density regime A_V < 4. ¹³CO emission is a more reliable tracer of dense gas ranging from 1000 to 7000 cm⁻³ than ¹²CO because it is optically thin for most conditions within a cloud and ¹³CO abundance variations are small for densities <5000 cm⁻³ and temperatures of >15 K (Bachiller & Cernicharo 1986; Duvert et al. 1986; Heyer & Ladd 1995; Caselli et al. 1999).

To estimate the column densities from the ¹³CO J = 1–0 integrated intensity maps, we assume LTE, optically thin ¹³CO J = 1–0, and that ¹²CO J = 1–0 and ¹³CO J = 1–0 have equivalent excitation temperatures. In order to derive column densities, we also determine an optical depth ($\tau _{^{13}\rm CO}$) and excitation temperature (T_ex) from comparison to the ¹²CO J = 1–0 line. We can derive this by assuming the ¹²CO J = 1–0 is optically thick; as τ → ∞,

$$\begin{equation} T_{\rm ex} = \frac{5.5}{{\rm ln}\left(1 + \frac{5.5}{T_{{\rm peak},^{12}\rm CO} + 0.82} \right)} \ (\rm K), \end{equation} \tag{ 11 }$$

where $T_{\rm peak,^{12}\rm CO}$ is the ¹²CO J = 1–0 peak main beam brightness temperature which we measure on a pixel-by-pixel basis.

The ¹³CO J = 1–0 optical depth is then

$$\begin{equation} \tau _{^{13}\rm CO} = -{\rm ln} \left(1 - \frac{T_{\rm peak,^{13}{\rm CO}}}{5.3}\left[\frac{1}{\exp ^{(5.3/T_{\rm ex})} -1} -0.16 \right]^{-1}\right), \end{equation} \tag{ 12 }$$

where $T_{\rm peak,^{13}\rm CO}$ is the ¹³CO J = 1–0 peak main beam brightness temperature measured in each pixel. We can then use the definition of ¹³CO optical depth and column density from Rohlfs & Wilson (1996) to estimate the column density of ¹³CO:

$$\begin{eqnarray} &&N_{^{13}\rm CO}\, {=}\, 2.6{\times} 10^{14} \left(\frac{\tau _{^{13}\rm CO}}{1 {-} \exp ^{-\tau _{^{13}\rm CO}}} \right)\!\left(\frac{ I_{^{13}\rm CO}}{1 {-} \exp ^{(-5.3/T_{\rm ex})}}\right) \ (\rm cm^{-2}).\nonumber\\ \end{eqnarray} \tag{ 13 }$$

Certain regions near A_V peaks in the clouds are optically thick and are affected by ¹²CO self-absorption. In these regions, we cannot accurately determine T_ex and therefore $\tau _{^{13}\rm CO}$. This problem affects ∼10% of the pixels in Oph (A_V>30 mag) and ∼5% of the pixels in Per (A_V>20 mag). Since most of the mass lies at low A_V, we mask out the pixels that have ¹³CO self-absorption and do not include them in determining the ¹³CO column density and mass estimate discussed below.

For comparison with extragalactic work, where clouds are not resolved, we derive ¹³CO column densities by using average spectra to determine T_ex and $\tau _{^{13}\rm CO}$ using a $T_{\rm peak,^{12}\rm CO}$ of 3 and 5.3 K and a $T_{\rm peak,^{13}\rm CO}$ of 0.7 and 1.5 K for Per and Oph, respectively. Comparing the two methods, we find that using peak temperatures from average spectra or a constant T_ex and $\tau _{^{13}\rm CO}$ will result in higher $N_{^{13}\rm CO}$ by a factor of ∼2 at A_V < 10 over the pixel-by-pixel measurements.

In order to convert the $N_{^{13}\rm CO}$ column density into H₂ column density, we use an H₂-to-¹³CO abundance ratio. The H₂/¹³CO abundance ratio for the Per cloud was determined by Pineda et al. (2008), who found a value of (3.98 ± 0.07) × 10⁵ for A_V < 5 mag. Dividing the cloud into separate regions, they found an average abundance ratio of 3.8 × 10⁵. Other values found in the literature range from 3.5 to 6.7 × 10⁵, with an average value of ∼4 × 10⁵ using both extinction maps (Pineda et al. 2008) and star counts (Bachiller & Cernicharo 1986; Duvert et al. 1986; Langer et al. 1989). We adopt the average H₂/¹³CO ratio from the literature of (4 ± 0.4)×10⁵ to convert ¹³CO-to-H₂ column densities using the relation

$$\begin{equation} N_{\rm H_{2}(^{13}\rm CO)} = (4\pm 0.4)\times 10^{5} N_{^{13} \rm CO} \ (\rm cm^{-2}) \end{equation} \tag{ 14 }$$

or in terms of surface densities:

$$\begin{eqnarray*} \Sigma _{\rm H_{2}(^{13}\rm CO)} &=& \bigl (2 m_{\rm H}\times N_{\rm H_{2}(^{13}\rm CO)} \times A_{\rm pixel} \bigr) / A_{\rm cloud}\nonumber \\ &\approx & 10^{-33}\times m_{\rm H} \times \biggl (\frac{N_{\rm H_{2}(^{13}\rm CO)}}{\rm cm^{-2}}\biggr)\times \biggl (\frac{A_{\rm pixel}}{\rm cm^{2}}\biggr)\nonumber \\ &&\times \biggl (\frac{{\it A}_{\rm cloud}}{{\rm pc}^{2}}\biggr)^{-1} \;({{\it M}}_{\odot } \;{\rm pc}^{-2}), \end{eqnarray*}$$

where we choose a factor of two instead of the mean molecular weight in order to consistently compare to ¹²CO. We show the ¹³CO integrated intensity versus the visual extinction derived from the 2MASS and Spitzer data for both Per (left) and Oph (right) in the bottom two panels of Figure 4. The average H₂/¹³CO ratio is shown by the dashed line. A turnover in ¹³CO is seen at A_V∼ 7 (Per) and ∼10 (Oph) that is likely due to an increase in optical depth. The amount of ¹³CO in Per follows the average abundance ratio out to A_V∼ 5, but in Oph, the ¹³CO integrated intensity is underproduced.

To test how well CO traces A_V as a function of M_gas or Σ_gas, we measure Σ_gas densities in our A_V maps and $\Sigma _{\rm H_{2}}$ in the CO maps in the overlapping area where there is a positive CO integrated intensity over five times the rms noise. In Figure 5, we plot the ratio of $\Sigma _{\rm H_{2}}$ and Σ_gas from A_V, which are effectively mass ratios, since the area measured is the same. The cyan squares and circles are points for the c2d and GB clouds ($\Sigma _{\rm H_{2}(^{12}\rm CO, cloud)}, \Sigma _{\rm H_{2}(^{13}\rm CO, cloud)}$) and the filled green and yellow squares and circles are measurements in contours of A_V using the same method as in Section 2.3 ($\Sigma _{\rm H_{2}(^{12}\rm CO, con)}, \Sigma _{\rm H_{2}(^{13}\rm CO, con)}$) for Oph and Per, respectively (Tables 4 and 5). A measurement for the Taurus cloud using both ¹²CO and ¹³CO above A_V = 2 from (Goldsmith et al. 2008) is also shown (cyan triangle). If CO traces the mass we find using extinction maps, we would expect the ratio of CO/A_V mass to be of order unity as shown by the solid black line in Figure 5. For ¹²CO, we find the total cloud measurement for Per to have $\Sigma _{\rm H_{2}}$ of ∼1.6Σ_gas at Σ_gas≲ 100 M_☉ pc⁻², but the ratio is close to unity within the errors. We find that ¹²CO traces A_V relatively well in the Oph cloud out to ∼200 M_☉ pc⁻². At Σ_gas ≳ 200 M_☉ pc⁻², ¹²CO underestimates the A_V mass in both Per and Oph by ∼30%, on average.

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Since ¹³CO should trace denser gas (Duvert et al. 1986; Bachiller & Cernicharo 1986), we also explore how it traces A_V as a function of Σ_gas. We plot this on Figure 5 for the total clouds (cyan points) and contour measurements (green points). We find a constant value of ¹³CO versus the surface density of extinction, but find that it underestimates Σ_gas by a factor of ∼4–5 and lies below measurements of ¹²CO by a factor of ∼5, on average. The difference we find between ¹³CO and H₂ measured by A_V, could be due to the LTE method we used to compute ¹³CO masses or the assumption that there is a constant abundance of CO relative to H₂. Heyer et al. (2009) explored the properties of galactic molecular clouds using ¹³CO emission and found that the assumption of equivalent excitation temperatures for both ¹²CO and ¹³CO in the LTE method may underestimate the true column density of ¹³CO in subthermally excited regions. As the ¹³CO density increases, the J = 1–0 transition becomes thermalized, and the column density estimates are more accurate. Also, both Heyer et al. (2009) and Goldsmith et al. (2008) found that if ¹³CO-to-H₂ abundance variations in LTE-derived cloud masses are not considered, they would underestimate the true column densities by factors of 2–3. Since we only include ¹³CO emission greater than five times the rms noise, we are likely measuring gas that is thermalized with little abundance variation. Assuming a constant abundance ratio will therefore not account for the difference between A_V and ¹³CO masses. ¹³CO, might therefore be a more consistent tracer of Σ_gas, but it may underestimate the mass by factors of ∼4–5, which can be corrected for.

Variations in the CO-to-H₂ conversion factor may impact the slope of the SFR–gas relations as measured by extragalactic studies. Since we are resolving molecular clouds, we cannot place constraints on gas densities lower than ∼50 M_☉ pc⁻², typical of spiral galaxies (Bigiel et al. 2008). However, if we consider the effects of using CO as a tracer of the total gas density, it underestimates the mass measured from A_V by ≳30% for Σ_gas ≳ 200 M_☉ pc⁻². This would effectively shift the extragalactic observed points to the right above 200 M_☉ pc⁻². This shift would flatten the slope slightly in the fitted Kennicutt (1998b) relation. These small factors, however, do not explain the large discrepancy between the extragalactic relations and the much higher SFR in the local clouds, seen both in the whole molecular clouds and looking at the youngest YSOs as a function of Σ_gas.

By studying nearby molecular clouds, we can obtain the most accurate measurement of Σ_gas and Σ_SFR, but it is regions of massive star formation that form the basis for extragalactic studies. Massive star-forming regions are the only readily visible regions forming stars at large distances and thus are the only probes of star formation in distant regions in the Milky Way and external galaxies. We can measure Σ_SFR and Σ_gas in individual massive star-forming regions to see if there is better agreement with extragalactic SFR–gas relations.

To investigate where individual regions of massive star formation fall on the SFR–gas relation, we use data from the molecular line survey of dense gas tracers in >50 massive dense (〈n〉 ∼ 10⁶ cm⁻³; e.g., Plume et al. 1997) clumps from Wu et al. (2010). The Wu et al. (2010) survey measured clump sizes, virial masses, and dense gas surface densities using HCN J = 1–0 as a tracer (Σ_HCN) at FWHM of the peak intensity. These are the sites of formation of clusters and massive stars. The most popular tracers of massive star formation include the ultraviolet, Hα, FIR, and singly ionized oxygen; however, due to high extinction toward and in these regions, we can use only the total IR luminosity to measure the SFR in these massive clumps.

Since HCN J = 1–0 gas has been shown to be tightly correlated with the total IR luminosity in clumps as long as L_IR>10^4.5L_☉ (Wu et al. 2005), as well as in both normal spiral and starburst galaxies (Gao & Solomon 2004a, 2004b), we can use it to compare gas and star formation from the total IR luminosity in both the Milky Way and external galaxies.

Σ_HCN is calculated using the mass contained within the FWHM size (R_HCN) of the HCN gas, following the methods used in Shirley et al. (2003):

$$\begin{equation} \Sigma _{\rm HCN} = \pi ^{-1} \times \biggl (\frac{M_{\rm vir}}{M_{\odot }} \biggr)\biggl (\frac{ R_{\rm HCN}}{{\rm pc}} \biggr)^{-2} (M_{\odot } \;{\rm pc}^{-2}), \end{equation} \tag{ 15 }$$

where M_vir is the virial mass enclosed in the source size at FWHM intensity. Uncertainties in the Σ_HCN are computed by adding in quadrature the errors in the FWHM size and the mass as discussed in Wu et al. (2010).

We compute the SFR for massive dense clumps following extragalactic methods using the total infrared (IR) luminosity (L_IR; 8–1000 μm) derived from the four IRAS bands. We assume the conversion SFR_IR (M_☉ yr⁻¹) ≈ 2 × 10⁻¹⁰L_IR (L_☉) from Kennicutt (1998a). $\Sigma _{\rm SFR_{\rm IR}}$ are computed using the FWHM source sizes:

$$\begin{equation} \Sigma _{\rm SFR_{\rm IR}} = \pi ^{-1} \times \biggl (\frac{\rm SFR_{\rm IR}}{M_{\odot } \;\rm yr^{-1}}\biggr) \biggl (\frac{R_{\rm HCN}}{\rm kpc} \biggr)^{-2} (M_{\odot } \;\rm yr^{-1} \;\rm kpc^{-2}). \end{equation} \tag{ 16 }$$

The uncertainties in the $\Sigma _{\rm SFR_{\rm IR}}$ density only include the error in FWHM size and a 30% error in SFR calibrations using the IR. In fact, the uncertainties are larger. The SFR calibration assumes that the observed far-IR emission is re-radiated by dust heated by O, B, and A stars (Kennicutt 1998a). For low SFR, heating by older stars of dust unrelated to star formation can contaminate the SFR signal, causing an overestimate of the SFR. This is not a problem for the regions of massive star formation in our Galaxy, where the heating is certainly due to the recent star formation. A bigger issue for the Galactic regions is that the full L_IR seen in the extragalactic studies is not reached unless the individual region forms enough stars to fully sample the IMF and is old enough that the stars have reached their full luminosity. These conditions may not be met during the time span of an individual massive clump (Krumholz & Thompson 2007; Urban et al. 2010). In particular, Urban et al. (2010) have calculated the ratio of luminosity to SFR in a simulation of a cluster forming clump. They find that $L/\rm SFR$ increases rapidly with time, but lies a factor of 3–10 below the relation in Kennicutt (1998a) when their simulations end at times of 0.7–1.4 Myr. Therefore, we may expect both large variations and a tendency to underestimate the SFR in individual regions. Unfortunately, despite these issues, L_IR remains the best measure of SFR available to us at present in these regions. While it would be suspect to apply a correction factor based on the Urban et al. simulation, increasing the SFR of the regions of massive star formation by 0.5–1 order of magnitude would bring them into better agreement with the highest surface density points from the nearby clouds.

For the sample of ∼50 massive dense clumps, 42 sources have corresponding IR measurements. The resulting gas surface densities and Σ_SFR for the sample of massive dense clumps are shown in Figure 6 and Table 6. The relation between SFR and dense gas mass, M_dense(H₂), for galactic clumps can be derived from $\langle L_{\rm IR}/L_{\rm HCN(1\hbox{--}0)} \rangle = 911\pm 227\; (\rm K \;\rm km \;\rm s^{-1} \;{\rm pc}^{2})^{-1}$, and $\langle M_{\rm dense}(\mbox{H}_{2})/L_{\rm HCN(1\hbox{--}0)} \rangle = 11\pm 2 \;M_{\odot }\; (\rm K \;\rm km \;\rm s^{-1} \;{\rm pc}^{2})^{-1}$ from Wu et al. (2005). These relations can be combined with the IR SFR conversion from above to obtain a relation for $\Sigma _{\rm SFR_{\rm IR}}$ and Σ_HCN:

$$\begin{equation} \Sigma _{\rm SFR_{\rm IR}} \sim 1.66\times 10^{-2}\left(\frac{\Sigma _{\rm HCN(1\hbox{--}0)}}{1 \;M_{\odot } \, {\rm pc}^{-2}} \right) (M_{\odot } \;\rm yr^{-1} \;\rm kpc^{-2}). \end{equation} \tag{ 17 }$$

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Table 6. Massive Clumps HCN J = (1–0)

Source	log ΣHCN	log $\Sigma _{\rm SFR_{\rm IR}}$
	(M☉ pc−2)	(M☉ yr−1 kpc−2)
(1)	(2)	(3)
W3(OH)	3.39 ± 0.13	1.35 ± 0.12
RCW142	3.40 ± 0.14	1.08 ± 0.13
W28A2(1)	3.66 ± 0.14	2.12 ± 0.13
G9.62+0.10	3.28 ± 0.13	1.45 ± 0.14
G10.60−0.40	3.32 ± 0.12	1.83 ± 0.12
G12.21−0.10	2.63 ± 0.24	0.28 ± 0.23
G13.87+0.28	2.28 ± 0.15	0.61 ± 0.13
G23.95+0.16	2.28 ± 0.25	0.79 ± 0.21
W43S	2.63 ± 0.14	1.51 ± 0.14
W44	2.79 ± 0.18	1.00 ± 0.16
G35.58−0.03	3.41 ± 0.24	0.89 ± 0.22
G48.61+0.02	1.98 ± 0.14	0.75 ± 0.13
W51M	3.14 ± 0.17	1.53 ± 0.17
S87	2.76 ± 0.16	0.97 ± 0.12
S88B	2.68 ± 0.19	1.31 ± 0.15
K3-50	3.26 ± 0.12	1.75 ± 0.13
ON1	2.83 ± 0.14	0.73 ± 0.13
ON2S	2.76 ± 0.16	1.44 ± 0.14
W75N	3.13 ± 0.13	1.50 ± 0.12
DR21S	3.20 ± 0.13	1.75 ± 0.12
W75(OH)	3.21 ± 0.13	0.62 ± 0.13
CEPA	3.44 ± 0.17	2.00 ± 0.13
IRAS20126	2.98 ± 0.18	1.22 ± 0.13
IRAS20220	2.63 ± 0.21	0.29 ± 0.18
IRAS23385	2.79 ± 0.19	0.43 ± 0.15
Clump average	3.12 ± 0.16	1.44 ± 0.15

Note. Columns are (1) source name; (2) surface gas density; and (3) SFR surface gas density.

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This equation is equivalent to that shown in Wu et al. (2005), which is the fit to both massive clumps and galaxies.

Wu et al. (2005) found a decline in the linear $L_{\rm }-L^{\prime }_{\rm HCN(1\hbox{--}0)}$ correlation at L_IR < 10^4.5 L_☉, where the clump is not massive or old enough to sample the IMF. Since the majority of the points with L_IR < 10^4.5 L_☉ in Figure 6 lie off this relation, we include only massive dense clumps with L_IR>10^4.5 L_☉. The resulting number of HCN J = 1–0 sources is 25 (Table 6). The HCN J = 1–0 clumps are found to range from ∼10² to 4.5 × 10³ M_☉ pc⁻² in Σ_HCN and from 2 to 130 M_☉ kpc⁻² yr⁻¹ in $\Sigma _{\rm SFR_{\rm IR}}$. These Σ_gas and $\Sigma _{\rm SFR_{\rm IR}}$ values are similar to those of circumnuclear starburst galaxies from Kennicutt (1998b), which range from ∼10² to 6 × 10⁴ M_☉ pc⁻² and 0.1 to 9.5 × 10² M_☉ kpc⁻² yr⁻¹ (see Figure 9). The average HCN J = 1–0 clump has Σ_HCN(1–0) of (1.3 ±  0.2) × 10³ M_☉ pc⁻² and $\Sigma _{\rm SFR_{\rm IR}}$ of 28 ± 6 M_☉ kpc⁻² yr⁻¹.

We also compare the relation we find for the massive dense clumps to known extragalactic relations in Figure 7. In this figure, we compare to the H₂ gas surface density relation from Bigiel et al. (2008) and the total (H i + H₂) gas surface density relation from Kennicutt (1998b). We find that most of the points for the HCN J = 1–0 line lie above both the Bigiel et al. (2008) and Kennicutt (1998b) extragalactic relations with the average clump lying a factor of ∼5–20 above the Kennicutt (1998b) and Bigiel et al. (2008) relations, respectively.

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In Figure 8, we plot the ratio of Σ_SFR/Σ_gas versus Σ_gas for c2d and GB clouds, YSOs, and massive clumps. We find a steep decline in Σ_SFR and Σ_SFR/Σ_gas at around ∼100–200 M_☉ pc⁻². We identify this steep change in Σ_SFR over ∼100–200 M_☉ pc⁻² as a star-forming threshold (Σ_th) between regions actively forming stars and those that are forming few or no low-mass stars.

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In Figure 9, we show points for the massive dense clumps, c2d and GB clouds, the youngest YSOs, and both the Wu et al. (2005) and extragalactic relations. We also show the range of gas surface densities for spiral and circumnuclear starburst galaxies from the sample of Kennicutt (1998b). Σ_gas,con for Class I and Flat SED YSOs lie intermediate between the regions where spiral galaxies and starburst galaxies are found on the Kennicutt (1998b) relation. At Σ_gas>Σ_th, the youngest Class I and Flat SED YSOs overlap with the massive clumps (Figure 9). Therefore, high-mass and low-mass star-forming regions behave similarly in the Σ_SFR–Σ_gas plane. The difference between extragalactic relations and c2d and GB clouds is not caused by the lack of massive stars in the local clouds. Also, the overlap with the massive clumps in Figure 9 suggests that L_IR provides a reasonable SFR indicator, as long as it exceeds 10^4.5 L_☉, though an upward correction would produce better agreement.

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A steep increase and possible leveling off in Σ_SFR at a threshold Σ_th∼100–200 M_☉ pc⁻² is seen in both Figures 8 and 9. We further constrain this steep increase and the possibility of Σ_SFR flattening at Σ_gas>Σ_th, by approximating it as broken power law with a steep rise that levels off in Section 3.2.1.

3.2.1. Star Formation Threshold

In order to determine a robust estimate of Σ_th, we fit the data using two models: a single power law (y = Nx + A, where y = log₁₀ Σ_SFR; x = log₁₀ Σ_gas) and a broken power law (y₁ = N₁x + A₁; y₂ = N₂x + A₂). We first fit Class I and Flat SED YSO points (Σ_SFR,con, Σ_gas,con) and massive clumps ($\Sigma _{\rm SFR_{\rm IR}}, \Sigma _{\rm HCN(1\hbox{--}0)}$) to a single power law. We do not include upper limits for YSOs or points for Sco and Cep clouds, which are co-added separate cloud regions and only provide a rough estimate of Σ_SFR and Σ_gas. The single power-law fit yields N = 1.57 ±  0.09 and A = −3.0 ±  0.2, and a reduced chi-square (χ²_r) of 3.7 (84 dof). We fit the data for both YSOs and massive clumps to a broken power law for the range of Σ_gas = 50–250 $M_{\odot }\rm \, pc^{2}$. We minimize the total χ² for the two segments of the broken power law using a simplex routine, which yields best-fit parameters: N₁ = 4.58 ± 0.5, A₁ = −9.18 ± 0.9, N₂ = 1.12 ± 0.07, A₂ = −1.89 ± 0.2 with a χ²_r of 3.04 (82 dof). We attribute the large χ²_r to the scatter and large errors in the data, but since the χ²_r is ∼18% lower for the broken power law compared to the single power law, we take it to be the best-fit model. Equating the broken power-law fits (y₁ = y₂), we obtain a power-law break at Σ_th = 129.2 M_☉ pc² (A_V = 8.6) with a statistical 1σ deviation in χ² of ±14 M_☉ pc² giving a range in Σ_th of ∼115–143 M_☉ pc². Figure 10 shows the broken power-law fit (cyan and magenta lines), Σ_th, and the 1σ statistical range of Σ_th (dashed black vertical line and gray shaded region, respectively). The slope of the broken power law changes from a steep relation at Σ_gas < Σ_th (slope of ∼4.6) to linear relation (slope of ∼1.1) at Σ_gas>Σ_th. We note however, that variations in cloud distances will change this threshold slightly. One example is for Ser, which has a recent distance estimate by Dzib et al. (2010) of 415 pc and another estimate by Straizys et al. (1996) of 260 pc (used in this paper). If we use the larger distance of 415 pc, this would change our star formation threshold slightly to 126 ± 12 M_☉ pc² (A_V = 8.4).

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This star-forming threshold we find is in agreement with the threshold found in a study of local molecular clouds by Lada et al. (2010) at A_V ∼ 7 or 116 M_☉ pc². Enoch et al. (2007) also found extinction thresholds for dense cores found in Bolocam 1.1 mm maps in the Perseus, Serpens, and Ophiuchus clouds of A_V∼ 8, 15, and 23, respectively (∼120, 225, and 350 M_☉ pc²), with a low probability of finding cores below these thresholds. Onishi et al. (1998) surveyed Taurus in C¹⁸O and found a star-forming column density threshold of $8\times 10^{21}\;\rm cm^{-2}$, which corresponds to a gas surface density of 128.1 M_☉ pc² (A_V = 8.5). Similarly, both Johnstone et al. (2004) and André et al. (2010) find thresholds of A_V∼ 10 (150 M_☉ pc²).

Mouschovias & Spitzer (1976) proposed the idea of a physical column density threshold corresponding to the central surface density above which the interstellar magnetic field cannot support the gas from self-gravitational collapse. This was later modified by McKee (1989) who considered the local ionization states owing to UV radiation. Mouschovias & Spitzer (1976) predicted that when clumps combine to form a large cloud complex, there exists a natural surface density threshold (Σ_crit) for a given magnetic field:

$$\begin{equation} \Sigma _{\rm crit} > \biggl (\frac{80}{M_{\odot } \,{\rm pc}^{2}}\biggr) \times \biggl (\frac{B}{30 \;\mu \rm G}\biggr). \end{equation} \tag{ 18 }$$

The total mean strength of the line-of-sight magnetic field (B_los) measured in molecular clouds (n ∼ 10³–10⁴ cm³) is ∼10–20 $\mu \rm G$ (Crutcher 1999; Troland & Crutcher 2008). Since statistically $B_{\rm los} \approx \frac{1}{2}B_{\rm tot}$ (Heiles & Crutcher 2005), the corresponding total magnetic field, B_tot, is ∼20–40 $\mu \rm G$. Using Equation (18), the corresponding Σ_crit>50–110 M_☉ pc². A similar idea of a threshold at a particular extinction was predicted by McKee (1989) for photoionization-regulated star formation. This model predicts that the rate of star formation is controlled by ambipolar diffusion and therefore depends on the ionization levels in the cloud. Star formation in a "standard" ionization case will occur at A_V ≳ 4–8 mag, which translates into a Σ_crit ≳ 60–120 M_☉ pc². Both of these predictions for a critical density of star formation are similar to Σ_th = 129 ± 14 M_☉ pc². We note, however, that both these models are for parts of clouds that are in a "quasi-static" or turbulence-supported state. Alternatively, these parts of clouds may never become bound and are transient (Vázquez-Semadeni et al. 2009). In this picture, the threshold would correspond to parts of molecular clouds that become gravitationally bound and form stars.

The third possibility is based on the fact that the extragalactic relations are averaging over large scales which do not resolve the regions where stars are forming. Current "spatially resolved" extragalactic measurements are still limited to scales of ∼0.2–2 kpc (Kennicutt et al. 2007; Bigiel et al. 2008; Blanc et al. 2009); therefore, we cannot directly measure extragalactic SFRs on scales of galactic star-forming regions. In any given spatially resolved extragalactic measurement of Σ_SFR and Σ_gas, the beam will contain a fraction of diffuse gas that does not trace star formation and a fraction of dense, star-forming gas (f_dense). As discussed in Section 3.1, the dense gas that is forming stars is not well traced by CO so there will be an excess of non-star-forming gas in each beam measurement. A local example of this is a study of the Taurus molecular cloud; Goldsmith et al. (2008) found that 33% of diffuse ¹²CO is contained in regions not associated with ¹³CO, which is a more reliable tracer of dense, star-forming gas (Section 3.1). Even in the regions with A_V>2 studied by the c2d and GB projects, star formation is highly concentrated to regions of high extinction (e.g., Figure 2). Extragalactic studies averaging over hundreds of pc scales would include this diffuse gas, causing an increase in the amount of CO flux that is being counted as star-forming gas. In order to better understand approximately how much gas is forming stars at present, a measurement of the fraction of gas that contains YSOs over a larger area on kpc scales in the Galaxy is needed.

Lada & Lada (2003) proposed the idea that clusters of stars which form in clumps located in giant molecular clouds are the fundamental building blocks of galaxies. The rate at which these stellar clusters form is set by the mechanisms that enable these clumps to condense out of their low density parent cloud. A similar idea was explored by Wu et al. (2005), who proposed that there is a basic unit of clustered star formation with the following typical properties: L_IR>10⁵ L_☉, R_dense ∼ 0.5 pc, and M_vir∼300–1000 M_☉. As more of these basic units are contained in a galaxy, the SFR increases linearly. This linear correlation between SFR_IR and the mass of dense gas (M_dense) from HCN J = 1–0 was seen in a sample of both spiral and luminous or ultra-luminous IR galaxies (LIRGs and ULIRGs; Gao & Solomon 2004b), and Wu et al. (2005) showed that the same relation fits the Galactic massive dense clumps. It is therefore the dense gas tracers, such as HCN, that directly probe the volume of gas from which stars form in dense clumps and produce the star formation in external galaxies.

If a linear relation between dense gas and the SFR is assumed at all Σ_gas and Σ_SFR densities, how can we explain the nonlinear behavior of the Kennicutt–Schmidt SFR–gas relation? Let us suppose that the underlying star formation relation is what we actually observe in regions forming massive stars: a threshold around 129 M_☉ pc⁻² and a roughly linear relation between dense gas and star formation above that threshold:

$$\begin{equation} \Sigma _{\rm SFR} \propto \Sigma _{\rm dense}. \end{equation} \tag{ 19 }$$

Let us also assume, the Kennicutt–Schmidt relation for the gas surface density averaged over large scales (〈Σ_gas〉):

$$\begin{equation} \Sigma _{\rm SFR} \propto \mbox{$\langle \Sigma _{\rm gas}\rangle $}^{1.4}. \end{equation} \tag{ 20 }$$

Then, the fraction of gas above the threshold (f_dense) would have to scale with mean surface density of all gas:

$$\begin{eqnarray} f_{\rm dense} &=& \Sigma _{\rm dense}/ \mbox{$\langle \Sigma _{\rm gas}\rangle $}\nonumber\\ &\propto & \Sigma _{\rm SFR}/ \mbox{$\langle \Sigma _{\rm gas}\rangle $}\nonumber\\ &\propto & \mbox{$\langle \Sigma _{\rm gas}\rangle $}^{0.4}. \end{eqnarray} \tag{ 21 }$$

Near the Σ_th of ∼129 M_☉ pc⁻², the average Σ_SFR measured on small scales is about ∼3 $M_{\odot } \;\rm yr^{-1}\;\rm kpc^{-2}$ (taking the average between Class I and Flat SED sources) and f_dense is about 1/40. When 〈Σ_gas〉 ∼ 300Σ_th, f_dense ∼ 1. At this point, all the gas is dense enough to form stars and star formation is most efficient, creating a maximal starburst. This is also where the dense gas (Wu et al. 2005) and CO (Kennicutt 1998a) relations cross (see Figure 8). Above ∼300Σ_th, the only way to increase star formation efficiency is to make it more efficient even in the dense gas. This is possible because even in dense gas, the SFR per free-fall time is less than unity (Zuckerman & Evans 1974; Krumholz & Tan 2007).

Is there local evidence for a preponderance of gas below the threshold? Complete maps of CO for the local 0.5 kpc are not readily available, but we can measure the mass below and the mass above the threshold of A_V = 8 for the 16 clouds with Spitzer coverage down to A_V = 2. The ratio of total mass lying below the threshold to the total mass above the threshold is 4.6. The massive star-forming region, Orion, also has a similar ratio of 5.1 for the mass below over the mass above A_V = 8 (M. Heyer, unpublished data). A few clouds have been mapped to still lower levels and a factor of two more mass is found in Taurus, for example, Goldsmith et al. (2008). Alternatively, if we assume Orion to contain the largest reservoir of molecular material within 0.5 kpc, we can derive the ratio of mass from ¹²CO and A_V maps to get 6.4 (M. Heyer 2010, private communication). Taken together, these factors make it plausible that there is 10 times more molecular mass than mass above the threshold. Furthermore, most of the gas within 0.5 kpc is atomic. If that is included, the predictions of the extragalactic Kennicutt–Schmidt relation for total gas agree reasonably with the local SFR surface density (see Evans 2008 and references therein).

Finally, if the underlying star formation law in the dense gas is linear, then the arguments invoking the density dependence of the free-fall time to get a power of 1.5 (see Section 1) are specious. In fact, the idea of a single free-fall time for a molecular cloud with an enormous range of densities is highly dubious in the first place.