THE SCHMIDT LAW IN SIX GALACTIC MASSIVE STAR-FORMING REGIONS

Data was taken with the Spitzer Space Telescope (Werner et al. 2004) Infrared Array Camera (Fazio et al. 2004) during the warm mission (PID 70157, PI Lori Allen). The dates of observation for each region are given in Table 2. We used IRAC's 12 s frame time High Dynamic Range mode, acquiring consecutive individual observations with exposure times of 0.4 and 10.4 s. The long exposures are used to probe the faint stellar population in each cloud, while the addition of the short exposure observations allows for the recovery of bright sources that are saturated in the long exposure frames. The individual basically calibrated frames (processed with the Spitzer IRAC pipeline version S18.18.0) for each exposure time were mosaicked together using the IRACproc package (Schuster et al. 2006) to perform outlier rejection and bad pixel masking. IRACproc is a PDL⁴ wrapper script for the standard Spitzer Science Center mosaicing software MOPEX (Makovoz & Khan 2005), which has been enhanced to improve cosmic-ray rejection. Each mosaic was resampled from the native IRAC resolution of 122 per pixel to 086267 per pixel, providing pixels with half of the native IRAC size. We projected each mosaic onto a common world coordinate system grid. The final mosaics combined a median of 5 dither positions per pixel for an effective integration time of 52 s per pixel in each long exposure mosaic and 2 s per pixel in each short exposure mosaic. Although all 5 of these star-forming regions were covered by GLIMPSE programs, these new deep data allow us to study the stellar population to 2–3 mag fainter than the GLIMPSE observations.

We also present near-infrared observations of these regions that we acquired with the NOAO Extremely Wide-Field Infrared Imager (NEWFIRM) camera on board the Blanco 4 m telescope at Cerro Tololo Inter-American Observatory during the 2010A semester between 2010 May 24 and 2010 June 1. The NEWFIRM camera, designed to quickly map large areas of the sky, contains 4 InSb 2048 × 2048 pixel arrays arranged in a 2 × 2 pattern with a 28' field of view and an approximately 1' gap between the CCDs. The detector has a pixel scale of 04 per pixel.

Each observation consists of 11 dither positions with 60 s exposures. Our observations were taken with a random dither offset large enough to fill in the 1' gap between CCDs. Each dither position was a combination of multiple co-added exposures to achieve longer exposure times while avoiding the non-linear regime of the detector. For the J band we used two co-added 30 s exposures, for the H band we used three co-added 20 s exposures, and for the K_s band we used six co-added 10 s exposures. The sky background was measured by taking three dithered sky observations 2° off field. The effective exposure time for each band in each star-forming region is 660 s.

Standard processing of dark frame subtraction, flat fielding, sky-subtraction, and bad pixel masking were performed by the NEWFIRM Science Pipeline (Dickinson & Valdes 2009; Swaters et al. 2009) to produce a stacked composite image for each near-infrared band. The final stacked images retain the detector pixel scale of 04 per pixel. Variable seeing conditions caused the final observed point-source FWHM in the stacked images to range from approximately 09 to 15. The difference in effective resolution at the different NEWFIRM bands has a minimal effect on later source-matching.

To study the structure of the molecular clouds we also used multiple archival data sets in the far-infrared. The Herschel  Infrared Galactic Plane Survey (Hi-GAL) key project (Molinari et al. 2010) observed the Galactic plane between 2010 January and 2012 November. The Hi-GAL program used Herschel's parallel, fast-scanning mode, resulting in simultaneous observation by the Photoconductor Array Camera and Spectrometer (PACS; Poglitsch et al. 2010) and the Spectral and Photometric Imaging Receiver (SPIRE; Griffin et al. 2010) instruments. The data, calibrated according to the Standard Product Generation 9.2.0 were obtained from the Herschel Science Archive.

The observations were recorded in two orthogonal scan directions that were combined into a single map using the Herschel Interactive Processing Environment (HIPE) versions 9.2 and above. Cross-scan combination and de-striping were performed using the standard HIPE tools over the Hi-GAL maps, divided into fields covering 2° × 2°.

Table 3 shows the typical wavelength (in microns) for each band of the Herschel PACS and SPIRE instruments, the angular resolution given by the FWHM of the point-spread function (PSF) calculated by Olmi et al. (2013), and the noise level of the final maps. Position uncertainty of the Hi-GAL maps is of the order of 3'', which is small relative to the PSF at most bands. The noise level given in the table is for SPIRE data taken in "nominal" mode.

Multiple bolometer readings cover each pixel and are combined in the map to determine the observed intensity at each position. Therefore, we can estimate the uncertainty associated with Hi-Gal images from the FWHM of the distribution of the difference between adjacent pixels. In this way, we filter the sky emission that is expected to vary on spatial scales larger than 1 beam-size. The high-pass filter applied to estimate these differences is described in Rank et al. (1999).

The 1σ point-source sensitivities derived from this method are typically 24 mJy for the PACS band at 160 μm, and 12 mJy for each of the three SPIRE bands at 250, 350, and 500 μm. These derived sensitivities are in agreement with those expected for the Hi-GAL survey (Molinari et al. 2010). In addition to the previously derived noise level, we adopt an independent uncertainty of 10% for the flux calibration.

We created the final source list catalog by spatially matching the detected sources across the five bands. Our cross-band correlation method first calculates whether there is an overall systematic shift in source centroid positions between the input frames and then looks for sources with shifted centroid positions matching between bands. Sources were considered a match if the difference between their shifted centroids was less than than 10.

To further increase the reliability of our point-source catalog, we require simultaneous detection in 2 adjacent bands (e.g., H and K_s on NEWFIRM, or 3.6 and 4.5 μm with IRAC). NEWFIRM sources that are identified only in the K_s band are rejected unless they were detected at both 3.6 and 4.5 μm with IRAC. The completeness limit for each band was determined by recovering fake sources added with DAOPHOT's addstar routine. We created a source list with 5000 fake stars with randomly generated positions, and magnitudes randomly sampled between 8.0 and 18.0 mag for 3.6 and 4.5 μm and between 8.0 and 24.0 mag for the J, H, and K_s bands. We used the derived PSF for each band to add these sources onto the original image and then repeated the original source identification and PSF-fitting steps. We repeated this process 20 times for each photometric band in each region.

We binned the fraction of fake sources that were recovered in bins 0.25 mag wide and took the median fraction of fake sources that were recovered in each bin. The faintest magnitude where the fraction of sources that were recovered was greater than 90% was adopted as the completeness limit. This value varies for each star-forming region and for each band depending on the intrinsic source crowding, the intensity of the nebular emission, and the fraction of the frame that is filled by the nebular emission. The derived completeness limit for each band in each star-forming region is summarized in Table 4. The limiting magnitude reported is the faintest source detected in the original images at the 5σ level or higher (a maximum photometric error of 0.217 mag). A summary of the 4th Quadrant Point-source Catalogs can be found in Table 5. Sample entries selected from the G305.1940 Point-source Catalog can be seen in Table 6 and the full version of the point-source catalog for each region is available in electronic format.

This complex lies at approximately 4 kpc and is one of the most massive (M > 6 × 10⁵ M_⊙, Hindson et al. 2010) and luminous (log(L_bol) = 5.94, Urquhart et al. 2013) star-forming regions in the Galaxy (Clark & Porter 2004). The complex displays a ring like morphology of bright mid-infrared emission approximately 30 pc in diameter surrounding two optically visible open clusters Danks 1 & 2 (Davies et al. 2012). The complex contains at least one Wolf–Rayet star but no red supergiants, placing the onset of star formation in the region around 3–5 million years ago (Faimali et al. 2012). Star formation appears to be continuing today in the ring structure surrounding Danks 1 & 2, indicated by results from the Spitzer GLIMPSE program that found an overdensity of young stars with infrared excess distributed along the the nebular emission surrounding the two massive clusters (Indebetouw et al. 2005). Based on the Spitzer GLIMPSE and Herschel Hi-GAL observations of this star-forming complex Faimali et al. (2012) derived a star-formation rate of 0.01–0.02 M_⊙ yr⁻¹.

We find that Class II YSOs are widely distributed throughout the outer ring and that the projected area corresponding to the inside of the ring has significantly fewer Class II YSOs. We find that the Class I YSOs are more clustered than the Class II YSOs, with at least six sites indicative of ongoing clustered star formation. We found very few infrared-excess sources that are spatial matches to either Danks 1 or Danks 2, despite their reported youth. Danks 1 is reported to be ${1.5}_{-0.5}^{+1.5}$ Myr based on isochrone fitting to cluster members observed by the Hubble Space Telescope. Danks 2 is reported to be ${3}_{-1}^{+3}$ Myr based on the same observing program (Davies et al. 2012). Studies of low-mass star-forming regions have determined the average disk half-life is approximately 2 Myr. At 1.5 Myr or even 3 Myr, we should still see a number of infrared-excess sources in these clusters.

The lack of detected sources may be the result of a combination of multiple factors: first, the clusters are very compact and even with PSF-fitting photometry we are unable to resolve many of the individual cluster sources. Second, the majority of the sources in these clusters that are brighter than our detection limit would be intermediate-mass or larger. The pre-main-sequence evolution timescale of stars is expected to be inversely correlated with the star's mass, so the disk dissipation timescale of an intermediate-mass star may be shorter than for low-mass stars. A third possibility is that the close proximity to the large complement of O and B stars may be accelerating the disk dispersal in the few low-mass stars belonging to these clusters that are sufficiently separated to be resolved.

Two of the star-forming regions in our sample lie at approximately l = 326°. The two fields we have observed, which we call G326-4 and G326-6, are centered at l = 326.4, b = 0.9, and l = 326.6, b = 0.5 respectively. These fields contain four bright IRAS point sources (IRAS 15384–5348, IRAS 15394–5358, IRAS 15408–5356, and IRAS 15411– 5352) with kinematic distances from CS(2-1) emission detected by Bronfman et al. (1996) placing them at 2.7–2.8 kpc. Each of the IRAS sources has far-infrared colors that match ultra-compact H ii regions. SIMBA measurements of the 1.2 mm flux estimate that each source contains approximately 10³ M_⊙ of molecular gas derived from dust continuum observations. The total luminosity of the four IRAS sources is estimated to be log(L_bol) = 5.6 (Faúndez et al. 2004). Observations with the Australian Telescope Compact Array at 4.8 GHz identified a linear feature extending westward from IRAS 15384–5348 that may indicate the presence of an ionized outflow from a protostellar source.

Three of the four IRAS sources lie on the edge of a roughly circular cavity centered at l = 326.675, b = 0.535. The cavity has a radius of r = 4.1', corresponding to approximately 6.4 pc in diameter, assuming a distance of 2.8 kpc. The circular cavity seen in the infrared corresponds to the H ii region RCW 95. A catalog of Spitzer-identified IRDCs identified several dozen sources in this field (Peretto & Fuller 2009). G326-6 hosts the largest H ii region in the field as well as most of the bright IRAS sources including IRAS 15408–5356. Most of G326-4 appears relatively quiescent in the near-infrared images, with extremely bright point sources clustered near IRAS 15394–5358. G326-4 contains both filamentary IRDCs as well as the smaller H ii region associated with IRAS 15384–5348.

The star-forming complexes G326-6 and G326-4 have very small projected spatial separations and distance estimates that are the same to within the typical margin of error (2.68 and 2.8 kpc, respectively). The Bronfman et al. (2000) CS(2-1) catalog lists another source (IRAS 15394–5358) that is spatially located between the major IRAS sources in G326-6 and G326-4, with a velocity that is intermediate between the other two sources. Given the similar distances, small spatial separation, and velocity profile it is possible that the combined G326 field may be one star-forming complex extending at least 60 or more parsecs, with different sub-regions in different evolutionary stages. However, since the Herschel column density maps do not clearly indicate filamentary structures linking the three regions, we treat G326-4 and G326-6 as two separate fields in the subsequent analysis and include IRAS 15394–5358 with the rest of the G326-4 field.

In G326-6 we identified 6 Class I YSO candidates within 05 of the far-infrared peak at IRAS 15412–5359. The H ii regions corresponding to the far-infrared sources IRAS 15408–5356 and IRAS 15411–5352 are also detected, with over a dozen YSO candidates each. We have determined a subcluster of YSOs centered at l = 326.672, b = 0.554 containing 13 Class I YSO candidates and 7 Class II YSO candidates within 05. This cluster was not detected by IRAS, indicating that the subcluster does not contain any high-mass stars. Both IRAS 15411–5352 and IRAS 15412–5359 have subclusters of Class II YSOs with small offsets from the peak of the far-infrared emission and the Class I YSO distributions.

G326-4 also has multiple IRAS sources in the field of view, although they are more widely separated than in G326-6 and with no clear indication of connection in the molecular material through filamentary structures. Over half of the YSO candidates in this field were identified toward the extended H ii region corresponding to IRAS source IRAS 15394–5358. Two additional subclusters of Class I YSOs were identified with significant separation from (115 and 112 from IRAS 15394–5358. One of these subclusters, located approximately 115 to the east of IRAS 15394–5358, has no IRAS counterpart. This cluster, located at l = 326.609, b = 0.799 contains 13 Class I YSO candidates. The other YSO cluster is located approximately 112 to the southeast of IRAS 15394–5358, corresponding to IRAS 15394–5358, with the molecular gas at intermediate an velocity between G326-4 and G326-6.

The star-forming region G333, also known as RCW 106, is another of the most luminous star-forming regions in our galaxy, with log(L_bol) = 6.28 at a distance of 3.6 kpc (Urquhart et al. 2013). The region has a backbone of molecular gas that extends at least 50 pc in a roughly northeast to southwest direction across the star-forming region, with significant diffuse material extending tens of parsecs off of the ridge. Massive star-formation activity is concentrated in pockets along this ridge, as indicated by the presence of compact H ii regions. SIMBA 1.2 mm observations suggest the mass of the giant molecular cloud complex is on the order of 10⁵ M_⊙ (Mookerjea et al. 2004). Our observations cover only the central 05 × 04 of this field centered at l = 333.3, b = −0.4. We use the distance estimate 3.45 kpc from (Bronfman et al. 2000) for comparison to earlier works.

Ultimately, our coverage of G333 limits our ability to comment on the global properties of this star-forming region. We have surveyed approximately 50% of the northeastern side of this complex. The Class I YSO population reveals multiple sites of clustered star formation. The two largest clusters of Class I objects correspond to IRAS 16177–5018 and IRAS 16172–5028, compact H ii regions. Both clusters have 19 Class I YSO candidates within 15 of the IRAS source. The widespread population of Class I and Class II YSOs reflects the extent of the nebular emission, suggesting that the extended distributions are due to the physical boundaries of the molecular cloud extending beyond the region observed from the J band to 4.5 μm rather than a high fraction of contamination of background sources.

The G351 cloud is the second-closest of the clouds in our sample, at a distance of 2.34 kpc. Early observations of G351 resolved the region into two H ii regions using radio recombination line observations at 5 GHz (Caswell 1972), one centered at l = 351.6, b = −1.3 and the other centered at l = 351.7, b = −1.2. Subsequent far-infrared observations using a balloon-borne detector operating at 150 microns, combined with IRAS and detected 7 sources with L_IR between 1.1 and 128 × 10⁴ L_⊙ (Ghosh et al. 1990). G351.6-1.3 corresponds to an embedded O4 star, while G351.7-1.2 is best fit by multiple sources rather than a single dominating heat source. The other five far-infrared sources Ghosh et al. (1990) that were detected could be heated by either small clusters of B0 or later zero-age main-sequence (ZAMS) stars or a single O6 star. Data from the 2MASS led to the identification of two additional infrared star clusters associated with the nebula (Bica et al. 2003).

The Class I YSOs in this region are primarily found along a northwest to southeast line stretching 20' from IRAS 17256–3631 to IRAS 17258–3637 to the IRDC toward the southeast. Many of these are condensed into two subclusters that appear to be associated with the far-infrared sources and their associated H ii regions. Only two Class I YSO candidates are detected in the IRDC itself. Either this IRDC is quiescent and currently lacks star formation, or the stars that are forming in this IRDC remain too heavily embedded to be detected at H and K_s. The lack of an IRAS source toward this IRDC is an indication that if star formation is active in this cloud, it is primarily forming low-mass stars. Since G351 lacked full Herschel  coverage, there is no currently existing data to confirm whether this IRDC has a detectable population of low-mass, embedded protostars, or whether the dark cloud in G351 is not currently active.

NGC 6334 and its young stellar content are described in more detail in Willis et al. (2013) and a relatively recent review article  by Persi & Tapia (2008). NGC 6334 contains molecular gas on the order of 10⁵ M_⊙ and stellar content on the order of 10,000–15,000 M_⊙, most of which has formed within the last 2 million years. Several subclusters have age estimates of 1 Myr or less (Getman et al. 2014). NGC 6334 also has a ridge of molecular material extending from the northeast to the southwest, with most of the mass concentrated toward the northeast. There are at least seven H ii regions of various sizes within approximately 10 pc, marking the region as a relatively dense complex of sites of massive star formation. The heavily-concentrated star formation, particularly toward the northeast portion of the cloud, mark NGC 6334 as a potential "mini-starburst" region.

We examine the column density structure of these regions using a combination of maps of the near-infrared extinction and their emission from 160 through 500 μm in the Herschel observations. The star-forming regions we have observed are located at very low Galactic latitudes. As a result, along the line of sight, toward each of these complexes, the majority of point sources observed are unassociated objects, such as background giant stars, background galaxies, and a limited number of foreground objects. The background objects are seen through a high level of extinction, both due to the star-forming regions of interest as well as due to the more diffuse material along the line of sight. Mapping the extinction in the near- and mid-infrared across the region leads to median extinction values of A_V ∼ 10 mag. However, many of these clouds contain high-density regions where even at IRAC wavelengths we cannot detect background sources behind the cloud. Toward these dense lines of sight we observe only a small handful of foreground sources. This results in maps that predict extinction values that are too high across the majority of the field where we would expect low to moderate extinction due to the diffuse component of the cloud, and extinction values that are too low toward the dense portions of the cloud that may contain 50% or more of the actual mass.

To reduce the effects of this foreground/background confusion we have used observations from the PACS and SPIRE instruments on board the Herschel Space Telescope to map the column density toward each region. These observations were taken as part of the Hi-GAL guaranteed time program (Molinari et al. 2010) to survey the inner Galactic plane at 70, 160, 250, 350, and 500 μm. The map at each waveband is convolved to the lowest resolution, that of the 500 μm map, using the kernels computed by Aniano et al. (2011). We calculate the median emission in each band within a box size of 30 arcmin in order to produce a map of the diffuse (background) emission, which is subtracted from each waveband. This also eliminates the need to apply an offset correction to the Herschel observations, but decreases the sensitivity of these maps to low column -ensity features that may be associated with the star-forming complexes.

The resulting background subtracted maps were aligned and stacked using MONTAGE.⁵ The resulting stack has a flux measurement at each pixel for 160, 250, 350, and 500 μm; the 70 μm fluxes are ignored because they are not well fit by a single-temperature modified blackbody. We assume β = 2 and a gas-to-dust ratio of 100. Temperature values below 8 K and column densities below log(N_H2) < 20 are unphysical and pixels with these values are ignored in the subsequent analysis.

We extracted column density maps matching the size of the fields for which we had five band coverage (shown in Figure 7) and summed the mass corresponding to the given column densities assuming μ = 2.37. Many of these maps indicate that the structure of these star-forming complexes extend beyond the area for which we have 5 band coverage. In addition, some unknown fraction of the diffuse component that was subtracted from each star-forming region may be diffuse material local to the star-forming region rather than unrelated background/foreground material. For this reason the mass estimates that we utilize here are likely lower limits for the true mass of these complexes, but we restrict our subsequent analysis to the mass densities within the A_V > 2 contour for an ease of comparison to Evans et al. (2014) and Padoan et al. (2014).

One region in our sample, G351, is centered around b = 15 and is therefore too far from the Galactic plane to have sufficient coverage in the Hi-GAL survey to generate a column density map. For this region we have generated a map of the density of point sources using our point-source catalog from J band through 4.5 μm. Following the procedure in Cambrésy (1999) we convert the star counts into an estimate of the visual extinction. The star density map provides a more realistic lower-limit of the extinction in high column density areas with A_V ∼ 40, although the true column densities may exceed A_V > 100. We assume the median level of extinction across the map is due to unrelated line-of-sight material and subtract the median value, A_V = 6.9 from the map. We use this background subtracted extinction map in place of the Herschel column density map that we have calculated for each of the other regions. To convert between column density and extinction we adopt N(H₂) = 1.37 × 10²¹ A_V. The resulting column density maps for each star-forming region are shown in Figure 7.

We combine the column density information in each region with the spatial distribution of number counts of Class I YSOs. For this analysis, we ignore the Class II population because we assume that the more evolved YSOs are less correlated with the gas structures responsible for their formation. There is some debate whether this offset is due to dynamic motions of the stars within the molecular cloud environment or whether the cloud has moved away from or dissipated in the vicinity of the older stars. Regardless of the origin, as Figures 3 and 4 showed, the spatial extent of the Class II YSOs is broader, and the Class I YSOs are both more clustered and located in closer proximity to the portions of each molecular cloud containing denser gas.

First, we examined the surface density of YSOs (Σ_YSO) within different column density contours beginning at A_V = 1 and continuing in steps of log(A_V) = 0.25. The highest column density contour to examine for each region was determined by requiring a minimum of 2 pixels in the column density map and a minimum of 3 YSOs within the contour. The Σ_YSO versus A_V is shown in Figure 8.

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Overall, at low column densities the star-forming regions G326-4 (blue) and G326-6 (green) show the highest YSO surface densities. This may be an indication that these regions have a higher relative fraction of contamination in the YSO sample from objects such as giant stars and background galaxies. However, all of the regions display a relatively low density of YSOs at low column densities, picking up substantially at column densities between A_V ∼ 5 and A_V ∼ 20. One common result of previous star formation studies has been the determination that A_V ∼ 8 is the threshold for star formation (e.g., Heiderman et al. 2010; Lada et al. 2010). Above this critical density star formation is expected to progress according to a Schmidt Law. Below this critical density, star formation will be weak or suppressed.

The extinction is much lower in the infrared than it is at optical wavelengths. Studies of the mid-infrared extinction law find that A_K ∼ 0.1A_V and A_3.6 ∼ 0.6A_K (Flaherty et al. 2007). This means that for every magnitude of visual extinction we have about 0.06 magnitudes of extinction at 3.6 μm. So when A_V = 15, we are already peering through approximately 1 magnitude of extinction at 3.6 μm. For A_V > 40 we may be looking through 3 or more magnitudes of extinction in the near- and mid-infrared. This means that in our magnitude-limited surveys we are likely to miss distinct portions of the YSO population toward both high column density (low surface brightness) regions as well as portions of the observable cloud that are dominated by the bright extended emission excited by the high-mass stars.

We see this effect in Figure 8. The observed surface density of the YSOs reaches a maximum around (or before) A_V = 40 in our six massive star-forming regions; above this point we do not detect sufficient numbers of YSOs or sufficiently resolve the column density contours in the majority of our star-forming regions. In NGC 6334, we have a large area with high column density, but we see the surface density of (near-infrared detected) Class I YSOs plummet beyond this peak. For NGC 6334 additional deep observations are available at 5.8 and 8.0 μm that enabled the selection of more embedded YSOs, and even longer wavelength observations from MIPS, Herschel, and the SMA indicate signficant populations of infrared-dark sources at the beginning of the star formation process (e.g., Russeil et al. 2013; Willis et al. 2013; Hunter et al. 2014). However, we use only the near-infrared selected YSOs in the subsequent analysis for direct comparison to the rest of the star-forming regions.

The decrease in YSO density at the higher column densities suggests that we need to account for extinction when we turn our YSO counts (Σ_YSO) into a star-formation rate estimate (Σ_SFR)—our samples will suffer from incompleteness as we increase to higher column densities. Our method for determining the star-formation rate based on the number of YSOs observed brighter than the magnitude at which the sample is complete includes a correction for the average extinction toward the stars. We used the method described in Willis et al. (2013) to determine the star-formation rate indicated by the YSOs observed at each A_V contour by assuming that each star was observed behind an extinction given by the minimum value of that extinction contour. As we increase to higher column densities the total number of stars (both observed and unobserved) decreases, increasing the relative error on these measurements.

In Figure 9 we examine log(Σ_SFR) versus log(Σ_gas) above ∼200 M_⊙ pc⁻², equivalent to the suggested A_V ∼ 8 threshold for star formation (e.g., Heiderman et al. 2010). We fit a linear relation to the data between Σ_gas ∼ 200 M_⊙ pc⁻² and Σ_gas ∼ 3000 M_⊙ pc⁻², shown as the black dotted line. The red dotted line extrapolates this fit out to higher column densities. We find α = 2.15 ± 0.41 above the assumed critical density A_V = 8. The high dispersion reflects the significant differences in the observed star-formation rate within the same minimum column density contour in different molecular clouds. The dashed line shows the extragalactic Kennicutt–Schmidt relation from Kennicutt (1998) for α = 1.4. As we extend our slope determination to lower minimum column densities we find that the fit becomes shallower. The slope for log(Σ_YSO) versus log(Σ_gas) above A_V > 2 (Σ_gas ∼ 50 M_⊙ pc⁻²) is α = 1.59 ± 0.26, and for A_V > 1 we find α = 1.37 ± 0.23.

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The slope for the power-law behavior above A_V > 1 is similar to the alpha = 1.4 slope observed for the galaxy-averaged view of Σ_SFR versus Σ_gas in Kennicutt (1998). However, the slope varies as a function of the column density range examined, flattening as we extend to lower minimum column densities. The variation of the slope as a function of the minimum column density is also consistent with the results from Bigiel et al. (2008), where the slope of the ${{\rm{\Sigma }}}_{\mathrm{SFR}}\sim {{\rm{\Sigma }}}_{\mathrm{gas}}^{\alpha }$ relation differed depending on the column density regime examined using both atomic and molecular gas. The total gas surface density probed by Bigiel et al. (2008) approximately extends to an order of magnitude below the level we can probe using the Herschel column density maps, which reflects only the presence of molecular gas, not the total HI + H ii gas component of the molecular clouds. Bigiel et al. (2008) found a shallower slope for the power law fit to Σ_SFR versus ${{\rm{\Sigma }}}_{{H}_{2}}$ at α = 0.96 ± 0.07. Although our data show that the slope becomes shallower at lower column densities, their fits were centered around ${{\rm{\Sigma }}}_{{H}_{2}}=10\;{M}_{\odot }$ pc⁻², which is a factor of 5 lower than the lowest column densities we probe with our maps, and very few of their sub-kiloparsec scale regions contain sufficient dense gas to reach into the Σ_gas range found in resolved star-forming clouds.

We fit Σ_SFR versus Σ_gas within each molecular cloud individually shown in 10, beginning with the A_V > 2 contour and continuing in steps of $\mathrm{log}{A}_{V}=0.25$ to the maximum density in the map. The power-law index for the sample of clouds is between 1.77 < α < 2.86, with the best fit for each cloud given in Table 8. The uncertainty reported for each index is purely based on the G326-4 and G326-6, which we previously identified as appearing to have a higher than expected surface density. YSOs at low column densities also appear to have a slightly shallower slope for Σ_SFR versus Σ_gas below A_V ∼ 8, but no significant difference is seen in the power-law slope above A_V > 8, compared to A_V > 2 for the full sample of clouds. However, the star-formation rates are particularly uncertain toward higher column densities where additional data from longer wavelength observations would assist in recovering the extremely embedded sources.

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Each individual cloud is well fit by a power law between Σ_gas and Σ_SFR, but the range in star-formation rates seen at the same column density for different clouds increases the scatter in the combined fit to all the star-forming regions. The Herschel column density maps do not provide velocity information that would enable disentangling line-of-sight components at different distances. The clouds in this study are all at very low Galactic latitudes ($b\lt 1$) and potentially signficant levels of foreground or background dust could contribute to the overall difference in star-formation rate at the same column density interval in different clouds. This effect would drive up the scatter when comparing the star formation between different clouds but not significantly affect the slope for the fit to a single cloud.

The average for our sample of molecular clouds finds ${{\rm{\Sigma }}}_{\mathrm{SFR}}={{\rm{\Sigma }}}_{\mathrm{gas}}^{2.15\pm 0.41}$ once the gas becomes dense enough for star formation to proceed. In Lada et al. (2013), a similar average slope of α = 2.04 ± 0.01 was found for Orion A, Taurus, and California and α = 3.30 ± 0.21 for Orion B. Combined with our sample, these star-forming regions range from extremely quiescent (California Molecular Cloud) to some of the most active star-forming regions in the Galaxy (G305, NGC 6334). The similarity of the power-law indexes indicates that this relation is likely independent of the overall star formation activity of a cloud. We demonstrate this in Figure 11, showing that for both our molecular clouds and the sample of Lada et al. (2013) there is a lack of correlation between the power-law index α and the total luminosity of the cloud at 70 μm, an indicator of the star-formation activity.

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This independence is seen both for low-mass clouds that form only a handful of stars, as well as giant molecular clouds that form thousands of stars, including some of the most active GMCs in our galaxy. The relative small spread in the power-law indexes and the lack of correlation between the power-law index and star formation tracers suggests that on the large scale, the process of star formation is occurring in the same manner from low density through high density portions of molecular clouds. Clouds that appear to have a high relative star-formation rate are therefore likely clouds that have a current or recent high fraction of gas sufficiently dense for star formation to progress.

It is interesting to note that the two mini-starburst regions from Motte et al. (2012), W43 and G035, fall more than 2σ above the empirical fit to the 4th Quadrant regions above A_V = 8. However, the star-formation rates and gas surface densities for these regions were derived using different methodlogies, so this finding may not be signficant. The 4th Quadrant sample includes clouds with highly concentrated star formation activity, such as the dense ridge that mimics mini-starburst conditions in NGC 6334 (Willis et al. 2013) and the rich young clusters in G305 that have presumably recently formed through mini-starburst-like structure. This result will need to be followed up by combining mid- and far-infrared detected point sources at these high column density ranges to see whether the reported mini-starburst regions show similar behavior to the 4th Quadrant sample. More data will be needed to determine if these apparent differences for the prototypical mini-starburst sources reflect a physical difference in the star formation process in the densest environments.

The star formation history of molecular clouds is not a simple case of either a constant star-formation rate at all times, or an instantaneous burst that immediately forms a large population of stars. Instead, the process of star formation begins and ends, and the total rate and efficiency of the activity vary while it is ongoing. The interplay between the stellar feedback and the diffuse material sculpts the cloud, compressing some gas and dispersing other portions, causing the star-formation rate to vary within a given cloud at a particular time. Both the gas and stars in molecular clouds are in motion, which can potentially lead to an observable offset between dense star-forming gas and young stars over relatively short timescales (Goodman & Arce 2004).

The dissection of galaxies on small spatial scales captures star formation over extended timescales that are much longer than the lifetime traced out by identifying populations of YSOs. The effects of radiative feedback that disperses clouds and the dynamical interactions experienced by young stars that move them relative to the gas are magnified over longer timescales probed by star-formation rate tracers applied in unresolved distant galaxies. This likely contributes to the scatter observed in measurements of the ${{\rm{\Sigma }}}_{\mathrm{SFR}}\sim {{\rm{\Sigma }}}_{\mathrm{gas}}^{\alpha }$ relation for resolved molecular-cloud sized structures in other galaxies.

Inside molecular cloud complexes in our galaxy we see a tight correlation between the surface density of early-stage YSOs (and their inferred star-formation rate) and gas density inside of a molecular cloud, agreeing with the Schmidt law. Numerous effects, including line-of-sight column density contamination and the relatively small number of observed YSOs contribute to the scatter in observed star-formation rates and gas densities in Milky Way clouds.

It is also important to remember that when we consider a critical density for star formation, we typically use extinction and column density measurements that trace the projected two-dimensional structure of the cloud. In reality, star formation takes place within the three-dimensional gas within the molecular clouds and it is the volume density that should be intrinsically associated with the star-formation activity.