The High-mass Protostellar Population of a Massive Infrared Dark Cloud

The catalog of Hi-GAL 70 μm point sources (Molinari et al. 2016) forms the basis of our sample. These sources were identified using the CUrvature Thresholding Extractor (CuTEx) algorithm, which finds pixels with high curvature by calculating the second derivatives in intensity profiles; areas above a certain curvature threshold indicate the location of a source. As described by Molinari et al. (2016), the source extraction threshold was chosen to be able to detect relatively faint sources, while minimizing false detections.

We obtained the coordinates of all 70 μm sources in the Hi-GAL catalog that overlap with the elliptical region defining IRDC C (Simon et al. 2006), identifying a total of 40 sources. We then inspected the Hi-GAL images of these sources, especially at 70 μm, to examine source crowding. We found that several sources were in locally crowded regions, such that it was not possible to resolve their emission at ∼160 μm near the peak of their SEDs. The angular resolution of Herschel at these wavelengths led us to set a minimum aperture size of ∼6'' in radius.

Thus, in these cases of source crowding we simply treat the region as a single source. The most prominent example of source crowding is that of Cp23, near the C9 region in Butler et al. (2014) (hereafter BTK14), which was marked as four different sources in the Hi-GAL catalog. We model it as a single, large source, with its coordinates chosen from the most central of the four Hi-GAL sources. There are then only two other cases of "crowding," involving close pairs of sources. Here, we set the strongest 70 μm source to be the source location, so all of the coordinates are still directly from the Hi-GAL catalog. After these steps, 35 sources remain in our sample, which we label Cp01, Cp02, Cp03, etc., i.e., protostellar candidate sources in IRDC C, based on increasing Galactic longitude (see Figure 1).

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We analyzed archival 70, 110, and 160 μm Herschel–PACS images of proposal ID "KPGT-okrause-1." These data were observed with medium scanning speed and had 6'', 7'', and 11'' angular resolution, respectively (Poglitsch et al. 2010). The data sets were obtained as product level 2.5 and the Standard Product Generation v14.2.0. We applied zero-level offset correction by following the method of Lim & Tan (2014), which we describe below. We adopted a model SED of the diffuse Galactic plane emission (Draine & Li 2007) from NIR to sub-mm. We fit this model to the observed median intensities from 90% to 110% size annuli compared to the major and minor axes of the IRDC defined by Simon et al. (2006). We considered data at 8 μm (Spitzer–IRAC), 24 μm (Spitzer–MIPS), 70, 160 μm (Herschel–PACS), 250, 350, and 500 μm (Herschel–SPIRE) (with these Herschel data from the Hi-GAL survey; Molinari et al. 2010) and then predicted the expected intensities in the archival Herschel–PACS 70, 110, and 160 μm band data. A single-value offset for each wavelength was then applied to each data set (760, 2615, and 3801 MJy sr⁻¹ for the 70, 110, and 160 μm bands, respectively). We found an astrometric difference of a few arcseconds between the Herschel and Spitzer maps. We corrected this by the average value of the mean positional offset of point sources seen at 8, 24, 70, 110, and 160 μm.

For the photometry of the sources at shorter wavelengths, we utilized the 24 μm Spitzer–MIPS images from the MIPSGAL survey (Carey et al. 2009). We also examined images from the Spitzer–IRAC Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (Churchwell et al. 2009). Most of our sources appear "dark" at the shortest IRAC wavelengths, ∼3 μm, and often even the 8 μm image only provides an upper limit on source flux. Given that the ZT protostellar models do not accurately predict fluxes at these wavelengths, where polycyclic aromatic hydrocarbon (PAH) emission can often be significant, we only utilized the 8 μm (IRAC Band 4) image to place upper limit constraints on source SEDs.

We used fixed aperture sizes that were determined by inspecting the morphology of the 70 μm images. The apertures were chosen to include as much of the emission coming from the source as possible, while avoiding the emission of nearby sources. Since the beam size for this image is 6'', the smallest aperture allowed for the sources was also set to a radius of 6'' in order to match the beam size. The majority of the sources had apertures slightly larger than the beam size, averaging about 10'' in radius. We also examined the sensitivity of our results to varying the aperture size by 30%. For a given aperture, the photometric flux of each source was then measured at 8, 24, 70, 100, 160, 250, 350, and 500 μm using the Python package PHOTUTILS.

Since the protostellar sources were embedded in a high-mass surface density protocluster clump, it was important to carry out background subtraction of flux from this surrounding material. We used an annular region extending to twice the aperture radius to measure this background emission, which follows the methods adopted previously by De Buizer et al. (2017). The level of the background was then assessed as the median intensity value in this annulus. We will examine the effects on the SEDs and other results of either carrying out (which is our fiducial case) or not carrying out this step of background subtraction.

The uncertainties in the fluxes receive a contribution from basic photometric/calibration uncertainties, which we set to 10%, combined in quadrature with those due to background subtraction, which can often be the dominant source of uncertainty. We assessed the level of background uncertainty by examining the level of the background fluctuations, σ_bg, measured as the standard deviation of flux densities patches in the annular background region that had an area equal to that of the aperture.

Once a source SED is derived, consisting of measured fluxes, including upper limits, and their estimated uncertainties, then these data are used to constrain the ZT protostellar SED models, under the assumption of fixed source distances of 5 kpc. The detailed method of the fitting procedure follows that of De Buizer et al. (2017) and Liu et al. (2019): in particular the short-wavelength IRAC 8 μm data point is used only as an upper limit, given the uncertainties of its possible contamination with PAH emission that is not treated in the ZT SED models.

The physical basis of the ZT protostellar models is the Turbulent Core Model (McKee & Tan 2003). While there are several other grids of protostellar radiative transfer models (e.g., Robitaille et al. 2006; Molinari et al. 2008; Robitaille 2017), these tend to be less physically self-consistent, especially for the high-pressure, high-density conditions of IRDCs, and have much larger numbers of free parameters. There are only three main parameters in the ZT models: initial core mass, M_c, with the current grid exploring a range from 10 to 480 M_⊙; surrounding clump mass surface density, Σ_cl, with a range from 0.1 to 3.2 g cm⁻² (which sets the bounding pressure of the core, so cores in high Σ_cl regions are smaller and denser); and the current protostellar mass, m_*, which sets the evolutionary stage of the collapse of a given core. The protostellar mass is sampled from masses from 0.5, 1, 2 ... M_⊙ up to masses that can be typically ∼50% of M_c, with this efficiency set by protostellar outflow feedback. In addition to these three primary parameters, the fitting procedure also returns an estimate of the inclination angle of the protostellar outflow axis to the line of sight and an estimate of the foreground extinction.

Here we illustrate results of the SED model fitting for three example sources: Cp23 selected as an example of a large, bright source, Cp15 as an example of a more typical source in the sample of moderate flux, and Cp03 as an example of a relatively faint source.

The left column of Figure 2 shows the 70 μm images of Cp23, Cp15, and Cp03 (top, middle, bottom), along with the fiducial choice of aperture for each source (middle circles). The second column then shows the derived source SEDs, both before background subtraction (dashed lines) and after (solid lines). One can see that background subtraction has a much greater effect for the fainter sources. The third column shows the effect of different aperture sizes, varying the radius by 30% to smaller and larger sizes, on the background subtracted SEDs. Finally, the fourth column shows the data for the fiducial SEDs and the ZT model fits to these data. Note that for Cp03 the two longest-wavelength measurements for the SED become negative after background subtraction in the fiducial case, and at these wavelengths we assume upper limits during the model fitting process, with the values set by the level of background fluctuations.

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The parameters of the 10 best SED models for each source are shown in Table 1 in order of decreasing goodness of fit (i.e., increasing value of reduced χ² (this is normalized by the number of data points, N); second column). The presentation here follows a similar format as that of De Buizer et al. (2017) for the fitting results of eight massive protostars in the SOMA survey. The other parameters presented are: initial core mass, M_c; mass surface density of the clump environment, Σ_cl; initial core radius, R_c, which is determined by M_c and Σ_cl and is listed in both parsecs and angular size that can be compared to the aperture size; the current protostellar mass, m_*; the viewing angle to the outflow axis, θ_view; the foreground extinction, A_V; the current remaining gas mass in the infall envelope, M_env, i.e., given what has been accreted and expelled by feedback; the opening angle of the outflow cavity, θ_w,esc; the accretion rate of the star, ${\dot{m}}_{* }$; the total luminosity of the source assuming isotropic emission given the received bolometric flux from the model, L_tot,iso; and the actual total bolometric luminosity of the protostar model, L_tot,bol.

Table 1.  Parameters of 10 Best ZT Models for Example Protostars: Cp23; Cp15; Cp03

Source	χ2	Mc	Σcl	Rc	m*	θview	AV	Menv	θw,esc	${\dot{m}}_{* }$	Ltot,iso	Ltot,bol
		(M⊙)	(g cm−2)	pc ('')	(M⊙)	(deg)	(mag)	(M⊙)	(deg)	(M⊙ yr−1)	(L⊙)	(L⊙)
Cp23	12.425	400	3.2	0.083 (3.42)	8.0	12.84	36	382	7	1.1 × 10−3	4.2 × 104	2.0 × 104
Rap = 28''	13.171	320	3.2	0.074 (3.05)	8.0	12.84	55.6	308	8	1.0 × 10−3	4.4 × 104	1.7 × 104
	14.311	480	3.2	0.091 (3.74)	8.0	12.84	17.2	462	6	1.1 × 10−3	4.0 × 104	2.2 × 104
	17.961	240	3.2	0.064 (2.65)	8.0	12.84	87.9	227	10	9.5 × 10−4	6.0 × 104	1.7 × 104
	22.821	200	3.2	0.059 (2.42)	8.0	12.84	100.0	184	11	9.0 × 10−4	7.8 × 104	2.0 × 104
	28.397	200	3.2	0.059 (2.42)	4.0	12.84	7.1	191	7	6.5 × 10−4	2.5 × 104	1.2 × 104
	29.639	160	3.2	0.052 (2.16)	8.0	22.33	0.0	146	13	8.5 × 10−4	1.9 × 104	2.0 × 104
	29.948	400	1.0	0.147 (6.07)	8.0	12.84	0.0	383	8	4.6 × 10−4	2.3 × 104	1.2 × 104
	31.283	480	0.3	0.287 (11.83)	12.0	22.33	72.7	459	10	2.5 × 10−4	3.8 × 104	4.0 × 104
	32.020	480	1.0	0.161 (6.65)	12.0	12.84	100.0	461	9	5.9 × 10−4	8.5 × 104	3.8 × 104
Averages	15.720	312	3.2	0.073 (3.02)	8.0	12.84	59.4	296	9	1.0 × 10−3	5.1 × 104	2.0 × 104
Cp15	0.662	80	0.1	0.208 (8.59)	1.0	88.57	100.0	77	8	1.9 × 10−5	1.7 × 102	1.9 × 102
Rap = 10''	0.662	60	0.1	0.180 (7.44)	1.0	88.57	100.0	57	10	1.8 × 10−5	1.7 × 102	2.0 × 102
	0.849	50	0.1	0.165 (6.79)	1.0	61.64	100.0	48	11	1.7 × 10−5	1.5 × 102	1.7 × 102
	0.906	200	0.1	0.329 (13.58)	0.5	12.84	48.5	200	3	1.7 × 10−5	1.5 × 102	1.3 × 102
	0.910	100	0.1	0.233 (9.60)	1.0	88.57	100.0	98	7	2.0 × 10−5	1.8 × 102	2.0 × 102
	1.024	10	0.3	0.041 (1.71)	0.5	28.96	78.8	9	18	1.9 × 10−5	1.4 × 102	1.9 × 102
	1.065	40	0.1	0.147 (6.07)	2.0	88.57	100.0	36	19	2.2 × 10−5	1.8 × 102	2.7 × 102
	1.211	40	0.1	0.147 (6.07)	1.0	54.90	100.0	38	12	1.6 × 10−5	1.4 × 102	1.7 × 102
	1.407	160	0.1	0.294 (12.14)	0.5	22.33	0.0	158	3	1.6 × 10−5	1.0 × 102	9.8 × 101
	1.435	200	0.1	0.329 (13.58)	1.0	85.70	84.8	197	4	2.5 × 10−5	1.7 × 102	1.8 × 102
Averages	0.980	69	0.1	0.183 (7.53)	0.9	62.06	81.2	66	10	1.9 × 10−5	1.5 × 102	1.8 × 102
Cp03	0.248	10	0.3	0.041 (1.71)	4.0	77.00	100.0	1	68	2.4 × 10−5	4.9 × 101	6.7 × 102
Rap = 9''	0.279	10	0.1	0.074 (3.04)	2.0	79.92	5.1	4	50	1.1 × 10−5	2.1 × 101	1.3 × 102
	0.331	40	0.1	0.147 (6.07)	12.0	88.57	100.0	2	82	9.5 × 10−6	5.7 × 101	1.1 × 104
	0.388	30	0.3	0.072 (2.96)	12.0	88.57	100.0	1	81	2.2 × 10−5	7.0 × 101	1.2 × 104
	0.539	10	0.1	0.074 (3.04)	1.0	88.57	100.0	7	31	1.0 × 10−5	4.4 × 101	1.1 × 102
	0.838	10	0.1	0.074 (3.04)	0.5	88.57	100.0	9	20	7.8 × 10−6	4.6 × 101	7.5 × 101
	3.444	10	0.3	0.041 (1.71)	2.0	88.57	100.0	5	43	3.0 × 10−5	5.8 × 101	2.8 × 102
	4.003	20	0.1	0.104 (4.29)	0.5	88.57	100.0	19	13	9.6 × 10−6	7.0 × 101	9.0 × 101
	5.088	30	0.1	0.127 (5.26)	0.5	88.57	100.0	29	10	1.1 × 10−5	7.6 × 101	9.0 × 101
	5.484	40	0.1	0.147 (6.07)	0.5	88.57	100.0	39	8	1.1 × 10−5	7.8 × 101	8.8 × 101
Averages	0.399	15	0.1	0.075 (3.08)	2.9	85.20	84.2	3	55	1.3 × 10−5	4.5 × 101	3.4 × 102

Note. The 10 best models of the three example sources with the 11th line for each source being the calculated average of "good" models (see the text), using the fiducial aperture with background subtraction.

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The last row for each source in Table 1 displays the average of each listed parameter for "good" model fits, using the following method. We have two different methods based on the distribution of χ² values. The first is for sources such as Cp23 that have all values of χ² greater than 1. Here, we take the geometric mean of the parameters of all the models with χ² less than or equal to twice the first, i.e., smallest, value of χ² value. This acts to exclude models with relatively high χ². For example, the average for Cp23 would include all of the models with χ² ≤ 2 × 12.425, which are the top five models. The second method is for sources like Cp15 and Cp03 which have a best χ² value smaller than 1. Here we set a limit of χ² < 2, and take the average of all the values of models from the best set of models, up to 10, that meet this limit. For both methods, we take the geometric mean for all quantities of the accepted models except ${A}_{V},\,{\theta }_{{\rm{v}}{\rm{i}}{\rm{e}}{\rm{w}}}$ and ${\theta }_{{\rm{w}},{\rm{e}}{\rm{s}}{\rm{c}}}$, where the arithmetic mean is used.

For Cp23, the best-fit model has χ² = 12.425, which is a relatively large value, i.e., the models do not fit particularly well. We discuss the reasons for this below. Still, considering the properties of the best model, we see it has an initial core mass of M_c = 400 M_⊙, current protostellar mass of m_* = 8 M_⊙, forming in a clump mass surface density of Σ_cl = 3.2 g cm⁻², and a total luminosity of 2 × 10⁴ L_⊙. The range of values of these parameters of the best models does not vary greatly, with the averages of "good" models being M_c = 312 M_⊙, Σ_cl = 3.2 g cm⁻², and m_* = 8 M_⊙.

A more complete view of the model parameter space for Cp23 is shown in Figure 3(a), which is a standard output of the ZT model fitting routines. The figure shows a series of 2D parameter space plots that illustrate all the models with χ² < 50 and with the best five models shown with crosses (the best model has a large cross). These plots show the correlations and degeneracies in the resulting model parameters that are constrained by the SED data.

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For Cp15, which has its model parameter space displayed in Figure 3(b), we see that the preferred models shift to lower core (≲100 M_⊙) and protostellar (∼1 M_⊙) masses. Lower clump mass surface densities also tend to be selected. There is relatively large dispersion in certain parameters, such as M_c and Σ_cl, i.e., they are not as tightly constrained as in the case of Cp23.

These trends continue for Cp03, which has its model parameter space displayed in Figure 3(c). However, now we also see the models with lowest core mass, i.e., M_c = 10 M_⊙, are quite strongly preferred, though not exclusively. Such values are at the lower boundary of the current model grid parameter space, so caution is needed in the interpretation of the results. In particular, it is possible that lower core masses could be reasonable fits to the data.

Next we investigate the effects of not carrying out background subtraction, and of varying aperture size when background subtraction is carried out, on the model fitting results. Table 2 shows the values of χ² and various model parameters of the best-fitting models and the average of "good" models (see above) for these cases. Focusing on average values, we see the general reduction of core mass, envelope mass, and luminosity following background subtraction, with relatively larger effects seen for the lower-luminosity sources Cp15 and Cp03 compared to Cp23. We also see the expected dependence of derived model properties on aperture size, i.e., smaller masses and luminosities when smaller apertures are used. The range in these values gives some guidance on the degree of systematic uncertainties that result from the process of background subtraction and choice of aperture size. Note that the size of these uncertainties depends on the source luminosity.

Table 2.  Effect of Aperture Size on Protostellar Model Results^a

Source	χ2	Mc	Σcl	m*	Menv	Ltot,iso	Ltot,bol
		(M⊙)	(g cm−2)	(M⊙)	(M⊙)	(L⊙)	(L⊙)
Cp23	19.866, ${12.425}_{2.587}^{34.817}$	480, ${400}_{160}^{480}$	3.2, ${3.2}_{3.2}^{3.2}$	8.0, ${8.0}_{4.0}^{8.0}$	462, ${382}_{153}^{462}$	40000, ${40000}_{32000}^{40000}$	22000, ${20000}_{12000}^{22000}$
${R}_{\mathrm{ap}}$ = 28''	27.028, ${15.720}_{3.839}^{54.884}$	414, ${312}_{221}^{348}$	2.4, ${3.2}_{1.8}^{2.4}$	8.9, ${8.0}_{6.2}^{9.8}$	398, ${296}_{209}^{331}$	50000, ${51000}_{36000}^{55000}$	29000, ${20000}_{14000}^{31000}$
Cp15	2.324, ${0.662}_{0.997}^{0.299}$	20, ${80}_{20}^{60}$	3.2, ${0.1}_{0.1}^{0.3}$	0.5, ${1.0}_{1.0}^{0.5}$	19, ${77}_{17}^{60}$	2800, ${170}_{94}^{300}$	860, ${190}_{150}^{180}$
Rap = 10''	3.026, ${0.980}_{1.364}^{0.470}$	126, ${69}_{34}^{111}$	0.4, ${0.1}_{0.1}^{0.1}$	1.7, ${0.9}_{0.7}^{0.9}$	123, ${66}_{31}^{109}$	1800, ${150}_{86}^{250}$	980, ${180}_{110}^{210}$
Cp03	0.054, ${0.248}_{0.378}^{0.239}$	60, ${10}_{10}^{10}$	1.0, ${0.3}_{0.1}^{0.1}$	0.5, ${4.0}_{2.0}^{1.0}$	59, ${1}_{4}^{7}$	570, ${49}_{20}^{44}$	400, ${670}_{130}^{110}$
Rap = 9''	0.092, ${0.399}_{0.378}^{0.520}$	88, ${15}_{10}^{18}$	0.3, ${0.1}_{0.1}^{0.1}$	1.0, ${2.9}_{2.0}^{1.6}$	86, ${3}_{4}^{6}$	520, ${45}_{20}^{49}$	380, ${340}_{370}^{340}$

Note.

^aWithin each column, the first value uses fiducial aperture size without background subtraction, the second value is the fiducial aperture with background subtraction, the subscripted value is the case with aperture 30% smaller than the fiducial, and the superscripted value is the aperture 30% larger than the fiducial. For each source, two lines are shown: the first line is the best-fitting model; the second line is the average of "good" models (see the text). The first column also lists the angular size of the fiducial aperture.

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We apply our fiducial analysis methods to all 35 identified sources in the IRDC and list the derived protostellar properties (best-fit model and average of "good" models) in Table 3.

Table 3.  Parameters of the Fitted Models for All Sources

Source	Rap	χ2	Mc	Σcl	Rc	m*	θview	AV	Menv	θw,esc	${\dot{M}}_{\mathrm{disk}}$	Ltot,iso	Ltot,bol
	(arcsec)		(M⊙)	(g cm−2)	pc (arcsec)	(M⊙)	(deg)	(mag)	(M⊙)	(deg)	(M⊙ yr−1)	(L⊙)	(L⊙)
Cp01	15	0.439	10	3.2	0.013 (0.54)	4.0	64.85	16.2	2	56	1.9 × 10−4	2.4 × 102	1.9 × 103
	10	0.908	23	0.3	0.063 (2.59)	${2.64}_{0.94}^{7.45}$	59.69	36.8	10	37	4.0 × 10−5	3.8 × 102	1.1 × 103
Cp02	6	0.955	10	0.3	0.041 (1.71)	4.0	88.57	100.0	1	68	2.4 × 10−5	2.9 × 101	6.7 × 102
	4	1.235	19	0.2	0.075 (3.11)	${5.83}_{2.72}^{12.50}$	88.57	100.0	1	70	1.6 × 10−5	3.9 × 101	4.7 × 102
Cp03	9	0.248	10	0.3	0.041 (1.71)	4.0	77.00	100.0	1	68	2.4 × 10−5	4.9 × 101	6.7 × 102
	6	0.399	15	0.1	0.075 (3.08)	${2.88}_{0.88}^{9.48}$	85.20	84.2	3	55	1.3 × 10−5	4.5 × 101	3.4 × 102
Cp04	12	0.235	20	0.3	0.059 (2.42)	4.0	82.82	0.0	11	38	5.4 × 10−5	3.1 × 102	1.1 × 103
	10	0.569	34	0.2	0.102 (4.22)	${2.46}_{1.58}^{3.84}$	64.63	23.7	26	24	3.5 × 10−5	3.3 × 102	5.6 × 102
Cp05	10	0.109	10	0.3	0.041 (1.71)	2.0	88.57	84.8	5	43	3.0 × 10−5	5.8 × 101	2.8 × 102
	10	0.311	22	0.1	0.103 (4.27)	${0.76}_{0.44}^{1.32}$	82.85	89.6	19	19	1.3 × 10−5	7.0 × 101	1.1 × 102
Cp06	9	0.014	10	0.3	0.041 (1.71)	4.0	82.82	100.0	1	68	2.4 × 10−5	3.4 × 101	6.7 × 102
	10	0.087	16	0.1	0.079 (3.28)	${1.89}_{0.60}^{5.93}$	87.71	91.3	5	43	1.4 × 10−5	5.2 × 101	3.7 × 102
Cp07	8	0.159	10	0.3	0.041 (1.71)	2.0	77.00	100.0	5	43	3.0 × 10−5	6.0 × 101	2.8 × 102
	10	0.307	20	0.1	0.093 (3.85)	${1.43}_{0.44}^{4.66}$	72.51	80.9	9	34	1.3 × 10−5	1.9 × 102	3.0 × 102
Cp08	10	0.445	10	0.3	0.041 (1.71)	0.5	85.70	100.0	9	18	1.9 × 10−5	1.2 × 102	1.9 × 102
	10	0.659	44	0.1	0.138 (5.68)	${0.76}_{0.48}^{1.20}$	69.79	67.9	40	12	1.7 × 10−5	1.1 × 102	1.5 × 102
Cp09	20	7.383	480	0.1	0.510 (21.03)	8.0	28.96	10.1	463	9	8.5 × 10−5	9.3 × 103	9.7 × 103
	6	10.082	376	0.1	0.451 (18.62)	${10.48}_{8.66}^{12.69}$	28.76	66.8	351	13	9.1 × 10−5	1.5 × 104	1.7 × 104
Cp10	6	1.040	10	0.1	0.074 (3.04)	2.0	88.57	100.0	4	50	1.1 × 10−5	2.0 × 101	1.3 × 102
	2	1.189	10	0.2	0.055 (2.28)	${2.83}_{2.00}^{4.00}$	88.57	100.0	2	59	1.7 × 10−5	2.4 × 101	3.7 × 102
Cp11	8	0.868	10	1.0	0.023 (0.96)	2.0	43.53	16.2	5	39	7.5 × 10−5	2.6 × 102	7.6 × 102
	7	1.291	16	0.4	0.048 (1.97)	${2.69}_{0.95}^{7.60}$	58.95	45.3	6	43	4.0 × 10−5	3.9 × 102	7.9 × 102
Cp12	19	0.265	60	1.0	0.057 (2.35)	24.0	88.57	42.4	5	71	1.9 × 10−4	2.1 × 103	9.3 × 104
	10	0.708	38	0.4	0.076 (3.13)	${9.80}_{4.71}^{20.40}$	68.88	56.9	10	54	7.8 × 10−5	2.5 × 103	1.4 × 104
Cp13	12	10.302	240	0.1	0.360 (14.87)	1.0	12.84	60.6	240	4	2.6 × 10−5	3.2 × 102	2.4 × 102
	10	11.740	173	0.1	0.306 (12.63)	${1.52}_{1.08}^{2.13}$	37.76	79.7	169	7	2.9 × 10−5	3.4 × 102	3.0 × 102
Cp14	10	0.013	10	0.3	0.041 (1.71)	4.0	88.57	78.8	1	68	2.4 × 10−5	2.9 × 101	6.7 × 102
	10	0.136	16	0.1	0.079 (3.28)	${1.89}_{0.60}^{5.93}$	88.57	89.1	5	43	1.4 × 10−5	5.1 × 101	3.7 × 102
Cp15	10	0.662	80	0.1	0.208 (8.59)	1.0	88.57	100.0	77	8	1.9 × 10−5	1.7 × 102	1.9 × 102
	10	0.980	69	0.1	0.183 (7.53)	${0.87}_{0.57}^{1.32}$	62.06	81.2	66	10	1.9 × 10−5	1.5 × 102	1.8 × 102
Cp16	6	8.522	10	1.0	0.023 (0.96)	4.0	88.57	100.0	1	59	7.7 × 10−5	1.1 × 102	1.1 × 103
	9	12.995	14	0.3	0.051 (2.12)	${1.59}_{0.71}^{3.53}$	83.23	62.4	7	35	2.8 × 10−5	1.1 × 102	2.9 × 102
Cp17	8	0.068	10	0.1	0.074 (3.04)	1.0	88.57	100.0	7	31	1.0 × 10−5	4.4 × 101	1.1 × 102
	9	0.608	16	0.1	0.077 (3.18)	${1.88}_{0.56}^{6.27}$	85.99	93.6	4	44	1.3 × 10−5	5.1 × 101	3.7 × 102
Cp18	10	1.308	60	0.3	0.101 (4.18)	0.5	12.84	84.8	60	5	3.0 × 10−5	3.0 × 102	1.8 × 102
	9	1.617	115	0.1	0.206 (8.50)	${0.79}_{0.57}^{1.10}$	32.13	96.1	113	5	2.5 × 10−5	2.5 × 102	2.1 × 102
Cp19	8	0.764	120	0.1	0.255 (10.52)	0.5	85.70	100.0	118	4	1.5 × 10−5	8.4 × 101	8.8 × 101
	10	1.029	53	0.1	0.160 (6.58)	${0.57}_{0.44}^{0.76}$	79.12	100.0	50	9	1.4 × 10−5	8.9 × 101	1.1 × 102
Cp20	10	0.118	120	0.1	0.255 (10.52)	0.5	85.70	100.0	118	4	1.5 × 10−5	8.4 × 101	8.8 × 101
	10	0.176	51	0.1	0.156 (6.45)	${0.54}_{0.44}^{0.66}$	87.71	100.0	48	9	1.4 × 10−5	8.3 × 101	1.0 × 102
Cp21	8	1.165	30	0.3	0.072 (2.96)	12.0	88.57	41.4	1	81	2.2 × 10−5	7.0 × 101	1.2 × 104
	5	1.378	16	0.2	0.075 (3.09)	${4.10}_{1.54}^{10.92}$	86.27	34.3	2	63	1.4 × 10−5	4.1 × 101	4.7 × 102
Cp22	8	0.009	80	0.1	0.208 (8.59)	1.0	12.84	92.9	77	8	1.9 × 10−5	4.6 × 102	1.9 × 102
	10	0.022	65	0.1	0.188 (7.75)	${1.23}_{0.90}^{1.69}$	21.73	66.3	61	12	2.0 × 10−5	2.6 × 102	2.1 × 102
Cp23	28	12.425	400	3.2	0.083 (3.42)	8.0	12.84	36.4	382	7	1.1 × 10−3	4.2 × 104	2.0 × 104
	5	15.720	312	3.2	0.073 (3.02)	${8.00}_{8.00}^{8.00}$	12.84	59.4	296	9	1.0 × 10−3	5.1 × 104	2.0 × 104
Cp24	8	1.066	10	0.1	0.074 (3.04)	0.5	54.90	100.0	9	20	7.8 × 10−6	5.0 × 101	7.5 × 101
	10	1.289	22	0.1	0.098 (4.03)	${1.04}_{0.36}^{3.04}$	72.82	100.0	11	29	1.3 × 10−5	1.2 × 102	2.0 × 102
Cp25	14	0.516	40	1.0	0.047 (1.92)	1.0	12.84	38.4	39	10	9.1 × 10−5	3.6 × 103	1.0 × 103
	10	0.970	54	0.7	0.064 (2.64)	${1.74}_{0.88}^{3.43}$	17.56	34.6	49	13	9.8 × 10−5	2.5 × 103	1.2 × 103
Cp26	10	0.282	10	0.3	0.041 (1.71)	1.0	88.57	25.3	8	28	2.5 × 10−5	1.1 × 102	2.6 × 102
	10	0.660	17	0.2	0.064 (2.65)	${1.41}_{0.70}^{2.88}$	74.50	78.1	11	28	2.5 × 10−5	1.3 × 102	3.0 × 102
Cp27	10	0.102	10	0.3	0.041 (1.71)	4.0	85.70	48.5	1	68	2.4 × 10−5	3.0 × 101	6.7 × 102
	10	0.165	14	0.3	0.054 (2.25)	${2.86}_{1.11}^{7.38}$	87.71	65.2	3	52	2.3 × 10−5	5.8 × 101	6.4 × 102
Cp28	8	0.108	10	0.3	0.041 (1.71)	4.0	88.57	100.0	1	68	2.4 × 10−5	2.9 × 101	6.7 × 102
	10	0.663	14	0.3	0.054 (2.25)	${2.49}_{0.84}^{7.35}$	87.71	99.1	3	50	2.1 × 10−5	5.7 × 101	5.9 × 102
Cp29	8	0.119	10	0.3	0.041 (1.71)	4.0	79.92	100.0	1	68	2.4 × 10−5	4.0 × 101	6.7 × 102
	10	0.282	14	0.3	0.054 (2.25)	${2.86}_{1.11}^{7.38}$	87.13	100.0	3	52	2.3 × 10−5	6.0 × 101	6.4 × 102
Cp30	11	0.016	10	3.2	0.013 (0.54)	4.0	82.82	9.1	2	56	1.9 × 10−4	1.6 × 102	1.9 × 103
	10	0.044	11	0.3	0.044 (1.83)	${1.62}_{0.76}^{3.48}$	55.30	27.7	6	37	2.9 × 10−5	1.2 × 102	3.5 × 102
Cp31	20	0.921	320	0.1	0.416 (17.17)	8.0	88.57	72.7	307	11	7.7 × 10−5	7.5 × 103	8.8 × 103
	8	1.234	139	0.4	0.134 (5.52)	${4.76}_{2.68}^{8.45}$	37.62	74.7	129	12	1.4 × 10−4	1.2 × 104	6.8 × 103
Cp32	11	0.409	10	0.1	0.074 (3.04)	0.5	88.57	100.0	9	20	7.8 × 10−6	4.6 × 101	7.5 × 101
	10	0.643	18	0.1	0.094 (3.86)	${1.20}_{0.46}^{3.12}$	80.42	100.0	10	31	1.2 × 10−5	8.2 × 101	1.9 × 102
Cp33	9	0.745	10	0.1	0.074 (3.04)	0.5	88.57	100.0	9	20	7.8 × 10−6	4.6 × 101	7.5 × 101
	5	1.068	10	0.2	0.058 (2.41)	${1.52}_{0.75}^{3.07}$	85.68	92.5	4	43	1.5 × 10−5	3.8 × 101	2.1 × 102
Cp34	10	6.589	10	1.0	0.023 (0.96)	4.0	12.84	0.0	1	59	7.7 × 10−5	3.4 × 103	1.1 × 103
	5	9.966	23	0.4	0.056 (2.29)	${3.78}_{1.96}^{7.28}$	21.25	0.0	5	46	4.4 × 10−5	3.2 × 103	1.7 × 103
Cp35	8	1.991	10	0.3	0.041 (1.71)	1.0	88.57	60.6	8	28	2.5 × 10−5	1.1 × 102	2.6 × 102
	2	2.592	10	0.3	0.041 (1.71)	${0.71}_{0.50}^{1.00}$	58.76	80.3	8	23	2.1 × 10−5	1.2 × 102	2.4 × 102

Note. For each source, two lines are shown: the first line is the best-fitting model; the second line is the average of "good" models (see the text). For the current protostellar masses of the good models, we also indicate with super- and subscripts the range of masses set by ±1σ, where σ is the standard deviation in log₁₀ m_*.

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Figure 4(a) and (b) show the luminosity functions, based on isotropic luminosity values, of the identified protostellar sources in the IRDC. The luminosities range from ∼3 × 10⁴ L_⊙, i.e., for Cp23, down to ∼30 L_⊙, i.e., sources similar to Cp03. We fit a power law to the observed distribution of the form dN/dlog L ∝ ${L}^{-{\alpha }_{L}}$, with the result being α_L ≃ 0.35 ± 0.09 for the averages of "good" models. The distributions are quite well fit by a single power law, with little evidence for any break in the distribution, e.g., due to incompleteness at low luminosities. Figures 4(c) and (d) show the same information, but now for the distributions of bolometric luminosities of the ZT models that are fit to the SEDs. The distributions can still be fit with declining power laws, although there now appears to be more deviation from simple, single power-law distributions. This is a reflection of the fact that the bolometric model luminosity can be significantly different from the isotropic luminosity, to both higher and lower values, due to beaming, i.e., "flashlight," effects (see ZT).

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Our derived power-law indices for the protostellar luminosity functions are broadly consistent with those found by Eden et al. (2018), who found the equivalent of α_L = 0.26 ± 0.05 in W49A and α_L = 0.51 ± 0.03 in W51 (however, note there are significant differences in their methods of source selection compared to our 70 μm-based method).

Figure 5 shows the distributions of the protostellar population in terms of their initial core masses (top row), current envelope masses (middle row) and current protostellar masses (bottom row), as derived from the model fitting, with best-fit model results shown in the left column and average of "good" model results shown in the right. The minimum core mass in the model grid is 10 M_⊙, which truncates the distribution at this point. This may lead to a "pile up" in the distribution at the lower boundary, which appears to be present in the distribution of best-fit values of M_c, but is not apparent for the average masses. We fit power laws to the observed distributions of the form dN/dlog M_c ∝ ${M}_{c}^{-{\alpha }_{M}}$, with the result being α_M ≃ 0.44 ± 0.18 for the averages of "good" models. For comparison, the standard Salpeter distribution of the stellar initial mass function has α_M = 1.35, with such values also being found for observed core mass functions (CMFs) in some regions (e.g., Alves et al. 2007; Ohashi et al. 2016; Cheng et al. 2018; Massi et al. 2019). We see that our derived result for the initial CMF in G028.37+00.07 is significantly shallower than the Salpeter value. There have been claims of CMFs that are shallower, i.e., more top heavy, than Salpeter in some star-forming regions: e.g., W43 by Motte et al. (2018), who find α_M = 0.90 ± 0.06, and dense IRDC clumps by Liu et al. (2018), where a value of α_M = 0.86 ± 0.11 has been reported. Still, our result of α_M ≃ 0.44 ± 0.18 is even flatter than these cases. It should be noted that it applies over a higher mass range than has been probed by the Liu et al. (2018) study. Also, our results here are based on indirect inference of the initial protostellar core masses, while observational CMF studies, including that of Liu et al., are based on direct observations of cores.

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To more closely connect with observational studies of the CMF, in Figures 5(c) and (d), we also show the core envelope mass function of the identified protostars. Here the models now span to masses below 10 M_⊙. For the average model results, the derived power-law index of the mass function is shallower than the initial CMF, which may be related to a larger mass range that is now being probed and thus potentially great levels of incompleteness affecting the lower-mass regime. Still, fitting cores with envelope masses ≳5 M_⊙, we still find a relatively shallow index of α_M = 0.47 ± 0.16.

Finally, the last two panels of Figure 5 show the current protostellar mass functions. The average results do not appear well described by a single power law, perhaps because of incompleteness and/or larger uncertainties at low masses. If we exclude the lowest-mass bin, we find α_M = 0.66 ± 0.22, while excluding the two lowest-mass bins yields α_M = 1.16 ± 0.30, which is consistent with the Salpeter value of 1.35. However, it is already known that deriving protostellar masses from SED fitting of massive protostellar sources can suffer from high degeneracy (De Buizer et al. 2017; Liu et al. 2019), and we see from the cases of Cp23, Cp15, and Cp03 reported in Table 1 and Figure 3 and from the dispersions of the protostellar masses listed in Table 3 that this problem persists also for the lower-luminosity sources that we fit here. Thus, caution is needed when considering the value of α_M that is found from this analysis, since it may be subject to change once more accurate methods of estimating individual protostellar masses become available, e.g., by dynamical means from study of their accretion disk gas kinematics.

For the core masses, several points also need to be considered. Core masses that are derived from SED model fitting results show quite a wide dispersion in values among the 10 best model fits. This is expected since the models are mostly being constrained by the luminosity of the source, much of which comes from warmer material that does not dominate the core mass. Most of the core mass is at larger distances from the source and thus at cooler temperatures and so mostly affects the longer-wavelength part of the SEDs. As shown in Figure 2, the model SEDs can often underpredict the long-wavelength part of the SED. This difficulty was already noted by De Buizer et al. (2017) and ZT. The cause may be imperfect background subtraction at the longer wavelengths, especially when model core radii are relatively small compared to source apertures.

A more direct mass estimate from a given SED can be made by carrying out a single-temperature graybody fit to just the longer-wavelength component of the SED, i.e., from 160 to 500 μm, following the methods of, e.g., Lim et al. (2016). Comparisons of these mass estimates, i.e., M_submm, with those resulting from the ZT model values for M_c and M_env are shown in Figure 6. This figure shows the large effect of background subtraction on M_submm. Also visible is the pile-up of M_c values at the minimum value of 10 M_⊙, which is simply an artifact of a limitation of the ZT model grid. The best agreement is expected between background-subtracted values of M_submm and M_env and indeed this is apparent in the lower right panel of Figure 6. Still, this comparison shows there is significant scatter in the correlation and with a modest systematic offset of M_submm being lower than M_env by a factor of a few on average.

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A recent study of molecular outflows in the central region of IRDC C was performed by Kong et al. (2019). A comparison of our 70 μm Herschel-defined sources with the ALMA-detected outflow-driving sources is shown in Figure 7.

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We see that in some of the Herschel-identified sources, i.e., Cp13, Cp18, and Cp20, there are actually several different ALMA-identified protostars present. There are two ALMA-protostars in Cp19 and only one in Cp21. Of the six Herschel-identified protostars that are fully covered by the ALMA data, only Cp10 does not have an ALMA-identified protostar. At the same time, the Herschel-identification method also misses a significant amount of protostellar activity: about 2/3 of the ALMA sources are not associated with a Herschel-identified source. It should be noted that this comparison has been done for just a small number of Herschel-identified sources and in a relatively small part of the IRDC. Also it is a particularly MIR and FIR (including 70 μm) dark region, with the Herschel sources being of relatively low luminosity.

These results indicate that the association of a Herschel-identified protostar, i.e., based on a relatively low angular resolution imaging of dust continuum emission, can often be problematic, at least for sources at ∼5 kpc distances, like IRDC G028.37+00.07. This is mitigated somewhat when there are just a few sources in the aperture and one of them is clearly the dominant source, which may be the case in Cp19 based on the intensity of the outflows.

Ultimately, more accurate protostellar SED characterization will require higher angular resolution observations that cover the peak wavelength range of the SED. Still, with these caveats in mind we proceed to derive the overall star formation activity that is implied by the population of Herschel-identified sources.

The SFR, i.e., the SFE per freefall time (_ff), is important to quantify, e.g., as a constraint on theoretical models of star cluster formation.

The total mass of IRDC C is measured from extinction mapping to be 68,300 M_⊙ as given by BTK14, with 50% overall uncertainty (dominated by systematics). Using a different method based on sub-mm dust emission, Lim et al. (2016) estimated the mass to be 72,000 M_⊙. We will adopt a total mass of IRDC C in the defined ellipse region of 70,000 M_⊙.

Summing all of the masses resulting from the ZT models (Table 3), the total mass of the sources in the cloud is 1642 M_⊙ of total initial protostellar core mass for best-fitting models and 1740 M_⊙ of total initial protostellar core mass for the average of good models. About 50% of this mass is expected to eventually go into stars (Tanaka et al. 2017). However, the results of Section 3.3 indicate a completeness correction factor with respect to the ALMA-detected population of about 3 needs to be accounted for (or potentially a smaller factor if the missing sources tend to be lower-mass protostars). However, the ALMA-identified population is also likely to be incomplete at some level. Thus by this method we estimate that the protostellar population that is forming in the IRDC will produce a mass of stars of about 2000 M_⊙. This represents 2.9% of the total IRDC mass.

However, since the ZT model grid is designed for higher-mass protostellar cores (i.e., >10 M_⊙), it is possible that the above results are biased toward too high core masses on average. As an alternative method, we can consider the current protostellar masses implied by the models. If we sum the current protostellar masses then we obtain 132.5 M_⊙ for the best models and 86.8 M_⊙ for average of good models, i.e., ∼100 M_⊙. Then with a factor of 2 correction between current and final protostellar mass and a factor of 3 correction due to incompleteness, we would estimate a total mass of stars that will form of ∼600 M_⊙, i.e., 0.86% of the total IRDC mass.

In the context of the Turbulent Core Model of McKee & Tan (2003), the protostellar formation time is approximately 37% of the mean freefall time of the clump, ${\bar{t}}_{\mathrm{ff},\mathrm{cl}}$ (Equation (37) of McKee & Tan 2003), based on the mass of a 10 M_⊙ star forming from a 20 M_⊙ core (the formation time scales weakly as m^1/4). Thus, assuming the protostellar population we have sampled traces the activity of protostars forming in the last 40% of t_ff,cl, we estimate an SFE per freefall time in the IRDC of between 2.1% and 7.3%, depending on the above methods of mass estimation.

We consider that the lower estimate here is more reliable, since it is tied more closely to the protostellar luminosities and avoids the expected bias of too high initial core masses that will be found from using the ZT grid. If we have included some already formed stars that are present in the IRDC and that are simply heating surrounding local IRDC material, then we will have overestimated the SFR. On the other hand, the uncertain ALMA incompleteness factor would boost the estimate. Overall, we consider that the data support an estimate of _ff ∼ 0.03 in IRDC C.

If the IRDC is to form a bound cluster, which may be a reasonable expectation since it is one of the most extreme IRDCs and does appear to be gravitationally bound and in approximate virial equilibrium at the moment (BTK14; Hernandez & Tan 2015), then an overall SFE of ≳30% is likely to be needed. At the current SFR this would then take ∼10 freefall times to be achieved.

The absolute value of the freefall time is measured as 1.3 × 10⁶ yr, using the equation t_ff = [3π/32Gρ]^1/2 and the properties measured by BTK14. Then the total SFR implied by _ff = 0.03 is 1.6 × 10⁻³ M_⊙ yr⁻¹. Thus, age spreads of at least 1 Myr are expected, even in fastest formation models, but closer to 10 Myr if a bound cluster is to form with our above estimate of _ff = 0.03. However, the age spread could be reduced if the protocluster clump evolves to a denser state that has a shorter local freefall time, which would then lead to an increasing absolute SFR (see, e.g., Palla & Stahler 2000).

The initial spatial distribution of stars within forming clusters, including degree of substructure, central concentration, and primordial mass segregation, is of interest to help constrain theoretical models of both massive star formation (i.e., are special conditions needed for massive star formation) and for star cluster formation. However, it is in general difficult to infer these properties from observations of already formed stars, because signatures of the initial conditions are erased by dynamical evolution. As far as we are aware, there are no measurements yet of these properties based on protostellar populations in massive (>10⁴ M_⊙) protoclusters.

One widely used parameter to measure substructure is the ${Q}$ parameter (Cartwright & Whitworth 2004), which is the ratio between the mean length of the edges of the minimum spanning tree (MST) of the cluster, $\bar{m}$, and the mean separation between stars in the cluster, $\bar{s}$. This parameter has the ability to distinguish between a substructured regime and a radially concentrated regime. A value ${Q}\lt 0.785$ means the cluster is relatively substructured with a lower value corresponding to more clumpiness. In contrast, ${Q}\gt 0.785$ means the cluster has an overall radial structure/concentration, with a higher ${Q}$ value indicating that it is more concentrated in the center.

We measure a value of ${Q}=0.667$ for the 35 protostellar sources of IRDC C. This value classifies the cluster as "substructured," i.e., comparable to a three-dimensional distribution with a fractal dimension D ∼ 2, i.e., considerably substructured and not centrally concentrated (see Cartwright & Whitworth 2004). As stated in Section 2, the total number of sources identified by the Hi-GAL catalog was 40, but then was reduced to 35 due to crowding and lack of resolution for our aperture photometry analysis. However, if we consider the case of all 40 sources, the ${Q}$ parameter changes only modestly to 0.640, resulting in the same basic classification for the degree of substructure.

This value of Q ≃ 0.67 falls within the middle of the range of values reported by Cartwright & Whitworth (2004) for lower-mass clusters, i.e., Taurus (0.47), IC 2391 (0.66), Chameleon (0.67), ρ Ophiuchus (0.85) and IC 348 (0.98). This may indicate that whatever process controls the initial distribution of protostars does not vary significantly across the star-forming clump mass spectrum.

Our observed value of Q can be compared to that seen in numerical simulations of cluster formation. For example, Wu et al. (2017) studied cluster formation from colliding and non-colliding giant molecular clouds (GMCs). In the colliding case, the simulations showed values of Q that fluctuated in the range from ∼0.3 to ∼1.5, but often settling at values near 0.6 (Figure 9 of Wu et al. 2017). However, in the non-colliding models the values of Q were typically much smaller at ∼0.2. It should be noted that these simulations did not include feedback from the forming stars and were based on particular initial conditions of quite idealized GMCs. Nevertheless, in the context of these models, the colliding cases were able to form more concentrated clusters that are closer to the observed systems, including our result for the massive protocluster forming in IRDC G028.37+00.07.

Considering the degree of mass segregation of the protostellar population, the simplest approach is to examine where the most massive stars are with respect to a defined center of the cluster. In the case of IRDC G028.37+00.07, while there is a center of the elliptical region that has been used for defining the IRDC, it must be noted that this definition, originally based on low-resolution Midcourse Space Experiment images of the Galactic plane (Simon et al. 2006), is somewhat arbitrary. Still, we consider the three sources with the highest current protostellar mass estimates (based on averages of "good" models; see Table 3), which are: Cp09 with m_* = 10.5 M_⊙, Cp12 with m_* = 9.8 M_⊙, and Cp23 with m_* = 8.0 M_⊙. The source Cp09 is located near the ellipctical boundary of the IRDC, Cp12 is at an intermediate distance from the center, while Cp23 is relatively close to the center of the cloud (at least in projection). These results do not support there being any strong preference for massive stars to form in the center of the protocluster, at least as defined by the Simon et al. (2006) ellipse.

Given the difficulty of defining a cluster center, an alternative approach to study mass segregation is to think of it as the tendency of massive stars to stay near other massive stars. This definition does not need a defined center and so is better suited to deal with a substructured protocluster. A popular method to measure mass segregation in this way is the Λ_MSR parameter, which also uses the MST (Allison et al. 2009). This parameter compares the total length of the MST of the N most massive stars in the cluster to the length of the MST of N stars in the cluster chosen randomly. To reduce variation caused by the random selection of stars, this parameter is measured multiple times, which also allows an estimate of its uncertainty (see Allison et al. 2009 for details). Then, a mass-segregated cluster has values of Λ_MSR > 1, a cluster with no mass segregation has Λ_MSR = 1, and a cluster with inverse mass segregation has Λ_MSR < 1, i.e., having the N most massive stars more separated in comparison with the average star. However, while this method does not require a defined cluster center, it does require defining a population of sources, which in our case has been done with the condition that they are inside the already defined IRDC boundary. We will return to this point below.

For calculation of Λ_MSR, we focus on the estimates of current protostellar masses based on average of "good" models (see above). Figure 8 shows Λ_MSR as a function of the number, N, of the most massive sources used to define the high-mass sample. The figure also shows the maximum and minimum possible values of Λ_MSR based on the locations of the protostars, but with the freedom to reassign the masses to achieve these extreme values. We see that Λ_MSR has values close to 1 for N ≥ 4, with a modest enhancement of Λ_MSR ≃ 1.4 when N ≤ 3. This is tentative evidence for a signature of mass segregation at these numerical scales. However, given the size of the uncertainties, this cannot be regarded as strong evidence for a signature of primordial mass segregation (i.e., enhanced clustering) of the massive protostars in the IRDC. Still, these results provide basic constraints with which to test theoretical and numerical models of star cluster formation.

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We have also investigated the sensitivity of these results to the choice of IRDC boundary location. In particular we examine whether the results change if we exclude the seven sources in the SE region of the IRDC, which are quite well separated from the main IRDC features in a relatively IR-bright region and may be part of another grouping of protostars seen just outside the IRDC boundary. We find that the dependence of Λ_MSR versus N shows very similar behavior when we repeat the analysis on the remaining sources. Thus the grouping of seven sources near the SE boundary was not significantly affecting these clustering results. However, of course the results for Λ_MSR versus N could change if the actual physical "protocluster" had a larger extent which included a population of massive protostars that dominated over those in the local IRDC region that we have focused on. However, in this case we would still conclude that any clustering of such protostars is not especially concentrated toward the IRDC and that the protostars in the IRDC region are themselves not especially clustered or centrally concentrated.

A graphical illustration of the minimum and maximum levels of mass segregation as measured by Λ_MSR is shown in Figure 9. This figure shows the protostars at their actual locations, but with the masses free to be swapped around to obtain the minimum and maximum levels of mass segregation (the area of the symbol is proportional to the mass of the protostar). The most mass-segregated case places the most massive stars together in the region of highest source areal density in the NW region of the IRDC. The most inverse mass-segregated case places the most massive stars in a ring near the outer boundary of the IRDC. These distributions are independent of N.

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Our result of an apparent lack of or limited level of enhanced clustering for more massive sources is similar to results presented by Román-Zúñiga et al. (2019) for density peaks identified in extinction maps of the Pipe nebula, but different from their results in the Orion region, where they do find evidence for Λ_MSR rising systematically as peak mass increases. There are important differences between our work and that of Román-Zúñiga et al. (2019), including: we have considered protostellar masses, rather than core (or peak) gas masses; our sources were identified by their 70 μm emission with the CuTEx algorithm (see Section 2), while Román-Zúñiga et al. used the clumpfind algorithm (Williams et al. 1994) to find sources in their column density map; our target cloud, IRDC G028.37+00.07, is much more massive and of higher velocity dispersion than the clouds studied by Román-Zúñiga et al. In general, it will be important to extend these types of studies, using uniform methods, to larger samples of clouds that probe wider ranges of physical conditions and Galactic environments.