THE SPATIAL STRUCTURE OF YOUNG STELLAR CLUSTERS. II. TOTAL YOUNG STELLAR POPULATIONS

The statistical relationships that traditionally underlay our understandings of star-formation processes have been scale-free relationships like the Salpeter (1955) stellar initial mass function (IMF) and the Kennicutt–Schmidt law relating global galactic star-formation rate to interstellar material (Schmidt 1959; Kennicutt & Evans 2012). Nevertheless, important preferred scales for star formation were later found. The power-law IMF for high stellar masses peaks around 0.2–0.3 ${{M}_{\odot }}$ and declines for lower mass stars (Chabrier 2003). And, in the Galactic neighborhood, a threshold for star formation was found at ${{A}_{V}}\approx 7$ magnitudes dust absorption ($\approx n[{{H}_{2}}]\sim 3\times {{10}^{4}}$ cm⁻³) associated with Galactic disk stability (Johnson et al. 2004; Lada 2010; Martin & Kennicutt 2001; Schaye 2004; Leroy et al. 2008), although this rule is not applicable within the inner 0.5 kpc of the Galaxy where star formation is comparatively suppressed (Longmore et al. 2013).

The surface density of stellar populations in star-forming regions, in units of stars per square parsec, has been a property of interest for the field of star-cluster formation (e.g., Carpenter 2000; Lada & Lada 2003; Allen et al. 2007; Jørgensen et al. 2008; Gutermuth et al. 2009). The surface densities of young stars can also have astrophysical implications, such as tidal truncation of protoplanetary disks (Pfalzner et al. 2005), binary star distributions (Bate 2009a; Moeckel & Clarke 2011), or the survival of clusters after molecular cloud dispersal (Kruijssen 2012). (Bressert et al. 2010, henceforth B10) have recently examined the shape of the distribution of stellar surface densities in star-forming regions. They use samples of disk/envelope-bearing stars identified through IR excess in regions within 0.5 kpc of the Sun, including the Gould Belt (Allen et al. 2006), the Orion A and B molecular clouds (Megeath et al. 2012), the Taurus molecular cloud (Rebull et al. 2010), and the regions from the cores-to-disks (c2d; Evans et al. 2003) project—thus their sample is dominated by low-mass young stellar object (YSO) environments. B10 argue that if "clustered" star formation and "distributed" star formation were two distinct star-formation processes, then "clustered" and "distributed" populations should appear as distinct modes in the surface-density distribution (cf. Gieles et al. 2012; Pfalzner et al. 2012), and that such a scenario could be tested by searching for a "scale" of star formation separating these two modes. Such a kink in the surface-density distribution has been used in some investigations of the structure of young stellar clusters, for example by Gutermuth et al. (2009), to separate distinct clusters of stars in star-forming regions. Given that B10 find a smooth distributions of surface density from their data, which they report is adequately fit by a log-normal distribution with a peak at 22 stars pc⁻², they conclude that no such "scale" exists, at least for low-mass YSO environments present in the solar neighborhood.

Nevertheless, the log-normal distribution from B10 is not scale free, but instead a peak at 22 stars pc⁻² and width of 0.85 dex suggests a density distribution weighted toward low-density environments. If this result is a general characteristic of most star formation in the Galaxy, rather than just the nearby regions investigated by B10, it would have implications for theories of star formation as numerous researchers have discussed. For example, Parmentier & Pfalzner (2013) find that their models of local-density-driven star formation from a single molecular clump could produce the 22 stars pc⁻² scale from B10. King et al. (2012) suggest that surface densities significantly greater than 22 stars pc⁻² could indicate that a cluster has undergone a "cool collapse phase." Kruijssen et al. (2012) present a model in which the low fraction of star formation that results in bound clusters is, in part, a result of a density spectrum weighted toward low surface densities. Parker et al. (2011) notes that dynamical processing of primordial binaries by clusters depends on whether most stars form in low density regions as suggested by B10, or whether higher density clusters are more common. And, de Juan Ovelar et al. (2012) investigate the threshold densities in star-forming regions where stellar interactions affect habitable planet formation—and the fraction of the stars born in environments above their 3 × 10³ stars pc⁻² threshold would depend on whether high-density regions are just a tail of the B10 log-normal or a different mode not seen in the B10 sample. B10 support this interpretation of the empirical results from their sample stating, "only a small fraction (<26%) of stars form in dense clusters where their formation and/or evolution is expected to be influenced by their surroundings."

It is important to investigate whether these results continue to hold for more MSFRs (those regions containing O-type stars). Nearly 70% of B10's sample comes from the Orion giant molecular clouds, which do contain O-type stars, but they note that the IR-excess methods used are ineffective at identifying young stars in the presence of nebulosity and crowding in this complex. Their other star-forming regions are lower mass. Given that studies of the mass function for star-forming regions favor the birth of stars in more massive complexes, investigation of these complexes in a way that is more effective at probing the densest regions could be helpful for determining the validity of B10's suggestion that clustered stars (${\Sigma }\gt 200$ stars pc⁻²) exist in the tail of the surface-density distribution, rather than being a dominant component. Nevertheless, a definitive study of surface-density distributions for star formation would require the construction of an unbiased survey of all star-forming environments. MYStIX neither includes the most massive star-forming environments in the Galaxy (such as W 43, Wd 1, NGC 3603, or the Arches Cluster) nor includes large angular area studies of diffuse molecular clouds necessary to capture the lowest surface density environments.

The MYStIX survey (Feigelson et al. 2013) differs from many previous studies in that it focuses on relatively MSFRs lying in nearby Galactic spiral arms, and supplements samples of IR-excess young stars with X-ray selected young stars and spectroscopically identified OB stars.

MYStIX is a survey of 20 of the nearest ($d\lt 3.6$ kpc) MSFRs that have been observed with NASA's Chandra X-ray Observatory, the Spitzer Space Telescope, and the United Kingdom Infra-Red Telescope (UKIRT) or the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006). The MYStIX regions include the Orion Nebula, the Flame Nebula, W40, RCW 36, NGC 2264, the Rosette Nebula, the Lagoon Nebula, NGC 2362, DR 21, RCW 38, NGC 6334, NGC 6357, the Eagle Nebula, M17, W 3, W 4, the Carina Nebula, the Trifid Nebula, NGC 3576, and NGC 1893, from which a sample of 31,784 MYStIX Probable Complex Members (MPCMs; Broos et al. 2013) is obtained. The MPCM catalog thus consists of young stars that are X-ray selected, IR excess selected, or OB stars from the literature. MYStIX provides the cleanest and largest lists of young stars for most of the 20 regions included in the study, so these catalogs should be scientifically useful for different purposes. One of the requirements of the MYStIX project was to use sensitive and homogeneous data analysis procedures for all 20 regions to facilitate inter-comparisons between regions. Special procedures had to be developed to deal with challenges working in the Galactic Plane, as described in the MYStIX technical-catalog papers: Kuhn et al. (2013a), Townsley et al. (2014), King et al. (2013), Kuhn et al. (2013b), Naylor et al. (2013), Povich et al. (2013), and Broos et al. (2013).

The spatial distributions of MPCMs in 17 of the MYStIX MSFRs are investigated in Kuhn et al. (2014, henceforth Paper I) and in this work. Three regions are omitted, NGC 3576, W 3, and W 4, because they lack JHK UKIRT photometry and have a low match rate between X-ray sources and sources from the 2MASS catalog. We use a subset of the MPCM sources (∼17,000 stars) produced after X-ray selected MPCMs are pruned to a uniform X-ray sensitivity within each region (Paper I). This eliminates artificial surface density gradients associated with differing X-ray exposure times in Chandra mosaics and the sensitivity variations within each Chandra field due to telescope coma and vignetting, the "egg-crate effect" (Townsley et al. 2011). To prune a region, we remove sources with X-ray photon fluxes (log_PhotonFlux_t; Broos et al. 2013) that are lower than the completeness limits provided in Table 1 of Paper I.

Table 1.  XLF Properties of MSFRs

Region	${{\alpha }^{*}}$	Ntot	${{N}_{ME\lt 1.5}}$	${{N}_{1.5\lt ME\lt 2.5}}$	${{N}_{ME\gt 2.5}}$
		(stars)	(stars)	(stars)	(stars)
(1)	(2)	(3)	(4)	(5)	(6)
Orion	−0.9	2600	700	1200	720
Flame	−1.0	800	55	250	500
W40	−1.1	520	0	250	270
RCW 36	−0.8	550	0	230	320
NGC 2264	−1.0	1900	1100	230	570
Rosette	−1.3	2500	840	1000	610
Lagoon	−1.1	3800	2000	870	930
NGC 2362	−1.1	600	570	30	0
DR 21	−1.0	2900	50	550	2253
RCW 38	−1.3	9900	70	1900	8000
NGC 6334	−1.2	9400	180	2900	6200
NGC 6357	−1.2	12000	440	7400	3700
Eagle	−1.3	8100	1300	4300	2500
M17	−1.3	16000	170	5400	11000
Carina	−1.5	34000	9500	17000	7000
Trifid	−1.0	3100	1500	1100	470
NGC 1893	−1.5	4600	1700	2400	590

Note. Column 1: name of MYStIX MSFR. Column 2: power-law index ($d{\rm log} N/d{\rm log} {{L}_{{\rm t},{\rm c}}}$) for the portion of the XLF tail shown in Figure 2 (uncertainties are ±0.1). Column 3: inferred intrinsic total population. Columns 4–6: inferred intrinsic numbers of stars with $ME\lt 1.5$ keV, $1.5\lt ME\lt 2.5$ keV, and $ME\gt 2.5$ keV, respectively. (Due to rounding, the sum of Columns 4, 5, and 6 is not always equal to Column 3.)

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Nevertheless the resulting observed surface densities, used by the analysis in Paper I, do not contain the entire intrinsic population, differ in sensitivity from MSFR to MSFR, and are affected by spatially variable N_H absorption and MIR sensitivity. As pre-main-sequence (PMS) X-ray luminosities strongly scale with stellar mass (Telleschi et al. 2007), inconsistent X-ray sensitivities due to differing Chandra exposure times and MSFR distances lead to different samplings of the cluster IMFs. We overcome these selection effects calibrating the observed X-ray luminosity function (XLF) and IMF distributions to the Orion Nebula Cluster (ONC) that serves as a template for young cluster populations, rather than attempting to model instrumental and observational effects.

The organization of this paper (Paper II) is as follows. We analyze the IMF and XLF to infer intrinsic populations from observed young stellar populations (Section 2). We derive intrinsic stellar surface density maps from these populations (Section 3), and investigate surface density distributions (Sections 4 and 5). The MYStIX sample of star-forming regions are typically richer than those in the sample studied by B10. The MYStIX MSFRs exhibit a large diversity in their surface density distributions (ranging from ∼10 to ∼30,000 stars pc⁻²), neither showing a tendency to follow a universal surface density distribution nor showing a convincing peak at some characteristic surface density. These results are discussed in Sections 4 and 5 and summarized in the conclusion (Section 6).

Young stars in the Chandra Orion Ultradeep Project (COUP; Getman et al. 2005b) had large numbers of X-ray counts, allowing X-ray luminosities (${{L}_{{\rm t},{\rm c}}}$; total 0.5–8.0 keV band and absorption corrected) to be obtained by parametric modeling of the X-ray spectrum using the XSPEC code (Arnaud 1996). As the MYStIX stars are mostly too faint for this procedure, X-ray luminosities for other MPCMs were computed using non-parametric calibrations for PMS stars (XPHOT; Getman et al. 2010) by Kuhn et al. (2013a), Townsley et al. (2014), Broos et al. (2011), and Kuhn et al. (2010).

The probability distribution of ${{L}_{{\rm t},{\rm c}}}$, called the XLF, is associated with the IMF due to the statistical link between X-ray luminosity and stellar mass M (e.g., Feigelson et al. 1993; Preibisch et al. 2005a; Telleschi et al. 2007). The assumption of a "universal XLF" (Feigelson & Getman 2005) has been used to estimate total populations in young stellar clusters, including Cep B (Getman et al. 2006), M17 (Broos et al. 2007), NGC 6357 (Wang et al. 2007), Rosette (Wang et al. 2008, 2009, 2010), W40 (Kuhn et al. 2010), Trumpler 15 (Wang et al. 2011), Trumpler 16 (Wolk et al. 2011), Sh 2–254/255/256/257/258 (Mucciarelli et al. 2011), NGC 1893 (Caramazza et al. 2012), and IC 1396 (Getman et al. 2012). During PMS stellar evolution, there is a weak relation between X-ray luminosity and age (e.g., Preibisch & Feigelson 2005; Pandey et al. 2014); however, ${{L}_{{\rm t},{\rm c}}}$ does not rapidly decline during the first 5 Myr, unlike the rapid decrease in bolometric luminosity L_bol during PMS evolution along the Hayashi track. Instead, ${{L}_{{\rm t},{\rm c}}}\sim M$ appears to be the fundamental relationship rather than ${{L}_{{\rm t},{\rm c}}}\sim {{L}_{{\rm bol}}}$ (Getman et al. 2014b). Thus, X-ray luminosity evolution appears to have little effect on the shape of the PMS XLF (e.g., Bhatt et al. 2013).

Following these previous studies, we use a sample of stars from COUP to approximate the probability distribution of the universal XLF. The COUP study contains a sample of 839 lightly absorbed stars (Feigelson et al. 2005), which are identified as the members of the ONC, while the more highly absorbed stars are identified as being embedded in the Orion Molecular Cloud (OMC) behind the ONC. These lightly obscured COUP stars are complete down to a mass of 0.1–0.2 ${{M}_{\odot }}$ (with partial coverage into the proto–brown-dwarf regime; Preibisch et al. 2005b) and show an XLF shape characterized by a falling distribution at high luminosities with a break to an approximately flat distribution at luminosities below ${{L}_{{\rm t},{\rm c}}}\approx {{10}^{30.4}}$ erg s⁻¹. Henceforth, we label this distribution the COUP XLF. The tail with X-ray luminosities greater than this turnover can be fit with a power-law (Pareto) distribution of slope α, with a minimum variance unbiased estimator ${{\alpha }^{*}}$ and variance ${\rm Var}({{\alpha }^{*}})$ given by the equations

$$\begin{eqnarray}\begin{array}{rcl} {{\alpha }^{*}} & = & -\left( 1-\frac{2}{n} \right)\frac{n}{\sum _{i=1}^{n}\left( {\rm ln} {{x}_{i}}-{\rm ln} {{x}_{m}} \right)}\quad {\rm and} \\ {\rm Var}({{\alpha }^{*}}) & = & \frac{{{\alpha }^{*2}}}{n-3}, \\ \end{array}\end{eqnarray} \tag{ 1 }$$

where x_m is the X-ray luminosity of the turnover point and x_i is the X-ray luminosity of the ith source in a sample of n sources in the distribution tail (Johnson et al. 1994). Thus, ${{\alpha }^{*}}=-0.9\pm 0.1$ with n = 61, while the distribution is roughly flat in logarithmic bins below the turnover point. Mucciarelli et al. (2011) have also found similar ${{L}_{{\rm t},{\rm c}}}$ Pareto distributions for the ONC and the Sh 2-254–258 regions.

The ${{L}_{{\rm t},{\rm c}}}$ distributions for the MYStIX regions can be compared to the empirical distribution function (EDF) of 839 lightly absorbed COUP sources to further investigate the universality of the XLF shape. For example, Wang et al. (2008, their Figure 9(b)) shows that the XLF cumulative distributions in the Rosette star-forming region agree with the COUP XLF above the X-ray luminosity completeness limit for the Rosette sample. For this analysis it is necessary to be cautious about how completeness limits are treated because differential absorption can change the apparent shape of the XLF if the sample of highly absorbed sources is incomplete. For example, a sample of sources that are more deeply embedded will have a higher mean luminosity than a sample of sources from the same observation that are unabsorbed, which could lead to a flattening of the power-law of the combined distribution.

In Figure 1, we show the COUP EDF running from unity at very low luminosities (${{L}_{{\rm t},{\rm c}}}\sim {{10}^{27}}$ erg s⁻¹) to zero for the most luminous PMS star in the Orion Nebula field. Note that spectroscopically identified OB stars have been removed from both the COUP XLF and from MYStIX samples considered here because MYStIX regions can differ widely in their massive stellar subpopulations. The ${{L}_{{\rm t},{\rm c}}}$ EDFs for the other regions are shown below, with arbitrary vertical spacings. Only the portion of the XLF where the sample is complete is shown. (The completeness limit for the full sample is set to the completeness limit for the most heavily obscured subpopulation in the region.) In general there is excellent agreement between the shapes of the different lines. Some curvature can be seen in the COUP XLF between ${{10}^{30.5}}$ erg s⁻¹ and ${{10}^{32.5}}$ erg s⁻¹, which is also reflected in the shapes of the XLFs for other regions. The nearest regions tend to probe a lower-luminosity section of the XLF, while the more distant regions tend to probe a higher-luminosity section of the XLF. Due to the XLF curvature, the XLF shape appears less steep for nearer regions and steeper for more distant regions. Table 1 (Column 2) gives the power-law indices⁵ (for the full sample) calculated over the regions shown in Figure 1. This confirms the trend in which more distant regions have steeper slopes—not because of differences in intrinsic XLF shape, but due to differences in the available portion of the XLF.

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2.1.1. Intrinsic Numbers of Stars

If we accept the assumption of a "universal XLF" with the COUP sample serving as a template, the total stellar population of a star-forming region may be extrapolated from the missing stars at low luminosities where the region's observed XLF has dropped to zero due to incompleteness, but where the existence of these stars may be inferred from the universal XLF shape. To perform this calculation, the COUP XLF histogram is scaled so that it matches the observed XLF in the section of the XLF where the observed sample is complete. The completeness limit for the full sample of young stars will be the completeness limit for the most absorbed subpopulation of stars in the region. However, few of the most deeply embedded protostars may be observed, so this population will be poorly characterized, and the inferred total populations may be considered lower limits that do not necessarily account for all undetected, embedded stars.

If we further assume that X-ray absorptions from the molecular cloud are independent of the intrinsic stellar X-ray luminosities, we can estimate the intrinsic numbers of stars with different amounts of absorption in the star-forming region—this is useful because it allows us to compare total populations in more extinguished parts of a star forming region to total populations in less extinguished areas. Using this assumption, any subset of stars selected by X-ray absorption will be drawn independently from the "universal XLF" and therefore have the same XLF shapes—allowing us to perform the XLF population analysis described above on the subset. This assumption may not be entirely true; for example, mass segregation may cause more X-ray luminous stars to lie preferentially in the center of a cluster where absorptions are often higher. There is also weak evidence for a factor of ∼2 systematic increase in X-ray luminosity from the younger Class I to the older Classes II/III systems (Prisinzano et al. 2008). Nevertheless, the absorption stratified MYStIX XLFs all show consistency with the COUP XLF, indicating that these are not major effects.

Interstellar medium absorption of MPCMs may be evaluated using $J-H$ color indices or X-ray Median Energy (ME; Getman et al. 2010) indicators, both of which increase as absorbing columns increase (Getman et al. 2014b). The spread in absorptions for subclusters of stars is provided in Paper I (their Figure 8), which shows the median $J-H$ and ME for clusters of young stars in the MYStIX regions. There is a clump of data points with $ME\approx 1.5$ keV (the unembedded population), while the absorbed population ranges over $1.5\lt ME\lt 5.0$ keV. The bulk of the absorbed groups of stars are moderately absorbed, with $1.5\lt ME\lt 2.5$ keV, and tend to have physical properties similar to the unabsorbed clusters. In contrast, the most absorbed groups of stars are much more compact (${{r}_{{\rm core}}}\approx 0.08$ pc) and tend to be centered on dense molecular filaments or clumps (Getman et al. 2014b).

We choose to stratify the MYStIX MPCM stars into three absorption strata using ME divisions at 1.5 and 2.5 keV. This captures different aspects of the spatial structure of the stars: roughly, the population of stars outside the molecular cloud, the population within the molecular cloud, and the population associated with dense molecular filaments and cores. Figure 2 shows the observed XLF histograms for these three subpopulations in NGC 6357 (black lines): the $ME\lt 1.5$ keV sources (left), the $1.5\lt ME\lt 2.5$ keV sources (center), and the $ME\gt 2.5$ kev sources (right). The completeness limits in this region vary from ${{10}^{30.3}}$, to ${{10}^{30.6}}$, to ${{10}^{31.0}}$, for these three strata, respectively. The template COUP XLF (gray lines) is scaled to these three populations, using a vertical shift that minimizes the area between the two EDFs in the range where the XLF is approximately complete. The XLFs of all three strata show agreement between the shape of the sample XLF and the COUP XLF. The intrinsic number of stars in each absorption stratum can be estimated by integrating the number of stars under the scaled universal XLF curve, i.e., the gray COUP line. Thus, the inferred intrinsic population in NGC 6357 is ∼440 lightly obscured stars, ∼7400 moderately obscured stars, and ∼3700 highly obscured stars.⁶ Similar plots of stratified XLFs are provided for the other 16 regions as a figure set in the electronic version of the article.

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As expected, for every region, the more highly absorbed strata have a higher ${{L}_{{\rm t},{\rm c}}}$ completeness limit than the less absorbed strata. Generally, there is good agreement with shape of the COUP XLF; however, in most cases the sample becomes incomplete before reaching the turnover point at ${{L}_{{\rm t},{\rm c}}}\approx {{10}^{30.4}}$ erg s⁻¹ in the XLF. The Flame Nebula is one of the few cases where the completeness limit is less luminous than the XLF turnover point in the lightly and moderately absorbed strata, so the "flat" portions of the XLFs can be compared—the figure indicates that the XLFs are consistent. There is also indication of this turnover in the lightly absorbed stratum for NGC 2264.

The extrapolated intrinsic stellar populations for each stratum in each region are provided in Table 1. These values are combined to produce estimates of the total intrinsic populations for each star-forming region (Column 3). These values of N_tot are the principal empirical results of this study. Effects of distance and observational sensitivity should be approximately corrected by this analysis, but inferred total populations for more distant, less complete regions will be less precise and may be affected by additional systematics. To investigate these effects, we simulate the X-ray sensitivity for the Orion Nebula region if it were at the distance of the Carina Nebula (∼2.3 kpc) with the pruned X-ray sensitivity (${\rm log} \_PhotonFlux\_t\gt -5.9$) for the Chandra Carina Complex Project (CCCP; Townsley et al. 2011). Getman et al. (2005a, their Table 3) report 21 close doubles with <2'' separations in COUP, which would be indistinguishable at the distance of Carina. For this test, we combine their X-ray luminosities into a single source for the purpose of X-ray sensitivity limits and XLF analysis. From the 120 X-ray sources in this reduced-sensitivity Orion sample (out of 1216 original X-ray sources), we infer a total of ∼2000 stars rather than ∼2600 stars. Thus our N_tot values may underestimate the true populations by up to ∼30%, with most of the missing stars from the highly obscured ME stratum. Effects of distance and observational sensitivity are mentioned in Sections 3 and 4.

2.1.2. Detection Fraction for X-ray Selected MPCMs

The MPCM X-ray detection fraction, $f={{N}_{{\rm obs}}}/{{N}_{{\rm tot}}}$, can be used as a correction factor to convert observed surface densities of young stars into intrinsic surface densities. However, dividing the surface density of a region by a single detection fraction does not account for spatial variation in sensitivity. The statistical sample of X-ray selected young stars provided in Paper I, is a pruned subset of the MPCMs to which a uniform photon-flux limit is applied to correct for observational sensitivity effects, including Chandra telescope vignetting, variation in point-spread function, and differing net exposure times in a mosaicked field. However, their samples do not control for differing luminosity completeness limits due to variable extinction.

An improved estimate of stellar surface densities can be made using the absorption-stratified samples from Section 2.1.1. Each ME stratum has a narrower range of absorptions than the full sample, so the spatial variation in sensitivity within these samples will be lower than for the full sample. The sources in each stratum are adaptively smoothed (described below) to produce surface density maps, and the detection fraction for each stratum given by the XLF analysis.

The spatially dependent detection fractions for the full Paper I X-ray selected samples are computed using the equation,

$$\begin{eqnarray}&&{{f}_{{\rm full}}}(\alpha ,\delta )=\frac{{{f}_{1}}{{{\Sigma }}_{1}}(\alpha ,\delta )+{{f}_{2}}{{{\Sigma }}_{2}}(\alpha ,\delta )+{{f}_{3}}{{{\Sigma }}_{3}}(\alpha ,\delta )}{{{{\Sigma }}_{1}}(\alpha ,\delta )+{{{\Sigma }}_{2}}(\alpha ,\delta )+{{{\Sigma }}_{3}}(\alpha ,\delta )},\end{eqnarray} \tag{ 2 }$$

where f_i is the detection fraction of the ith stratum and ${{{\Sigma }}_{i}}(\alpha ,\delta )$ is the adaptively smoothed observed surface density of that stratum. Detection fraction tends to be roughly constant across the fields of view, with dips found where molecular clouds and cores are located. But, detection fraction varies strongly between regions, so it is essential that scientific comparison between regions be based on the intrinsic young stellar surface densities rather than observed MPCM surface densities.⁷

Figure 3 shows detection fraction maps for the Carina, Eagle, and NGC 6334 fields of view. The larger distance and shorter Chandra exposure times for the Carina Nebula compared to the Eagle Nebula result in a lower overall detection fraction for Carina compared to Eagle. In the Eagle Nebula, the bubble around the main NGC 6611 cluster results in a relatively high detection fraction, while the embedded subclusters to the northeast have lower detection fractions. A molecular filament passes through NGC 6334 from northeast to southwest, producing a notable trough in the detection fraction map of this region. Overall, detection fractions for MYStIX MSFRs range between 1–60%.

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Table 2 lists the intrinsic stellar populations inferred from the X-ray MPCM populations for the 142 subclusters in Paper I, counting stars out to four times the subcluster core–radius.⁸ The detection fractions for these subclusters were obtained by interpolating the detection fraction at the location of each star assigned to a subcluster and taking the mean value, weighted to correct for incompleteness of the sample. The intrinsic population of a subcluster is the number of observed stars in that subcluster from Paper I, multiplied by the fraction of those stars that are X-ray selected, and divided by the detection fraction of the X-ray selected sample.

Table 2.  Intrinsic Population Estimates from the XLF and IMF

	Properties from Paper I
Subcluster	α	δ	${{r}_{4,{\rm major}}}$	${{r}_{4,{\rm minor}}}$	PA	${\rm log} {{N}_{{\rm XLF}}}$	${\rm log} {{N}_{{\rm IMF}}}$
	(J2000)	(J2000)	(arcminutes)	(arcminutes)	(deg)	(stars)	(stars)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
Orion A	83.8110030	−5.3752777	0.42	0.32	85	1.66 ± 0.13	...
Orion B	83.8154178	−5.3897248	1.93	1.35	28	2.17 ± 0.10	...
Orion C	83.8195378	−5.3761802	10.17	5.19	5	3.21 ± 0.02	3.37 ± 0.09
Orion D	83.8242661	−5.2763330	7.60	1.20	12	1.94 ± 0.12	...
Flame A	85.4270870	−1.9037960	5.14	3.24	146	2.74 ± 0.03	2.91 ± 0.43
W40 A	277.8614542	−2.0940426	4.54	4.37	107	2.48 ± 0.04	2.66 ± 0.09
RCW 36 A	134.8623491	−43.7555688	3.47	2.32	122	2.73 ± 0.04	2.27 ± 0.06
RCW 36 B	134.8634966	−43.7571938	1.01	0.15	23	1.66 ± 0.11	1.94 ± 0.31
NGC 2264 A	100.1312445	9.8311532	1.35	1.16	136	1.21 ± 0.15	0.94 ± 0.12
NGC 2264 B	100.1545919	9.7918914	0.56	0.31	124	1.12 ± 0.21	...

Note. Column 1: subcluster name from Paper I. Columns 2–3: celestial coordinates (J2000) for the subcluster center. Columns 4–5: semi-major and semi-minor axes for an ellipse four times the size of the subcluster core defined in Paper I. Column 6: position angle of the subcluster ellipse in degrees east from north. Column 7: intrinsic number of stars projected within four subcluster core radii estimated from the XLF analysis. Column 8: intrinsic number of stars projected within four subcluster core radii estimated from the IMF analysis. A value of "..." indicates that a subcluster has too few stars with good JH photometry to estimate a mass-completeness limit and/or the IMF scaling. Estimates of uncertainty include the $\sqrt{N}$ Poisson uncertainty, but do not include the multiple sources of systematic error in the XLF and IMF analysis.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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The J versus $J-H$ diagram has been used to obtain mass estimates of young stars, and thereby IMFs for young clusters, in a number of multiwavelength studies (e.g., Getman et al. 2008; Kuhn et al. 2010; Wright et al. 2010). For low mass stars, individual inferred masses typically have 30% systematic errors (Kuhn et al. 2010, their Appendix A); however, this uncertainty is relatively small compared to the range of stellar masses in these samples and unimportant when the scientific questions concern their collective distributions. The method may be biased for the youngest stars: the spectral energy distribution modeling by Povich et al. (2013) indicates that NIR absorption from a heavy disk or envelope may substantially increase the J-band magnitude, causing a star to appear to have a lower mass than it truly does. This method does not account for the mass of circumstellar material, which is substantial for protostars. And, furthermore, this method is insensitive to very young protostars undetected in NIR bands.

For each MYStIX subcluster, we adopt the median age from Getman et al. (2014b)—their Age_JX estimates where these exist, otherwise their Age_JH estimates—and use the Siess et al. (1997) numerical pre-main-sequence evolutionary models to estimate absorption and mass for each star. A completeness mass-limit is estimated empirically from the inferred stellar masses for each subcluster. The Maschberger (2013) IMF is then scaled to the complete end of the observed mass functions, and the number of missing stars down to 0.1–0.2 ${{M}_{\odot }}$ is extrapolated.

Table 2 lists the intrinsic numbers of stars calculated from IMF analysis for each of the 142 Paper I subclusters. While there are numerous potential sources of error in this analysis, at a minimum there is a counting-statistics uncertainty of ${{N}_{{\rm tot}}}/\sqrt{{{N}_{M\gt M{\rm lim} }}}$, where N_tot is the inferred intrinsic population and ${{N}_{M\gt M{\rm lim} }}$ is the number of stars used to scale the IMF.

Figure 4 plots the intrinsic population from IMF analysis versus the intrinsic population from XLF analysis on a log-log plot—the points fall along the y = x line with a root-mean-square of ∼0.25 dex, although some points may deviate by up to a factor of ∼3. IMF-inferred populations for sparser subclusters are slightly systematically higher than the XLF-inferred populations, but, overall, there is little systematic shift or tilt in the relation. Subclusters that are more highly absorbed tend to have fewer members, but they do not show any more or less deviation from the y = x line than the unabsorbed subclusters.

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Povich et al. (2011) find that IR-derived populations estimates, which include the IR excess selected stars that lack detected X-ray counterparts, produce results ∼20% higher than the estimates that just use the X-ray selected sources. However, this offset is small relative to other sources of uncertainty in population estimation, and is not apparent on the log-log plot in Figure 4. Another source of uncertainty in Figure 4 is the comparison of the completeness limit in L_tc to a mass completeness limit. The sample of 839 COUP stars is complete down to 0.1–0.2 ${{M}_{\odot }}$, so there is some uncertainty in determining which mass limit to use for best comparison between XLF and IMF inferred populations.

The structures seen in the maps in Figure 5 are the projected distributions of stars in star-forming regions, so physically discrete groups of stars may overlap each other on the map. This is known to be the case for the Orion region—this field of view includes, in order of distance along our line of sight, the periphery of the older NGC 1980 cluster (Alves & Bouy 2012), the ONC, a dense subcluster containing massive stars (BN/KL; Becklin & Neugebauer 1967; Kleinmann & Low 1967), and stars embedded in the OMC, including the OMC1-S subcluster (Grosso et al. 2005). The clumpiness seen in the MYStIX maps hints at such structure in other regions.

The stellar surface density maps in Figure 5 show that most regions have small clumps of stars with extremely high surface density, but most of the area of the star-forming regions have surface densities well below the maxima. The highest peaks in surface density include the core of the RCW 38 cluster (∼34,000 stars pc⁻²), Orion (∼17,000 stars pc⁻² for the ONC and ∼22,000 stars pc⁻² for BN/KL), M17 (∼12,000 stars pc⁻²), the Tr 14 cluster in Carina (∼10,000 stars pc⁻²), and RCW 36 (∼10,000 stars pc⁻²). In the M17 region, a projected area of 9.8 arcmin² ($=3.3$ pc²) has surface densities greater than 1000 stars pc⁻²—substantially larger than for any other MYStIX MSFR. While, at the other extreme, the Rosette Nebula has an overall low stellar surface density.

The ONC plays a paradigmatic role in our understanding of young stellar clusters (e.g., Hillenbrand 1997; Getman et al. 2005b; Bally et al. 2000), and the surface-density maps show that it is similar in size and density to many of the densest clumps of stars in other MSFRs, even regions in which a much larger physical area has been surveyed. An example of this is the NGC 6357 region, which has three dense clusters similar to the ONC. Nevertheless, not all of the MYStIX MSFRs regions contain structures comparable to the ONC; neither the Eagle Nebula nor the Lagoon Nebula are as centrally concentrated as the ONC, despite having larger total young stellar populations. Older MSFRs like NGC 1893 and NGC 2362 entirely lack dense cluster cores.

The simulation from Section 2.1.1 of the degradation on the Orion Nebula data if this region were observed with the distance and exposure time for Carina (reducing the number X-ray sources from 1216 to 120) reveals how results for more distant MYStIX regions might be affected by lower sensitivity. The smoothed surface-density map for the simulated observation has a broadened central core, and, hence, the maximum surface density at the center of the ONC is reduced. Most of the highly embedded stars around BN/KL were removed, so this subcluster is not seen. However, the range of surface densities outside these peaks is not changed significantly. Thus, in other MSFRs, one may expect that some of the small but dense subclusters may be missing from our surface-density maps.

Care must be taken when comparing the results here with the results from B10 due to different selection effects on the young stellar samples used for each study. The sky coverage for both MYStIX and B10 is the result of multiple projects with different objectives, so neither represents an unbiased survey of star-forming environments in the Galaxy. Perhaps most important, the B10 survey is based only on IR-excess stars, and thus misses the large X-ray selected disk-free population. In MPCM samples, the stars with no detected IR excess typically outnumber the disk-bearing subpopulation by 2–3:1, and sometimes more than 7:1 (Broos et al. 2013, their Table 1). It is therefore not surprising that stellar surface densities derived from the MYStIX survey are higher than those derived by B10.

One of the main observations of B10 is that the ${{{\Sigma }}_{\star }}$ distribution of their sample has no discrete modes (peaks in the logarithmically binned histogram) corresponding to "distributed" star formation or "clustered" star formation. As a result, they concluded that the histogram of surface densities alone is not sufficient for determining which stars are clustered and which stars are not. This assertion is supported by Gieles et al. (2012) who demonstrate that even if all stars are born in clusters, cluster expansion can yield a variety of different surface density distributions, including the log-normal distribution of B10. Pfalzner et al. (2012) additionally demonstrate that for King (1962) cluster density profiles, the low density portion of the cluster outside the dense core can make it difficult to identify distinct peaks in the ${{{\Sigma }}_{\star }}$ distribution, even if they do exist. Our results show that different MSFRs have radically different ${{{\Sigma }}_{\star }}$ distributions (Figure 7), rather than following a universal distribution. It is thus difficult or impossible to identify any dividing surface density threshold from the empirical ${{{\Sigma }}_{\star }}$ data. B10 also see shifts in the peaks of the distributions for the different surveys they use, ∼7 stars pc⁻² for the Gould Belt+Taurus and ∼30 stars pc⁻² for the c2d regions.

Another conclusion of B10 is that the unimodal log-normal distribution—peaked at 22 stars pc⁻² with standard deviation width of 0.85 dex—means that most stars form outside clusters, while densities like 100, 1000, 10,000 stars pc⁻² are in the tail of the distribution.¹² They choose a surface-density threshold of 200 stars pc⁻² to be their definition of a cluster because this is where stars become likely to interact with their neighbors. Gieles et al. (2012) and Pfalzner et al. (2012) caution about using projected surface-densities to estimate the fraction of stars undergoing local dynamical interactions because analysis of the empirical surface-density histograms for a sample does not take into account cluster evolution, radial cluster structure, and superposition of discrete clusters. Thus, they argue that stellar interaction could be important even if the empirical histograms show most stars lie in environments below an astrophysically selected threshold like 200 stars pc⁻².

Both MYStIX and B10 investigate star formation in the Orion giant molecular clouds, including the Orion A and B molecular clouds, containing the Orion Nebula and Flame Nebula, respectively, making this a useful region to directly compare results. The combination of the B10 and the MYStIX surface-density distributions for this star-forming region provides a less biased estimate of the full surface-density distribution. We have emulated the B10 analysis for the larger-scale Orion star-forming region in the areas outside the MYStIX fields of views, using the Megeath et al. (2012) catalog of YSOs, with a magnitude limit at IRAC $[3.6]=14$ mag. The objects excluded due to overlap with the MYStIX fields of view include 19% of the B10 Orion sample. The distribution of these sources on the sky is shown in the left panel of Figure 8, with polygons cut out representing the Flame Nebula MYStIX field of view in the north and the Orion Nebula MYStIX field of view in the south. To compute the ${{{\Sigma }}_{\star }}$ histogram for the Orion A and B region (excluding the MYStIX fields for the Orion and Flame nebulae), we follow B10 by assuming a uniform disk fraction of 65%, so all densities and bin heights are increased by a factor of 1.35.

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The right panel of Figure 8 shows these two histograms, labeled "B10 method" for the Orion A and B molecular clouds (excluding the MYStIX Orion and Flame Nebulae fields of view) and labeled "MYStIX" for the coadded Orion Nebula and Flame Nebula histograms. Both methods attempt to account for intrinsic numbers of stars: in B10's case, by accounting for missing Class III stars with a uniform correction; in our case, through the XLF and IMF methods described in Section 3. At 5 Myr and a distance of 0.414 kpc, a PMS star of 0.1 ${{M}_{\odot }}$would have an L-band (≈ the IRAC [3.6] band) photospheric magnitude of ∼13 mag (Siess et al. 2000); so B10's sample should be sensitive to YSOs down to the hydrogen-burning mass limit. Although the "B10 method" histogram is missing 19% of the B10 stars due to overlap with the MYStIX regions, the location of its peak is consistent with the histogram in Figure 1 of B10.

The full distribution for the Orion giant molecular clouds in Figure 8 is clearly very different from the results in Figure 1 of B10. Furthermore, based on the total numbers of stars in these histograms, clustered stars make up one of the dominant components of the ${{{\Sigma }}_{\star }}$ distribution, rather than just being an extreme value "tail" of the distribution. Underestimation of the number of stars in high density regions was also commented on by Pfalzner et al. (2012). The coadded B10 and MYStIX histograms (the dashed line) has some bimodal structure, with peaks around 100 and 1000 stars pc⁻² and a shoulder around 20 stars pc⁻², but this may be an artefact of selection effects in both studies.

Figure 9 shows ${{{\Sigma }}_{\star }}$ histograms for just the Flame Nebula region, which are obtained using the B10 method with only IR-excess selected members from Megeath et al. (2012) and the histogram obtained from MYStIX using both IR-excess and X-ray selected members. Visual inspection of the Spitzer IRAC images reveals that IR sources are likely lost in the wings of bright sources and in regions with bright nebulosity. As a result, the IR excess only method underestimates both the number of stars in this cluster and the stellar surface densities.

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Limitations that affect the IR-excess census of YSOs (outside the dense Flame Nebula and Orion Nebulae) include variations in Spitzer sensitivity due to the presence of nebulosity (e.g., Muench et al. 2007) and variations in the disk fraction (e.g., Getman et al. 2009; Kuhn et al. 2010; Povich et al. 2011; Getman et al. 2012; Getman et al. 2014a). If a significant numbers of young stars were missed in the outer portions of the star-forming complex, it might increase the low-density component of the combined histogram to the point where it is comparable to the high-density component. Results from a large XMM survey of the Orion A molecular cloud (e.g., Pillitteri et al. 2013) will be able to address some of these limitations. Limitations to the MYStIX censuses include reduced sensitivity of X-ray selection to protostars (Prisinzano et al. 2008) and highly absorbed clumps of stars and assumptions about a uniform XLF and IMF.

While improved information about the typical environments in which stars form would be useful for theoretical models of star formation and cluster formation and evolution; it is difficult to generalize results like the ones here or from B10 to star formation in the Galaxy as a whole. Even within the MYStIX sample, surface density distributions from one region provide little information about other regions (Figure 7). We have demonstrated that B10's results are likely not valid for the Orion molecular clouds (∼70% of their sources). But their results are likely to be reasonable for other nearby star-forming clouds that are less affected by crowding and nebulosity, although scalings to include disk-free subpopulations may vary. Nevertheless, the high surface densities seen in other MSFRs indicate that the results for the Orion molecular clouds are not anomalous, and may indeed be typical for most stars, in the context of Galactic star formation. The initial cluster mass function described by a power-law with index of approximately -2 over a large cluster mass range (Lada & Lada 2003; Chandar et al. 2010; Portegies Zwart et al. 2010; Lada 2010; Ryon et al. 2014; Fouesneau et al. 2014), would imply that stars are roughly equally likely to form in complexes of different masses. This would mean that star formation is evenly spread between small Taurus/Chamaelion-type clouds with tens-to-hundreds of stars, smaller OB star-forming regions like NGC 2264 containing hundreds to thousands of stars, giant molecular clouds like the Orion complex thousands to 10⁴ stars, and major star-forming complexes like Carina containing 10⁴–10⁵ stars. Since massive stars appear in the second group, most stars are likely born in regions containing or influenced by massive stars.