THE BOLOCAM GALACTIC PLANE SURVEY. XIII. PHYSICAL PROPERTIES AND MASS FUNCTIONS OF DENSE MOLECULAR CLOUD STRUCTURES

A simple estimate of a dense molecular cloud structure's mass may be computed from optically thin millimeter dust-continuum data via

$$\begin{eqnarray}&&M=\frac{\;{{S}_{\nu ,\operatorname{int}}}\;d_{\odot }^{2}}{{{\kappa }_{\nu }}{{B}_{\nu }}({{T}_{d}})},\end{eqnarray} \tag{ 2 }$$

where ${{S}_{\nu ,\operatorname{int}}}$ is the source-integrated flux density, ${{d}_{\odot }}$ is the estimated heliocentric distance, ${{\kappa }_{\nu }}$ is the opacity per mass of dust, $\equiv ({{m}_{{\rm g}}}/{{m}_{d}})$ is the gas-to-dust mass ratio, and ${{B}_{\nu }}({{T}_{d}})$ is the Planck function evaluated at dust temperature T_d. For the specific case of the BGPS,

$$\begin{eqnarray}&&M=13.1\;{{M}_{\odot }}\left( \frac{{{e}^{13.0\,{\rm K}/{{T}_{d}}}}-1}{{{e}^{13.0\,{\rm K}/20.0\,{\rm K}}}-1} \right)\left( \frac{{{S}_{_{1.1}}}}{{\rm Jy}} \right){{\left( \frac{{{d}_{\odot }}}{{\rm kpc}} \right)}^{2}},\end{eqnarray} \tag{ 3 }$$

where ${{S}_{_{1.1}}}$ is the $\lambda =1.1$ mm source-integrated flux density, ${{\kappa }_{\nu }}=1.14$ cm² g⁻¹ of dust (Ossenkopf & Henning 1994, Table 1, Column 5),¹² $\sf {X}$ = 100 (Hildebrand 1983), and the equation has been normalized to T_d = 20 K. Standard error-propagation methods may be used to determine the mass uncertainty, but the asymmetric nature of many DPDF error bars (EB15) complicates the issue. The uncertainty in the distance may be marginalized over to produce the expectation value of the mass via

$$\begin{eqnarray}&&\langle M\rangle =\frac{\;{{S}_{\nu ,\operatorname{int}}}}{{{\kappa }_{\nu }}{{B}_{\nu }}({{T}_{d}})}\int _{0}^{\infty }{\rm DPDF}\;d_{\odot }^{2}\;d{{d}_{\odot }},\end{eqnarray} \tag{ 4 }$$

where $\langle M\rangle$ is proportional to the second moment of the DPDF. The bimodal nature of many posterior DPDFs in the inner Galaxy, however, causes this to be a biased estimator of the mass; in much the same way $\bar{d}$ (the first-moment distance) can be a biased estimator of the distance. For instance, an object with a well-constrained distance estimate may have up to $\approx 20\%$ of the integrated DPDF in the non-favored kinematic distance peak; the difference between $\bar{d}$ and the maximum-likelihood distance (${{d}_{{\rm ML}}}$) can be greater than 1 kpc, or $\geqslant 2\;\times$ the uncertainty in ${{d}_{{\rm ML}}}$ (see Ellsworth-Bowers et al. 2013 for a complete discussion).

This bias, coupled with the measured source flux density and assumed dust temperature uncertainties, suggests using a Monte Carlo approach for estimating an object's mass and associated uncertainty. To determine the maximum-likelihood mass for a given catalog object, Equation (2) is computed many times with each realization drawing a distance, flux density, and temperature from the suitable probability density functions. The resulting (properly normalized) mass probability density function (MPDF) may then be used to estimate the maximum-likelihood mass (${{M}_{{\rm ML}}}$) and the 68.3% error bars in the same fashion as was done with DPDFs in EB15.

Being the probability of an object having a mass between M and $M+\delta M$, the MPDF of a given object is created from the histogram of masses produced by many ($\sim {{10}^{6}}$) Monte Carlo realizations of Equation (2). Given the fractional nature of the uncertainties in the derived mass, we chose to use logarithmic binning for the computation of MPDFs and the probability density functions for other physical properties. Note that the computation of DPDFs (EB15) is done with respect to a fixed linear distance scale and is not subject to the need to create histograms from Monte Carlo calculations.

In addition to the masses of dense molecular cloud structures, the catalog of heliocentric distances enables the computation of the physical sizes and number densities of these objects. Not only can these physical properties help to disentangle the various populations of objects (i.e., cloud, clump, core) detected by the BGPS and others, but they can also test the extension of various empirical relationships for GMCs down to more dense substructures.

The physical radius is computed from

$$\begin{eqnarray}&&R=0.29\;{\rm pc}\left( \frac{{{\theta }_{R}}}{{\rm arcmin}} \right)\left( \frac{{{d}_{\odot }}}{{\rm kpc}} \right),\end{eqnarray} \tag{ 5 }$$

where ${{\theta }_{R}}$ is the deconvolved angular radius of a detected object computed from the geometric mean of the deconvolved major (${{\sigma }_{{\rm maj}}}$) and minor (${{\sigma }_{{\rm min} }}$) axes of the flux-density distribution,

$$\begin{eqnarray}&&{{\theta }_{R}}=\eta {{\left[ \left( \sigma _{{\rm maj}}^{2}-\sigma _{{\rm bm}}^{2} \right)\left( \sigma _{{\rm min} }^{2}-\sigma _{{\rm bm}}^{2} \right) \right]}^{1/4}}\end{eqnarray} \tag{ 6 }$$

(Rosolowsky et al. 2010). For the BGPS, the rms size of the beam is given by ${{\sigma }_{{\rm bm}}}={{\theta }_{{\rm FWHM}}}/\sqrt{8{\rm ln} 2}=14^{\prime\prime}$, and $\eta =2.4$ is a factor relating the rms size of the emission distribution to the angular radius of the object.¹³ Objects whose measured minor axis dispersion is smaller than ${{\sigma }_{{\rm bm}}}$ do not have a real solution to Equation (6); actual physical objects should not fall into this category, but the effects of the cataloging process lead to 29% of the full BGPS V2.1 catalog and 20% of the Distance Catalog having a complex computed ${{\theta }_{R}}$. These objects are excluded from the analyses of physical radius and number density in Section 4.

As a means for classifying dense molecular cloud structures, the mean number density offers a possible insight into the physical processes, such as structure formation via turbulence or gravitational collapse, that may be at play in a given catalog object. The number density computed with assumed cylindrical geometry (i.e., a circle on the sky with radius R, and depth R),¹⁴ $n=M/\pi \mu {{m}_{H}}{{R}^{3}}$, may be parameterized in terms of mass and radius or observable quantities as

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} n & = & 4.60\times {{10}^{2}}\;{\rm c}{{{\rm m}}^{-3}}\left( \frac{M}{100\;{{M}_{\odot }}} \right){{\left( \frac{R}{{\rm pc}} \right)}^{-3}} \\ {} & = & 2.47\times {{10}^{3}}\;{\rm c}{{{\rm m}}^{-3}}\left( \frac{{{e}^{13.0\,{\rm K}/{{T}_{d}}}}-1}{{{e}^{13.0\,{\rm K}/20.0\,{\rm K}}}-1} \right) \\ {} & {} & \times \left( \frac{{{S}_{1.1}}}{{\rm Jy}} \right){{\left( \frac{{{\theta }_{R}}}{{\rm arcmin}} \right)}^{-3}}{{\left( \frac{{{d}_{\odot }}}{{\rm kpc}} \right)}^{-1}}. \\ \end{array}\end{eqnarray} \tag{ 7 }$$

The first parameterization relates the physical quantities in units appropriate to dense molecular cloud structures, while the second is specific to the observational quantities of the BGPS and the assumptions used in Equation (3). Note that the number density is not as strong a function of heliocentric distance as is the mass; incorrect KDA resolution will have a smaller effect on the derived value.

In its simplest form, the mass distribution of dense molecular cloud structures may be compiled from the direct application of Equation (2) to each object with a well-constrained distance estimate, where ${{d}_{\odot }}$ is either the maximum-likelihood distance or the first-moment distance, as discussed in Ellsworth-Bowers et al. (2013). This form of the distribution for the Distance Catalog sources is shown as the red histogram in Figure 1 (left), and represents a naïve, but effective, realization of the mass distribution.

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While the simple mass distribution utilizes the best distance estimate from the DPDF formalism, it discards much of the information contained in the posterior DPDFs. Given the asymmetric nature of many DPDFs, the most straightforward way to capitalize on this pool of information is through Monte Carlo realizations utilizing the MPDFs. Plotted in black in Figure 1 (left) is the mass distribution computed using the ${{M}_{{\rm ML}}}$ from the Monte Carlo generated MPDFs. While there are noticeable deviations in the wings of the distributions, a two-sample K-S test between these two distributions reveals that they are indistinguishable to better than 99.9%. Despite the negligible differences between the distributions, the power of the MPDF-based mass distributions lies in the ability to create many Monte Carlo realizations of the mass distribution, randomly drawing a mass from each object's MPDF for each trial.

If the aggregate mass distributions between the two methods are indistinguishable, what about the masses of individual objects? The comparison of the mass derived from a simple application of Equation (2) using the single-value distance estimate from EB15 (M_sim) with the maximum-likelihood value from the MPDF (${{M}_{{\rm ML}}}$) is shown in the middle panel of Figure 1, where the dashed line marks the 1:1 relationship. The histogram of the ratio of these masses (${{M}_{{\rm ML}}}$ $/{{M}_{{\rm sim}}}$) is plotted in the right panel along with a Gaussian fit. The mean of the distribution is consistent with unity, and has a standard deviation of $\approx 2\%$. Differences between the masses are more significant for low-mass objects ($M\lesssim 100\;{{M}_{\odot }}$), with ${{M}_{{\rm ML}}}$becoming larger than the simple mass by up to 60% in the most extreme case.

Going beyond constructing a mass distribution for the Distance Catalog using a single mass for each object, a more complete Monte Carlo approach is shown as the hashed cyan histogram in Figure 1 (left). Rather than taking the maximum-likelihood value from the MPDF of each object (as in the black histogram in that panel), each object's MPDF was randomly sampled 20,000 times to produce a mass distribution containing 34.2 million entries. The resulting histogram was then normalized to represent the N = 1710 objects of the Distance Catalog. The resulting distribution is much smoother at both mass extrema because of the lack of small-number statistics. Interestingly, the mode (peak) of the distribution is shifted to slightly higher mass, which is likely due to the $d_{\odot }^{2}$ dependence in Equation (2). This Monte Carlo composite distribution is discussed further in Section 4.2.1.

(Sub-)Millimeter continuum surveys of thermal dust emission are flux-limited, so Malmquist bias must be addressed when analyzing the mass distribution of detected objects. It is relatively straightforward to identify flux-density completeness levels, but since higher mass (and therefore brighter) objects are detectable through a larger Galactic volume than low-mass objects, it is imperative to define the mass criteria at which a survey is complete to some level. While small surveys of isolated regions at known distances (e.g., Enoch et al. 2006; Ridge et al. 2006; Rosolowsky et al. 2008) can directly estimate strict mass completeness levels, heterogenous surveys of the entire Galactic plane must be content to have a more complex criterion.

We cast the issue of completeness levels in terms of a mass completeness function, which describes the fraction of objects in the survey area at any given mass that should be detected. Mass completeness begins with flux-density completeness, which is usually defined in terms of the rms noise in survey maps. For instance, the BGPS V1 catalog is 99% complete at $5\times$ the rms noise flux density for any given field (Rosolowsky et al. 2010), a value that is unchanged in version 2 (G13). While surveys that do not have uniform noise properties across the Galactic plane pose an additional complication to the process, variable backgrounds throughout the Galactic plane (especially at shorter wavelengths) will also affect the completeness calculation. The flux-density completeness level must, therefore, be estimated on a field-by-field (or, better, source-by-source) basis.¹⁵ Estimates of rms noise in survey images, however, usually do not take the confusion limit into account, so these completeness levels should be viewed as lower limits.

To begin, we compute the local flux-density completeness level for each catalog object as $5\times$ the mean of the BGPS noise maps within the object's catalog boundary (see Rosolowsky et al. 2010). The minimum complete flux density for each object implies that any source at this location in the survey map brighter than this value will be detected $\geqslant 99\%$ of the time. Approximately 13% of the objects in the Distance Catalog have flux densities below their own completeness value, meaning that they are $\lt 5\sigma$ detections. These objects represent a tip of the iceberg; similar objects exist in those same fields but were not recovered.

Combining the completeness flux density with the object's distance information allows an estimate of the "minimum complete mass" for each object, meaning that any object at the specified ($\ell ,b,\;$ ${{d}_{\odot }}$) with mass higher than this value will be detected $\geqslant 99\%$ of the time. Repeating this process on a source-by-source basis allows for the effective marginalization over the variable noise properties across the Galactic plane. The mass completeness function is therefore the cumulative distribution of the minimum complete mass for all objects in the sample.

Figure 2 illustrates the mass completeness functions computed using the two techniques described in Figure 1. The completeness function for the single distance estimate from EB15 for each object is plotted in cyan, and an aggregation of 10³ Monte Carlo realizations pulling distances from each object's DPDF is shown in black. The curves are similar, but a two-sample K-S test reveals that they are not drawn from the same population at greater than 99.5% significance. Shown as horizontal dotted lines are the 50% and 90% completeness levels, and the vertical lines mark the intersections of the curves with these levels. The 50% completeness level is $\approx 70\;{{M}_{\odot }}$ for both curves, and the 90% completeness level is $\approx 400\;{{M}_{\odot }}$. The importance of these values is in limiting conclusions about the mass distributions (or other derived physical properties) to larger masses; extending conclusions to smaller-mass objects is not supported by the available data.

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The use of power-law functions for describing the mass distributions of stars and their precursors goes back to the original studies of the stellar initial mass function by Salpeter (1955). The mass function is defined as the number of objects either per logarithmic mass interval,

$$\begin{eqnarray}&&\xi ({\rm log} M)=\frac{dN}{d{\rm log} M}\propto {{M}^{-x}},\end{eqnarray} \tag{ 8 }$$

or per linear mass interval,

$$\begin{eqnarray}&&\xi (M)=\frac{dN}{dM}=\frac{\xi ({\rm log} M)}{M\;{\rm ln} 10}=\left( \frac{1}{M{\rm ln} 10} \right)\frac{dN}{d{\rm log} M},\end{eqnarray} \tag{ 9 }$$

where $\xi (M)\propto {{M}^{-\alpha }}$ and $x=\alpha -1$ (Chabrier 2003). Within this framework, the Salpeter power-law index is x = 1.35, $\alpha =2.35$. For a detailed description of this parameterization and its consequences, see Swift & Beaumont (2010), Olmi et al. (2013), and references therein. The mathematical nature of a power law requires a finite lower limit on the mass range over which it is a valid descriptor of the data in addition to the observational constraint that the mass distribution turns over at $M\lesssim 1\;{{M}_{\odot }}$ (e.g., Kroupa 2001).

To fit a power-law function to the observed mass distribution, we use the maximum-likelihood method described by Clauset et al. (2009) to estimate the power-law index and range over which it is valid.¹⁶   Briefly, assuming $\alpha \gt 1$ and ${{M}_{{\rm max} }}\gg {{M}_{{\rm min} }}$, the probability (or likelihood) that the observed masses (M_i) are drawn from a power-law distribution is proportional to

$$\begin{eqnarray}&&{{p}_{{\rm pl}}}(M\;|\;\alpha )=\mathop{\prod }\limits_{i=1}^{n}\ {{\frac{{{\alpha }^{{}}}-1}{{{M}_{{\rm min} }}}}^{{}}}{{\left( \frac{{{M}_{i}}}{{{M}_{{\rm min} }}} \right)}^{-\alpha }},\end{eqnarray} \tag{ 10 }$$

where α is defined as in Equation (9). The maximum-likelihood value is computed by taking the derivative of the logarithm of the likelihood ($\mathcal{L}$) with respect to α and finding the root (i.e., $\partial \mathcal{L}/\partial \alpha =0$).¹⁷ The power-law index estimator is

$$\begin{eqnarray}&&\hat{\alpha }=1+n{{\left[ \mathop{\sum }\limits_{i=1}^{n}\;{\rm ln} \frac{{{M}_{i}}}{{{M}_{{\rm min} }}} \right]}^{-1}}.\end{eqnarray} \tag{ 11 }$$

The minimum mass (M_min) for which a power law is a good descriptor of the mass distribution is computed from the data. For each value in the mass distribution, Equation (11) is computed using that mass as M_min. The cumulative distribution of $M\geqslant {{M}_{{\rm min} }}$ is compared with the analytic power law with index $\hat{\alpha }$, and the maximum difference between them is computed (i.e., the K-S test D-statistic); the value of M_min that minimizes D is returned by the algorithm (see Clauset et al. 2009 for a complete description of the algorithm). The results presented here were computed using a python translation of the PLFIT routine from Clauset et al.¹⁸

Predicated on the properties of supersonic turbulence and the turnover in the IMF at $M\lesssim 1\;{{M}_{\odot }}$, the lognormal form of the mass function is also commonly fit to the mass distributions of dense molecular cloud structures (e.g., Swift & Beaumont 2010; Olmi et al. 2013). Analogous to the power-law case, we use a maximum-likelihood method for computing the best fit to the mass distribution. The likelihood of the observed masses being drawn from a lognormal distribution is proportional to

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} {{p}_{{\rm ln} }}(M|\mu ,\sigma ) & = & \mathop{\prod }\limits_{i=1}^{n}\ {{\frac{{{C}_{{\rm ln} }}}{M}}^{{}}}{\rm exp} \left[ -\frac{{{({\rm ln} M-\mu )}^{2}}}{2{{\sigma }^{2}}} \right] \\ {} & = & \mathop{\prod }\limits_{i=1}^{n}\ {{\frac{{{C}_{{\rm ln} }}}{M}}^{{}}}{\rm exp} \left[ -{{x}^{2}} \right], \\ \end{array}\end{eqnarray} \tag{ 12 }$$

where the normalization factor

$$\begin{eqnarray}&&{{C}_{{\rm ln} }}=\sqrt{\frac{2}{\pi {{\sigma }^{2}}}}{{\left[ {\rm erfc}({{x}_{{\rm min} }})-{\rm erfc}({{x}_{{\rm max} }}) \right]}^{-1}},\end{eqnarray} \tag{ 13 }$$

and erfc $(x)$ is the complementary error function evaluated at x. The parameters [$\mu ,\sigma$] are the mean and width of the lognormal Gaussian, respectively, and [${{x}_{{\rm min} }},{{x}_{{\rm max} }}$] are related to the minimum and maximum masses, respectively, over which the lognormal fit is valid. The maximum-likelihood estimators $[\hat{\mu },\hat{\sigma }]$ are found by numerically solving ${{\nabla }_{\mu ,\sigma }}\mathcal{L}=0$ (i.e., numerically maximizing $\mathcal{L}$), where

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} \mathcal{L} & = & {\rm ln} \;{{p}_{{\rm ln} }}(M|\mu ,\sigma ) \\ {} & = & n\;{\rm ln} \;{{C}_{{\rm ln} }}-\mathop{\sum }\limits_{i=1}^{n}\;{\rm ln} \;{{M}_{i}}-\mathop{\sum }\limits_{i=1}^{n}\;x_{i}^{2}. \\ \end{array}\end{eqnarray} \tag{ 14 }$$

As with the power-law case, the limits $[{{M}_{{\rm min} }},{{M}_{{\rm max} }}]$ over which the lognormal fit best describe the data must be estimated from the data themselves. We employ the same K-S test D-statistic analysis as described previously, but use a numerical minimizer to explore the two-dimensional parameter space including M_max. Following the algorithmic structure of PLFIT (Section 3.3.1), we wrote a routine in python to maximize Equation (14) using Powell's Method inside a numerical minimization of the K-S test D-statistic using a downhill simplex optimization.¹⁹

We present a physical properties catalog for the 1710 sources in the Distance Catalog. The mass, physical radius, and number density for each object were computed from the Monte Carlo probability density functions as described in Section 3.1, where a dust temperature was drawn for each realization from a lognormal distribution with a mean of 20 K and FWHM of 8 K (Battersby et al. 2011). This choice of temperature distribution was recently affirmed by the analysis of analogous dense molecular cloud structures from the ATLASGAL survey by Wienen et al. (2015). We note that the study of Merello et al. (2015) combining $\lambda =1.1$ mm and $\lambda =350$ μm data finds a lower mean temperature for BGPS sources (T_d = 13.3 K for dust spectral index $\beta =2.0$, and T_d = 16.3 K for $\beta =1.7$). Using a mean T_d = 20 K would lead to an underestimation of the mass of each object by a factor of 1.3–1.8 for $\beta =1.7$ and $\beta =2.0$, respectively, of Merello et al. (2015). While the computed mass and number density values are affected by this uncertainty in temperature, Monte Carlo simulations of BGPS source mass distributions by Schlingman et al. (2011) showed that the power-law slope of the mass function (Section 4.2.2) is not very sensitive to the choice of underlying dust temperature distribution.

The pdfs for mass, physical radius, and number density were constructed in log space to capture the spread in values over orders of magnitude, and error bars were computed to enclose 68.3% of the total probability, where the endpoints occur at equal probability (see the discussion of DPDF error bars in EB15). The error bars approximate the Gaussian $\pm 1\sigma$ region, but account for the asymmetric nature of the pdfs. The catalog of physical properties is shown in Table 1, along with relevant information from the BGPS V2.1 catalog (G13) and heliocentric distance (EB15).

Table 1.  Physical Properties of BGPS Distance Catalog Sources

BGPS V2.1 Catalog Properties						Derived from the PDFsa
Catalog	ℓ	b	Sintb	${{\theta }_{R}}$ c	${{d}_{\odot }}$ d	M	R	n
Number	(°)	(°)	(Jy)	('')	(kpc)	(${\rm log} \;{{M}_{\odot }}$)	(pc)	(log cm−3)
2235	7.993	−0.268	$6.23(0.49)$	74.3	$11.94_{-0.42}^{+0.58}$	$4.07_{-0.11}^{+0.12}$	$4.31_{-0.16}^{+0.19}$	$2.81_{-0.10}^{+0.12}$
2254	8.187	0.482	$0.61(0.15)$	35.9	$1.64_{-1.06}^{+0.84}$	$1.47_{-0.71}^{+0.43}$	$0.36_{-0.19}^{+0.11}$	$3.58_{-0.28}^{+0.39}$
2256	8.207	0.190	$1.44(0.22)$	56.7	$2.46_{-1.16}^{+0.84}$	$2.09_{-0.50}^{+0.33}$	$0.73_{-0.28}^{+0.19}$	$3.24_{-0.23}^{+0.28}$
2261	8.249	0.180	$2.29(0.29)$	77.6	$3.16_{-0.86}^{+0.64}$	$2.48_{-0.30}^{+0.22}$	$1.23_{-0.29}^{+0.19}$	$2.93_{-0.17}^{+0.19}$
2262	8.263	0.168	$2.82(0.32)$	79.8	$2.68_{-0.96}^{+0.74}$	$2.43_{-0.38}^{+0.27}$	$1.09_{-0.37}^{+0.24}$	$3.05_{-0.19}^{+0.22}$
2265	8.281	0.164	$1.00(0.15)$	23.8	$3.06_{-0.88}^{+0.66}$	$2.09_{-0.32}^{+0.23}$	$0.36_{-0.09}^{+0.06}$	$4.13_{-0.18}^{+0.19}$
2295	8.545	−0.342	$1.00(0.20)$	35.5	$4.30_{-0.78}^{+0.60}$	$2.36_{-0.18}^{+0.16}$	$0.75_{-0.09}^{+0.06}$	$3.44_{-0.22}^{+0.21}$
2296	8.551	−0.296	$0.25(0.10)$	⋯	$12.20_{-0.40}^{+0.48}$	$2.72_{-0.21}^{+0.19}$	⋯	⋯
2297	8.579	−0.344	$0.86(0.18)$	37.3	$4.04_{-0.56}^{+0.46}$	$2.25_{-0.21}^{+0.18}$	$0.73_{-0.10}^{+0.08}$	$3.35_{-0.16}^{+0.17}$
2300	8.669	−0.406	$0.44(0.11)$	⋯	$11.80_{-0.36}^{+0.44}$	$2.92_{-0.16}^{+0.16}$	⋯	⋯

^aSee Section 3.1. ^bThe source-integrated $\lambda =1.1$ mm flux density; uncertainties in parentheses. ^cThe deconvolved radius of the catalog object computed using Equation (6). ^dThe appropriate distance estimate from EB15; $\bar{d}$ for objects near the tangent point and ${{d}_{{\rm ML}}}$ otherwise.

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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Of particular note are the 20% of sources in the BGPS Distance Catalog whose angular extents on the sky do not yield real solutions to Equation (6) for ${{\theta }_{R}}$. A brief analysis of these objects in the context of the flux-density distribution for all catalog sources reveals a strong trend whereby objects with non-physical ${{\theta }_{R}}$ are preferentially dim (Figure 3). The gray and blue histograms represent the unresolved sources in the full V2.1 catalog and the Distance Catalog, respectively, whereas the black and red histograms represent the resolved sources in the same groups.

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The obvious correlation in the data points in the top panel (Spearman rank correlation coefficient $\rho =0.753$) is a direct consequence of a universal scaling relationship in the dense interstellar medium first identified by Larson (1981), whereby the mass of a molecular cloud structure is roughly proportional to the square of its radius. Section 5.2 explores this relationship in more detail, culminating in Equation (17), which is plotted as green lines in Figure 3 for three different heliocentric distances.

The Larson relationships in the figure extend to low flux density, but since the cataloging process truncates source boundaries where the flux density meets the local rms value (Rosolowsky et al. 2010), source masks do not extend to the theoretical zero-flux-density isophot. The resulting catalog source may have an extent in the map that is smaller than the beam size, leading to a complex solution to Equation (6). To further aggravate the situation, the beam deconvolution step assumes that sources are spherical (or at least have circular projections on the sky), which is invalid for many of the fainter filamentary structures throughout the Galactic plane (e.g., Molinari et al. 2010a). Future cataloging routines may be constructed that divide the emission between filamentary and compact sources, and may decrease the number of BGPS sources with invalid deconvolved angular extent. For the present, however, unresolved sources are excluded from the following analyses that rely on physical radius or number density, leaving a sample of N = 1369 objects with a tabulated physical size.

The ensemble physical property distributions from Table 1 are shown in Figure 4 to illustrate the range of BGPS objects in the Distance Catalog. This subsample is generally representative of the entire BGPS catalog, except that detection in a molecular transition line that traces dense gas (e.g., HCO⁺(3–2)) is strongly correlated with continuum flux density; dim sources are largely absent from the Distance Catalog (see EB15 for a complete discussion). In all three panels, sources above the 50% (70 ${{M}_{\odot }}$) and 90% (400 ${{M}_{\odot }}$) mass completeness levels are indicated by gray solid and magenta hashed histograms, respectively.

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The mass distribution (top panel) peaks around 100 ${{M}_{\odot }}$, but the completeness levels indicated by the filled histograms imply that the location of the distribution peak is observationally biased. The location of the 50% completeness level demonstrates that the predicted turnover at low mass is not constrained by the current observations. Physical radius is illustrated in the middle panel of Figure 4, with vertical dashed lines marking plausible boundaries between molecular clouds (large R), clumps (intermediate R), and individual cores. We place approximate radius boundary markers at 1.25 pc between cloud and clump and 0.125 pc between clump and core, which are the same boundaries used by D11, which were based on the general guidelines from Bergin & Tafalla (2007). These divisions, while rooted in physical distinctions based on the virial ratio and capacity for Jeans fragmentation, are flexible and the boundaries are easily blurred by the continuum of structure in molecular cloud complexes (see Section 5.2), and by variations in Galactic environment. Note, however, the near complete lack of Distance Catalog objects with radii $\lesssim 0.1$ pc. From the relationship in Figure 3, smaller sources have lower flux density and are therefore less likely to be included in the Distance Catalog (EB15).

Definitions notwithstanding, the typical size scale of BGPS-identified dense molecular cloud structures is clearly in the vicinity of 1 pc and may encompass multiple true but unresolved molecular cloud clumps. The mass completeness subsets illustrate the effects of Malmquist bias (coupled with the Larson relationship connecting mass and physical radius), whereby the 90% complete sample is nearly all cloud-scale ($R\gtrsim 1$ pc) objects, and the BGPS does not detect any large ($R\geqslant 2$ pc), lower-mass objects ($M\leqslant 400\;{{M}_{\odot }}$). This places a rough constraint on the minimum number density ($n\gtrsim 230$ cm⁻³, Equation (7)) detectable by the BGPS for large-scale objects.

The bottom panel of Figure 4 depicts the number density distribution of resolved sources, which spreads across a wide dynamic range. The approximate number density boundaries at n = 750 cm⁻³ between cloud and clump, and $n={{10}^{4}}$ cm⁻³ between clump and core (D11) are indicated by vertical dashed lines. Whereas there is nearly complete overlap between high-mass and large-radius objects ("clouds"; middle panel), there exist a handful of low-mass ($M\leqslant 70$ ${{M}_{\odot }}$) objects with low computed number density ("clouds"; bottom panel). These objects, along with the spread of higher-mass objects ($M\geqslant 400$ ${{M}_{\odot }}$) across classifications in the bottom panel, insinuate that number density computed via Equation (7) for BGPS sources is not a reliable indicator of physical classification. It is possible that the non-uniform geometry of sources is causing the spread of the completeness samples over a large range, or that the algorithm by which the cataloging routine decomposes emission into discrete sources is doing so in a manner that is inconsistent with the underlying physical structure in molecular clouds (e.g., by assuming spherical clouds when computing N(H₂)).

A major challenge in characterizing the ensemble physical property distributions of detected molecular cloud structures is separating observational and data-processing systematic effects from underlying physical relationships, as systematic effects on the measurement of physical properties can lead to significant biases in the derived quantities. Understanding the physics of dense molecular cloud structures requires well-constrained distance estimates. It is instructive, therefore, to analyze the functional dependences.

The left panels of Figure 5 illustrate the dependence on the heliocentric distance of the mass, physical radius, and number density for sources in the Distance Catalog. Sources are plotted as gray dots, with the median value per 2 kpc bin marked by black squares. The systematic dependence on heliocentric distance can clearly be seen in the median values as $M\sim d_{\odot }^{2}$, $R\sim {{d}_{\odot }}$, and $n\sim d_{\odot }^{-1}$, following Equations (2), (5), and (7), respectively. For the distribution of mass (top left panel), the hashed region identifies the mass associated with the mean $\leqslant 5\sigma$ flux-density completeness level (${{S}_{{\rm complete}}}=0.30$ Jy, Section 3.2.2). Especially given the variable rms noise from field to field in the BGPS, sources may be detected in this region, but completeness is not assured. The horizontal dashed lines mark the 50% and 90% completeness levels (as in Figure 4). The large collection of sources around ${{d}_{\odot }}$= 5 kpc has a median mass $\approx 300\;{{M}_{\odot }}$, whereas the median grows to $\approx 2000\;{{M}_{\odot }}$ for objects $\gtrsim 10$ kpc.

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The physical radius plot (middle left panel) shows similar effects, with the hashed regions marking ${{\theta }_{R}}\leqslant 10^{\prime\prime}$ (typical minimum angular radius) and ${{\theta }_{R}}\geqslant 150^{\prime\prime}$ (approximate maximum recoverable radius; see G13). The horizontal dashed lines represent the R = 0.125 pc and R = 1.25 pc divisions between the physical object categories discussed above. Finally, the number density distribution (bottom left panel) displays the expected trend, except that the median values remain somewhat constant between 500 cm⁻³ $\leqslant \;n\;\leqslant \;1000$ cm⁻³ over a large range of heliocentric distance. Because this flattening is not observed for physical radius (which also depends linearly on ${{d}_{\odot }}$), this range of n may represent a typical value for BGPS dense molecular cloud structures, which is subject to the geometric and cataloging caveats discussed above.

The right panels of Figure 5 illustrate the dependence of these physical properties on Galactocentric radius. The vertical dashed line identifies our adopted ${{R}_{0}}=8.34$ kpc (Reid et al. 2014) for reference. Black squares mark the median values in 1 kpc bins, and roughly show the systematic effects of heliocentric distance (i.e., objects far from R_gal = R₀ are at large ${{d}_{\odot }}$). Gaps in the black function represent bins with $N\leqslant 20$ sources, where the median value is skewed by outliers. The gap in sources (gray dots) at R_gal = 8.5–9.5 kpc is pointed out in EB15 as arising from a unique combination of the BGPS coverage region and a kinematic avoidance zone (where kinematic distances are unreliable in the face of noncircular motions about the Galactic center), and is not likely a true feature of the underlying Galactic dense molecular gas distribution. Many of the dense molecular cloud structures observed in the Molecular Ring / Scutum–Centarus Arm feature at R_gal = 4–5 kpc, ${{d}_{\odot }}$= 3–6 kpc, correspond the physical properties typical of molecular cloud clumps.

The systematic effects of the heliocentric distance will obscure some of the underlying physical trends in properties as a function of Galactocentric radius. For instance, the spike in median number density near R_gal = R₀ is due to the population of nearby (${{d}_{\odot }}$≲ 2 kpc) core-scale objects. The observed decrease in mass for dense molecular cloud structures from R_gal = 4 kpc to R_gal = 7 kpc is consistent with the trend observed for GMCs by Roman-Duval et al. (2010), but is likely primarily an observational bias.

The full information available from the MPDFs (and ultimately the DPDFs from EB15) allow for the utilization of a large number of Monte Carlo trials to fit functional forms to the dense molecular cloud structure mass distribution. For each trial, a single value was drawn from the MPDF of each object (Section 3.1.1) to create an independent realization of the mass distribution of the Distance Catalog. Both a power law (Section 3.3.1) and lognormal (Section 3.3.2) function were fit to each realization, and the fit parameters were recorded. This process was repeated N = 20,000 times, a value that was chosen to balance appropriately sampling the parameter spaces for the functional fits with computation time. The collected fit parameters from the Monte Carlo process were then plotted as two-dimensional histograms to identify joint confidence intervals. Below, we discuss the results of the power-law (Section 4.2.2) and lognormal (Section 4.2.3) fits to the entire data set. Attempts to subdivide the Distance Catalog into astrophysically meaningful subsets are described in the Appendix, but most fits resemble those of the entire sample. Maximum-likelihood fit parameter values for all fits are listed in Table 2.

Table 2.  Mass Distribution Functional Fit Parameter Values

				Power-law Fit			Lognormal Fit
Cut	Subseta			$\hat{\alpha }$	Mmin		$\hat{\mu }$	$\hat{\sigma }$	Mminb	Mmaxb
Type	Name	N			(${\rm log} \;{{M}_{\odot }}$)		(${\rm ln} \;{{M}_{\odot }}$)	(${\rm ln} \;{{M}_{\odot }}$)	(${\rm log} \;{{M}_{\odot }}$)	(${\rm log} \;{{M}_{\odot }}$)
	Distance Catalog	1710		$1.94_{-0.10}^{+0.34}$	$2.98_{-0.06}^{+0.46}$		$5.30_{-0.20}^{+0.15}$	$2.00_{-0.08}^{+0.16}$	$1.84_{-0.16}^{+0.08}$	$4.08_{-0.12}^{+0.24}$
${{d}_{\odot }}$	Nearby	1328		$2.01_{-0.09}^{+0.11}$	$2.79_{-0.19}^{+0.20}$		$4.85_{-0.08}^{+0.15}$	$1.72_{-0.02}^{+0.08}$	$1.84_{-0.12}^{+0.12}$	$4.14_{-0.24}^{+0.18}$
	Distant	382		$1.93_{-0.05}^{+0.19}$	$3.33_{-0.29}^{+0.01}$		$7.10_{-0.05}^{+0.05}$	$1.48_{-0.08}^{+0.08}$	$1.84_{-0.08}^{+0.12}$	$4.14_{-0.18}^{+0.24}$
Rgal	Sct-Cen Arm	866		$1.98_{-0.11}^{+0.09}$	$2.96_{-0.16}^{+0.18}$		$5.38_{-0.25}^{+0.18}$	$1.90_{-0.20}^{+0.12}$	$1.84_{-0.08}^{+0.12}$	$3.96_{-0.18}^{+0.30}$
	Sgr / Local Arms	610		$2.02_{-0.28}^{+0.02}$	$3.19_{-0.42}^{+0.04}$		$4.90_{-1.08}^{+0.88}$	$2.82_{-0.64}^{+0.52}$	$1.84_{-0.08}^{+0.12}$	$3.96_{-0.18}^{+0.18}$
	Sgr Arms	330		$1.97_{-0.24}^{+0.01}$	$3.17_{-0.35}^{+0.03}$		$4.90_{-1.08}^{+0.88}$	$2.82_{-0.64}^{+0.52}$	$1.84_{-0.08}^{+0.12}$	$3.96_{-0.18}^{+0.18}$
	Per / Outer Arms	234		$1.94_{-0.10}^{+0.03}$	$2.56_{-0.13}^{+0.04}$		$5.40_{-0.15}^{+0.25}$	$1.70_{-0.22}^{+0.06}$	$1.84_{-0.16}^{+0.12}$	$3.96_{-0.18}^{+0.30}$
R	Clump / Core	807		$1.90_{-0.06}^{+0.26}$	$3.24_{-0.44}^{+0.01}$		$5.32_{-0.32}^{+0.25}$	$2.12_{-0.18}^{+0.26}$	$1.84_{-0.16}^{+0.12}$	$4.14_{-0.24}^{+0.18}$
	Cloud	562		$1.86_{-0.24}^{+0.05}$	$2.90_{-0.59}^{+0.18}$		$5.25_{-0.55}^{+0.30}$	$2.22_{-0.24}^{+0.34}$	$1.84_{-0.16}^{+0.12}$	$4.14_{-0.18}^{+0.36}$
n	Clump / Core	993		$1.87_{-0.10}^{+0.17}$	$2.91_{-0.10}^{+0.41}$		$5.35_{-0.50}^{+0.15}$	$2.22_{-0.20}^{+0.26}$	$1.84_{-0.12}^{+0.12}$	$4.14_{-0.24}^{+0.12}$
	Cloud	376		$1.76_{-0.06}^{+0.18}$	$3.03_{-0.48}^{+0.02}$		$5.35_{-0.55}^{+0.27}$	$2.12_{-0.38}^{+0.26}$	$1.84_{-0.16}^{+0.12}$	$4.14_{-0.24}^{+0.24}$
Σ	Mean $\leqslant 120\;{{M}_{\odot }}$ pc−2	1444		$2.38_{-0.36}^{+0.03}$	$3.04_{-0.38}^{+0.03}$		$5.10_{-0.05}^{+0.12}$	$1.68_{-0.10}^{+0.04}$	$1.84_{-0.20}^{+0.12}$	$4.00_{-0.24}^{+0.30}$
	Mean $\geqslant 120\;{{M}_{\odot }}$ pc−2	358		$2.06_{-0.38}^{+0.02}$	$3.44_{-0.45}^{+0.04}$		$7.20_{-0.05}^{+0.10}$	$1.72_{-0.08}^{+0.18}$	$1.84_{-0.08}^{+0.16}$	$4.18_{-0.18}^{+0.30}$
	Peak $\leqslant 120\;{{M}_{\odot }}$ pc−2	392		$2.15_{-0.25}^{+0.01}$	$2.14_{-0.34}^{+0.01}$		$3.62_{-0.20}^{+0.85}$	$1.70_{-0.40}^{+0.02}$	$1.84_{-0.16}^{+0.12}$	$4.00_{-0.06}^{+0.30}$
	Peak $\geqslant 120\;{{M}_{\odot }}$ pc−2	1410		$1.93_{-0.08}^{+0.36}$	$3.47_{-0.50}^{+0.01}$		$5.85_{-0.12}^{+0.07}$	$1.80_{-0.08}^{+0.10}$	$1.84_{-0.12}^{+0.12}$	$4.18_{-0.24}^{+0.18}$
${{v}_{{\rm LSR}}}$ Type	Dense Gas	1372		$1.98_{-0.08}^{+0.31}$	$3.07_{-0.07}^{+0.36}$		$5.32_{-0.15}^{+0.15}$	$1.96_{-0.08}^{+0.14}$	$1.84_{-0.12}^{+0.12}$	$4.14_{-0.18}^{+0.24}$
	13CO	422		$1.97_{-0.08}^{+0.05}$	$2.59_{-0.17}^{+0.07}$		$4.55_{-0.28}^{+0.22}$	$1.72_{-0.04}^{+0.14}$	$1.84_{-0.20}^{+0.12}$	$3.96_{-0.06}^{+0.36}$
"Protocluster"	Blind ($\ell \leqslant 90{}^\circ$)	1513		$1.94_{-0.09}^{+0.34}$	$2.97_{-0.02}^{+0.47}$		$5.38_{-0.22}^{+0.15}$	$2.08_{-0.14}^{+0.16}$	$1.84_{-0.12}^{+0.12}$	$4.18_{-0.24}^{+0.12}$
	$R\geqslant 0.2$ pc	1331		$1.92_{-0.12}^{+0.11}$	$2.99_{-0.20}^{+0.19}$		$5.30_{-0.30}^{+0.18}$	$2.16_{-0.14}^{+0.22}$	$1.84_{-0.12}^{+0.12}$	$4.18_{-0.24}^{+0.12}$
	Mixed Protocluster	579		$1.92_{-0.07}^{+0.12}$	$2.79_{-0.13}^{+0.21}$		$6.45_{-0.12}^{+0.38}$	$0.66_{-0.10}^{+0.40}$	$2.12_{-0.12}^{+0.24}$	$2.98_{-0.24}^{+0.36}$
	$M\geqslant 2000\;{{M}_{\odot }}$	216		$2.23_{-0.06}^{+0.03}$	$3.45_{-0.04}^{+0.02}$		$8.38_{-0.02}^{+0.05}$	$0.82_{-0.12}^{+0.06}$	$1.80_{-0.04}^{+0.04}$	$4.30_{-0.12}^{+0.30}$

^aSee the Appendix for definitions of these subsets. ^bThe lognormal fitting routine tends to find mass bounds very near the initial guess (M_min =10² M_☉, M_max = 10⁴ M_☉), but parameter correlations show that the exact bounds have little impact on the [$\hat{\mu },\hat{\sigma }$] returned.

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The collected power-law fits to the complete mass distribution are shown in Figure 6. The left panel illustrates the mass distribution both as the complementary cumulative distribution function (CCDF, solid black) and as the differential mass distribution (gray histogram).²⁰ Both distributions represent the aggregate of 20,000 Monte Carlo realizations of a single mass distribution, normalized so the sum equals 1710 objects. To smooth the effects of small sample size and utilize the full amount of information encoded in the collected set of MPDFs, the aggregate is shown rather than any single one of the Monte Carlo trials. The power-law fits (derived from the data in the right panel) are shown for the CCDF (cyan line) and the differential mass distribution (black dashed line).

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The power-law fitting algorithm (PLFIT) returns two parameters, the power-law index $\hat{\alpha }$ and the minimum mass over which the data may be described by a power law (M_min). The two-dimensional histogram of the 20,000 parameter pairs from the Monte Carlo trials are shown in the right panel of Figure 6. Color scale intensity indicates the number of points in each pixel of the histogram, and contours enclose 68.3% (highest contour), 95.4%, and 99.7% of the parameter pairs, which are meant to illustrate joint confidence regions. The high degree of correlation between $\hat{\alpha }$ and M_min is indicative of the continuously steepening nature of the mass distribution shown in the left panel (where a power law is a straight line). The yellow $\sf {X}$ in the right panel marks the peak of the two-dimensional histogram, or the joint maximum-likelihood parameter values ($\hat{\alpha }=1.95,{{M}_{{\rm min} }}=966\;{{M}_{\odot }}$). This minimum mass is above the 90% completeness level described in Section 3.2.2 ($\approx 400\;{{M}_{\odot }}$), meaning the power-law fit is not affected by Malmquist bias, although other selection effects may be at play. Note the secondary peak of points near $\hat{\alpha }=2.3,{{M}_{{\rm min} }}=3000\;{{M}_{\odot }}$ in the right panel; there appears to be a sharp steepening of the mass distribution at the high-mass end (see Section 5.3.2 for discussion). The distribution truncates around ${{M}_{{\rm min} }}=3500\;{{M}_{\odot }}$ due to a requirement in the fitting algorithm for a minimum number of data points required for a robust fit (see Clauset et al. 2009). This truncation is a general feature of the astrophysical subset fits discussed in the Appendix.

The continuously steepening nature of the mass CCDF makes fitting or interpreting a power law function for the entire data set difficult at best. Furthermore, distance inhomogeneities in large-scale surveys such as the BGPS tend to drive the composite observed mass distribution toward a lognormal (see Section 5.3.3). We applied our lognormal fitting routine to the N = 20,000 Monte Carlo realizations of the mass distribution, and the resulting fit parameter distributions are shown in Figure 7. The CCDF and differential mass distributions in the left panel are identical to Figure 6 (left). The functional fits are drawn from the right panel as was done with the power law, where the cyan line fits the CCDF and the dashed black line fits the differential mass distribution.

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Whereas the power-law fit returns two parameters, the lognormal fitting optimizes the functional fit over four parameters: the lognormal characteristic [$\mu ,\sigma$], as well as both ends of the mass range [${{M}_{{\rm min} }},{{M}_{{\rm max} }}$] over which the lognormal fit is valid. For computational expediency, we divided the fitting into a nested structure following the recipe from Olmi et al. (2013, Appendix B). The lognormal characteristic is fit for a given pair of mass range values, then the mass range is adjusted to find the best match between the model CCDF and the CCDF constructed from the source masses. Shown in the right panel of Figure 7 are the distributions of lognormal best-fit parameters paired in this fashion. The [$\mu ,\sigma$] plane, shown in the background, is more fundamental, because these define the position and shape of the function. As with the power-law fit, the two-dimensional histogram of the 20,000 Monte Carlo trial fit parameters is shown, with the contours representing the same enclosed fractions of points. Both parameters are tightly constrained, with $\hat{\mu }=5.30_{-0.20}^{+0.15}$ ($160\;{{M}_{\odot }}\leqslant {{M}_{\mu }}\leqslant 230\;{{M}_{\odot }}$) and $\hat{\sigma }=2.00_{-0.08}^{+0.16}$, where the uncertainties represent the marginalized 68.3% confidence intervals. The joint maximum-likelihood values (marginalized over [${{M}_{{\rm min} }},{{M}_{{\rm max} }}$], yellow $\sf {X}$) occur at $\mu =5.30,\sigma =2.01$, and are reflected in the solid cyan and dashed black curves in the left panel. Since the peak occurs at ${{M}_{\mu }}={{e}^{5.30}}=200$ ${{M}_{\odot }}$, which is below the 400 ${{M}_{\odot }}$ 90% completeness level, it may be biased and should be viewed as an upper limit.

Inset in the right panel is the two-dimensional histogram of the [${{M}_{{\rm min} }},{{M}_{{\rm max} }}$] plane, which shows the relationship between the bounding parameters. There is scant correlation, except that the spread along the diagonal from top left to bottom right represents the expansion and contraction about a central value. The optimum values of these parameters (marginalized over [$\mu ,\sigma$]) are indicated by the yellow $\sf {X}$ in the inset (${{M}_{{\rm min} }}=70\;{{M}_{\odot }},{{M}_{{\rm max} }}=12,000\;{{M}_{\odot }}$), and are marked in the left panel by vertical gray dashed lines. This minimum mass extends down to the 50% mass completeness level ($\approx 70\;{{M}_{\odot }}$), and its location may not be entirely reliable because the peak of the fitted lognormal function always falls below the 90% completeness level ($\mu \lt 6.0={\rm ln} \;400\;{{M}_{\odot }}$).

Comparison of the lognormal fit in Figure 7 with the power-law fit in Figure 6 yields some interesting findings. As expected, the minimum mass for the power-law fit is well above the peak of the lognormal distribution. In fact, the 99.7% contour in Figure 6 (right) terminates at about this value. Furthermore, the curvature of the mass distribution is well-matched to a lognormal function over nearly 2.5 orders of magnitude. The continual curve of the mass distribution may be an effect of observational and cataloging biases against the highest mass sources (see Section 5.3.2).

Theories of star formation hold that a strong power-law behavior should be evident for dense molecular cloud structures, particularly leading toward the stellar IMF (Krumholz et al. 2005; Hennebelle & Chabrier 2011, 2013; Padoan & Nordlund 2011; Federrath & Klessen 2012). The lack of a clear single power law fit to the complete Distance Catalog prompted us to fit functional forms to astrophysically motivated subsets of the data to identify homogenous sets of objects predicted by theory. Whereas the fits to many of the subsets did not reveal any strongly compelling behavior (see the Appendix), there are two whose power-law fits are fairly well constrained. The set of nearby objects (${{d}_{\odot }}$ $\leqslant \;6.5$ kpc) may be described by an $\hat{\alpha }=2.0\pm 0.1$ power law, and the set of criteria aimed at isolating molecular cloud clumps (2 kpc $\leqslant$ ${{d}_{\odot }}$ $\leqslant$ 10 kpc and $M\geqslant 300$ ${{M}_{\odot }}$) is fit by a $\hat{\alpha }=1.9\pm 0.1$ power-law. The heliocentric distance cut removes cloud-scale objects that are more strongly affected by the angular filtering function of the BGPS data pipeline, leaving principally molecular cloud clumps that are similar to the mixed criteria.

The overall lack of clean fits to either the complete Distance Catalog or most of the subsets indicates that the sample is largely homogenous and the interstellar medium is not cleanly or tightly described by simple relations. These effects are discussed in depth in Section 5.3. The results of the studied astrophysical subsets are presented in the Appendix, along with the accompanying figures, which are similar to Figures 6 and 7. The fit results are all shown in Table 2.

Ellsworth-Bowers et al. (2015, their Figure 15) presented a map of the locations of dense molecular cloud structures in the Galactic disk with respect to a rendering of Galactic structure based on Spitzer data. Since these objects span several orders of magnitude in mass (Figure 4), their locations alone do not necessarily indicate the mass distribution of dense molecular gas. To estimate the Galactic disk mass surface density of the objects in the BGPS Distance Catalog, we constructed a grid consisting of 0.25 kpc × 0.25 kpc bins into which masses computed for individual sources were placed. Since typical distance uncertainties are ∼0.5 kpc, using 0.25 kpc bins approximately Nyquist samples the data. Taking advantage of the information contained in source DPDFs, we produced a map (Figure 8, left) by randomly drawing 10³ distances for each object, and adding the mass derived using Equation (3), divided by 10³ to reflect the probabilistic contribution of that mass to the appropriate grid cell. This use of the full DPDF leads to some of the smearing effects along the line of sight toward isolated regions in the outer Galaxy (e.g., $\ell \approx 110{}^\circ ,130{}^\circ$), where peculiar motions have a larger effect on kinematic distances. Furthermore, given that BGPS coverage for $\ell \gt 90{}^\circ$ is neither blind nor contiguous, a lack of mass in the outer Galaxy in Figure 8 should not be construed to imply a void in the Galactic plane.

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Mass surface density contours are plotted atop an image of the Galaxy produced using Spitzer near-infrared data (R. Hurt: NASA/JPL-Caltech/SSC) in the right panel of Figure 8. As with the corresponding map from EB15, there are many caveats in the interpretation of this image. First, there are virtually no BGPS sources with well-constrained distances at R_gal ≲ 4 kpc, owing to noncircular motions induced by the Galactic bar,²¹ with the exception of objects associated with trigonometric parallax measurements of maser transitions in regions of high-mass star formation (e.g., Reid et al. 2014). Second, given that the typical uncertainty for well-constrained distance estimates from EB15 is ∼0.5 kpc, the map and contours have a ∼1 kpc effective resolution, which is significantly coarser than the underlying image, although commensurate with Herschel maps of nearby galaxies (mean distance ≈10 Mpc; Kennicutt et al. 2011).

Third, there is a significant concentration of mass near $({{x}_{{\rm gal}}},{{y}_{{\rm gal}}})=(-4,3)$ kpc that appears to lie in a void in the underlying image. As discussed in EB15, this could represent either molecular gas at that location not identified in the Spitzer data (which is a "best guess" picture of the Galaxy), or sources in the W43 star-forming complex at the end of the Galactic bar erroneously placed at the far kinematic distance. Comparison of Figure 8 with the Galactocentric locations of GMCs in the ¹³CO Galactic Ring Survey (GRS; Jackson et al. 2006) reveals a similar large collection of gas at this location (Roman-Duval et al. 2010, their Figure 5). Most of the BGPS sources in this region are associated with one or more H ii regions with robust KDA resolution from the H ii Region Discovery Surveys (HRDS; Bania et al. 2010, 2012), and the distances to GRS clouds were primarily resolved using the same H i absorption techniques as the HRDS (EB15).

The BGPS-GRS correlation implies either the existence of a large collection of mass in an apparently empty region of the Spitzer image, or a fundamental limitation in the application of H i absorption techniques for distance resolution. Evidence in support of the former comes from both Russeil et al. (2011), who find interarm clumps for this region in the Hi-GAL $\ell =30{}^\circ$ SDP field, and a study of the M51 spiral galaxy, which reveals a population of GMCs downstream of a major spiral arm (Egusa et al. 2011). Furthermore, the appearance of interarm gas could also be the effect of a spiral feature possessing a systematic noncircular velocity, biasing kinematic distances.

Identified in the right panel of Figure 8 are the high-mass star-forming regions (SFRs) W43, W51, and W49, which stand out in the mass diagram as high surface density areas. Despite the rough tracing of some spiral structures, this map is not a definitive distribution of the dense molecular gas in the Milky Way. First of all, objects used to produce this map represent only 20% of the full BGPS catalog (by number, although 33% by flux density). Second, given the coarse resolution of the map, any conclusions about the detected spiral structure should be used cautiously because the map serves to show where the detected mass lies, and not a complete census of mass in the Galaxy. The gap observed between $60{}^\circ \leqslant \ell \leqslant 75{}^\circ$, however, is real and corresponds to a similar minimum in ¹²CO column seen in the Dame et al. (2001) maps. The expansion of robust distance determination to other dust-continuum surveys, however, will have high utility for tracing the Galactic dense molecular gas distribution across the entire disk.

The azimuthally integrated vertical and radial distributions of the mass of star-forming clouds provide an alternate view of Galactic star-formation distribution, and are shown in Figure 9. The Galactocentric radius and vertical position were computed using the $(\ell ,b,{{d}_{\odot }})\to ({{R}_{{\rm gal}}},\phi ,z)$ conversion matrix from Appendix C of Ellsworth-Bowers et al. (2013), which accounts for the 25 pc vertical solar offset above the Galactic midplane. As in Figure 8, masses and Galactocentric positions were derived from 10³ independent samples from the DPDF for each object. The color image at the center represents the amount of mass in each 0.25 kpc (radial) × 10 pc (vertical) bin. Because this is an azimuthal integration, the color scale intensity of the image is physically meaningless other than to identify overdense regions; pixels at larger R_gal represent a significantly larger volume than those nearer the Galactic center. Identified in the image are the W43, W51, and W49 regions, which stand out in the face-on image of the Galaxy. These three regions dominate the mass distribution, but there are still clear collections of mass in the 4–5 kpc range and around 6 kpc. The warping of the disk to large R_gal is visible in the mass distribution, but we caution that BGPS observations in the outer Galaxy ($\ell \gt 90{}^\circ$) were targeted toward regions of known star formation, and do not represent the same blind sample as the inner Galaxy coverage.

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To the right of the image is the marginalization over R_gal, or the the vertical histogram of the mass distribution, shown as total mass per bin. A Gaussian fit to this mass-weighted histogram ($\mu =-2.4\pm 1.8$ pc, FWHM = 63.7 ± 3.6 pc) is consistent with the distribution derived from the number-count histogram from EB15. This implies that the mass is roughly evenly distributed among catalog sources regardless of vertical position, and that there is no particular bias in the mass of dense molecular cloud structures near the Galactic midplane.

The physical properties of molecular clouds and their substructures appear to follow a series of scaling relationships known as Larson's Laws, following their introduction in a literature study of molecular clouds by Larson (1981). The most frequently used of these scaling relationships is the size-linewidth relationship, which holds that virial and/or turbulent motions across a molecular cloud structure increase with the square root of the size of that structure. This relationship is valid for single molecular line tracers (e.g., Solomon et al. 1987; Heyer et al. 2009), but the linewidths of different molecular transitions observed in the same dense molecular cloud structure may vary by an order of magnitude. The heterogenous nature of the spectroscopic data gathered for BGPS distance estimation therefore prevents their use in deriving such a relationship.

The first two Larson relationships (the size-linewidth and mass-linewidth relationships) may be combined to form a mass-size relationship, sometimes called Larson's Third Law.²² Using the physical radius $R=L/2$ rather than the overall length of the molecular feature, for consistency with the remainder of our analysis, we derive

$$\begin{eqnarray}&&\frac{M}{{{M}_{\odot }}}=460{{\left( \frac{R}{{\rm pc}} \right)}^{1.90}}\end{eqnarray} \tag{ 15 }$$

as the form of the Larson mass–radius relationship. This power-law relationship has been reproduced by simulations of isothermal supersonic turbulence (Kritsuk et al. 2013).

While Equation (15) appears to have theoretical backing, it is important to note the effect of observational systematics on the derived relationship. Combining Equations (2) and (5) leads to a systematic $M\propto {{R}^{2}}$ relationship, which has been advanced as the source of Larson's Third Law (e.g., Ballesteros-Paredes et al. 2012, and references therein). Owing to the similarity between Equation (15) and the systematic relationship, we investigate the deviation from $M\propto {{R}^{2}}$ for both BGPS sources and measurements of GMCs from the ¹³CO(1–0) data of the GRS.

To remove the effect of heliocentric distance, we compare the mean mass surface density (Σ) with the mass of each object. Figure 11 plots the 1,369 BGPS Distance Catalog objects with real ${{\theta }_{R}}$ as black circles, where the vertical axis is computed via Σ = ${{M}_{{\rm ML}}}$ $/\pi {{R}^{2}}$. The wide scatter in points is likely due in part to the non-uniform geometry of objects (as discussed earlier). To evaluate the correspondence of the relationship for the BGPS objects (i.e., clump-scale structures) with larger structures, we also plot the results of two studies of ¹³CO clouds from the GRS in Figure 11. GMCs studied by Roman-Duval et al. (2010, hereafter RD10) are shown as cyan squares, and the clouds containing BGPS sources from Battisti & Heyer (2014, hereafter BH14) are depicted with orange triangles. While both of these studies used the GRS ¹³CO data, they used different source identification algorithms, resulting in the differences shown in Figure 11. For each data set, we fitted a linear function in logarithmic space that is robust against outliers to the data as log Σ = log a + b log M,²³ and the resulting power-law fit values are listed in Table 3. The individual fits for each data set are shown in the same color as the data points. Furthermore, we also fit a power law through the entire collection of points (red) to span a further order of magnitude in each dimension than any one data set alone.

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Table 3.  Power-law Fit Values for Larson-like Surface Density–Mass Relationship

	${\Sigma }=a\;{{M}^{b}}$
Data Set	a	b	ρa
BGPS	$43_{-9}^{+12}$	0.15 (0.04)	−0.96
RD10	$29_{-9}^{+14}$	0.16 (0.04)	−0.98
BH14	$35_{-20}^{+50}$	0.15 (0.08)	−0.99
Combined	$46_{-6}^{+7}$	0.12 (0.02)	−0.95
Larson (1981)	202	−0.053	⋯

^aParameter correlation coefficient $\rho ={{\sigma }_{{\rm ab}}}/{{\sigma }_{{\rm a}}}{{\sigma }_{{\rm b}}}$.

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Casting Equation (15) in the same form as the data of Figure 11 yields

$$\begin{eqnarray}&&\frac{{\Sigma }}{{{M}_{\odot }}\;{\rm p}{{{\rm c}}^{-2}}}=202{{\left( \frac{M}{{{M}_{\odot }}} \right)}^{-0.053}},\end{eqnarray} \tag{ 16 }$$

indicating a nearly flat mean mass surface density as a function of mass, and is shown as the dotted-dashed gray line in Figure 11. The slope is so shallow as to cause only a factor of two difference in Σ over some six orders of magnitude of mass.

Strikingly, the data from both continuum (BGPS) and ¹³CO (RD10, BH14) observations consistently show a trend of increasing Σ with object mass, with the fits to the three individual sets returning nearly identical power-law indices. The similarity of these fit values indicates no significant difference between physical properties computed from the dust-continuum and molecular transition line observations of dense molecular cloud structures. These fit values significantly differ from the value in Equation (16). Although contrasting with the Larson slope, these fits still represent less than an order of magnitude change in Σ over six orders of magnitude in mass.

Caution should be taken when drawing conclusions from this result. While the intrinsic scatter within each data set is large, none of the data sets could be consistent with the predicted Larson relationship ($b=-0.053$) within the fit uncertainties. However, both the Larson relationship and the fits shown in Table 3 deviate only slightly from the systematic M ∼ R². The fact that the properties of dense molecular cloud structures from three separate studies all reveal nearly the same, slight departure from the systematic b = 0 indicates a real, although slight, trend in mass surface density as a function of mass. In its simplest interpretation, this relation seems to imply that higher-mass GMCs are also denser and therefore could possibly form stars more efficiently. This interpretation must be tempered, however, by the inability of current survey data-reduction techniques and cataloging algorithms to accurately represent large-scale, low-surface-brightness structures in the interstellar medium.

The continuum of structure within molecular cloud complexes is a fundamental property whose connection with the intimate processes of star formation requires further study. Much of the literature discusses objects as clouds, clumps, or cores, and assigns specific properties and physical significance to each (e.g., Bergin & Tafalla 2007; McKee & Ostriker 2007; Kennicutt & Evans 2012, and references therein). As seen in Figure 11, however, observed dense molecular cloud structures exhibit a continuum of properties across many orders of magnitude in scale; here we discuss the implications of this continuum.

The near uniformity of the mean mass surface density of dense molecular cloud structures provides a crucial postulate for the construction of the Galactic distribution of dense molecular gas. Whether an artifact of observational systematics or a consequence of supersonic turbulence (e.g., Hopkins 2012), the observed uniform relationship indicates that the fractal quality of the molecular interstellar medium is perhaps more fundamental and a stronger driver of molecular cloud complex evolution than the individual classes of objects (i.e., clouds, clumps, cores) working independently.²⁴ There exist real physical divisions as the fractal nature of molecular clouds is driven by the the dominance of turbulence (Kritsuk et al. 2013); regimes dominated by other physical processes (i.e., gravitational collapse) obey other scaling relationships.²⁵

Aside from questions about the evolution of the ISM, what are the implications of these scaling relations on observational data? As a basic reference point, we start from the Larson result of Equation (15) and convert the physical properties to observable quantities via Equations (3) and (5). The resulting relationship between observed flux density, deconvolved angular radius, and heliocentric distance is given by

$$\begin{eqnarray}&&\left( \frac{{{S}_{1.1}}}{{\rm Jy}} \right)=3.34{{\left( \frac{{{\theta }_{R}}}{{\rm arcmin}} \right)}^{1.90}}{{\left( \frac{{{d}_{\odot }}}{{\rm kpc}} \right)}^{-0.10}}.\end{eqnarray} \tag{ 17 }$$

This equation implies that for dense molecular cloud structures, the observed flux density is nearly independent of heliocentric distance, and depends only on the angular radius on the sky, which is the same as saying that the mass surface density is independent of mass (as seen in Figure 11). This relationship is plotted in Figure 3 (top) as green lines for ${{d}_{\odot }}$= 1 kpc (solid), 5 kpc (dotted), and 20 kpc (dashed), and matches the slope of data quite well. Therefore, in maps from (sub-)millimeter continuum surveys, the physical scales of the detected objects are indistinguishable, meaning that a core looks like a clump, which looks like a cloud. It is only through robust distance estimation or application of the size-linewidth relationship that the scales of molecular cloud structures may be interpreted. This is an important point because the use of ancillary data is essentially the only means for deriving physically meaningful results from these surveys.

In EB15, it was noted that the surface brightness (SB) distributions for the full BGPS catalog and various subsets (including the Distance Catalog) were largely independent of the cataloged flux density. Indeed, by combining the SB equation from EB15 (their Equation (3)) with Equation (17), we obtain

$$\begin{eqnarray}&&{\rm SB}=13.3\ {\rm MJy}\;{\rm s}{{{\rm r}}^{-1}}{{\left( \frac{{{S}_{_{1.1}}}}{{\rm Jy}} \right)}^{-0.05}}{{\left( \frac{{{d}_{\odot }}}{{\rm kpc}} \right)}^{-0.105}},\end{eqnarray} \tag{ 18 }$$

which indicates that the SB of dense molecular cloud structures, regardless of size scale and heliocentric distance, should be nearly identical. Indeed, Solomon et al. (1987) comment that for the standard ${{\sigma }_{v}}\propto {{R}^{0.5}}$ size-linewidth relationship, the virial equilibrium translates into constant average surface density, and hence constant average SB. The observed SB along any line of sight is then proportional to the number of structures, or total mass, along that line of sight. If one assumed an axisymmetric model and sampled a large fraction of the plane, the radial dependence (and normalization) of the molecular gas along the line of sight could be solved for.

The aggregate mass distribution shown in the left panels of Figures 6 and 7 represents an inhomogenous set of dense molecular cloud structures across the Galactic plane. The entire collection of sources appears to be fitted well by a lognormal function over more than two orders of magnitude, and displays one or two power laws at the high-mass end. Prompted by theoretical predictions for a power-law form of the so-called clump mass function (e.g., Donkov et al. 2012; Veltchev et al. 2013), we made a series of astrophysically motivated cuts (see the Appendix) in an attempt to isolate a more-homogenous population of sources.

While the classifications of molecular cloud versus clump versus core are intuitively appealing, the mass distributions of the BGPS sources divided into core/clump and cloud groups (Table 2) return functional fits very similar to the aggregate set. Three possible explanations exist, likely working in concert: (1) the dividing lines between object classes are somewhat arbitrary, (2) uncertainties in the heliocentric distances mix objects across classification boundaries, and (3) the distinctions themselves are artificial in light of the continuum of structure in molecular cloud complexes. It is likely that a survey with improved angular resolution may be able to disentangle mass function parameters as a function of source classification or physical properties. Furthermore, use of multi-band Hi-GAL data will obviate the need to assume a dust temperature (or to marginalize over the dust temperature, as was done with the Monte Carlo trials), eliminating a source of uncertainty in derived properties.

The best power-law fits were for subsets primarily limited in heliocentric distance, and are characterized by nearly symmetric [$\hat{\alpha },{{M}_{{\rm min} }}$] joint confidence intervals (similar to the right panel of Figure 6). These subsets were the "nearby" heliocentric distance and "mixed protocluster" cuts, which exclude the most distant objects, with the latter additionally excluding nearby, low-mass objects. While both are strongly power-law, the nearby subset has a steeper power-law index ($\hat{\alpha }=2.04\pm 0.09$ versus $1.92_{-0.06}^{+0.13}$), although the two are marginally consistent with each other.

The lack of a strong power-law fit for the distant sample or the two cloud-scale subsets (in R and n) may be partially caused by the apparent steepening in the mass distribution at high mass, as discussed below. The subsets that do exhibit power-law structure, however, indicate that the presence of a true underlying power law (if present) is not obscured and converted to a lognormal by sample inhomogeneity (see Section 5.3.3).

One curious aspect of the mass function fits is the propensity for the high-mass end to display a fairly steep power-law tail (α ≈ 2.2–2.3). While this power-law index is similar to the high-mass end of the Salpeter (1955) IMF, objects detected by the BGPS in this mass range are generally cloud-scale in physical radius. Since Galactic GMCs tend to follow a power-law mass function with α ≈ 1.6–1.8 (Blitz 1993), the present result is somewhat contradictory.

A steep power-law function (i.e., $\alpha \gt 2$) implies that most of the total mass is in low-mass objects and that high-mass objects are rarer than more shallow functions (i.e., there are fewer per mass bin; Equation (9)). Could the apparent steepening be caused by missing high-mass sources in the present Distance Catalog? Simple simulations show that removal of sources from a shallow power-law distribution does have the effect of imitating a steeper power law, given that a progressively larger fraction of sources are removed with increasing mass.

Between the atmospheric subtraction in the BGPS data-reduction pipeline removing emission on large angular scales (≳300''; G13) and the tendency of the cataloging routine to break up large-scale emission into smaller chunks based on intermediate peaks in the flux-density distribution, the net effect is to remove the largest scale objects and convert single large structures into multiple smaller ones. What effect do these have on the derived mass distribution? Converting the mass-size scaling relationship of Equation (15) into observational quantities via Equation (5) yields

$$\begin{eqnarray}&&M=43.8\;{{M}_{\odot }}{{\left( \frac{{{\theta }_{R}}}{{\rm arcmin}} \right)}^{1.90}}{{\left( \frac{{{d}_{\odot }}}{{\rm kpc}} \right)}^{1.90}}.\end{eqnarray} \tag{ 19 }$$

The effective ${{\theta }_{R}}\lesssim 2\buildrel{\,\prime}\over{.} 5$ limit of the BGPS means that clouds at ${{d}_{\odot }}$= 5 kpc (near the peak of the heliocentric distance distribution) with $M\gtrsim 5000$ ${{M}_{\odot }}$ should be either truncated or broken into multiple smaller sources. At a distance of 10 kpc, the filtering affects sources with $M\gtrsim 2\times {{10}^{4}}$ ${{M}_{\odot }}$. Indeed, in the Physics Catalog of Table 1, there are only seven objects with ${{M}_{{\rm ML}}}$ above this value. In terms of the mass distribution, the breaking up of large structures into smaller ones adds objects to the low-mass end of the mass distribution, steepening the power law. The power-law indices presented in Table 2 should, therefore, be viewed as upper limits. Accurate measurement of the high-mass power-law distribution of dense molecular cloud structures, therefore, requires retaining some amount of large-scale structures in survey images and catalogs.

Observationally, the aggregate mass distribution, as shown in Figure 6 (left), should be expected to be well fitted by a lognormal function with a power-law tail regardless of the actual underlying distribution of source mass. Given the finite angular resolution and flux-limited nature of the BGPS (and similar surveys), it is possible to simulate the observed mass distribution of an inhomogenous sample spread across the Galactic plane.

We conducted a series of Monte Carlo trials whereby we observed 100 SFRs whose distances were drawn from the ${{d}_{\odot }}$ distribution of the Distance Catalog. Each SFR consisted of 10³ objects drawn purely from either a power-law (Trial 1) or lognormal (Trial 2) mass distribution. A physical radius was assigned using Equation (15) to estimate whether any given object would be resolved by the BGPS, and the extent to which its flux density (computed from the mass) would be diluted in the beam. Using an average BGPS noise level, sources that would not be detected (${{S}_{\nu }}\lt 3\sigma$) were culled from the list. The aggregate distribution of detected sources was recorded, and the process was repeated for N = 100 Monte Carlo realizations.

The results of Trial 1, where the underlying power-law mass distribution was allowed to range over $1.8\leqslant \alpha \leqslant 2.2$ with ${{M}_{{\rm min} }}=1$ ${{M}_{\odot }}$, are shown in the left panel of Figure 12. The aggregate shows a strong power-law tail with $\alpha =1.9$ and a lognormal distribution at lower mass, peaking at ${{M}_{\mu }}\approx 15\;{{M}_{\odot }}$. While the location of the lognormal peak is well below the completeness levels of the BGPS data set, these simple simulations do not account for real features in the data, such as source confusion and hierarchical (fractal) structure, and likely overestimate the number of low-mass sources visible. Trial 2 (right panel) constructed individual lognormal mass distributions using the ranges $4.5\leqslant \mu \leqslant 6.5$ and $1.5\leqslant \sigma \leqslant 2.5.$ The resulting aggregate mass distribution shows a steeper ($\alpha \approx 2.3$) power-law tail at high mass, but has a lognormal fit that is much closer to that of the full BGPS set in Figure 7. These simulations are not meant to definitively show the form of the underlying mass distributions, but rather to demonstrate the inevitability of a lognormal distribution with a power-law tail at high mass for inhomogenous data sets such as the BGPS, regardless of the underlying source distribution. The inhomogeneity should be reduced in the astrophysically motivated subsets (Appendix), hence we do not necessarily expect them to approach lognormal distributions with muted high-mass power laws.

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There has been much discussion in the literature about dense molecular cores and their relationship with the stellar IMF (see Shirley et al. 2003), but it is only with the recent advent of large-scale Galactic plane surveys that large, statistically robust samples of clump-scale objects have become available. For comparison with the present results, we discuss the recent studies of clump-scale objects by Olmi et al. (2013) from the science demonstration fields from the Herschel Hi-GAL survey (Molinari et al. 2010b) and by Wienen et al. (2015) from the $\lambda =870$ μm ATLASGAL survey (Schuller et al. 2009).

Turning first to the Olmi et al. (2013) analysis, direct comparison with the BGPS results is not fully straightforward, as the source-finding algorithm used by those authors is based on a Mexican Hat Wavelet (e.g., Barnard et al. 2004) and typically finds structures on scales that are smaller than those identified in the BGPS catalog (which used a seeded-watershed algorithm). Furthermore, Hi-GAL sources were identified as well-separated sources visible in each of the $\lambda =70$, 160, and 250 μm bands; the beam size at 250 μm is 17'', which is approximately half the width of the BGPS 33'' beam. These two features imply that the Olmi et al. data set will find more, smaller sources than the BGPS.

Comparing source lists in the $\ell =30{}^\circ$ field, the BGPS V2 catalog identifies 492 sources, whereas Olmi et al. identify 1950 sources in the same longitude range. Although the Hi-GAL latitude range extends twice as far as the nominal BGPS coverage, the bulk of the Herschel-measured emission lies within the boundaries of the BGPS mosaics. For the $\ell =59{}^\circ$ field, the differences are much more astounding; the BGPS catalog contains 17 sources, whereas Olmi et al. find 3402. A visual comparison of the data sets reveals that the Hi-GAL tile contains significant filamentary structures at modest flux density, whereas the BGPS mosaic exhibits nearly uniform noise, indicating that atmospheric removal has obliterated much of this low-level signal. The vast majority of the Hi-GAL sources in this tile lie along these filaments (see Molinari et al. 2010a, their Figure 3). Careful comparison of Hi-GAL and BGPS maps, however, shows that much of what appears to be atmospheric 1/f noise actually corresponds to the cirrus-like filamentary structure made choppy by the filtering.

With these differences between the two data sets in mind, we compare the power-law fits to the mass distribution of sources. Since Olmi et al. (2013) use a uniform distance for each of the two fields ($\ell =30{}^\circ$ is assigned a distance of 7.6 kpc and $\ell =59{}^\circ$ is assigned to 3.6 kpc), it is possible to compare their mass function fits to our subsets in heliocentric distance. Taking first the $\ell =30{}^\circ$ field, the large distance assigned to the Hi-GAL sources corresponds to the set of distant BGPS objects (${{d}_{\odot }}$ $\;\gt \;6.5$ kpc; see Appendix A.1). While our sample is not fitted well by a power-law, the maximum-likelihood parameters are $\hat{\alpha }=1.93_{-0.05}^{+0.19}$ for $M\gtrsim 2000$ ${{M}_{\odot }}$. This is in comparison to the steeper Olmi et al. fit values $(\hat{\alpha },{{M}_{{\rm min} }})=(2.15,212\;{{M}_{\odot }})$. The lognormal fit presented by those authors has a peak near ${{M}_{\mu }}={{e}^{4.58}}=100\;{{M}_{\odot }}$, which is somewhat smaller than the fit to the entire BGPS set (Table 2), following the trend of their power-law fit.

The $\ell =59{}^\circ$ field is placed at a distance corresponding to our nearby sample in heliocentric distance (${{d}_{\odot }}$ $\;\lt \;6.5$ kpc). The nearby sample is fitted well by a power-law with $\hat{\alpha }=2.01_{-0.09}^{+0.11}$ for $M\gtrsim 620$ ${{M}_{\odot }}$. As with the $\ell =30{}^\circ$ field, Olmi et al. (2013) find a steeper power-law that extends to lower mass $(\hat{\alpha },{{M}_{{\rm min} }})=(2.20,7.3\;{{M}_{\odot }})$. Because this power-law fit extends to such low mass, it is tempting to conclude that this Hi-GAL tile is imaging core-scale rather than clump-scale objects. However, since sources must be identified in the $\lambda =250$ μm images whose angular resolution is ≈25'', this corresponds to a minimum observable size of $\approx 0.4$ pc at the assigned 3.6 kpc distance.

The more recent Wienen et al. (2015) analysis of clump-scale dense molecular cloud structures in the ATLASGAL data provides a more direct comparison set with the BGPS Distance Catalog. Whereas the Olmi et al. results derive from two specific regions, ATLASGAL sources span the entire inner Galactic plane ($-60{}^\circ \leqslant \ell \leqslant 60{}^\circ$). As such, it suffers from the same spread in heliocentric distance (and hence resolving power) as the BGPS, and sweeps up objects across the spectrum of molecular cloud structures. We compare the BGPS Distance Catalog distribution with the ATLASGAL sources from the first Galactic quadrant, which is a largely overlapping population. Wienen et al. (2015) note that only a 70% distance matching rate (within 2 kpc) exists between their distance assignments and those from EB15. This relatively large disagreement seems to arise primarily from the fraction of objects placed at the far kinematic distance (44% of ATLASGAL sources versus 22% of BGPS sources). Those authors dismiss the significance of the distance mismatch by claiming that distance errors average out by virtue of the large ATLASGAL sample size. However, if the population of true distances is similar for the two surveys, the larger fraction of ATLASGAL sources placed at the far kinematic distance would systematically shift the mass distribution and affect the mass function fits. While a detailed distance catalog comparison is beyond the scope of this paper, the significant overlap between the ATLASGAL and BGPS survey data merits a careful comparison of source identification schemes, derived distances, and implications for mass distributions.

Notwithstanding the discrepancies between the two distance catalogs, differences in source properties arise from the angular transfer function of the ATLASGAL data, which ranges from the beam-scale 192 to a maximum recoverable scale of ≈150'' (Schuller et al. 2009). This beam size is $\approx 60\%$ that of the BGPS, meaning that smaller-scale structure is resolved in the ATLASGAL mosaics. The corresponding physical size at the 4 kpc heliocentric distance peak in the source distribution is $\approx 0.4$ pc, meaning that the source catalog is still dominated by clump-scale objects (Bergin & Tafalla 2007). On the large-scale end, the ATLASGAL data-reduction pipeline effectively removes uniform emission on scales larger than $\sim 2\buildrel{\,\prime}\over{.} 5$, whereas the BGPS angular transfer function is $\gtrsim 0.8$ for scales smaller than ∼5' (G13), leading to the recovery of larger objects.

The aforementioned differences may play a role in the comparison between the mass function fits described in Section 4.2 and those presented by Wienen et al. (2015). While those authors experimented with fitting power laws to binned data, we compare only the results obtained using the maximum-likelihood method of (Clauset et al. (2009). A power-law function fit to the entire ATLASGAL source collection yields $\hat{\alpha }=1.82\pm 0.02$, which is somewhat flatter than our value of $\hat{\alpha }=1.94_{-0.10}^{+0.34}$. The source of the difference is unclear, unless there exist sources of significant mass in the ATLASGAL catalog within the R_gal = 4 kpc kinematic avoidance zone (EB15) for which kinematic distances are not computed for BGPS sources. Furthermore, the very small error bars reported by those authors conflict with the wider spread in the power-law index resulting from the Monte Carlo realizations of the BGPS mass distribution shown in Figure 6, and likely do not take into account the effects of distance uncertainty. The only subset of the data to which Wienen et al. fit mass functions is the cut 2 kpc $\leqslant$ ${{d}_{\odot }}$ $\leqslant$ 5 kpc, encompassing the peak of the heliocentric distance distribution. For this subset, they find a power-law index of $\hat{\alpha }=2.26\pm 0.05$, which is somewhat steeper than either our set of nearby sources (${{d}_{\odot }}$ $\leqslant \;6.5$ kpc; $\hat{\alpha }=2.0\pm 0.1$) or the mixed protocluster criterion (2 kpc $\leqslant$ ${{d}_{\odot }}$ $\leqslant$ 10 kpc, $M\geqslant 300$ ${{M}_{\odot }}$; $\hat{\alpha }=1.9\pm 0.1$). One possible explanation for the steeper ATLASGAL fits is the filtering of large-scale emission into smaller sources due to the angular transfer function (Section 5.3.2). Also included in the Wienen et al. (2015) analysis is a lognormal fit to this distance subset, yielding a peak near 700 ${{M}_{\odot }}$, which is considerably larger than the lognormal fits to BGPS Distance Catalog sources (Section 4.2.3).

For both these comparison studies, a better understanding of the discrepancies between the mass function fits will come when all data sets are analyzed using the same cataloging method to remove systematic effects introduced by source identification. Furthermore, a more detailed study of the filtering effects of the data pipeline angular transfer functions on source identification within the large-scale structure may provide a more solid basis of comparison between results from different (sub-)millimeter Galactic plane surveys.