THE MASS–SIZE RELATION FROM CLOUDS TO CORES. II. SOLAR NEIGHBORHOOD CLOUDS

2.1.1. Basic Map Analysis

Our basic analysis approach is summarized in Section 2.1 of Paper I, and illustrated in Figure 1 of the same paper. In essence, starting from a set of local maxima, we contour a given column density map at all levels possible. For every contour, we derive the enclosed mass and area, A. Following the terminology of Peretto & Fuller (2009), we define "cloud fragments" in the maps as such regions enclosed by a continuous column density contour. The area is used to derive effective radii,

$$\begin{equation} r = (A / \pi)^{1/2} \,. \end{equation} \tag{ 2 }$$

Subsequent contours are usually nested in the map. This defines a relation between measurements. This essentially yields series of mass–size measurements. In the plots shown in this paper, such series are drawn using continuous lines (e.g., Figure 1).

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In practice, the processing is implemented using the dendrogram map analysis technique introduced by Rosolowsky et al. (2008). As a bonus, this also yields diagnostic diagrams on the cloud hierarchy. We do not use this feature in the present paper, though.

In our analysis, we reject all cloud fragments that have a diameter (i.e., 2r) smaller than the map resolution. Further, we require a minimum contrast between the local maxima used to seed the contouring. Here, this limit is set to the noise level times a factor 3. Further, we do not characterize column densities below a certain threshold. As discussed below, Table 1 lists these parameters for each map studied here.

Table 1. Data Included in This Study

Region/Distance	Data	Resolution	Noise	Rejection Threshold	Reference
			(mag)	(mag)
Clouds only forming low-mass stars
Taurus	Extinction	⩽6'	0.4	2.0	Rowles & Froebrich (2009)
d = 140 pc	Dust emission (10.0 K)	20''	1.4	4.3	Kauffmann et al. (2008)
Perseus	Extinction	5'	0.4	2.0	Ridge et al. (2006)
d = 260 pc	Dust emission (12.5 K)	30''	1.4	4.3	Enoch et al. (2006)
Ophiuchus	Extinction	5'	0.6	2.0	Ridge et al. (2006)
d = 120 pc	Dust emission (12.5 K)	30''	2.5	7.6	Enoch et al. (2007)
Pipe Nebula	Extinction	60''	0.5	4.0	Lombardi et al. (2006)
d = 130 pc
B59	Extinction	20''	1.2	3.6	Román-Zúñiga et al. (2009)
d = 130 pc
B68	Dust emission (10.0 K)	15''	1.1	3.3	J. Alves (2010, private communication)
d = 130 pc
Clouds also forming high-mass stars
Orion A	13CO	17	⋅⋅⋅	⋅⋅⋅	Bally et al. (1987)
d = 414 pc	Dust emission (20.0 K)	14''	⋅⋅⋅	⋅⋅⋅	Nutter & Ward-Thompson (2007)
G10.15 − 0.34	Dust emission (20.0 K)	14''	6.4	19.2	M. A. Thompson et al. (2010, in preparation)
d = 2100 pc
G11.11 − 0.12	Dust emission (15.0 K)	14''	6.6	19.7	Carey et al. (2000)
d = 3600 pc	Dust emission (15.0 K)	4''	4.9	14.6	T. Pillai et al. (2010, in preparation)

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2.1.2. Calculation of Mass–Size Slopes

Paper I demonstrates that mass–size data can conveniently be characterized using power laws of the form

$$\begin{equation} m(r) = m_0 \, (r / {\rm pc})^b \,. \end{equation} \tag{ 3 }$$

These are characterized by slope, b, and intercept, m₀. As shown in Paper I, slopes can be derived using various methods. As demonstrated in Paper I, b ⩽ 2 in our work, since b > 2 would imply an increase of the mean column density with radius.

We define global slopes to capture trends between small and large spatial scales (Section 4.3 in Paper I). To calculate these, we derive the maximum fragment masses, m_max, observed at radii of r_sm = 0.05 pc and r_lg = 5.0 pc. (These radii are chosen to permit comparison between clouds, as becomes more obvious below.) Based on these, we derive the global slope,

$$\begin{equation} b_{\rm glob} = \frac{ \ln [m_{\rm max}(r_{\rm lg}) / m_{\rm max}(r_{\rm sm})] }{ \ln [r_{\rm lg} / r_{\rm sm}] } \,. \end{equation} \tag{ 4 }$$

As illustrated in Figure 1, this slope is defined such that $m(r) \propto r^{b_{\rm glob}}$ connects the mass and size measurements at 0.05 pc and 5.0 pc radius (for appropriate intercept).

To characterize trends at a given spatial scale, we use tangential slopes (Section 4.4 in Paper I). These are derived infinitesimally at a given radius,

$$\begin{equation} b(r) = \left. \frac{{d} \ln (m[r^{\prime }])}{{d} \ln (r^{\prime })} \right|_{r^{\prime } = r} \,. \end{equation} \tag{ 5 }$$

As illustrated in Figure 7 of Paper I, the measured slopes are smoothed and filtered to improve the data quality.

2.1.3. Reference Mass–Size Relations

Paper I describes, in detail, how dust extinction and emission maps can be used to derive column density maps for molecular cloud complexes. Once calibrated to a common mass conversion scale (Section 4.2 of Paper I), dust extinction and emission data for a given cloud can be combined to probe the masses and sizes across a vast range of spatial scales. In Paper I, we did this for Perseus only. Here, we combine our Perseus data with maps of other clouds. We do this by repeating the analysis already carried our for Perseus.

We start with a survey of a sample of well-studied nearby clouds that are essentially devoid of high-mass stars. Besides the Perseus molecular cloud, those in Taurus, Ophiuchus, and the Pipe Nebula are examined here. As a general reference to clouds also forming high-mass stars, we include the Orion A cloud. Then, the more remote G10.15 − 0.34 (hereafter G10; ∼2.1 kpc) and G11.11 − 0.12 complexes (hereafter G11; ∼3.6 kpc) are studied to build a first bridge toward the study of relatively distant sites of high-mass star formation (see Pillai et al. 2007 for distances and references; G10 is further discussed by Wood & Churchwell 1989 and Thompson et al. 2006, while Pillai et al. 2006b study G11). Note that G11 is an Infrared Dark Cloud (IRDC; see Menten et al. 2005 and Beuther et al. 2007 for reviews).

A data summary for our sample is provided in Table 1. This includes parameters used for the source extraction.

2.2.1. Data and Analysis

2.2.2. Mass Estimates

As a first characterization of the cloud sample, we look at the maximum radius-dependent mass of cloud fragments. We begin with clouds not forming massive stars. In the 0.01 ≲ r/pc ≲ 10 radius range, essentially all fragments in such clouds have a mass smaller than some limiting law,

$$\begin{equation} m(r) = 870 \, M_{\odot } \, (r / {\rm pc})^{1.33} \,. \end{equation} \tag{ 6 }$$

Figure 2 gives an illustration. This limit thus gives the typical mass range for structure in clouds like Taurus, Perseus, Ophiuchus, and the Pipe Nebula.

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In detail, this relation excludes a bright fragment in NGC 1333, which we remove from our analysis as a possibly unphysical outlier (its estimated mass exceeds those of other fragments of similar size by a factor 2, possibly because of neglected protostellar heating). The limiting law is derived by searching for the smallest intercept for which m₀(r/pc)^b does still provide an upper limit to the data. Practically, this is done by varying b until max [m(r)/(r/pc)^b] is minimized.

Interestingly, our sample clouds with massive star formation exceed the limiting relation (Equation (6)). This is shown in Figure 2(b). For the Orion A cloud and G10, we derive an excess of up to a factor 10 in the 0.01 ≲ r/pc ≲ 2 radius range. This suggests that these clouds have a structure significantly different from what is found for clouds not containing such stars (Figure 2(a)). In this light, Equation (6) may approximate a limit for massive star formation: it could be that only clouds also containing fragments that exceed Equation (6) are able to form massive stars. Larger samples of clouds forming massive stars must be screened to prove this point. Note, however, that most fragments in Orion A do fulfill Equation (6). This cloud does therefore also contain objects that have masses and sizes not distinguishable from those found for clouds not forming massive stars.

As a mass–size limit for clouds not forming massive stars, Equation (6) is probably uncertain to just a few 10%. If we, e.g., not use Ophiuchus data in the derivation of Equation (6), then Ophiuchus would exceed the resulting mass–size limit by 30%. If we do the same with Perseus data, the excess is 15%. It is plausible to expect similar changes for other regions not forming massive stars. Observational uncertainties are most likely of a similar order, if one adopts the mass measurement techniques used here (i.e., dust emission and extinction). Then, several uncertainties (e.g., dust opacities) are simply removed by calibration to the same standard. In an absolute sense (i.e., when considering the true masses), Equation (6) is as accurate as the mass conversion standards used here (e.g., relation of dust emission and mass). These are probably uncertain to less than a factor 2.

We stress that, excluding observational uncertainties, Equation (6) provides a strict upper mass limit to the solar neighborhood sample as we have defined it here. Clouds violating Equation (6) are not similar to the clouds in the solar neighborhood sample discussed here. This statement is sufficient for many purposes.

Inspection of Figures 1 and 2 suggests that all solar neighborhood clouds have similar masses at given size for radii ≳1 pc. Specifically, the most massive fragments at given radius are essentially all within ±30% of

$$\begin{equation} m(r) = 400 \, M_{\odot } \, (r / {\rm pc})^{1.7} \end{equation} \tag{ 7 }$$

in the 1 ⩽ r/pc ⩽ 4 radius range, when considering Taurus, Ophiuchus, Perseus, and the Pipe Nebula. Also, our preliminary Orion A data is, at 11 pc radius, within 40% of this law—with the caveat that we cannot examine whether Orion A follows Equation (7) down to ∼1 pc radius. It thus appears that—with Orion A as a possible exception—all local clouds are similar in their large-scale mass structure.

This mass–size relation is very similar to the one originally derived by Larson (1981), m(r) = 460 M_☉(r/pc)^1.9 (see Equation (1) above). Our study thus confirms his result—but only for spatial scales ≳1 pc. As we show throughout this paper (e.g., Equation (6) and Figures 4 and 5), no single mass–size relation describes all structural aspects of our observational data.

Some cloud fragments ≳1 pc are, however, much more massive than solar neighborhood clouds of similar size. G10, for example, violates Equation (7) by about an order of magnitude. The similarity between local clouds implied by Equation (7) might thus only hold for the solar neighborhood.

It is obvious that the formation of a cluster requires a larger mass reservoir than necessary to form a single isolated star. Thus, one might naively expect that regions containing clusters are more massive than those devoid of stellar groups. If true, one would thus expect that, within a given cluster-forming cloud, the regions containing clusters are more massive than cluster-less regions of similar size. This hypothesis is tested in Section 3.3.1. Also, one would expect that cluster-forming cloud fragments are more massive than all similar-sized fragments in clouds not containing clusters. Section 3.3.2 investigates this issue.

In our sample, the Pipe Nebula and Taurus serve as examples of regions dominated by isolated star formation. Actually, except for B59, the Pipe Nebula does hardly form stars at all (Forbrich et al. 2009). Perseus and Ophiuchus serve as examples for cluster-forming regions. They do contain clusters much more significant than any stellar aggregate found in Taurus and the Pipe Nebula⁷. Orion A is another example of a cluster-forming cloud.

3.3.1. Cluster-forming Fragments Dominate Their Host Cloud

3.3.2. Cluster-forming Fragments Exceed Fragments without Clusters in Mass

Figure 3 compares cluster-forming cloud fragments to clouds devoid of significant clusters. For Taurus and the Pipe Nebula, the mass–size data is presented as done before (e.g., Figure 1). We stress again that the Taurus MAMBO data for small spatial scales do not cover the entire cloud. At r ≲ 0.2 pc, the data only characterize the conditions in regions devoid of significant stellar groups. For Perseus and Ophiuchus, we choose a different plotting scheme. Here, we only plot the data for fragments containing the most massive fragment found at small radii (NGC 1333 in Perseus and L1688 in Ophiuchus).

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We find that, at given radius, cluster-forming Ophiuchus cloud fragments (i.e., toward L1688) are significantly more massive than those in Taurus and Pipe. The naive expectation (i.e., that cluster-forming fragments are more massive) is thus confirmed. For Perseus (i.e., toward NGC 1333), however, the result is more nuanced. At r ≲ 0.3 pc, cluster-forming fragments are significantly more massive than any structure in Taurus or the Pipe Nebula. For 0.4 ≲ r/pc ≲ 2, however, the extinction-derived mass toward NGC 1333 is similar to the maximum extinction-based masses for Taurus and the Pipe Nebula. If true, this would suggest that some cluster-forming regions at r ≳ 0.4 pc have a structure similar to clouds only forming isolated stars (or no stars at all).

It may be, though, that the Perseus extinction-based masses are biased toward lower values. The Perseus map has relatively poor physical resolution (5' at 260 pc distance), and intrinsic colors of stars in NGC 1333 can bias extinction measurements. These problems are amplified by high column densities toward NGC 1333. Section 2.2.1 demonstrates these problems for B59. Thus, we speculate that, at given size, the region containing NGC 1333 is indeed more massive than any structure in Taurus and the Pipe Nebula. This remains to be proven, though.

The fragments in the Orion Nebula Cluster, as well as Orion A as an entire cloud, are more massive than any feature in Taurus and the Pipe Nebula (Figure 2). These data are thus consistent with the aforementioned trend.

In summary, fragments containing clusters appear to be more massive than all structure in clouds devoid of clusters. Larger samples, and better data, are needed to ultimately establish this trend.

As we just have seen in Section 3.3.1, cluster-forming regions dominate the mass reservoir of their host cloud at any given spatial scale. This suggests a tight correlation between the properties of the cluster-forming fragments and the large-scale cloud structure. This can, for example, be characterized by "global" slopes of connection lines between mass–size measurements of the most massive fragments at large and small scale (Equation (5)). Since we lack comprehensive data on all spatial scales, we are unfortunately presently unable to derive global slopes for G10 and G11. To do so, one must also carefully characterize flux losses due to spatial filtering, as they occur in bolometer and interferometer maps. This is beyond the scope of the present paper.

Specifically, the cluster-forming clouds are probed both at ∼0.05 pc and ∼5.0 pc radius (e.g., Figure 1). We calculate their global slopes by connecting the most massive fragments detected within 10% of these reference radii (30% radius deviation for the sparsely sampled Orion data). For Orion, where only very large scales are probed reliably, we use a mass measurement at 11 pc radius for slope calculations. Also, in Orion, the ≫100 M_☉ fragments at ∼0.05 pc radius (right in the center of the Orion Nebula) are rejected as outliers. This rejection of objects is a regrettable move, since cloud fragments with extreme properties could be the actual sites where the most massive stars are born. However, we feel that a more detailed and careful analysis than possible here, including a detailed consideration of the temperature structure, is warranted. We hope to do this in one of our future studies. The resulting slopes range from 1.10 to 1.33, as shown in Figure 4. Note that a much smaller slope (as small as ∼0.7) would hold for Orion A, if we do not reject the outliers. The uncertainties indicated in Figure 4 hold for an uncertainty of a factor 2 in the mass ratio (factor 4 for the less reliable Orion data). This error budget presents the worst expected scenario; the true errors are probably smaller. In Ophiuchus, we have sufficient resolution to also measure masses at ∼0.02 pc radius. If we use such higher-resolution data for this cloud, the slope decreases to 1.30.

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Note that we experimented with different approaches to define global slopes to characterize the core-cloud structure. We then decided to use the current method using masses only at 0.05 pc and 5.0 pc because we find no significant differences compared with the other methods. For example, following the construction of Equation (6), we measure for each cloud the slope, b, that minimizes max [m(r)/(r/pc)^b]. This value is then derived by utilizing data at all spatial scales. Slopes derived in this fashion only differ by ±0.04 from those shown in Figure 4, though. In the end, Equation (5) appears to provide the simplest approach that can also be repeated by other researchers.

The uncertainty-weighted mean global slope for the three regions is 1.27 ± 0.11. As shown in Figure 4, our data are consistent with the hypothesis that all cluster-forming clouds have the same global slope. This suggests that mass–size relations m ∝ r^1.27±0.11 provide a crude tool to use mass measurements at large spatial scale (∼5 pc) to gauge the star formation conditions of the most massive embedded regions (∼0.05 pc). Unfortunately, the extreme SCUBA cores in Orion A, with masses ≫100 M_☉, suggest that much smaller slopes may hold in some clouds. The data are incompatible with larger slopes, though. Thus, the uncertainties shown in Figure 4 suggest that the slopes are not larger than 1.5.

Figure 5 presents tangential slopes for the entire sample. Because of too strong spatial filtering, we exclude maps taken with interferometers, SCUBA, and Bolocam from this calculation. For MAMBO, Taurus fragments larger r = 0.1 pc might be affected by filtering (Kauffmann et al. 2008). Their slopes are thus not drawn.

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A first trend noted in Figure 5 is that the infinitesimal slopes change with radius. Also, they have no obvious relation to the global slopes shown in Figure 4. Both slope definitions are thus complementary.

At given spatial scale ≳0.5 pc (in this domain, we only have data for Perseus, Taurus, and Ophiuchus), all clouds have comparable infinitesimal slopes. Those for Ophiuchus are a bit lower than for the other regions, indicating that the observed clouds differ in their structure at r ≳ 0.5 pc. In all clouds, the slopes decrease with increasing radius. As we detail in Section 4.3, this behavior resembles the mass–size trends for model clouds with finite extent (i.e., density vanishes outside a finite radius).

In Taurus and the B59 and B68 regions of Ophiuchus, we derive very small infinitesimal slopes for r ≲ 0.1 pc. (These regions are the only ones well probed at such small scales.) Near r = 0.1 pc, some fragments are observed to have slopes as low as d ln(m)/d ln(r) ∼ 1. This result is not entirely unexpected. Spheres supported by isothermal pressure, for example, obey d ln(m)/d ln(r) = 1 at intermediate radii (Section 4.3). Such (Bonnor–Ebert) spheres are commonly believed to constitute good idealizations of dense core structure. In particular, B68 is today quoted as the textbook model of a Bonnor–Ebert sphere (Alves et al. 2001). As expected, B68 shows a slope approaching 1 for large radii. We caution that the spatial filtering of bolometer maps, as well as the removal of extended features in extinction maps, might bias the slope measurements toward lower values. However, at least for cores with simple geometries, like B68, slopes ∼1 are expected for larger radii. This is required for Bonnor–Ebert-like density profiles. If such slopes were not observed, the paradigm of Bonnor–Ebert-like dense cores (e.g., Bergin & Tafalla 2007) would not be consistent with our data.

Note that the Taurus and Perseus infinitesimal slopes at r ∼ 0.5 pc are much larger than the slopes expected for the embedded dense cores of simple geometry (∼1 for every core at some radius). If this dichotomy is real, it has interesting implications. Basically, the slope must decrease from ≳1.5 at r ≳ 0.5 pc to d ln(m)/d ln(r) ∼ 1 for radii ≲0.1 pc. In other terms, there appears to be a transition in slope from the diffuse cloud structure to self-gravitating dense cores, when considering decreasing radii. Again we must caution that these trends might partially reflect artifacts in the data. As demonstrated before (Section 2.2.1), the mass in extinction maps is sometimes increasingly underestimated toward small scales. This biases slopes toward lower values. It is not likely, though, that all features in our extinction maps are massively biased. In particular for Perseus, strong biasing would be inconsistent with comparisons between CO and extinction maps (Goodman et al. 2009; also see Paper I). Still, it is discomforting that the slope transition does apparently exactly occur in the spatial domain that is not well probed by any tracer.

Interestingly, a cloud-to-core slope transition might not exist in Ophiuchus: note that slopes ∼1 are observed for radii ≳0.3 pc, so that no slope transition between cloud and core might be necessary. Specifically, at any given radius, the most massive fragment containing L1688 follows tightly the line connecting the most massive fragments at 0.05 pc and 5 pc radius. If this is true, a clean division between the L1688 dense cores and the surrounding diffuse cloud structure might not exist.

Larson (1981) carried out one of the first studies of mass–size relations of molecular clouds (in his Figure 5) and obtained m(r) ∝ r^1.9. (Actually, he studied size–density relations. Since he assumed spherical cloud geometries, these can be turned into mass–size laws.) Today, this "third Larson law" is usually quoted as m(r) ∝ r² and considered one of the fundamental properties of cloud structure (e.g., McKee & Ostriker 2007).

We find a nuanced relation between our data and the detailed result of Larson (1981),

$$\begin{equation*} m(r) = 460 \, M_{\odot } \, (r / {\rm pc})^{1.9} \end{equation*}$$

(our Equation (1)). In the 1 ⩽ r/pc ⩽ 4 radius range, his and our results are compatible for several solar neighborhood clouds, as shown by Equation (7). However, some clouds, like G10, can deviate by about an order of magnitude in mass from this relation. Further, some trends within individual clouds are not described well by one single mass–size law. Slope and intercept of Equation (6), and the slopes shown in Figures 4 and 5, may serve as examples for deviant mass–size relations. The Larson (1981) mass–size law thus fails to describe a lot of the cloud substructure visible in state-of-the-art cloud maps. Deviations from this relation have also been noted before (e.g., McKee & Ostriker 2007 for references). Also, note that Larson (1981) never intended to match all cloud substructure; as seen in his Figure 5, the mass–size relation only provides an order-of-magnitude fit to his data. In this sense, the mass–size laws derived here do not really supersede Larson's results; molecular clouds are very complex structures, and every one has to be considered individually. However, for certain mass–size trends, Equation (6) and the global slopes in Figure 4 do provide a more detailed description than offered by Equation (1).

Beyond these quantitative differences, Larson's technique differs fundamentally from ours. He plotted all cloud and dense core data available to him, and attempted to fit this ensemble data by a common law, independent of whether the cores and clouds were residing in the same area of the sky. Here, however, we usually handle data separately for each cloud, to, e.g., calculate the global and tangential slopes shown in Figures 4 and 5. Only Equations (7) and (6) stem from an analysis of data combined for several clouds.

Ensemble-based mass–size relations of the kind considered by Larson have been studied by many other groups; here, we can only present excerpts from such work. Lada et al. (2008), for example, derive b = 2.6 for dense cores in the Pipe Nebula extinction map used here. For a sample of IRDCs, Ragan et al. (2009) derive even steeper slopes, b ≈ 3, much in excess of any slope derived here. Their approach is very different from ours, though; they extract dense cores, plot their masses and sizes, and fit the data with a single line. We, instead, measure a core's mass at various contours, and derive tangential slopes by only considering data for a single dense core. Their ensemble slopes thus serve a different purpose than our global and tangential slopes. In this respect, it may be helpful to consider the Type 1–4 line width–size relations of Goodman et al. (1998; replace line width with mass in their discussion): there are many different ways to define slopes, and they will characterize different properties. The large-scale structure of molecular clouds, as originally addressed by Larson (1981), is in principle discussed by Solomon et al. (1987) and Heyer et al. (2009). Their data has, however, not been exploited to derive mass–size relations.

Note that Stüwe (1990) uses a technique similar to ours to probe the spatial domain considered here. As discussed in Section 4.3, his work is consistent with ours.

It is thought since long that clouds need to achieve a high column density in order to produce dense cores and stars. Johnstone et al. (2004), in particular, introduced the concept of extinction (or column density) thresholds for dense core formation (also see Onishi et al. 1998; Hatchell et al. 2005; Enoch et al. 2007). Similarly, Lombardi et al. (2006) and Lada et al. (2009) argue that a low fraction of mass at high column density yields low star formation activity (also see Kainulainen et al. 2009). This is in line with the Kennicutt–Schmidt law between star formation rate and mass surface density, Σ_SFR ∝ Σ^p_gas (e.g., Kennicutt 1998). Since p ∼ 1 for star-forming clouds (Evans et al. 2009), this relation predicts an increase of star formation activity with increasing column density.

The analysis in Section 3 shows that these laws do also manifest in our data. Sections 3.1 and 3.3 suggest that the ability to form clusters and massive stars increases with increasing mass, when considering a given radius. Since $\langle N_{\rm H_2} \rangle \propto m / r^2$, one could also say that cloud fragments do appear to only form massive stars and clusters when they have a high mean column density—just as suggested by the aforementioned laws.

Mass–size studies do, however, also provide information not available from the aforementioned plain column density studies. First, note that "extinction threshold" studies (e.g., Johnstone et al. 2004) do only consider a single spatial scale, i.e., the beam used to construct the extinction map. This is a major difference to mass–size studies, where many spatial scales are considered. Second, observe that probability density distribution studies of column density (e.g., Lombardi et al. 2006) do consider all scales of a map, but do not register which part of the signal shown on the histogram originates in which part of the cloud. (Depending on the analysis, this is not necessarily a problem.) Mass–size studies, in contrast, treat individual cloud fragments separately.

Mass–size data sets give a new twist to discussions of extinction thresholds. To see this, consider the solar neighborhood clouds examined here. These are all similar at large spatial scale (Equation (7)). Still, at smaller scale, they differ significantly in mass and star formation activity (Section 3.3). The processes determining the star formation activity must thus operate on spatial scales smaller than the entire cloud. In this sense, the star formation activity depends (at least in our sample clouds) on a cloud's ability to create, from a given mass reservoir, a small number of fragments that dominate the mass reservoir and concentrate it into a small volume. The presence of high column densities are then a consequence of the cloud structure, but not the governing reason for the formation of dense cores and stars.

The observed power-law-like mass–size relations,

$$\begin{equation*} m(r) = m_0 \, (r / {\rm pc})^b \,, \end{equation*}$$

are characterized by slopes, b, and intercepts, m₀. As we show here, slopes and intercepts can be used to gauge densities and their gradients. At the same time, it is possible to constrain the absolute level of pressure and the nature of its origin.

Consider, for example, an infinite equilibrium sphere with power-law density profile that is supported by isothermal pressure from gas at temperature T_g. Then,

$$\begin{equation} m(r) = 2.6 \, M_{\odot } \, \left(\frac{T_{\rm g}}{10 \,{\rm K}} \right) \, \left(\frac{r}{0.1 \,{\rm pc}} \right) \end{equation} \tag{ 8 }$$

(Kauffmann et al. 2008, Equation (11), in their case → π/2). Thus, if this model holds, the intercept will encode information on the gas temperature supporting the cloud. Conversely, the slope predicted by the model, b = 1, can be used to validate the model; if observations yield b ≠ 1, then the model will not apply.

It has to be kept in mind that we consider mass–size laws derived from two-dimensional maps. These are related to, but not identical with, mass–size laws obtained from three-dimensional density maps. This is illustrated by the experiments conducted by Shetty et al. (2010) who use the fragment identification technique also used by us. Their analysis is based on three-dimensional numerical simulations of turbulent clouds. As part of their experiments, they derive mass–size slopes from their data. For their particular set of simulations, the power-law slopes derived in this fashion are similar to the number of dimensions used for mass measurements (i.e., three when based on density, and two when based on column density). This underlines that the number of dimensions considered has to be kept in mind.

4.3.1. Density Laws

The above discussion can be extended to include many more models of cloud structure. We do this in Appendices A and B. The results of this analysis are summarized here.

First, let us examine the connection between mass–size laws and cloud density profiles. Consider a sphere with about constant density for radii smaller than some flattening radius, s₀, a power-law drop at intermediate radii, n(s) ∝ s^−k (where s is the distance from the center), and vanishing density beyond some outer truncation radius, R. Such profiles provide a good match to observations of dense cores (Tafalla et al. 2002). They are a good approximation to the structure of isothermal equilibrium spheres (Dapp & Basu 2009). As we show in Appendix A.1, for apertures of radius r, this yields mass–size relations of the form

$$\begin{equation} m(r) \propto n_{\rm c} \, r^{3 - k} \quad {\rm for} \quad s_0 \ll r \ll R \,, \end{equation} \tag{ 9 }$$

where n_c is the density for s = 0, and mass–size slopes

$$\begin{equation} {d} \ln (m) / {d} \ln (r) = 3 - k \quad {\rm for} \quad s_0 \ll r \ll R. \end{equation} \tag{ 10 }$$

Both relations apply only if k < 3. As the equations show, the intercept contains information on the central density, and the mass–size slope depends on the slope of the density law, k. Both does, of course, only hold at intermediate radii, s₀ ≪ r ≪ R. For reference, we note that the column density obeys N ∝ r^1−k.

A generalized version of power-law spheres are triaxial ellipsoids. Appendix A.3 considers the case in which n(s) ∝ (s/s₀)^−k along any main axis, but with s₀ depending on the direction chosen (Equation (A9)). Detailed analysis shows that such ellipsoids follow the same mass–size relations as spheres, when r = (A/π)^1/2. Thus, the laws listed above apply.

In a next step, one may wish to consider models of cylindrical clouds of length ℓ. Here, we adopt density drops n(s) ∝ s^−k perpendicular to the cylinder axis for intermediate values of s. At intermediate radii s₀ ≪ r ≪ R, such clouds obey

$$\begin{equation} m(r) \propto n_{\rm c} \, r^{4 - 2 k} \quad \Rightarrow \quad {d} \ln (m) / {d} \ln (r) = 4 - 2 k, \end{equation} \tag{ 11 }$$

if their major axis is perpendicular to the line of sight, and

$$\begin{equation} m(r) \propto n_{\rm c} \, r^{2 - k} \quad \Rightarrow \quad {d} \ln (m) / {d} \ln (r) = 2 - k \,, \end{equation} \tag{ 12 }$$

if the axes are aligned. (Intermediate angles are not considered here.) Meaningful relations are only obtained for k < 2. In both relations, the radius is defined as r = (A/π)^1/2.

Thus, just as one may naively expect, the mass–size slope gauges the slope of the density profile. Further, the intercepts of mass–size relations constrain the absolute density of cloud fragments. There is, however, one less obvious fact that calls for attention: the exact relations between mass–size slopes, intercepts, and density law depend on the cloud model and viewing angle. It is therefore not possible to derive the true density profile without further information on the cloud geometry. Such information may, e.g., be derived by studying the elongation of cloud fragments. Also, the above power-law relations do only apply at intermediate radii, s₀ ≪ r ≪ R. This domain might not exist in actual observed clouds. Then, the central density plateau and the finite size have to be considered. These give

$$\begin{equation} m(r) \left\lbrace \begin{array}{@{}llll} \propto n_{\rm c} \, r^2 & {\rm for} & r \ll s_0 &{\rm and}\\ \approx M & {\rm for} & r \gtrsim R. & \\ \end{array} \right. \end{equation} \tag{ 13 }$$

4.3.2. Polytropic Equilibria

The density slopes themselves depend on the processes shaping the model cloud. As a first example, here we consider static equilibrium models in which pressure gradients are in balance with self-gravity. We assume a polytropic equation of state, $P \propto n^{\gamma _P}$, in which pressure and density are related by the polytropic exponent, γ_P.

In Appendix B, we show that

$$\begin{equation} k = \frac{2}{2 - \gamma _P} \end{equation} \tag{ 14 }$$

for polytropic equilibrium spheres (if γ_P < 4/3) and cylinders (if γ_P < 1). The density and mass–size slopes are, thus, related to the polytropic exponent. Isothermal pressure, for which γ_P = 1, implies k = 2 in spheres, for example. Then, d ln(m)/d ln(r) = 1 in spherical model clouds; laws too complex to be considered here apply to cylinders. As seen in Figures 4 and 5, such a model can explain some, but not most slope measurements.

Polytropic exponents γ_P = 1/2 are sometimes suggested to describe "turbulent" pressure within clouds, as, e.g., arising from Alfvén waves (McKee & Zweibel 1995). In this case, k = 4/3, and so d ln(m)/d ln(r) assumes values of 5/3 ≈ 1.67 (spheres and ellipsoids), 4/3 ≈ 1.33 (perpendicularly viewed cylinder), and 2/3 ≈ 0.67 (end-on cylinder) for the different models. Among these, spheres, ellipsoids, and cylinders viewed from the side provide an acceptable match to the observed mass–size slopes b > 1. Cylinders viewed along their major axis yield too shallow mass–size laws (and such a viewing direction is highly unlikely).

For a given physical model, the intercept can be used to gauge a cloud's stability against collapse, respectively suggest the level of supporting pressure. Here, we limit ourselves to the isothermal case, i.e., γ_P = 1. Stability considerations (e.g., of Bonnor–Ebert-type; Ebert 1955; Bonnor 1956) imply

$$\begin{equation} M \le M_{\rm cr} \approx 2 \, \frac{\sigma ^2(v) \, R}{G} \end{equation} \tag{ 15 }$$

for the total mass, where σ(v) is the characteristic one-dimensional velocity dispersion (Equations (B3) and (B5)). For spheres, R is the radius, while R → ℓ in cylinders. If σ(v) is known (e.g., for thermal pressure), M > M_cr implies collapse of the object considered. Conversely, depending on the situation, σ(v) can be inferred by requiring that M = M_cr. Required values of σ(v) significantly in excess of the thermal velocity dispersion of the mean free particle might, e.g., suggest the presence of significant non-thermal pressure. If we only require that pressure balances gravity, and drop the constraint that the object is to be stable against perturbations, the above law yields Equation (8).

As particular example, consider B68. It is thought that this dense core has a structure very similar to a Bonnor–Ebert sphere (Alves et al. 2001). Thus, one would expect the mass and size of B68 to obey Equation (8), when considering large enough radii. This is indeed the case, as seen in Figure 1.

For intuitive communication, it may be helpful to report synoptic density slopes,

$$\begin{equation} \left[ - \frac{{d} \, \ln (n)}{{d} \, \ln (s)} \right]_{\rm syn} = 3 - \frac{{d} \, \ln (m)}{{d} \, \ln (r)} \,, \end{equation} \tag{ 16 }$$

i.e., the density slope a sphere of the same mass–size slope would have when observed at intermediate radii. The synoptic slopes give a good first idea of the true density slopes. First, recall that the model mass–size laws do not sensitively depend on the assumption of exact spheres; the same relation holds for ellipsoids. Also, in the observed range 1 ≲ d ln(m)/d ln(r) < 2, spheres (or ellipsoids) and perpendicularly viewed cylinders (the end-on view is statistically insignificant) imply similar slopes; for these geometries, the synoptic slopes exceed the true ones by less than 0.5, assuming intermediate radii. Thus, we derive

$$\begin{equation*} \left[ - {d} \, \ln (n) / {d} \, \ln (s) \right]_{\rm syn} = 1\,{\rm to}\,2 \end{equation*}$$

for mass–size slopes 1 ≲ d ln(m)/d ln(r) < 2.

A limited comparison of these density–size slopes with previous results is possible. Tafalla et al. (2002), e.g., study the dust emission of five starless dense cores, and derive density–size slopes of 2.0 to 2.5 for four of them. This is a typical result for cloud fragments of ≲0.1 pc size (Bergin & Tafalla 2007), a spatial domain not too well covered by our data.

Concerning the analysis method and spatial range considered, the extinction study by Stüwe (1990) might provide the best match to our work. Based on star counts, he derives −d ln(n)/d ln(s)>1.0 ± 0.4 on scales of up to ∼1 pc. This is consistent with our results, also given the differences in map construction (he uses optical star counts).

Integration along the line of sight gives the column density at offset u,

$$\begin{equation} N(u) = 2 \, \int _0^{[R^2 - u^2]^{1/2}} n([u^2 + w^2]^{1/2}) \, {d} \, w \,, \end{equation} \tag{ A2 }$$

where the integration stops at the boundary radius. Evaluation yields

$$\begin{equation} N(u) \propto u^{1 - k} \quad (R \rightarrow \infty) \end{equation} \tag{ A3 }$$

if k > 1, a relation that can be used for offsets s₀ ≪ u ≪ R. Integration of the column density over a circular peak-centered aperture of maximum offset u gives the mass,

$$\begin{equation} m(u) = 2 \pi \mathcal {C} \int _0^u N(w) \, w \, {d} \, w \,, \end{equation} \tag{ A4 }$$

where $\mathcal {C} = \Sigma / N_{\rm H_2} = \mu _{\rm H_2} m_{\rm H}$ is the conversion factor to mass surface density. Evaluation in the limit R → ∞ gives

$$\begin{equation} m(u) \propto u^{3 - k} \quad \Rightarrow \quad m(r) \propto r^{3 - k} \end{equation} \tag{ A5 }$$

for s₀ ≪ r ≪ R. We use u = r in the transition from offsets to aperture sizes since, in spheres, the offset from the center, u, is equal to the radius defined via the aperture area, r = (A/π)^1/2.

The meaningful range of this relation is limited to k < 3; for k ⩾ 3, the mass would decrease with size, which is not physical. In essence, spheres with k ⩾ 3 have a finite mass, even for R → ∞, so that power-law mass–size relations cannot hold in any radius domain.

Homogeneous spheres, in which the density is spatially constant, constitute a special case of the aforementioned spherical models. Consider an offset u from the sphere's center, where the density drops to zero beyond the sphere's outer radius, R. Then, the column density is the product of the density, n₀, and the length of the line of sight,

$$\begin{equation} N(u) = 2 n_0 (R^2 - u^2)^{1/2} \,. \end{equation} \tag{ A6 }$$

Integration following Equation (A4) yields (with the substitution u → r)

$$\begin{equation} m(r) = \frac{4}{3} \pi \varrho _0 [ R^3 - (R^2 - r^2)^{3/2} ] \,, \end{equation} \tag{ A7 }$$

in which $\varrho _0 = \mathcal {C} n_o$ is the mass density corresponding to n₀. For r → R, we derive m = 4/3πϱ₀R³, just as expected for a homogeneous sphere truncated at radius R. The slope obeys

$$\begin{equation} \frac{{d} \ln (m)}{{d} \ln (r)} = 3 \, \frac{ (R^2 - r^2)^{1/2} \, r^2 }{ R^3 - (R^2 - r^2)^{3/2} } \end{equation} \tag{ A8 }$$

and does monotonously decrease from d ln(m)/d ln(r) = 2 at r = 0 to d ln(m)/d ln(r) = 0 at r = R.

Another variation of spherical power-law density profiles are triaxial density distributions with ellipsoidal iso-density surfaces. These can be derived from spherical power laws by substituting s/s₀ → [(x/x₀)² + (y/y₀)² + (z/z₀)²]^1/2 (where a subscript "0" indicates reference properties for normalization). The coordinates x, y, and z give the projection of a given position on the three axes of the ellipsoidal density distribution. Then, the density law reads

$$\begin{equation} n = n_0 \, \left[ \left(\frac{x}{x_0} \right)^2 + \left(\frac{y}{y_0} \right)^2 + \left(\frac{z}{z_0} \right)^2 \right]^{-k/2} \,. \end{equation} \tag{ A9 }$$

The chosen coordinates form an orthogonal coordinate system. Any position along a straight line of sight, w, is therefore linearly related to the coordinates chosen for the density distribution:

$$\begin{equation} x = m_x + n_x \, w \, ; \quad y = m_y + n_y \, w \, ; \quad z = m_z + n_z \, w \,. \end{equation} \tag{ A10 }$$

For convenience, here we choose that w = 0 at the position of highest density along the line of sight. Substitution of Equations (A10) into Equation (A9) yields

$$\begin{eqnarray} n &=& n_0 \Bigg[ \frac{m_x^2}{x_0^2} + \frac{m_y^2}{y_0^2} + \frac{m_z^2}{z_0^2} + 2 \, \Bigg(\frac{m_x n_x}{x_0} + \frac{m_y n_y}{y_0} + \frac{m_z n_z}{z_0} \Bigg) \, w\nonumber\\ &&+ \Bigg(\frac{n_x^2}{x_0^2} + \frac{n_y^2}{y_0^2} + \frac{n_z^2}{z_0^2} \Bigg) \, w^2 \Bigg]^{-k/2} \,, \end{eqnarray} \tag{ A11 }$$

i.e., some terms that do not contain w, some that include w linearly, and a few that contain w to its square. Analysis shows that the sum of the terms with a linear dependence on w must be zero. To see this, consider the requirement that the density reaches its maximum where w = 0. For this to happen, the derivative of the sum within the square brackets of Equation (A11) with respect to w must vanish at w = 0. This is only the case if all terms linear in w cancel out. Equation (A11) can thus be written

$$\begin{equation} n = n_0 \, \left(\frac{u^2 + w^2}{r_0} \right)^{-k / 2} \,, \end{equation} \tag{ A12 }$$

if one chooses u² = (m²_x/x²₀ + m²_y/y²₀ + m²_z/z²₀) · (n²_x/x²₀ + n²_y/y²₀ + n²_z/z²₀)⁻¹ and r²₀ = (n²_x/x²₀ + n²_y/y²₀ + n²_z/z²₀)⁻¹. Formally, Equation (A12) is identical to the spherical density law substituted in Equation (A2). Therefore, we can use the calculations in Appendix A.1 to conclude that, again,

$$\begin{equation} N(u) \propto u^{1 - k} \quad (R \rightarrow \infty) \end{equation} \tag{ A13 }$$

if k > 1. Thus, the spatial column density distribution does only depend on u.

A detailed look at the above analysis does actually show that we did not use the exact nature of the density law to this point. Instead, we exploited that the density depends on (x/x₀)² + (y/y₀)² + (z/z₀)². Thus, we can make the conclusion that all density distributions with ellipsoidal iso-density surfaces yield column density distributions only depending on u.

Thus, u can be interpreted as an offset, just as in the spherical situation. Inspection of the equation defining u reveals that the three-dimensional offsets from the distribution's center, m_i, form a triaxial ellipsoid too, when u is kept constant. Projection of the u = const. surface into a plane corresponds to a cut through this ellipsoid. Since cuts though ellipsoids yield ellipses, the u = const. curve in a plane is an ellipse too. In this case, spatial integration of the column density can be conveniently executed in polar coordinates. Let v and w be the minor and major axis of the projected ellipse. For a given viewing perspective, v/w is constant for all contours of constant column density. Then, the mass follows from

$$\begin{equation} m(u) = 2 \pi \, (v/w) \, \mathcal {C} \int _0^u N(w) \, w \, {d} w \,, \end{equation} \tag{ A14 }$$

where we use that the ellipse's area increases with w as 2π(v/w) w d w. This integral becomes

$$\begin{equation} m(u) \propto u^{3 - k} \quad \Rightarrow \quad m(r) \propto r^{3 - k} \end{equation} \tag{ A15 }$$

for s₀ ≪ r ≪ R. In this, we use that u is proportional to the effective radius of the ellipsoid formed by N = const. (i.e., u = const.) contours. The latter holds since, for a given ellipsoid, v/w is a constant, and its area is πvw = π(v/w)w². Thus, the effective radius becomes r = (v/w)^1/2w ∝ w.

In summary, we retrieve the relations already derived for spheres, including the limits on reasonable values of k. As expected, spheres form a particular case of ellipsoids with x₀ = y₀ = z₀.

In cylinders with the main axis perpendicular to the line of sight, the integration in Equation (A2) must be evaluated with the integration path perpendicular to the cylinder axis. Then

$$\begin{equation} N(u) \propto u^{1 - k} \quad (R \rightarrow \infty) \,, \end{equation} \tag{ A16 }$$

as long as k > 1. This provides, again, a convenient description for s₀ ≪ u ≪ R. Since the column density is constant for given offset, u, apertures along a given column density contour have a size 2uℓ, where ℓ is the cylinder length. Such an aperture contains a mass

$$\begin{equation} m(u) = 2 \ell \mathcal {C} \int _0^u N(w) \, {d} w \,. \end{equation} \tag{ A17 }$$

Substitution yields

$$\begin{equation} m(u) \propto u^{2 - k} \quad \Rightarrow \quad m(r) \propto r^{4 - 2 k} \,. \end{equation} \tag{ A18 }$$

This relation is applicable if k < 2, following the discussion of finite masses for spheres. In the transition from u to r, we use that r = (2ℓu/π)^1/2, which gives u ∝ r².

In cylinders with the main axis aligned with the line of sight, the integration in Equation (A2) must be evaluated along the cylinder axis. In this case,

$$\begin{equation} N(u) = n(u) \cdot \ell \,. \end{equation} \tag{ A19 }$$

This geometry further implies

$$\begin{equation} m(u) = 2 \pi \mathcal {C} \int _0^u N(w) \, w \, {d} \, w \,, \end{equation} \tag{ A20 }$$

which gives

$$\begin{equation} m(u) \propto u^{2 - k} \quad \Rightarrow \quad m(r) \propto r^{2 - k} \end{equation} \tag{ A21 }$$

in the usual s₀ ≪ u ≪ R limit, with the further constrain that k < 2. As in spheres, u = r for contours of constant column density.

For central flattening of the density profile, the column density becomes constant for u ≪ s₀, independent of the model adopted. Then, the mass in a given aperture is just given by the product of the mass surface density, which scales with n_cR, and the aperture area. For large radii, the mass is equal to the total mass when considering apertures larger than the object's size. Thus

$$\begin{equation} m(r) \left\lbrace \begin{array}{@{}llll} \propto n_{\rm c} \, R \, r^2 & {\rm for} & r \ll s_0, \quad {\rm and}\\\\[-6pt] \approx M & {\rm for} & r \gtrsim R, \\ \end{array} \right. \end{equation} \tag{ A22 }$$

independent of the adopted model geometry.

Hydrostatic equilibrium spheres supported by polytropic pressure have recently been summarized by McKee & Holliman (1999) and Curry & McKee (2000). In spherical symmetry, $\vec{\nabla } \rightarrow \partial / \partial s$, Φ(s) = −Gm(s)/s, and m(s) = 4π∫^s₀ϱ(w) w d w. This differential equation can easily be solved for density profiles ϱ(s) ∝ s^−k. Solutions are, however, only physical for k < 3; the power law implies m(s) ∝ s^3−k, which only increases with radius if k is small enough. The hydrostatic equation implies $s^{- \gamma _P k -1} \propto s^{1 - 2 k}$. This can only be fulfilled if both exponents to s are identical, which implies

$$\begin{equation} k = \frac{2}{2 - \gamma _P} \quad (\gamma _P < 4/3) \,, \end{equation} \tag{ B2 }$$

where the limit on γ_P reflects the condition k < 3. Stability against perturbations implies that the mass of the equilibrium is smaller than some critical mass (McKee & Holliman 1999). For γ_P = 1, this critical mass is identical to the Bonnor–Ebert mass,

$$\begin{equation} M_{\rm cr} = 2.4 \, \frac{\sigma ^2(v) \, R}{G} \,, \end{equation} \tag{ B3 }$$

where the numerical constant follows the discussion by McKee & Holliman (1999). Stable clouds have M < M_cr.

Horedt (1987) presents a summary of hydrostatic equilibrium cylinders (as well as sheets and spheres) supported by polytropic pressure. It builds on discussions of the case γ_P ⩾ 1 by Ostriker (1964), and analysis of polytropes with γ_P < 1 by Viala & Horedt (1974).

The Poisson equation can be tackled using the divergence theorem, $\int _V \vec{\nabla } \vec{F} \, {d} \, V = \int _{\partial V} \vec{F} \, {d} \, \vec{A}$. When analyzing this equation for the volume of an infinitely long cylinder, the ends can be neglected in the integration over the volume's surface, ∂V. Thus, if the component of $\vec{F}$ perpendicular to the cylinder surface is constant, and assumes the value F_⊥ along it, then $|\! \int _{\partial V} \vec{F} \, {d} \, \vec{A} | = 2 \pi \ell s F_{\perp }$ for cylinders of length ℓ and radius s. To apply this analysis to the Poisson equation, we substitute $\vec{F} \rightarrow \vec{\nabla } \Phi$. In cylindrical coordinates, the component of the potential's gradient that is perpendicular to the cylinder surface implies the substitution F_⊥ → ∂Φ/∂s. In essence, combination of the Poisson equation and the divergence theorem yields 4πG ∫_Vϱ d V = ∫_∂V∂Φ/∂s d A, which evaluates to ∂Φ/∂s = 2Gm(s)/(sℓ), where m(s) is the mass within the radius s. For ℓ → ∞, we further have $\vec{\nabla } \rightarrow \partial / \partial s$, and so the hydrostatic equation becomes ∂P/∂s = −ϱ ∂Φ/∂s.

A simple solution is provided by ϱ(s) ∝ s^−k, which implies $s^{- \gamma _P k -1} \propto s^{1 - 2 k}$, just as found for the spherical case. Analysis of the integral for the mass implies k < 2, similar to what is found for spheres. Thus, solutions must fulfill

$$\begin{equation} k = \frac{2}{2 - \gamma _P} \quad (\gamma _P < 1) \,, \end{equation} \tag{ B4 }$$

where the limit enforces k < 2. A perturbation analysis analog to the spherical Bonnor–Ebert case gives a critical limiting mass

$$\begin{equation} M_{\rm cr} = 2 \, \frac{\sigma ^2(v) \, \ell }{G} \end{equation} \tag{ B5 }$$

for γ_P = 1, as, e.g., demonstrated by Ostriker (1964).