Are Massive Dense Clumps Truly Subvirial? A New Analysis Using Gould Belt Ammonia Data

A key parameter in the study of molecular clouds and their substructures is the virial ratio:

$$\begin{eqnarray}&&\alpha \equiv \displaystyle \frac{2{{ \mathcal T }}_{\mathrm{cl}}}{| {{ \mathcal W }}_{\mathrm{cl}}| }.\end{eqnarray} \tag{ 1 }$$

Here, ${{ \mathcal T }}_{\mathrm{cl}}$ is the object's kinetic energy in its center-of-mass frame, and ${{ \mathcal W }}_{\mathrm{cl}}$ is its self-gravitational energy. The value of α as a diagnostic tool arises from the fact that it compares prominent opposing terms in the virial theorem (Chandrasekhar & Fermi 1953; McKee & Zweibel 1992)—that is, in the competition of forces (expressed as energies) that cause inward or outward accelerations of an object's radius (expressed as the trace of its moment-of-inertia tensor). Because any significant imbalance leads to rapid change, there is good reason to expect that some chosen collection of dense interstellar structures is close to a state of equilibrium, at least in a statistical sense, especially if the structures live for at least a single crossing time. ¹⁷ Insofar as this is true, one can read the value of α as an indication of the other, less easily observed, forces or energetic terms. A state in which α > 1 suggests the importance of the kinetic surface term that represents confinement by external thermal or turbulent pressure, or by the ram pressure due rapid inflow or outflow (see Goldbaum et al. 2011), or due to colliding flows. For example, α = 1.683 in the critical state of an isothermal, unmagnetized, pressure-bounded sphere (Ebert 1955; Bonnor 1956). If α ≫ 1, then gravity is negligible in comparison to external pressure: an equilibrium object in this state is pressure confined.

In contrast, α < 1 indicates the importance of an additional positive term, corresponding to an outward force that opposes the combination of self-gravity and external pressure. Magnetic fields supply one such force, although for molecular clumps and cores, both Zeeman measurements (Crutcher 2012) and estimates based on the Davis–Chandraskhar–Fermi method (Myers & Basu 2021) indicate median mass-to-flux ratios about twice the critical value. This implies that the quasi-static portion of the magnetic force is rarely sufficient to fully offset gravity. The fluctuating portion of the magnetic energy is also limited in its impact, as it tends to be in equipartition with the turbulent portion of ${{ \mathcal T }}_{\mathrm{cl}}$ (McKee & Zweibel 1992; Federrath 2016). Another, often-overlooked outward force is the momentum injected by protostellar outflows or photoionized regions when star formation is especially active. These must be included as an inner surface term when their kinetic energies are not included in ${{ \mathcal T }}_{\mathrm{cl}}$. The importance of this term is evident in the fact that the kinetic energy in protostellar outflows can be comparable to ${{ \mathcal T }}_{\mathrm{cl}}$ (e.g., Graves et al. 2010). However, their effect cannot overwhelm $2{{ \mathcal T }}_{\mathrm{cl}}$ in the virial theorem, for the simple reason that star-driven flows go on to stir turbulence within the medium (Matzner 2002, 2007; Nakamura & Li 2007). Similar arguments apply to stellar radiation forces (McKee & Zweibel 1992), which in any case are only significant in the presence of vigorous massive star formation (Krumholz & Matzner 2009; Murray et al. 2010; Raskutti et al. 2016; Jumper & Matzner 2018). Yet another often-overlooked term comes from the gravity of matter outside the clump boundary (Ballesteros-Paredes 2006); however, this should be small compared to ${{ \mathcal W }}_{\mathrm{cl}}$ for clumps that are overdense and not bounded by tidal forces.

For these reasons it is difficult to envision a scenario in which any collection of interstellar structures would be strongly subvirial—that is, characterized by α ≪ 1.

It is very puzzling, therefore, that observational studies of dense substructures within molecular clouds often find that α is well below unity for the most massive of these objects (see, for instance, Kauffmann et al. 2013; Urquhart et al. 2014; Traficante et al. 2018a, 2018c). Although the selection of objects varies from one study to another (as does the specific correlation between estimates of α and mass), the substructures in question are all molecular clumps: objects intermediate in scale between molecular clouds and the compact cores from which individual star systems are born. If massive molecular clumps are truly subvirial, this has important implications for the initial conditions for star cluster formation.

Could the strongly subvirial appearance of massive molecular clumps in fact be an artifact of the way that observations are taken or interpreted? Traficante et al. (2018b) advance one reason that it might be. They point out that the data used to determine ${{ \mathcal T }}_{\mathrm{cl}}$ and ${{ \mathcal W }}_{\mathrm{cl}}$ tend to weight different regions of a clump, due to the influence of a critical density on molecular line excitation, and that this may lead to a systematic offset in α.

Here, we explore another possibility: that choices involved in the method used to estimate α may themselves introduce systematic errors. Because it depends on an object's three-dimensional density, temperature, and velocity fields, α cannot be determined from projected data. One must construct a proxy, such as the virial parameter introduced by Bertoldi & McKee (1992, hereafter BM92) or the estimate presented by Singh et al. (2019, hereafter SMJ19). We find that at several steps of this process, common analysis choices have the cumulative effect of suppressing the derived value of α, especially for high-mass clumps.

For our exploration we employ NH₃ data from the Green Bank Ammonia Survey (GAS; Friesen et al. 2017) and column densities derived from a new analysis of dust optical depth in the Herschel data (A. Singh & P.G. Martin, 2021, in preparation). These provide sensitive, uniform, well-calibrated, and well-resolved information for an investigation such as ours. As a by-product, we present a catalog of properties for 85 clumps in the Gould Belt, using these high-quality data. Our catalog overlaps previous virial analyses, based on a subset of the same data, conducted by Kirk et al. (2017a), Redaelli et al. (2017), Keown et al. (2017), Chen et al. (2019), and Kerr et al. (2019). However, we use a somewhat different algorithm to define clump boundaries; this allows us to focus on the influence of analysis choices on estimates of α.

In Section 2, we review methods for estimating α. We introduce the data for our study in Section 3, and present the details of our technique in Section 4. As a case study, we highlight the analysis of a single clump in Section 4.1. In Section 5, we present results for our full sample of Gould Belt clumps. Finally, in Section 6, we draw conclusions about the impact of biases on the apparent physical state of massive clumps.

The most widely used method for estimating α involves the virial parameter introduced by BM92:

$$\begin{eqnarray}&&{\alpha }_{\mathrm{BM}92}=\displaystyle \frac{5{\sigma }_{\mathrm{cl}}^{2}{R}_{\mathrm{cl}}}{{{GM}}_{\mathrm{cl}}}.\end{eqnarray} \tag{ 2 }$$

Here, M_cl and σ_cl are the mass and one-dimensional velocity dispersion, respectively, of the object under study (a clump, in our case), and R_cl is its effective radius, usually defined so the projected area of the clump is $\pi {R}_{\mathrm{cl}}^{2}$. We adopt that definition as well.

The value of α_BM92 derives from the fact that the gravitational energy of an interstellar object is usually similar to that of a uniform sphere with the same mass and effective radius; hence, the correction factor a, defined by ${{ \mathcal W }}_{\mathrm{cl}}=-3{{aGM}}_{\mathrm{cl}}^{2}/5{R}_{\mathrm{cl}}$, is of order unity. BM92 consider the class of spheroidal clumps with power-law density profiles ρ ∝ r^−k in spheroidal radius coordinate r as an example. For these, a can be decomposed (a = a₁ a₂) into a stratification factor a₁ = (1 − k/3)/(1 − k/2.5), and a geometric factor a₂ whose angle average is close to unity except in the case of very prolate spheroids (see Figure 2 of BM92).

Along with the definition of a, Equations (1) and (2) imply

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{{\alpha }_{\mathrm{BM}92}}{a}\left(\displaystyle \frac{2{{ \mathcal T }}_{\mathrm{cl}}}{3{M}_{\mathrm{cl}}{\sigma }_{\mathrm{cl}}^{2}}\right),\end{eqnarray} \tag{ 3 }$$

showing that either α_BM92, or the refined quantity α_BM92/a, can be used to estimate α. An important detail is that σ_cl must be defined so that $3{M}_{\mathrm{cl}}{\sigma }_{\mathrm{cl}}^{2}$ is a valid estimate for $2{{ \mathcal T }}_{\mathrm{cl}};$ we return to this point below.

As an alternative to α_BM92 we will consider the method proposed by SMJ19, who reexamine the process of estimating α and suggest a procedure that is robust for non-spheroidal structures despite the effects of projection.

The SMJ19 method begins with how a clump is identified and extracted from projected data. One must start by defining a projected clump boundary, which encompasses a peak in the column density Σ. The projected clump boundary should also contain reliable molecular line data from which to obtain the line-of-sight radial velocity v_z , the one-dimensional thermal velocity dispersion σ_th, and the total line-of-sight velocity dispersion σ_z , from which the nonthermal portion σ_NT,z can be derived.

The next step is to extract the cloud or clump column density Σ_cl from any external material projected within the cloud boundary. Foreground and background removal is especially important when Σ is derived from submillimeter dust emission; note that spatially filtered observations (such as chopped or interferometric ones) accomplish this in an approximate way. SMJ19 demonstrate that an approximation based on the Abel transform is more successful than either keeping all material within the cloud boundary, or using simple interpolation to clip a background level. Abel reconstruction uses the fact that there is an exact relationship between any axisymmetric density distribution and its projection. Applied to the column density, it provides an estimate for the component interior to a three-dimensional surface whose projection is the two-dimensional clump boundary. The removed component, Σ_env = Σ − Σ_cl, is always lower toward the core of a clump than toward its edge, thanks to a projection effect that is analogous to limb brightening.

As only one component of the cloud's kinetic energy is visible in projection, the quantity

$$\begin{eqnarray}&&{{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}={{ \mathcal T }}_{\mathrm{bulk}}+\displaystyle \frac{3}{2}\int {\sigma }_{\mathrm{eff}}{(x,y)}^{2}{{\rm{\Sigma }}}_{\mathrm{cl}}(x,y){dx}\,{dy},\end{eqnarray} \tag{ 4 }$$

where x and y are sky coordinates at the cloud distance, provides an estimate for ${{ \mathcal T }}_{\mathrm{cl}}$ that is unbiased, in the sense that its average over viewing angles (denoted $\left\langle \cdots \right\rangle$) is exact:

$$\begin{eqnarray}&&\left\langle {{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}\right\rangle ={{ \mathcal T }}_{\mathrm{cl}}.\end{eqnarray} \tag{ 5 }$$

We note that BM92 adopt the same quantity, defining ${\sigma }_{\mathrm{cl}}^{2}=2{{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}/3{M}_{\mathrm{cl}}$ in their Appendix C. In Equation (4),

$$\begin{eqnarray*}&&{\sigma }_{\mathrm{eff}}{(x,y)}^{2}={\sigma }_{\mathrm{th}}{(x,y)}^{2}+{\sigma }_{\mathrm{NT},z}{(x,y)}^{2}\end{eqnarray*}$$

is the square of the effective one-dimensional velocity dispersion along each line of sight, and

$$\begin{eqnarray}&&{{ \mathcal T }}_{\mathrm{bulk}}=\displaystyle \frac{3}{2}\int {[{v}_{z}(x,y)-{v}_{\mathrm{CM},z}]}^{2}\,{{\rm{\Sigma }}}_{\mathrm{cl}}(x,y){dx}\,{dy}\end{eqnarray} \tag{ 6 }$$

is the contribution from resolved variations in the line-of-sight velocity v_z across the cloud, which we will refer to as bulk kinetic energy. Note that this may include a portion due to rotation. The center-of-mass velocity v_CM,z is computed from M_cl v_CM,z = ∫Σ_cl(x, y)v_z (x, y)dx dy.

It is worth noting that ${{ \mathcal T }}_{\mathrm{bulk}}$ will be resolution-dependent, in practice, because σ_cl,z (x, y) can only be defined at the resolution of a given experiment. In the limit that a cloud is too small to be resolved, it would be described by the single centroid velocity v_z = v_CM,z , and thus have ${{ \mathcal T }}_{\mathrm{bulk}}=0$. However, its observed kinetic energy should still be captured well by Equation (4) because its line width σ_cl,z will include these unresolved velocity gradients.

For the denominator of α, SMJ19 define a quantity ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}$, derived from Σ_cl, with the desirable property that

$$\begin{eqnarray}&&\left\langle {{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}\right\rangle ={{ \mathcal W }}_{\mathrm{cl}}.\end{eqnarray} \tag{ 7 }$$

SMJ19 show that ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}$ can be computed by collapsing the clump mass profile to a sheet in the plane of the sky, obtaining the sheet's gravitational self-energy, and correcting the result by a factor 2/π. By using information from the resolved column density map, this procedure avoids the need to choose R_cl and estimate a. It therefore provides a means to calibrate the characteristic value of a for an ensemble of clouds.

With these definitions for ${{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}$ and ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}$, SMJ19's quantity

$$\begin{eqnarray}&&{\alpha }_{\mathrm{SMJ}19}=\displaystyle \frac{2{{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}}{| {{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}| }\end{eqnarray} \tag{ 8 }$$

is a valid estimate for α, in the sense that both the numerator and denominator are exact when averaged over viewing angles.

It is important to note that α_SMJ19 and α_BM92/a are equivalent if evaluated under the following conditions: (1) they are derived from the same model of the clump column density profile Σ_cl (and furthermore, to give valid estimates of α, this must have been been cleaned of foreground and background contamination); (2) the quantity σ_cl is defined so that $2{{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}=3{M}_{\mathrm{cl}}{\sigma }_{\mathrm{cl}}^{2}$, thus incorporating $2{{ \mathcal T }}_{\mathrm{bulk}};$ and (3) the BM92 correction factor a is evaluated in a way that properly reflects the clump profile.

As we shall see, other choices for M_cl, σ_cl, and a tend to introduce biases when α_BM92/a (sometimes called M_vir/M_cl) is used as an estimate for α.

3.1. GAS Molecular Line Data

3.2. Herschel-derived Column Densities

We identify objects for our analysis as identifiable peaks in NH₃ emission and total column density. We draw projected clump and region boundaries around these according to the following procedure.

To isolate structures for which we have complete and well-resolved data, we start by creating a version of the NH₃ column density map that we convolve with a bounded parabolic (Epanechnikov) kernel of 88ʺ radius. Contours of this smoothed map are candidates for the clump boundary, from which we choose the largest for which two criteria are met: First, we must have complete data coverage to determine Σ, v_z , and σ_z , and nearly complete coverage for T_k , within the boundary. This requires that NH₃ (1,1) and (2,2) transitions are both well detected. (We fill any gaps in the temperature data using linear interpolation, which adds a negligible uncertainty to ${{ \mathcal T }}_{\mathrm{cl}}$.) Second, we require that the inferred mass density $3{M}_{\mathrm{cl}}/4\pi {R}_{\mathrm{cl}}^{3}$ implies n_H > 3.6 × 10³ cm⁻³, the critical density of the NH₃ (1,1) transition, which exceeds that of the (2,2) transition. Because NH₃ has a reasonably consistent abundance at these densities (e.g., Redaelli et al. 2017) and is not affected by freeze-out or significant line optical depth, this choice ensures that our kinematic information can be used to calculate ${{ \mathcal T }}_{\mathrm{cl}}$ in a reasonably unbiased way. Nevertheless, abundance variations (e.g., Crapsi et al. 2007) do inevitably inject some uncertainty.

Once a trial boundary has been identified, it is modified if necessary to exclude regions in which v_z falls more than 4 standard deviations outside its overall distribution for the clump. Velocity outliers probably represent errors in the line fitting process, or possibly NH₃ emission from unrelated structures along the line of sight, and so it is important to exclude the spurious additional kinetic energy they would imply. We find that this approach affects only a small fraction of the clump area and mass (a few pixels at most). Figure 1 provides one example, for clump L1251-1 in Cepheus.

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As the figure shows, it is possible for our procedure to produce a clump boundary that includes a hole, indicating where NH₃ data is lacking. L1251-1 has by far the largest such hole, ¹⁸ covering 2% of its area. A number of other clumps have holes that span a few pixels; we indicate these cases with asterisks in Table 3. Because the lack of NH₃ data tends to coincide with a region of low to zero Σ_cl, it makes no practical difference to ${{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}$ or ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}$ whether we fill these holes with interpolated data. In computing radii we use the outer area of the mask; excluding the holes would only slightly lessen R_cl.

For every clump, we must also identify the boundary of an enclosing region for the purposes of removing dust emission from foreground and background material, thereby creating a map with compact support for Abel-transform analysis. While the exact choice of region boundary is not significant, it should be separated well enough from the clump boundary to contain the envelope physically associated with the clump, yet also close enough to the clump to sample a similar column density of background and foreground material (or at least, the component of this material on long spatial scales).

To accomplish this, we generate a trial map and choose one of its contours to be the region boundary. We create a version of the NH₃ column density smoothed with a Gaussian kernel of width R_cl/8. We then multiply this by a vignetting function that is equal to the clump mask (unity within the clump boundary, zero outside), convolved with the function $1/{({r}_{k}+{R}_{\mathrm{cl}})}^{1/4}$, where r_k is the distance from the center of the kernel as measured in the plane of the sky. From this trial map, we choose the lowest contour that comes within R_cl/4 of the clump boundary. An example region boundary is displayed in Figure 2.

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Given the clump and region contours, we use the SMJ19 prescription to obtain Σ_cl in two steps. First we subtract from Σ a bi-cubic interpolation of its value from the region boundary; this is meant to extract an estimate of the emission from foreground/background dust. We then apply the Abel transform, in the manner discussed by SMJ19, to subtract the emission from the clump's envelope, leaving Σ_cl. The lower panels of Figure 2 provides an illustration of the decomposition, for the case of Clump 3 in Perseus B1.

From the clump column density profile Σ_cl we derive an estimate for self-gravitational energy ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}$ according to Equation (4) of SMJ19; we ensure that ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}$ is accurate to within a fraction of 1% using numerical refinement and a correction for discretization errors,which we describe in the Appendix.

In addition to our fiducial procedure, we consider several alternatives to identify what affects the trend of α with M_cl. One such choice involves employing α_BM92/a, but assuming a value for a (such as unity). Others involve evaluating Σ_cl differently. For instance, one might replace the Abel transform with a simple interpolation, leaving a different clump column profile, or one might adopt Σ_cl = Σ, forgoing any foreground and background removal. We shall also consider the effect of neglecting ${{ \mathcal T }}_{\mathrm{bulk}}$ when evaluating σ_cl and ${{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}$, to demonstrate that the velocity dispersion derived from individual beams does not capture the entire kinetic energy along the line of sight.

Note that our definition of R_cl, which derives from the projected area of NH₃ emission, differs from definitions involving fitted profiles or moments of the column density distribution. This should be kept in mind when comparing radii among catalogs. Moreover, because radius enters into the evaluation of α_BM92, the method used to determine R_cl can introduce a new and potentially significant source of systematic error, which we do not attempt to evaluate here.

4.1. Case Studies: Cepheus L1251-1 and Perseus B1-3

As our primary case study we consider the clump shown in Figure 1, Clump 1 within L1251 in Cepheus. Applying our fiducial analysis we obtain ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}=-5.51\times {10}^{44}$ erg and ${{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}=5.75\times {10}^{44}$ erg, of which ${{ \mathcal T }}_{\mathrm{bulk}}$ is 71%. These values imply α_SMJ19 = 2.05.

In Table 2 we provide additional parameters for this particular clump, comparing our fiducial (SMJ19) evaluation of α against the BM92 analysis, and against alternative reconstructions of Σ_cl based on removing an interpolated column density level from the clump boundary ("Edge Interp."), or making no correction for foreground and background contamination. We also consider the effect of neglecting ${{ \mathcal T }}_{\mathrm{bulk}}$.

Table 2. Impact of Analysis Choices for Cepheus L1251-1

Quantity	Abel	Edge Interp.	No Bg./Fg. Corr.
Mcl [M⊙]	56.2	43.2	82.6
Rcl [pc]	0.32	0.32	0.32
σcl [km s−1]	0.59	0.6	0.58
$| {{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}|$ [erg]	5.68 × 1044	3.62 × 1044	1.08 × 1045
${{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}$ [erg]	5.83 × 1044	4.59 × 1044	8.2 × 1044
${{ \mathcal T }}_{\mathrm{bulk}}$ [erg]	3.99 × 1044	3.16 × 1044	5.55 × 1044
αSMJ19	2.05	2.54	1.52
αSMJ19 (No ${{ \mathcal T }}_{\mathrm{bulk}}$)	0.65	0.79	0.49
αBM92	2.3	3.05	1.49
αBM92 (No ${{ \mathcal T }}_{\mathrm{bulk}}$)	0.73	0.95	0.48

Note. Columns indicate method used to remove foreground and background emission from Σ_cl: our fiducial Abel-transform reconstruction, interpolation from the clump boundary, or no correction at all. Rows marked "No ${{ \mathcal T }}_{\mathrm{bulk}}$" indicate that resolved kinetic energy is omitted from ${{ \mathcal T }}_{\mathrm{cl}}$ and σ_cl.

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Several aspects of the analysis affect the result. The method used to extract the clump's column density profile has a significant effect on its derived mass: simple interpolation from the clump boundary yields a 25% lower estimate, whereas making no correction for foreground and background emission attributes 49% more mass to the clump. Of the observed kinetic energy, ${{ \mathcal T }}_{\mathrm{bulk}}$ constitutes 68% in this case. Equating ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}$ and $-3{{aGM}}_{\mathrm{cl}}^{2}/5{R}_{\mathrm{cl}}$ (or equivalently, α_SMJ19 and α_BM92/a) requires a = 1.12; notably, this is quite close to unity. Taken together, we see that overestimating the mass, excluding ${{ \mathcal T }}_{\mathrm{bulk}}$, and adopting a value for a significantly above unity (e.g., a = a₁ = 5/3, corresponding to k = 2, as chosen by Roman-Duval et al. 2010 and Csengeri et al. 2017) can lead one to underestimate α for this object by as much as a factor of 7.3.

How representative is Cepheus L1251-1? We note that Perseus B1-3, the clump depicted in Figure 2, yields an identical estimate for a (i.e., a = 1.12) but contains only 22% of its kinetic energy in bulk form. Our estimate for its virial ratio is α_SMJ19 = 0.58, the lowest of our entire sample. In its case, neglecting to correct for foreground or background material, omitting ${{ \mathcal T }}_{\mathrm{bulk}}$, and adopting a = 5/3 would lead one to underestimate α by the more moderate factor of 2.7.

In the next section, we place these examples into context, and show that Cepheus L1251-1 and Perseus B1-3 are not outliers, but do bracket the range of massive clump properties.

We list our findings for all 85 of our Gould Belt NH₃ clumps in Table 3, and plot them for various choices of the analysis method in Figures 3 and 4.

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Table 3. Clump Properties and Virial Ratios Derived from Abel Reconstruction

Clump	R.A.	Decl.	Mass	Radius	αSMJ19	αSMJ19	αBM92	αBM92
			[M⊙]	[pc]		(No ${{ \mathcal T }}_{\mathrm{bulk}}$)		(No ${{ \mathcal T }}_{\mathrm{bulk}}$)
B1-1	03h33m36s	+31d18m49s	2.97	0.11	8.49	8.23	8.82	8.55
B1-2	03h33m18s	+31d18m31s	2.89	0.13	3.34	3.08	4.32	3.99
B1-3	03h33m18s	+31d04m44s	74.93	0.31	0.58	0.45	0.66	0.51
B1-4	03h32m43s	+30d58m41s	12.32	0.21	2.03	1.50	2.12	1.57
B1-5	03h32m24s	+30d48m01s	5.48	0.11	1.62	1.44	2.25	2.00
B1-6	03h31m26s	+30d44m00s	3.99	0.10	3.47	2.52	3.89	2.83
L1448-1	03h25m41s	+30d42m49s	58.84	0.28	1.15	0.54	1.30	0.61
L1451-1	03h25m35s	+30d20m17s	2.56	0.11	2.23	1.99	2.55	2.28
L1451-2*	03h24m33s	+30d22m09s	0.16	0.03	11.35	11.32	10.74	10.71
L1455-1	03h27m50s	+30d10m00s	18.41	0.25	2.25	1.42	2.56	1.63
L1455-2	03h28m07s	+30d04m55s	3.97	0.14	2.40	1.99	2.44	2.02
L1455-3	03h27m36s	+29d57m10s	0.55	0.06	5.64	5.34	6.17	5.85
NGC 1333-1	03h30m00s	+31d37m39s	2.76	0.12	1.99	1.79	2.52	2.26
NGC 1333-2	03h29m29s	+31d34m32s	0.82	0.07	4.12	3.97	4.78	4.60
NGC 1333-3	03h29m29s	+31d31m51s	1.77	0.09	2.31	2.20	3.11	2.97
NGC 1333-4	03h29m28s	+31d26m37s	7.44	0.20	2.27	1.82	2.37	1.91
NGC 1333-5	03h29m05s	+31d18m36s	20.97	0.19	1.76	1.35	2.11	1.61
NGC 1333-6	03h29m10s	+31d12m55s	80.46	0.29	1.29	0.73	1.63	0.93
NGC 1333-7	03h28m42s	+31d13m29s	15.24	0.24	3.83	1.56	4.58	1.87
NGC 1333-8	03h28m44s	+31d04m06s	8.00	0.17	3.08	1.64	3.50	1.86
Perseus-1	03h30m40s	+30d25m04s	0.95	0.06	3.62	3.46	4.38	4.18
Perseus-2	03h30m22s	+30d21m52s	1.28	0.10	3.60	3.37	4.36	4.08
IC 348-1	03h45m23s	+32d03m24s	0.92	0.09	4.68	4.40	5.67	5.33
IC 348-2	03h45m07s	+31d59m09s	1.02	0.09	5.08	4.95	5.37	5.23
IC 348-3	03h44m57s	+31d59m03s	1.14	0.08	5.26	5.01	5.84	5.57
IC 348-4	03h44m43s	+31d56m54s	0.95	0.07	8.74	6.63	9.67	7.34
IC 348-5	03h44m26s	+31d57m31s	5.68	0.17	2.96	2.33	3.25	2.56
IC 348-6	03h44m03s	+32d00m42s	36.46	0.28	1.26	0.87	1.42	0.98
B18-1	04h35m46s	+24d07m49s	3.62	0.08	1.81	1.29	2.17	1.55
B18-2	04h32m53s	+24d22m50s	4.53	0.09	1.32	1.13	1.45	1.25
B18-3	04h32m04s	+24d30m20s	2.33	0.08	2.01	1.52	2.57	1.94
B18-4	04h30m12s	+24d24m10s	0.36	0.04	4.82	4.69	5.12	4.98
B18-5	04h29m30s	+24d33m22s	2.68	0.06	1.28	1.21	1.41	1.33
B18-6	04h27m06s	+24d39m52s	0.31	0.05	7.73	7.62	7.76	7.64
HC 2-1	04h41m39s	+25d59m46s	4.77	0.10	1.00	0.89	1.05	0.93
HC 2-2	04h41m35s	+25d44m15s	10.01	0.16	2.04	1.28	1.67	1.05
HC 2-3	04h40m38s	+25d28m03s	0.08	0.04	36.92	35.29	36.74	35.12
HC 2-4	04h39m36s	+26d24m56s	1.70	0.07	1.83	1.75	2.01	1.92
HC 2-5	04h39m24s	+25d50m26s	0.51	0.05	8.53	7.61	9.76	8.71
HC 2-6*	04h39m45s	+25d39m39s	1.18	0.08	3.81	3.15	4.03	3.34
IC 5146-1	21h47m31s	+47d31m17s	65.38	0.43	1.51	1.11	2.00	1.47
IC 5146-2	21h47m15s	+47d31m37s	7.19	0.17	3.87	3.61	4.28	4.00
IC 5146-3	21h46m08s	+47d34m15s	17.70	0.27	1.71	1.60	2.06	1.93
IC 5146-4	21h45m14s	+47d32m02s	23.69	0.34	1.37	1.20	1.48	1.30
IC 5146-5	21h45m06s	+47d38m21s	38.07	0.36	1.39	1.12	1.76	1.42
Cepheus L1228-1	20h58m07s	+77d42m03s	3.07	0.10	1.76	1.66	2.00	1.88
Cepheus L1228-2	20h57m09s	+77d39m51s	0.57	0.06	5.39	5.36	5.90	5.86
Cepheus L1228-3	20h57m44s	+77d34m26s	11.11	0.16	1.88	1.20	2.34	1.50
Cepheus L1251-1*	22h39m09s	+75d09m56s	56.32	0.32	2.05	0.65	2.30	0.73
Cepheus L1251-2*	22h36m15s	+75d17m16s	4.25	0.14	1.77	1.48	2.26	1.88
Cepheus L1251-3	22h30m55s	+75d12m09s	26.25	0.26	1.01	0.53	1.00	0.53
Cepheus L1251-4	22h28m25s	+75d12m29s	4.66	0.16	1.67	1.49	1.91	1.70
CrA west-1	19h01m55s	−36d58m00s	7.64	0.09	3.01	2.22	3.30	2.43
CrA east-1	19h10m26s	−37d09m45s	1.84	0.05	1.81	1.69	2.11	1.97
L1688-1	16h29m05s	−24d22m08s	0.85	0.05	3.11	3.00	3.61	3.48
L1688-2	16h28m38s	−24d19m58s	0.94	0.06	3.23	2.98	3.70	3.42
L1688-3	16h28m29s	−24d37m48s	0.45	0.05	5.21	5.03	6.71	6.48
L1688-4	16h28m06s	−24d35m03s	1.04	0.06	3.08	2.91	4.02	3.79
L1688-5	16h27m42s	−24d43m57s	2.48	0.09	8.29	5.39	8.11	5.28
L1688-6	16h27m26s	−24d29m39s	13.28	0.12	2.47	1.89	2.64	2.02
L1688-7	16h27m02s	−24d34m14s	6.89	0.12	1.74	1.56	2.07	1.85
L1688-8	16h26m35s	−24d25m00s	5.83	0.07	1.75	1.54	2.54	2.25
L1689-1	16h32m34s	−24d30m10s	11.67	0.09	0.83	0.53	1.12	0.72
L1689-2	16h31m48s	−24d51m46s	5.23	0.08	1.45	1.24	1.66	1.43
L1689-3*	16h32m03s	−24d58m59s	4.68	0.11	3.12	1.99	3.38	2.15
Serpens MWC 297-1	18h28m15s	−03d48m49s	16.44	0.19	1.03	0.84	1.24	1.00
Orion A-1	05h35m11s	−04d57m12s	8.95	0.16	2.19	2.00	2.42	2.22
Orion A-2	05h35m11s	−05d37m50s	46.92	0.28	4.15	1.56	4.59	1.72
Orion A-3	05h35m14s	−05d53m21s	7.09	0.16	5.22	2.63	6.76	3.41
Orion A-4	05h36m17s	−06d12m15s	8.60	0.14	2.82	2.31	3.97	3.25
Orion A-5	05h35m17s	−06d15m18s	13.53	0.15	6.31	3.86	7.54	4.62
Orion A S-1	05h39m12s	−07d13m22s	10.76	0.22	4.83	1.60	5.52	1.83
Orion A S-2	05h39m37s	−07d24m32s	5.58	0.23	4.26	1.95	6.46	2.97
Orion A S-3	05h39m19s	−07d24m15s	36.26	0.33	1.09	0.84	1.28	0.98
Orion A S-4	05h40m02s	−07d28m15s	19.81	0.19	2.03	0.90	2.33	1.04
Orion A S-5	05h40m27s	−07d37m43s	20.12	0.23	1.32	0.76	1.43	0.82
Orion A S-6	05h40m30s	−07d44m18s	14.89	0.17	1.02	0.89	1.26	1.10
Orion B NGC 2023-2024-1	05h41m27s	−01d43m38s	0.72	0.07	14.87	14.05	15.51	14.65
Orion B NGC 2023-2024-2	05h41m17s	−01d47m55s	0.76	0.08	11.70	11.07	12.74	12.06
Orion B NGC 2023-2024-3	05h41m49s	−01d55m38s	30.73	0.10	1.22	1.16	1.65	1.56
Orion B NGC 2023-2024-4	05h41m51s	−01d58m00s	73.42	0.11	0.79	0.76	1.45	1.40
Orion B NGC 2023-2024-5	05h41m56s	−02d01m10s	1.22	0.09	8.51	7.57	9.49	8.44
Orion B NGC 2023-2024-6	05h41m38s	−02d19m08s	50.29	0.32	1.97	1.30	2.00	1.32
Orion B NGC 2023-2024-7	05h41m37s	−02d25m49s	4.42	0.15	7.77	6.82	7.89	6.94
Orion B NGC 2068-2071-1	05h46m10s	−00d16m31s	8.35	0.18	2.86	2.67	3.73	3.49

Note. R.A. and decl. values quoted refer to the geometric center of each clump. Asterisks denote clumps whose maps include regions missing NH₃ data, most of which have very small covering fraction.

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In Figure 3, we adopt our fiducial method (Abel reconstruction) for removing foreground and background matter, and compare the outcomes of other choices in the estimation of α. Two trends are immediately clear. First, α_BM92 and α_SMJ19 are very similar across our entire sample. Because we use the same data to derive both quantities, the comparison provides a statistical calibration of BM92's self-gravity parameter a, which our analysis shows is only slightly above unity (geometric mean value 1.125) in our Gould Belt clumps. For more detail, examining the ratio of the fits plotted in Figure 3, we see that a grows from unity to a maximum of 1.15 across the range of M_cl.

Second, excluding ${{ \mathcal T }}_{\mathrm{bulk}}$ suppresses α by an amount that changes with M_cl. The difference is insignificant for M_cl < M_⊙, where clumps harbor subsonic motions and are not highly resolved. However, the influence of ${{ \mathcal T }}_{\mathrm{bulk}}$ grows with mass and amounts to 0.26 dex (a factor of 1.81) for M_cl ∼ 100 M_⊙. This is important, as most analyses adopt for α a typical value of the total beam-wise line width σ_eff(x, y), which has the effect of neglecting ${{ \mathcal T }}_{\mathrm{bulk}}$ in the estimate of ${{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}$. (There do exist counterexamples, however: for instance, Roman-Duval et al. 2010's Equation (7) includes all of ${{ \mathcal T }}_{\mathrm{cl},\ 2{\rm{D}}}$.)

Methods for extracting Σ_cl from the column density map Σ also influence the virial parameter. In Figure 4, we compare several methods: removing emission from around the clump using our default method of Abel reconstruction, removing an interpolated value from the clump boundary, or making no correction (in which case Σ_cl = Σ within the boundary). The trends in α(M_cl) are shown for these different choices.

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Simple interpolation removes the most mass, while Abel reconstruction, by accounting for projection effects, attributes less material to the envelope. These differences are greatest for the least massive clumps, whose column density contrasts the least with their surroundings. The choice of extraction method affects the inferred virial parameter, mostly through the relation $\alpha \propto {M}_{\mathrm{cl}}^{-1}$ that applies when Σ_cl is multiplied by a constant. However, the effect on α diminishes for higher clump masses. The effect is also roughly in the direction of the trend line α(M_cl), that is, it is partly degenerate with the original trend. These facts diminish the influence of the extraction method on the trend of α with clump mass.

In summary, although α drops with M_cl within our collection of clumps, it is consistently below unity at high masses only when one ignores spatially resolved component of the bulk kinetic energy, when one overestimates the gravitational self-energy by adopting a > 1.1, or when one does not correct at all for foreground and background matter projected within the clump boundary. Using our most complete analysis, which addresses these shortcomings, we find no evidence that the massive clumps in our sample are consistently subvirial.

Note that the clumps in our sample typically have α_SMJ19 ∼5 for M_cl = 1 M_⊙, while studies of dense cores, such as that of Johnstone et al. (2000), find that cores approach the critical state of a Bonnor–Ebert sphere (in which α = 1.683) at around a solar mass. The discrepancy is not surprising, considering that our algorithm defines clumps that include as much NH₃ emission as possible. Any compact cores in our sample are therefore enclosed within larger clumps.

5.1. Sources of Error

As first stressed by Kauffmann et al. (2013), many studies have found massive clumps to be moderately to strongly subvirial (Roman-Duval et al. 2010; Pillai et al. 2011; Wienen et al. 2012; Ragan et al. 2012; Li et al. 2013; Tan et al. 2013; Urquhart et al. 2014; Friesen et al. 2016; Svoboda et al. 2016; Kirk et al. 2017b; Keown et al. 2017; Redaelli et al. 2017; Csengeri et al. 2017; Traficante et al. 2018a; Keown et al. 2019; Chen et al. 2019; Kerr et al. 2019; Billington et al. 2020; Traficante et al. 2020). In our fiducial analysis, which follows SMJ19, we do not. We trace the difference to several key features.

First, we account for the component of kinetic energy associated with spatial variations of the line-of-sight velocity in the plane of the sky, which we call ${{ \mathcal T }}_{\mathrm{bulk}}$. BM92 include this component (see their Appendix C), but many later studies exclude it by adopting a typical value of σ_eff(x, y) for σ_cl in the evaluation of α_BM92. There do exist exceptions, such as Roman-Duval et al. (2010), who include bulk energy in their Equation (7), and works like Colombo et al. (2019) in which σ_cl,z is defined in terms of the intensity-weighted variance of velocity (as adopted, for instance, by Solomon et al. 1987). Including ${{ \mathcal T }}_{\mathrm{bulk}}$ matters most for massive clumps: these contain supersonic internal motions, but their dense substructures tend to have narrow internal line widths with subsonic nonthermal components. When the clump is well resolved, these substructures will tend to set the line width within each beam. The effect therefore varies with resolution. This has been observed, for instance, in B5 by Pineda et al. (2010, 2015) and in Orion by Friesen et al. (2017) and Monsch et al. (2018).

Second, we compute the gravitational self-energies of our clumps using the robust method of SMJ19. Applied to an ensemble of clumps, this provides a sensitive calibration of the BM92 parameter a. An average over our clump sample indicates a ≃ 1.13, while fits to the mass dependence indicate that a grows from unity for low clump masses to ∼1.15 for massive clumps. Because our clumps have generally modest projected aspect ratios (with the exception of HC 2-2), these represent low values of central concentration: the low-mass result a ≃ 1 is consistent with these cores being essentially uniform-density, pressure-confined objects, and our entire sample is less centrally concentrated than a critical Bonnor–Ebert sphere (for which a = 1.22). Other analyses often ignore a, effectively taking a = 1 (e.g., Billington et al. 2020); or they adopt a specific value, such as a = 1.25 (corresponding to k = 3/2: e.g., Keown et al. 2019) or a = 1.67 (corresponding to k = 2: e.g., Roman-Duval et al. 2010; Csengeri et al. 2017). We note that using the definition of M_vir in Rohlfs & Wilson (2004) to estimate α = M_vir/M_cl (e.g., Wienen et al. 2012) amounts to adopting a = 5/6. We also note that our sample does not support the suggestion by Ballesteros-Paredes et al. (2018) that cases in which clumps are inferred to be unbound (α > 2) might result from their values of a being significantly underestimated.

Third, while we use dust emission to trace mass, we are careful to account for foreground and background emission because overestimating M_cl leads one to underestimate α, roughly according to the scaling $\alpha \propto {M}_{\mathrm{cl}}^{-1}$. For this we prefer the Abel reconstruction method advanced in SMJ19.

On this point, we note that prior studies determine clump masses and profiles using a wide range of data sources and analysis techniques (see Kauffmann et al. 2013), not all of which exclude foreground and background contamination. Common approaches include subtracting an interpolated background, matching and extracting a template profile, or making no correction. Dendrogram analysis (Rosolowsky et al. 2008) can employ any of these, although by default it makes no correction. Approaches tailored to extract compact cores tend to fit templates (e.g., Johnstone et al. 2000) or subtract an estimated background, as in Men'shchikov et al. (2012). As we mentioned in Section 2, the spatial filtering afforded by interferometric or chopped observations accomplishes background subtraction in an approximate way.

In Figure 5, we review the effect of each of the three key features of our analysis mentioned above. We note that while foreground/background subtraction can dominate the error for an individual clump, this effect is partly degenerate with the trend in α(M_cl), as shown in Figure 4. Focusing on the mass scale 100 M_⊙, we see that neglecting ${{ \mathcal T }}_{\mathrm{bulk}}$ depresses one's estimate for α_SMJ19 by about a factor of 1.8 (depending slightly on the clump extraction method). Including foreground and background emission suppresses it by a further factor of 1.4. Adopting the extreme choice a = 1.67 over the more accurate value of 1.15 reduces α further by a factor of 1.45. Compounding these leads to an underestimate of α by a factor of 3.6.

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This analysis indicates that, while all three factors have comparable impact, bulk kinetic energy is the most important effect at the reference mass of 100 M_⊙, and the only one that is growing more important with increasing clump mass. Although its impact is clearly resolution-dependent, this behavior indicates that bulk kinetic energy might dominate at masses higher than those included in our sample.

A further and potentially significant difference is that we associate each clump with a definite boundary on the sky, which we derive from the emission of a reliable dense gas tracer (NH₃) and which we use to infer quantities like the clump radius and gravitational energy. Compared to approaches in which radii are defined by moments of the emission, or by fitting to predetermined model templates, ours has the advantage that the derived quantities reflect those of the three-dimensional clump in question in a mathematically rigorous way (Equations (5) and (7)).

In summary, we conclude that it is easy to underestimate α by a significant margin as a result of choices in the analysis. Addressing these points carefully, as we do here, alleviates the physical puzzle that a finding of consistently subvirial clumps would pose—namely, that while a mildly subvirial population may be close to equilibrium (McKee & Zweibel 1995; Tan et al. 2013) for the observed degree of magnetization (Crutcher 2012), this is not true of the strongly subvirial state. And, although one could prepare initial conditions far from equilibrium in which α ≪ 1, this state would be ephemeral, persisting only for a fraction of a freefall time, as noted, for instance, by Kauffmann et al. (2013). Moreover, it is not clear how a dense clump could be assembled in this state without concomitantly developing velocities at the virial scale.

Within our Gould Belt sample, we find that clump motions are, in fact, virial in magnitude, with α ≃ 1.34 and little dependence on mass at the upper end of our sample. This impacts any conclusions that depend on the virial ratio, such as the expected mode of protostellar accretion. It remains to be seen whether conclusions drawn from this sample will be relevant to samples that include more massive clumps.

Our result that the virial parameter levels off at a value of order unity, rather than continuing to decline with increasing clump mass, points to a change in physical or evolutionary state for the most massive clumps in our sample. Could this reflect a greater role for dynamical feedback from star formation? This would be consistent with the fact that protostellar outflows are active in a number of our massive clumps (Kun et al. 2008; Devine et al. 2009; Arce et al. 2010; Curtis et al. 2010; Feddersen et al. 2020). It is also consistent with the fact that class 0 protostars are highly segregated toward these massive clumps and the structures that host them, as is evident in Orion A (e.g., Stutz & Kainulainen 2015) and Perseus (e.g., Mercimek et al. 2017).

It is already well established that the local star formation rate has a threshold-like (Onishi et al. 1998) or power-law (Heiderman et al. 2010) dependence on Σ, and that the recent history of star formation correlates with the slope of the local probability density function of Σ (Sadavoy 2013; Stutz & Kainulainen 2015). We find that the clump-averaged values of Σ and Σ_cl both correlate with M_cl and anticorrelate with α, so the former rule, at least, is qualitatively consistent with and anticorrelation between α and star formation intensity. We consider this a promising avenue for future work.

We thank our referee and Christopher McKee for useful feedback and important suggestions. A.S. thanks Tina Peters for advice on data analysis techniques. The research of A.S. and C.D.M. is supported by an NSERC Discovery Grant. A.P. acknowledges the support from the Russian Ministry of Science and Higher Education via the State Assignment Project FEUZ-2020-0038. A.P. is a member of the Max Planck Partner Group at the Ural Federal University. We acknowledge that this work was conducted on the traditional land of the Huron-Wendat, the Seneca, and the Mississaugas of the Credit; we are grateful for the opportunity to work here.

Software: Astropy (Astropy Collaboration et al. 2013, 2018), Scikit-learn (Pedregosa et al. 2011).

We describe here our procedure for managing discretization errors in the computation of ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}$.

SMJ19 suggest computing ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}$ from Σ_cl(x, y) via

$$\begin{eqnarray}&&{{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}=\displaystyle \frac{1}{\pi }\int {\rm{\Psi }}(x,y){{\rm{\Sigma }}}_{\mathrm{cl}}(x,y){dx}\,{dy},\end{eqnarray} \tag{ A1 }$$

where

$$\begin{eqnarray}&&{\rm{\Psi }}(x,y)=-\displaystyle \frac{G}{\sqrt{{x}^{2}+{y}^{2}}} \circledast {{\rm{\Sigma }}}_{\mathrm{cl}}(x,y).\end{eqnarray} \tag{ A2 }$$

This procedure is perfectly valid in the continuous limit, but requires a small modification to account for the fact that Σ_cl is sampled on a discrete grid at some resolution δ x. The most important consequence of discreteness is that, because the Green's function $-G/\sqrt{{x}^{2}+{y}^{2}}$ diverges at the origin, its central element must be replaced with some finite value − ε G/δ x, where ε parameterizes the self-energy of a single pixel. We adopt

$$\begin{eqnarray}&&\varepsilon =4\left[{\sinh }^{-1}(1)-\displaystyle \frac{\sqrt{2}-1}{3}\right]=2.973,\end{eqnarray} \tag{ A3 }$$

the value that correctly yields the self-energy of a square of width δ x in the case that Σ_cl(x, y) is uniform. We test this choice using discretized projections of structures of known self-gravitational energy, such as uniform-density spheres, and find that it reduces the error in ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}$ by a factor of about 2.4 relative to the outcome of choosing ε = 0. Significantly smaller improvements can be obtained by adjusting the noncentral elements of the Green's function to account for the finite sizes of pixels. However, we do not implement these because the error can further be reduced by interpolating Σ_cl to a finer grid before evaluating ${{ \mathcal W }}_{\mathrm{cl},2{\rm{D}}}$. Finding that the relative error when using ε is ∼ δ x/2R_cl, we adopt a strategy in which each clump's column density is resampled so that R_cl > 100δ x, implying errors well below 1%.