The Green Bank Ammonia Survey: Dense Cores under Pressure in Orion A

NH₃ observations were obtained through the GAS, a large project to map the ammonia (1, 1), (2, 2), and (3, 3) rotation-inversion transitions across the high-extinction regions of nearby Gould Belt molecular clouds using the Green Bank Telescope's K-band Focal Plane Array. The survey strategy, data reduction procedure, and basic data properties are described in detail in Friesen et al. (2017). The spatial resolution is 32'' (0.064 pc), while the spectral resolution is 0.07 km s⁻¹. The northern portion of the Orion A cloud is one of the first four areas mapped, and observations of it are presented in Friesen et al. (2017) as "Data Release 1" (DR1).¹⁵ The DR1 observations of Orion A cover the ISF and areas slightly southward of it. Additional data further south were obtained at a later date and will be included in a second data release. Figure 1 shows an overview of the area mapped overlaid on the JCMT SCUBA-2 850 μm image of the region, with the dense cores (see Section 2.2) and protostars (see Section 2.3) overlaid in each panel.

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Although NH₃ is an excellent tracer of dense gas, defining dense cores from its emission alone can be complicated by varying abundance levels. At the distance of Orion (∼415 pc; Menten et al. 2007; Kim et al. 2008), the ∼32'' resolution of our GAS data also presents a challenge in identifying individual dense cores in such a complex and clustered environment. Submillimeter continuum emission offers an alternative method for identifying dense cores and estimating their sizes and masses. There are, however, several separate significant sources of uncertainty in the conversion between flux density and mass, as well as the potential for chance alignments of lower-density structures giving rise to apparent cores. The JCMT GBS (Ward-Thompson et al. 2007b) observed 6.2 square degrees around the Orion A molecular cloud at 850 μm and 450 μm with SCUBA-2, with resolutions of 146 and 98 (Dempsey et al. 2013), respectively. Observations of the northern and southern portions of Orion A were first published in Salji et al. (2015) and Mairs et al. (2016), respectively. For our analysis, we use the dense core catalog presented in Lane et al. (2016), which covers the entire Orion A complex.

Lane et al. (2016) analyzed the JCMT GBS 850 μm data for the entire Orion A molecular cloud, using a map that reaches the full survey depth but is slightly poorer at recovering the largest-scale structures than the current best reduction method. Lane et al. (2016) identified dense cores using two independent methods, getsources (Men'shchikov et al. 2012) and FellWalker (Berry 2015), and analyzed their clustering properties. The getsources algorithm is a multi-scale, multi-wavelength source extraction algorithm. It first decomposes emission from each wavelength into a variety of scales, and then uses the combined information across wavelengths to create a Gaussian-based model that characterizes the small-scale sources and separates them from larger-scale emission features. In contrast, the FellWalker algorithm separates peaks based on local gradients, assigning each pixel to the peak that the local gradient points toward. FellWalker therefore does not assume any particular source geometry, and in the Lane et al. (2016) catalog, no background subtraction was performed to remove large-scale structures. Since Orion A is home to quite complex emission structures, Lane et al. (2016) found that in general getsources performed better in isolating dense and compact structures. For our main analysis, we therefore adopt the getsources-based catalog of Lane et al. (2016), although in Appendix A, we present an analysis using instead the FellWalker-based catalog. In Appendix A, we demonstrate that our results are robust against changes in the core catalog used.

The Orion A catalog of Lane et al. (2016) provides the sizes, total fluxes, and peak positions of the dense cores. We approximate the dense core radius as the geometric mean of the major and minor axis FWHMs fit by getsources, noting that at a radius equal to the FWHM, a Gaussian profile is at one eighth of its full height, and is therefore a reasonable estimate of the full core extent. We assume a distance of 415 pc to Orion A, and correct the catalog values of Lane et al. (2016) from their assumed distance of 450 pc. We also correct the core sizes for the telescope beam; for cores with sizes less than half of the 146 FWHM of the SCUBA-2 850 μm beam, deconvolution is likely to be unreliable, and so we report this value as the upper limit to their true size. We convert the total flux of each core measured by getsources to a mass using

$$\begin{eqnarray}M & = & 1.30\left(\displaystyle \frac{{S}_{850}}{1\,\mathrm{Jy}}\right){\left(\displaystyle \frac{{\kappa }_{850}}{0.012{\mathrm{cm}}^{2}{{\rm{g}}}^{-1}}\right)}^{-1}\\ & & \times \left(\exp \left(\displaystyle \frac{17\,{\rm{K}}}{{T}_{d}}\right)-1\right){\left(\displaystyle \frac{D}{450\mathrm{pc}}\right)}^{2}{M}_{\odot },\end{eqnarray} \tag{ 1 }$$

where S₈₅₀ is the total flux density at 850 μm, ${\kappa }_{850}$ is the dust opacity at 850 μm, T_d is the dust temperature, and D is the distance. We assume a constant dust grain opacity at 850 μm of 0.012 cm² g⁻¹, consistent with previous JCMT and Herschel Gould Belt analyses, and which includes the standard dust-to-gas ratio of 0.01 (e.g., Kirk et al. 2013; Pattle et al. 2015). (Note that the popular model 5 of Ossenkopf & Henning (1994) for icy grains would give a larger opacity of 0.018 cm² g⁻¹ at 850 μm.) For each core, we adopt a dust temperature equal to the kinetic temperature derived from our GAS NH₃ observations (see Section 2.4). For cores where a kinetic temperature could not be measured, we assume a value of 18 K, which is the mean temperature measured for the cores in NH₃. Previous works, including Lane et al. (2016), assumed a constant dust temperature of 15 K, which would give masses roughly 30% larger than those quoted here for a mean dust temperature of 18 K. Similarly, assuming instead a higher temperature of 21 K would give masses that were 20% smaller. We note that while a 20%–30% change in estimated mass can be significant for a detailed study of a single object, this level of difference has relatively little impact on our population study because of the broad range in properties measured. For example, most of the cores would not change their status of being gravitationally bound or unbound in Section 3.2 with only a 20%–30% change in their mass. Much larger variations in the mass estimated for cores arise in regions of complex emission structure such as Orion A through the choice in algorithm used to identify cores. In Appendix A, the cores identified using FellWalker tend to be more massive and larger than the getsources cores. Highlighting the challenges of core identification, there is not always a one-to-one correspondence between objects in the two catalogs, because getsources tends to split emission structures more finely than FellWalker does. Despite these large uncertainties in defining individual cores and accurately measuring their properties, our overall conclusions about the typical virial state of cores are similar using either algorithm, suggesting that our overall conclusions about bulk virial properties are robust.

Finally, we note that some of the dense cores in Lane et al. (2016), although they are clearly detected, have poor Gaussian fits. While this was not important for the original clustering analysis of Lane et al. (2016), poor estimates of the core radius could significantly impact our virial analysis. We therefore apply several cuts to the getsources catalog of Lane et al. (2016), requiring dense cores in our present analysis to have parameters:

• gs_sig_glob ≥ 1
• gs_sig_mono8 ≥ 7
• gs_good > 0
• gs_relp850 > 3
• gs_relt850 > 3.

In short, these criteria imply "significant" fits (gs_sig_glob, gs_sig_mono8) with fitted peak fluxes of signal-to-noise ratio (S/N) ≥ 3 (gs_relp850) and fitted total fluxes of S/N ≥ 3 (gs_relt850), and no getsources flags (gs_good) associated with the fits. These fitting criteria reduce the original 919 dense cores in Lane et al. (2016) to 610, primarily excluding the faintest cores with the poorest fits. In the culled catalog, the lowest mass core identified is 0.09 ${M}_{\odot }$. Of the 610 reliably fit dense cores, roughly half (311) lie within the area mapped by GAS. The median mass of the cores is 0.7 ${M}_{\odot }$ for those with reliable getsources sizes and reliable temperature and kinematic measures (see Section 2.4).

Figure 1 shows the GBS 850 μm emission over the area observed with GAS, with the culled getsources catalog overlaid. The dearth of getsources cores in the center of the ISF visible in Figure 1 is caused by two effects. First, getsources excludes large-scale emission modes from its core-fitting routine, and hence a significant amount of flux within the center of the ISF is not attributed to any getsources core. Second, the larger-scale emission within the center of the ISF increases the difficulty of fitting Gaussian models to the compact emission features present, leading to a greater fraction of cores being culled in this particular area. We note, however, that our overall results are similar even when the full getsources catalog from Lane et al. (2016) is used.

In Table 1, we summarize the properties of the cores identified by getsources for which there are measurements of their GAS-derived properties (see following sections).

Table 1.  Properties of the Dense Cores

IDa	R.A.a	Decl.a	Ma	Reffa	Ca	Pr?a	${\sigma }_{\mathrm{obs}}$ (km s−1)b			Tkin(K)b			$\mathrm{log}({P}_{c}/{k}_{{\rm{B}}})$ c	Virial	$-{{\rm{\Omega }}}_{G}/{{\rm{\Omega }}}_{K}$ d	${{\rm{\Omega }}}_{G}/{{\rm{\Omega }}}_{P,c}$ d
	(J2000)	(J2000)	(${M}_{\odot }$)	(pc)			mean	err	std	mean	err	std	log(K cm−3)	Ratiod
5	83.84726	−5.02459	8.28	0.017	0.64	Y	0.311	0.002	0.041	17.5	0.1	0.8	7.3	1.4E+00	2.6E+00	2.0E+01
9	83.86135	−5.16620	11.10	0.023	0.48	Y	0.435	0.001	0.067	23.3	0.1	1.5	7.3	8.2E-01	1.5E+00	1.1E+01
10	83.82113	−5.32181	4.92	0.017	0.51	N	0.542	0.001	0.075	25.2	0.0	0.8	7.5	3.9E-01	6.2E-01	4.2E+00
12	83.81898	−5.32483	3.84	0.017	0.58	N	0.517	0.001	0.084	25.5	0.0	1.0	7.5	3.7E-01	5.2E-01	2.5E+00
16	83.81604	−5.33532	2.06	0.025	0.69	N	0.477	0.001	0.084	28.1	0.0	1.5	7.5	8.5E-01	2.1E-01	1.4E-01
17	83.82469	−5.31807	2.93	0.017	0.54	N	0.563	0.001	0.064	25.2	0.0	0.9	7.5	2.9E-01	3.5E-01	1.6E+00
18	83.81026	−5.31189	3.64	0.026	0.62	N	0.450	0.001	0.109	23.7	0.0	0.8	7.4	6.1E-01	4.1E-01	5.0E-01
19	83.86445	−5.15861	3.11	0.027	0.63	Y	0.423	0.001	0.074	23.7	0.1	1.7	7.3	5.8E-01	3.8E-01	4.8E-01
21	83.84284	−5.01978	3.61	0.017	0.56	Y	0.272	0.001	0.052	17.0	0.1	0.7	7.3	8.5E-01	1.3E+00	3.7E+00
23	83.82478	−5.00483	2.49	0.017	0.62	N	0.341	0.001	0.033	17.8	0.0	0.4	7.2	5.2E-01	6.9E-01	1.9E+00

Notes.

^aCore properties based on the getsources catalog of Lane et al. (2016). Only cores that have kinematic properties measured by GAS are listed. Columns are the core ID in Lane et al. (2016), central position, mass (estimated using Equation (1) from the total flux), effective radius (geometric mean of the major and minor axis lengths), concentration as derived using Equation (11) and whether the core has an associated protostar. ^bVelocity dispersion (${\sigma }_{\mathrm{obs}}$) and kinetic temperature (T_kin) measured for the cores, averaging over all core pixels where sufficient signal was present. The value reported for each quantity is the mean weighted by the inverse square of the uncertainty. The formal error in the weighted mean value is also reported (second column), as is the weighted standard deviation (third column). This latter quantity is more reflective of the variation between fitted values across the core. ^cEstimated pressure on the core boundary due to the weight of the overlying cloud material. See Section 3.2 for more information. ^dVirial parameters estimated according to Section 3.2. The virial ratio is given by $-({{\rm{\Omega }}}_{G}+{{\rm{\Omega }}}_{P,c})/2{{\rm{\Omega }}}_{K}$. $-{{\rm{\Omega }}}_{G}/{{\rm{\Omega }}}_{K}$ reflects the balance of gravitational binding versus thermal pressure, while ${{\rm{\Omega }}}_{G}/{{\rm{\Omega }}}_{P,c}$ reflects whether gravity or external cloud pressure dominates the confinement of the cores.

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

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Lane et al. (2016) further classify their dense cores as being protostellar or starless based on whether or not they are spatially coincident with a protostar identified by Megeath et al. (2012) using Spitzer observations or by Stutz et al. (2013) using Herschel data. Due to the highly clustered nature of Orion A, Lane et al. (2016) required a separation of less than one beam radius (725) from the location of the dense core's peak for a core to be classified as protostellar. Note that this requirement of a small separation ensures that any given protostar can be associated with a maximum of one dense core. Meingast et al. (2016) recently identified YSOs in Orion based on near-infrared observations using VISTA. We searched their catalog for evidence of additional protostellar cores, but found only three possible new associations with our core catalog, all of which were listed in Meingast et al. (2016) as previously known class II sources. For simplicity, we did not reclassify these three cores. Protostellar and starless dense cores are shown in Figure 1 in the right panel, while the protostars themselves are shown in the left panel.

The GAS NH₃ (1, 1) and (2, 2) observations were used to estimate the properties of the dense gas, as described in more detail in Friesen et al. (2017). Our interest here is in the velocity dispersion, σ, and the kinetic temperature, T_kin. Figure 2 shows a comparison of T_kin overlaid with the SCUBA-2 850 μm emission of Orion A, highlighting the well-known fact that the most active part of the ISF tends to be noticeably warmer than its surroundings. We note that for the GAS DR1, all spectra are fit with a single velocity component. A visual inspection of the spectra suggests that a single velocity component is sufficient to describe the majority of NH₃ spectra, even in Orion A; however, several per cent of the spectra require a second velocity component to be well described (Friesen et al. 2017). Fits with multiple velocity components are planned for a future GAS data release.

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Using the GAS NH₃-based property maps, we calculate the weighted mean kinetic temperature and velocity dispersion of each dense core. For the getsources-based dense core catalog used for the bulk of our analysis, we consider all GAS pixels that lie within a radius of one core FWHM from each core's peak (i.e., the same area as the core's full extent, as discussed in Section 2.2), and calculate the weighted mean of each property (weighting by the inverse square of the uncertainty in the fitted parameter). We calculate both the error in the weighted mean as well as the weighted standard deviation, to measure the variation in property values across each core. These values are all listed in Table 1. We also measured the kinetic temperature and line width at the peak position for all of the cores, and found largely similar results to those presented here. A total of 29 protostars and 282 starless cores lie within the GAS observing footprint. Of these, 26 protostars and 211 starless cores have kinematic properties measured, i.e., there was sufficient sensitivity in the ammonia data to estimate line widths and kinetic temperatures. The velocity dispersions are similar for the starless and protostellar cores, with a mean and standard deviation of 0.38 km s⁻¹ ± 0.20 km s⁻¹ and 0.37 km s⁻¹ ± 0.17 km s⁻¹, respectively. The kinetic temperatures are also similar. After removing less reliable measures where ${T}_{\mathrm{kin}}\gt 30$ K, we find typical kinetic temperatures of 18 K ± 4 K for the starless cores, and 17 K ± 3 K for the protostellar cores. Previous observations have tended to find smaller kinetic temperatures for starless cores than for protostellar cores—for example, Jijina et al. (1999) compiled all available NH₃ observations in the literature and found median values of the kinetic temperature of 12.4 K and 15 K for apparently starless and protostellar cores respectively, while more recent work in the Perseus molecular cloud reveals typical kinetic temperatures of 10.6 K and 11.9 K for starless and protostellar cores respectively (Foster et al. 2009). Both Jijina et al. (1999) and Foster et al. (2009) also note that environment appears to play a stronger role than protostellar content in determining the kinetic temperature. Jijina et al. (1999) found median kinetic temperatures of 20.5 K and 12.0 K for cores within and outside clusters, while Foster et al. (2009) measured typical kinetic temperatures of 12.9 K and 10.8 K, for clustered and unclustered cores respectively, in Perseus. Since all of the dense cores we analyze in Orion A lie within a highly clustered environment, we might not expect to see cores with lower kinetic temperatures. This general trend of higher kinetic temperatures being present in more clustered environments is also obvious across the four GAS DR1 regions (Friesen et al. 2017), where the isolated B18 (Taurus) has the lowest kinetic temperatures, and the moderately clustered NGC 1333 (Perseus) and L1688 (Ophiuchus) have intermediate kinetic temperatures. It is possible that our estimated T_kin values in Orion A are also are slightly elevated due to contamination from the warmer envelope material surrounding the dense cores.

To estimate the ambient pressure due to the weight of surrounding molecular cloud material on the dense cores, we require a map of the total column density in Orion A. Lombardi et al. (2014) derived such a map by performing point-by-point modeling of the spectral energy distribution of flux measured by both the Planck and Herschel Space Telescopes. We use the 850 μm map of optical depth derived in Lombardi et al. (2014), and convert into column density using the equations and constants given in their Equations (12) and (16):

$$\begin{eqnarray}&&{A}_{K}=\gamma {\tau }_{850}+\delta \end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Sigma }}}{{A}_{K}}=\mu {\beta }_{K}{m}_{p},\end{eqnarray} \tag{ 3 }$$

where A_K is the extinction in the K-band, $\gamma =2640$ mag, $\delta =0.012$ mag, Σ is the column density (in mass units), $\mu =1.37$, ${\beta }_{k}=1.67\times {10}^{22}$ cm⁻² mag⁻¹, and ${m}_{p}=1.67\,\times {10}^{-24}$ g.¹⁶ Note that the factors γ and δ are fitted by Lombardi et al. (2014), and they obtain slightly different values in Orion B than in Orion A. We adopt the Orion A values here. The left panel of Figure 3 shows the map of column density from Lombardi et al. (2014) with the scalings above adopted.

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Every position in the map of column density may have contributions from structures on a variety of size scales. To estimate the pressure on the cores from the weight of the overlying molecular cloud, we need to consider only the portion of the column density that is attributable to larger-scale structure. In particular, the 36'' angular resolution of the map of column density (matching the Herschel resolution at 500 μm) is comparable to the size of the dense cores, suggesting that in the vicinity of the dense cores, a non-negligible portion of the column density measured could arise from the cores themselves and not from larger-scale structures. We therefore filter out the smallest scale structures from the map of column density when estimating the pressure of the cloud weight. To perform this filtering, we use a similar method to that in Kainulainen et al. (2014). Namely, we use the à Trous wavelet transform to measure the amount of structure on various scales.¹⁷ Structures are measured on scales of ${2}^{N}$ pixels, by smoothing and subtracting the smoothed image from the remainder, going from the largest to the smallest scale. The large-scale structure can then be represented as the sum of all single-scale structure maps at sizes above the desired smoothing scale.

In the maps of Lombardi et al. (2014), each pixel is 15'', corresponding to 0.03 pc at a distance of 415 pc. Dense cores tend to have sizes between 0.03 pc and 0.2 pc (Bergin & Tafalla 2007), while larger-scale clumps and clouds span sizes from several tenths of a parsec to more than 10 pc. For a conservative estimate, we assume that the structures of order 0.5 pc and larger belong to the cloud rather than the dense core. The closest smoothing scale to 0.5 pc is 16 pixels, which corresponds to 0.48 pc. For our nominal best estimate of the column density attributable to the cloud, we sum all structures of sizes ∼0.5 pc and larger generated by the à Trous algorithm. In Appendix B, we show that our qualitative conclusions remain similar even when we change the limiting size scale for the features of the cloud column density. Figure 3 shows the original map of column density of Lombardi et al. (2014), as well as the 0.5 pc filtered version of the map, which we use for the majority of our analysis.

Within the large-scale cloud structure, dense cores are also usually found to reside within filaments. Observations from Herschel suggest that filaments have widths of order 0.1 pc (Arzoumanian et al. 2011; André et al. 2014), similar to the size of the dense cores, which makes separating cores and filaments unfeasible using the simple filtering method described above. In our analysis in Section 3.3, we approximate the bounding pressure of filaments using a different method than that discussed here. Finally, we note that not all of the column density may belong to structures that the core is associated with, i.e., some of the column density may belong to additional structures along the line of sight. This is especially important to acknowledge in a region as complex as Orion A. While multiple structures along the line of sight are likely more common at intermediate size scales than at the largest cloud scale, we note that our estimate of the cloud column density may be slightly too high.

We first examine the relationship between the line widths and sizes of the dense cores. The observed line width, ${\sigma }_{\mathrm{obs}}$, has contributions from both turbulent and thermal energy, i.e.,

$$\begin{eqnarray}&&{\sigma }_{\mathrm{obs},{\mathrm{NH}}_{3}}^{2}={\sigma }_{\mathrm{turb}}^{2}+{\sigma }_{\mathrm{therm},{\mathrm{NH}}_{3}}^{2}.\end{eqnarray} \tag{ 4 }$$

We assume that the kinetic temperature for the mean gas is the same as we measure for NH₃, and calculate the total line width by subtracting the thermal contribution from NH₃ and adding in the assumed total gas thermal contribution, i.e.,

$$\begin{eqnarray}&&{\sigma }_{\mathrm{tot}}=\sqrt{{\sigma }_{\mathrm{obs}}^{2}-\displaystyle \frac{{k}_{{\rm{B}}}{T}_{\mathrm{kin}}}{{\mu }_{{\mathrm{NH}}_{3}}{m}_{{\rm{H}}}}+\displaystyle \frac{{k}_{{\rm{B}}}{T}_{\mathrm{kin}}}{{\mu }_{\mathrm{mean}}{m}_{{\rm{H}}}}},\end{eqnarray} \tag{ 5 }$$

where k_B is the Boltzmann constant, ${\mu }_{{\mathrm{NH}}_{3}}$ is the molecular weight of NH₃ (17), ${\mu }_{\mathrm{mean}}$ the mean molecular weight, and m_H the mass of a hydrogen atom. Following Kauffmann et al. (2008), we take ${\mu }_{\mathrm{mean}}=2.37$.

Bulk motions within a core can also contribute to the total velocity dispersion, and are usually quantified through measurement of the standard deviation in the centroid velocity of the gas measured across the core. This contribution would then be added to the total velocity dispersion in quadrature. We find that including this term typically has a small effect on the resulting total velocity dispersions, adding less than 0.1 km s⁻¹ to the total velocity dispersion calculated without its inclusion for more than 90% of the cores. The mean difference in the total velocity dispersion between including and excluding the standard deviation in centroid velocity is 0.04 km s⁻¹, while the median difference is 0.02 km s⁻¹. We therefore exclude the contribution from the change in line centroid to the total velocity dispersion that we use for our analysis.

Figure 4 shows the total line width and size measured for each core, using Equation (5). All of the cores have significantly supersonic velocity dispersions—the velocity dispersion expected for a purely thermal gas at the mean T_kin value observed is shown by the horizontal dashed line, and it lies about a factor of two lower than the smallest total velocity dispersions measured. These line widths are larger than is typical for the nearest molecular clouds, such as those in Perseus, where cores tend to have roughly transsonic total line widths (e.g., Foster et al. 2009). Orion, however, has long been recognized as an environment where cores have larger non-thermal motions present in dense gas tracers such as NH₃ (e.g., Caselli & Myers 1995; Jijina et al. 1999). These larger line widths do not appear to be an artefact of the larger distance to Orion as compared to nearby clouds such as Perseus. We compared the NH₃ line widths reported by Li et al. (2013) using VLA observations of higher angular resolution (∼5'' versus our 32'') and see no evidence that our measured line widths are systematically larger. On the contrary, our measurements tend to be slightly smaller, likely due to the poorer velocity resolution of the observations of Li et al. (2013).

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The dotted diagonal line in Figure 4 shows the standard line width–size relationship of Larson (1981) measured for larger-scale structures using CO observations. Clearly our population of cores does not follow Larson's scaling law, but instead it shows no variation as a function of core size, and only a large amount of scatter. This lack of change in line width as a function of size is exactly what is expected for the behavior of dense gas within cores, because the material lies in a zone of coherence (e.g., Goodman et al. 1998; Pineda et al. 2010), although the zone of coherence is typically also characterized by a thermally dominated velocity dispersion, which is not the case here.

We next compare the amounts of thermal and non-thermal support available to each dense core to their self-gravity. The thermal Jeans criterion is

$$\begin{eqnarray}&&{M}_{{\rm{J}},{\rm{T}}}=\displaystyle \frac{5R}{2G({\mu }_{\mathrm{mean}}{m}_{{\rm{H}}}/{k}_{{\rm{B}}}{T}_{\mathrm{kin}})}\end{eqnarray} \tag{ 6 }$$

(e.g., Bertoldi & McKee 1992) where ${M}_{{\rm{J}},{\rm{T}}}$ is the (thermal) Jeans mass, R is the radius, and G is the gravitational constant. If non-thermal support is included, we can rewrite the equation as

$$\begin{eqnarray}&&{M}_{{\rm{J}},\mathrm{tot}}=\displaystyle \frac{5R{\sigma }_{\mathrm{tot}}^{2}}{2G}\end{eqnarray} \tag{ 7 }$$

where ${\sigma }_{\mathrm{tot}}$ is the total line width given by Equation (5).

Figure 5 shows the mass and size measured for each dense core based on SCUBA-2 850 μm data compared to various Jeans stability criteria. Here, we use the mean values of T_kin and ${\sigma }_{\mathrm{nt}}$ (the non-thermal component of the velocity dispersion) obtained for dense cores where a good fit to these parameters was obtained. Unsurprisingly, turbulent motions dominate over thermal motions in all cores. When non-thermal motions are considered in addition to the thermal pressure supporting the dense cores, nearly all of the dense cores lie below the Jeans line (dashed line in the figure). This behavior implies that the cores are gravitationally unbound, i.e., they have insufficient self-gravity due to their mass to counteract the internal (turbulent) gas motions.

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In Figure 6, we show ${\sigma }_{\mathrm{tot}}$ versus ${\sigma }_{\mathrm{grav}}$, the dispersion needed to balance gravity alone, to make a similar comparison on a core-by-core basis and account for variations in the T_kin and ${\sigma }_{\mathrm{tot}}$ measured for each core. Here too, the final result is similar: most of the cores do not have sufficient mass to remain bound.

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The cores that are gravitationally bound tend to have larger masses. The virial ratio, defined as

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{5{\sigma }_{\mathrm{tot}}^{2}R}{{GM}},\end{eqnarray} \tag{ 8 }$$

represents the degree to which a core is gravitationally bound, with bound cores satisfying $\alpha \lt 2$ (e.g., Bertoldi & McKee 1992). Figure 7 shows the virial parameter as a function of core mass, showing that the most massive cores are the ones most likely to be gravitationally bound. Similar behavior has been seen in other studies of populations of dense cores, such as the Pipe Nebula (Lada et al. 2008).

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Although the trend that we observe in virial ratios decreasing with increasing core mass is consistent with previous studies, it is worth noting that most nearby dense core populations such as Perseus tend to have overall lower virial ratios than what we measure (e.g., Kirk et al. 2007; Foster et al. 2009). There are several reasons why dense cores in Orion could appear to be less self-gravitating than in nearby clouds: (1) the larger distance to Orion could lead to a greater confusion of multiple velocity components within the telescope's beam, artificially increasing the measured core velocity dispersions; (2) the higher mean density in Orion could imply that NH₃ traces kinematics beyond the core, implying that the NH₃ velocity dispersion is larger than should be attributed to the core; or (3) Orion does truly harbour more non-self-gravitating cores.

We can rule out the first possibility through comparison with Li et al. (2013). Li et al. (2013) use a combination of VLA and GBT NH₃ observations to assess the virial state of dense cores in the OMC2 and OMC3 regions of Orion A. There, the angular resolution is 5'' and the velocity resolution is 0.6 km s⁻¹. We compared the reported velocity dispersions for cores at nearly coincident positions in Li et al. (2013) and our own work, and found that our velocity dispersions tended to be slightly smaller. The relatively poor velocity resolution in the observations of Li et al. (2013) likely causes slight overestimates in their measurements of velocity dispersion. The fact that we observe slightly lower core velocity dispersions, however, suggests that the spatial resolution of our GBT observations does not cause a significant increase to the velocity dispersions that we report. Further comparisons with the results of Li et al. (2013) are discussed in Section 3.6.

The second possibility requires observations using a molecular tracer of higher effective density (e.g., Shirley 2015). Ideally, such a tracer would be known to trace unambiguously much denser gas than NH₃, such as a deuterated form of ammonia (e.g., Crapsi et al. 2007). The closest observations that we were able to find to satisfy this condition use N₂H⁺. N₂H⁺ has a higher effective density than NH₃ (Shirley 2015), and NH₃ appears to be sensitive to gas at lower densities than N₂H⁺. For example, the lower-density filament B216 in Taurus is visible in NH₃ but not N₂H⁺ (Seo et al. 2015). Due to a combination of chemical and optical depth effects, however, NH₃ also traces gas to higher densities than N₂H⁺—e.g., see the higher concentration of core emission in NH₃ relative to N₂H⁺ in Tafalla et al. (2002) and the simulations of Gaches et al. (2015). While a full understanding of the chemistry behind the creation and destruction of both N₂H⁺ and NH₃ remains elusive (see Caselli et al. 2017, for a recent test of models of the latter), observations such as those described here are strong evidence that NH₃ traces a wider range of densities than N₂H⁺. Therefore, while N₂H⁺ is not strictly a tracer of denser gas than NH₃, comparison between the two should allow us to test whether or not a significant amount of the NH₃ emission originates from the lowest densities of material to which it is sensitive. We note that if the mean density in Orion is sufficiently high that even N₂H⁺ is sensitive to intercore gas, as has been found in some infrared dark clouds (IRDCs; e.g., Henshaw et al. 2013), then this comparison between NH₃ and N₂H⁺ will not rule out the possibility of contamination in the NH₃ spectra from non-core gas.

Tatematsu et al. (2008) observed the N₂H⁺ (1−0) emission line in OMC2 and OMC3 using the Nobeyama Radio Observatory, with a spatial resolution of 17'' (with a spacing between observations of 411) and a velocity resolution of 0.12 km s⁻¹. We compared the N₂H⁺ velocity dispersion reported by Tatematsu et al. (2008) with the NH₃ velocity dispersion that we measure at the same locations of a sample of their cores with a range of declinations and find very good agreement, always within 0.02 km s⁻¹. This close correspondence in velocity dispersions implies that the NH₃ and N₂H⁺ are tracing similar zones of material. Furthermore, a visual comparison of on-core and nearby off-core NH₃ spectra suggests that the core emission is not being significantly biased to higher widths by non-core material. While these tests suggest that the NH₃ emission is not dominated by material at the lowest densities to which it is sensitive, observations of a tracer of unambiguously higher density, such as emission from deuterated NH₃, should be used to verify this behavior.

With the data available, we argue that it is reasonable to assume that the cores in Orion A represent a truly less self-gravitating population than is typically observed in nearby molecular clouds. The less self-gravitating cores certainly do not appear to be due to observational biases caused by the greater distance to Orion A. Observations of a high-density molecular tracer on small spatial scales are necessary, however, to verify that the large line widths originate in gas associated with the dense cores.

3.3.1. Cloud Weight

We next modify our analysis to include the fact that the ambient molecular cloud material provides an additional confining pressure on the cores. Under the assumption that the large-scale cloud can be roughly approximated by a sphere, the column density at the position of each core then provides an estimate for how deeply embedded the core is within the cloud. Although the Orion A cloud is clearly not spherical, we argue that the approximation is acceptable for the large-scale column density distribution, since, as we discussed in Section 2.5, we already exclude small-scale column density features from consideration. The pressure exerted on the ambient cloud is often expressed as

$$\begin{eqnarray}&&{P}_{\mathrm{cloud}}=\pi G\bar{{\rm{\Sigma }}}{\rm{\Sigma }},\end{eqnarray} \tag{ 9 }$$

where P is the pressure, $\bar{{\rm{\Sigma }}}$ is the mean cloud column density, and Σ is the column density at the location of the dense core (see, for example, McKee 1989; Kirk et al. 2006). As we show in Appendix C, this equation is strictly appropriate only for a spherical cloud with density varying as $\rho \propto {r}^{-k}$ with k = 1. Other values of k in the density scaling lead to a two-term expression for the pressure, given in Equation 19. When the cloud's density distribution has an exponent of $1\lt k\lt 3$—a reasonable range for large-scale clouds—this second term is positive, so Equation (9) provides a lower limit to the full pressure. Additional considerations, such as using ${\rm{\Sigma }}/2$ as a proxy for how deeply embedded a dense core is within the cloud, may introduce some bias to the estimate of the pressure of the cloud weight when $k\ne 1$. See Appendix C for further discussion of the caveats associated with estimating the pressure of the cloud weight.

To calculate the mean cloud column density used in Equation (9), we consider only the area within the large-scale map of column density that was covered in the GAS observations, yielding a value of $\bar{N}=3.9\times {10}^{22}$ cm⁻² or $\bar{{\rm{\Sigma }}}=8.9\times {10}^{20}$ g cm⁻².

To compare the relative strengths of cloud pressure, core self-gravity, and thermal plus non-thermal support, we use the formalism introduced in Pattle et al. (2015), expressing each in terms of its energy density in the virial equation:

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{P}=-4\pi {{PR}}^{3},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{G}=\displaystyle \frac{-1}{2\sqrt{\pi }}\displaystyle \frac{{{GM}}^{2}}{R},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{K}=\displaystyle \frac{3}{2}M{\sigma }_{\mathrm{tot}}^{2},\end{eqnarray} \tag{ 12 }$$

where ${{\rm{\Omega }}}_{P}$ is the pressure term, ${{\rm{\Omega }}}_{G}$ is the gravitational term, and ${{\rm{\Omega }}}_{K}$ is the kinetic term. The expression for ${{\rm{\Omega }}}_{G}$ is derived in Pattle (2016) for a core with an infinitely extending Gaussian density distribution. For a core with Plummer-like density distribution (e.g., Whitworth & Ward-Thompson 2001)¹⁸ with an exponent of 4, the factor of $\sqrt{\pi }$ becomes π instead, i.e., implying slightly smaller ${{\rm{\Omega }}}_{G}$ values than we estimate. For an object of constant density, as is assumed when deriving the standard virial parameter or Jeans mass discussed earlier, ${{\rm{\Omega }}}_{G}$ is a factor of ∼3.5 larger. A dense core is in virial equilibrium when $-({{\rm{\Omega }}}_{G}+{{\rm{\Omega }}}_{P})=2{{\rm{\Omega }}}_{K}$.

In Figure 8, we show the energy density ratio for the dense cores, compared to the ratio of the gravitational and pressure energy densities. The latter ratio expresses whether gravity or pressure is the dominant element binding the core, and is termed the "confinement ratio" in Pattle (2016). As can be seen in Figure 8, the majority of dense cores are bound, with energy density ratios exceeding one. As already illustrated in Figure 6, self-gravity alone contributes relatively little to this binding; pressure dominates gravity for the vast majority of the cores. A trend of decreasing confinement ratio with increasing energy density ratio is seen in Figure 6. We investigated the cause of this trend, and found no correlation between ${{\rm{\Omega }}}_{P,\mathrm{cloud}}$ and either ${{\rm{\Omega }}}_{K}$ or ${{\rm{\Omega }}}_{G}$. ${{\rm{\Omega }}}_{G}$, however, varies approximately as ${{\rm{\Omega }}}_{K}^{2}$. This latter trend appears to be driven by the mass term in ${{\rm{\Omega }}}_{G}$ and ${{\rm{\Omega }}}_{K}$, which follows the same relative scaling. As Figure 4 has already shown, there is no relationship between R_eff and ${\sigma }_{\mathrm{tot}}$, and both have a relatively small range in values (approximately an order of magnitude). The range of masses is just over three orders of magnitude, and thus appears to be driving the correlation between ${{\rm{\Omega }}}_{K}$ and ${{\rm{\Omega }}}_{G}$, which in turn is responsible for the trend seen in Figure 6.

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3.3.2. Filament Pressure

As discussed in Section 2.5, cores are typically found embedded within filaments in the larger molecular cloud structure. Like the larger molecular cloud, these filaments will also help to confine the dense cores. Since it is difficult to disentangle the column density belonging to the filaments from that belonging to the dense cores with simple techniques due to their similar size scales, we instead use previous observations to estimate a lower limit to the confining pressure due to filaments. Tafalla & Hacar (2015) observed filaments and cores in the Taurus L1495/B213 region using a combination of observations of the dust continuum and of molecular emission lines. They used these observations to model the density profile of the filaments in locations where cores were not present, using a cross-sectional profile of

$$\begin{eqnarray}&&n(r)=\displaystyle \frac{{n}_{0}}{1+{(r/{r}_{0})}^{\alpha }}\end{eqnarray} \tag{ 13 }$$

where n is the density at radial separation r, n₀ is the central density, r₀ is the characteristic size, and α is the power-law dependence (Whitworth & Ward-Thompson 2001). Across eight different filaments in Taurus, Tafalla & Hacar (2015) find ${n}_{0}\sim 6.5\times {10}^{4}$ cm⁻³, ${r}_{0}\sim 45^{\prime\prime}$ (at an assumed distance of 140 pc), and $\alpha \sim 3$, while a characteristic temperature of 10 K was assumed.

The pressure induced by the weight of overlying material in an isothermal filament can be expressed as

$$\begin{eqnarray}&&{P}_{{\rm{fil}}}=\displaystyle \frac{2}{\pi }G{{\rm{\Sigma }}}_{\perp }^{2}\end{eqnarray} \tag{ 14 }$$

(see Appendix C), where ${{\rm{\Sigma }}}_{\perp }$ is the column density through the center of the filament. The binding pressure on cores from the filaments measured in Taurus can therefore be estimated using the model estimates of Tafalla & Hacar (2015). Filaments in Orion, however, are likely to be much more dense than those found in Taurus, and therefore the Taurus estimate provides us with a lower limit to the true filament pressure. Generally, the pressure exerted by the cloud on cores is about a factor of 10 higher than the pressure exerted by these model filaments. Figure 9 shows the resulting energy density ratios with this additional pressure included (see also the following section for a further pressure term added).

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3.3.3. Turbulent Pressure

In addition to the weight of overlying material, turbulent pressure, i.e., pressure induced from the material of higher velocity dispersion and lower column density surrounding the cores, may also be present. We estimate the magnitude of this pressure in an approximate way. Shimajiri et al. (2014) obtained ¹³CO (1−0) and C¹⁸O (1−0) observations of the northern portion of Orion A, extending slightly southward of the main ISF. According to their Figure 2, the C¹⁸O (1−0) emission has a velocity dispersion <1.5 km s⁻¹ everywhere, with most of the gas having velocity dispersions <0.75 km s⁻¹. They assume a mean density of material traced by C¹⁸O (1−0) emission of 5 × 10³ cm⁻³, based on previous analysis by Ikeda & Kitamura (2009). Using these estimates, the turbulent pressure on the dense cores can be written as

$$\begin{eqnarray}&&{P}_{\mathrm{turb}}={\rho }_{\mathrm{turb}}\,{\sigma }_{v,\mathrm{turb}\,\mathrm{gas}}^{2}\end{eqnarray} \tag{ 15 }$$

(e.g., Pattle et al. 2015). A more careful measurement of both the mean gas density and typical CO velocity dispersion around each dense core (or even the velocity dispersion of the surrounding fainter NH₃ emission) would be required to obtain a more precise estimate of the turbulent confining pressure. Nonetheless, using the present equation to approximate the influence of turbulent pressure in combination with the pressure of the filament weight discussed above, Figure 9 shows that all cores have slightly increased energy density ratios, with an additional six dense cores now appearing to be bound. We emphasize that more precise, core-specific, estimates for both the turbulent pressure and the pressure of the filament weight could yield additional binding, especially as the latter pressure estimate is a lower limit.

3.3.4. Bonnor–Ebert Sphere Pressure

Cores are often approximated as Bonnor–Ebert (BE) spheres (Ebert 1955; Bonnor 1956), a spherically symmetric, isothermal equilibrium model where self-gravity and an external binding pressure balance internal thermal motions. In this model, the maximum external binding pressure for a stable BE sphere model can be expressed as

$$\begin{eqnarray}&&{P}_{\mathrm{BE},\mathrm{crit}}=1.40\displaystyle \frac{{c}_{s}^{8}}{{G}^{3}{M}^{2}}\end{eqnarray} \tag{ 16 }$$

(Hartmann 1998). While we do not find any correlation between the critical binding pressure of a BE sphere and the estimated pressure on each core from the weight of the overlying cloud material, the values span a similar range. The mean and standard deviation of $\mathrm{log}({P}_{\mathrm{BE},\mathrm{crit}}/{k}_{{\rm{B}}})$ is 6.5 ± 0.8 (where P_BE,crit/k_B is in K cm⁻³). This rough similarity in pressures between the BE sphere model and those estimated from the weight of the cloud suggests that the BE sphere model may provide a reasonable representation of the cores, although this conjecture should be tested more carefully through measurements of the radial column density profile of the cores.

When only information on the dust continuum is available for cores, their stability is often assessed in terms of their concentration, or peakiness. Following Johnstone et al. (2001), the concentration, C, can be written as

$$\begin{eqnarray}&&C=1-\displaystyle \frac{1.13{B}^{2}{S}_{\mathrm{tot}}}{\pi {R}^{2}{F}_{\mathrm{pk}}},\end{eqnarray} \tag{ 17 }$$

where B is the telescope beam size, S_tot is the total flux observed, and F_pk is the peak flux density observed. Typically, B is expressed in arcsec, S_tot in Jy, R in arcsec, and F_pk in Jy beam⁻¹. In the BE sphere model, the minimum concentration is 0.33 when observed in 2D, representing a sphere of uniform density, while the maximum concentration for a stable configuration is 0.72 (Johnstone et al. 2000). In Figure 10, we show the concentration measured for each dense core compared to its respective energy density ratio. Cores of higher concentration have more mass within a smaller size, and therefore would be expected to have stronger self-gravity, and hence a higher energy density ratio. As plotted in Figure 10, there is an absence of low-concentration, strongly bound cores and high-concentration, strongly unbound cores, both of which represent states that would be difficult to populate or maintain. There is no further correlation visible between the two quantities.

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Some of the protostellar cores have low concentration values. Previous studies, such as van Kempen et al. (2009), have found low-concentration protostellar cores when the core evolution is more advanced, i.e., much of the core material has already been accreted onto the central protostar or dispersed by protostellar outflows, making the remaining envelope traced by the submillimeter continuum observations appear more diffuse. Angular resolution and the core identification algorithm used also play a role in the concentrations observed, since both of these influence the peak and total flux associated with each core.

We additionally checked whether or not there are any bulk differences in the virial properties of dense cores associated with the ISF and the remainder of the field. Lane et al. (2016) identified clusters of dense cores across Orion A using minimal spanning trees. One of these clusters roughly encompasses the ISF, and therefore provides a simple way to identify dense cores either associated or not associated with the ISF. In all of the previous figures showing the properties of dense cores, filled symbols denote cores associated with the ISF, while open symbols show cores not associated with the ISF. Dense cores associated with the ISF tend to have slightly smaller energy density ratios than the remaining cores, due to their typically smaller sizes and therefore much smaller ${{\rm{\Omega }}}_{P}$ values, but the two populations largely overlap. The starless cores in the ISF have typical energy density ratios of $\mathrm{log}(-{{\rm{\Omega }}}_{G}-{{\rm{\Omega }}}_{P,\mathrm{tot}})/2{{\rm{\Omega }}}_{K}\,=0.39\pm 0.42$, while the starless cores outside the ISF have typical values of 0.62 ± 0.42 (median and median absolute deviation quoted for both). The non-ISF population that we analyze is small at present (39 starless cores), but the full GAS map of Orion A will extend much further south than the DR1 map analyzed here, and will allow for a more stringent test of whether or not virial properties vary between the ISF and the rest of Orion A.

As noted in Section 3.2, Li et al. (2013) observed NH₃ in cores in the OMC2 and OMC3 regions of Orion A and assessed their virial nature. While Li et al. (2013) also find that many of their dense cores are not self-gravitating ($\alpha \gt 2$), they do not report virial ratios as high as those found in our analysis. The relative lack of high virial ratios in Li et al. (2013) can be attributed to our greater sensitivity to lower mass cores. For example, almost none of the cores of Li et al. (2013) have masses below 1 ${M}_{\odot }$, while more than half of our cores (65%) do. As discussed earlier, higher-mass cores tend to be more self-gravitating, so it is reasonable that the lower mass cores in our sample have larger virial ratios than those reported in Li et al. (2013).

In their full virial analysis, Li et al. (2013) include a turbulent pressure term estimated assuming a mean gas density of 10⁴ cm⁻³ and a mean velocity dispersion of 1 km s⁻¹. This implies a slightly larger turbulent pressure than the one we estimate in Section 3.3.3, which we noted is typically smaller than the pressure binding provided by the overlying cloud material that we estimated for each core individually. Li et al. (2013) also include a magnetic support term in their analysis, assuming a typical field strength of 0.1 mG based on nearby observations of the Zeeman effect in CN. Including both the turbulent pressure binding and magnetic support reduces the number of supervirial cores in their analysis, although nearly half (47%) of their cores still appear to be unbound, with a "critical mass ratio" above one. The overall conclusions of Li et al. (2013) therefore seem consistent with our own results, given the different core samples and different assumptions used for the virial analysis.

New results from the BISTRO survey (Ward-Thompson et al. 2017) suggest that the total magnetic field strength may be substantially larger than assumed in the analysis of Li et al. (2013). Using the Chandrasekhar–Fermi method (Chandrasekhar & Fermi 1953), Pattle et al. (2017b) estimate a magnetic field strength in the OMC-1 region, i.e., the central densest portion of the ISF) to be 6.6 ± 4.4 mG in the plane of the sky. At this level, the magnetic field would contribute significantly to the energy density of each core. Inclusion of the magnetic pressure would serve to increase the amount of internal support against the gravitational collapse of the cores, thus points in Figure 9 would move up and to the left. The magnetic pressure may not be well approximated by a constant field strength value over the large area of Orion A included in our analysis, because the field strength derived only for the densest part of the cloud may not be representative of the entire region. Therefore, while we exclude the magnetic support term from our virial analysis, we note that this should be revisited when estimates of magnetic field strength are available for a larger extent of Orion A.

Our conclusion that dense cores in Orion A are significantly bounded by the pressure of the surrounding cloud material is a result that has been found in other nearby molecular clouds as well, spanning a range of environments. One example is the nearby Pipe Nebula, a star-forming environment significantly different from Orion A. Dense cores in the Pipe have typical volume densities of less than 10⁴ cm⁻³, sizes of ∼0.2 pc, non-thermal velocity dispersions of ∼0.2 km s⁻¹, and inhabit a cloud with a mean extinction of ${{\rm{A}}}_{V}\sim 4$ mag and a total mass of $\sim {10}^{4}$ ${M}_{\odot }$ that is located a mere 130 pc away (Alves et al. 2007; Lada et al. 2008, and references therein). Lada et al. (2008) used pointed NH₃ or C¹⁸O observations, in combination with dense core sizes and masses estimated using extinction mapping, to demonstrate that most dense cores in the Pipe have insufficient mass to be gravitationally bound. Furthermore, the vast majority of the internal pressure support in Pipe cores was found to be from thermal pressure rather than non-thermal motions. They also found a surprising similarity in the total internal pressures of Pipe cores across the cloud and argued that the most likely source of pressure on Pipe core boundaries is that caused by the weight of the overlying cloud material. Indeed, typical internal core pressures tend to be slightly larger in the "bowl" of the Pipe than in the 'stem'; the former area is also associated with higher mean cloud column densities. Magnetic pressure may also play an important role in confining cores in the Pipe, as argued by Alves et al. (2008). In contrast to the Pipe, Orion A has a mean cloud column density contributing to the cloud external pressure that is much higher (a factor of >5), and non-thermal motions and perhaps additional sub-core bulk motions provide significant contributions to the internal pressures of its dense cores.

Several other studies examining the role of cloud pressure have been performed in star-forming regions that have properties intermediate to the Pipe and Orion A (i.e., cloud mass, volume density of dense cores, size, and non-thermal velocity dispersion). In the Ophiuchus molecular cloud, three independent analyses using different data sets all came to the conclusion that external cloud pressure is important in confining many cores in the cloud. Specifically, Johnstone et al. (2000) argued this point based on Bonnor–Ebert sphere modeling of dust continuum observations at 850 μm and the level of pressure expected from ambient cloud material, Maruta et al. (2010) used Nobeyama H¹³CO⁺ observations, while Pattle et al. (2015) used a combination of SCUBA-2 850 μm dust continuum data and IRAM 30 m N₂H⁺ and C¹⁸O observations. In the Perseus molecular cloud, Kirk et al. (2007) analyzed the dense cores, while Sadavoy et al. (2015) analyzed the larger B1-E structure, both also finding evidence that pressure confinement is important. Note, however, that Foster et al. (2009) also analyzed dense cores in Perseus and found instead that self-gravity alone was sufficient to confine them. This difference is primarily due to the larger masses derived for the dense cores by Foster et al. (2009) using CSO Bolocam data, rather than higher resolution JCMT SCUBA data of Kirk et al. (2007). Foster et al. (2009) furthermore did not estimate the magnitude of cloud pressure to see how it compares with core self-gravity, and it is likely that such an estimate would provide a larger amount of binding than they estimated from self-gravity alone. In the Taurus L1495/B218 region, Seo et al. (2015) analyzed the virial properties of filaments and smaller structures using NH₃ observations, and found that most of the small NH₃ structures are gravitationally unbound when considering just the balance between self-gravity and thermal plus non-thermal support. They argued that many of these structures, especially the youngest ones, appear to be pressure-confined, and that cloud pressure is a reasonable candidate for the source of that pressure, with ram pressure from material inflowing to the system being their other main candidate. In the Cepheus clouds, Pattle et al. (2017a) performed a virial analysis using a combination of JCMT SCUBA-2 850 μm dust continuum and archival ¹³CO observations to argue that cloud pressure is required to bind most of the dense cores. Similarly, Kirk et al. (2016) find suggestive evidence that cloud pressure is important for dense cores in Orion B, although there, the analysis lacks kinematic information to provide precise estimates of the dense cores' temperatures and non-thermal motions.¹⁹

For more distant (non-Gould Belt) star-forming regions, Kauffmann et al. (2013) performed a self-consistent virial analysis combining observations from a variety of previous surveys. While external pressure was not included in their analysis, a comparison of self-gravity and non-thermal motions showed that the majority of structures were unbound. Structures that were bound were either much larger entities (masses above 100 ${M}_{\odot }$) or were in regions of high-mass star formation, or both. Kauffmann et al. (2013) speculate that the structures, which do appear to be gravitationally unbound, may in fact be pressure-bound. Additionally, Barnes et al. (2016) analyze a large sample of massive molecular clumps observed with the CHaMP survey in HCO⁺ and ¹²CO emission to demonstrate that the less dense gas traced by ¹²CO appears to provide significant pressure binding to the more compact clumps traced by HCO⁺, at a sufficient level that most of the clumps are near virial equilibrium. Thus, while it may be the case that high-mass stars form under a different mode, dense cores that form low- to intermediate- mass stars appear to be well represented by a large virial parameter, requiring the presence of some additional factor such as external pressure to remain bound.

Some numerical simulations have also illustrated the importance of cloud pressure to the formation and evolution of dense cores. Lucas et al. (2017) show that, for otherwise identical simulations of star-forming regions, either the absence of a cloud envelope at the beginning of formation, or the removal of such an envelope during evolution, leads to less dense substructure forming and to markedly lower star formation efficiencies.

Despite the differences in the analyses described above, it is striking that all of these studies come to a similar conclusion: pressure from the ambient cloud material on dense cores is important and often is required to keep the cores bound. The fact that this conclusion holds across such different star-forming environments from analyses using different techniques suggests that this phenomenon is a universal feature of star formation.

The one major component missing from most virial analyses of core populations is the support provided by magnetic fields. The magnetic field strength remains challenging to observe directly, although estimates can be made based on the field alignment, as measured by polarization data. Polarization observations are beginning to be available over larger areas of molecular clouds at lower resolution through telescopes such as Planck (e.g., Planck Collaboration et al. 2015) and Blast-Pol (e.g., Fissel et al. 2016), and at higher resolution with the JCMT's POL-2 (through the BISTRO survey; Pattle et al. 2017b; Ward-Thompson et al. 2017).

As discussed in Section 2.2, Lane et al. (2016) presented two dense core catalogs for Orion A, one using getsources, which we adopted for our main analysis, and also one using FellWalker, which we examine here. FellWalker identifies peaks based on local gradients, effectively associating any pixel lying along an upward path to a peak with that peak. There is no assumed shape for each core, and each pixel can be assigned to a maximum of one peak. When calculating the NH₃-based properties of each core, analogously to Section 2.4, we use the full core footprint. Otherwise, we follow an identical procedure to that presented for the getsources-based dense core catalog.

In Figure 11, we show the results from the virial analysis using instead the FellWalker dense core catalog. As in our main analysis, we see that nearly all of the dense cores are bound (energy density ratios greater than one). Only seven of the cores have sufficient self-gravity to be entirely bound without the pressure of the cloud (i.e., $-{{\rm{\Omega }}}_{G}/{{\rm{\Omega }}}_{K}\geqslant 2$), so our conclusion that pressure binding is required for the majority of dense cores holds.

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Salji et al. (2015) analyzed the region around the ISF in Orion A using an early JCMT GBS data release that did not reach the final survey depth and an early reduction method that is less sensitive to larger-scale structure in the cloud ("Internal Release 1"). To identify dense cores, Salji et al. (2015) developed a new structure detection algorithm, tuned to identify only compact and roughly round sources. Sources identified with protostars were also removed from their final published catalog. The remaining cores were classified as either starless or prestellar based on a thermal stability analysis using only SCUBA-2 data (and no kinematic information). To derive the NH₃-based properties for each core, we follow a similar procedure to that in the main analysis: we consider all pixels within the 1σ contour of the peak core position. Since Salji et al. (2015) did not publish axial ratios of dense core or a rotation angle, we assume the cores are well described by a single radius.

The right panel of Figure 11 shows the virial analysis for the dense cores of Salji et al. (2015). Here, the cores appear to follow a much more linear trend, suggesting that higher self-gravity and higher cloud pressure values tend to go together. The distinction that Salji et al. (2015) made between starless and prestellar cores does appear to be largely borne out in our virial analysis, with the prestellar cores tending to lie higher (more bound by both self-gravity and cloud pressure) than the starless cores. Although the qualitative appearance of the plot differs dramatically from our main analysis, we find that our main conclusion still holds: the majority of dense cores are bound (94%), and none would appear to be bound if the pressure from the cloud were ignored.

What is the most reasonable range of pressures to associate with a specific core, given a map of the projected column Σ? A natural choice is to associate ${\bar{{\rm{\Sigma }}}}_{\mathrm{cl}}$ with the mean of Σ within the cloud boundaries, and ${{\rm{\Sigma }}}_{s}$ with one-half of Σ at the location in question, but there are several ambiguities. First, the external pressure P_s must be guessed at. Second, our Σ map may be contaminated by emission unrelated to the structure of interest. Third, if the density distribution is not isotropic, Σ is affected by inclination. Fourth, one usually does not know how much of Σ is in front of the core and hence what value to associate with ${{\rm{\Sigma }}}_{s}/{\rm{\Sigma }}$. Finally, observations may suffer from intrinsic uncertainties involving finite resolution, spatial filtering, emission-to-mass conversions, and the like.

We concentrate on the fourth ambiguity (foreground and background). The natural assumption that half of Σ is in the foreground tends to maximize P, but it is more rigorous to adopt a model and some prior expectations regarding the occurrence of cores within clouds, and to derive a probability distribution from P. Let us assume: (i) no cores exist with ${{\rm{\Sigma }}}_{s}\lt {{\rm{\Sigma }}}_{s,\min }$, a situation corresponding to extinction thresholds observed in various regions (Pak et al. 1998; Johnstone et al. 2004) perhaps because of far-UV photoionization (McKee 1989). (ii) The probability of finding a core scales as ${\rho }^{\alpha }$ per unit volume, a scenario accommodating a constant star formation rate per unit mass ($\alpha =1$) and a dynamically limited rate $(\alpha =2)$.

The differential cumulative probability is then $d{\mathscr{P}}\propto {\rho }^{\alpha }{dz}$, where z is distance along the line of sight, except $d{\mathscr{P}}=0$ where $P\lt P({{\rm{\Sigma }}}_{s,\min })$.

For the special case $\alpha =1$, $d{\mathscr{P}}\propto d{\rm{\Sigma }}$ so the column toward the core is uniformly distributed within its allowed range. In the case of a planar sheet, ${{\rm{\Sigma }}}_{s}$ is uniformly distributed from ${{\rm{\Sigma }}}_{s,\min }$ to ${{\rm{\Sigma }}}_{\perp }/2$. The median pressure within this distribution is lower than the midplane value by ${(\pi /2)G({{\rm{\Sigma }}}_{\perp }/2-{{\rm{\Sigma }}}_{s,\min })}^{2}.$

To go further, we require relations among P, ρ, and z Hence, we adopt here the spherical model: then $P\propto {\rho }^{{\gamma }_{p}}$ where ${\gamma }_{p}={k}_{P}/k=2-2/k$ is the polytropic exponent. Also, $P\propto {r}^{-{k}_{P}}\propto {({z}^{2}+{\varpi }^{2})}^{-{k}_{P}/2}$ if ϖ is the projected offset from the cloud center and z = 0 at the cloud center. Working out dz in terms of dP,

$$\begin{eqnarray}&&d{\mathscr{P}}\propto \displaystyle \frac{{P}^{(\alpha /2-1)k/(k-1)}}{| z(P)| }\,{dP}\propto \displaystyle \frac{{P}^{(\alpha /2-1)k/(k-1)}}{{[{(P/\max P)}^{-2/{k}_{P}}-1]}^{1/2}}\,{dP}.\end{eqnarray} \tag{ 20 }$$

Note that for the dynamical star formation rate ($\alpha =2$) the numerator is constant and $d{\mathscr{P}}\propto | z(P){| }^{-1}\,{dP}$ so long as $k\ne 1$.

The next step is to determine the maximum possible pressure by determining the maximum value of ${{\rm{\Sigma }}}_{s}$ along the line of sight, given the observed value of Σ, i.e., $\max P=P(\max {{\rm{\Sigma }}}_{s}$). Often we can ignore the cloud boundary and extend a line-of-sight integration to infinity. This assumption is especially reasonable if $P({{\rm{\Sigma }}}_{s,\min })\gg {P}_{s}$, which is usually the case for giant molecular clouds. We then obtain

$$\begin{eqnarray}\displaystyle \frac{{\rm{\Sigma }}}{\max {{\rm{\Sigma }}}_{s}} & = & {\pi }^{1/2}(k-1)\displaystyle \frac{{\rm{\Gamma }}[(k-1)/2]}{{\rm{\Gamma }}(k/2)}\\ & \simeq & k+1+(\pi -3)(k-1)(3-k).\end{eqnarray} \tag{ 21 }$$

The middle expression comes from integrating through an infinite sphere, and the quadratic approximation is exact for $k=1,2,3$ and errs by only 0.6% over the entire range. It shows that the usual assumption ($\max {{\rm{\Sigma }}}_{s}={\rm{\Sigma }}/2$) is correct when k = 1, but amounts to an overprediction in steeper density profiles.

Once $\max {{\rm{\Sigma }}}_{s}$ and $\max P$ have been determined, Equation (20) provides the probability density. In Figure 14, we plot the factor relating the median overpressure to the maximum value.

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