The Green Bank Ammonia Survey: A Virial Analysis of Gould Belt Clouds in Data Release 1

We use dust emission maps from the JCMT GBS (Ward-Thompson et al. 2007) to identify the dense cores studied in our virial analysis. The JCMT GBS obtained large-area maps of dust continuum emission at 850 and 450 μm across all nearby visible star-forming regions, including the subset for which we have GAS kinematic data. Here, we used the most recent maps available at the time of our analysis, from Data Release 2 (Kirk et al. 2018). For our main analysis, we use the Getsources algorithm (Men'shchikov et al. 2012) to identify dense cores, which form part of the JCMT GBS full core catalog to be presented in K. Pattle et al. (2019, in preparation). Getsources allows users to identify compact structures using information available at multiple wavelengths, and we find it does a good job of identifying visually apparent dense cores in the three regions we analyze. We discuss the details of the Getsources algorithm and its application in Appendix A.1.1, and compare this to the results obtained with a second core-identification algorithm, FellWalker, in Appendix A.1.2. Including all types of cores (starless cores and protostellar objects), we identify 134, 78, and 17 cores in L1688, NGC 1333, and B18, respectively, using Getsources. Figures 1–3 show the dense cores identified in each of the three regions overlaid on the GAS-based NH₃ emission. A sample source identification using both Getsources and FellWalker on the GBS 850 μm maps is shown in Figure 9, focused on Oph B in L1688.

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Measurements of the core size and total flux from Getsources are obtained for our virial analysis. We define the effective radius as the geometric mean of the full diameter of the major and minor axes of the elliptical Getsources cores, which we take to represent the full extent of the cores in our analysis. To account for the JCMT beam size, we furthermore deconvolve the core diameters by subtracting in quadrature the beam size of 141 at 850 μm (Dempsey et al. 2013). Deconvolution becomes highly uncertain for sources of sizes similar to the beam, so for cores with calculated deconvolved diameters of less than the JCMT beam size, we instead use a value of half of the beam as an upper limit to the true radius. These unresolved cores represent 19% of those detected.

We convert the total flux measured at 850 μm to a mass assuming the mean kinetic temperature for each core (see Section 2.3), using the equation

$$\begin{eqnarray}&&M=1.06\,{S}_{850}\,\exp \left(\displaystyle \frac{17}{T}-1\right){\left(\displaystyle \frac{{\kappa }_{850}}{0.0125}\right)}^{-1}{\left(\displaystyle \frac{D}{415}\right)}^{2}\end{eqnarray} \tag{ 1 }$$

(Hildebrand 1983; Kirk et al. 2017), where M is the core mass in M_⊙, S₈₅₀ is the total flux at 850 μm in jansky, T is the dust temperature in kelvin, κ₈₅₀ is the dust opacity in square centimetres per gram, and D is the distance to the cloud in parsecs. Following previous JCMT GBS studies (e.g., Kirk et al. 2017), we assume κ₈₅₀ = 0.0125 cm² g⁻¹. Furthermore, we assume that the dust temperature is equal to the gas temperature estimated from the GAS NH₃ observations (see Section 2.3). Finally, we adopt new distance estimates where available. For L1688, we use the Very Long Baseline Interferometry-based Gould's Belt Distances Survey (GOBELINS) distance of 137 pc (Ortiz-León et al. 2017), noting that this is nearly identical to previous values (e.g., Mamajek 2008). For NGC 1333, we adopt a distance based on recent Gaia observations, which suggest a distance of 271 pc (Herczeg et al. 2017). This distance is somewhat larger than the standard 235 pc adopted by Hirota et al. (2008), but is also smaller than that in another recent study combining Gaia and GOBELINS data (Ortiz-León et al. 2018). For B18, we adopt a distance of 140 pc (Loinard et al. 2008). The GOBELINS team recently published results for the Taurus molecular cloud (Galli et al. 2018); however, none of the targets with measured distances appear to be associated with B18. We note that the GOBELINS results show that the Taurus molecular cloud has a large inclination relative to the plane of the sky, with line-of-sight distances ranging from 127 to 163 pc along its extent. Our distance estimate of 140 pc for B18 lies comfortably in the middle of the GOBELINS range.

The uncertainty in the core masses estimated using Equation (1) is expected to be dominated by the assumed dust grain opacity: the distances and temperatures are typically accurate to 10%–20% or better, and the flux calibration factor at 850 μm is also accurate to about 10% (Dempsey et al. 2013). The dust opacity at 850 μm could vary by a larger amount. For example, Chen et al. (2016) found that the slope of the dust emissivity spectral index (β) varies between 1.0 and 2.7 within the Perseus molecular cloud, while a constant value of 2 was used for our estimation of the dust opacity at 850 μm. Similarly, Chacon-Tanarro et al. (2019) found variations of roughly a factor of 2 in the opacity for the dense core L1544. Opacity values under a variety of conditions are provided by Ossenkopf & Henning (1994). Within star-forming environments, these values can vary by up to an order of magnitude, but normally they do not vary by more than a factor of 2. In addition to the formal errors associated with the mass estimation, the choice of the source identification method can also have a noticeable effect on both the total fluxes and sizes of objects identified. We explore the implications of adopting different source identification algorithms on our virial analysis in Appendix A.1.

We use data from the Spitzer Space Telescope to search for protostars associated with each of the dense cores. For L1688 and NGC 1333, we use the young stellar object (YSO) catalog from Dunham et al. (2015). The Dunham et al. (2015) catalog, however, does not include Taurus, so for B18, we use the catalog of Rebull et al. (2010). The observations used for the Dunham et al. (2015) catalog are discussed extensively in terms of completeness by Evans et al. (2009), who estimate 90% completeness down to 0.05 L_⊙, and 50% completeness down to 0.01 L_⊙ for the highly crowded Serpens region, then assumed to be at a distance of 260 pc. As the result of a combination of distance and crowdedness, Evans et al. (2009) presented the Serpens completeness as a lower limit for the remainder of the data set, including L1688 and NGC 1333. Therefore, the vast majority of protostars should be detected in both L1688 and NGC 1333. While a full completeness analysis of the Taurus protostars does not appear to be published, Luhman et al. (2009) used a comparison of Spitzer data and X-ray observations from the XMM-Newton Extended Survey of the Taurus molecular cloud (Güdel et al. 2007) to suggest that the YSO census is complete down to ∼0.02 M_⊙ and, therefore, also covers nearly all of the protostars present.

We define as protostellar all dense cores with peaks located less than 705 from the nearest protostar, i.e., within half the JCMT SCUBA-2 850 μm beam size. For our analysis, we exclude dense cores associated with any class of protostar or YSO. Especially for the more evolved protostars, we expect that most of the material associated with the YSO is on scales smaller than can be resolved with the JCMT or GBT (e.g., Sadavoy et al. 2018), and, hence, a kinematic analysis on these larger scales could be misleading; our focus here is on the starless dense cores.

The first results from GAS were recently published as "Data Release 1" (DR1) (Friesen & Pineda et al. 2017), which covered L1688 in Ophiuchus, B18 in Taurus, NGC 1333 in Perseus, and the northern section of Orion A (which is covered on its own in Kirk et al. 2017). Observations were carried out using the K-Band Focal Plane Array (KFPA) on the GBT, using the On-The-Fly mapping mode with a typical noise level of 0.1 K in the main beam scale. The NH₃ (1, 1) and (2, 2) transitions (23.7 GHz) were observed with spectral resolution of 5.7 kHz (0.07 km s⁻¹) and spatial resolution of 32''. Further information on GAS is available in Friesen & Pineda et al. (2017), including information on data reduction and a general look at the properties of the data. The final GAS data products that are vital for this analysis are the property maps for kinetic temperature (T_k) and velocity dispersion (σ), which were both derived by fitting the ammonia (1, 1) and (2, 2) transitions simultaneously. Because sufficiently bright lines are required to attain reliable property data, the coverage of the T_k and σ maps is less extensive than the total GAS fields, covering just under three quarters of identified cores in the L1688 and B18 GAS fields and about 90% of the cores in the NGC 1333 field.

We use the full extents derived from the JCMT Getsources extraction as pixel selection regions for the GAS data. While it is true that the large beam size of GBT compared to JCMT (32'' versus 141) extends the area over which core emission appears, the same effect will make the edge of a core more contaminated by external emission, therefore suggesting that a better representation of the core's properties can be found by adopting the original Getsources extent, rather than a reconvolved core area. We note that we tested our analysis using both smaller and larger pixel selection radii for each core and found similar results. All pixels that overlap with the core boundary are included, although core-overlapping pixels whose centers are not located within the core's extent are assigned a weighting of one quarter that of an interior pixel, to minimize the effects of pixels tracing more turbulent gas that borders cores. This criterion for pixel selection is also employed for the column density (Section 2.4) and CO maps (Section 2.5).

Within this core extent, we calculate the weighted average of each kinematic property (e.g., total velocity dispersion), weighted by the uncertainty of that property. Pixels within core extents that did not have a value determined for the given property were excluded from the average. Any cores containing no informative pixels for any single property were dropped from the analysis entirely, leaving 74 cores in L1688, 51 in NGC 1333, and nine in B18. Thermal broadening of the NH₃ emission is much lower than that of the majority H₂ gas that dominates these regions, and, therefore, we calculate the total velocity dispersion expected using the standard formulation:

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

where k_B is Boltzmann's constant, μ_NH3 is the molecular weight of NH₃ (i.e., 17), m_H is the mass of a hydrogen atom, and μ_mn is the mean molecular weight of the gas, which we take to be 2.37, following Kauffmann et al. (2008).

In theory, the total velocity dispersion within each core could have one additional component that we have not yet considered, namely a variation in the centroid velocity of NH₃ across the core's area due to, for instance, rotation or shear. Using the v_lsr maps from GAS DR1 (Friesen et al. 2017), we calculated the standard deviation in centroid velocity across each core (${\sigma }_{{v}_{\mathrm{lsr}}}$) and found that it was, on average, an order of magnitude smaller than the averaged line widths. There were, however, a few cases in which the magnitude of ${\sigma }_{{v}_{\mathrm{lsr}}}$ was much larger (at least 40% of the line width), including six cores in NGC 1333, eight in L1688, and, most notably, four of the nine cores in B18. A visual inspection of this subset of cores revealed that some show clear centroid velocity gradients, which are typically attributed to rotation or shear (Goodman et al. 1993), and are expected to contribute to the internal support term. A similar fraction of these high-${\sigma }_{{v}_{\mathrm{lsr}}}$ cores, however, have centroid velocity variations driven by pixels exclusively near core edges, where very different v_lsr values often appear. The most extreme ${\sigma }_{{v}_{\mathrm{lsr}}}$ value appears in a faint core in B18, where noisy edge pixels have greater influence in our uncertainty-weighted statistics. Many other cases show evidence of contamination from overlapping structures at different centroid velocities, such as two B18 cores with emission from nearby protostellar sources on their edges. Similar contamination effects are also evident in some NGC 1333 and L1688 cores. Because our visual checks reveal that most, if not all, cores have minimal ${\sigma }_{{v}_{\mathrm{lsr}}}$ components that can be confidently attributed to potential sources of internal core support, we omit this term from all further analysis.

Figure 4 shows the distribution of the total velocity dispersions measured for the dense cores in each of the three clouds. The Ophiuchus A region within L1688 (in the Ophiuchus molecular cloud) is shown separately, as we notice very different kinematic properties there compared to the rest of the cloud. Oph A is notable for its proximity to the B star HD 147889 and other young, hot protostars (Pattle et al. 2015), which are thought to raise the values of T_k in the region, approaching ∼30–40 K in some places. Meanwhile, gas in the rest of L1688 has temperatures typically in the range of 10–20 K. The bimodality in the velocity dispersion distribution in Oph A is caused by the structure of the region, in which a dense central filament is surrounded by less dense outlying material. The lower line widths are usually associated with the central region, whereas the higher line widths generally correspond to outlying cores. Pattle et al. (2015) labeled this lower-density, high-line-width region surrounding central Oph A as Oph A'; however, for the sake of simplicity, we present Oph A as a single region during our analysis.

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We also include total H₂ column density maps in our analysis to later estimate the pressure on the cores from the weight of surrounding cloud material. These maps are presented in A. Singh et al. (2019, in preparation), in which a pixel-by-pixel fit to the spectral energy distribution of Herschel and Planck data is run to estimate the column density at 36'' resolution.

Finally, we include 45'' resolution ¹³CO (1–0) data maps from the Five College Radio Astronomy Observatory (FCRAO), which cover a wide region around each of the three clouds. ¹³CO typically traces gas with densities between 10 and 10⁴ cm⁻³, as the line becomes optically thick at higher densities (Pineda et al. 2008).¹² This density range makes ¹³CO an ideal tracer for estimating the turbulent pressure of the medium surrounding the dense cores. The ¹³CO observations of NGC 1333 and L1688 were obtained by the Co-Ordinated Molecular Probe Line Extinction Thermal Emission (COMPLETE) Survey of Star-forming Regions (Ridge et al. 2006), while B18 was covered by the Taurus Molecular Cloud Survey (Narayanan et al. 2012). A summary of the kinematic, virial, and Getsources-derived properties for all cores with complete data is provided in Tables 1 (L1688), 2 (NGC 1333), and 3 (B18).

Table 1.  Dense Core Properties in L1688

IDa	R.A.a	Decl.a	Ma	Reffa	σobs (km s−1)b			Tk(K)b			Ωkc	−Ωgc	−Ωp,wc	−Ωp,tc	Vir.d
	(J2000)	(J2000)	(M⊙)	(pc)	mn	err	std	mn	err	std	(erg)	(erg)	(erg)	(erg)	Rat.
2	246.61529	−24.39922	2.60	0.014	0.263	0.001	0.016	17.9	0.1	0.4	9.5e+42	2.4e+43	1.4e+41	9.8e+40	1.3e+00
3	246.61417	−24.39099	1.95	0.013	0.242	0.001	0.027	17.3	0.1	0.8	6.4e+42	1.5e+43	9.2e+40	6.6e+40	1.2e+00
9	246.59075	−24.38132	0.08	0.009	0.674	0.016	0.051	26.1	0.7	1.0	1.2e+42	3.4e+40	3.0e+40	2.2e+40	3.5e-02
12	246.61650	−24.38225	0.38	0.014	0.264	0.001	0.052	17.8	0.1	0.8	1.4e+42	5.2e+41	1.2e+41	8.5e+40	2.6e-01
13	246.62512	−24.40843	2.71	0.024	0.244	0.001	0.050	18.4	0.1	1.1	9.3e+42	1.6e+43	6.2e+41	3.9e+41	9.2e-01
16	246.63204	−24.41403	0.13	0.008	0.211	0.002	0.038	17.8	0.2	0.8	3.7e+41	1.0e+41	2.6e+40	1.4e+40	1.9e-01
21	246.87321	−24.46245	0.14	0.011	0.329	0.005	0.077	15.5	0.2	0.5	6.5e+41	9.2e+40	6.6e+40	5.1e+40	1.6e-01
25	246.87129	−24.45139	0.15	0.009	0.269	0.002	0.065	13.6	0.1	0.4	5.2e+41	1.3e+41	3.7e+40	2.7e+40	1.9e-01
26	246.86696	−24.45261	0.43	0.015	0.367	0.002	0.056	14.0	0.1	0.6	2.3e+42	6.4e+41	1.6e+41	1.2e+41	2.0e-01
27	246.62238	−24.37677	0.17	0.014	0.320	0.003	0.054	19.6	0.2	1.4	8.3e+41	1.1e+41	1.2e+41	7.5e+40	1.8e-01
28	246.60938	−24.37360	0.10	0.014	0.327	0.003	0.078	21.1	0.2	1.2	5.3e+41	4.0e+40	1.2e+41	8.9e+40	2.4e-01
38	246.87213	−24.44358	0.21	0.013	0.327	0.003	0.061	13.9	0.2	0.4	9.4e+41	1.8e+41	1.1e+41	8.2e+40	2.0e-01
39	246.89050	−24.43856	0.91	0.027	0.286	0.002	0.078	14.2	0.1	0.9	3.4e+42	1.6e+42	8.9e+41	7.7e+41	4.8e-01
40	246.54250	−24.38684	0.04	0.012	0.571	0.011	0.018	24.5	0.5	1.6	4.9e+41	7.6e+39	6.1e+40	4.6e+40	1.2e-01
41	246.60004	−24.36541	0.03	0.011	0.308	0.003	0.064	21.5	0.2	1.6	1.3e+41	3.4e+39	5.2e+40	3.8e+40	3.7e-01
44	246.87275	−24.65492	0.01	0.008	0.630	0.026	0.009	21.5	1.1	0.7	1.5e+41	7.1e+38	2.4e+40	2.0e+40	1.5e-01
45	246.99508	−24.55845	0.58	0.023	0.125	0.000	0.014	11.1	0.1	0.4	8.5e+41	7.5e+41	5.0e+41	2.6e+41	8.9e-01
46	246.80067	−24.49384	0.49	0.029	0.420	0.001	0.033	15.7	0.1	0.6	3.3e+42	4.3e+41	1.3e+42	9.5e+41	4.1e-01
48	246.77500	−24.61921	0.02	0.008	0.577	0.021	0.075	15.4	1.0	1.2	2.4e+41	2.9e+39	2.7e+40	2.7e+40	1.2e-01
50	246.85188	−24.46204	0.27	0.014	0.427	0.003	0.029	13.5	0.1	0.5	1.8e+42	2.6e+41	1.3e+41	1.0e+41	1.4e-01
51	246.76937	−24.65370	0.55	0.020	0.134	0.001	0.032	12.3	0.1	0.7	9.0e+41	7.5e+41	3.9e+41	3.6e+41	8.4e-01
52	246.88671	−24.45047	0.14	0.013	0.188	0.002	0.035	12.4	0.1	0.8	3.1e+41	7.6e+40	1.1e+41	8.9e+40	4.6e-01
53	246.85487	−24.44933	1.28	0.032	0.413	0.001	0.052	13.6	0.1	0.8	8.1e+42	2.6e+42	1.6e+42	1.2e+42	3.4e-01
57	246.62862	−24.36973	0.09	0.015	0.364	0.008	0.050	22.7	0.6	1.7	5.5e+41	2.9e+40	1.5e+41	8.4e+40	2.4e-01
58	246.83371	−24.45318	0.11	0.015	0.405	0.004	0.052	14.7	0.2	0.9	6.9e+41	4.3e+40	1.7e+41	1.4e+41	2.5e-01
59	246.54125	−24.32932	0.12	0.019	0.608	0.015	0.078	24.4	0.7	2.9	1.5e+42	3.6e+40	2.6e+41	2.4e+41	1.7e-01
60	246.81371	−24.51046	0.44	0.021	0.308	0.001	0.042	14.7	0.1	1.1	1.8e+42	4.7e+41	5.2e+41	3.3e+41	3.6e-01
61	246.59042	−24.39326	0.02	0.013	0.605	0.036	0.057	31.1	2.1	1.4	2.7e+41	1.6e+39	9.7e+40	7.6e+40	3.2e-01
62	246.79958	−24.63337	0.11	0.013	0.485	0.017	0.097	11.9	2.5	0.0	8.7e+41	4.7e+40	1.0e+41	9.8e+40	1.4e-01
63	246.87675	−24.43000	0.04	0.009	0.380	0.012	0.097	14.6	0.9	0.1	2.1e+41	8.1e+39	3.6e+40	3.0e+40	1.7e-01
65	246.84021	−24.66555	0.38	0.022	0.450	0.003	0.163	15.0	0.1	1.0	2.8e+42	3.3e+41	4.9e+41	4.4e+41	2.2e-01
66	246.86075	−24.65661	0.01	0.009	0.615	0.022	0.057	20.9	1.3	0.6	1.3e+41	5.6e+38	2.8e+40	2.6e+40	2.1e-01
69	246.62213	−24.35786	0.01	0.011	0.183	0.004	0.034	18.8	0.6	0.9	3.2e+40	7.1e+38	5.2e+40	3.1e+40	1.3e+00
70	246.54708	−24.41278	0.02	0.013	0.836	0.017	0.075	27.2	0.6	3.3	4.8e+41	1.7e+39	8.8e+40	7.5e+40	1.7e-01
71	246.54267	−24.40085	0.01	0.008	0.555	0.016	0.040	26.0	0.9	2.1	8.9e+40	3.7e+38	2.3e+40	1.9e+40	2.4e-01
72	246.53204	−24.34111	0.09	0.019	0.501	0.009	0.064	21.3	0.4	2.5	8.1e+41	2.0e+40	2.5e+41	2.1e+41	3.0e-01
73	247.11954	−24.32155	1.18	0.022	0.102	0.001	0.006	8.4	0.7	0.6	1.3e+42	3.3e+42	2.1e+41	1.4e+41	1.4e+00
75	246.63654	−24.43725	0.02	0.012	0.570	0.014	0.099	25.5	0.7	4.3	2.8e+41	2.3e+39	8.4e+40	3.9e+40	2.3e-01
76	246.75813	−24.57900	0.30	0.013	0.144	0.001	0.012	10.5	0.1	0.2	4.6e+41	3.5e+41	1.1e+41	1.4e+41	6.5e-01
77	246.67837	−24.43469	0.09	0.030	0.290	0.004	0.092	18.7	0.2	1.7	4.0e+41	1.5e+40	1.5e+42	6.9e+41	2.7e+00
78	246.74692	−24.57280	5.87	0.042	0.133	0.000	0.026	10.7	0.0	0.7	8.7e+42	4.2e+43	4.0e+42	4.2e+42	2.9e+00
79	246.82092	−24.46278	0.05	0.013	0.405	0.003	0.040	14.5	0.1	0.7	3.2e+41	1.1e+40	1.1e+41	8.8e+40	3.3e-01
83	246.57800	−24.42014	0.02	0.011	0.663	0.016	0.070	26.4	0.8	1.3	2.9e+41	1.7e+39	5.8e+40	5.4e+40	1.9e-01
84	246.59396	−24.41963	0.01	0.010	0.630	0.027	0.033	25.2	1.4	2.8	1.5e+41	6.0e+38	4.8e+40	4.2e+40	3.0e-01
87	246.79425	−24.65705	0.20	0.023	0.483	0.006	0.226	15.9	0.3	1.1	1.7e+42	9.3e+40	5.6e+41	5.3e+41	3.5e-01
88	246.56358	−24.43950	0.02	0.012	0.638	0.051	0.022	25.0	2.7	0.0	2.3e+41	1.1e+39	6.6e+40	4.9e+40	2.6e-01
90	246.85067	−24.67744	0.17	0.019	0.130	0.001	0.048	12.5	0.1	1.0	2.8e+41	7.7e+40	3.1e+41	2.5e+41	1.2e+00
91	246.50921	−24.53763	0.03	0.017	0.333	0.008	0.039	23.1	0.5	1.9	1.6e+41	2.8e+39	1.6e+41	9.3e+40	7.7e-01
92	246.68467	−24.55352	0.09	0.022	0.148	0.001	0.030	12.1	0.1	1.1	1.7e+41	2.1e+40	5.7e+41	4.1e+41	3.0e+00
93	246.61654	−24.44259	0.03	0.023	0.750	0.046	0.075	30.1	2.1	2.8	5.2e+41	1.6e+39	5.6e+41	3.2e+41	8.4e-01
94	246.84054	−24.41269	0.06	0.014	0.123	0.002	0.007	12.4	0.3	0.5	8.6e+40	1.1e+40	1.3e+41	1.3e+41	1.6e+00
95	247.24146	−24.34838	1.80	0.035	0.120	0.001	0.009	10.2	0.1	0.6	2.4e+42	4.8e+42	6.0e+41	3.4e+41	1.2e+00
96	246.72017	−24.61907	0.02	0.011	0.186	0.004	0.065	12.2	0.6	1.1	3.8e+40	1.5e+39	7.0e+40	5.7e+40	1.7e+00
99	246.90942	−24.71049	0.09	0.024	0.142	0.001	0.026	13.1	0.1	1.0	1.7e+41	1.9e+40	5.2e+41	2.5e+41	2.4e+00
101	246.80104	−24.43306	0.04	0.021	0.426	0.009	0.052	18.2	0.6	1.4	2.9e+41	4.2e+39	5.0e+41	4.6e+41	1.6e+00
102	246.56946	−24.38237	0.01	0.012	0.692	0.016	0.090	22.5	0.7	2.8	1.5e+41	3.4e+38	7.2e+40	5.8e+40	4.5e-01
104	246.70058	−24.39840	0.03	0.022	0.333	0.014	0.071	22.7	1.2	2.6	1.5e+41	1.8e+39	5.5e+41	2.8e+41	2.8e+00
105	246.93021	−24.71032	0.05	0.016	0.235	0.003	0.043	14.2	0.2	0.9	1.3e+41	6.7e+39	1.5e+41	7.8e+40	8.9e-01
107	246.70508	−24.44994	0.12	0.025	0.346	0.005	0.096	17.2	0.4	2.5	6.2e+41	3.1e+40	8.2e+41	4.7e+41	1.1e+00
108	246.86646	−24.41034	0.22	0.034	0.134	0.005	0.046	11.7	2.2	0.0	3.4e+41	7.1e+40	1.8e+42	1.6e+42	5.1e+00
109	246.73371	−24.61132	0.06	0.022	0.181	0.002	0.104	14.3	0.2	1.7	1.4e+41	8.5e+39	5.1e+41	5.0e+41	3.7e+00
110	246.73967	−24.52867	0.71	0.028	0.123	0.000	0.023	10.5	0.1	0.8	9.9e+41	9.2e+41	1.2e+42	1.0e+42	1.6e+00
112	246.72262	−24.42202	0.02	0.018	0.336	0.010	0.050	19.8	1.3	2.1	1.0e+41	1.2e+39	3.0e+41	1.8e+41	2.3e+00
113	246.56167	−24.38003	0.02	0.020	0.613	0.008	0.082	22.8	0.4	2.6	2.6e+41	1.0e+39	3.3e+41	2.8e+41	1.2e+00
115	246.63904	−24.41661	0.06	0.011	0.204	0.002	0.029	17.7	0.2	0.9	1.8e+41	1.9e+40	6.3e+40	3.3e+40	3.2e-01
116	246.69417	−24.48982	0.64	0.046	0.263	0.002	0.114	18.1	0.1	2.9	2.3e+42	4.5e+41	5.3e+42	3.0e+42	1.9e+00
117	247.13250	−24.30284	0.50	0.030	0.094	0.000	0.007	10.0	0.1	0.7	5.8e+41	4.3e+41	5.3e+41	4.5e+41	1.2e+00
118	246.81575	−24.40339	0.07	0.022	0.201	0.004	0.051	15.3	0.4	1.1	1.7e+41	1.1e+40	5.4e+41	4.4e+41	2.8e+00
123	246.71762	−24.58159	0.13	0.026	0.354	0.003	0.056	14.8	0.2	0.9	6.7e+41	3.4e+40	9.4e+41	7.9e+41	1.3e+00
124	246.59242	−24.37514	0.03	0.014	0.575	0.007	0.100	23.5	0.3	1.3	3.1e+41	2.5e+39	1.1e+41	8.1e+40	3.1e-01
126	246.91358	−24.68730	0.07	0.024	0.376	0.005	0.047	15.6	0.3	2.2	4.1e+41	1.2e+40	5.5e+41	2.4e+41	9.7e-01
127	246.74225	−24.38574	0.50	0.062	0.231	0.003	0.056	16.1	0.3	3.1	1.5e+42	2.1e+41	1.2e+43	8.0e+42	6.5e+00
129	246.74675	−24.55165	0.24	0.023	0.163	0.001	0.023	12.0	0.1	0.5	4.5e+41	1.3e+41	6.7e+41	6.9e+41	1.7e+00
133	246.93429	−24.61873	0.25	0.058	0.438	0.003	0.049	17.3	0.1	1.8	1.8e+42	5.7e+40	8.3e+42	4.9e+42	3.6e+00

Notes.

^aThese core properties are based on our Getsources source extraction, which was performed on the JCMT GBS continuum maps. Only starless cores with coverage from GAS are listed. These columns display the core ID, location of the core center, mass (estimated using Equation (1)), and effective radius (geometric mean of the major and minor axes reported by Getsources). ^bObserved velocity dispersion (σ) and kinetic temperature (T_k) are estimated using a weighted average of GAS kinematic data. We also report the error in the weighted mean and the weighted standard deviation, the latter being more reflective of the variation in GAS values across extent of the core. ^cThese virial parameters are derived according to the methods presented in Section 4. ^dThe virial ratio, which indicates whether the core is bound by the combined forces of gravity and pressure, is given by $-({{\rm{\Omega }}}_{g}+{{\rm{\Omega }}}_{p,w}+{{\rm{\Omega }}}_{p,t})/2{{\rm{\Omega }}}_{k}$.

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Table 2.  Dense Core Properties in NGC 1333

IDa	R.A.a	Decl.a	Ma	Reffa	σobs (km s−1)b			Tk(K)b			Ωkc	−Ωgd	−Ωp,wd	−Ωp,td	Vir.d
	(J2000)	(J2000)	(M⊙)	(pc)	mn	err	std	mn	err	std	(erg)	(erg)	(erg)	(erg)	Rat.
5	52.26300	31.26464	1.86	0.016	0.310	0.001	0.035	17.0	0.1	0.4	8.1e+42	1.1e+43	1.5e+41	3.6e+41	7.1e-01
7	52.25846	31.26069	0.92	0.016	0.384	0.002	0.073	16.9	0.1	0.6	5.5e+42	2.7e+42	1.5e+41	3.7e+41	3.0e-01
11	52.25717	31.34161	1.98	0.028	0.261	0.001	0.015	15.8	0.1	0.8	6.8e+42	7.1e+42	7.5e+41	1.4e+42	6.8e-01
15	52.25900	31.33671	0.47	0.016	0.234	0.002	0.011	15.5	0.1	0.2	1.4e+42	7.2e+41	1.4e+41	2.7e+41	3.9e-01
16	52.32925	31.38748	0.34	0.017	0.176	0.006	0.034	15.0	0.7	0.5	7.8e+41	3.5e+41	1.1e+41	1.9e+41	4.2e-01
17	52.24992	31.35978	1.24	0.021	0.328	0.002	0.029	14.3	0.1	0.4	5.6e+42	3.8e+42	2.8e+41	4.5e+41	4.0e-01
19	52.27758	31.26116	0.85	0.020	0.383	0.002	0.039	14.7	0.1	0.6	4.8e+42	1.9e+42	2.7e+41	5.8e+41	2.8e-01
22	52.26150	31.25393	0.32	0.021	0.547	0.002	0.064	17.0	0.1	0.4	3.3e+42	2.4e+41	3.4e+41	7.8e+41	2.1e-01
23	52.26713	31.34976	0.63	0.021	0.257	0.002	0.009	14.4	0.1	0.3	2.1e+42	9.6e+41	3.0e+41	4.7e+41	4.2e-01
25	52.27925	31.29047	0.74	0.020	0.224	0.002	0.018	12.9	0.1	0.4	2.0e+42	1.4e+42	2.8e+41	8.5e+41	6.5e-01
26	52.22392	31.24904	0.77	0.024	0.477	0.002	0.023	14.1	0.1	0.4	6.2e+42	1.3e+42	4.8e+41	1.0e+42	2.2e-01
27	52.27150	31.35081	0.69	0.021	0.250	0.002	0.012	14.7	0.1	0.5	2.2e+42	1.1e+42	3.0e+41	4.4e+41	4.3e-01
28	52.28754	31.25473	2.03	0.031	0.427	0.001	0.053	12.7	0.0	0.6	1.3e+43	6.7e+42	1.1e+42	2.2e+42	3.8e-01
29	52.32633	31.41929	0.70	0.020	0.246	0.005	0.051	12.8	0.4	0.6	2.1e+42	1.3e+42	1.5e+41	2.1e+41	4.0e-01
30	52.35658	31.47147	0.75	0.018	0.209	0.002	0.023	10.6	0.2	0.3	1.7e+42	1.6e+42	8.1e+40	3.9e+41	6.2e-01
32	52.29475	31.36245	1.33	0.034	0.328	0.019	0.245	18.5	2.4	7.5	6.5e+42	2.7e+42	1.1e+42	1.5e+42	4.1e-01
33	52.30683	31.31146	0.98	0.032	0.235	0.001	0.021	13.4	0.1	0.5	2.8e+42	1.5e+42	1.1e+42	3.2e+42	1.0e+00
34	52.31754	31.21038	1.51	0.038	0.352	0.002	0.036	11.2	0.1	0.6	7.1e+42	3.1e+42	1.5e+42	3.2e+42	5.5e-01
35	52.30712	31.30276	0.96	0.028	0.224	0.002	0.028	13.0	0.1	0.3	2.6e+42	1.7e+42	7.1e+41	2.1e+42	8.7e-01
37	52.28554	31.28593	0.88	0.029	0.247	0.002	0.033	13.7	0.1	1.0	2.7e+42	1.4e+42	8.8e+41	2.4e+42	8.7e-01
39	52.34996	31.55555	1.20	0.028	0.129	0.001	0.003	10.6	0.1	0.3	1.7e+42	2.7e+42	2.4e+41	2.3e+41	9.0e-01
41	52.27354	31.27033	0.21	0.019	0.280	0.002	0.040	16.6	0.1	0.5	7.9e+41	1.1e+41	2.5e+41	6.0e+41	6.1e-01
42	52.27150	31.37014	0.15	0.026	0.539	0.020	0.152	21.8	1.9	1.7	1.6e+42	4.3e+40	5.0e+41	8.7e+41	4.5e-01
43	52.33325	31.39885	0.39	0.029	0.170	0.004	0.027	15.0	0.4	1.0	8.5e+41	2.7e+41	5.0e+41	9.2e+41	9.9e-01
44	52.29983	31.24761	0.19	0.020	0.243	0.002	0.034	12.5	0.1	0.3	5.4e+41	8.9e+40	2.8e+41	5.1e+41	8.2e-01
45	52.16458	31.30774	2.39	0.034	0.208	0.001	0.022	11.8	0.0	0.3	5.6e+42	8.6e+42	1.1e+42	1.5e+42	1.0e+00
46	52.26054	31.19716	0.23	0.020	0.187	0.002	0.024	12.5	0.2	0.3	4.9e+41	1.3e+41	2.2e+41	5.3e+41	9.0e-01
47	52.25308	31.23985	0.12	0.021	0.512	0.009	0.128	17.6	0.4	1.5	1.1e+42	3.3e+40	3.2e+41	6.9e+41	4.7e-01
48	52.27067	31.31327	0.52	0.032	0.234	0.002	0.024	13.8	0.1	0.6	1.5e+42	4.4e+41	1.1e+42	3.8e+42	1.8e+00
50	52.28117	31.24738	0.46	0.023	0.421	0.001	0.096	12.2	0.1	0.2	3.0e+42	4.9e+41	4.1e+41	8.1e+41	2.9e-01
52	52.21137	31.24105	0.21	0.025	0.410	0.002	0.016	13.7	0.1	0.5	1.3e+42	8.8e+40	4.9e+41	9.4e+41	5.9e-01
53	52.17858	31.10492	0.27	0.021	0.132	0.001	0.016	10.5	0.1	0.1	3.9e+41	1.8e+41	1.4e+41	3.9e+41	9.0e-01
54	52.29508	31.29623	0.42	0.022	0.208	0.002	0.023	12.3	0.2	0.4	1.0e+42	4.3e+41	3.5e+41	1.0e+42	8.8e-01
55	52.17567	31.29345	1.98	0.069	0.203	0.000	0.022	11.9	0.0	0.4	4.6e+42	2.9e+42	9.6e+42	1.4e+43	2.9e+00
56	52.23917	31.36732	0.15	0.021	0.463	0.005	0.051	15.1	0.2	0.5	1.2e+42	5.9e+40	2.6e+41	3.7e+41	2.9e-01
57	52.26900	31.22556	0.08	0.020	0.435	0.010	0.043	16.2	0.5	1.0	5.4e+41	1.5e+40	2.5e+41	5.1e+41	7.2e-01
58	52.13675	31.08251	0.20	0.023	0.147	0.001	0.015	10.6	0.1	0.3	3.2e+41	9.1e+40	1.6e+41	7.3e+41	1.5e+00
60	52.19796	31.25258	0.22	0.026	0.174	0.001	0.020	11.3	0.1	0.6	4.1e+41	9.2e+40	5.7e+41	9.1e+41	1.9e+00
61	52.18950	31.09689	0.11	0.016	0.111	0.001	0.003	10.6	0.2	0.2	1.4e+41	3.5e+40	6.1e+40	1.7e+41	9.7e-01
62	52.30725	31.34255	0.64	0.040	0.367	0.006	0.044	16.1	0.5	1.4	3.5e+42	5.1e+41	2.0e+42	4.1e+42	9.4e-01
63	52.25246	31.35077	0.11	0.023	0.314	0.002	0.022	16.4	0.1	0.7	4.7e+41	2.6e+40	3.7e+41	6.4e+41	1.1e+00
64	52.13550	31.25533	0.21	0.028	0.184	0.002	0.047	11.2	0.1	0.7	4.3e+41	8.3e+40	4.8e+41	8.1e+41	1.6e+00
65	52.12367	31.16582	0.26	0.022	0.099	0.001	0.003	9.6	0.2	0.4	3.0e+41	1.5e+41	1.8e+41	1.0e+42	2.3e+00
66	52.13075	31.23895	0.34	0.030	0.166	0.001	0.023	10.5	0.2	0.3	5.9e+41	1.9e+41	5.8e+41	1.2e+42	1.7e+00
67	52.34829	31.60201	0.63	0.031	0.115	0.001	0.010	9.4	0.1	0.6	7.8e+41	6.4e+41	3.0e+41	3.4e+41	8.3e-01
68	52.33888	31.22258	0.14	0.031	0.389	0.010	0.047	13.9	0.8	2.5	8.0e+41	3.1e+40	7.9e+41	1.7e+42	1.6e+00
69	52.23613	31.34918	0.05	0.020	0.381	0.005	0.024	18.2	0.3	0.9	3.0e+41	6.2e+39	2.6e+41	4.2e+41	1.2e+00
70	52.34258	31.42884	0.34	0.041	0.103	0.001	0.017	12.1	0.1	0.3	4.7e+41	1.4e+41	1.2e+42	2.4e+42	3.9e+00
73	52.32963	31.19548	1.81	0.049	0.310	0.002	0.067	10.8	0.1	0.8	6.9e+42	3.4e+42	2.8e+42	6.0e+42	8.8e-01
75	52.23483	31.32137	0.18	0.034	0.199	0.001	0.026	15.9	0.1	1.5	4.7e+41	4.8e+40	1.4e+42	3.9e+42	5.7e+00
76	52.16329	31.31961	0.70	0.040	0.261	0.001	0.039	13.1	0.1	0.8	2.2e+42	6.3e+41	1.6e+42	3.1e+42	1.2e+00

Notes.

^aThese core properties are based on our Getsources source extraction, which was performed on the JCMT GBS continuum maps. Only starless cores with coverage from GAS, FCRAO, and the column density maps (A. Singh et al. 2019, in preparation) are listed. These columns display the core ID, location of the core center, mass (estimated using Equation (1)), and effective radius (geometric mean of the major and minor axes reported by Getsources). ^bObserved velocity dispersion (σ) and kinetic temperature (T_k) are estimated using a weighted average of GAS kinematic data. We also report the error in the weighted mean and the weighted standard deviation, the latter being more reflective of the variation in GAS values across extent of the core. ^cThese virial parameters are derived according to the methods presented in Section 4. ^dThe virial ratio, which indicates whether the core is bound by the combined forces of gravity and pressure, is given by $-({{\rm{\Omega }}}_{g}+{{\rm{\Omega }}}_{p,w}+{{\rm{\Omega }}}_{p,t})/2{{\rm{\Omega }}}_{k}$.

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Table 3.  Dense Core Properties in B18

IDa	R.A.a	Decl.a	Ma	Reffa	σobs (km s−1)b			Tk(K)b			Ωkc	−Ωgd	−Ωp,wd	−Ωp,td	Vir.d
	(J2000)	(J2000)	(M⊙)	(pc)	mn	err	std	mn	err	std	(erg)	(erg)	(erg)	(erg)	Rat.
8	67.34042	24.55539	0.13	0.019	0.118	0.002	0.013	12.8	0.4	1.1	2.0e+41	4.6e+40	6.3e+40	1.3e+41	5.8e-01
9	68.90600	24.15507	4.85	0.044	0.133	0.000	0.025	9.9	0.0	0.3	6.9e+42	2.8e+43	4.6e+41	8.1e+41	2.1e+00
11	67.98225	24.54840	1.43	0.031	0.103	0.000	0.009	8.8	0.0	0.3	1.6e+42	3.4e+42	2.1e+41	4.7e+41	1.3e+00
12	66.64562	24.69573	0.14	0.029	0.116	0.003	0.029	14.7	0.9	3.8	2.4e+41	3.5e+40	1.3e+41	2.4e+42	5.3e+00
13	68.18908	24.36316	0.49	0.028	0.151	0.001	0.022	9.8	0.2	0.7	7.6e+41	4.4e+41	1.7e+41	3.2e+41	6.1e-01
14	67.34754	24.57879	0.66	0.043	0.111	0.000	0.007	10.0	0.0	0.5	8.3e+41	5.2e+41	7.2e+41	1.4e+42	1.6e+00
15	68.19529	24.42679	0.81	0.036	0.097	0.000	0.010	9.3	0.0	0.5	9.0e+41	9.3e+41	3.7e+41	5.7e+41	1.0e+00
16	68.18333	24.38401	0.66	0.032	0.122	0.000	0.009	9.3	0.1	0.5	8.4e+41	6.9e+41	2.7e+41	4.7e+41	8.5e-01
17	68.96121	24.16152	0.58	0.037	0.118	0.001	0.023	10.6	0.3	1.6	8.0e+41	4.7e+41	2.7e+41	5.9e+41	8.3e-01

Notes.

^aThese core properties are based on our Getsources source extraction, which was performed on the JCMT GBS continuum maps. Only starless cores with coverage from GAS, FCRAO, and the column density maps (A. Singh et al. 2019, in preparation) are listed. These columns display the core ID, location of the core center, mass (estimated using Equation (1)), and effective radius (geometric mean of the major and minor axes reported by Getsources). ^bObserved velocity dispersion (σ) and kinetic temperature (T_k) are estimated using a weighted average of GAS kinematic data. We also report the error in the weighted mean and the weighted standard deviation, the latter being more reflective of the variation in GAS values across extent of the core. ^cThese virial parameters are derived according to the methods presented in Section 4. ^dThe virial ratio, which indicates whether the core is bound by the combined forces of gravity and pressure, is given by $-({{\rm{\Omega }}}_{g}+{{\rm{\Omega }}}_{p,w}+{{\rm{\Omega }}}_{p,t})/2{{\rm{\Omega }}}_{k}$.

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Kirk et al. (2017) found that dense cores in Orion A appear to be confined primarily because of the pressure from the weight of overlying molecular cloud material. Following this work, we estimate pressure from the weight of the surrounding cloud using (McKee 1989; Kirk et al. 2006):

$$\begin{eqnarray}&&{P}_{w}=\pi G{\rm{\Sigma }}\bar{{\rm{\Sigma }}}\end{eqnarray} \tag{ 7 }$$

where P_w is the pressure from cloud weight, Σ is the surface mass density (column density multiplied by mean particle mass) along the line of sight of the core, and $\bar{{\rm{\Sigma }}}$ is the mean surface mass density across the entire cloud. This equation assumes that the large-scale structure of the molecular cloud is well approximated by a sphere and that the density profile is similar to $\rho \propto {r}^{-k}$ with k = 1. Somewhat more centrally concentrated density profiles in the range of k = 1–3 introduce a second positive term to these relationships, so in that case Equation (7) would provide a lower limit (Kirk et al. 2017). Additional information on the derivation of this cloud weight pressure equation and relevant geometry can be found in Kirk et al. (2017).

As in Kirk et al. (2017), we apply an á Trous wavelet decomposition to the column density maps to estimate the contributions of cloud material versus core material. In our source extraction, cores are generally less than 0.1 pc across (comparable to the typical core sizes of 0.03–0.2 pc given in Bergin & Tafalla 2007; Di Francesco et al. 2007). For our main analysis, we conservatively assign scales of approximately 0.5 pc and above as belonging to the cloud. The á Trous algorithm decomposes structures on scales of 2^N pixels, while the column density maps have a pixel scale of 14'' per pixel. We therefore adopted a scale of 64 pixels for L1688 and B18 and a scale of 32 pixels for NGC 1333, corresponding to physical scales of approximately 0.6 pc in all three regions. The effects of choosing a different scale are further explored in Appendix A.3. Aside from changing the magnitude of the cloud weight pressure term, the filter scale choice has a relatively small effect on the qualitative results of this paper, largely because of the relatively modest strength of cloud weight pressure in the regions studied.

For all three regions that we analyze, we find that the pressure attributed to cloud weight produces a relatively small virial term (Ω_p,w) relative to the findings of Kirk et al. (2017) in Orion A. This difference is likely attributable to the typical column density associated with each of these regions: Orion A is a Giant Molecular Cloud, with a mean column density of $\bar{N}=3.9\times {10}^{22}$ cm⁻² over the area analyzed in Kirk et al. (2017). Meanwhile, the mean column densities estimated for the three clouds that we analyze here are $\bar{N}=4.6\times {10}^{21}$ cm⁻², 2.7 × 10²¹ cm⁻², and 2.0 × 10²¹ cm⁻² for L1688, NGC 1333, and B18, respectively, i.e., roughly an order of magnitude smaller than Orion A. Because the pressure attributable to cloud weight is relatively small, it does not dominate the forces contributing to core confinement as seen in Orion A (Kirk et al. 2017). We therefore estimate additional pressure sources before presenting the results of our virial analysis.

A separate form of pressure is exerted on each core by turbulent motions of the surrounding cloud. These motions can be traced by transitions from molecules that are common at lower densities, such as ¹³CO, which will be used in this paper. Line widths are found by fitting a Gaussian distribution to the ¹³CO lines, and the result is put into the following equation, which follows from the ideal gas law:

$$\begin{eqnarray}&&{P}_{t}={n}_{{}^{13}\mathrm{CO}}{\mu }_{{mn}}{m}_{{\rm{H}}}\left({\sigma }_{\mathrm{tot}{,}^{13}\mathrm{CO}}^{2}\right)\end{eqnarray} \tag{ 8 }$$

where ${n}_{{}^{13}\mathrm{CO}}$ is the density of the gas that ¹³CO is probing. Because ¹³CO freezes out at densities above 10⁵ cm⁻³ (Di Francesco et al. 2007), we conservatively select a less dense value of 3 × 10³ cm⁻³ to represent the typical density of the ambient cloud. This value is somewhat lower than the 5 × 10³ cm⁻³ employed in Kirk et al. (2017), which reflects the lower mean densities in these clouds compared to Orion A, and the lower-density gas traced by ¹³CO compared to the C¹⁸O used in Kirk et al. (2017). The resulting value for P_t can be used in Equation (6) to generate a value for an external turbulent pressure term, Ω_p,t. When this value is added to the cloud weight pressure term, a total external pressure term is generated. We discuss the implications of changing some of the assumptions made in computing the turbulent pressure term in Appendix A.4.

The result of a more complete virial analysis, including both pressure terms, is shown in Figure 7. The vertical axis shows the ratio between the gravitational and pressure terms, which describes which of the two terms dominates the binding of cores. This ratio is plotted against the virial ratio, $-({{\rm{\Omega }}}_{g}+{{\rm{\Omega }}}_{p,t}+{{\rm{\Omega }}}_{p,w})$/2Ω_k. A virial ratio of one indicates virial equilibrium, while a ratio greater than one indicates that the core is bound. With the addition of turbulent and cloud weight pressure, 36% of cores in L1688, 27% of cores in NGC 1333, and 56% of B18's cores qualify as bound. Furthermore, over 75% of cores across all regions have a pressure term, Ω_p, that is larger than the corresponding gravity term, Ω_g, implying that external pressure dominates the binding of the dense cores. We also find that this pressure is dominated by turbulent pressure in both NGC 1333 and B18, with a median turbulent pressure term of roughly double the cloud weight pressure in both regions. L1688 has more equal contributions from the two pressure sources, with turbulent pressure contributing roughly 0.75 times that of the cloud weight to the total pressure term in L1688. This balance of pressures is in stark contrast to Orion A (Kirk et al. 2017), which has a median cloud weight pressure term that is 15 times higher than the corresponding turbulent pressure term. Comparisons to the Kirk et al. (2017) results in Orion A are discussed more extensively in Section 5.3. More information comparing the relative strengths of forces binding cores can be found in Table 4, which includes Orion A.

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Table 4.  A Summary of the Virial Terms

	L1688	NGC 1333	B18	Orion Ac
$\mathrm{log}| {{\rm{\Omega }}}_{k}|$ a,b	${41.58}_{-0.37}^{+0.38}$	${42.19}_{-0.30}^{+0.41}$	${41.90}_{-0.38}^{+0.04}$	${42.46}_{-0.36}^{+0.46}$
$\mathrm{log}| {{\rm{\Omega }}}_{g}|$ a,b	${40.30}_{-0.89}^{+0.91}$	${41.86}_{-0.72}^{+0.43}$	${41.67}_{-0.46}^{+0.24}$	${41.35}_{-0.44}^{+0.65}$
$\mathrm{log}| {{\rm{\Omega }}}_{p,w}|$ a,b	${41.05}_{-0.47}^{+0.66}$	${41.45}_{-0.27}^{+0.36}$	${41.31}_{-0.34}^{+0.19}$	${43.16}_{-0.34}^{+0.31}$
$\mathrm{log}| {{\rm{\Omega }}}_{p,t}|$ a,b	${40.92}_{-0.42}^{+0.60}$	${41.81}_{-0.29}^{+0.38}$	${41.67}_{-0.33}^{+0.17}$	${41.94}_{-0.31}^{+0.34}$
$\mathrm{log}| {{\rm{\Omega }}}_{g}/{{\rm{\Omega }}}_{k}|$ a	−1.21${}_{-0.53}^{+0.62}$	−0.41${}_{-0.38}^{+0.13}$	−0.20${}_{-0.03}^{+0.22}$	$-{0.96}_{-0.38}^{+0.33}$
$\mathrm{log}| {{\rm{\Omega }}}_{p,w}/{{\rm{\Omega }}}_{k}|$ a	−0.47${}_{-0.29}^{+0.59}$	−0.48${}_{-0.37}^{+0.24}$	−0.49${}_{-0.16}^{+0.11}$	${0.69}_{-0.45}^{+0.40}$
$\mathrm{log}| {{\rm{\Omega }}}_{p,t}/{{\rm{\Omega }}}_{k}|$ a	−0.56${}_{-0.29}^{+0.54}$	−0.12${}_{-0.52}^{+0.24}$	−0.21${}_{-0.17}^{+0.08}$	$-{0.46}_{-0.44}^{+0.47}$
$\mathrm{log}({{\rm{\Omega }}}_{p,t}/{{\rm{\Omega }}}_{p,w})$ a	−0.12${}_{-0.09}^{+0.05}$	${0.31}_{-0.08}^{+0.07}$	${0.28}_{-0.02}^{+0.04}$	$-{1.18}_{-0.12}^{+0.14}$
$\mathrm{log}({{\rm{\Omega }}}_{g}/({{\rm{\Omega }}}_{p,t}+{{\rm{\Omega }}}_{p,w}))$ a	−1.20${}_{-0.74}^{+1.00}$	−0.51${}_{-0.54}^{+0.61}$	−0.05${}_{-0.56}^{+0.04}$	$-{1.68}_{-0.80}^{+0.58}$
$\mathrm{log}| ({{\rm{\Omega }}}_{p,t}+{{\rm{\Omega }}}_{p,w}+{{\rm{\Omega }}}_{g})/{{\rm{\Omega }}}_{k}|$ a	−0.35${}_{-0.29}^{+0.46}$	−0.08${}_{-0.29}^{+0.11}$	−0.02${}_{-0.10}^{+0.18}$	${0.74}_{-0.39}^{+0.39}$

Notes.

^aFor each virial term listed, we provide the median value and the range spanned by the upper and lower quartiles in the distribution. ^bVirial terms are given in units of erg. ^cValues calculated from Kirk et al. (2017) with their protostellar cores excluded for consistency with our analysis.

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In the Appendix, we examine the sources of uncertainty in our virial parameter estimates, including the dust-based core catalog, the NH₃-based core line widths, the large-scale cloud weight pressure, and the ambient turbulent pressure. Of particular note is our examination of the core line widths in Appendix A.2, where we find evidence that the NH₃ observations may be biased to larger values than should be attributed to the dense core gas. We discuss the implications of this conclusion in Appendix A.2.4.

Magnetic fields constitute a potentially important additional factor in the virial state of cores, working internally to support cores against collapse from gravity and external pressure. Subsequently, a strong magnetic term could change our conclusions about the virial state of cores. The importance of magnetic fields in the binding of cores, however, is not clear. In L1688, Pattle et al. (2015) used magnetic field strength data from Crutcher et al. (1993) and Troland et al. (1996) to argue that the magnetic support term was small, with a ratio of Ω_M to Ω_k,NT (i.e., the ratio of the magnetic field to nonthermal kinetic energy terms) of only 0.11 on average.

The recent JCMT B-fields In Star-forming Region Observations (BISTRO) survey (Ward-Thompson et al. 2017) provides polarization maps for many JCMT GBS clouds, including NGC 1333 and L1688, but not B18. These data allow updated magnetic field strength estimates to be made for various star-forming regions. Using the Chandrasekhar–Fermi method (Chandrasekhar & Fermi 1953), Soam et al. (2018) estimated the mean plane-of-the-sky magnetic field strength in Oph B2 to be 630 ± 410 μG, while a similar study in Oph A (Kwon et al. 2018) found a slightly smaller plane-of-the-sky magnetic field strength of between 200 and 500 μG. Measurements were also recorded for Oph B1, but the magnetic field there was found to be disorganized on the cloud scale, so an overall estimate could not be produced (Soam et al. 2018). Assuming that the magnetic field strengths are uniform across each separate region, these values would correspond to dramatically higher magnetic terms than those used in Pattle et al. (2015), with a Ω_M to Ω_k,NT ratio of 1.9 on average in Oph A and 2.0 in Oph B2, which would certainly have a significant impact on the virial state of many cores in the sample.

BISTRO-based estimates of magnetic field strength have been made for only a handful of regions (Pattle et al. 2017a; Kwon et al. 2018; Soam et al. 2018) so far. Additional research will be necessary to provide sufficiently detailed estimates in the regions of interest, which must also take regional variations into account in clouds as diverse as L1688. Therefore, because high-resolution estimates of the magnetic field are not available in all regions explored, we exclude the term for the sake of maintaining a self-consistent analysis across all regions. Our results for the fraction of bound cores are subsequently best interpreted as providing an upper limit to the bound fraction of cores, as any nonzero magnetic term would make cores appear less bound. Once more extensive and detailed studies on the magnetic field strength become available for many regions, future virial analyses should include this term, as the early BISTRO magnetic field strength estimates (e.g., Soam et al. 2018) suggest that the impact of magnetic fields may be significant.

In Appendix A.2, we investigate the unusually large dense core velocity dispersions noted in Section 3. We find evidence suggesting that the observed NH₃ line widths may be larger than should be attributed solely to the dense core material. While corrections are not possible for each individual core, in Appendix A.2, we derive representative correction factors for the velocity dispersion of cores in each of the regions. Figure 8 shows the virial ratios after these correction factors have been applied. Using these approximate correction factors, the number of unbound cores is reduced to 55 of the total 134 cores. Higher resolution observations of a higher-density gas tracer for each core in the sample are needed to verify the true dense core velocity dispersions.

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Our results present a primarily pressure-dominated view of cores in L1688, NGC 1333, and B18, a conclusion shared by many previous analyses of dense cores (e.g., Maruta et al. 2010; Barnes et al. 2011). While virial analyses of dense cores have traditionally focused on comparisons between gravitational confinement and internal support, the importance of pressure has recently become a much more consistent theme. Bertoldi & McKee (1992) were among the first to consider this term, concluding that of their sample of cores in Ophiuchus, Orion, the Rosette, and Cepheus OB3, pressure, primarily rooted in cloud turbulence, is required to bind a significant population of mainly smaller cores. Later studies such as that by Maruta et al. (2010) built on this research by demonstrating that external pressure is consistently quite large relative to other forces in these systems, making it potentially critical to the virial state of cores. Maruta et al. (2010) also provided a virial analysis, confirming that in their sample of Ophiuchus cores, the external pressure term was more important than the gravitational term in most cases.

Many more recent papers have supported a similar narrative. An analysis of the CHaMP survey region (Barnes et al. 2011), which focused on a diverse set of more distant clouds between 1.1 and 6.8 kpc away, also showed generally small gravitational terms, with evidence suggesting that external pressure, mainly due to the turbulence of the surrounding cloud, may bind these cores in most cases. Studies making use of the JCMT GBS continuum data in both Ophiuchus and the Cepheus Flare region (Pattle et al. 2015, 2017b) also concluded that pressure is the dominant force in many cores, even more so in the Cepheus Flare region than Ophiuchus. Papers adopting newer core-identification methods have reached similar conclusions. How-Huan Chen et al. (2018) concluded that most objects in their unique set of droplets were pressure dominated, while the dendrogram analysis of Seo et al. (2015) found that almost half of the gravitationally unbound cores identified by their survey can be bound with the consideration of this pressure term.

While pressure does appear to be important in many virial analyses, some studies identify cores whose binding is dominated by self-gravity. In the analysis of dendrogram leaves in Cepheus-L1251 presented by Keown et al. (2017) and in the B59 virial analysis by Redaelli et al. (2017), for example, many of the cores appear to be dominated by self-gravity. Furthermore, in dendrogram analyses of Perseus L1448 (Rosolowsky et al. 2008) and Serpens South (Friesen et al. 2016), it is suggested that larger-scale structures may be more likely to be self-gravitating. Similarly, in our present analysis, we identified 13 cores bound by gravity alone, and these cores tended to be the most massive of the group.

We further conclude from our analysis that external turbulent pressure is dominant over cloud weight pressure in NGC 1333 and B18, while the two pressure terms share a similar level of importance in L1688. An interesting contrast can be made between our conclusions and those of Kirk et al. (2017), who observed the much more massive Orion A cloud using essentially the same data sets as this paper: Getsources catalogs derived from JCMT SCUBA-2 data, GAS DR1 data for internal kinetic properties, and cloud weight pressure estimates from column density maps derived using Planck and Herschel data (Lombardi et al. 2014). That paper concluded that cloud weight pressure is the dominant force in binding Orion A cores, contributing a binding term 15 times greater than the corresponding turbulent pressure term on average. This situation is primarily due to the high column density of the region. In our analysis, cloud weight pressure was either roughly the same size as or smaller than the turbulent pressure, ranging from a cloud weight pressure to turbulent pressure ratio of 1.33 in L1688 to roughly 0.5 in NGC 1333 and B18. Like in L1688, the turbulent pressure term derived in Orion A is also smaller relative to the internal kinetic energy term than it is in NGC 1333 and B18, suggesting that the strength of turbulent pressure may correlate loosely with the level of clustering in the parent cloud.

This small sample of observations provides evidence that cloud weight pressure may tend to dominate the pressure budget of cores in larger, more clustered systems, rather than turbulent pressure, which may dominate in sparser regions. It is difficult, however, to make definitive statements about these trends given the current sample size of clouds that have been analyzed in an identical manner. While some virial analyses do argue that cloud pressure (e.g., Lada et al. 2008; Kirk et al. 2017) or turbulent pressure (e.g., Barnes et al. 2011; Pattle et al. 2015) dominates, many analyses treat the different forms of pressure as a single combined force, and therefore our ability to compare these different pressure terms across a diverse array of regions is extremely limited. Virial analyses considering both sources of pressure on different star-forming regions will be critical to the development of our knowledge of the role that cloud weight and turbulent pressure have in a diverse range of star-forming environments. Future GAS observations, which cover numerous additional regions, will represent an important contribution to the study of these forces.

The method used to identify dense cores has long been a point of discussion, and many different techniques have been used in past virial analyses. A common technique, which is applied in this paper, defines cores according to their extent in dust emission. There are many ways to extract sources in this manner, though they all conceptually work in the same way—by identifying peaks in the emission map and assigning an emission region to that peak. Cores can also be identified via gas emission, however. Because the total core mass can be more difficult to estimate using only a single gas tracer, gas data are often supplemented with dust continuum observations, which can be applied at the location of the identified structures to compute masses. Finally, hybrid approaches also exist, drawing from gas, dust, and temperature data. These approaches attempt to identify cores on the basis of their expected properties, such as lower line widths, lower temperature, and higher dust intensity (e.g., How-Huan Chen et al. 2018).

A recent study by Keown et al. (2017) highlights the importance of the core-identification method on virial analyses. Using Cepheus-L1251 observations from GAS and Herschel, Keown et al. (2017) performed two complementary virial analyses. Dense cores were identified both in the GAS map using a dendrogram analysis technique (Rosolowsky et al. 2008) and also in the Herschel dust map using Getsources. Remaining portions of the analysis were very similar to those of this paper: Herschel observations were used to estimate core masses, GAS NH₃ data was used to estimate core temperatures and nonthermal velocity dispersions, ¹³CO observations were used to determine external turbulent pressure, and H₂ column density maps were used to determine cloud weight pressure. Keown et al. (2017) found that, while most Herschel-identified cores appear pressure confined, similar to our results, the majority of the GAS-based dendrogram leaves were instead gravitationally bound. This result suggests that the core-identification method may strongly influence the subsequent virial analysis. Other gas-based dendrogram virial analyses, such as Friesen et al. (2016) in Serpens South and Rosolowsky et al. (2008) in Perseus L1448, also tend to conclude that self-gravity plays a greater role than we find, but without a direct comparison using the same regions and data sets, it is not possible to attribute this difference to the core-identification method.

How-Huan Chen et al. (2018) uses a more complicated, hybrid technique for core identification to perform a virial analysis on L1688 and B18 GAS data, in which a very specific subset of dense cores, (referred to as "droplets" in that paper) are identified using their observed kinematic properties. Droplets are identified by dividing up contiguous intersections of regions with 10σ integrated intensity emission and subsonic velocity dispersion into smaller regions with one peak each. These are then compared to Herschel column density and dust temperature maps, to ensure that each core contains a minimum in dust temperature and a maximum in column density. If an object satisfies these criteria and is resolved by the GAS beam (32''), is it included. Hasenberger et al. (2018) also use minima in dust temperatures in their identification of cores, so there is a slowly expanding body of literature exploring the detection of coherent cores through their kinematic properties. Similar to our conclusions, How-Huan Chen et al. (2018) conclude that turbulent pressure dominates the binding of cores. An important difference, however, is that the techniques in How-Huan Chen et al. (2018) target a specific subset of dense cores, which resemble smaller variants of "coherent cores" (Goodman et al. 1998), with much lower levels of nonthermal support than we estimate in our dense core sample. Because of the focus on cores of this variety, these methods are not applicable to as wide of a variety of sources as most other core-identification methods.

At present, it is unclear which source identification technique or techniques is best for virial analyses and related studies. Fortunately, surveys such as GAS are allowing progress to be made in this regard. With a consistent data set across a variety of clouds, GAS data allow for a multiregion analysis, highlighting similarities and differences in core populations that can be measured with any single core-identification technique. Furthermore, the application of multiple core-identification techniques across one or more regions in a consistent manner will provide a way to test and understand the biases and advantages of each technique.

As part this paper, we run our analysis on dense cores in all three regions using two different source-finding algorithms: Getsources and FellWalker. Here, we describe the implementation of both algorithms, as well as the resulting core catalogs. An example comparison of the results of both source extraction methods is shown in Figure 9, in one of the more crowded regions of our sample, Oph B. We note that well-separating emission in crowded areas is a known challenge (Pineda et al. 2009) and that employing and comparing the results from multiple algorithms helps to reduce uncertainties in the potential biases associated with any single technique. We also compare these results to those from the Pattle et al. (2015) L1688 dense core catalog, which uses another separate source-finding algorithm called CuTEx.

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A.1.1. Getsources

A.1.2. FellWalker

We repeated our main analysis using instead the FellWalker algorithm to identify cores. FellWalker is an algorithm that identifies significant peaks according to the number of pixels visited on all possible paths to the peak. FellWalker has a number of user-adjustable parameters. We kept most of these at the default recommended values but changed several settings where testing suggested that the final core catalog was improved, typically by providing cores that were better separated from surrounding filamentary structure. All of these nondefault parameters are summarized in Table 5. We applied the optional "Findback" function (Currie et al. 2014), which subtracts larger-scale background emission from the individual cores. Finally, we culled the resulting core catalog to remove nonrobust detections. First, we required that at least one third of the pixels in each core have values of greater than five times the local noise level. To rule out contamination by noise spikes, we also slightly smoothed the image (using a Gaussian smoothing kernel of σ = 2 pixels) and required that at least one third of the pixels in the smoothed core have a value of greater than two times the local noise level. Roughly one third of cores were eliminated through the culls, leaving 38 in L1688, 37 in NGC 1333, and 12 in B18. The cores identified by FellWalker in each region are shown in Figures 10–12.

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Table 5.  Nondefault Settings Used For FellWalker

Parameter	Value	Description
FellWalker.AllowEdge	0	Do not include cores that touch the image boundary
FellWalker.CleanIter	6	Apply smoothing to boundaries of clumps
FellWalker.MaxJump	2	The maximum distance (in pixels) that a path can skip
FellWalker.Noise	3.2*rms	The minimum value from which to start climbing
FellWalker.rms	0.05	The rms Noise level of the data

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We ran our virial analysis on the FellWalker catalog and show the results in Figure 13. About 40% of the cores in L1688, 50% in NGC 1333, and over 80% of B18 cores are identified as bound by the combination of gravity and pressure, which are all higher than the values of 36%, 27%, and 56%, respectively, that we estimated in our main Getsources analysis. Furthermore, all cores except three have their binding dominated by pressure, not gravity, which, as in the Getsources results, underlines the importance of pressure in these cores. The fact that we find a higher fraction of bound cores using FellWalker, especially in NGC 1333 and B18, may be due to the nature of the FellWalker cores, which are often larger than those found by Getsources. Larger effective radii increase the area over which the pressure acts, therefore increasing the pressure terms.

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Because of its ability to separate cores with closely spaced peaks using additional higher-resolution but noisier 450 μm data, Getsources detects significantly more cores in crowded regions compared to FellWalker. As can be seen in Figure 9, FellWalker sometimes groups together visually distinct faint emission structures with nearby brighter features. Furthermore, because FellWalker does not model and subtract larger-scale structures during core detection, it often has difficulty separating cores from bright, larger-scale structures. As such, in regions with complex emission structures, Getsources appears to perform better than FellWalker in separating nearby peaks from large-scale emission. While we conclude that Getsources is preferable to FellWalker for the purposes of our analysis, it is reassuring to see that the two algorithms provide qualitatively similar conclusions in a virial analysis.

A.1.3. Pattle et al. (2015) Catalog (CuTEx)

As noted in Section 3, the line widths measured by GAS for the dense cores are often significantly nonthermal, particularly in L1688 and NGC 1333. This basic observation runs contrary to most previously published studies, where dense cores are typically found to be thermally dominated structures (e.g., Jijina et al. 1999; Kirk et al. 2007; Foster et al. 2009). The consequence of the large line widths in our subsequent analysis is to imply that the dense cores are mostly gravitationally unbound, and, therefore, significant pressure binding is required to prevent the cores from expanding. In this section, we use comparisons with other data sets to determine whether the GAS NH₃ data are biased toward systematically higher line widths than should be attributed to the dense cores. Specifically, there are two issues that we wish to test. First, because of the moderately low resolution of GAS (32''), there may be cases where GAS measures emission from more than one dense structure along the line of sight, thus increasing the observed line width. Second, NH₃ is sensitive to gas at both high and moderate densities, and it may therefore be tracing not only the dense core but also the surrounding lower-density envelope, which is typically more turbulent. Outflows may also contaminate the NH₃ data, as outflows have been shown to host significantly more NH₃ emission compared to denser gas tracers (e.g., Tafalla & Bachiller 1995). For each of the three regions analyzed, we compare the GAS data to available complementary data to establish whether either of these biases might be present and estimate the degree to which they affect our results.

A.2.1. NGC 1333

N₂H⁺ is a frequently observed molecular tracer of dense cores, and, subsequently, diverse N₂H⁺ data exist across the GAS DR1 regions. The N₂H⁺(1–0) transition is not strictly a tracer of denser gas than NH₃(1, 1); the latter likely traces a larger range of densities than the former (see, e.g., the discussion in Kirk et al. 2017). Subsequently, NH₃ (1, 1) and N₂H⁺ (1–0) often have very similar emission profiles (Johnstone et al. 2010; Gaches et al. 2015). Nonetheless, the effective critical density probed by N₂H⁺(1–0) is higher than that of NH₃(1, 1) (Shirley 2015), and, therefore, N₂H⁺(1–0) is likely a tracer of higher-density gas in some conditions.

Kirk et al. (2007) used the Institut de Radioastronomie Millimétrique (IRAM) 30 m telescope to observe N₂H⁺(1–0) in single pointings at the peaks of 157 dense core candidates across the Perseus molecular cloud. The angular resolution of 25'' is similar to that of GAS (32''). We identified 21 observations where the IRAM N₂H⁺observed position was within one IRAM beam (25'') of one of our cores. For our comparison of line widths, we separately examine cores located in the most clustered part of NGC 1333 and the remainder of the sample. Here, we define the clustered area as a 4'-wide square centered on J2000 (03:29:07, 31:14:42) that roughly encapsulates the filament surrounding the protostar IRAS 4. Figure 14 shows a comparison of the total line widths measured with GAS NH₃ and IRAM N₂H⁺, where the latter were calculated assuming the same gas temperature as observed with GAS.

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We find that the ratio between the GAS NH₃ and IRAM N₂H⁺ line widths in the clustered center of NGC 1333 is 1.12 ± 0.12 (mean and standard deviation), i.e., the NH₃ line width is typically 12% higher than the N₂H⁺ line width. The results outside of the central cluster on average agree quite closely between the two studies, with a ratio of 0.99 ± 0.08. Excluding one outlier, outside of the central cluster, no NH₃ line widths exceed the N₂H⁺ results by more than 2%. Using GAS line widths and kinetic temperatures taken directly at the location of the N₂H⁺ pointings rather than averaged over the core returns similar ratios, albeit with a notably wider spread in values, reaching ratios of nearly 1.5 at most, despite having a somewhat lower average. These relatively subtle differences under most circumstances are consistent with the results in Johnstone et al. (2010), which also compared pointed GBT NH₃ to pointed IRAM N₂H⁺ in the Perseus Molecular Cloud. The similarity between the pointed IRAM N₂H⁺ and current GAS core-averaged NH₃ also illustrates the similarity between pointed and core-averaged line width measurements.

We next compare our measurements with CARMA observations of N₂H⁺(1–0) at 10'' resolution from Walsh et al. (2007). We require matched cores to have peak positions separated by less than the JCMT beam size (141), finding a total of 23 matches, of which roughly half (12) lie within the central dense portion of NGC 1333. In the outskirts of NGC 1333, we find relatively small variations in the line width; the ratio of GAS to CARMA line widths is 1.14  ± 0.26, and only three cores matched here have GAS line widths that are more than 10% larger than their CARMA counterparts. In the central part of NGC 1333, the ratio between the GAS and CARMA line widths is much larger, at 1.49 ± 0.27. Several of the cores have ratios of close to 2.

We additionally run a test within the GAS data itself. Friesen & Pineda et al. (2017) ran a visual check of all GAS DR1 spectra and found that no more than 10% to 15% of the spectra in zones with more complex emission appeared to contain more than one velocity component. The hyperfine structure of NH₃(1, 1) emission can, nonetheless, make the identification of multiple velocity components difficult, particularly if the second component is much fainter. M. Chen et al. (2019, in preparation) recently pioneered a new technique for extracting such additional velocity components. In NGC 1333, M. Chen et al. (2019, in preparation) find that about 25% of the pixels that are bright enough to enable reliable line-fitting are better fit with two velocity components, where the second velocity component is typically associated with faint and broad emission. Dense cores identified using Getsources, which are generally found in more crowded regions, are much more likely than the rest of the cloud to have multiple components, with 68% of the dense cores containing pixels with a second velocity component within their extent. Assuming that the brightest of the two velocity components corresponds with the true dense core emission, we compare the line width measured by M. Chen et al. (2019, in preparation) with the single-component DR1 line width. For 50% of these comparisons, the two line width measurements agree within 10%, i.e., the second velocity component identified by M. Chen et al. (2019, in preparation) is so faint that it did not influence the single-component fit used in DR1. Yet 27% of the sample has significant discrepancies of 25% or more (corresponding to a change in Ω_k of at least 50%) between the original GAS DR1 and two-component line width results. In the most extreme case, the DR1 line width is almost double the brighter two-component velocity fit from M. Chen et al. (2019, in preparation). These cores with discrepant line widths represent about 18% of the total population in NGC 1333, i.e., of the 68% of all cores best fit by multiple velocity components, 27% have significant discrepancies. This result suggests that, in a nonnegligible number of cases, the NH₃ line widths derived for the cores according to the GAS DR1 fits are significantly influenced by the presence of multiple reservoirs of dense gas along the line of sight.

The M. Chen et al. (2019, in preparation) two-component line widths in the central filament of NGC 1333 are, on average, larger than those in the Walsh et al. (2007) CARMA data, which may be due to the presence of additional, unfit components. Slightly narrower line widths are found in the two-component GAS line widths compared to CARMA in outlying regions, which is somewhat unexpected, as these regions are not expected to contain much crowded emission. This difference, however, is small relative to the standard deviation of the line width distribution, so it can likely be explained by the significantly larger sample size available for the two-component comparison, which contains more cores in the marginally more crowded outer filaments of the cloud.

Through our two comparisons to other catalogs and additional check within GAS, we find that the line widths of cores in the outskirts of NGC 1333 are minimally affected by both the angular resolution and the dense gas tracer, while the most clustered cores often have noticeably elevated line widths in GAS. We find that the smallest line widths are found in the Walsh et al. (2007) CARMA N₂H⁺ observations for the central filament and the M. Chen et al. (2019, in preparation) two-component fits for outlying regions. The average ratios between GAS DR1 line widths and these lowest-line width data sets (1.18 in outlying regions and 1.49 in the clustered center) correspond to an Ω_k term that is, on average, a factor of 1.4 larger in outlying regions and 2.2 in the clustered center—significant effects. We applied these two ratios in line widths to the full population of isolated and clustered cores in NGC 1333 when recalculating Ω_k in Appendix A.2.4, thereby providing a rough correction to the biases that may be present in the GAS DR1 single-component fits used for our main analysis.

A.2.2. L1688

A.2.3. B18

A.2.4. Revised Virial Estimates

We next estimate the influence of the possible NH₃-based core line width estimates discussed in the previous sections on our virial analysis. To do so, we multiply σ_tot for each core by the ratio found between GAS data and reference data in the core's parent region as stated in the preceding subsections (see Table 6). We note that we did not use any core-specific correction ratios, as not all cores in our GAS analysis had other kinematic data available for a direct comparison. The resulting virial plot is shown in Figure 8, and some important statistics are shown in Table 8.

Table 6.  Line Width Ratios between GAS and Other Dense Gas Tracers

	L1688 Oph Aa	L1688 Oph Ba	L1688 Outa	NGC 1333-Cena	NGC 1333-Outa
Denser Tracer Onlyb	1.13 ± 0.09	1.20 ± 0.37	1.02 ± 0.39	1.12 ± 0.12	0.99 ± 0.08
Denser Tracer and Higher Resolutionc	1.11 ± 0.05	1.46 ± 0.38	1.04 ± 0.45	1.49 ± 0.27	1.14 ± 0.26
Multicomponent Fitting on GASd	1.20 ± 0.20	1.24 ± 0.16	1.12 ± 0.20	1.35 ± 0.25	1.18 ± 0.27

Notes.

^aThe line width ratios between GAS and other dense gas observations. "Out" indicates the outlying portions of each region, while "Cen" indicates the central portion. B18 is excluded, as no direct comparisons were possible there. The boldface entries indicate the corrective ratios applied in Appendix A.2.4. ^bComparison with a denser gas tracer at a similar resolution to GAS. For L1688, the denser tracer is N₂D⁺ (from Punanova et al. 2016), and for NGC 1333 it is N₂H⁺ (from Kirk et al. 2007; Walsh et al. 2007). ^cThe ratios use the same dense gas tracer as above, but at 163 in L1688 and 10'' in NGC 1333. ^dA comparison with the multiple velocity component fitting by M. Chen et al. (2019, in preparation).

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After applying these ratios, we find that only about 40% of cores in both L1688 and NGC 1333 are identified as unbound, corresponding to a total of 55 of 134 cores across all three regions. Oph B and the clustered center of NGC 1333 both show a majority of bound cores under this correction, despite containing all unbound cores in the original uncorrected result. A total of 19 cores remain identified as unbound Oph A, however. The relatively large fraction of apparently unbound cores in Oph A compared to the other clustered regions merits additional research, especially at higher resolution, to determine whether this difference is physical, perhaps relating to the somewhat warmer, active nature of the region (e.g., Stamatellos et al. 2007; Friesen & Pineda et al. 2017), or just a side effect of observational limitations. We emphasize that the above calculation provides a statistical correction to the potential overestimate in the core line widths, and the results are not expected to be exactly correct for any individual core. Nonetheless, this estimated correction illustrates two important points. First, it suggests that many of the dense cores that we identify as being unbound in our analysis may, in fact, be classified as bounded by the combination of pressure and gravity, if we were instead using a higher-density tracer observed at higher angular resolution. Second, our main conclusion, that pressure is the dominant factor in binding the cores, remains true even accounting for potential biases in the reported core line widths; in Figure 8, the vast majority of the cores lie below the horizontal dotted line.

As discussed in Section 4.1, the binding of dense cores due to the weight of overlying cloud material is another factor in our virial calculation for the cores. The calculation of this parameter relies on several assumptions, including approximate spherical symmetry of the cloud, which are discussed in more detail in Kirk et al. (2017). One assumption that is poorly constrained and might vary from cloud to cloud is the size scale of core versus cloud material. In our simple model, we associate the column density at any location with an effective cloud pressure and must, therefore, take care to remove the contribution of the column density associated with the dense core itself beforehand. We separated the core and cloud material using a simple spatial filtering algorithm, the á Trous wavelet decomposition. In this scheme, the size scale dividing core and cloud must be represented as 2^N pixels. In our main analysis, we conservatively selected a scale of approximately 0.6 pc (64 pixels in L1688 and B18 and 32 pixels in NGC 1333) to represent the division between cores and the larger structures that they inhabit. Here, we examine the influence of that selection by rerunning our analysis using a suite of different filtering scales.

Table 7 shows the typical (mean) estimated cloud weight pressure for different assumptions about the filter scale. Increasing or decreasing the filter scale by a factor of 2 results in pressure estimates that vary by a similar factor, roughly 2 or less (<0.3 in logarithmic units), while not applying any filtering (i.e., attributing all material with the cloud) increases the cloud pressure by a factor of about 3 in L1688 and NGC 1333 and a factor of 6 in B18. The case of no filtering represents an extreme unphysical case, because the dense cores must have some material associated with them. Still, only L1688 seems to show significant changes in the virial ratio as a result of applying a somewhat higher or lower filtering scale choice, with a virial ratio that varies by up to 60% (0.2 in logarithmic units) under a doubling or halving of the filter scale. Both NGC 1333 and B18 show much more modest variations under these filter scale changes, with typical variations in derived core virial parameters of less than 25% (<0.1 in logarithmic units). As noted in Section 4.2, cloud weight pressure provides a minor contribution to the binding of the dense cores in both B18 and NGC 1333, hence its lack of impact on the virial ratios. Subsequently, because L1688 has a cloud weight pressure term with a magnitude that is more equal to the contribution from turbulent pressure, changes to the filter scale have a more significant impact.

Table 7.  Variation in Cloud Pressure with Filter Scale

Filter Scale (pc)a	L1688	NGC 1333	B18
	$\overline{\mathrm{log}({P}_{w}/{k}_{B})}\,\,(\mathrm{log}\,{\mathrm{cm}}^{-3}\,{\rm{K}})$
0	6.47	6.23	5.94
0.3	6.10	5.90	5.39
0.6	5.95	5.74	5.16
1.2	5.71	5.50	4.94
	$\overline{\mathrm{log}(-({{\rm{\Omega }}}_{g}+{{\rm{\Omega }}}_{p,w}+{{\rm{\Omega }}}_{p,t})/2{{\rm{\Omega }}}_{{\rm{k}}})}$
0	−0.04	0.03	0.73
0.3	−0.39	−0.18	0.23
0.6	−0.55	−0.26	0.20
1.2	−0.77	−0.35	0.13

Note.

^aApproximate filtering scale applied. Exact values vary slightly by region because of the differing distances to each cloud.

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The turbulent pressure estimate required several assumptions. Here we examine the likely range of uncertainty introduced into our final Ω_p,t values as a result. In our main analysis in Section 4.2, we estimated the turbulent pressure using ¹³CO(1–0) to measure the random motions in the medium surrounding each core and assumed a representative density of 3 × 10³ cm⁻³. Both of these assumptions may cause us to overestimate the true turbulent pressure. Indeed, the ¹³CO emission may be tracing a much larger volume along the line of sight, and the mean density that this emission represents may be different than assumed.

We first present a likely underestimate of the turbulent pressure, to illustrate the maximum range of potential values. Previous research (e.g., Pineda et al. 2010; How-Huan Chen et al. 2018) suggests that the transition between quiescent dense cores and their turbulent surroundings can be observed directly in NH₃(1, 1) observations. As an initial, coarse approximation, we take the median value of all line widths in the GAS maps that are outside of all core boundaries, which yields a value of 0.37 km s⁻¹ in L1688, 0.31 km s⁻¹ in NGC 1333, and 0.18 km s⁻¹ in B18. This result would suggest significantly lower turbulent pressure terms in all three regions. These small line widths, however, likely underestimate the true intercore line width, as there are likely additional quiescent gas structures that have not been adequately captured by the GBS-based core catalog.

For a more accurate estimate of the intercore line width, we visually examine the GAS line width values surrounding a subset of more isolated cores, where the transition to coherence is clearer. We find line widths that are often upward of 0.8 km s⁻¹ in both NGC 1333 and L1688, significantly larger than the median line width estimates listed above and, critically, only about 20% smaller than the typical line widths found using ¹³CO. This comparison suggests that our original estimate using the ¹³CO line widths may not be substantially overestimated. This suggestion is also in line with a comparison of the ¹³CO line widths measured in L1688 and the C¹⁸O line widths presented in Pattle et al. (2015). C¹⁸O also traces denser material than ¹³CO, making it less susceptible to overestimating the turbulence in the local core environment. We find, however, that the C¹⁸O line widths are typically only 10% smaller than the ¹³CO line widths. While these somewhat smaller line widths would imply a smaller effect from turbulent pressure, no major qualitative conclusions would be changed by a line width reduction of 20%. Even the importance of turbulent pressure relative to cloud weight pressure in NGC 1333 and B18 remains evident, as shown in Table 8.

Table 8.  Virial Parameters Estimated for Different Analysis Methods

	$\overline{({{\rm{\Omega }}}_{p,t}/{{\rm{\Omega }}}_{p,w})}$	$\overline{\mathrm{log}({{\rm{\Omega }}}_{g}/({{\rm{\Omega }}}_{p,t}+{{\rm{\Omega }}}_{p,w}))}$	$\overline{\mathrm{log}(-({{\rm{\Omega }}}_{g}+{{\rm{\Omega }}}_{p,w}+{{\rm{\Omega }}}_{p,t})/2{{\rm{\Omega }}}_{{\rm{k}}})}$	% Bound Cores
	L1688
Original Getsources Results	0.75	−2.34	−0.55	36
FellWalker Core Cataloga	0.75	−2.28	−0.55	39
Corrected GAS Core Line Widthsb	0.75	−2.34	0.26	57
Pt Uses GAS NH3 Propertiesc	0.19	−1.96	−0.84	27
20% Lower 13CO Line Widthsd	0.48	−2.18	−0.68	32
Pt Uses ${n}_{{}^{13}\mathrm{CO}}=1\times {10}^{3}$ e	0.25	−2.01	−0.81	28
Pt Uses ${n}_{{}^{13}\mathrm{CO}}=1\times {10}^{4}$ e	2.50	−3.03	0.00	46
Filter Scale = 0.3 pcf	0.54	−2.54	−0.39	41
Filter Scale = 1.2 pcf	1.32	−2.07	−0.77	30
	NGC 1333
Original Getsources Results	2.16	−1.17	−0.26	27
FellWalker Core Cataloga	1.78	−2.45	−0.09	49
Corrected GAS Core Line Widthsb	2.16	−1.17	0.17	63
Pt Uses GAS NH3 Propertiesc	0.25	−0.24	−0.83	6
20% Lower 13CO Line Widthsd	1.38	−0.89	−0.44	20
Pt Uses ${n}_{{}^{13}\mathrm{CO}}=1\times {10}^{3}$ e	0.72	−0.57	−0.64	10
Pt Uses ${n}_{{}^{13}\mathrm{CO}}=1\times {10}^{4}$ e	7.19	−2.11	0.44	67
Filter Scale = 0.3 pcf	1.58	−1.29	−0.18	35
Filter Scale = 1.2 pcf	3.71	−1.03	−0.35	22
	B18
Original Getsources Results	3.47	−0.35	0.20	56
FellWalker Core Cataloga	4.26	−3.64	0.74	83
Corrected GAS Core Line Widthsb	3.47	−0.35	0.20	56
Pt Uses GAS NH3 Propertiesc	0.40	0.59	−0.36	22
20% Lower 13CO Line Widthsd	2.22	−0.07	0.04	44
Pt Uses ${n}_{{}^{13}\mathrm{CO}}=1\times {10}^{3}$ e	1.15	0.27	−0.14	44
Pt Uses ${n}_{{}^{13}\mathrm{CO}}=1\times {10}^{4}$ e	11.5	−1.31	0.81	100
Filter Scale = 0.3 pcf	2.06	−0.55	0.31	56
Filter Scale = 1.2 pcf	6.16	−0.22	0.13	44

Notes.

^aThese results use the Fellwalker core catalog described in Appendix A.1.2. ^bEmploys the corrections described in Appendix A.2.4. No corrective ratio is employed for B18. ^cThis provides an extreme lower limit to the estimate of external turbulent pressure by applying the mean GAS NH₃ line widths outside of identified cores to trace the kinematics of the ambient cloud. See Appendix A.4 for details. ^dThis provides a rough approximation to possible overestimates resulting from our use of ¹³CO rather than denser gas tracers in our estimate for P_t. See Appendix A.4 for details. ^eThe value used for ${n}_{13\mathrm{CO}}$ in the main analysis is 3 × 10³ cm⁻³. See Appendix A.4 for more information. ^fThe scale used in the main analysis is 0.6 pc. See Appendix A.3 for details.

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Finally, we examine the influence of the mean gas density that we assumed. We recompute the turbulent pressure assuming instead mean densities of 10³ cm⁻³ and 10⁴ cm⁻³, which are a factor of ∼3 lower and higher than the value used in our main analysis, respectively. Assuming ${n}_{{}^{13}\mathrm{CO}}\sim {10}^{3}$ cm⁻³ decreases the turbulent pressure on each core by a factor of 3, which then increases the relative importance of the cloud weight pressure in binding the cores, with cloud weight pressure becoming the dominant source of binding in both L1688 and NGC 1333.

Table 8 shows that, for the most plausible range of assumptions in calculating the turbulent pressure and other quantities, our final conclusions are qualitatively unchanged: pressure plays a dominant role in binding the majority of dense cores in all three clouds. We emphasize that the relative importance of each virial term in the three clouds is even more robust, because the same data sets and analysis methods were applied to each of them.