Star Formation in a Strongly Magnetized Cloud

The observations were conducted with the ALMA 7 m array in Bands 6 and 7 in Cycle 6 (Project ID 2018.1.00227.S, PI: J. C. Tan), during the period of 2019 March to April. The entire field (10' × 4.5') was divided into four strips, each about 150'' wide and 270'' long.

For the Band 6 observation we set the central frequency of the correlator sidebands to be the rest frequency of the ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$(3-2) line for SPW0 with a velocity resolution of 0.046 km s⁻¹. The second baseband SPW1 was set to $231.00\$ GHz, i.e., 1.30 mm, to observe the continuum with a total bandwidth of $1.875\$ GHz, which also covers CO(2-1) with a velocity resolution of 1.46 km s⁻¹. SPW2 was split to cover the ¹³CO(2-1) and ${{\rm{C}}}^{18}{\rm{O}}$(2-1) lines, both with a velocity resolution of 0.096 km s⁻¹. The frequency coverage for SPW3 ranged from 215.85–$217.54\$ GHz to observe DCN(3-2), DCO⁺(3-2), SiO(5-4), and ${\mathrm{CH}}_{3}\mathrm{OH}({5}_{\mathrm{1,4}}-{4}_{\mathrm{2,2}})$.

For Band 7 we set the central frequency to be the rest frequency of the ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$(3-2) line for SPW0 with a velocity resolution of 0.038 km s⁻¹. The central frequencies of SPW1 and SPW2 were set to 278.88 and 291.10 GHz, respectively, and each band had a bandwidth of $1.875\$ GHz to observe continuum emission. SPW3 was split equally to observe two lines, i.e., DCN(4-3) and DCO⁺(4-3), with 58.59 MHz (61 km s⁻¹) bandwidth and resolution of 0.073 km s⁻¹.

The raw data were calibrated with the data reduction pipeline using Casa 5.4.0. The continuum visibility data were constructed with all line-free channels. We performed imaging with the tclean task in Casa and during cleaning we combined data for all four strips to generate a final mosaic map. The 7 m array data were imaged using a Briggs weighting scheme with a robust parameter of 0.5, which yields a resolution of $7\buildrel{\prime\prime}\over{.} 00\times 4\buildrel{\prime\prime}\over{.} 29$ for Band 6 and $5\buildrel{\prime\prime}\over{.} 92\times 3\buildrel{\prime\prime}\over{.} 47$ for Band 7. The 1σ noise levels in the continuum image are 1.3 and 1.8 mJy beam⁻¹ for Band 6 and Band 7, respectively. The resolutions and sensitivities for the spectral lines are summarized in Table 1.

We have retrieved archival data to provide auxiliary information at infrared wavelengths. The 3.5 and 4.5 μm maps are from the Spitzer Heritage Archive hosted in the NASA/IPAC Infrared Science Archive. For 12 and 22 μm we use Wide-field Infrared Survey Explorer (WISE) archival data. Continuum images in the wavelengths of Herschel PACS (70 and 160 μm) and SPIRE (250, 350, and 500 μm) were obtained from the Herschel Science Archive. For this, Vela C was observed on 2010 May 18, as part of the Herschel imaging survey of OB young stellar objects (Motte et al. 2010) guaranteed time key program.

We also obtained the total hydrogen column density ${N}_{{\rm{H}}}$ (in units of hydrogen nuclei per square centimeter) and temperature map (see Figure 1), which were first presented in Section 5 of Fissel et al. (2016). These maps are based on dust spectral fits to four FIR/submillimeter dust emission maps: Herschel-SPIRE maps at 250, 350, and 500 μm; and a Herschel-PACS map at 160 μm. These maps have the same spatial resolutions as the 500 μm map, i.e., 35.2''.

Figure 2 illustrates the Band 6 (1.3 mm) and Band 7 (1.05 mm) continuum of the Vela C CR1 clump. Overall there are about 10 clearly visible cores sparsely distributed over the field. The detections at 1.05 mm are similar to those at 1.3 mm. The two brightest cores are located in the southern part of the field, with a linear filament or bridging feature connecting them. This bridge is about 0.27 pc long and appears more prominent at 1.3 mm. As shown in Figure 1, the orientation of this bridging feature is close to the POS direction of the magnetic field derived in the BLASTPol survey, with an offset of ∼18°.

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We used the dendrogram algorithm (Rosolowsky et al. 2008) implemented with astrodendro to carry out an automated, systematic search for cores in the continuum images following the method used in Cheng et al. (2018) and Liu et al. (2018). We defined the identified leaves (the base element in the hierarchy of dendrogram that has no further substructure) as cores. We set the minimum flux density threshold to 4σ, the minimum significance for structures to 1σ, and the minimum area to half the size of the synthesized beam. We tried dendrogram identification on the continuum maps of both bands and found almost equivalent results. Hereafter, we define the positions and boundaries of cores based on the 1.3 mm data, which have slightly better signal-to-noise ratios, as shown in Figure 3. The cores are named as CR1c1, CR1c2, etc., with the numbering order from highest to lowest integrated flux. There is an additional core (CR1c11) that is located at the bridging feature and not identified as a core from the 1.3 mm data, but it does appear as an independent condensation in 1.05 mm continuum, and moment 0 maps of some lines like ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$(3-2) and ${\mathrm{DCO}}^{+}$(3-2). So we also include CR1c11 in our sample and adopt a core boundary defined using the ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ moment 0 map (by running dendrogram with the same setup). Then the regions of CR1c1 and CR1c2 that overlap with CR1c11 are excluded from the definition of CR1c1 and CR1c2 when deriving their properties.

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We then estimated the masses of cores assuming the 1.3 mm emission comes from optically thin thermal dust emission with a uniform temperature of 15 K following the methods and assumptions used in the study of Cheng et al. (2018), with the only difference being that this previous study adopted a fiducial temperature of 20 K. Our reason for choosing a slightly lower temperature is the availability in Vela C of a relatively high resolution temperature map (though not high enough to resolve individual cores themselves) that indicates temperatures closer to 15 K. The estimated masses range from 0.17–6.7 ${M}_{\odot }$. If temperatures of 10 or 20 K were to be adopted, then the mass estimates would differ by factors of 1.85 and 0.677, respectively. In Figure 4 we plot the CMF of the detected sample. Given the small numbers of detected cores, it is difficult to make meaningful comparison with the CMFs of other regions.

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The core radii are evaluated as ${R}_{c}=\sqrt{A/\pi }$, where A is the projected area of each core returned by the dendrogram algorithm. The median radius is 0.016 pc (i.e., 3300 au), similar to the spatial resolution of 5100 au that is achieved by the ∼5.5'' angular resolution observations. This indicates that most cores are not well resolved. Given masses and radii, the mass surface densities and volume number densities of the cores can be estimated. These properties are summarized in Table 2.

Table 2. Core Properties

Core	R.A.	Decl.	Mc	Area	${R}_{c}(^{\prime\prime} )$	Rc	${{\rm{\Sigma }}}_{c}$	${n}_{{\rm{H}},c}$	$\tfrac{{f}_{1.05\,\mathrm{mm}}}{{f}_{1.30\,\mathrm{mm}}}$	Tc a	${\alpha }_{\mathrm{vir}}$ b
	(°)	(°)	(${M}_{\odot }$)	(arcsec2)	(arcsec)	(0.01 pc)	(g cm−2)	(105 cm−3)		(K)
1	134.85254	−43.53361	6.69	301	9.47	4.27	0.228	6.00	1.83 ± 0.14	${7.8}_{-1.9}^{+3.6}$	1.30
2	134.83645	−43.52111	4.86	206	7.83	3.53	0.242	7.70	1.59 ± 0.13	${5.1}_{-0.8}^{+1.2}$	1.01
3	134.88394	−43.49805	2.52	117	5.90	2.66	0.222	9.35	1.90 ± 0.16	${9.4}_{-2.8}^{+7.6}$	0.98
4	134.85906	−43.43667	1.19	76	4.77	2.15	0.161	8.41	2.02 ± 0.21	${13.6}_{-6.1}^{+116.6}$	1.58
5	134.84758	−43.44444	0.88	68	4.50	2.03	0.133	7.35	1.37 ± 0.19	${3.8}_{-0.7}^{+1.0}$	1.47
6	134.89812	−43.50666	0.88	92	5.23	2.36	0.098	4.66	1.36 ± 0.21	${3.8}_{-0.8}^{+1.2}$	3.00
7	134.84488	−43.51500	0.61	52	3.95	1.78	0.120	7.59	1.59 ± 0.25	${5.1}_{-1.4}^{+2.9}$	...
8	134.88242	−43.51444	0.35	37	3.33	1.50	0.097	7.30	1.63 ± 0.35	${5.5}_{-2.0}^{+6.3}$	...
9	134.88358	−43.56583	0.34	37	3.33	1.50	0.094	7.04	1.63 ± 0.36	${5.4}_{-2.0}^{+6.7}$	...
10	134.86173	−43.43500	0.17	20	2.44	1.10	0.087	8.88	2.48 ± 0.68	>7.4	...
11	134.84373	−43.52667	0.88	110	7.72	2.58	0.082	3.56	1.28 ± 0.22	${3.5}_{-0.7}^{+1.0}$	4.84

Notes.

^a Estimated from ratio of ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ assuming optically thin thermal emission from dust and dust opacities of the moderately coagulated thin ice mantle model of Ossenkopf & Henning (1994). ^b For each core the virial parameter is derived with a deconvolved core radius, and velocity dispersions combining measurements with different tracers, i.e., the same as panel (d) in Figure 8.

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In Figure 2(c) we present the map of 1.05 mm/1.3 mm flux ratio for positions with fluxes greater than 3σ in both bands (after convolving 1.05 mm data to the angular resolution of the 1.3 mm map). This ratio ranges from 1.0—2.5 over the map. We use this ratio to give more constraints on the dust temperature. To do this, we compare the observed ratio ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ with that predicted from models of optically thin thermal dust emission, i.e.,

$$\begin{eqnarray}&&\displaystyle \frac{{f}_{{\nu }_{1}}}{{f}_{{\nu }_{2}}}=\displaystyle \frac{{B}_{{\nu }_{1}}({T}_{\mathrm{dust}})}{{B}_{{\nu }_{2}}({T}_{\mathrm{dust}})}\cdot \displaystyle \frac{{\kappa }_{{\nu }_{1}}}{{\kappa }_{{\nu }_{2}}}=\displaystyle \frac{{B}_{{\nu }_{1}}({T}_{\mathrm{dust}})}{{B}_{{\nu }_{2}}({T}_{\mathrm{dust}})}\cdot {\left(\displaystyle \frac{{\nu }_{1}}{{\nu }_{2}}\right)}^{\beta },\end{eqnarray} \tag{ 2 }$$

where f_ν is the dust emission flux at frequency ν, ${B}_{\nu }({T}_{\mathrm{dust}})$ is the Planck function with dust temperature ${T}_{\mathrm{dust}}$, ${\kappa }_{\nu }$ is the dust opacity, and β is the dust opacity index. For fiducial dust opacity we adopt the same model that we have used for our mass estimates, i.e., the thin ice mantle model of Ossenkopf & Henning (1994) with 10⁵ yr of coagulation at a density of ${n}_{{\rm{H}}}={10}^{6}\,{\mathrm{cm}}^{-3}$. At submillimeter wavelengths, this model exhibits ${\kappa }_{\nu }=0.1{(\nu /1000\ \mathrm{GHz})}^{\beta }\,{\mathrm{cm}}^{2}{{\rm{g}}}^{-1}$ with $\beta \simeq 1.8$. As shown in Figure 5, ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ increases from 1.5 at ${T}_{\mathrm{dust}}=5$ K to about 2.1 at ${T}_{\mathrm{dust}}=20$ K, and grows asymptotically to 2.2 at higher temperatures. For comparison, we also present the predicted ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$–${T}_{\mathrm{dust}}$ relation for the equivalent bare grain and thick ice mantle models of Ossenkopf & Henning (1994). We see that for the models with ice mantles (thin/thick), which are expected to be the most relevant for prestellar and early stage protostellar cores, the choice of dust opacity model does not strongly affect the derived ${T}_{\mathrm{dust}}$ for a given flux ratio. More generally, our derived ${T}_{\mathrm{dust}}$ estimates are valid for dust opacity models that have a spectral index β close to 1.8 in the millimeter wavelength regime.

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Given the observed values of ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$, we estimate ${T}_{\mathrm{dust}}$ by looking for the corresponding values on the predicted relation, as shown in Figure 5. The uncertainties in ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ are also transferred into the uncertainties in ${T}_{\mathrm{dust}}$. Table 2 lists these derived temperatures. The fluxes of cores are measured by integrating over the region defined by dendrogram and for flux uncertainties we consider both the rms error and a flux calibration uncertainty of about 5%, and combine them in quadrature.

The measured ${T}_{\mathrm{dust}}$ values range from 3.5–13.6 K. For CR1c10 the ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ is 2.48 ± 0.68, leading to an unrealistic ${T}_{\mathrm{dust}}$ of ${957.0}_{-949.6}^{+\infty }$ K, so we only conservatively list the lower limit of 7.4 K. In general the derived core temperatures appear to be relatively low compared to canonical estimates of temperatures in molecular clouds, i.e., typically found to be in the range ∼10–20 K. On the larger scales probed by the Herschel submillimeter observations (see Figure 1), the CR1 clump is estimated to have dust temperatures of ∼12–16 K. Still, we note that the centers of some prestellar cores have been measured to have temperatures as low as about 6 K from NH₃ observations (Crapsi et al. 2007). We further note that there are several potential sources of systematic uncertainties in the temperature estimation from ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$. The effects of choice of dust model have already been described in Figure 5. In addition, since the core boundaries are defined based on the 1.3 mm data, we expect that the estimated flux ratio ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ and correspondingly ${T}_{\mathrm{dust}}$ could be systematically underestimated. Differences in recovered flux fractions could also introduce systematic uncertainties, with a smaller flux recovery fraction generally expected at 1.05 mm. Another potential source of uncertainty is if the cores (or part of the cores) become optically thick, which would occur first at 1.05 mm. This would tend to lower the flux received at 1.05 mm, again causing an underestimation of ${T}_{\mathrm{dust}}$. For example, if a core is moderately optical thick at 1.05 mm with ${\tau }_{1.05\,\mathrm{mm}}=1$, then the resulting ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ will be ∼13% lower than the case assuming optical thin. However, most cores in our sample should be optically thin judging from the observed low brightness temperatures T_B . In Band 7 the median T_B seen at the continuum peaks of different cores is about 0.02 K, i.e., even assuming a very low ${T}_{\mathrm{dust}}$ of ∼5 K, it is still about a factor of 250 lower. Such a big difference cannot be explained purely by a beam filling effect, since it requires a small source size of ≲20 au (a factor of 250 smaller than the spatial resolution, i.e., around 4.5'', or ∼4200 au in Band 7). Thus it is more likely due to a small optical depth, i.e., $\tau \ll$ 1 in the observed bands, at least averaged on the core scale, although a small inner region that is optically thick is still possible in some cores. These results motivate future work on radiative transfer models of protostellar cores to predict these Band 6 to Band 7 flux ratios.

Figure 6 shows the moment 0 maps of C¹⁸O(2-1), ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$(3-2), ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$(3-2), ${\mathrm{DCO}}^{+}$(3-2), $\mathrm{DCN}$(3-2), and SiO(5-4). Other transitions described in Section 2 (${\mathrm{DCO}}^{+}$(4-3), $\mathrm{DCN}$(4-3), and ${\mathrm{CH}}_{3}\mathrm{OH}({5}_{\mathrm{1,4}}-{4}_{\mathrm{2,2}})$) do not have detection above 5σ and hence are not included here. The maps of both ¹³CO(2-1) and C¹⁸O(2-1) appear strongly affected by missing large scale information due to the interferometric nature of the observations. It is likely that there exists significant CO line emission from nearby regions that are outside of the field of view, which hinders the performance of the cleaning process, and leads to strong sidelobes. In light of this we only include C¹⁸O(2-1) here for quantitative analysis, which is more optically thin and relatively less affected. ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$(3-2) has strong detections and appears closely associated with the dust continuum. ${\mathrm{DCO}}^{+}$(3-2) is also associated with the dust continuum but slightly more extended. ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$(3-2) and $\mathrm{DCN}$(3-2) have more limited detection compared with ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$(3-2), and are only seen clearly toward a few cores. SiO(5-4) is only detected at the position of CR1c1, possibly tracing shocks related with accretion or outflows.

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To investigate the kinematic and dynamical properties of cores, we extract the average spectra of each core, as shown in Figure 7. Among the four tracers, ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ and ${\mathrm{DCO}}^{+}$ have clear detections for almost all cores, while other lines are relatively weak and only detected for part of the core sample. The C¹⁸O profiles appear to be relatively complicated for some cores, like CR1c3 and CR1c5. To measure the centroid velocity and velocity dispersion of each core we perform a fitting on spectra with well defined profiles, i.e., those with a peak greater than a certain threshold value. Here, we adopt a 5σ criterion for this threshold. Since the noise levels of the average spectra vary for different cores (depending on the pixel numbers in the core, etc.), we estimate the rms noise separately for each core and each line using the signal-free channels. This signal-to-noise criterion gives six cores for analysis with C¹⁸O(2-1), 11 for ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$(3-2), six for ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$(3-2), 11 for ${\mathrm{DCO}}^{+}$(3-2), and two for $\mathrm{DCN}$(3-2).

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We characterize the C¹⁸O(2-1) spectra with Gaussian fitting using the curve_fit function in the Scipy.optimize Python module, i.e., the brightness temperature at velocity v, T_B (v), is given by

$$\begin{eqnarray}&&{T}_{B}(v)={T}_{0}\exp \left[-\displaystyle \frac{{\left(v-{v}_{\mathrm{cen}}\right)}^{2}}{2{\sigma }^{2}}\right],\end{eqnarray} \tag{ 3 }$$

where ${T}_{0}\simeq {\tau }_{0}{T}_{\mathrm{ex}}$ when the line is optically thin. CR1c2 and CR1c6 can be well described with a single Gaussian component. In general, we expect that C¹⁸O(2-1) traces somewhat lower density envelope gas surrounding the dense core and thus could be more affected by multiple components along the line of sight. In CR1c1, CR1c4, CR1c5, and CR1c9, where the spectra have more complex profiles and hence cannot be well approximated by a single Gaussian, we also allow for a second Gaussian component. The component closest to the velocity determined from other dense gas tracers is assumed to be associated with the core. For the ${\mathrm{DCO}}^{+}$(3-2) and $\mathrm{DCN}$(3-2) lines we also perform the Gaussian fitting with the curve_fit function.

On the other hand, ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ and ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ lines have blended hyperfine components and cannot be approximated with a simple Gaussian. We adopt the frequencies and relative optical depths of ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ and ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ taken from Pagani et al. (2009). We further assume the line emission is optically thin to limit the number of free parameters, i.e.,

$$\begin{eqnarray}&&{T}_{B}(v)={T}_{0}\sum _{i}{R}_{i}\exp \left[-\displaystyle \frac{{\left(v-{v}_{i}-{v}_{\mathrm{cen}}\right)}^{2}}{2{\sigma }^{2}}\right],\end{eqnarray} \tag{ 4 }$$

where R_i and v_i are the relative intensity and velocity for the ith hyperfine component, respectively. For CR1c1 the signal-to-noise ratio is very high (∼20) and three hyperfine groups are clearly detected, so we attempted to include the excitation temperatures (${T}_{\mathrm{ex}}$) and opacities (${\tau }_{\mathrm{tot}}$) as free parameters to fit the profile, which is described in Appendix A. The best-fit parameters of centroid velocity and velocity dispersion are displayed along with the spectral lines in Figure 7. As can be seen, the centroid velocities range from 5.7–9.2 km s⁻¹ and the velocity dispersions ${\sigma }_{\mathrm{obs}}$ range from 0.15–0.5 km s⁻¹ for all species, in which the nonthermal component can be derived via

$$\begin{eqnarray}&&{\sigma }_{\mathrm{nth}}^{2}={\sigma }_{\mathrm{obs}}^{2}-{\sigma }_{\mathrm{th},\mathrm{obs}}^{2}={\sigma }_{\mathrm{obs}}^{2}-\displaystyle \frac{{kT}}{{\mu }_{\mathrm{obs}}{m}_{{\rm{H}}}},\end{eqnarray} \tag{ 5 }$$

where ${\mu }_{\mathrm{obs}}$ is the mass of the particular tracer species. At a temperature of 15 K, the thermal dispersion ${\sigma }_{\mathrm{th}}=\sqrt{{kT}/\mu {m}_{{\rm{H}}}}$ is 0.23 km s⁻¹, with μ = 2.33 assuming ${n}_{\mathrm{He}}=0.1{n}_{{\rm{H}}}$. Thus, the Mach number is measured to range from 0.61–2.2, with a median of 1.4 for ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$, 0.77 for ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$, and 1.0 for ${\mathrm{DCO}}^{+}$.

To further examine the dynamical state of the dense cores, we calculate the virial parameter (Bertoldi & McKee 1992), defined as

$$\begin{eqnarray}&&{\alpha }_{\mathrm{vir}}\equiv 5{\sigma }_{c}^{2}{R}_{c}/({{GM}}_{c})=2{{aE}}_{K}/| {E}_{G}| ,\end{eqnarray} \tag{ 6 }$$

where ${\sigma }_{c}$ is the intrinsic 1D velocity dispersion of the core and R_c is the core radius. The dimensionless parameter a accounts for modifications that apply in the case of nonhomogeneous and nonspherical density distributions and we adopt a fiducial value of $a=5/4$ following McKee & Tan (2003), which corresponds to a radial density profile of $\rho \propto {r}^{-1.5}$. For a self-gravitating, unmagnetized core without rotation, a virial parameter above a critical value ${\alpha }_{\mathrm{vir},\mathrm{cr}}=2a$ indicates that the core is unbound and may expand, while one below ${\alpha }_{\mathrm{vir},\mathrm{cr}}$ suggests that the core is bound and may collapse.

Following the procedures in Cheng et al. (2020), we calculate the virial parameters separately using each tracer, i.e., ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$, ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$, ${\mathrm{DCO}}^{+}$, and $\mathrm{DCN}$. The intrinsic velocity dispersion ${\sigma }_{c}$ is derived from the observed dispersion ${\sigma }_{\mathrm{obs}}$ following

$$\begin{eqnarray}&&{\sigma }_{c}={\left({\sigma }_{\mathrm{nth}}^{2}+{\sigma }_{\mathrm{th}}^{2}\right)}^{1/2}={\left({\sigma }_{\mathrm{obs}}^{2}-\displaystyle \frac{{kT}}{{\mu }_{\mathrm{obs}}{m}_{{\rm{H}}}}+\displaystyle \frac{{kT}}{\mu {m}_{{\rm{H}}}}\right)}^{1/2},\end{eqnarray} \tag{ 7 }$$

where μ = 2.33 is the mean molecular weight assuming ${n}_{\mathrm{He}}=0.1{n}_{{\rm{H}}}$ and ${\mu }_{\mathrm{obs}}$ is the molecular weight of different observed species. For the core masses, we use the values from Table 2, i.e., derived based on millimeter continuum emission. For the core radius, we attempt two methods. The first is to use the effective radius calculated from the dendrogram-returned area in Section 3.1. For the second method, we adopt a deconvolved size defined as ${R}_{c}=\sqrt{(A-{A}_{\mathrm{beam}})/\pi }$ for cores with $A\gt 1.5{A}_{\mathrm{beam}}$, where A and ${A}_{\mathrm{beam}}$ are the core area and synthesized beam size, respectively. Figures 8(a) and (b) display the virial parameters measured with different tracers versus core mass for the two methods described above. In Figures 8(c) and (d), we combine the measurements from different tracers by taking the linear average of their nonthermal velocity dispersion in the virial parameter derivation. We note that this procedure may potentially introduce some bias, since cores can vary in the number and type of chemical species that have detected line emission.

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We see virial parameters ranging from 1 –20 as measured by individual dense gas tracers. There is a trend for more massive cores to have smaller virial parameters. The scatter is reduced for the deconvolved size method, suggesting some data points with virial parameter $\gt 5$ in panel (a) could arise from an overestimation of the core radius. There are no significant systematic differences between different tracers. For example, with the deconvolved size method (panel (b)), the median values are 1.56, 1.14, and 1.46 for ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$, ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$, and ${\mathrm{DCO}}^{+}$. The virial parameters estimated by averaging all the available dense gas data for each core show a further reduction in the scatter. For the second method with deconvolved sizes that focus on the larger cores, we obtain a median value of 1.45, with two out of the seven cores exceeding the critical value of ${\alpha }_{\mathrm{vir},\mathrm{cr}}=2.5$. For comparison, the virial parameters of the cores appear to be similar to those of the 76 cores in G286, which have a median value of 1.22 (Cheng et al. 2020). Thus, we see that most cores have a virial parameter that is consistent with a value expected in virial equilibrium. Note that the derivation of virial ratios relies on the assumption of temperature, which strongly affects the mass estimation. Here, an uniform temperature of 15 K has been assumed. If we use a temperature of 20 K, then the median virial parameter rises to 2.58.

Following similar discussions in Cheng et al. (2020), the absolute uncertainties in the derived virial parameters, including uncertainties in measured 1D line dispersion, mass, and temperature, can be as high as a factor of 2.5. Therefore, it is difficult to be more certain about whether the dense cores are actually closer to a supervirial or subvirial state. For example, CR1c11 has the highest virial parameter of 4.8, but if a lower temperature of 10 K is adopted, then ${\alpha }_{\mathrm{vir}}=2.3$, i.e., below the critical value of 2.5, so it is still likely to be gravitational bound. However, we note that the uncertainty factor includes systematic effects, some of which are not expected to vary that much from core to core, so the cores with smallest virial parameters, like CR1c1, CR1c2, and CR1c3, are more likely to be gravitationally bound and collapsing.

For optically thin lines, following Mangum & Shirley (2015), the column density is calculated from the line integrated intensity by

$$\begin{eqnarray}\begin{array}{rcl}{N}_{\mathrm{tot}}^{\mathrm{thin}} & = & \left(\displaystyle \frac{3h}{8{\pi }^{3}S{\mu }_{\mathrm{dm}}^{2}{R}_{i}}\right)\left(\displaystyle \frac{{Q}_{\mathrm{rot}}}{{g}_{J}{g}_{K}{g}_{I}}\right)\displaystyle \frac{\exp \left(\tfrac{{E}_{u}}{{{kT}}_{\mathrm{ex}}}\right)}{\exp \left(\tfrac{h\nu }{{{kT}}_{\mathrm{ex}}}\right)-1}\\ & & \times \displaystyle \frac{1}{({J}_{\nu }({T}_{\mathrm{ex}})-{J}_{\nu }({T}_{\mathrm{bg}}))}\displaystyle \int \displaystyle \frac{{T}_{B}{dv}}{f},\end{array}\end{eqnarray} \tag{ 8 }$$

where ${J}_{\nu }(T)\equiv \tfrac{h\nu /k}{\exp (h\nu /[{kT}])-1};$ S is the transition line strength; ${\mu }_{\mathrm{dm}}$ is the molecular dipole moment, R_i is the relative transition intensity (for hyperfine transitions), ${Q}_{\mathrm{rot}}$ is the rotational partition function, T_B is the measured brightness temperature; f is the filling factor, and g_J , g_K , and g_I are the rotational degeneracy, K degeneracy, and nuclear spin degeneracy, respectively. In our calculations we assume a fiducial excitation temperature of 10 K, i.e., moderately cooler than the fiducial dust temperature of 15 K. Such subthermal excitation conditions are motivated in part by the results of Kong et al. (2016) who derived and/or considered excitation temperatures from about 4–7 K for ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ and ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ in massive cores in IRDCs, with these being significantly lower than gas temperature estimates of ∼10–$15\ {\rm{K}}$ from NH₃ observations of the same regions (Kong et al. 2018). Since our observations of ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ and ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ are mostly in protostellar core envelopes that are smaller-scale, denser, and warmer than the IRDC regions studied by Kong et al. (2016), we consider ${T}_{\mathrm{ex}}=10\ {\rm{K}}$ to be the most appropriate fiducial choice. However, we will discuss, below, the effects of variation of this choice.

For the derivation of the column densities, we assumed a unity filling factor for all sources. The column density of different species are summarized in Table 3. For the ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ line emission of CR1c1, since the opacity can be determined from the spectral line fitting, the column density is corrected by

$$\begin{eqnarray}&&{N}_{\mathrm{tot}}={N}_{\mathrm{tot}}^{\mathrm{thin}}\displaystyle \frac{\tau }{1-\exp (-\tau )}.\end{eqnarray} \tag{ 9 }$$

Furthermore, with the derived column densities the deuteration ratio for each core is estimated as ${D}_{\mathrm{frac}}=N({{\rm{N}}}_{2}{{\rm{D}}}^{+})/N({{\rm{N}}}_{2}{{\rm{H}}}^{+})$. The results are listed in Table 3. Figure 9 shows the ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ and ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ column density measurements of the dense cores. The ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ column densities are in the range of $3\times {10}^{11}\mbox{--}3\,\times {10}^{13}\,{\mathrm{cm}}^{-2}$, while the ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ column densities are in the range of ${10}^{11}\mbox{--}6\times {10}^{11}\,{\mathrm{cm}}^{-2}$. The values of ${D}_{\mathrm{frac}}$ are between 0.011 and 0.85, with a median value of 0.16. This is similar to the value found by Crapsi et al. (2005) in their sample of low-mass starless cores.

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Table 3. Estimated Column Densities, Deuteration Fractions, CO Depletion Factors, and Infrared Detections for the Cores

Core	${N}_{{\rm{H}}}$	$N({{\rm{C}}}^{18}{\rm{O}})$	$N({{\rm{N}}}_{2}{{\rm{H}}}^{+})$	$N({{\rm{N}}}_{2}{{\rm{D}}}^{+})$	${D}_{\mathrm{frac}}$	fD	12 μm	70 μm
	(${10}^{22}\,{\mathrm{cm}}^{-2}$)	(${10}^{14}\,{\mathrm{cm}}^{-2}$)	(${10}^{11}\,{\mathrm{cm}}^{-2}$)	(${10}^{11}\,{\mathrm{cm}}^{-2}$)
1	9.75	4.79	310.57	3.32	0.011	62.4	Y	Y
2	10.36	9.54	20.18	3.35	0.17	33.3	Y	Y
3	9.48	1.00	12.46	2.74	0.22	290.7	N	Y
4	6.87	4.79	8.49	<1.47	<0.17	43.9	Y	Y
5	5.68	3.72	3.15	<1.24	<0.39	46.8	N	N
6	4.19	3.39	21.05	2.95	0.14	37.8	Y	Y
7	5.13	<1.42	17.40	2.82	0.16	>110.7	N	Y
8	4.16	2.87	4.29	<2.10	<0.49	44.5	N	Y
9	4.02	1.24	6.60	<1.99	<0.30	99.2	Y	Y
10	3.72	2.04	24.80	3.77	0.15	55.8	N	N
11	3.50	1.62	6.54	5.57	0.85	66.2	N	N

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The uncertainties in the column density estimation mainly result from the assumption of the excitation temperature ${T}_{\mathrm{ex}}$. If temperatures of 7 or 15 K were adopted, then $N({{\rm{N}}}_{2}{{\rm{H}}}^{+})$ would vary by factors of 2.3 and 0.59, respectively, and $N({{\rm{N}}}_{2}{{\rm{D}}}^{+})$ would vary by factors of 1.9 and 0.69, respectively. Nevertheless, assuming the species have the same excitation temperature, the deuteration ratio ${D}_{\mathrm{frac}}$ is relatively robust and differs only by factors of 0.83–1.17 from the low to the high temperature limits of this range. The uncertainties in flux measurement are typically <10% for ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$, with only a few exceptions for the cores with weaker ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ emission, i.e., CR1c5, CR1c8, and CR1c9, that have ≲30% uncertainties. For ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ the uncertainties in flux measurement are all <20%. Additionally, there are flux calibration uncertainties of about 10% for Bands 6 and 7, respectively.

The CO depletion factor, f_D , is defined as the ratio between the expected abundance of CO and the observed value:

$$\begin{eqnarray}&&{f}_{D}=\displaystyle \frac{{X}_{{{\rm{C}}}^{18}{\rm{O}}}^{\exp }}{{X}_{{{\rm{C}}}^{18}{\rm{O}}}^{\mathrm{obs}}}.\end{eqnarray} \tag{ 10 }$$

In the abundance calculation we derive the column density of hydrogen nuclei, ${N}_{{\rm{H}}}$ from the mass surface density ${{\rm{\Sigma }}}_{c}$ listed in Table 2 by ${N}_{{\rm{H}}}={{\rm{\Sigma }}}_{c}/{\mu }_{{\rm{H}}}{m}_{{\rm{H}}}$, where ${\mu }_{{\rm{H}}}{m}_{{\rm{H}}}=1.4{m}_{{\rm{H}}}$ is the mean mass per H nucleus. To compute ${X}_{{{\rm{C}}}^{18}{\rm{O}}}^{\exp }$ we adopt the abundance ratios of ${n}_{16{\rm{O}}}/{n}_{18{\rm{O}}}=327$ from Wilson & Rood (1994) and ${n}_{12\mathrm{CO}}/{n}_{{{\rm{H}}}_{2}}=2\times {10}^{-4}$ from Lacy et al. (1994). Thus, our assumed abundance ratio of C¹⁸O to ${{\rm{H}}}_{2}$ is 6.12 × 10⁻⁷. The results are listed in Table 3. All the cores have f_D measured to be ≳40. Note that the imperfect cleaning due to incomplete uv sampling may have affected the C¹⁸O flux measurement, and CO depletion factor accordingly. As mentioned in Section 3.2 the moment 0 map of C¹⁸O(2-1) in Figure 6 does have some artificial ringing features and some cores are not clearly associated with enhanced C¹⁸O emission. It is difficult to quantify the uncertainties introduced from the cleaning process, but it may have affected the CO depletion factor by factors of a few for specific cores.

We examined the CO(2-1) data toward this region to see if protostellar outflows are detectable. Figure 10(a) illustrates the low velocity CO(2-1) emission integrated over relative velocities ranging from 4–12 km s⁻¹ (compared to ${v}_{\mathrm{sys}}\,\approx 7$ km s⁻¹) for blueshifted and redshifted emission, and Figure 10(b) illustrates the high velocity CO(2-1) emission integrated over relative velocities ranging from 12–20 km s⁻¹. There is a clear bipolar outflow associated with CR1c1, which has an orientation roughly perpendicular with the filamentary bridging feature seen in the continuum. The outflow has a biconical shape with a half opening angle of ∼30°. In the vicinity of CR1c2 there appears to be some blueshifted and redshifted CO emission, possibly resulting from a weak outflow, which is also perpendicular to the bridging filament. CR1c7 appears to host a relatively collimated outflow in east–west orientation. The blueshifted lobes has a knotty appearance with a bending feature extending to the northeast direction. There is also a tentative detection of CO outflow from CR1c4 at relatively low velocities in the redshifted lobe, suggesting CR1c4 may also host a protostar.

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We also examined the CO channel maps centered on CR1c3, CR1c5, CR1c6, CR1c8, CR1c9, CR1c10, and CR1c11 and did not find evidence for outflows. The strong CO emission from the molecular cloud and the spatial filtering, however, make these nondetections questionable, and observations with higher signal-to-noise are required to properly establish the presence or lack of CO outflows from these sources.

In the continuum map there is an interesting linear filament in which CR1c1, CR1c2, and CR1c11 are located. CR1c1 and CR1c2 are located at the ends of this filament and connected by extended emission seen in 1.3 mm continuum. CR1c11 lies in between CR1c1 and CR1c2 and is further identified from the moment 0 maps of ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$, ${\mathrm{DCO}}^{+}$, and 1.05 mm continuum. These three cores exhibit signatures of different evolutionary stages: both CR1c1 and CR1c2 are associated with outflows and have relatively larger values of ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ (1.83 and 1.59, respectively), indicating that they already host a protostar that is actively accreting and heating up the surroundings. CR1c11 shows no sign of star formation activity and has a low value of ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ of 1.28. As discussed in Section 3.3, CR1c11 could be gravitationally bound if a lower temperature of ≲10 K is assumed. If true, then CR1c11 may be a prestellar core. The chemical properties including ${D}_{\mathrm{frac}}$ are also consistent with these differences in evolutionary stage.

We further divide the bridging filament into 20 strips to derive properties along its length, as shown in Figure 11. Each strip has a size of 7'' × 3.5''. The column densities of ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$and ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ are calculated following the procedures in Section 3.2 and are also shown in Figure 11. Note that in addition to the uncertainties discussed in Section 3.2, spatially filtering of interferometer observations may also lead to an underestimation of flux measurements along the bridge. For example, if there is a more diffuse cocoon component surrounding the bridge we are probably not able to detect it with the current observations. The hydrogen column density ${N}_{{\rm{H}}}$ is calculated from the continuum emission assuming a uniform ${T}_{\mathrm{dust}}$ of 15 K as in Section 3.1. We plot the derived column densities, as well as ${D}_{\mathrm{frac}}$ in Figure 12. The evolutionary differences are better illustrated in the ${D}_{\mathrm{frac}}$ profile, which exhibits a plateau around ${D}_{\mathrm{frac}}$ ≈0.8 from 30''–60'', i.e., covering the bridging region between CR1c1 and CR1c2. It can also be seen that CR1c1, CR1c2, and CR1c11 have similar N(${{\rm{N}}}_{2}{{\rm{D}}}^{+}$), but there is a lack of ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ for CR1c11, thus leading to a high ${D}_{\mathrm{frac}}$. Therefore, CR1c11 is expected to be in an early stage before the onset of star formation.

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To investigate the kinematic properties, we check the line spectra along the bridging filament. ${\mathrm{DCO}}^{+}$(3-2) is the best tracer for this purpose, since it is clearly detected throughout the bridge and has a better signal-to-noise ratio compared to ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$(3-2). We fit the ${\mathrm{DCO}}^{+}$ spectra with the same routine as used in Section 3.2. Figure 12 illustrates the variation of centroid velocity and velocity dispersion along the bridge. The ${\mathrm{DCO}}^{+}$ velocity dispersion ranges from 0.15–0.45 km s⁻¹. With gas temperatures of 10–20 K, the thermal line broadening is 0.05–0.07 km s⁻¹ for ${\mathrm{DCO}}^{+}$, so the observed line width is dominated by the nonthermal component. The thermal sound speed of molecular gas is 0.23 km s⁻¹ at 15 K, and so the Mach number ranges from 0.6–2. The filament appears mildly subsonic at the relative quiescent part, i.e., at offsets from 30''–50''. Note that there could be multiple velocity components along the filament that are unresolved in the current observation. There is a clear peak in line dispersion at the position of CR1c1, possibly resulting from an increase in temperature or enhanced nonthermal motions, such as infall and/or outflow due to star formation activity. The case of CR1c2 and CR1c11 is less clear. We see an increase in ${\sigma }_{{\mathrm{DCO}}^{+}}$ from 50''–80'' in offset, which is roughly in between CR1c2 and CR1c11.

For ${v}_{\mathrm{cen}}$ there is a decreasing trend from 6.7 km s⁻¹ at 20'', to 6.0 km s⁻¹ at around 80'', indicating a global velocity gradient of about 2.6 ${\mathrm{kms}}^{-1\,}{\mathrm{pc}}^{-1}$. Velocity gradients along filaments have been observed in both nearby low-mass star-forming clouds (e.g., Hacar & Tafalla 2011) and massive clouds (e.g., Henshaw et al. 2013; Peretto et al. 2014), and often interpreted as flows along filaments, feeding gas into dense cores. However, the global velocity gradient in the filaments may also be attributed to the motion of the filaments themselves (e.g., rotation or oscillation along the line of sight) rather than accretion flows. Interestingly, the positions of CR1c1 and CR1c2 seem to coincide well with local maxima or minima on the ${v}_{\mathrm{cen}}$ profile, possibly suggesting gas infall is taking place in the vicinity of the cores. Figure 13 illustrates a possible scenario to explain the observed ${v}_{\mathrm{cen}}$ variations, in which the local bending feature of the velocity profile is caused by infall and/or rotational motion around CR1c1 and CR1c2, while the global velocity gradient between CR1c1 and CR1c2 may arise from other mechanisms like rotation. Summarizing the results, it is likely that the bridging feature is a remnant of a larger filament. CR1c1 and CR1c2 have been accumulating gas material from this filament and have formed protostars, while CR1c11 has condensed from the gas reservoir more recently and is still in a very early, starless evolutionary stage.

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To investigate the dynamic state of the filament we perform a filamentary virial analysis following Fiege & Pudritz (2000). As shown by Fiege & Pudritz (2000), a pressure-confined, nonrotating, self-gravitating, filamentary (i.e., length ≫width) magnetized cloud that is in virial equilibrium satisfies

$$\begin{eqnarray}&&\displaystyle \frac{{P}_{e}}{{P}_{f}}=1-\displaystyle \frac{{m}_{f}}{{m}_{\mathrm{vir},{\rm{f}}}}\left(1-\displaystyle \frac{{M}_{f}}{| {W}_{f}| }\right),\end{eqnarray} \tag{ 11 }$$

where P_f is the mean total pressure in the filament, P_e is the external pressure at its surface, m_f is its mass per unit length, ${m}_{\mathrm{vir},{\rm{f}}}=2{\sigma }_{f}^{2}/G$ is its virial mass per unit length, and M_f and W_f are the gravitational energy and magnetic energy per unit length, respectively. Here, because of the observational difficulties of measuring the surface pressure and magnetic fields, we ignore the surface term and magnetic energy term, i.e., only considering the balance between gravity and internal pressure support.

The 100'' length of the filament corresponds to 0.45 pc at an assumed distance of 0.93 kpc. Without direct observational constraints, we further assume the filament axis is inclined by an angle $i=60^\circ$ to the line of sight (90° would be in the POS). If an inclination angle of 90° or 30° were to be adopted, then the length estimates would differ by factors of 1.15 and 0.577, respectively. Thus, the actual length of the filament is assumed to be 0.52 pc. In Figure 12 we plot the ratio ${m}_{{\rm{f}}}/{m}_{{\rm{f}},\mathrm{vir}}$. The masses are calculated from the 1.3 mm continuum flux, assuming a temperature of 15 K and other dust properties as in Section 3.1. ${m}_{{\rm{f}},\mathrm{vir}}$ is calculated using the velocity dispersion measured from ${\mathrm{DCO}}^{+}$. The values of ${m}_{f}/{m}_{\mathrm{vir},{\rm{f}}}$ along the filament range from 0.2–2.0. ${m}_{f}/{m}_{\mathrm{vir},{\rm{f}}}$ clearly peaks at the positions of CR1c1 and CR1c2, with peak values of 1.4 and 2.0, respectively, and it is relatively small (∼0.2–0.6) in regions between the two cores, suggesting that the filament may only be gravitationally bound around the positions of CR1c1 and CR1c2. However, since the ALMA 7 m array observations only probe scales up to ∼19'', they may be missing some flux from the filament leading to an underestimation of the masses. Furthermore, if a temperature of 10 K instead of 15 K is adopted, which is probably more realistic for the less evolved region between CR1c1 and CR1c2, the estimated mass will be larger by a factor of 1.85, thus bringing the ${m}_{f}/{m}_{\mathrm{vir},{\rm{f}}}$ ratio to ∼0.4–1.2. Also given other systematic uncertainties in measuring lengths of the structure, it is still likely that the majority of the filament is in approximate virial equilibrium, even without accounting for surface pressure and magnetic support terms.

An obvious feature in the continuum map of Vela C CR1 clump is an overall deficit of compact substructures at a few 0.01 pc scales. We have identified 11 cores from the 1.3 mm continuum, which add up to a total mass of only 19.4 ${M}_{\odot }$. Alternatively, if we sum up the fluxes above 4σ in the 1.3 mm map and convert to masses following the same assumptions as in Section 3.1, it yields 20.7 ${M}_{\odot }$, suggesting the bulk of the ALMA 1.3 mm emission is included in our identified dense cores. For comparison, the total clump mass in the field of view estimated from the Herschel column density map is about 2300 ${M}_{\odot }$, leading to a dense gas fraction, ${f}_{\mathrm{dg}}$, of only 0.84% (or 0.90%, using the total integrated flux). Therefore, only a very small fraction of gas mass is currently contained in compact prestellar and protostellar cores. The estimation of dense core masses depends on dust opacity, gas-to-dust mass ratio, temperatures, and dust emission fluxes, as well as the distance to the region. The major uncertainty of mass estimation arises from the assumption of temperature. For example, if we assume a higher temperature of T = 20 K, the total mass will be a factor of 0.677 smaller, leading to a dense gas fraction of 0.57% or 0.61%. Note that for estimating ${f}_{\mathrm{dg}}$, some of these uncertainties cancel out, i.e., those due to distance and gas-to-dust mass ratio, so we expect the dense gas fraction in Vela C CR1 clump is ≲2%.

We compare the CR1 clump with another well studied region, G286.21 + 0.17 (G286), which is a protocluster at a distance of 2.5 kpc (Cheng et al. 2018). G286 has a total Herschel-estimated mass of around 2900 ${M}_{\odot }$ in a 2.6' × 1.7' elliptical aperture (Cheng et al. 2020), leading to an average column density, ${N}_{{\rm{H}}}$, of $\sim 4\times {10}^{22}\,{\mathrm{cm}}^{-2}$, similar to the Vela C CR1 clump (∼$5\times {10}^{22}\,{\mathrm{cm}}^{-2}$). For the compact gas mass we adopt two methods. For method 1 we simply sum up the masses of cores listed in Cheng et al. (2020), which follows the same assumptions as in Section 3.1. For method 2 we integrate the fluxes for pixels above 4σ using the 1.3 mm continuum image made with only the 12 m array, and then convert to masses following the same assumptions. The 12 m array data of G286 have a maximum recoverable scale of 11'', corresponding to 0.13 pc at the distance of 2.5 kpc, which is close to the 7 m array observation of Vela C (sensitive to structures up to 29'', ∼0.13 pc, in Band 6). Methods 1 and 2 yield ${f}_{\mathrm{dg}}$ of 7.3%, and 14.3% in G286, respectively, so both estimations are an order of magnitude higher compared with the Vela C CR1 clump.

One possible explanation for these differences is that the formation of dense substructures in the Vela C CR1 clump has been suppressed by its strong magnetic field. Alternatively, the CR1 clump could simply be in a very early evolutionary stage of collapse, but with core formation not particularly influenced by the B-field. Follow-up observations to constrain the dynamical and chemical history of Vela C CR1, e.g., to measure infall speeds and chemical ages, can help distinguish these possibilities.

There have been a number of other studies of dense gas fractions in the literature. Direct comparison with our results is generally more difficult given the variety of methods used to estimate masses for both the large scale cloud and the dense (or compact) component. For example, Battersby et al. (2020) studied the dense gas fractions of the central molecular zone (CMZ) and compared to similar studies of clouds in the Galactic disk, finding that ${f}_{\mathrm{dg}}\sim 0.1 \%$–2% in most CMZ clouds (even though these clouds have relatively high column densities), while typical star-forming Galactic clouds have ${f}_{\mathrm{dg}}\sim 2 \%$–20%. The measured ${f}_{\mathrm{dg}}$ of Vela C CR1 clump appears similar to the CMZ clouds, and lower than typical Galactic disk clouds. But note that the maximum recoverable scale of our observation (29'', 0.13 pc) is smaller than the scales probed with the Submillimeter Array (SMA) observations in Battersby et al. (2020) for most sources.

The study of deuterated molecules is an important probe of the physical conditions in star-forming regions. Prior to the formation of a star, the cold ($T\lt$ 20 K) and dense (${n}_{{\rm{H}}}\gt {10}^{5}\ {\mathrm{cm}}^{-3}$) conditions within star-forming molecular cloud cores drive a cold-gas chemistry that has been well studied in recent years. Many molecular species, including CO and its isotopologues, become depleted in the gas phase by freezing out onto dust grains. Unlike CO, N-bearing species, in particular ${\mathrm{NH}}_{3}$ and ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$, better trace dense and cold gas (e.g., Caselli et al. 1999; Bergin et al. 2002). This is due to the fact that CO, largely frozen out, is unable to effectively destroy their molecular ion precursors. These physical/chemical properties are commonly observed in prestellar cores, where the deuteration fraction (i.e., ${D}_{\mathrm{frac}}$) of non-depleted molecules, defined as the column density ratio of one species containing deuterium to its counterpart containing hydrogen, is orders of magnitude larger than the average interstellar [D/H] abundance ratio, which is ∼${10}^{-5}$ (Oliveira et al. 2003). Therefore, deuterated species, like ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ are better suited to probe the physical conditions of the earliest stages of star formation. The ${D}_{\mathrm{frac}}$(${{\rm{N}}}_{2}{{\rm{H}}}^{+}$) ratio has been found to be a good evolutionary indicator in both low- and high-mass star formation (Friesen et al. 2010; Fontani et al. 2011). In addition, ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ is probably the best tracer of prestellar cores, e.g., compared to ${D}_{\mathrm{frac}}$ of HNC and ${\mathrm{NH}}_{3}$(Fontani et al. 2015).

The ${D}_{\mathrm{frac}}$(${{\rm{N}}}_{2}{{\rm{H}}}^{+}$) in the Vela C CR1 clump is found to be in the range of 0.011–0.85. Our observed values are consistent with measurements made in other low-mass star-forming regions (e.g., Caselli et al. 2002; Crapsi et al. 2005; Daniel et al. 2007; Emprechtinger et al. 2009; Friesen et al. 2013). For four out of 11 cores, no significant ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ is detected and only an upper limit of the ${D}_{\mathrm{frac}}$ is given. These cores also have relatively low ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ column densities and the upper limit on ${D}_{\mathrm{frac}}$ (≲0.5) is a rather loose constraint. The extreme value of 0.85, measured toward CR1c11, is among the highest levels of ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ deuteration reported so far (e.g., Miettinen et al. 2012), indicating the prestellar nature of CR1c11 in a very dense and cold condition. A caveat is that CR1c11 is defined based on the ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$ moment 0 map, which thus biases toward a higher ${D}_{\mathrm{frac}}$ estimation. We note that ${{\rm{N}}}_{2}{{\rm{H}}}^{+}$ could have a greater degree of missing flux compared with ${{\rm{N}}}_{2}{{\rm{D}}}^{+}$, given the properties of the observations in Band 6 and Band 7, however, we do not expect significant flux losses on the scales of the observed cores. In Figure 14 we plot the ${D}_{\mathrm{frac}}$ ratio against other core properties to look for potential correlations. As discussed in Section 3.1, the ratio ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ can be interpreted as a temperature indicator, with higher ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ suggesting higher temperature. There appears to be a weak anticorrelation between ${f}_{1.05\,\mathrm{mm}}/{f}_{1.30\,\mathrm{mm}}$ and ${D}_{\mathrm{frac}}$, which is consistent with our expectation, since CO will be released from dust grains at higher temperatures as cores evolve, thus leading to a lower deuteration level.

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Given the current available information on core properties it is difficult to assign a precise evolutionary stage for each one, but we do see groups of cores in different evolutionary stages. CR1c1 is probably the most evolved source in CR1. This core has the lowest ${D}_{\mathrm{frac}}$ and drives a powerful, wide-angle CO outflow. CR1c2, CR1c4, and CR1c7 also have associated outflow detections, indicating their protostellar nature. CR1c11 has the highest ${D}_{\mathrm{frac}}$ and is likely a prestellar core that is on the verge of collapsing, although more sensitive observations, especially better temperature measurements, are required to confirm its nature as a gravitationally bound core. A measurement of deuteration on the larger clump scale using single dish observations would be important for understanding the initial astrochemical conditions of prestellar core formation.

The auxiliary infrared data provide extra constraints on the evolutionary stages. Here we focus on two infrared wavelengths, i.e., 12 and 70 μm. A more complete investigation of other infrared wavelengths in presented in Appendix B. The 12 μm emission usually suggests thermal dust emission heated by protostars and 70 μm data could reveal deeply embedded protostars that are undetected at shorter wavelengths. The detection status is summarized in Table 3. As can be seen, CR1c5 and CR1c11 are not detected at 70 μm, suggesting that they are in a very early evolutionary stage, likely prestellar. The detection of CR1c10 is confused by another adjacent bright source (see Appendix B). All other cores should have formed a protostar. While CR1c3, CR1c7, and CR1c8 are detected at 70 μm, they are still very faint and not seen at 12 μm and hence they should be in a relatively earlier stage compared with other cores (i.e., CR1c1, CR1c2, CR1c4, CR1c6, and CR1c9), which are bright in both 12 μm and 70 μm and hence more evolved.