The Initial Conditions of Clustered Core Collapse: Multiwavelength Analysis of Oph A SM1N and N6 at 100 au Resolution

We present new Atacama Large Millimeter/submillimeter Array (ALMA) continuum and NH2D, N2D+, and H2D+ line emission at matched, ∼100 au resolution toward the dense star-forming cores SM1N and N6 within the Ophiuchus molecular cloud. We determine the density and temperature structure of SM1N based on radiative transfer modeling and simulated observations of the multiwavelength continuum emission at 0.8, 2, and 3 mm. We show that SM1N is best fit by either a broken power-law or Plummer-like density profile with high central densities (n ∼ 108 cm−3), and an inner transition radius of only ∼80–300 au. The free-fall time of the inner region is only a few ×103 yr. The continuum modeling rules out the presence of an embedded first hydrostatic core (FHSC) or protostar. SM1N is therefore a dynamically unstable but still starless core. We find that NH2D is likely depleted at high densities within SM1N. The nonthermal velocity dispersions increase from NH2D to N2H+ and H2D+, possibly tracing increasing (but still subsonic) infall speeds at higher densities as predicted by some models of starless core contraction. Toward N6, we confirm the previous ALMA detection of a faint, embedded point source (N6-mm) in 0.8 mm continuum emission. NH2D and N2D+ avoid N6-mm within ∼100 au, while H2D+ is not strongly detected toward N6. The distribution of these tracers is consistent with heating by a young, warm object. N6-mm thus remains one of the best candidate FHSCs detected so far, although its observed (sub)millimeter luminosity remains below predictions for FHSCs.


Introduction
Low-mass stars form from the collapse of cold, dense molecular cores in star-forming clouds (for a review, see Pineda et al. 2023).While starless, the radial density profile determined observationally for gravitationally or pressure-bound dense cores is often flat at radii less than some value, and then decreases proportional to r −2 at larger radii (e.g., Ward-Thompson et al. 1999;Bacmann et al. 2000;Alves et al. 2001).Cores apparently in equilibrium have been well described as Bonnor-Ebert spheres (BES), with the density structure constrained by the balance between thermal pressure and self-gravity and external pressure, resulting in this radial density form.
As an unstable core contracts, the density profile remains similar in shape, where the central density increases while the transition radius decreases (Larson 1969;Foster & Chevalier 1993), thus approaching the ρ ∝ r −2 density profile of a singular isothermal sphere (SIS; Keto & Caselli 2010).This progression remains true in global-gravitational-collapse models, where core-density profiles evolve toward the SIS solution (Gómez et al. 2021).Post-formation of an accreting object, the SIS model predicts a transition radius that moves outward with time, shifting back to a more shallow, ρ ∝ r −3/2 density profile at small r as collapse proceeds from the inside-out (Shu 1977).The protostellar source MMS-1 in L1521F shows such a transition (Tokuda et al. 2016).Cores with density profiles that show transitions at small radii are thus highly evolved, either pre-or just post-formation of an accreting object at their center.One such starless object is the isolated core L1544, with a central density n  10 6 cm −3 within a transition radius of only r ∼ 1400 au (Caselli et al. 2019).Determining accurately the density profiles of such objects are thus the first steps to examining the earliest stages of protostellar formation.
Another such highly evolved object is the dense core SM1N within the Ophiuchus molecular cloud.The L1688 region in the Ophiuchus molecular cloud is one of the nearest young stellarcluster-forming environments (d = 137 pc; Ortiz-León et al. 2017).Within L1688, the Oph A clump is a ridge-like feature that contains ∼23 M e of dust and gas, multiple compact and dense cores (Motte et al. 1998), and at least six protostars (Friesen et al. 2018), including the prototypical Class 0 protostar VLA 1623 (André et al. 1990).Oph A itself is thus forming a small group or cluster of protostars, within the general L1688 cluster.Friesen et al. (2018) proposed that the radial density profile of the SM1N core in the Ophiuchus A star-forming filament was proportional to r −1.3 at radii r  200 au.Determined via Atacama Large Millimeter/submillimeter Array (ALMA) continuum observations at ∼80 au (0 6) resolution, this suggests that the apparently starless core is nevertheless gravitationally unstable and likely undergoing collapse.An additional evolved core in Oph A, N6, was previously thought to be starless (Di Francesco et al. 2004;Bourke et al. 2012), but was shown by Friesen et al. (2018) to contain a faint point source in ∼1 mm continuum emission.Neither source showed evidence for the bipolar outflows that would be expected from a protostellar object, although outflow identification is complicated toward N6 due to its proximity to the substantial, narrow outflow originating from VLA 1623.Given their observed structure, lack of strong detections at short wavelengths, and lack of clear bipolar outflows with velocities greater than a few kilometers per second, SM1N and N6 were identified as the best candidates of first hydrostatic cores (FHSCs) of the handful of proposed sources yet detected (Young et al. 2019), along with several more recent candidates (Fujishiro et al. 2020;Maureira et al. 2020).
At high densities (n  10 6 cm −3 ) and low temperatures (T  20 K), molecular species such as CO are depleted from the gas phase due to freezing out onto dust grains, forming icy mantles (e.g., Bergin & Tafalla 2007).Concurrently, the abundance of H 2 D + is increased via reactions between + H 3 and HD, which occur preferentially in the forward direction at these temperatures (Roberts & Millar 2000).Enhanced abundances of deuterated forms of typical dense-gas-tracing molecules like NH 3 and N 2 H + follow via reactions with H 2 D + .At very high densities, H 2 D + and its multiply deuterated isotopologues may be the most abundant species remaining in the gas phase, as molecules containing heavier elements deplete onto dust grains (Walmsleyet al. 2004), including NH 3 and NH 2 D (Pineda et al. 2022;Caselli et al. 2022).Once a protostar forms and warms its surroundings above T ∼ 20 K, however, the abundances of H 2 D + and other deuterated species decrease as increased temperatures allow the desorption of CO from dust grains.Deuterated species, particularly H 2 D + , are therefore key tracers of the physical conditions of dense cores on the cusp of gravitational collapse.
Here, we present ALMA observations toward SM1N and N6 of emission from the dust continuum emission at 3, 2, and 0.8 mm, as well as from dense-gas-tracing molecular lines, including the deuterated species NH 2 D, N 2 D + , and H 2 D + at matched, ∼0 85 × 0 65 (∼100 au) resolution.Critical densities for the lines chosen range from 6.5 × 10 4 cm −3 (NH 2 D; Daniel et al. 2014) through ∼10 5 cm −3 (H 2 D + ; Hugo et al. 2009) to 8.7 × 10 5 cm −3 (N 2 D + ; Redaelli et al. 2019).We thus use the line-emission distribution and spectral-fitting results to constrain the density, temperature, and kinematics of the cores at high densities and small radii.Based on the ALMA-detected continuum emission from ∼3 to ∼850 μm wavelengths, we model the radial volume density and temperature structure of SM1N using the 3D Monte Carlo radiative transfer code RADMC-3D (Dullemond et al. 2012) to determine the radial density profile, and assess whether internal heating (i.e., protostellar) sources are needed to explain the observed emission.
In Section 2, we discuss the ALMA observations and imaging of the continuum and line data.We present the resulting continuum images and initial discussion in Section 3. We fit the observed line data in Section 4, using hyperfine modeling for the NH 2 D and N 2 D + lines.We perform radiative transfer modeling of the continuum in Section 5, where we use a grid of density profiles and a range of central densities to determine the predicted temperature profiles.We then simulate ALMA observations of the results, using the telescope configuration and uv coverage of our actual observations presented here.In Section 6, we analyze jointly the continuum and line results.We show that SM1N is a starless core undergoing dynamic, gravitational collapse, while N6 contains a compact source that remains one of the best FHSC candidates thus discovered.We summarize our findings in Section 7.

Observations
The ALMA data newly presented here were obtained as part of the Cycle 6 project 2018.1.00935.S and Cycle 5 project 2017.1.01006.S (PI: R. Friesen).Observations were performed over a range of dates between 2018 January and 2019 April.We also make use of observations toward the same sources from the Cycle 2 project 2013.1.00937.S (PI: R. Friesen), published in Friesen et al. (2018).
Each source was observed with a single pointing in continuum and line emission at ALMA Bands 3, 4, and 7.The central frequency, spectral resolution, and dense-gas-tracing lines targeted are listed in Tables 1 and 2. Observations were performed over a range of ALMA configurations to provide matched angular resolution and matched sensitivity to larger angular scales.Additional carbon-bearing species were also observed simultaneously; these will be presented and discussed in a later paper.
At Bands 3 and 4, two 12 m configurations were used.To achieve the same sensitivity to extended structure at Band 7 as at Bands 3 and 4, observations were performed with one 12 m configuration and separately with the 7 m ALMA Compact Array.We list in Table 1 the final synthesized beam and maximum recoverable scale (θ MRS ) at each band.The synthesized beam size depends on choices made in the deconvolution and imaging process, described further below, while θ MRS is calculated from the minimum baselines in final measurement sets.

Continuum Emission
Calibration and imaging were done using the Common Astronomy Software Applications (CASA) package (CASA Team et al. 2022).All data were pipeline reduced in CASA v5.4.
Continuum images for each target were made with the tclean task in CASA v5.4, using multifrequency synthesis to combine data from both 12 m configurations (or in the case of Band 7, the 12 and 7 m arrays), and all 1.875 GHz wide continuum spectral windows (two windows each at Bands 3 and 4, and three windows at Band 7).For all data sets, a Briggs weighting with a robustness parameter of 0.5 was used.We additionally performed multiscale cleaning on the SM1N data with scale sizes of 0, 1, and 3 times the synthesized beam size to better recover extended emission.The resulting synthesized beams are roughly 0 85 × 0 65 across all bands (∼115 × 90 au at the distance to L1688).Details of the noise level achieved and the beam parameters for each source as a function of band are provided in Table 1.The resulting continuum images are shown in Figure 1.The figure also includes the ALMA Band 6 continuum images from Friesen et al. (2018) toward SM1N and N6, which have similar resolution (∼0 6) but more limited sensitivity to larger-scale structures.
Hereafter, we will use the approximate wavelength of the continuum setup to describe the observations and analysis of the continuum emission at the different bands: 3 mm (Band 3), 2 mm (Band 4), and 0.8 mm (Band 7).

Line Emission
Details of the lines targeted in the ALMA observations are given in Table 2. To image both sources in all lines, the continuum emission was identified in line-free spectral channels and subtracted from the spectral windows in the uv plane.The continuum-subtracted visibilities were then imaged over the spectral range of interest with the same robustness parameter used for the continuum images.We used multiscale cleaning for both sources and all lines to recover better the extended emission.We additionally applied CASA's automated masking algorithm to identify more robustly the regions containing line emission in each spectral channel (Kepley et al. 2020).We found that the automated masking algorithm correctly identified regions with emission in each frequency channel with some adjustment of the default algorithm parameters for different lines and sources.Final velocity resolutions varied by line and are listed in Table 2, but are generally ∼0.1 km s −1 , and the emission lines toward both sources are spectrally resolved.The final synthesized beams match closely those for the continuum maps at similar frequencies.
The resulting cubes were corrected for the primary beam, and the amplitudes converted to brightness temperature.We show in Figure 2 the emission from the dense-gas tracers listed above, integrated over the extent of the emission in velocity (including hyperfine components, where present; we discuss this further in Section 4).We additionally overlay contours showing the 0.8 mm continuum, noting that the primary beam at this wavelength covers only the central component in some of the line maps.Toward N6, we furthermore tapered the H 2 D + data to a final angular resolution of 1 5 to improve sensitivity, as the H 2 D + emission was undetected toward this source at the original resolution.

Continuum Results
Here, we discuss the continuum emission of both compact and extended sources detected at the observed wavelengths.More detailed radiative transfer analysis of the extended emission toward SM1N is described in Section 5.

(Sub)millimeter Continuum Detections
Figure 1 shows the continuum emission across all observed ALMA wavelengths toward SM1N and N6.We include the observations at 3, 2, and 0.8 mm newly described here, as well as continuum data at Band 6 (hereafter 1 mm) from Friesen et al. (2018).Both compact and extended continuum emission is detected within the primary beam toward both sources and across all frequencies observed with ALMA.
The main continuum feature toward SM1N, seen at all wavelengths, is an elongated structure with a compact, but not point-like, emission peak near the northwestern end.In the larger fields of view, fainter, extended emission follows the line of the Oph A ridge to the northwest and southeast.The contrast between the emission peak and extended emission increases with decreasing wavelength.In the 3 mm observations, the wider field of view contains two bright compact sources: SM1 to the south, and GY30 to the northwest (labeled in Figure 1).Previous ALMA observations of SM1 suggest the continuum emission reveals a compact disk around a young protostar (Friesen et al. 2014(Friesen et al. , 2018;;Kawabe et al. 2018).GY30 (also Oph-emb9) has been identified as a Class I object (Greene & Young 1992;Enoch et al. 2009).SM1 and GY30 are outside of the field of view at the other observed wavelengths.
Toward N6, continuum emission is detected near the pointing center at 2, 1, and 0.8 mm.At 2 and 0.8 mm, faint extended emission surrounds a point-like source, N6-mm, first detected at 1 mm in Friesen et al. (2018).Extended 1 mm emission of ∼3000 × 1400 au extent was previously detected at 300 au resolution (Bourke et al. 2012).This is more extended than in our data, but in good location agreement.The extended emission was also previously detected at 1 mm when the data were tapered to a slightly larger beam (Friesen et al. 2018, not shown here).At 3 mm, extended continuum is seen to the northwest, and coincides with the larger continuum ridge structure of Ophiuchus A. In addition, a compact continuum source is detected at 3 and 2 mm, also to the northwest but offset from the extended emission.This compact source, which we label N6-w in Figure 1, has not been identified in previous observations of N6 or Oph A, and is outside of the field of view at 1 and 0.8 mm.N6-w lies within the Oph A ridge as traced by single-dish continuum emission, but is offset from the dense gas as traced by, for example, N 2 H + (Di Francesco et al. 2004).

Gaussian Fitting of N6-mm and Other Compact Continuum Sources
For the compact continuum emission sources, including N6mm, we fit the emission peaks in the images with 2D Gaussians using the CASA task imfit, and determine the total flux in an elliptical aperture around each source matching the Gaussian fit parameters here and in previous papers.For N6-mm, we first remove the extended background emission by subtracting a median flux value in an elliptical annulus with inner and outer limits of 3 and 5 times the aperture axes, oriented with the position angle of the Gaussian fit.We list in Table 3 the fit results at each wavelength, including the location in R.A. and decl., the total flux S ν and peak flux density I ν , and the major and minor axes and position angle of the best-fit Gaussian model.For SM1 and N6-mm, the source locations agree with those found in Friesen et al. (2018).
For the compact sources detected at at least two wavelengths, we show in Figure 3 the resulting fluxes as a function of wavelength.The source fluxes span 2 orders of magnitude across the observed wavelengths.We include compact sources  In each panel, the area imaged is limited to where the primary-beam gain is greater than 0.2.The resulting scale of each panel is shown by the scale bar.SM1N and N6 are centered in each panel.The synthesized beam is shown in the lower left of each panel.Black contours show the ALMA 0.8 mm continuum toward each source.Toward SM1N, only N 2 H + and H 2 D + show integrated intensity peaks at the continuum peak, whereas both NH 2 D and N 2 H + appear optically thick at this location.Toward N6, the integrated intensity peaks for all lines are offset from the continuum point source, but are colocated with the extended continuum emission.H 2 D + is only marginally detected toward N6 in a tapered beam.
from Friesen et al. (2018) for reference.With the exception of N6-mm, the spectral energy distributions (SEDs) for the compact sources are more shallow than expected for typical interstellar medium (ISM) dust, with low values of the dust spectral index β that are likely the result of either grain growth or high-optical-depth effects.This includes the new source N6-w.Given the data in-hand, we cannot make a strong claim as to the nature of N6-w but note that it has similar size and emissivity spectral index as the known protostellar sources in the region, and is brighter than N6-mm by at least a factor of a few.
The dust spectral index toward N6-mm appears steeper and more consistent with ISM values, but we caution that the extended emission is much more prominent at shorter wavelengths and may not be removed perfectly by the annulus method used here, potentially resulting in a steeper spectral index.
If we assume the dust emission is optically thin, we can estimate the mass M for both the compact and extended continuum emission following ) .We assume the dust is characterized by a spectral index β = 1.7, and a gas-to-dust mass ratio of 100, with κ 0 = 0.1 cm 2 g −1 at a frequency ν 0 = 1000 GHz.
For the compact sources, we assume dust temperatures T d = 15 K.We calculate masses based on the 3 mm flux, which is least likely to be optically thick, apart from N6-mm, where we use the 0.8 mm flux since we do not detect the source at 3 mm.The compact-source masses are given in Table 3.For SM1, the resulting estimated mass is in agreement with Kirk et al. (2017), but is greater than in Friesen et al. (2018) by a factor of 2 (accounting for the different T d ∼ 30 K used), likely due to the uncertainties in β and the dust opacity at shorter wavelengths.The compact-source masses are generally consistent with disk masses in Class 0/I objects (Jørgensen et al. 2009;Enoch et al. 2011), with known protostellar sources GY30 and SM1 at the high end of typical mass range, and sources N6-mm and N6-w at the low end (Andersen et al. 2019).
Toward N6, we also estimate the mass in the more extended emission.We do not have direct measurements of the dust temperature at ∼100 au resolution, but at 15″-30″ (2000-4000 au) scales, Pon et al. (2009) showed that the dust and gas temperatures are ∼15 − 20 K, potentially increasing toward the continuum peak.Allowing for T d to vary between 30 and 15 K, we find the total mass of the extended 0.8 mm emission toward N6 is only M ∼ 0.01-0.02M e .
We give a more detailed analysis of the density and temperature profile of the extended SM1N emission in Section 5, but a similar calculation for SM1N gives a total mass M ∼ 0.16 M e , given the emission at 0.8 mm and a dust temperature of 15 K, matching previous mass measurements (Friesen et al. 2014).

Integrated Intensity Maps
We show in Figure 2 the integrated intensity of NH 2 D, N 2 D + , H 2 D + , and N 2 H + emission toward SM1N and N6.We detect all emission lines toward both sources; for N6, we detect the H 2 D + emission in the tapered map only.The observed transitions of NH 2 D, N 2 H + , and N 2 D + contain hyperfine structure (hfs), and for these species we calculate the integrated intensity over all line components.While all the above species are typically considered good tracers of cold, dense gas, we find a clear progression in how well the emission matches the continuum data in the integrated intensity maps.Toward SM1N, H 2 D + and N 2 D + show integrated intensity peaks at the continuum peak location, although N 2 D + shows an additional peak to the southeast, and more extended emission than H 2 D + .Overall the integrated H 2 D + emission matches exceptionally well the 0.8 mm  9.9(1)  1.8(1)  1.9(1)  1.4(1)  130(8)  0  continuum, and we see no evidence of the H 2 D + offset reported in Cycle 0 ALMA observations (Friesen et al. 2014).This may be due to the improvement in uv coverage and sensitivity in the newer data.The southeast N 2 D + peak overlaps an extension from the peak of the SM1N core in the 2 mm continuum.While the continuum extends to this point, no additional continuum emission peak is seen, suggesting this is a region of enhanced N 2 D + abundance.Further south of the SM1N emission peak, additional N 2 D + emission overlaps with faint extended continuum emission visible at 3 and 2 mm.Two peaks are also seen in the NH 2 D integrated intensity around the SM1N continuum, but the northwest NH 2 D maximum is offset from the continuum peak, in contrast to N 2 D + .Further emission extends to the south of the continuum peak, overlapping and extending that seen in N 2 D + .In both the lines and the continuum, there is a narrow region of fainter emission between the more compact emission associated with SM1N and the extended emission that continues to the south and follows the extension of the Oph A ridge.
N 2 H + 4-3 emission is a clear outlier, where the emission appears to line the northeast side of the core.The N 2 H + line emission is significantly optically thick, affecting the spatial appearance in the integrated intensity map.The N 2 H + emission extends toward the southern tip of the continuum source, although does not show an additional peak matching that detected in the N 2 D + and NH 2 D maps.
Toward N6, the integrated intensity of NH 2 D, N 2 D + , and N 2 H + appear to correlate well, but the integrated intensity peaks in all cases are offset from the compact continuum peak (N6mm), and extend along the direction of the lower-level continuum emission.Similar offsets between the continuum and N 2 D + 3-2 emission were identified at 15″ resolution (Pon et al. 2009).Two peaks are seen in the N 2 D + and NH 2 D integrated intensity maps, although the second peak is fainter in N 2 D + .The same two peaks were seen in N 2 D + 3-2 at 300 au resolution (Bourke et al. 2012).A tentative detection of H 2 D + offset from the southwest edge of the continuum may be real, but we note again that this data has been tapered to a larger beam to improve the signal.The N 2 H + 4-3 line appears optically thick toward N6 (as is the N 2 H + 3-2 line; Bourke et al. 2012), similar to SM1N, but the integrated intensity is in better agreement with the extended continuum and other molecular lines than in SM1N.The dense structures traced by N 2 D + and NH 2 D toward N6 are much more evident than in the continuum emission, where the extended emission near N6-mm is only barely detected at 2 mm and not at all at 3 mm.

Line Fitting
We show in Figure 4 the observed spectra for each line toward the continuum emission peaks of SM1N and N6, and their best-fit spectra as described below.
For H 2 D + , we fit a single Gaussian to the spectrum at each pixel across the map, and from this determine the line amplitude, the local standard of rest line-of-sight velocity, v LSR , and the velocity dispersion, σ v , at each pixel where H 2 D + is detected.
For NH 2 D and N 2 D + , we fit the line emission, including the line hfs, using the pyspeckit package (Ginsburg & Mirocha 2011;Ginsburg et al. 2022).With pyspeckit, synthetic spectra are modeled assuming a constant excitation temperature T ex for all hyperfine line components, and that the components are additionally characterized with a single v LSR and velocity dispersion σ v .The difference between the model and observed spectrum is minimized, and we obtain maps of the resulting T ex , line opacity τ summed over all hyperfine components, v LSR , and σ v .For NH 2 D, the line fitting takes into account the hyperfine splitting caused by the deuterium nucleus as calculated by Daniel et al. (2016).Line fits for all species were assessed and masked based on the uncertainties returned for each parameter.
We show the resulting v LSR and σ v distributions for H 2 D + , N 2 D + , and NH 2 D toward SM1N and N6 in Figures 5 and 6, respectively.We also list in Table 4 the mean and standard deviation of all fit parameters within contours defined by the 0.8 mm continuum toward each source.
Toward both sources, the velocity patterns are very similar across all detected dense-gas-tracing species.Toward SM1N, the H 2 D + and N 2 D + in particular show the same v LSR pattern: roughly constant velocity along the long axis of the continuum structure, with an apparent velocity gradient perpendicular to the long axis of the core, across the continuum peak.Additional changes in the v LSR seen at the south and southwestern edges of the core in H 2 D + and N 2 D + are revealed to be part of largerscale velocity gradients detected more broadly in NH 2 D. Similar agreement is seen in N 2 D + and NH 2 D kinematics toward N6.In addition, both N 2 D + and NH 2 D show an apparent periodic variation of order ∼0.1 km s −1 in v LSR in narrow stripes across N6.The shift in v LSR values appears correlated with edges in the N 2 D + integrated intensity (and similarly variations in N(N 2 D + ), derived below), where higher v LSR is associated with lower integrated intensity values.This pattern is not seen at lower resolution (Bourke et al. 2012).The range of v LSR is greater by a factor of 2 in SM1N compared with N6.
The observed velocity dispersions are also similar across all three species toward both sources, and are consistently narrow, with N6 showing more narrow line widths on average than SM1N (σ v ∼ 0.1 km s −1 and σ v ∼ 0.15 km s −1 , respectively, for N 2 D + and NH 2 D).This is consistent with single-dish observations, which identified N6 as a site of extremely narrow line widths within Oph A (Di Francesco et al. 2004), and is in agreement with previous high-resolution observations (Bourke et al. 2012).Intriguingly, although the integrated intensity maps show both N 2 D + and NH 2 D avoid the compact continuum peak N6-mm, there is no clear increase in σ v seen in either line in the vicinity.
Toward SM1N, we found an offset between the NH 2 D moment map peak and the continuum peak.We find that the NH 2 D emission is optically thick toward the continuum peak and in the emission to the northwest.Total NH 2 D line opacities (summed over the hyperfine components) are ∼15 at the continuum peak, and excitation temperatures T ex ∼ 12 K with small variation across the source.N 2 D + also shows moderate optical depths, with total opacities of a few toward the continuum peak, and excitation temperatures T ex ∼ 9 K, slightly lower than found for NH 2 D.
In N6, the peak line opacity and excitation for both N 2 D + and NH 2 D are offset to the northeast from the compact continuum peak N6-mm, matching the offset seen in the integrated intensity maps.In contrast with SM1N, we also find higher excitation temperatures overall for N 2 D + than NH 2 D, with a maximum T ex ∼ 12 K for N 2 D + .Measured T ex values toward N6 agree well with N 2 D + 3-2 observations at 300 au resolution (T ex ∼ 10 K; Bourke et al. 2012).While the values found are similar to or slightly less than previous estimates of the dust temperature at high spatial resolution in SM1N (Friesen et al. 2014), they are greater than predictions of the core dust temperature based on radiative transfer modeling of the expected radiation field in Oph A (Stamatellos et al. 2007).The lines thus originate in high-density gas where n > n crit for each line, and constrain the gas (and dust) temperature at high densities within SM1N.(2014), we first determine the line opacity, τ, assuming an excitation temperature, T ex .We then determine the total column density, and estimate the partition function by reducing the H 2 D + level structure to a two-level system as in Caselli et al. (2008).Both the line opacity and the partition function thus depend on our assumption for T ex .Friesen et al. (2014) assumed T ex = 12 K.As we will show further in Section 5, the best-fit density model for SM1N gives dust temperatures T d ∼ 9 K within several hundred au, in agreement with the excitation temperatures for N 2 D + near the continuum peak (T ex ∼ 9 K).Furthermore, at the high densities previously estimated for SM1N (n  10 7 K; Friesen et al. 2018), H 2 D + should be very close to thermalized (Hugo et al. 2009).We thus assume T ex = 10 K for H 2 D + in this analysis.We note that, given the line brightness, we obtain unphysical results for the line opacity if we assume T ex  9.5 K.With T ex = 10 K, we find a median line opacity τ = 0.9, with a maximum τ = 1.8.The median H 2 D + column density is then N(H 2 D + ) = 3.2 × 10 13 cm −2 , with a maximum N(H 2 D + ) = 7.6 × 10 13 cm −2 that is offset by ∼0 8 from the SM1N continuum peak.The maximum H 2 D + column density is a factor of ∼6 greater than found in Friesen et al. (2014).This is partly due to the lower T ex assumption in this work, and partly due to the improved sensitivity, in both line and spatial scale, of the newer data.
Here, ν is the transition frequency, g u is the degeneracy of the upper level, A ul is the Einstein A coefficient, E u is the energy of the upper level, and the integral of the opacity over the line ) over the first 10 energy levels, where the state degeneracy g j = 2J + 1, given the fit T ex at each pixel.
We determine N(oNH 2 D), rather than N(NH 2 D), as the orthoto-para ratio is dependent on the physical conditions where the molecule is formed and can thus vary.Following Caselli et al. (2022), we take the values for E u , g u , and A ul from the Leiden Atomic and Molecular Database (LAMDA; Schöier et al. 2005;van der Tak et al. 2020) and calculate the partition function up to the first 30 energy levels.We show the resulting column density maps across SM1N and N6 for all species in Figures 5 and 6, respectively.Note that the noisy appearance of the N(NH 2 D) map toward N6 is due to the high opacity of the line, where small variations in the T ex and τ fits create jumps in the column density.
Toward SM1N, both N(H 2 D + ) and N(N 2 D + ) peak toward the continuum peak.N 2 D + shows a second column density peak to the southeast of similar magnitude to that at the continuum peak, along the extension seen in the continuum.This second peak is also seen in the integrated intensity.In contrast, N(NH 2 D) shows less variation across the core, with slightly lower values at the continuum peak.Higher NH 2 D column densities are found to the south of SM1N (seen as an integrated intensity peak in Figure 2).Toward N6, a clear drop in N(N 2 D + ) is seen toward N6-mm, and the greatest N(oNH 2 D) values are also offset from the compact source.For both N 2 D + and NH 2 D, the largest column densities are found northwest of the compact continuum source and coincident with the extended continuum emission.Additional regions of higher N(N 2 D + ) and N(oNH 2 D) extend to the southeast and northwest of the continuum.

Radiative Transfer Modeling of SM1N
We next analyze the radial density and temperature profiles of SM1N, using radiative transfer modeling to predict the continuum emission profile at the wavelengths observed with ALMA.

Radial Density Profiles
We create analytic density profiles following three different physically motivated models, and use the 3D Monte Carlo radiative transfer code RADMC-3D (Dullemond et al. 2012) to determine the radial temperature profile of the core, and the expected dust continuum flux density at each wavelength. 6To simplify the problem, we assume a spherical model geometry and thus investigate radial density and temperature profiles only.While SM1N is elongated overall, this approach is reasonable given that the brightest emission at all wavelengths is centrally concentrated.In all but one case we assume the core is starless and include no internal heating source.
We investigate three types of radial density profiles.The first is a spherically symmetric Plummer model (Plummer 1911), which was first applied to starless cores by Whitworth & Ward-Thompson (2001).This centrally condensed density profile is constant with a value of n 0 inside some radius r 0 , and then declines at r > r 0 following a power law with exponent p/2: This model is in agreement with single-dish observations which show that starless cores are centrally condensed, with a density profile that flattens at inner radii (e.g., Ward-Thompson et al. 1994;Alves et al. 2001;Tafalla et al. 2002).
a For H 2 D + , a constant T ex was assumed; see text.
The second radial density model is a singular power-law profile that follows where n 0 is the gas density at radius r 0 .This is the expected density profile for a SIS, and represents the envelopes of protostellar cores, where n(r) ∝ r −2 for all r (Larson 1969;Shu 1977).We choose a range of central densities such that the maximum density at 5 au is less than the density at which a FHSC would form (n ∼ 2 × 10 −11 cm −3 ; Larson 1969).Lastly, we test a broken power-law radial density profile that follows n ∝ r −2 at large radii, and is shallower inside some radius r 0 (but not flat, in contrast to the Plummer model).We again vary the central density n 0 , and test a range of power-law exponents inside r 0 , following The physical motivation for this density profile is that predicted by a collapsing SIS.In Friesen et al. (2018), a simple, isothermal calculation suggested that the density profile of SM1N followed a broken power law, where in the inner part of the core n(r) ∝ r −1.3 , transitioning to r −2 in the outer part at r  200 au.Here, we expand that analysis to model a broken power-law density profile self-consistently with the consequent dust temperature as a function of radius, and test several transition radii, 100 au < r 0 < 300 au, and two inner power-law indices, p = 1 and p = 1.5, for the same range of n 0 as for the Plummer model above.

Dust Opacities and Radiation Sources
To model the dust temperature and emission, we use dust opacities based on the OH5 model from Ossenkopf & Henning (1994), for grains with thin ice mantles corresponding to a coagulation time t = 10 5 yr at density n = 10 6 cm −3 .We follow Svoboda et al. (2019) to extrapolate the dust opacities at wavelengths λ < 1 μm and λ > 1.3 mm.We include dustscattering efficiencies and albedos based on Young & Evans (2005) and Weingartner & Draine (2001), respectively.
We assume that the cores are irradiated by an isotropic interstellar radiation field (ISRF).We use the ISRF model described by Black (1994), with the addition of the Draine (1978) description at ultraviolet wavelengths.We set the strength parameter, G 0 = 1.7.As stated above, however, Oph A and its embedded cores are also irradiated by the nearby star S1, which has a luminosity L = 1100 L e (Bontemps et al. 2001) and an effective temperature T = 16,000 K.We model the expected flux of S1 as a function of wavelength as a blackbody at temperature T at the projected distance from SM1N.
The ALMA data resolve out emission on scales greater than ∼17″ (Table 1).We therefore include dust opacity and scattering effects as a function of wavelength, applying the above dust absorption and scattering cross sections for a column density N = 5 × 10 19 cm −2 , which corresponds to a density of n ∼ 10 3 cm −3 over a distance of ∼0.01 pc, or roughly the thickness of the Oph A ridge.With these radiation field parameters, the dust temperatures at the core edges are ∼20-25 K for all density models, consistent with dust temperatures measured on larger scales (André et al. 1993;Santos et al. 2019) and dust temperature modeling of Oph A (Liseau et al. 2015).
For all models listed above, we assume the core is starless, and include no additional internal source of luminosity.We additionally create, however, a single model of a core with a power-law density profile as described above, where n = 2 × 10 11 cm −3 at a radius of 5 au.At this density, the innermost regions of the core become optically thick, and the temperature rises until the thermal pressure supports the core against gravitational collapse (the FHSC).We add to this model an internal blackbody source of radius r = 5 au, mass M = 0.05 M e , and temperature T = 170 K, based on theoretical models of FHSCs (Larson 1969;Masunagaet al. 1998;Saigo et al. 2008).We note that FHSCs may have radii up to several tens of au in the case where the initial core is rapidly rotating, and the mass and temperature will both increase as the FHSC accretes more material from the infalling envelope.

Eliminating Models Based on Radial Temperature Profiles
For each density profile, the resulting radial temperature profile was determined via radiative transfer by RADMC-3D.The density and temperature profiles for a selection of parameters for the three classes of density profiles are shown in Figure 7.
Before simulating the ALMA observations, we further constrain the models by requiring the core temperatures to agree with measurements of the gas temperature in Oph A on both large and small spatial scales, and for predicted densities to match expected densities given the extent of the emission of the deuterated molecular lines.The initial models ensure dust temperatures are ∼20-25 K at the core edges, ∼10 4 au.At scales ∼2500-4000 au, gas temperatures T K ∼ 15-18 K based on NH 3 modeling (Friesen et al. 2009(Friesen et al. , 2018)).Here, densities must be n  10 3 cm −3 , given the effective critical density of NH 3 , and the gas and dust temperatures should be coupled.On the ∼100-2500 au scales sampled by our ALMA observations, the predicted dust temperatures must be consistent with the measured excitation temperature of the dense-gas tracers, T ex  8-9 K, based on the hyperfine N 2 D + fits.These temperatures are slightly greater than typically assumed in isolated starless cores, where at the highest densities gas temperatures may become as low as ∼5-7 K (Crapsi et al. 2007;Pagani et al. 2007), due to the enhanced ISRF impacting Oph A. Pineda et al. (2022) find a similar result toward the core H-MM1, also in Ophiuchus, which they also ascribe to the strong radiation field.Densities must be n  10 6 cm −3 at radii of a few thousand au, given the critical densities of the N 2 H + and N 2 D + lines and their observed extents.
Examples of density and temperature profiles that match well the above requirements are shown in Figure 7. Overall, models with lower central densities show smaller drops in dust temperature toward the core interior (i.e., at small r).The radial temperature profiles are qualitatively similar for the Plummer and broken power-law density profiles, where models with higher central densities show sharp drops in T d at the core edges, with a turnover to a more flat distribution at smaller core radii.The transition from steeply decreasing T d to flatter T d with r occurs at larger r for larger central n.Power-law density models show no such turnover in the relationship between T d and r, and instead T d consistently decreases toward smaller r, albeit more rapidly for models with higher central densities.

ALMA Simulations of Model Images
To compare directly the models with observations, synthetic images were made at 0.8 and 2 mm of each model that met the above requirements for dust temperature and overall density, and the cores were projected to a distance d = 137.3pc.
We simulate ALMA observations of the predicted core emission that match the real integration and uv coverage achieved by the actual observations.We use the simulator toolkit within CASA to use the observed measurement sets at each wavelength to predict the observed visibilities for each model.At each wavelength, we predict the model visibilities for each ALMA configuration separately.We then combine the simulated measurement sets, applying a correction for the data weights when combining the 7 and 12 m measurement sets at 0.8 mm.We imaged the simulated data using tclean in CASA with the same parameters as were used for the observed data, with the exception of setting a fixed circular clean mask with a diameter of 8″, which is slightly smaller than the expected maximum angular scale achieved by the observations.
Figure 8 shows the observed continuum emission toward SM1N at 0.8 and 2 mm compared with the simulated emission maps at the same wavelengths that best fit the observed emission for each of the Plummer and broken power-law density profiles.Fit goodness is determined based on a comparison of the simulated and observed radial emission profiles, discussed further in Section 5.5.We furthermore show the simulated images of the model containing the warm blackbody source.We focus on the shorter-wavelength emission for this comparison, as no model reproduces well the observed 3 mm emission observed toward SM1N with ALMA.
A visual examination of the simulated observations shows clearly that only models with high central densities within a small central radius can match the actual core observations, particularly at longer wavelengths.
Comparing the simulated ALMA images for our simplistic FHSC model with more detailed simulations of FHSCs by Commerçon et al. (2012), we find that our predicted flux is a factor ∼2 greater than their simulations, which used a similar ALMA configuration.The authors show that the magnetic field strength influences the resulting expected flux densities of the formed FHSCs, but even their faintest, quasi-hydrostatic model predicts that a FHSC should be easily detected in our data as a compact source, given the depth of our 3 mm observations.

Observed and Simulated Radial Emission Profiles
We next compare directly the radial emission profiles of the model starless cores at the observed wavelengths with the ALMA data.We show in Figure 9 the radial profile of the observed emission toward SM1N at 0.8, 2, and 3 mm, centered on the peak emission location at 0.8 mm (Band 7), and averaged over radial annuli of 3 pixels, or ∼0 3. We create similar radial emission profiles from the simulated emission maps for each model.We calculate the reduced chi square χ 2 between the model and the observations at each observed wavelength.We limit the calculation to within 5″ (700 au), where we are comparing primarily the models against the compact central emission observed toward SM1N and omitting the outer, extended regions where our spherically symmetric model is not expected to agree well with observations.We show in Figure 9 the observed radial emission profiles at 3, 2, and 0.8 mm toward SM1N (black and gray lines), along with predictions from the best-fit broken power-law and Plummer density models.We note that while we show the observed 3 mm radial emission profile, no model fits the observations well at this wavelength while matching emission profiles at the shorter wavelengths.As stated previously, this is likely due to the slightly greater sensitivity to larger-scale structures in the 3 mm observations, including the extended Oph A ridge, which is not included in our simple core model.Another possible reason for this discrepancy, however, could be a mismatch between the dust properties assumed in our modeling, particularly the assumption of an MRN dust size distribution (Mathis et al. 1977).If dust grains were able to grow in size within SM1N, we might expect to see elevated emission at 3 mm due to the impact of grain growth on the dust spectral index, β (as introduced in Section 3.2).Grain growth would lower the dust spectral index below the β = 1.7 value used here.Dust spectral indices have been observed to decrease in protostellar cores (β ∼ 1) relative to starless cores (β ∼ 2; Bracco et al. 2017), while models of observed dust emission require grain growth in the inner ∼2000 au of the isolated starless core L1544 (Chacón-Tanarro et al. 2019).A comparison of the peak flux between 2 and 0.8 mm is consistent with β ∼ 1.7-2.3,assuming T d ∼ 10 K, however making grain growth on several hundred au scales unlikely to be the cause of the discrepancy.
Considering only the shorter-wavelength emission, the best-fit density model is a broken power-law model with an inner radius r 0 = 300 au, a transition density n 0 = 10 7 cm −3 at r 0 , and a powerlaw exponent p = 1.0 inside r 0 .A slightly steeper p = 1.5 also fits well the 0.8 mm radial profile, but fits the 2 mm profile less well than the p = 1.0 model.Overall, the Plummer models agree less well with the observed data, with the best-fit model having r 0 = 80 au, and a slightly higher central density n 0 = 10 8 cm −3 compared with the best-fit broken power-law models.The set of models with single power-law density profiles were consistently poorer fits to the data than those with a turnover in n, and none fit the observations well.We thus do not show their simulated images or radial profiles here.
In the best-fitting models to the observed 0.8 and 2 mm emission, the dust temperature reaches a minimum of ∼9-10 K within several hundred to ∼1000 au (see Figure 7).Within ∼500 au, the enclosed mass for both models is M enc = 0.02 M e , in agreement with previous mass calculations for SM1N from ALMA data at 1 and 0.8 mm that gave M ∼ 0.02-0.03M e ALMA images at 2 mm (left) and 0.8 mm (right) of SM1N for the best-fit starless core models following the Plummer and broken power-law density profiles.In these rows the color scale is set to match the observations.The bottom row shows the power-law density profile model with an internal blackbody radiation source as described in Section 5.4, but the color scale extends a factor of 10 in flux density beyond the other rows.
Figure 9.The observed radial emission profile toward SM1N at all observed ALMA wavelengths (black and gray lines and data points; the brightest emission is at 0.8 mm, followed by 2 mm and then 3 mm).Blue and green curves show the predicted radial emission profiles for the best-fit broken power-law (BPL) and Plummer density models, respectively.For all curves, shaded regions show the dispersion in flux values at that annulus band.As noted in Section 5.4, neither best-fit model predicts sufficient flux at 3 mm, and the predicted flux is below the limits of this plot.(Friesen et al. 2014(Friesen et al. , 2018) ) within ∼500 au.The enclosed mass within 3 × 10 3 au is M enc = 0.21 M e (broken power law) and M enc = 0.18 M e (Plummer).This is less than the total core mass for SM1N, M ∼ 1.3 M e , as determined from single-dish continuum data (Motte et al. 1998).This is not surprising, however, as we resolve out emission on scales 2400 au that would add to the overall core mass.The density profile should be considered accurate on scales less than this.

SM1N: Comparison with Core-collapse Models
The density model that best fits the observed radial emission profiles at 0.8 and 2 mm toward SM1N is a broken power-law model with an inner radius r 0 = 300 au, a density n 0 = 10 7 cm −3 at r 0 , and a power-law exponent p = 1.0 inside r 0 .At this density, the free-fall time is only a few × 10 3 yr.Here, we compare SM1N's radial density profile and observed kinematics with predictions from several core-collapse models.We note that typical collapse models describe the evolution of a core in isolation, whereas SM1N exists within the Oph A ridge.In cluster-forming regions like Oph A, dense cores are typically more compact than cores found in more isolated environments (Ward-Thompson et al. 1999), and are more likely to be affected by, for example, protostellar feedback from nearby forming young stellar objects.SM1N is thus an example of the initial conditions of core collapse in a clustered environment, and may not be perfectly represented by the isolated collapse models described here; however, clustered collapse models in similar detail are not yet available.
A density profile n ∝ r −1 , transitioning to n ∝ r −2 , is similar to the predicted density profile of an SIS after collapse has begun, where n ∝ r −3/2 in the central region r < r 0 undergoing free-fall collapse (Shu 1977).The transition radius moves outward as the collapse proceeds ("inside-out" collapse).In this model, however, an accreting source has already formed at the center of the cloud, whereas our modeling of the continuum emission and analysis of the molecular-line data shows no significant source of luminosity can exist within SM1N.
The density model is also similar to that predicted for a gravitationally contracting, unstable BES.Here, the inner radius describes the transition between a flat inner-density profile and the steep-outer density profile, and is equal to the Jeans length given by the sound speed and central density (Keto & Caselli 2010), as we find in our best-fit density model.Over time, the inner region increases in density and decreases in radius, such that the outer, singular density profile dominates more of the core and the density profile approaches that of the SIS.This contraction predates the formation of an accreting source.This model was found to fit the more isolated starless core L1544 (with a lower central density n c = 10 7 cm −3 ; Keto et al. 2015).The broken power-law model continues to increase in density as r → 0, however, in contrast to the BES model.If we require a flatter density profile for SM1N inside of the transition region to better match a BES, as in the Plummer model, we find that the best-fit model has a central density n c = 10 8 cm −3 and a transition radius of only 80 au, less than the Jeans length by a factor of 3 and thus smaller than predicted from these quasi-static evolutionary models.Gómez et al. (2021) model core collapse in a constant-density environment, such that the collapsing core continues to accrete from its surroundings.Such a model may be more applicable to cores within a dense molecular filament or ridge like Oph A. This global-gravitational-collapse model predicts a similar density profile, where p < 2 at earlier stages of collapse, and the inner Jeans-stable region shrinks over time as the density increases to eventually match an SIS-type density profile at the time of protostar formation.
The evolution of a collapsing BES predicts that the infall velocity, v in , is linear with r to near r 0 , and then decreases with increasing r (Foster & Chevalier 1993).Similarly, Gómez et al. (2021) show that for p 2 in the global-gravitational-collapse model, infall speeds should also decrease toward the core center.In a scale-free model of free-fall gravitational collapse, Li (2018) find that =v f GM r r ff ( ) , where M(r) is the mass internal to some radius r, and f is a constant that depends on the scale of turbulence driving.Post-protostar formation, the SIS model predicts v in increases as r −1/2 .Since H 2 D + and N 2 D + are largely optically thin, any infall motion will present as increased line widths.In the collapsing BES model, Keto & Caselli (2010) predict that the line FWHM for N 2 H + (1-0) reaches ∼0.35 km s −1 when n c = 10 8 cm −3 , giving a nonthermal velocity dispersion σ NT = 0.15 km s −1 at T = 5 K. Toward SM1N, velocity dispersions σ v for H 2 D + and N 2 D + appear to increase near the continuum peak, with smaller values found at the core edges (Figure 5).We show in Figure 10 the radial profiles of σ NT for H 2 D + , N 2 D + , and NH 2 D. The nonthermal velocity dispersion was calculated assuming a constant gas temperature of 10 K.For H 2 D + , we additionally determine the intrinsic velocity dispersion given the optical depth of the line following Hacar et al. (2016) prior to calculating σ NT .For H 2 D + and N 2 D + , σ NT is essentially constant to r ∼ 200 au, and then decreases at larger radii, albeit with larger scatter due to sensitivity and the appearance of a second N 2 D + emission peak toward SM1N.In particular, σ NT ∼ 0.15 km s −1 for N 2 D + , and slightly greater for H 2 D + .
Figure 10 furthermore highlights a trend of increased σ NT from NH 2 D through N 2 D + to H 2 D + .NH 2 D has a lower effective critical density than N 2 D + and H 2 D + , however.We thus find a potential increase in σ NT with increasing density in the core, assuming a constant gas temperature and that NH 2 D, N 2 D + , and H 2 D + trace progressively denser gas.This may be in conflict with the collapsing BES model, as infall speeds should become smaller interior to the transition radius, where densities are higher.If, however, NH 2 D traces gas outside the transition radius, then the observations are not in conflict with the model.
To examine further how well the different molecules may trace the highest-density gas, we show in Figure 10 (bottom) the radially averaged column densities of the three molecular species toward SM1N, centered on the continuum peak and normalized to the value at r = 1″.As expected from the column density maps, both N(H 2 D + ) and N(N 2 D + ) peak with the continuum, while N(oNH 2 D) decreases by a factor ∼30% compared with its highest value within 100 au of the continuum peak.The NH 2 D abundance relative to H 2 thus also decreases toward the continuum peak.
Given the projected N(H 2 ) as a function of radius from the best-fitting density model for SM1N, we find roughly constant radial H 2 D + and N 2 D + abundances toward SM1N within several hundred au, where the mean X(H 2 D + ) = 9.1 × 10 −11 and X(N 2 D + ) = 6.8 × 10 −12 .The mean X(NH 2 D) = 9 × 10 −10 , and decreases slightly at the continuum peak.The N (NH 2 D) and X(NH 2 D) profiles suggest that NH 2 D (and consequently NH 3 ) are depleted from the gas phase via freeze-out onto dust grains at the highest densities in the SM1N core.Similar results were found via chemical modeling of the evolved starless core L1544 (Caselli et al. 2022), as well as for the starless core H-MM1 (Pineda et al. 2022).In the case of L1544, the central density is slightly lower than found for SM1N, and the transition radius is significantly larger.This result is further evidence of the presence of a "complete depletion zone" within cores prior to the formation of a protostar (Walmsleyet al. 2004).Additional modeling of the chemistry as a function of radius may reveal whether N 2 D + is also depleted at small scales in SM1N, but is beyond the scope of this paper.Regardless, given the apparent depletion of NH 2 D, the smaller nonthermal gas motions traced by this line originate in regions with lower density, and σ NT therefore increases with core density (but remains subsonic) in SM1N.
As noted above, this is not necessarily incompatible with the collapsing BES model, as long as the lower-density gas traced by NH 2 D is outside of the transition radius where v in begins to decrease with increasing radius.Our best-fit r 0 ∼ 80-300 au for the broken power-law and Plummer models, respectively.This is similar to the radius where we see the drop in N(NH 2 D) in Figure 10, so our observations are consistent with the BES predictions.It is not clear whether the global-collapse model also predicts a turnover in v in ; if not, the kinematics of SM1N are not consistent with this model.We thus argue that the starless core SM1N is undergoing either inside-out gravitational collapse or is in a highly evolved, contracting BES state.

N6: Comparison with First Hydrostatic Core Predictions
What is the nature of the compact source N6-mm?We first want to rule out the serendipitous detection of a faint submillimeter galaxy (SMG) in our field of view.Tamura et al. (2015) argue that high-redshift SMGs can appear similar to FHSCs at millimeter wavelengths.Using the background source counts at 1.1 mm from Hatsukade et al. (2016), we estimate that we may detect ∼0.3 sources per 1.1 mm field with flux densities similar or greater than that detected toward N6-mm.Deep radio searches with the Very Large Array, however, find no corresponding 6 cm source (Leous et al. 1991;Gagné et al. 2004).Chen et al. (2011) argue that a lack of radio detection implies that the SMG must be at much higher redshift than assumed by Hatsukade et al. (2016), z 2.5 ¯.In this case, we would expect a rising SED toward shorter wavelengths and a detection by Herschel or Spitzer, but no sources are detected toward N6-mm at these wavelengths.Furthermore, we detect extended continuum emission around N6-mm, and the distribution of H 2 D + , N 2 D + , and NH 2 D avoid the continuum peak, suggesting that the dense gas is affected by N6-mm.We thus argue that it is very unlikely that N6-mm is a background, high-redshift SMG.
FHSC candidates are generally identified via compact continuum sources, coupled in some cases with the detection of low-velocity bipolar outflows, and either a lack of pointsource emission at wavelengths shorter than ∼160 μm or a very low luminosity L bol < 0.1 L e (e.g., Hirano & Liu 2014;Gerin et al. 2015).In some cases, compact continuum detections were not present in follow-up observations (e.g., Per-bolo 45 and L1448 IRS2E; Maureira et al. 2020), and these sources were ruled out as FHSC candidates.Fujishiro et al. (2020) identify a compact source, MC35-mm, in Taurus, with a tentative lowvelocity (∼2-4 km s −1 ) bipolar outflow extending only a few × 10 3 au, but at ∼800 au resolution are not able to identify a point source similar to N6-mm.
In contrast to SM1N, the distribution of dense-gas tracers near N6-mm is suggestive of (very compact) heating by an embedded source.Any embedded source must have L bol < 0.1 L e , the upper limit for a very-low-luminosity object (VeLLO), based on Herschel 70 μm data and following the tight relationship between 70 μm emission and L bol for protostars (Dunham et al. 2008).Unfortunately, the 70 μm-bright Oph A ridge precludes a more stringent upper limit.The expected luminosity for a FHSC at 70 μm is ∼200-300 mJy at the distance to Ophiuchus (Boss & Yorke 1995;Masunagaet al. 1998;Enoch et al. 2010), just below the observable 70 μm limit near N6-mm.N6-mm is also fainter than predictions for FHSCs at longer wavelengths.Enoch et al. (2010) model the SED of FHSC candidate Per-Bolo 58, finding a good fit with a source with L = 0.012 L e and T bol = 100 K within R = 1.7 au.At 850 μm (0.85 mm), the flux density of Per-Bolo 58 is 920 mJy, however, Figure 10.Top: radial σ NT profiles toward SM1N from Gaussian (H 2 D + ) and hyperfine (N 2 D + , NH 2 D) fits to the emission lines, centered on the continuum peak location.The shaded areas show the 1σ standard deviation around the mean value.The red line shows the sound speed at 10 K.The nonthermal component of the velocity dispersion is subsonic for all lines, and appears to increase toward higher densities given the molecular tracers used.Bottom: radial column density profiles of SM1N for H 2 D + , N 2 D + , and NH 2 D, normalized to their value at r = 1″.Shaded areas show the 1σ dispersion around the mean value.The radial profiles are centered on the continuum peak.Toward SM1N, the peak N(H 2 D + ) and N(N 2 D + ) values are found near the continuum peak and are closely correlated, whereas N(NH 2 D) peaks are offset from the continuum.
nearly 2 orders of magnitude greater than N6-mm (11 mJy), and Maureira et al. (2020) argue that this source is in fact a Class 0 protostar.Indeed, the observed flux density at 0.8 mm toward N6-mm is much less than predicted by simulated ALMA observations of FHSC models, unless we are viewing N6-mm nearly edge-on through a forming disk or pseudo-disk structure (Commerçon et al. 2012).In this case, the 0.8 mm emission would be optically thick, obscuring the embedded FHSC.This possibility seems unlikely, however, as the observed dust spectral index is consistent with typical ISM values, rather than the shallower slope expected for high optical depth (although deeper observations at 1.1 mm would constrain this better, as described in Section 3.2).
Similar to arguments used to constrain the temperature and density structure for SM1N, N 2 D + and NH 2 D both show high T ex values that indicate that the line emission originates from gas with temperatures 9-10 K, and densities n  10 5 -10 6 cm −3 .In contrast to SM1N, however, we find slightly larger NH 2 D σ NT values compared with N 2 D + , with typical values σ NT ∼0.09 and σ NT ∼0.07 km s −1 , respectively, such that the nonthermal component of the velocity dispersion decreases rather than increases toward higher densities.Furthermore, we do not detect compact significant H 2 D + emission toward N6-mm or the extended continuum structure, implying that H 2 D + is depleted in the gas phase.This may be due to a further deuterium fractionation from + H 3 and H 2 D + toward D 2 H + and + D 3 .Alternatively, gas temperatures may be enhanced toward N6mm such that the backwards direction of the H 2 D + -formation reaction is again allowed to occur, destroying H 2 D + and consequently N 2 D + and NH 2 D. As noted in Section 3.2, singledish temperature measurements toward N6 are warmer than in isolated starless cores, with T K ∼ 15-20 K.Only a small increment in temperature due to an embedded, warm object would thus be needed to allow for the decrease in H 2 D + (and hence N 2 D + and NH 2 D) abundance.Aikawa et al. (2008) show that the CO sublimation radius, where T  20 K, is only ∼100 au at the time of protostar formation, similar to the resolution of our ALMA observations.Deuterium fractionations can remain high in the envelopes of young protostellar sources (Giers et al. 2023), in agreement with the observed N 2 D + and NH 2 D within N6 (but offset from N6-mm).
The typical mass of a FHSC is M ∼ 0.05 M e (Boss & Myhill 1992;Masunagaet al. 1998), but can be lower in, for example, weakly magnetized cores (Commerçon et al. 2012).If the continuum emission is optically thin, we calculate a mass M = 0.0015 M e (1.6 M Jup ) in the compact source, giving an average n ∼ 3 × 10 6 cm −3 , embedded within extended emission with mass M ∼ 0.01-0.02M e (Section 3.2).N6-mm could thus potentially be a FHSC if the continuum emission is optically thick, or alternately a proto-brown dwarf.Its final mass will depend on how efficiently it can accrete mass from the larger core (M = 0.29 M e ; Bourke et al. 2012).Assuming a ∼30% mass accretion efficiency (Alves et al. 2007), the final source mass would fall on the border between a brown dwarf and stellar object.The N6-mm mass and luminosity upper limit are similar to values found for proto-or Class 0 brown dwarfs identified in Lupus (Santamaría-Miranda et al. 2021).
Outflows have been detected toward brown-dwarf candidates (Kim et al. 2019), in some cases with extents of ∼10″-20″.FHSCs have been predicted to drive compact (∼hundreds of au), low-velocity (∼1-4 km s −1 ) outflows (Commerçon et al. 2012;Hincelin et al. 2016), where the values depend on the magnetization of the contracting core.Outflow searches toward N6 are challenging due to the presence of the VLA 1623 outflow to the southwest of the N6 core, but previous studies would have been sensitive to outflows on 10″ scales.Friesen et al. (2018) identify a potential compact, low-velocity red outflow lobe in CO emission to the northeast of N6-mm.Young et al. (2019) argue, however, that any outflows from FHSCs would be too compact and low-mass to be detectable in emission like CO with ALMA, ruling out a number of FHSC candidates in the literature, including promising candidates L1451-mm (Maureira et al. 2020;Wakelam et al. 2022) and Barnard 1b-N and 1b-S (Young et al. 2019).The upper L bol limit shows that, at most, N6-mm may be a VeLLO, while the lack of large-scale outflows limits the dynamical age of any embedded, accreting source to 10 3 yr.Thus, N6-mm remains one of the best FHSC candidates yet identified.

On the Frequency of Compact Structures in Oph A
Previous surveys to detect compact structure in starless cores at high resolution have detected between zero and a few objects out of tens of targets (Dunham et al. 2016;Kirk et al. 2017).At somewhat lower resolution than this study (900 au spatial resolution), Tokuda et al. (2020) identify compact emission toward a third of prestellar targets in the Taurus molecular cloud, and argue that the lifetime of prestellar cores approaches the free-fall time as the core density increases.This implies that it should be unlikely to detect two objects like SM1N and N6-mm at these transitional stages, with only ∼10 3 -10 4 yr timescales, in such a small region as Oph A. Indeed, Oph A also contains several protostellar sources discovered with ALMA observations that remain undetected at shorter wavelengths (Friesen et al. 2014(Friesen et al. , 2018;;Kawabe et al. 2018), implying at least six low-mass stars have formed or are in the process of forming within the expected Class 0 lifetime of ∼0.1-0.16Myr (Evans et al. 2009).
As noted previously, Oph A is a dense ridge within a clustered star-forming environment.Dust and gas temperatures are greater than in typical cores in more isolated environments due to heating by nearby stars (Liseau et al. 2015;Chen & Hirano 2018).For example, Oph A lies between two B-type stars, S1 and HD 147889, resulting in known photodissociation regions (PDRs) along both the western and eastern edges (e.g., Liseau et al. 1999;Kamegai et al. 2003), although there does not seem to be a strong pressure gradient at the PDR border (Larsson & Liseau 2017).More generally, it has been argued that the morphology of L1688 may be explained by external compression from the Sco OB2 association (Loren & Wootten 1986).Furthermore, there is evidence for multiple outflows from nearby protostars, including the extended VLA 1623 outflow (Nakamura et al. 2011), that may impinge on the dense gas itself.Oph A and its embedded sources are thus evolving in a dynamic environment, which may be expected to produce a burst of star formation activity within a shorter time period than in isolated environments.Such a burst would produce more objects in a dynamically evolved state than expected for models of, for example, constant star formation.

Summary
Here, we have investigated in detail continuum emission and dense-gas-tracing molecular-line emission at ∼100 au resolution toward the previously identified SM1N and N6-mm sources in Ophiuchus A. We have also identified another compact continuum source within the larger Oph A ridge, which we label N6-w.The continuum SED and measured fluxes for N6-w are consistent with previously detected protostellar sources in Oph A, but additional, targeted continuum and line observations are needed to confirm its classification.
Previous analyses of observational data have consistently confirmed the starless nature of SM1N, despite its centrally peaked density profile and subsequent apparent instability to gravitational collapse.Here, we have tested a range of radial density profiles, determining via radiative transfer analysis their resulting radial temperature profiles, and predicted emission as observed by ALMA.Our results show that SM1N cannot contain a low-luminosity protostar.The predicted emission for theoretical FHSC models are also bright compared with the ALMA-detected emission from SM1N, even if the FHSC is only ∼5 au in radius (compared with our resolution of ∼100 au).We thus conclude that SM1N remains starless, but in a highly unstable dynamical state.
The observed emission from the dense-gas tracers, H 2 D + , N 2 D + , and NH 2 D, are consistent with this conclusion via the dust continuum analysis.Both H 2 D + and N 2 D + emission peaks are coincident with the continuum peak, with no evidence of decreasing abundance as would be expected if a warm region existed at small radii around a FHSC or young protostar.We show that NH 2 D is likely depleted toward the core center due to freeze-out onto dust grains, rather than by destruction via heating and CO release.The velocity dispersions at small radii (300 au) are relatively constant, with nonthermal motions similar to the sound speed at 10 K. We find a potential increase in σ NT with increasing density in the core, assuming a constant gas temperature and that NH 2 D, N 2 D + , and H 2 D + trace progressively denser gas.This increase of nonthermal width with increasing density is consistent with SM1N being in a state of dynamic collapse right before the formation of a central protostellar system.
Toward N6, we confirm the presence of a compact continuum source, N6-mm, within the core (Friesen et al. 2018).In contrast to starless SM1N, however, the dense-gas tracers avoid the continuum peak, consistent with the presence of an embedded source warming the immediate surroundings but retaining high deuterium fractions in the envelope.The estimated mass, assuming optically thin dust emission, is less than predicted for a FHSC, and its (sub)millimeter flux is similarly low compared with FHSC models.The measurements are consistent with several proto-brown-dwarf candidates.We do not detect an increase in the velocity dispersion with increasing density toward N6, and instead NH 2 D σ NT values are slightly greater than those measured for N 2 D + , although for both species σ NT remains subsonic.The lack of detected outflows indicates that the dynamical age of a potential embedded, accreting source is 10 3 yr.
For both sources, additional modeling of the expected molecular abundances and kinematics would provide further tests of prestellar core infall models.

Figure 1 .
Figure 1.Continuum emission toward Oph A SM1N (left) and N6 (right) at 0.8, 1, 2, and 3 mm (top to bottom; ALMA).In each panel, the area imaged is limited to where the primary-beam gain is greater than 0.2.The resulting scale of each panel is shown by the scale bar.SM1N and N6 are centered in each panel.Additional continuum point sources are seen in the 3 mm images toward both sources, and additionally in the 2 mm image toward N6; at other wavelengths they are outside of the imaged area.

Figure 2 .
Figure2.Integrated line intensity for the four molecular transitions discussed here toward SM1N (left) and N6 (right).In each panel, the area imaged is limited to where the primary-beam gain is greater than 0.2.The resulting scale of each panel is shown by the scale bar.SM1N and N6 are centered in each panel.The synthesized beam is shown in the lower left of each panel.Black contours show the ALMA 0.8 mm continuum toward each source.Toward SM1N, only N 2 H + and H 2 D + show integrated intensity peaks at the continuum peak, whereas both NH 2 D and N 2 H + appear optically thick at this location.Toward N6, the integrated intensity peaks for all lines are offset from the continuum point source, but are colocated with the extended continuum emission.H 2 D + is only marginally detected toward N6 in a tapered beam.
137.3 pc is the source distance, κ ν is the dust opacity at the frequency ν of the observations, and B ν (T d ) is the Planck function at frequency ν and dust temperature T d .The dust opacity k

. 0015 Notes.
Angular sizes have not been deconvolved by the beam.All angular size measurements and mass estimates are calculated from 2D Gaussian fits to Band 3 (3 mm) images, apart from N6-mm, where the Band 7 (0.8 mm) images were used.a Total flux and peak flux density fromFriesen et al. (2014).

Figure 3 .
Figure3.Total flux as a function of wavelength for all compact sources identified at at least two wavelengths in ALMA observations toward Oph A (this paper;Friesen et al. 2014Friesen et al. , 2018)).The gray line shows the expected slope for β = 1.

Figure 4 .
Figure 4. Examples of H 2 D + , N 2 D + , and NH 2 D spectra toward the continuum emission peak of SM1N (top three panels) and N6 (bottom three panels).Best single Gaussian (H 2 D + ) and hfs fits (N 2 D + , NH 2 D) are overlaid in blue.Note that for N6, the line-emission peaks are offset from the continuum.
4.3.2.NH 2 D and N 2 D +For NH 2 D and N 2 D + , where uncertainties in the fit T ex , σ Tex < 3 K, we use the fit parameters T ex , τ, and σ v to determine the total column density N of each species(Mangum & Shirley 2015):

Figure 7 .
Figure7.Here we show a selection of the core radial density models (bottom) and resulting radial temperature profiles (top) derived using RADMC-3D following the discussion in Section 5.The three columns show results from different radial density model families: the Plummer model (left), broken power law (BPL; middle), and power law (PL; right).Within each column linestyles and colors show different values of n 0 , r 0 , and p where varied.For the full range of parameters tested, see the text discussion.In the Plummer models, temperature spikes are due to photon packet limits and are not physical (and do not impact the final simulated images).

Figure 8 .
Figure 8.Comparison of the observed (top) and simulated (second to third rows)ALMA images at 2 mm (left) and 0.8 mm (right) of SM1N for the best-fit starless core models following the Plummer and broken power-law density profiles.In these rows the color scale is set to match the observations.The bottom row shows the power-law density profile model with an internal blackbody radiation source as described in Section 5.4, but the color scale extends a factor of 10 in flux density beyond the other rows.