Droplets I: Pressure-Dominated Sub-0.1 pc Coherent Structures in L1688 and B18

We present the observation and analysis of newly discovered coherent structures in the L1688 region of Ophiuchus and the B18 region of Taurus. Using data from the Green Bank Ammonia Survey (GAS; Friesen et al. 2017), we identify regions of high density and near-constant, almost-thermal, velocity dispersion. Eighteen coherent structures are revealed, twelve in L1688 and six in B18, each of which shows a sharp"transition to coherence"in velocity dispersion around its periphery. The identification of these structures provides a chance to study the coherent structures in molecular clouds statistically. The identified coherent structures have a typical radius of 0.04 pc and a typical mass of 0.4 Msun, generally smaller than previously known coherent cores identified by Goodman et al. (1998), Caselli et al. (2002), and Pineda et al. (2010). We call these structures"droplets."We find that unlike previously known coherent cores, these structures are not virially bound by self-gravity and are instead predominantly confined by ambient pressure. The droplets have density profiles shallower than a critical Bonnor-Ebert sphere, and they have a velocity (VLSR) distribution consistent with the dense gas motions traced by NH3 emission. These results point to a potential formation mechanism through pressure compression and turbulent processes in the dense gas. We present a comparison with magnetohydrodynamic simulations, and we speculate on the relationship of droplets with larger, gravitationally bound coherent cores, as well as on the role that droplets and other coherent structures play in the star formation process.


INTRODUCTION
In the early 1980s, NH 3 was identified as an excellent tracer of the cold, dense gas associated with highly extinguished compact regions. These regions were named "dense cores" by , and their properties were studied and documented in a series of papers throughout the 1980s and 1990s whose titles began with "Dense Cores in Dark Clouds" Myers 1983;Fuller & Myers 1992;Goodman et al. 1993;Benson et al. 1998;Caselli et al. 2002). Essentially since the start of that series, astronomers have used the "dense core" paradigm as a way to think about the small (0.1 pc; , prolate but roundish (aspect ratio near 2; Myers et al. 1991), quiescent (velocity dispersion nearly thermal ;Fuller & Myers 1992), blobs of gas that can form stars like the Sun. Whether these individual cores also exist in clusters where more massive stars form (Evans 1999;Garay & Lizano 1999;Tan et al. 2006;Li et al. 2015), how long-lived and/or transient these cores might be (Bertoldi & McKee 1992;Ballesteros-Paredes et al. 1999;Elmegreen 2000;Enoch et al. 2008), and how they relate to the ubiquitous filamentary structure inside star forming regions (McKee & Ostriker 2007;André et al. 2014;Padoan et al. 2014;Hacar et al. 2013;Tafalla & Hacar 2015) are still open questions. Nonetheless, a gravitationally collapsing "dense core" remains the central theme in discussions of star forming material.  made observations of NH 3 hyperfine line emission of four "dense cores" and found that the linewidths in the interior of a dense core are roughly constant at a value slightly higher than a purely thermal linewidth, and that the linewidths start to increase toward the edge of the maps of the dense cores. Using observations of OH and C 18 O line emission, Goodman et al. (1998) proposed that there exists a characteristic radius where the scaling law between the linewidth and the size changes, marking the "transition to coherence." Goodman et al. (1998) found that the characteristic radius is ∼ 0.1 pc and that within ∼ 0.1 pc from the center of a dense core the linewidth is virtually constant. This gave birth to the idea of the existence of "coherent cores" at the densest part of the "dense core" previously known and described by the "Dense Cores in Dark Clouds" series of papers. The coherent core, sitting at the end of the scaling law between the linewidth and the size, can provide the needed "calmness," or low turbulence, environment for further star formation dominated by gravitational collapse.
Using GBT observations of NH 3 hyperfine line emission, Pineda et al. (2010) made the first direct observation of a coherent core, resolving the transition to coherence across the boundary from a "Larson's Law"-like (turbulent) regime to a coherent (thermal) one. The observed coherent core sits in the B5 region in Perseus and has an elongated shape with a characteristic radius of ∼ 0.2 pc. The linewidths within the interior of the coherent core are almost constant and subsonic but not purely thermal. Later VLA observations by Pineda et al. (2011) of the interior of B5 show that there are finer structures inside the coherent core, and Pineda et al. (2015) found that these sub-structures are forming stars in a free-fall time of ∼ 40,000 years. The gravitationally collapsing sub-structures inside the coherent core are consistent with the picture of star formation within the "calmness" of the coherent core.
The coherent core in B5 has remained the only known case of a coherent structure where the transition to coherence is spatially resolved with a single tracer. In search of other coherent structures in nearby molecular clouds, we follow the same procedure adopted by Pineda et al. (2010) and identify a total of 18 coherent structures, 12 in the L1688 region in Ophiuchus and 6 in the B18 region in Perseus, using data from the Green Bank Ammonia Survey (GAS; Friesen et al. 2017). Although many of these structures may be associated with previously known cores or density features, this is the first time "transitions to coherence" are captured in a single tracer for these structures. The 18 coherent structures identified within a total projected area on the plane of the sky of ∼ 0.6 pc 2 suggest the ubiquity of coherent structures in nearby molecular clouds. The identification of these coherent structures allows statistical analyses of coherent structures for the first time.
In the analyses presented in this paper, we find that these newly identified coherent structures have small sizes, ∼ 0.04 pc, and masses, ∼ 0.4 M 1 . Unlike previously known coherent cores, the coherent structures identified in this paper are mostly gravitationally unbound and are instead predominantly bound by pressure provided by the ambient gas motions, in spite of the subsonic velocity dispersions found in these structures 2 . We term this newly discovered 1 The word "coherent" is used in this paper to indicate a structure with a spatially resolved change in velocity dispersion, from turbulencedominated values outside the structure to thermally dominated values inside the structure. Like many of the dense cores observed by Myers (1983), a coherent region thus has a thermally dominated velocity dispersion. The identification of these coherent structures are "new" in the sense that "transitions to coherence" are captured in a single tracer for the first time for many of these structures and that the identified coherent structures form a previously omitted population of gravitationally unbound and pressure confined coherent structures, as shown in the analyses below. We acknowledge that many of the coherent structures examined in this paper might be associated with previously known cores or density features. See Appendix C for discussion. 2 The adjectives "supersonic," "transonic," and "subsonic" in this paper indicate levels of turbulence. A supersonic/transonic/subsonic velocity dispersion has a turbulent (non-thermal) component larger than/comparable to/smaller than the sonic velocity. See Equation 3 below for a definition of the thermal and non-thermal components of velocity dispersion population of gravitationally unbound and pressure confined coherent structures "droplets" and examine their relation to the gravitationally bound and likely star forming coherent cores and other dense cores.
In this paper, we present a full description of the physical properties of the droplets and discuss the potential formation mechanism of the droplets. In §2, we describe the data used in this paper, including data from the GAS DR1 ( §2. 1;Friesen et al. 2017), maps of column density and dust temperature based on SED fitting of observations made by the Herschel Gould Belt Survey ( §2. 2;André et al. 2010), and the catalogues of previously known NH 3 cores ( §2. 3;Goodman et al. 1993;Pineda et al. 2010). In §3, we present our analysis of the droplets, including their identification ( §3.1), basic properties ( §3.2), and a virial analysis including an ambient gas pressure term ( §3.3). In the discussion, we further examine the nature of their pressure confinement in §4.1, by comparing the radial density and pressure profiles to the Bonnor-Ebert model ( §4.1.1) and the logotropic spheres ( §4. 1.2). We examine the relation between the droplets and the host molecular cloud by looking into the velocity distributions ( §4. 1.3). We then demonstrate that formation of droplets is possible in a magnetohydrodynamic (MHD) simulation and speculate on the formation mechanism of the droplets in §4.2, and we discuss their relation to coherent cores and their evolution in §4. 3. Lastly in §5, we summarize this work and outline future projects that might shed more light on how the droplets form, their relationship with structures at different size scales, and the role they might play in star formation. The Green Bank Ammonia Survey (GAS; Friesen et al. 2017) is a Large Program at the Green Bank Telescope (GBT) to map most Gould Belt star forming regions with A V ≥ 7 mag visible from the northern hemisphere in emission from NH 3 and other key molecules 3 . The data used in this work are from the first data release (DR1) of GAS that includes four nearby star forming regions: L1688 in Ophiuchus, B18 in Taurus, NGC1333 in Perseus, and Orion A.
To achieve better physical resolution, only the two closest regions in the GAS DR1 are used in our present study. L1688 in Ophiuchus sits at a distance of 137.3 ± 6 pc , and B18 in Taurus sits at a distance of 126.6 ± 1.7 pc (notice this is updated from the distance adopted by Friesen et al. 2017, which was taken from Schlafly et al. 2014;Galli et al. 2018). At these distances, the GBT FWHM beam size of 32 at 23 GHz corresponds to ∼ 4350 AU (0.02 pc). The GBT beam size at 23 GHz also matches well with the Herschel SPIRE 500 µm FWHM beam size of 36 (see §2.2 and discussions in Friesen et al. 2017). The GBT observations have a spectral resolution of 5.7 kHz, or ∼ 0.07 km s −1 at 23 GHz.

Fitting the NH3 Line Profile
In the GAS DR1, a (single) Gaussian line shape is assumed in fitting spectra of NH 3 (1,1) and (2,2) hyperfine line emission (see §3. 1 Friesen et al. 2017). The fitting is carried out using the "cold-ammonia" model and a forwardmodeling approach in the PySpecKit package (Ginsburg & Mirocha 2011), which was developed in Friesen et al. (2017) and built upon the results from Rosolowsky et al. (2008a) and Friesen et al. (2009) in the theoretical framework laid out by Mangum & Shirley (2015). No fitting of multiple velocity components or non-Gaussian profiles was attempted in GAS DR1, but the single-component fitting produced good quality results in 95% of detections in all regions included in the GAS DR1. From the fit, we obtain the velocity centroid of emission along each line of sight (Gaussian mean of the best fit) and the velocity dispersion (Gaussian σ), where we have sufficient signal-to-noise in NH 3 (1,1) emission. For lines of sight where we detect both NH 3 (1,1) and (2,2), the model described in Friesen et al. (2017) provides estimates of parameters including the kinetic temperature and the NH 3 column density. Figs. 1 to 4 show the parameters derived from the fitting of the NH 3 hyperfine line profiles.

Herschel Column Density Maps
The Herschel column density maps are derived from archival Herschel PACS 160 and SPIRE 250/350/500 µm observations of dust emission, observed as part of the Herschel Gould Belt Survey (HGBS André et al. 2010). We establish the zero point of emission at each wavelength using Planck observations of the same regions (Planck Collaboration et al. 2014). The emission maps are then convolved to match the SPIRE 500 µm beam FWHM of 36 and passed to a least squares fitting routine, where we assume that the emission at these wavelengths follow a modified blackbody emission function, I ν = (1 − exp −τν )B ν (T ), where B ν (T ) is the blackbody radiation, and τ is the frequency-dependent Figure 1. L1688 in Ophiuchus: Maps of (a) peak NH3 (1,1) brightness in the unit of main-beam temperature, T peak , and (b) kinetic temperature, T kin . The colored contours mark the boundaries of droplets, and the black contours mark the boundaries of droplet candidates. Because L1688-c1E and L1688-c1W overlap with L1688-d1, they are not shown in this figure or Fig. 2 (see §3.1). The stars mark the positions of Class 0/I and flat-spectrum protostars from Dunham et al. (2015). The scale bar at the bottom right corner corresponds to 0.5 pc at the distance of Ophiuchus. The black circle at the bottom left corner of each panel shows the beam FWHM of the GBT observations at 23 GHz. See §2.1.1 for details on the fitting of NH3 hyperfine lines, and see Appendix B for a gallery of the close-up views of the droplets.   Taurus, showing maps of (a) peak NH3 (1,1) brightness in the unit of mainbeam temperature, T peak , and (b) kinetic temperature, T kin . Here, the stars mark the positions of Class 0/I and flat-spectrum protostars with a reliability grade of A-or higher from Rebull et al. (2010). The scale bar at the bottom right corner corresponds to 0.5 pc at the distance of Taurus. opacity. The opacity can be written as a function of the mass column density, τ ν = κ ν Σ, where κ ν is the opacity coefficient. At these wavelengths, κ ν can be described by a power-law function of frequency, κ ν = κ ν0 ν ν0 β , where β is the emissivity index, and κ ν0 is the opacity coefficient at frequency ν 0 . Here we adopt κ ν0 of 0.1 cm 2 g −1 at ν 0 = 1000 GHz (Hildebrand 1983) and a fixed β of 1.62 (Planck Collaboration et al. 2014). The resulting I ν is a function of the temperature and the dust column density, the latter of which can be further converted to the total number column density by assuming a dust-to-gas ratio (100, for the maps we derive) and defining a mean molecular weight 4 (2.8 u; µ H2 in Kauffmann et al. 2008). The resulting column density map has an angular resolution of 36 (the SPIRE 500 µm beam FWHM), which matches well with the GBT beam FWHM at 23 GHz (32 ). In the following analyses, we do not apply convolution to further match the resolutions of the Herschel and GBT observations, before regridding the maps onto the same projection and gridding (Nyquist-sampled). Resulting maps column density and dust temperature are shown in Fig. 5 and Fig. 6 for L1688 in Ophiuchus and B18 in Taurus, respectively.

Source Catalogs
To understand droplets in context, we need compilations of the physical properties of previously identified dense cores. Goodman et al. (1993) (see §2.3.1) present a summary of cores from the observational surveys described in Benson & Myers (1989) and Ladd et al. (1994). The cores in Goodman et al. (1993) have low, nearly thermal velocity 4 In this paper, we use the mean molecular weight per H 2 molecule (2.8 u; µ H 2 in Kauffmann et al. 2008) in the calculation of the mass and other density related quantities, and we use the mean molecular weight per free particle (2.37 u;µp in Kauffmann et al. 2008) in the calculation of the velocity dispersion and pressure. Both numbers are derived assuming a hydrogen mass ratio of M H /M total ≈ 0.71, a helium mass ratio of M He /M total ≈ 0.27, and a metal mass ratio of M Z /M total ≈ 0.02 (Cox & Pilachowski 2000). See Appendix A.1 in Kauffmann et al. (2008). dispersions, and some of them are known to be "coherent" based on an apparent abrupt spatial transition from supersonic (in OH and C 18 O) to subsonic (in NH 3 ) velocity dispersion Caselli et al. 2002). We also include the coherent core in the B5 region in Perseus, as observed in NH 3 , the only coherent structure known before this work where the spatial change in linewidth is captured in a single tracer. Goodman et al. (1993) presented a survey of 43 sources with observations of NH 3 line emission (see Table 1 and Table 2 in Goodman et al. 1993; see also the SIMBAD object list), based on observations made by Benson & Myers (1989) and Ladd et al. (1994). The observations were carried out at the 37 m telescope of the Haystack Observatory and the 43 m telescope of the National Radio Astronomy Observatory (NRAO), resulting in a spatial resolution coarser than the modern GBT observations by a factor of ∼ 2.5. The velocity resolution of observations done by Benson & Myers (1989) and Ladd et al. (1994) ranges from 0.07 to 0.20 km s −1 . For comparison with the kinematic properties of the droplets measured using the GAS observations of NH 3 emission (Friesen et al. 2017), we adopt values that were also measured using observations of NH 3 hyperfine line emission, presented by Goodman et al. (1993). We correct the physical properties summarized in Goodman et al. (1993) with the modern measurement of the distance to each region. The updated distances are summarized in Appendix A.

Dense Cores Measured in NH3
The updated distances affect the physical properties listed in Table 1 in Goodman et al. (1998). The size scales with the distance, D, by a linear relation, R ∝ D. Since the mass was calculated from the number density derived from NH 3 hyperfine line fitting, it scales with the volume of the structure, and thus M ∝ D 3 . The updated distances also affect the velocity gradient and related quantities listed in Table 1 and Table 2 in Goodman et al. (1998), which we do not use for the analyses presented in this work.  Besides the updated distances, we combine the measurements of the kinetic temperature and the NH 3 linewidth, originally presented by Benson & Myers (1989) and Ladd et al. (1994), to derive the thermal and the non-thermal components of the velocity dispersion. See Equation 3 below for the definitions of the velocity dispersion components.
Among the 43 sources examined by Goodman et al. (1993), eight sources were later confirmed by Goodman et al. (1998) and/or Caselli et al. (2002 to be "coherent cores," using a combination of gas tracers of various critical densities (OH, C 18 O, NH 3 , and N 2 H + ). The interiors of these eight sources show signs of a uniform and nearly thermal distribution of velocity dispersion. However, unlike B5 and the newly identified coherent structures in this paper, the "transition to coherence" was not spatially resolved with a single tracer for these eight coherent cores. For the ease of discussion, we refer to the entire sample of 43 sources as the "dense cores," as they were originally referred to by Goodman et al. (1993). However, note that some of the 43 sources have masses and sizes up to ∼ 100 M and ∼ 1 pc, respectively. These larger-scale structures do not strictly fit in the definition of a dense core (with a small size and a nearly thermal velocity dispersion; see §4.3 for more discussions) and might be better categorized as "dense clumps" (as in McKee & Ostriker 2007).

Coherent Core in B5
Using GBT observations of NH 3 hyperfine line emission with a setup similar to GAS, Pineda et al. (2010) observed a coherent core in the B5 region in Perseus and spatially resolved the "transition to coherence"-NH 3 linewidths changing from supersonic values outside the core to subsonic values inside-for the first time. The coherent core sits in the eastern part of the molecular cloud in Perseus, at a distance of 315 ± 32 pc (the quantities measured by Pineda et al. 2010 assuming a distance of 250 pc are updated according to the new distance measurement; Schlafly et al. 2014). At 315 pc, the GBT resolution at 23 GHz corresponds to a spatial resolution of ∼ 0.05 pc. The coherent core has an elongated shape, with a size of ∼ 0.2 pc. Pineda et al. (2010) identified the coherent core in B5 as a peak in NH 3 brightness surrounded by an abrupt change in NH 3 velocity dispersion (∼ 4 km s −1 pc −1 ). In the following analysis, we search the new GAS data for coherent structures reminiscent of the B5 core, looking for abrupt drops in NH 3 linewidth to nearly thermal values around local concentrations of dense gas traced by NH 3 (see §3.1 for details). Below in the comparison between B5 and the newly identified coherent structures, we consistently follow the same methods adopted by Pineda et al. (2010) to derive the basic physical properties using GBT observations of NH 3 hyperfine line emission and Herschel column density maps derived from SED fitting ( §2.2; see also §3.2 for details on the measurements of the physical properties).

Identification of the Droplets
In this work, we look for coherent structures defined by abrupt drops in NH 3 linewidth and an interior with uniform, nearly thermal velocity dispersion 5 , reminiscent of previously known coherent cores examined by Goodman et al. (1998), Caselli et al. (2002, and Pineda et al. (2010). We identify the coherent structures using data from the Green Bank Ammonia Survey (see §2. 1 Friesen et al. 2017) and the Herschel maps of column density and dust temperature derived in §2.2, to enable a statistical analysis of coherent structures in two of the closest molecular clouds, Ophiuchus and Taurus.
By eye, one can already recognize many small plateaus of subsonic velocity dispersion associated with NH 3 -bright structures throughout L1688 and B18 in the maps of observed velocity dispersion (σ NH3 ) and NH 3 brightness. To identify these coherent structures quantitatively, we follow the procedure adopted by Pineda et al. (2010) to identify the coherent core region in B5. Consistent with the criteria used by Pineda et al. (2010), the set of criteria we use in this work to define the boundaries of coherent structures starts with the transition in velocity dispersion, σ NH3 , from supersonic to subsonic value, and continues with the spatial distribution of NH 3 brightness, T peak , and the velocity centroid, V LSR . A set of quantitative prescriptions for defining the boundary of a coherent structure is given below as a step-by-step procedure: 1. We start with the intersection of areas enclosed by two contours: one of the NH 3 velocity dispersion and one of the NH 3 brightness. First, we find the contour where the NH 3 velocity dispersion (σ NH3 ) has a non-thermal component equal to the thermal component at the median kinetic temperature measured in the targeted region. (See §3.2 and Equation 2 for details on the definition of velocity dispersion components.) Second, we select the contour that corresponds to the 10-σ level, where the NH 3 brightness (T peak ) is equal to 10 times the local rms noise, to match the extents of the contiguous regions where successful fits to the NH 3 (1, 1) profiles were found in Friesen et al. (2017). The intersection of the areas enclosed by these two contours is then used to define an initial mask. By this definition, the initial mask encloses a region where we have subsonic velocity dispersion and a signal-to-noise ratio larger than 10.
2. We expect the pixels within the mask defined in Step 1 to have a continuous distribution of velocity centroids (V LSR ). In this step, we remove pixels with V LSR that leads to local velocity gradients (between the targeted pixel and its neighboring pixels within the mask) larger than the overall velocity gradient found for all pixels within the mask by a factor of ∼ 2. This procedure generally removes pixels with local velocity gradients greater than 20 to 30 km s −1 pc −1 , which is larger than the velocity gradients known to exist because of realistic physical processes in these regions. The mask editing is done with the aid of Glue 6 .
3. We then check whether the mask from Step 2 contains a single local peak in NH 3 brightness. If there are more than one NH 3 brightness peaks, we find the contour level that corresponds to the saddle point between the peaks. This contour level is then used to separate the mask from Step 2 into regions, each of which has a single NH 3 brightness peak. However, if a region has an NH 3 brightness peak no more than 3 times the local rms noise level above the saddle point, the region is excluded, and only its sibling region with the brighter peak is kept. We examine and categorize the regions excluded in this step as candidates (see below).
4. The Herschel maps of column density and dust temperature are then used to make sure that the defined structure (a region from Step 3) is centered around a local rise in column density and a dip in dust temperature, consistent with the expectation of dense cores (Crapsi et al. 2007).
5. Lastly, we make sure that the resulting structure is resolved by the GBT beam at 23 GHz (32 ). We impose two criteria: 1) the projected area needs to be larger than a beam, and 2) the effective radius (the geometric mean of the major and minor axes; see Equation 1) needs to be larger than the beam FWHM.
Using these criteria, we identify 12 coherent structures in L1688 and 6 coherent structures in B18. In Figs. 1 to 6, the boundaries of the identified coherent structures in L1688 and B18 are shown as colored contours. Although the criteria are consistent with those used by Pineda et al. (2010) to define the coherent core in B5 and do not impose any limits on size, the newly identified coherent structures in L1688 and B18 are generally smaller than previously known coherent cores (see §3.2). As mentioned in §1, we refer to the newly identified coherent structures as "droplets" for ease of discussion.
As the criteria indicate, each droplet has a high NH 3 peak brightness and a subsonic velocity dispersion, in contrast to the ambient region, where if NH 3 emission is detected, we find a mostly supersonic velocity dispersion and a moderate distribution of NH 3 brightness. Fig. 7 shows the distributions of NH 3 linewidths and peak NH 3 brightness in main-beam units, for all pixels where there is significant detection of NH 3 emission and for pixels within the droplet boundaries (see Friesen et al. 2017, for criteria used to determine the significance of detection). We observe an overall anti-correlation between the observed NH 3 linewidth and NH 3 brightness, and the relation between the two quantities flattens toward the high NH 3 brightness end when the NH 3 linewidth approaches a thermally dominated value. The droplets are found in this regime of high NH 3 brightness and thermally dominated NH 3 linewidths. Figure 7. Distributions of NH3 linewidths and peak NH3 brightness in main-beam units, for every pixel with significant detection of NH3 (1, 1) emission (a) in L1688 and (b) in B18. The 2D histogram in each panel shows the distribution of pixels in the entire map, with the pixel frequency defined as the percentage of pixels on the map falling in each 2D bin in the 2D histogram. The colored dots are individual pixels inside droplets, with colors matching the contours in Figs. 1, 2, and 5 for L1688, and Figs. 3, 4, and 6 for B18. The horizontal lines are the expected NH3 linewidths when the non-thermal component of velocity dispersion is respectively equal to the sonic speed (thicker line) and half the sonic speed (thinner line), for a medium with an average particle mass of 2.37 u and a temperature of 10 K. Fig. 8 shows the radial profile of NH 3 velocity dispersion; the virtually constant NH 3 velocity dispersion in the interiors is consistent with what Goodman et al. (1998) found for coherent cores (see also Pineda et al. 2010). See Appendix B for a gallery of the close-up views of the droplets.
Two of the 18 droplets, L1688-d11 and B18-d4, are found at the positions of the dense cores analyzed by Goodman et al. (1993), L1696A and TMC-2A, respectively. The two droplets correspond to the central parts of corresponding Figure 8. The NH3 velocity dispersion as a function of distance from the center of each droplet. The dark green dots represent individual pixels inside the boundary of each droplet, as defined in §3.1. The transparent green band shows the 1-σ distribution of pixels in each distance bin, with a bin size equal to the beam FWHM of GAS observations. The dashed and dotted lines show the expected NH3 linewidths when the velocity dispersion non-thermal component is equal to the sonic speed and half the sonic speed, respectively. The vertical black line marks the effective radius, R eff , assigned to each droplet in §3.2 (Equation 1), and the gray vertical band marks the uncertainty in R eff , which takes into account both the resolution and the non-circular shape of the droplet boundary (see Appendix D for details). A red asterisk indicates that the droplet has an elongated shape with an aspect ratio larger than 2 that could bias the measurements using equidistant annuli (L1688-d1, L1688-d6, and B18-d5), and a blue asterisk indicates that the droplet sits near the edge of the region where NH3 emission is detected, resulting in the measurements at larger radii being dominated by fewer pixels (L1688-d2 and L1688-d5). dense cores and have radii a factor of ∼ 0.7 times the radii measured for these dense cores (Benson & Myers 1989;Goodman et al. 1993;Ladd et al. 1994). See Appendix C for a comparison of measured properties. Table 1 is a list of the positions and basic physical properties (see §3.2 below for details) of the droplets and the droplet candidates (see §3.1.1 below) identified in this work. In the following analyses, when we talk about the properties of the droplets or, together with previously known coherent cores, the coherent structures, we exclude the droplet candidates. The droplet candidates are included on the plots to show the distributions of physical properties of potential coherent structures at even smaller scales, which are only marginally resolved by the GAS observations. In Figs. 1 to 6, we also plot the positions of Class 0/I and flat spectrum protostars in the catalogues presented by Dunham et al. (2015) and Rebull et al. (2010), for L1688 and B18, respectively. Within the boundaries of six (out of 18) droplets-L1688-d4, L1688-d6, L1688-d7, L1688-d8, L1688-d10, and B18-d6, we find at least one protostar along the line of sight. However, this does not necessarily indicate actual associations of these six droplets with protostars. Among the six droplets where we find protostar(s) within the boundaries, none of them shows a strong signature of increased T kin or σ NH3 around the protostar(s). We also note that both L1688 and B18 are active star forming regions, and it is possible that the coincident positions of droplets and YSOs are results of larger numbers of YSOs in these regions, seen in projection. See below in §4.3 for more discussion on the association between cores and YSOs and how it might be used as a way to define subsets of cores.

Droplet Candidates
Besides the total of 18 droplets identified in L1688 and B18, we also include 5 droplet candidates in L1688 (black contours in Fig. 1, 2, and 5). Each droplet candidate is identified by a spatial change from supersonic velocity dispersion outside the boundary to subsonic velocity dispersion inside. However, they do not meet at least one criterion listed above. The detailed reasons why each of these coherent structures is identified as a droplet candidate, instead of a droplet, are listed below: 1. L1688-c1E and L1688-c1W: These two droplet candidates are the eastern and western parts of the droplet L1688-d1, each of which has a local peak in NH 3 brightness. However, neither peak is more than 3 times the local rms noise level above the saddle point between them, i.e., neither satisfies the criterion described in Step 3. Thus, we identify the entire region as a single droplet, L1688-d1, and include the eastern and the western parts of L1688-d1 as two droplet candidates.
2. L1688-c2: This droplet candidate shows a local dip in NH 3 velocity dispersion and a local peak in NH 3 brightness. However, the local peak in NH 3 brightness cannot be separated from the emission in the droplet L1688-d3 by more than 3 times the local rms noise in NH 3 (1, 1) observations. Nor do we find an independent local peak corresponding to L1688-c2 on the Herschel column density map. (That is, L1688-c2 does not meet the criteria described in Steps 3 and 4 above.) 3. L1688-c3: Similar to L1688-c2, L1688-c3 shows a local dip in NH 3 velocity dispersion and a local peak in NH 3 brightness. However, the local peak in NH 3 brightness cannot be separated from the emission in the droplet L1688-d4 by more than 3 times the local rms noise in NH 3 (1, 1) observations. Nor do we find an independent local peak corresponding to L1688-c3 on the Herschel column density map. While the projected area of L1688-c3 is larger than a beam, its effective radius is only ∼2.6 times the beam FWHM. (That is, L1688-c3 does not meet the criteria described in Steps 3, 4, and 5 above.) 4. L1688-c4: While L1688-c4 does show a significant dip in NH 3 velocity dispersion and an independent peak in NH 3 brightness, it sits close to the edge of the region where we have enough signal-to-noise of NH 3 (1, 1) emission to obtain a confident fit to the hyperfine line profile (Friesen et al. 2017). We do not find a strong and independent local peak corresponding to L1688-c4 on the Herschel column density map, either. Thus, we classify L1688-c4 as a droplet candidate. (That is, L1688-c4 does not meet the criterion described in Step 4 above.) The Oph A region (marked by the red rectangles in Fig. 1, 2, and 5) could potentially host more droplets/droplet candidates. However, Oph A is known to also host a cluster of young stellar objects (YSOs), and as Fig. 2b and 5b show, the extent of cold and subsonic dense gas identifiable on the maps of dust temperature and NH 3 velocity dispersion is limited. No coherent structure that satisfies the above criteria can be identified. f A value of "Y" means that there is at least one YSO within the droplet boundary defined on the plane of the sky (see §3.1), and a value of "N" means that there is no YSO within the droplet boundary. The YSO positions are taken from the catalogue presented by Rebull et al. (2010) for B18 and the catalogue presented by Dunham et al. (2015) for L1688. Since we are interested in the association between cores/droplets and the YSOs potentially forming inside, only Class 0/I and flat spectrum protostars are considered here. g The eastern part of L1688-d1. h The western part of L1688-d1.

Contrast with Velocity Coherent Filaments
We note that Hacar et al. (2013) and Tafalla & Hacar (2015) used the term "coherent" to describe continuous structures in the position-position-velocity space, with continuous distributions of line-of-sight velocity (V LSR ). The method they adopted is a friend-of-friend clustering algorithm and does not impose any criteria on the velocity dispersion. Since in Step 2, we require a coherent structure to have a continuous distribution of V LSR , the newly identified coherent structures could theoretically be parts of "velocity coherent filaments," but the same can be said of any structures that are identified to have continuous structures on the plane of the sky and continuous distributions of line-of-sight velocity. We do not recommend equating the coherent structures, including the newly identified droplets in this work and the coherent cores previously analyzed by Goodman et al. (1998), Caselli et al. (2002), and Pineda et al. (2010), to "velocity coherent filaments" identified by Hacar et al. (2013). Specifically, the droplets and other coherent structures are defined by abrupt drops in velocity dispersion from supersonic to subsonic values around their boundaries, which none of the "velocity coherent filaments" examined by Hacar et al. (2013) show. Moreover, in contrast to the elongated shapes of the "velocity coherent filaments" examined by Hacar et al. (2013), the droplets are mostly round, with aspect ratios generally between 1 and 2 (with the exceptions of L1688-d1 with an aspect ratio of ∼ 2.50, L1688-d6 with an aspect ratio of ∼ 2.52, and B18-d5 with an aspect ratio of ∼ 2.03; these exceptions are marked with red asterisks on Fig. 8).

Mass, Size, and Velocity Dispersion
With the droplet boundary defined in §3.1, we calculate the mass of each droplet using the column density map derived from SED fitting of Herschel observations (see §2.2). To remove the contribution of line-of-sight material, the minimum column density within the droplet boundary is used as a baseline and subtracted off. The mass is then estimated by summing column density (after baseline subtraction) within the droplet boundary. Fig. 9 shows how subtraction of the baseline can be a reasonable way to remove the contribution from line-of-sight material outside the targeted structure. This baseline subtraction method is similar to the "clipping paradigm" studied by Rosolowsky et al. (2008b), and has been applied by Pineda et al. (2015) to estimate the mass of structures within the coherent core in B5. The clipping method produces a mass estimate that would correspond better to the mass calculated from fitting the NH 3 emission, which traces only the material within the targeted structure and was used by Goodman et al. (1993) to estimate the masses of the dense cores (see §2.3.1). In an ideal scenario where the structure is a spherical structure in the 3D space, the implementation of the clipping method also allows analyses of physical properties at different layers within the structure without having to use numbers calculated for unrealistic cylindrical volumes (i.e., annuli integrated over the lines of sight; see analyses in §3.3.3 and §4.1). For the droplets, we find a typical mass 7 of 0.4 +0.4 −0.3 M . Table 1 lists the mass of each droplet. In Appendix E, we discuss the reasons for adopting the clipping method and the uncertainty therein, and in Appendix F, we examine the uncertainty in mass measurements due to the potential bias in SED fitting.
We define the radius of each droplet based on the geometric mean of NH 3 brightness weighted second moments along the major and minor axes. We designate the major axis direction as the one with the greatest dispersion in T peak according to a principal component analysis (PCA), and the minor axis is oriented perpendicular to the major axis 8 . We calculate the NH 3 brightness weighted second moments along the major and the minor axes, σ maj and σ min , from which we derive the sizes along the major and minor axes, r maj and r min , by multiplying the second moment by a scaling factor, 2 √ 2 ln 2, which is the scaling factor between the second moment and the full width at half maximum (FWHM) for a Gaussian shape. That is, The effective radius is then the geometric mean of r maj and r min , R eff = √ r maj r min . The multiplication of the scaling factor, 2 √ 2 ln 2, is done in the same way as the method applied by Benson & Myers (1989) and Goodman Figure 9. This cartoon shows the corresponding "layers" of material along a cut on the 2D column density map (left; with the vertical axis corresponding to the column density) and in a top-down view in the 3D space (right; with the line of sight along the vertical axis). The solid shaded area (in black/dark gray/gray) corresponds to materials inside a schematic spherical "droplet," while the hatched area corresponds to the material outside the droplet. On the right hand side, the dashed line marks the boundary of the droplet in the 3D space. On the left hand side, the dashed line shows how a constant column density baseline, corresponding to the minimum value of the solid shaded area, can be a reasonable estimate of the contribution from material outside the droplet (hatched area). The baseline subtraction method is similar to the "clipping paradigm" analyzed by Rosolowsky et al. (2008b) and applied to mass measurements of sub-0.1 pc density features by Pineda et al. (2015).
et al. (1993) to estimate the radii of dense cores and is applied to approximate the "true radius" of the droplet. Fig. 8 shows that the effective radius, R eff , plotted on top of the radial profile of velocity dispersion, σ NH3 , of each droplet, well characterizes the change from supersonic to subsonic velocity dispersion. In Appendix B, we show that a circle on the plane of the sky with a radius equal to R eff is also a good approximation of the observed shape of a droplet. The resulting effective radii of droplets are listed in Table 1 and have a typical value of 0.04 ± 0.01 pc. The effects of the resolution and the irregular shape of the boundary are included in the uncertainties listed in Table 1. See Appendix D for details on estimating the uncertainty and for a discussion on another common way to derive the "effective radius," using the radius of the circle that has an area equal to the projected area of the core/structure. From the GAS observations, we derive the NH 3 velocity dispersion, σ NH3 , and the gas kinetic temperature, T kin (Figs. 1 to 4; see §2.1.1 for details). Assuming that the NH 3 linewidth is a result of the thermal and non-thermal motions of the NH 3 particles along the line of sight, we can estimate the non-thermal component, σ NT : where σ T,NH 3 is the thermal component of the NH 3 velocity dispersion, which is a function of the kinetic temperature, T kin , the Boltzmann constant, k B , and the molecular weight of the species under discussion-in this case NH 3 , m NH3 . Assuming that the bulk molecular component is in thermal equilibrium with the NH 3 component and has the same kinetic temperature and assuming also that the non-thermal component of the velocity dispersion is independent of the chemical species observed, we can estimate a total velocity dispersion, σ tot , from the thermal component, σ T , and the non-thermal (turbulent) component, σ NT : where the thermal component is a function of the kinetic temperature, T kin : where m ave is the mean molecular weight in molecular clouds. By definition, the thermal component, σ T , is equal to the sonic speed, c s , in a medium with a particle mass of m ave at a temperature of T kin . Following Kauffmann et al. (2008), we use the mean molecular weight per free particle of 2.37 u (µ p in Kauffmann et al. 2008 For each droplet, we obtain characteristic values of the NH 3 velocity dispersion, σ NH3 , and the kinetic temperature, T kin , by taking the median value for the pixels within the droplet boundary on the parameter maps. Then, following Equation 2 to Equation 5, we estimate the characteristic values of σ NT , σ T , and σ tot , for each droplet. Note that σ tot is sometimes referred to as the "1D velocity dispersion," concerning the motions along the line of sight, as opposed to the "3D velocity dispersion," which cannot be observed but can be estimated by multiplying the 1D velocity dispersion by a factor of √ 3 assuming isotropy. We find a typical value of σ tot of 0.22 ± 0.02 km s −1 for the droplets (see Table  1). For reference, the purely thermal velocity dispersion at 10 K is 0.19 km s −1 . Fig. 10 shows the distributions of mass, M , and total velocity dispersion, σ tot , plotted against the effective radius, R eff , of droplets/droplet candidates in comparison with previously known coherent cores as well as other dense cores (see §2.3 for details on how the physical properties were estimated for the dense cores). Fig. 10a shows that droplets seem to fall along the same mass-radius relation as the dense/coherent cores. Using a gradient-based MCMC sampler to find a power-law relation between the mass and effective radius, M ∝ R p eff , for all the previously known dense/coherent cores (including B5) and the droplets (excluding droplet candidates), we find a power-law index, p = 2.4 ± 0.1 9 . This exponent lies between those expected for structures with constant surface density, M ∝ R 2 , and structures with constant volume density, M ∝ R 3 . As a reference, Larson (1981) found a scaling law, M ∝ R 1.9 , for larger-scale molecular structures (with sizes of 0.1 to 100 pc and masses of 1 M to 3×10 5 M ), using a compilation of observations of molecular line emission from species including 12 CO, 13 CO, H 2 CO, and for a few objects, NH 3 and other N-bearing species. Fig. 10b shows the relationship between σ tot and R eff . At scales below 0.1 pc, all structures shown have a subsonic velocity dispersion. The continuity of the distribution of M , R eff , and σ tot between the newly identified coherent structures-droplets-and the previously known coherent cores as well as other dense cores suggests that the identification of droplets is robust, and that droplets fall toward the small-size end of a potentially continuous population of coherent structures across different size scales. We discuss this continuity in details in §4.3.

Virial Analysis: Kinetic Support, Self-Gravity, and Ambient Gas Pressure
To investigate the stability of the coherent structures, we follow Pattle et al. (2015) to consider the balance between internal kinetic energy, self-gravity, and the ambient gas pressure, with respect to the equilibrium expression: where Ω K is the internal kinetic energy; Ω G is the gravitational potential energy; and Ω P is the energy term representing the confinement provided by the ambient gas pressure acting on the structure. The "external pressure" comes from thermal and non-thermal (turbulent) motions of the ambient gas (see the analysis below in §3.3.3). Since we do not have the observations needed to estimate magnetic energy, the magnetic energy term, Ω M , is omitted (compared to Equation 27 in Pattle et al. 2015). Here we focus on pressure exerted on a structure by thermal and non-thermal (turbulent) motions of the ambient gas for Ω P , and we ignore any contribution of ionizing photons to pressure (see discussions in Ward- Thompson et al. 2006;Pattle et al. 2015). In the case where the left-hand side of Equation 6 is larger than the right-hand side, 2Ω K > −(Ω G + Ω P ), a structure would be unbound; while in the case where the left-hand side of Equation 6 is smaller than the right-hand side, 2Ω K < −(Ω G + Ω P ), a structure would be bound. Figure 10. (a) The mass, M , plotted against the effective radius, R eff , for dense cores (green circles), the coherent core in B5 (a green circle marked with a black edge), and the newly identified coherent structures: droplets (filled blue circles) and droplet candidates (empty blue circles). The black line shows a power-law relation between the mass and the effective radius, found for both the dense cores (including B5) and the droplets (excluding droplet candidates) by a gradient-based MCMC sampler. Randomly selected 10% of the accepted parameters in the MCMC chain are plotted as transparent lines for reference. The solid gray line shows the empirical relation based on observations of larger-scale structures examined by Larson (1981).
The total velocity dispersion, σtot, plotted against the effective radius, R eff , for the same structures as in (a). The horizontal lines show σtot expected for structures where the non-thermal component is equal to the sonic speed (cs; thicker line) and half the sonic speed (thinner line) of a medium with an mean molecular weight of 2.37 u at a temperature of 10 K. The gray line shows an empirical relation adopted from Larson (1981). Notice that the original relation between the velocity dispersion and the size presented by Larson (1981) was for linewidths observed in various molecular lines (mostly from 12 CO and 13 CO). Here we convert the linewidth in the relation presented by Larson (1981) to σtot by assuming that the linewidth was measured from the CO (1-0) line emission with a gas temperature of 10 K. See §3.2 for details on how σtot is derived from the NH3 velocity dispersion and the kinetic temperature.

Internal Kinetic Energy, ΩK
The internal kinetic energy, Ω K , is given by: where M is the mass and σ tot is the total velocity dispersion, estimated from the observed NH 3 velocity dispersion, σ NH3 , and gas kinetic temperature, T kin , following Equation 5 (see §3.2 for details). The factor of 3 stands for the correction applied to the "1D velocity dispersion," σ tot , to obtain an estimate of the 3D velocity dispersion, assuming isotropy (see §3.2). For droplets, we measure a typical kinetic energy of 4.5 +5.8 −2.8 × 10 41 erg. Table 2 gives results for each droplet.

Gravitational Potential Energy, ΩG
Assuming spherical geometry, gravitational potential energy, Ω G , can be estimated from total mass and an effective radius; we adopt a gravitational potential energy expression: where we assume that the sphere of material has a uniform density distribution. In comparison, a sphere of material with a power-law density distribution, ρ ∝ r −2 , has an absolute value of gravitational potential energy, |Ω G |, a factor of ∼ 1.7 larger than that expressed in Equation 8, and a sphere with a Gaussian density distribution has |Ω G | a factor of ∼ 2 smaller than that expressed in Equation 8 (Pattle et al. 2015;Kirk et al. 2017a). In the following analysis, we  f The eastern part of L1688-d1.
g The western part of L1688-d1.
include the deviation in Ω G due to different assumptions of density distributions in the estimated errors. In §4.1.1, we show that the density distributions in droplets are nearly uniform at small radii with relatively shallow drops toward the outer edges, validating the assumption of a uniform density distribution used to derive Equation 8.
For droplets, we measure a typical gravitational potential energy of 1.3 +5.0 −1.1 × 10 41 erg (absolute value; see Table  2). Before a full analysis on the virial equilibrium expressed by Equation 6, we first examine the relation between gravitational potential energy, Ω G , and the internal kinetic energy, Ω K . Fig. 11a shows that most of the dense cores, including previously known coherent cores such as the one in B5, are close to an equilibrium between the gravitational potential energy and the internal kinetic energy. This indicates that the self-gravity of these coherent cores is substantial and may provide the binding force needed to keep the cores from dispersing. On the other hand, gravity in the newly identified droplets appears to be less dominant compared to the internal kinetic energy. For most of the droplets, the internal kinetic energy is close to an order of magnitude larger than the gravitational potential energy.
That larger structures have more dominant gravitational potential energies than smaller structures is expected for structures with a nearly flat σ tot -size relation and a steep mass-size relation (Fig. 10). For the coherent structures under discussion, we observe a power-law mass-size relation, M ∝ R 2.4 eff , and with a constant σ tot , we would expect a power-law relation between the gravitational potential energy and the size, |Ω G | ∝ R 3.8 eff , and a power-law relation between the internal kinetic energy and the size, Ω K ∝ R 2.4 eff . Consequently, a smaller coherent structure would have a smaller ratio between the gravitational potential energy and the internal kinetic energy, |Ω G | /Ω K . For reference, structures with a constant |Ω G | /Ω K are expected to have a mass-size relation of M ∝ R eff .
The above comparison between the gravitational potential energy and the internal kinetic energy is analogous to an analysis of stability using a virial parameter, conventionally defined as: where the leading factor, a, varies according to the assumption of the density distribution (e.g., a = 5 for a spherical structure with a uniform density, and a = 3 for a spherical structure with a power-law density profile with an index of 2, ρ ∝ r −2 ; see Bertoldi & McKee 1992). The virial parameter is a measure of the ratio between the mass which the internal gas motions can support against self-gravity and the observed mass, M , in the absence of ambient turbulent pressure. Conventionally, structures with α vir ≤ 2 would be considered "gravitationally bound." This corresponds to the parameter space above the black line in Fig. 11a (marked with α vir = 2). By this measure, only the most massive droplets (with masses on the order of 1 M ) along with most of the dense cores are "gravitationally bound." Figure 11. (a) Gravitational potential energy, ΩG, plotted against internal kinetic energy, ΩK (Equation 6), for dense cores (green circles), the coherent core in B5 (a green circle marked with a black edge), and the newly identified coherent structures: droplets (filled blue circles) and droplet candidates (empty blue circles). The red band from the lower left to the top right marks the equilibrium between ΩG and ΩK (solid red line) within an order of magnitude (pink band), according to the virial equation (Equation 6; omitting the pressure term). Structures in the parameter space above the red line (equilibrium) are expected to be dominated by self-gravity. The black line marks where the conventional virial parameter, αvir, has a value of 2 (Equation 9). αvir ≤ 2 (above the black line) is conventionally used to indicate that a structure is "gravitationally bound" (see §3.3.2).
(b) The energy term representing the confinement provided by the ambient gas pressure, ΩP, plotted agains the internal kinetic energy, ΩK (Equation 6), for the same structures shown in (a). Similarly, the red band from the lower left to the top right marks an equilibrium between ΩP and ΩK (solid red line) within an order of magnitude (pink red band), according to the virial equation (Equation 6; omitting the gravitational term). Structures in the parameter space above the red line (equilibrium) are expected to be dominated by the ambient gas pressure.

Energy Term Representing Ambient Pressure Confinement, ΩP
The pressure term, Ω P , in the virial equation (Equation 6) is characteristic of the pressure exerted on a structure by thermal and non-thermal (turbulent) motions of the ambient gas. To avoid the impression that there is a clear-cut boundary between the interior and the exterior of the targeted structure, we call the pressure provided by the ambient gas motions the "ambient gas pressure," P amb , which is sometimes called the "external pressure" and denoted by P ext in previous works (Ward-Thompson et al. 2007;Pattle et al. 2015;Kirk et al. 2017a).
For a spherical structure with a radius of R eff , the pressure term is given by: where P amb is the ambient gas pressure, and V is the volume of the structure under discussion (Ward- Thompson et al. 2006;Pattle et al. 2015). The pressure exerted on the structure can be estimated from: where ρ amb is the volume density of the ambient gas, and σ tot,amb is the total velocity dispersion, including both thermal and non-thermal motions of the ambient gas (same as σ tot defined in Equation 5 for the gas in the core). The leading factor of 3 in Equation 10 is applied to estimate the effects of gas motions in the 3D space, since for σ tot,amb , we use the "1D (line-of-sight) velocity dispersion" measured from observations. See the discussion in §3.2.
We base our calculation of the pressure, P amb , on the maps of σ NH3 and T kin from fitting the NH 3 hyperfine line profiles (for estimating σ tot,amb ; Figs. 1 to 4) and the Herschel column density maps (for estimating ρ amb ; Fig. 5 and Fig. 6). The former is possible, because there is significant detection of NH 3 (1, 1) emission in regions surrounding the droplets and the coherent core in B5, which appear embedded in the dense gas components of the clouds (see Fig. 2 and Fig. 4). We use the region (on the plane of the sky) immediately outside the targeted structure but within (R eff + 0.1) pc from the center of the structure to obtain an estimate of the ambient gas pressure. Since the typical sonic scale in nearby molecular clouds is roughly 0.1 pc (Federrath 2013), the hope is that the selected region represents the projection of the volume within a sonic scale from the surface of the structure and that the estimated pressure is from the motions of the gas relevant in confining the structure. The volume density of the ambient gas is estimated in the same fashion as demonstrated above in §3.2 and Fig. 9, by taking the difference between the mass measured within the core boundary and the mass measured within (R eff + 0.1) pc from the core center, ∆M = M (r < (R eff + 0.1 pc)) − M core , and dividing it by the difference in volume assuming a spherical geometry, ∆V = 4 3 π((R eff + 0.1 pc) 3 − R 3 eff ). Note that the density estimated this way would correspond to the average density within a shell-shaped volume surrounding the structure if the structure is spherical (see discussions in Appendix E). The total velocity dispersion of the ambient gas, σ tot,amb , is estimated by taking the median value of σ tot measured at pixels within the same projected region (outside the core, but within (R eff + 0.1) pc from the core center). For cores where we do not have significant detection toward every pixel within this projected region, we estimate an uncertainty up to 50%. We emphasize that the measurement of the ambient gas pressure (and thus the energy term representing the ambient gas pressure, Ω P , as expressed in Equation 10) using this method is independent of the measurement of the kinetics within the core (including σ tot , and thus the internal kinetic energy, Ω K ), since non-overlapping projected regions are used for the measurements. We also note that, in contrast to previous works, it is possible to measure the local variation in ambient gas pressure through this method with the GAS observations (Friesen et al. 2017, see also discussions in Kirk et al. 2017a).
Plugging the measured ρ amb and σ tot,amb in Equation 11, we get a typical value of P amb /k B ≈ 2.7 +4.7 −1.8 × 10 5 K cm −3 for the droplets (see Table 2 for the result of each droplet) and P amb /k B ≈ 1.2 × 10 5 K cm −3 for the coherent core in B5. Following Equation 10, we then estimate the virial energy term corresponding to the ambient pressure confinement of the droplets to be |Ω P | ≈ 6.8 +3.0 −6.3 × 10 41 erg and that of the coherent core in B5 to be |Ω P | ≈ 6.3 × 10 43 erg. See Table 2 for the estimated P amb /k B and Ω P of each droplet.
Since the 1980s, there have been efforts to find predominantly pressure confined structures and to estimate the magnitude of such pressure confinement. The earlier works focused on estimating the magnitude of "inter-clump" pressure based on models of pressure-confined clumps (Keto & Myers 1986;Bertoldi & McKee 1992). These models of pressure-confined clumps often presumed an equilibrium between the internal kinetic energy, the gravitational potential energy, and the energy terms representing pressure confinement through various physical processes. For example, using observations of molecular line emission and extinction to estimate the kinetic energy and the gravitational potential energy of dense clumps, Keto & Myers (1986) estimated that an inter-clump pressure, P/k B , between 10 3.5 and 10 4.5 K cm −3 was needed to keep the dense clumps at virial equilibrium. In a similar fashion, Bertoldi & McKee (1992) estimated that the "molecular cloud pressure" acting on the dense clumps within the molecular cloud ranged from 1.2 × 10 4 K cm −3 in Cepheus to 1.1 × 10 5 K cm −3 in Ophiuchus, in both cases balancing the observed internal pressure. Because of the relatively coarse resolution available at that time, these works focused on clumps with sizes between ∼ 0.5 to 1.0 pc.
At smaller size scales, work has been done to estimate the core confining pressure using direct observations of velocity dispersion in the host molecular clouds (see an incomplete summary in Table 3; for example, Johnstone et al. 2000;Lada et al. 2008;Maruta et al. 2010;Kirk et al. 2017a). In these works, observations of molecular line emission were devised to estimate the velocity dispersion. Then, by assuming that the molecular line emission traces a certain (range of) density, the pressure was estimated by equations similar to Equation 11. While these works found a large range of gas pressure from P amb /k B ≈ 5 × 10 4 K cm −3 to 2 × 10 7 K cm −3 for structures with sizes from 0.006 to 0.26 pc, they similarly concluded that a substantial portion of targeted structures was pressure confined. However, these works were limited by the lack of observations suitable for estimating the variation in the confining pressure from structure to structure.
Notably, previous analyses done by Pattle et al. (2015) of structures in Ophiuchus with sizes slightly smaller than the droplets gave an estimate of the ambient pressure two orders of magnitude larger than that estimated for the droplets. However, Pattle et al. (2015) found |Ω P | ≈ 9 × 10 41 erg for the same structures, which was comparable to the typical value found for the droplets, |Ω P | ≈ 7.6 × 10 41 erg. This is because the estimation of the virial energy term, Ω P , representing the confinement provided by the ambient gas pressure, is dominated by the size of the targeted structure, Ω P ∝ R 3 (Equation 10), and so a size difference of a factor of 2 amounts to roughly an order of magnitude difference in Ω P . Similarly, Johnstone et al. (2000) found a larger ambient gas pressure, P amb /k B ≈ 2 × 10 7 K cm −3 , and a comparable energy term, |Ω P | ≈ 2.2 × 10 41 to 1.3 × 10 44 erg, for even smaller structures with sizes between 0.006 and 0.05 pc. On the other hand, Maruta et al. (2010) found both an ambient pressure larger than that estimated for the droplets, P amb /k B ≈ 3 × 10 6 K cm −3 , and a pressure energy term larger than that estimated for the droplets, |Ω P | ≈ 1.6 × 10 42 to 5.0 × 10 43 erg, for structures in Ophiuchus with sizes of 0.022 to 0.069 pc. To some extent, the difference between the ambient gas pressure estimated in this work for the droplets and the gas pressure estimated for structures in the same region given by previous works can be attributed to the effects of a large uncertainty in the assumed critical density. Moreover, in previous works, the tracer used for estimating the gas pressure is usually different from the tracer used to define the structures themselves. This could result in the estimated gas pressure deviating from the actual local ambient gas pressure that is relevant in confining the structures under discussion. Fig. 11b shows a comparison between the kinetic energy and the energy term representing the ambient pressure confinement. Before including the gravitational potential energy (due to self-gravity acting as a confining force; see Equation 6), it already seems that the ambient gas pressure is substantial in both the droplets and the dense cores compared to the kinetic energy. Here for the dense cores, due to the lack of molecular line observations of the ambient gas, we follow Kirk et al. (2017a) and adopt a single value of P amb /k B = 9.5 × 10 5 K cm −3 based on observations of C 18 O (1-0) emission in nearby molecular clouds. The result is consistent with the conclusion drawn by Johnstone et al. (2000) that the ambient gas pressure is "instrumental" in confining the dense structures in the Ophiuchus cloud.
It is worth mentioning that a similar effort to obtain the local turbulent pressure structure-by-structure is done by Seo et al. (2015) for cores identified in the B218 region in Taurus. Seo et al. (2015) used the velocity dispersion and column density measurements at the circumference of the targeted core to estimate the work done by the ambient gas pressure, W amb ≈ 5 × 10 40 to 1 × 10 42 erg, and by assuming that the density distribution of the core follows the density profile of a critical Bonnor-Ebert sphere, Seo et al. (2015) estimated that the pressure at the surface of the core is P/k B ≈ 8 × 10 5 K cm −3 . Both numbers are similar to the numbers we get for the droplets, and similarly, Seo et al. (2015) conclude that some of the cores in the B218 region are pressure confined. A similar value of the ambient pressure, P/k B ≈ 2 × 10 4 K cm −3 , is found structure-by-structure for filamentary structures in molecular clouds by Fischera & Martin (2012), by modeling Herschel surface brightness profiles with near-equilibrium cylinders. See discussion below in §4.3.

Full Virial Analysis
Combining the estimates of Ω K , Ω G , and Ω P , we can assess the balance between the internal kinetic energy and the sum of "confining forces" in the form of the gravitational potential energy and the energy term representing the confinement provided by the ambient gas motions (Equation 6). Fig. 12a shows the distribution of the sum of the here based on the ambient gas pressures and the radii of corresponding structures. d For each of the droplets and the coherent core in B5, the density of the ambient gas is estimated based on the Herschel column density map. Other works derived the ambient gas density by assuming a "critical density" that the velocity dispersion tracer traces. The number density assumed to be traced by the ambient gas tracer is listed for reference. energy terms on the right-hand side of Equation 6 (Ω G and Ω P ) plotted against the internal kinetic energy, Ω K . Both the newly identified droplets and the dense cores appear to be virially bound (by self-gravity and the ambient gas pressure combined) or at least within an order of magnitude around an equilibrium. The dense cores appear to sit slightly deeper within the parameter space corresponding to a "bound" state, with the sum of Ω G and Ω P being roughly half an order of magnitude larger than Ω K . By contrast, the newly identified droplets and droplet candidates appear to be slightly closer to an equilibrium between the internal kinetic energy and the sum of energy terms representing the confining forces (self-gravity and the confinement provided by the ambient gas motions; i.e., Equation 6 holds for the droplets, within an order of magnitude).
In Fig. 12b, we examine the equipartition between the gravitational potential energy, Ω G , and the energy term measuring the confinement provided by the ambient gas pressure, Ω P . Most of the coherent cores, including the droplets, have |Ω P | ≥ |Ω G |, showing that even for dense cores which are often gravitationally bound, the ambient gas pressure is substantial. The full results from the virial analysis are listed in Table 2, and below in §4.1, we discuss the nature of the confinement provided by the ambient gas pressure. The fact that the newly identified coherent structures, droplets, are dominated by the ambient gas pressure but relatively less so by self-gravity ( §3.3; see also Fig. 11 and Fig. 12) seems to suggest that the confinement of the droplets is primarily provided by the ambient gas pressure. Understanding the nature of such pressure confinement and the related velocity structures is key to understanding the formation of the droplets and also to understanding the potential role the droplets, as well as the coherent structures, play in star/structure formation in nearby molecular clouds. Figure 12. (a) The sum of gravitational potential energy, ΩG, and the energy term representing the confinement provided by the ambient gas, ΩP, plotted against the internal kinetic energy, ΩK, for dense cores (green circles), the coherent core in B5 (a green circle marked with a black edge), and the newly identified coherent structures: droplets (filled blue circles) and droplet candidates (empty blue circles). The red band from the lower left to the top right marks the equilibrium between the sum of potential energy and the internal kinetic energy (solid red line) within an order of magnitude (pink red band), according to the virial equation (Equation 6). Structures in the parameter space above the red line (equilibrium) are expected to be bound, while the structures in the parameter space below the red line are expected to be unbound. (b) The gravitational potential energy, ΩG, plotted against the ambient pressure energy, ΩP, for the same structures as in (a). The red band from the lower left to the top right marks an equipartition between the two energy terms, ΩG and ΩP, shown in this plot.

Comparison to the Bonnor-Ebert Sphere
The droplets are likely confined by the pressure exerted on the surface by the ambient gas ( §3.3), and the subsonic velocity dispersion in the droplets indicates that the internal kinetic energy is largely provided by the thermal motions ( §3.2). The interior of each droplet has a virtually uniform distribution of the velocity dispersion dominated by the thermal motions, with the non-thermal component being roughly half of the thermal component (see Fig. 8 and Fig.  10). These results prompt us to compare the droplets to the Bonnor-Ebert model, which describes an isothermal core embedded in a pressurized medium (Ebert 1955;Bonnor 1956;Spitzer 1968).
By a similar approach described in §3.2 and Fig. 9, we derive the radial profiles of volume density, assuming a spherical geometry. To estimate the volume density at each radius, we first calculate the mass of the "layer" at that radial distance corresponding to an area of a certain shade of gray in Fig. 9. We then divide it by the volume of the layer to obtain an estimate of the volume density. This approach is similar to what we apply in §3.3.3 to calculate the ambient gas density, and in the analysis below, we repeat the procedure for layers of regions at different distances to obtain the radial density profile. We use one half of the GBT beam FWHM as the bin size in the radial direction (the "thickness" of the layer). The resulting radial density profiles are shown in Fig. 13. The typical uncertainty in the density measurement due to the assumption of spherical geometry is ∼ 25%, estimated based on the variation in column density at pixels within each radial distance bin. Note again that for a spherically symmetric distribution of density, the method would give the average density of a shell-shaped volume at each distance (see discussions in Appendix E).
We then compare the resulting density profiles of the droplets to the density profile of a Bonnor-Ebert sphere (Fig. 13). A Bonnor-Ebert sphere describes an isothermal sphere of gas in a pressurized medium, which satisfies the Lane-Emden equation: where r, ρ, and P are the radial distance from the center, the density as a function of the radius, and the pressure at r, respectively (Ebert 1955;Bonnor 1956). The Lane-Emden equation is the result of a continuous distribution of mass and an equilibrium between the pressure gradient and self-gravity. For a Bonnor-Ebert sphere, the pressure gradient of the isothermal sphere of gas comes from the gradient in density. Since the velocity dispersion is purely thermal, the equation of state of a Bonnor-Ebert sphere is equivalent to the ideal gas law: where c s is the (constant) sonic speed of the isothermal medium. Combining Equation 12 and 13, we have A non-singular numerical solution with a finite and differentiable density distribution at the center (r = 0) can be found in dimensionless units. Following analyses presented by Ebert (1955), Bonnor (1956), and Spitzer (1968), we normalize the density, ρ, by the density at the center. In practice, we use the average density estimated for materials enclosed within one beam FWHM from the center of the droplet, ρ cen : which is essentially the density "contrast" between the density at each distance and the density at the center. Again, following analyses presented by Ebert (1955), Bonnor (1956), and Spitzer (1968), the radius is normalized by a characteristic size scale, r c , defined as: The dimensionless radius is then: Notice that r c is the same order of magnitude as the distance traveled by a sound wave, r ff , during the free-fall time of a medium with a density equal to the central density: The corresponding distance in dimensionless units is shown as vertical lines in Fig. 13, which conventionally also serves as the radius outside of which the Bonnor-Ebert profile can be well approximated by the analytical Singular Isothermal Sphere (SIS) solution, ρ ∝ r −2 (Shu 1977). With the dimensionless variables defined above, the Lane-Emden equation can be rewritten as a function of x (Equation 17) and y (Equation 15): The numerical solution to Equation 20 is shown as the thick black curve in Fig. 13a in dimensionless units, x and y. The resulting Bonnor-Ebert sphere has a critical minimum radius for which the sphere is stable, x crit = 6.5, corresponding to a critical density contrast of y crit = 1/14.1 (the horizontal dashed line in Fig. 13a; see discussions in Bonnor 1956 andEbert 1955 for more details). The analytical solution represents a critical Bonnor-Ebert sphere, where the kinetic support and self-gravity is at a critical equilibrium. The non-critical, stable solutions to the Lane-Emden equation form a set of density profiles shallower than the critical Bonnor-Ebert sphere, where a core with a density profile steeper than that of the critical Bonnor-Ebert sphere would collapse under self-gravity (in the setting of a Bonnor-Ebert sphere in a pressurized medium). Notice that the analysis carried out in these dimensionless units allows comparisons between droplets of different sizes, independent of the constant terms in the solution to the original second-order differential equation, Equation 14 (see Fig. 13). See Appendix G for the radial profiles in physical units. Fig. 13 shows the density profiles of the droplets in normalized units (Equation 15 and 17), using the kinetic temperature from the NH 3 hyperfine line fitting, T kin , to derive the sonic speed, c s , in the normalization factor (Equation 16). The density profiles at r r ff appear to be near-constant, while the density profiles at r r ff appear to be shallower than the critical Bonnor-Ebert sphere (but steeper than a uniform density profile). On the outer edge, the density profiles of the droplets approach ρ ∝ r −1 , which can arise from structures having a constant column density and thus following a mass-size relation of M ∝ R 2 . This mass-size relation has been observed for cloud-scale structures (see examples in Larson 1981 and discussions in Kauffmann et al. 2010a,b). The non-critical, shallow density profiles can be consistent with the virial analysis presented in §3.3, where the droplets are found to be bound by ambient pressure but not self-gravity. For reference, we also compare the radial density profiles of the droplets to previously observed starless cores (Tafalla et al. 2004), and we find that the droplets have shallower density profiles than starless cores (Fig. 13b). and 17), compared to a critical Bonner-Ebert density profile. Each curve is the average radial profile of a droplet, color coded according to the ratio between gravitational potential energy and internal kinetic energy. The redder curves correspond to droplets with lower ΩG/ΩK ratios (less dominated by self-gravity), and the bluer curves correspond to droplets with higher ΩG/ΩK ratios (more dominated by self-gravity; §3.3.2). The thick black curve plots the density profile of a critical Bonor-Ebert sphere, and the lower horizontal line marks the critical contrast in volume density, when a critical Bonnor-Ebert sphere can hold a maximum mass through the balance between the pressure gradient and self-gravity. The light gray band shows the slope of a density profile as a power-law function of the radius, ρ ∝ r −1 . The vertical gray line marks the critical radius (equal to the free-fall length scale; Equation 18) in a dimensionless normalized unit, when the density profile of a critical Bonnor-Ebert sphere changes from approximately uniform to approximately a power-law function of radius, ρ ∝ r −2 . (b) Same as (a), the radial profile of volume density in normalized units of each droplet, this time plotted against previous observations of starless cores. The radial density profiles are color coded according to the ratio between gravitational potential energy and internal kinetic energy, ΩG/ΩK). The dark gray band shows the observed range of density profiles of typical starless cores observed by Tafalla et al. (2004). The light gray band again shows the slope of a density profile with a power-law dependence on the radius, ρ ∝ r −1 . Since the dumbbell shape of L1688-d1 affects this analysis which assumes spherical geometry, L1688-d1 is not included in these plots. The typical uncertainty for each volume density measurement along a density profile is ∼ 25%.
Since the Bonnor-Ebert sphere describes a thermal (no turbulent motions) and isothermal (uniform temperature) sphere, the radial profile of the gas pressure, derived from the ideal gas law, P = ρc 2 s , in the Bonnor-Ebert model, is the same as the density profile of a Bonnor-Ebert sphere in dimensionless units. In Fig. 14a, we compare the observed radial profiles of the gas pressure (due to the turbulent and thermal motions of the gas) in droplets to the pressure profile of a critical Bonnor-Ebert sphere. Intriguingly, L1688-d2, L1688-d5, and L1688-d6 have pressure profiles increasing outwards, and these droplets also appear to be less gravitationally bound (redder curves in Fig. 14). However, note that L1688-d2 and L1688-d5 sit near the edge of the region where NH 3 emission is detected, such that the profiles at larger radii are dominated by fewer pixels. Also note that the assumption of spherical geometry could break down due to the elongated shape of L1688-d6. Fig. 14b shows that the increases in velocity dispersion across the edges of the droplets are usually more abrupt than the change in the density profiles (Fig. 13). See Appendix G for the radial profiles of density and pressure in physical units. and 17). Since L1688-d2 and L1688-d5 sit near the edge of the regions with significant detection of NH3 (1, 1) emission, the profiles at larger radii could be dominated by fewer pixels, and thus the corresponding curves are specifically marked. L1688-d6 is also marked due to its highly elongated shape, for which the measurements using equidistant annuli could be biased. Note that the radial pressure profile of a Bonnor-Ebert sphere has the same shape as its radial density profile in normalized units-the black curve in Fig. 13a. (b) The radial profile of total velocity dispersion, σtot, relative to the value at the center of the droplet, σtot,cen, in Mach numbers (ratios to the sonic velocity). Same as (a), the curves are color coded according to ΩG/ΩK. The horizontal line marks when the change in σtot with respect to σtot,cen is equal to the sonic speed. L1688-d2, L1688-d5, and L1688-d6 are specifically marked, either because the droplets are elongated or because they sit near the edge of the regions where NH3 (1, 1) is significantly detected, both of which render the profiles potentially biased. Since the dumbbell shape of L1688-d1 affects this analysis, which assumes spherical geometry, L1688-d1 is not included in these plots.
Using a free parameter-the "effective temperature," T BE,eff -instead of the observed kinetic temperature, T kin , to derive c s in the equation of state (Equation 13), we can fit the critical Bonnor-Ebert profile to the observed density profiles of the droplets. Fig. 15 shows the resulting critical Bonnor-Ebert spheres at best-fit effective temperatures for droplets where we have reliable measurements of radial density profiles beyond the characteristic size scale. As Fig. 15 shows, most of the droplets have an excess in density compared to the best-fit critical Bonnor-Ebert profile at larger distances, approaching a power-law like density profile. And, for most droplets, the best-fit effective temperature, T BE,eff , is unreasonably higher than the kinetic temperature measured from NH 3 line fitting. Again, the results suggest that density and pressure profiles of the droplets cannot be well modeled with a critical Bonnor-Ebert sphere.

Comparison to the Logotropic Sphere
Based on the observational results obtained in the 1990s that 1) the density distribution at large radial distances from the center of a core is close to a power-law expression, ρ ∝ r −1 (instead of the singular isothermal solution, ρ ∝ r −2 ; Shu 1977), 2) the core is supported by both thermal and non-thermal (turbulent) velocity distributions, and 3) the total velocity dispersion is close to being purely thermal at the center and increases outwards, McLaughlin & Pudritz (1996, 1997 proposed that a dense core has a velocity dispersion distribution with a constant (isothermal) thermal component and a purely logotropic non-thermal component, i.e., P T ∝ ρ and P NT ∝ ln ρ/ρ cen in terms of pressure distribution, respectively. The resulting solution, known as the logotropic sphere, has an equation of state P = ρ cen c 2 s 1 + A ln ρ ρ cen , Figure 15. Individual radial profiles of normalized volume density, compared to critical Bonnor-Ebert spheres at best-fit effective temperatures. Each panel shows the radial density profile of a droplet, with the ID labeled at the top right of the panel.
The observed radial density profile in each panel is plotted as thick curves, color coded by the ratio between the gravitational potential energy and the kinetic energy, same as in Fig. 13. The radial density profile of a Bonnor-Ebert sphere at the best-fit effective temperature are shown as black curves (see surrounding text for details). The radial density profile of a Bonnor-Ebert sphere at the observed T kin , corresponding to the radial density profile of the critical Bonnor-Ebert sphere shown in Fig. 13, is plotted as a light gray curve in each panel. Same as Fig. 13, the gray band corresponds to a power-law density profile, ρ ∝ r −1 . Density profiles of droplets other than the one highlighted in each panel are plotted as transparent curves, similarly color coded by the ratio between the gravitational and kinetic energies.
where A > 0 is an adjustable parameter of the logotropic component. With a normalization factor similar to Equation 16: the Lane-Emden equation (Equation 12) with the logotropic equation of state (Equation 21) can be solved analytically. Similar to the Bonnor-Ebert sphere, the resulting equation can be written in dimensionless units: where x and y are the radial distance and the pressure in dimensionless units, respectively (see Equation 15 and 17). The resulting logotropic sphere has a nearly constant density near the center and a power-law density distribution, ρ ∝ r −1 , at larger radial distances (see Fig. 2 in McLaughlin & Pudritz 1996). We notice that the observational results listed above that prompted the examination of the logotropic spheres by McLaughlin & Pudritz (1996, 1997 are qualitatively true for the droplets (see Fig. 8 and Fig. 13). In Fig. 16, we compare the observed density and pressure profiles of droplets to a logotropic sphere with A = 0.2 (Equation 21, also used by McLaughlin & Pudritz 1996). While Fig. 16a shows that a logotropic sphere has a density profile generally matching the droplet density profiles, the observed pressure profiles of the droplets decrease faster at increasing distances than the pressure profile of a logotropic sphere (see Fig. 16b). The result suggests that the logotropic solution cannot describe the droplets, either.
In summary, we find that neither a critical Bonnor-Ebert sphere or a logotropic sphere describes the density and pressure profiles of the droplets well. Instead, the shallow radial density and pressure profiles of the droplets can be approximated by a uniform density at smaller radii and a power-law density distribution approaching ρ ∝ r −1 at larger radii, the latter of which has also been observed for cloud-scale structures. Figure 16. (a) The radial profile of volume density in normalized units of each droplet, same as the color-coded curves shown in Fig. 13, compared to a logotropic density profile. Each curve is the radial profile of a droplet, color coded according to the ratio between gravitational potential energy and internal kinetic energy. As in Fig. 13, the redder curves correspond to droplets with lower ΩG/ΩK ratios (less dominated by self-gravity), and the bluer curves correspond to droplets with higher ΩG/ΩK ratios (more dominated by self-gravity; §3.3.2). The thick black curve plots the density profile of a logotropic sphere (Equation 21), and the gray curve shows the density profiles of a critical Bonnor-Ebert sphere. As in Fig. 13, the light gray band shows the slope of the power-law density profile, ρ ∝ r −1 . The typical uncertainty for each volume density measurement along a density profile is ∼ 25%. (b) The radial profile of pressure in normalized units of each droplet, same as the color-coded curves shown in Fig. 14a. As in (a), each curve corresponds to a droplet, color coded according to the ratio between gravitational potential energy and internal kinetic energy, ΩG/ΩK. The black curve plots the radial profile of normalized pressure for a logotropic sphere, and the gray curve plots the radial pressure profiles of the critical Bonnor-Ebert sphere. Since L1688-d2 and L1688-d5 sit near the edge of the regions with significant detection of NH3 (1, 1) emission, the profiles at larger radii could be dominated by fewer pixels, and thus the corresponding curves are specifically marked. L1688-d6 is also marked due to its highly elongated shape, for which the measurements using equidistant annuli could be biased.

Velocity Distribution of the Droplet Ensemble
The virial analysis presented in §3.3 suggests that the confinement of the droplets is primarily provided by the ambient gas pressure. Consistently, we find that the droplets have non-critical and relatively shallow density profiles approaching ρ ∝ r −1 at the outer edges. Both results point to a close relation between the droplets and the local cloud environment. Below, to investigate this relationship between the droplets and the surrounding cloud, we examine the distribution of emission in the position-position-velocity (PPV) space. Fig. 17 shows the PPV distribution of the best fits to the NH 3 hyperfine line profiles observed at the pixels shown in Fig. 2b, with the locations along the velocity axis equal to the velocity centroids of the best fits. With each data point (the location of the Gaussian peak) color-coded by σ NH3 , several low linewidth features stand out having different line-of-sight velocities from the system velocity of the cloud, by ∼ 0.5 km s −1 . Overall, we find that roughly half of the total 12 droplets in L1688 sit at the local extremes in V LSR , while the other half of the 12 droplets appear more embedded in the main cloud component in the PPV space. Note that the distribution of emission in the PPV space does not correspond to the distribution of material in the position-position-position (PPP) space (Beaumont et al. 2013), and the deviation in V LSR from the main cloud component does not necessarily suggest that the droplet is separated from the cloud in the PPP space. Figure 17. The position-position-velocity (PPV) distribution of the best Gaussian fits to the NH3 hyperfine line profiles, color-coded by the NH3 velocity dispersion, σNH 3 , using the same color scale as that used in Fig. 2 and 4. Each dot corresponds to a pixel in the plane of the sky with a line-of-sight velocity equal to the best-fit velocity centroid. The map at the top right corner shows the projected point of view on the plane of the sky. The droplets that are distinguishable in PPV space from the distribution of the bulk material in the cloud (as traced by NH3 hyperfine line emission; usually the darker points) are marked by solid circles, while the approximate positions of the droplets that are more embedded in the cloud in PPV space are marked by dashed circles. The numbers correspond to the droplet IDs in Table 1, with the header "L1688-" removed for better visualization. The visualization is made with the aid of Glue.
Notably, the typical V LSR difference of ∼ 0.5 km s −1 between the V LSR of droplets found at local velocity extremes and the system velocity of the cloud component traced by the NH 3 emission is comparable to half of the median FWHM linewidth of the NH 3 (1, 1) emission, ∼ 0.46 km s −1 (shown as a vertical line along the velocity axis in Fig. 17; FWHM NH3 ≈ 0.92 km s −1 , measured for pixels outside the droplet boundaries-dark blue regions in Fig.  2b). A more detailed comparison shows that the dispersion in the velocity centroids of the droplets (analogous to the "core-to-core velocity" examined by Kirk et al. 2010) agrees well with the median NH 3 velocity dispersion measured at pixels outside the droplet boundaries (see Table 4). In Fig. 18, we compare the distribution of droplet V LSR to the average "deblended" spectrum of the entire L1688 region 10 and show that the distribution of droplet V LSR has a shape similar to the deblended NH 3 line profile. Given that the NH 3 velocity dispersion, σ NH3 , is associated with the thermal c Measured by taking the standard deviation of the VLSR distribution (see Table 1); the velocity resolution of the observations is ∼ 0.07 km s −1 . and turbulent motions of the dense gas, the results suggest that the droplets are traveling in the dense component of the cloud at velocities on par with the thermal and turbulent motions of the dense gas traced by NH 3 emission. The result further suggest that the velocities of the droplets are inherited from the velocity dispersion of materials in the environment.
For reference, we also compare the distribution of droplet V LSR to the average 13 CO (1-0) spectrum 11 and find that 13 CO (1-0) has a line profile 2 to 3 times as broad as the droplet-to-droplet velocity distribution. The result is consistent with what Kirk et al. (2010) observed in Perseus. Using the N 2 H + emission to trace the dense core motions in the molecular cloud, Kirk et al. (2010) found that the core-to-core velocity dispersion is about half of the total 13 CO velocity dispersion in the region.
In the analyses presented in §3.3 and §4.1, we find that 1) the droplets generally appear not bound by self-gravity and predominantly confined by the ambient gas pressure and that 2) there is a close relation between the droplets and the local cloud component traced by the NH 3 emission. Together, the results point to the possibility that the droplets, primarily defined by their subsonic and uniform interiors, are the result of compression due to the relatively more turbulent motions in the dense gas component of the cloud. Below in §4.2, we look for similar structures in a magnetohydrodynamic (MHD) simulation and speculate on the potential formation mechanism of the droplets.

Comparison with Hydrodynamic Models
Before looking for droplet-like structures in a magnetohydrodynamic (MHD) simulation, we first summarize the measured properties of the 18 droplets identified and examined in this paper in Table 5. The newly identified sub-0.1 pc coherent structures are smaller in size and mass than previously known coherent cores ( §3.2; Fig. 10). The droplets appear to be gravitationally unbound and confined mainly by the ambient gas pressure in a virial analysis ( §3.3; Fig. 11 and Fig. 12). Consistent with being gravitationally unbound, the droplets have density structures generally shallower than a critical Bonnor-Ebert sphere and previously observed starless cores ( §4. 1.1; Fig. 13). The population of droplets in L1688 appear to have a velocity distribution matching the dense gas motions traced by the NH 3 emission, suggesting that the velocities of the droplets are inherited from the velocity dispersion of materials in the environment.
Simple analytical models could hint at the formation mechanism of the droplets. For example, by extending the Jeans model (Jeans 1902), Myers (1998) proposed a "kernel" model, in which a condensation with a mass of 1 M and a size of 0.03 pc can exist within a dense core under ambient pressure provided by the thermal and turbulent motions. Below, we demonstrate that formation of droplets is also possible in an MHD simulation of a turbulent cloud with self-gravity and sink particles.
We analyze an MHD simulation of a star-forming turbulent molecular cloud (Smullen et al. in prep). The simulation is carried out with the ORION2 adaptive mesh refinement (AMR) code (Li et al. 2012). The domain represents a piece of a molecular cloud 5 pc on a side with physical parameters and initialization identical to those of the W2T2 Figure 18. Distribution of velocity centroids of the droplets and droplet candidates (blue histogram; lighter parts correspond to droplet candidates), plotted against the average spectra of NH3 (dark red curve) and 13 CO (1-0) emission (light red curve). The NH3 average spectrum is calculated from a "deblended" data cube created based on the results of NH3 line fitting, such that each spectrum is a Gaussian with a center the same as the velocity centroid and a spread (σ) the same as the velocity dispersion. The structure IDs of the structures included in each bin of the histogram are noted, with the leading "L1688-" removed for better visualization. The spectra are shown in relative units wherein the peak has a value of 1. b The standard deviation of the droplet VLSR distribution is 0.39 km s −1 (see Table 4). For droplets that sit at local velocity extremes, the typical value of the VLSR difference is ∼ 0.5 km s −1 . c Sim-d1 is identified in the synthesized NH3 spectral cube following the same procedure described in §3.1 (Fig. 19; Smullen et al. in prep). The naming follows the convention used for droplets identified in observations. See §4.2. d Sim-c1 is found to associate with a shock-induced structure not unlike the one associated with Sim-d1. While Sim-c1 also has a subsonic velocity dispersion, it is less clear whether a transition to coherence happens at its periphery (see Fig. 19). Thus, it is categorized as a "droplet candidate." simulation in Offner & Arce (2015). The mean gas density is ρ = 440 cm −3 (2.04 × 10 −21 g cm −3 ), the initial gas temperature is 10 K, the plasma beta is initially 0.1 and changed to 0.02 after 2 crossing times of driving, and the gas has a velocity dispersion of 1.98 km s −1 . The latter is set such that the cloud falls on the observed linewidth-size relation. The calculation has 5 AMR levels with a maximum resolution of 125 AU. We analyze a snapshot at 0.52 Myr or 0.35 t ff as measured from when the initial driving setup phase ends and self-gravity is turned on. At this time 1.3% of the gas is in stars.
We use RADMC-3D 12 to calculate the NH 3 emission given the simulated gas density and temperature distribution. We adopt a uniform NH 3 abundance of 2 × 10 −9 n H . We adopt the collisional parameters from the Leiden atomic and molecular database (Schöier et al. 2005) and compute the radiative transfer using the non-local thermodynamic equilibrium large velocity gradient approximation (Shetty et al. 2011). To look for structures that show 1) a sharp change in velocity dispersion, and 2) locally concentrated emission, we derive the moment maps based on the synthesized NH 3 spectral cube. Based on the moment maps, we follow the same identification procedure described in §3.1 and select a droplet-like structure that shows clear signs of a change in velocity dispersion and coincides with concentrated synthetic NH 3 emission.
The identified droplet-like structure, Sim-d1, has a radius of 0.036 pc, a mass of 0.96 M , and a subsonic velocity dispersion, σ tot ∼ 0.24 km s −1 (derived from the synthesized NH 3 spectral cube in the same way as the observed σ tot ; Fig. 19). Sim-d1 is also found to have a line-of-sight velocity different from the median line-of-sight velocity of the "cloud" (the simulation cube), with a difference of ∼ 0.63 km s −1 . Table 5 shows the properties of Sim-d1 alongside the typical values of the 18 droplets examined using the GAS observations. Figure 19. Top Row. Sim-d1, a droplet identified in the MHD simulation following the procedure described in §3. 1 (Smullen et al. in prep), with identifiable rise in integrated emission (0th moment; left panel) and a sharp drop in velocity dispersion (2nd moment; middle panel) near its edge. The right-hand most panel shows an overlay of the 0th and 2nd moment maps. Bottom Row. Same as the top row, but showing Sim-c1, another structure associated with an isolated shock-induced feature which has a subsonic velocity dispersion but where signs of a transition to coherence are less clear. Note that each panel in this figure shows only a 0.7 pc by 0.7 pc region near the density structures, out of the whole spectral cube which has a size of 5 pc by 5 pc on the "plane of the sky." Following the virial analysis presented in §3.3, we derive the internal kinetic energy, the gravitational potential energy, and the energy term representing the confinement provided by the ambient pressure for Sim-d1 (Equation 7, 8, and 10). Fig. 20c and Fig. 20d show that, similar to droplets observed in L1688 and B18, Sim-d1 is gravitationally unbound and likely confined by the pressure provided by the ambient gas motions. The ambient pressure measured using the synthetic NH 3 spectral cube created from the simulation is P amb /k B ≈ 4.7 × 10 5 K cm −3 , comparable to the typical value of P amb /k B ≈ 2.7 +4.7 −1.8 × 10 5 K cm −3 for the droplets. The results demonstrate that not only does Sim-d1 have mass, size, and the internal velocity dispersion similar to the droplets, it also sits in a similar pressurized environment. Figure 20. Plots showing properties of Sim-d1, a structure that strictly satisfies the identification criteria described in §3.1 (yellow circle) and another isolated shock-induced structure, Sim-c1 (red circle) found in the MHD simulation against the dense cores (green circles) and the droplets (blue circles; droplet candidates as empty blue circles) in: (a) the mass-size relation (Fig.  10a), (b) the linewidth-size relation (Fig. 10b), (c) the relation between gravitational potential energy and the internal kinetic energy (Fig. 11a), and (d) the relation between the energy term representing the confinement provided by ambient pressure and the internal kinetic energy (Fig. 11b).
By following the evolution of Sim-d1, a droplet identified in the MHD simulation, we find that Sim-d1 corresponds to a relatively isolated shock-induced feature in the MHD simulation, moving generally toward the viewer along the line of sight on which we "observe" the synthesized NH 3 cube (see Fig. 21 and the video showing the evolution of the MHD cube linked in the caption). Meanwhile, material seems to accumulate at the converging point of the shock-induced feature as the simulated cube evolves. The general movement of Sim-d1 toward the viewer is consistent with the relatively high line-of-sight velocity difference observed in the synthesized NH 3 cube, ∼ 0.63 km s −1 . The association between a droplet and a shock-induced feature, viewed from different angles, might explain why the observed droplets are sometimes found to sit at local line-of-sight velocity extremes.
We leave a systematic analysis of droplet-like structures in MHD simulations to a future work, but at first glance, at least several other similar shock-induced features can be identified (e.g., the structure marked by the red square in Fig.  21). These structures generally have subsonic internal velocity dispersions but do not necessarily show clear signs of a transition to coherence. We select one of these features, Sim-c1, and observe it in the synthesized NH 3 cube (Fig. 19). We find that Sim-c1 has properties consistent with the physical properties of the droplets identified in observations (see Fig. 20 Table 5).
Notably, the most active star forming regions are found around a series of converging points of multiple shocks in the MHD simulation (see Fig. 21). While a more detailed analysis is needed to understand the effects of converging/colliding shocks, it seems possible that the droplet-like features associated with isolated shock features might evolve into star forming cores through continuing accumulation of material and/or through converging with other shock-induced features. See more discussion below in §4.3. Figure 21. The integrated density along three different viewing directions of the MHD simulation examined in §4.2. The right-hand most panel shows the view based on which the synthesized NH3 cube is derived and used to calculate the moment maps shown in Fig. 19. The yellow square marks the droplet-like structure identified following the same procedure described in §3.1 and shown in Fig. 19. The red square marks another shock-induced feature, around which signatures of a subsonic velocity dispersion and a concentrated density distribution are found. The white dots mark the positions of the sink particles. The full evolution of the simulated cube can be found at https://goo.gl/PEd9Pd, where the droplet-like structure (in the yellow box) can be found moving toward the right-side in the central panel and moving toward the viewer in the panel on the right.

Cores & Droplets
Like coherent cores, the newly identified coherent structures, droplets, are defined by transitions to coherence-a change in velocity dispersion from turbulent values outside a droplet to near-constant, thermally dominated values inside. As Pineda et al. (2010) pointed out, resolving the transition to coherence with a single tracer requires a sufficient physical resolution  described a characteristic size scale of ∼ 0.1 pc) and a substantial depth of the observations (with significant detection of the line emission both inside and outside the boundary of a coherent structure, the latter of which is expected to be less bright than the former). The Green Bank Ammonia Survey (GAS, see §2. 1;Friesen et al. 2017), aiming at mapping the Gould Belt star forming regions with A V ≥ 7 mag, provides both the resolution (with a beam size at 23 GHz corresponding to 0.016 pc at the distance of 100 pc and 0.076 pc at the distance of 500 pc) and the depth (complete in detecting NH 3 (1, 1) emission above A V = 7 mag). By following the same procedure adopted by Pineda et al. (2010), we find a total of 18 coherent structures in a total projected area of ∼ 0.6 pc 2 where the GAS is complete in detecting the NH 3 hyperfine line emission (Friesen et al. 2017). While even more coherent structures could be "hidden" due to the assumption of a single velocity component along each line of sight during the NH 3 line fitting process, the results from the identification of coherent structures in Ophiuchus and Taurus suggest that the coherent structures are ubiquitous in nearby molecular clouds. a core needs to meet the criterion in order to be assigned to a certain category. A value of "No" means that a core needs to satisfy the negation of the criterion in order to be assigned to a certain category. A value of "Neutral" means that the definition of a certain category does not concern the criterion. b The thermal component of the velocity dispersion is larger than the non-thermal (turbulent) component.
c Observation of "transition to coherence," as described by Goodman et al. (1998). The observation of "transition to coherence" may be done by observing the same core with multiple tracers and focusing on the change in the linewidth-size relation going from one tracer to the next (Type 4 in Fig. 9 of Goodman et al. 1998, later used by Caselli et al. 2002 in observations of coherent cores). Another way to observe the "transition to coherence" is spatially resolve the transition with a single tracer. Examples include observations of NH3 emission in B5 , and the droplets in this work. Note that the criterion of "transition to coherence" is stricter than "thermally dominated linewidths." The thermally dominated linewidths concern the overall measurement of velocity dispersion in the core, but not the spatial change in velocity dispersion. d The canonical examples of dense cores are described by the "Dense Cores in Dark Clouds" series (starting with ). In the original literatures, a dense core is simply defined by a centrally concentrated density distribution. Based on observations of NH3 emission and emission from other higher density molecular line tracers, Myers (1983) found that most of the dense cores examined by  had velocity dispersions approaching transonic or subsonic values. The gravitational boundedness was less clear, oftentimes because of a lack of necessary observations to accurately estimate the boundedness of these structures. The dense cores analyzed by Goodman et al. (1993) and included in this paper are mostly gravitationally bound, as shown in Fig. 11. e As described by Goodman et al. (1998), and later observed by Caselli et al. (2002) and Pineda et al. (2010).
f In literatures, the starless cores and the prestellar cores both point to cores that are not associated with any YSOs. A criterion often used to distinguish between the two categories is the gravitational boundedness. The prestellar cores are cores that are gravitationally bound, and the starless cores are those that are not (e.g., Tafalla et al. 2004). In some cases, density features at smaller scales within the cores are used to further investigate the "starlessness" of the starless cores (e.g., Kirk et al. 2017b). g By definition, the protostellar cores are the cores associated with YSOs.
In this work, we examine the physical properties of two closely related populations of structures: the droplets and the dense cores (among which many were found to be coherent cores). In the analyses presented above, we essentially use the two terms, the droplets and the dense cores, to indicate structures identified in this work and those examined by Goodman et al. (1993), respectively. Although the two populations indeed have different physical properties, we are not satisfied with this rather arbitrary use of terminology. Thus, we provide a more physical set of definitions for different groups of structures discussed in this paper below.
We define the droplets to be the gravitationally unbound and pressure confined coherent structures, wherein the coherent structures include any structures that have subsonic velocity dispersions and show transitions to coherence. In comparison, a coherent core is a gravitationally bound coherent structure, and a dense core is a centrally concentrated density feature with a velocity dispersion approaching a transonic or subsonic value (not necessarily showing a transition to coherent at its boundary). Just like a coherent core may correspond to the densest region of a dense core, a droplet may be the inner most region of a larger density structure. The definitions are summarized in Table 6.
Due to the lack of observations needed to determine whether the cores examined by Goodman et al. (1993) show any signs of a transition to coherence, we simplify the set of criteria and recategorize the structures identified in this work and those examined by Goodman et al. (1993) into two categories: 1) the "droplets": structures not virially bound by self-gravity and with subsonic velocity dispersions, and 2) the "dense cores": structures virially bound by self-gravity and with subsonic velocity dispersions. In the discussions below, we retain the quotation marks around these terms to differentiate them from the terms used throughout the analyses above. Fig. 22 shows how this recategorization would change the groupings of structures examined in the analyses in this paper. To proceed with caution and to avoid uncertainty in using a virial analysis to determine the equilibrium state of a structure, we categorize those structures with subsonic velocity dispersions and within an order of magnitude of a virial equilibrium between the gravitational potential energy and the kinetic energy (see details in §3.3) as "dense core candidates." Notice in Fig. 22, most of these "dense core candidates" have virial parameters ≤ 2 and would conventionally be considered virially bound by self-gravity. In this recategorization, we temporarily omit structures with supersonic velocity dispersions, although the largest turbulent Mach number (the ratio between the turbulent component of velocity dispersion and the sonic speed) found in these structures is 1.5, i.e., not anywhere close to the turbulence measured for the entire molecular cloud. For example, using observations of the 13 CO (1-0) emission, we would measure a turbulent Mach number of ∼ 10 for the entire molecular cloud in Ophiuchus. In total, three out of 43 dense cores are recategorized as "droplets," and three out of 18 droplets are recategorized as "dense core candidates." Figure 22.
(a) Total velocity dispersion, σtot, plotted against the ratio between the gravitational potential energy, ΩG, and the kinetic energy, ΩK (more to the left-more bound by self-gravity; more to the right-less bound by self-gravity). The droplets and droplet candidates are plotted as solid and empty blue circles, and the dense cores analyzed by Goodman et al. (1993) are plotted as green circles. The horizontal lines show the total velocity dispersions expected for structures where the non-thermal component is equal to the sonic speed (thicker, black line) and half the sonic speed (thinner, gray line) of a medium with a mean molecular weight of 2.37 u at a temperature of 10 K. The vertical red band marks an equilibrium between the gravitational potential and the kinetic support (solid red line) within an order of magnitude (pink band), according to Equation 6 (omitting the pressure term). The vertical black line marks αvir = 2, which is conventionally used in virial analysis as the threshold for gravitational boundedness. (b) Same as (a), but with the data points color coded according to the proposed recategorization (see §4.3). According to the recategorization, all structures with subsonic velocity dispersions are grouped into three categories: 1) the "droplets" (dark blue) that are not virially bound by self-gravity, 2) the "dense cores" (dark green) that are virially bound by self-gravity, and 3) the "dense core candidates" (dark yellow) that are within an order of magnitude from an equilibrium state and would conventionally be considered gravitationally bound according to a virial parameter (αvir ≤ 2). Structures that have supersonic velocity dispersions are omitted in this recategorization and plotted as gray circles. Fig. 23 shows the distributions of the recategorized structures in parameter spaces. It is evident that different populations-"droplets" and "dense cores"-categorized according to their physical properties are mingled and form a continuous distribution in various parameter spaces. The continuity might suggest that different populations of coherent structures emerge from the same set of physical processes, although they differ in gravitational boundedness. Based on the MHD simulation examined in §4.2, a scenario is that both the gravitationally bound coherent structurescoherent cores-and the gravitationally unbound and pressure confined coherent structures-droplets-arise from shock-induced overdensities. In this scenario, we would find likely star forming coherent cores around the converging points of multiple shocks, and we would find droplets around more isolated shock-induced structures and/or isolated pairs of colliding shocks.
Based on a rough examination of the MHD simulation ( §4.2), a shock induced formation mechanism of coherent structures might point to an evolutionary sequence connecting the droplets and the coherent cores. If the droplets are formed as isolated shock-induced structures, they might still evolve into star forming and gravity dominated coherent cores in the future if the isolated shocks converge/collide with other shock-induced structures. Similarly, based on observations of core structures in the B218 region in Taurus, Seo et al. (2015) find that it is more likely to find Class 0/I YSOs associated with gravitationally bound cores than with pressure confined structures. Thus, Seo et al. (2015) suggest that there exists an evolutionary sequence connecting the pressure confined structures to the gravitationally bound cores. The evolutionary sequence suggested by Seo et al. (2015) might conform with the conventional evolutionary sequence observed toward the prestellar and the protostellar cores (see Table 6).
As discussed in §3.1, since the droplets are found within active star forming regions, the line of sight projection effects make it difficult to determine without a large uncertainty whether or not the droplets are associated with any YSOs. If we simply look at the existence of YSO(s) within the droplet boundary projected on the plane of the sky, we find that five out of 12 droplets in L1688 and one out of 6 droplets in B18 coincide with at least one YSO within each of their boundaries. This roughly gives a 40% chance of finding at least one YSO within the droplet boundary. Nevertheless, as discussed in §3.1, we do not find significant rises in the gas kinetic temperature, T kin , in any droplets. Furthermore, we do not find a significant difference in physical properties between the two subsets of droplets-ones with YSOs within their boundaries and ones without (see Fig. 24). Thus, we cannot clearly conclude if there is an association between the YSOs and (any of) the droplets.
In conclusion, the droplets, defined as the gravitationally unbound and pressure confined coherent structures and identified using observations of the NH 3 (1, 1) hyperfine line emission in this work, are a previously omitted subpopulation of coherent structures. Although imminent star formation within droplets is unlikely because of the gravitational unboundedness, the droplets might form from the same set of physical processes that lead to the formation of star forming coherent cores. Since the subsonic velocity dispersion within a coherent core is expected to be disturbed by ongoing formation of stars, the droplets may provide a precious chance to examine the internal kinematics and the formation of coherent structures. Furthermore, there could exist an evolutionary sequence connecting the pressure dominated droplets to the star forming coherent cores, but this cannot be confirmed with present data. More works to systematically find and examine droplets in simulations and to compare them with droplets and other cores identified in observations are needed to answer the following questions: How do the droplets form? Do the droplets evolve into star forming cores, and if so, how? What is the relation between the coherent structures, including both star forming coherent cores and pressure dominated droplets, and other populations of cores (e.g., starless and protostellar cores) and structures (e.g., filaments and bundles Hacar et al. 2013)? Are there observable velocity gradients and potentially associated rotational and/or shear motions in the interiors of coherent structures? Would the coherent structures fragment into smaller features in the future? We will address some of these questions in subsequent papers of this series.

CONCLUSION
In search of coherent structures defined by a change in velocity dispersion from supersonic to nearly constant subsonic values ( §3. 1;Figs. 1, 2, and 5 for L1688, and Figs. 3, 4, and 6 for B18), we identify a total of 18 coherent structures in the L1688 region of Ophiuchus and the B18 region of Taurus, using data from the first data release of the Green Bank Ammonia Survey (see §3. 1;Friesen et al. 2017). The 18 coherent structures newly identified within a total projected area of ∼ 0.6 pc 2 suggest that the coherent structures are ubiquitous in nearby molecular clouds and allow statistical analyses of coherent structures for the first time.
The newly identified coherent structures have a typical radius of 0.04 pc and a typical mass of 0.4 M ( §3.2; Table  1) and appear to follow the same mass-linewidth-size relation as the dense cores previously examined by Goodman et al. (1993), many of which are later found to be coherent cores (see Fig. 10; Goodman et al. 1998;Caselli et al. 2002;Pineda et al. 2010). In a virial analysis, we find that the newly identified coherent structures are not virially bound by self-gravity and are instead confined by the pressure provided by the ambient gas motions (see §3.3). This clearly differentiates the newly identified coherent structures from previously known coherent cores, which have been found to be gravitationally bound and sometimes hosting ongoing star formation (Pineda et al. , 2015. We term this newly discovered population of gravitationally unbound and pressure confined coherent structures the droplets. The mass-size distribution of the "dense cores" (dark green circles), the "dense core candidates" (dark yellow circles), and the "droplets" (dark blue circles). As in Fig. 10a, the black line shows a power-law relation between the mass and the effective radius, and randomly selected 10% of the accepted parameters in the MCMC chain used to find the power-law fit are plotted as transparent lines for reference. The solid gray line shows the empirical relation based on observations of larger-scale structures examined by Larson (1981). Structures that have supersonic velocity dispersions are omitted in this recategorization and plotted as gray circles. (b) The σtot-size distribution of the same structures shown in (a). As in Fig. 10b, the horizontal lines show σtot expected for structures where the non-thermal component is equal to the sonic speed (cs; thicker line) and half the sonic speed (thinner line) of a medium with a mean molecular weight of 2.37 u at a temperature of 10 K. (c) Gravitational potential energy, ΩG, plotted against internal kinetic energy, ΩK (Equation 6), for the same structures shown in (a). As in Fig. 11a, the red band from the lower left to the top right marks the equilibrium between ΩG and ΩK (solid red line) within an order of magnitude (pink band), according to the virial equation (Equation 6; omitting the pressure term). The black line marks where the conventional virial parameter, αvir, has a value of 2 (Equation 9). (d) The energy term representing the confinement provided by the ambient gas pressure, ΩP, plotted agains the internal kinetic energy, ΩK (Equation 6), for the same structures shown in (a). As in Fig. 11b, the red band from the lower left to the top right marks an equilibrium between ΩP and ΩK (solid red line) within an order of magnitude (pink red band), according to the virial equation (Equation 6; omitting the gravitational term). Structures in the parameter space above the red line (equilibrium) are expected to be dominated by the ambient gas pressure. Figure 24. (a) The mass-size distribution of the droplets with YSOs within the boundaries (yellow star marks) and the droplets without YSOs within boundaries (blue circles), in comparison to the dense cores (green circles) and the coherent core in B5 (the green star mark; the coherent core in B5 has at least one YSO within its boundary). As in Fig. 10a, the black line shows a power-law relation between the mass and the effective radius, and randomly selected 10% of the accepted parameters in the MCMC chain used to find the power-law fit are plotted as transparent lines for reference. The solid gray line shows the empirical relation based on observations of larger-scale structures examined by Larson (1981). (b) The σtot-size distribution of the same structures shown in (a). As in Fig. 10b, the horizontal lines show σtot expected for structures where the non-thermal component is equal to the sonic speed (cs; thicker line) and half the sonic speed (thinner line) of a medium with a mean molecular weight of 2.37 u at a temperature of 10 K. (c) Gravitational potential energy, ΩG, plotted against internal kinetic energy, ΩK (Equation 6), for the same structures shown in (a). As in Fig. 11a, the red band from the lower left to the top right marks the equilibrium between ΩG and ΩK (solid red line) within an order of magnitude (pink band), according to the virial equation (Equation 6; omitting the pressure term). The black line marks where the conventional virial parameter, αvir, has a value of 2 (Equation 9). (d) The energy term representing the confinement provided by the ambient gas pressure, ΩP, plotted agains the internal kinetic energy, ΩK (Equation 6), for the same structures shown in (a). As in Fig. 11b, the red band from the lower left to the top right marks an equilibrium between ΩP and ΩK (solid red line) within an order of magnitude (pink red band), according to the virial equation (Equation 6; omitting the gravitational term). Structures in the parameter space above the red line (equilibrium) are expected to be dominated by the ambient gas pressure.
The radial density and pressure profiles of the droplets cannot be well described by either a critical Bonnor-Ebert sphere or a logotropic sphere (see §4. 1.1 and §4.1.2). The droplets have relatively shallow density profiles (e.g., compared to previously observed starless cores; see Fig. 13b), and their density profiles can generally be approximated by a constant density at smaller radial distances and a power-law density distribution approaching ρ ∝ r −1 at larger distances, the latter of which has been observed toward cloud-scale structures (Fig. 13). While the droplets are sometimes found at local extremes of the line-of-sight velocity, the V LSR distribution of the droplets has a shape similar to that of the average NH 3 line profile (see §4.1.3; see also Fig. 17 and Fig. 18). Both the power-law density profiles (ρ ∝ r −1 ) and the distribution of V LSR suggest a close relation between the droplets and the natal cloud environment.
By identifying droplet-like structures in the synthesized NH 3 cube, we demonstrate that the formation of droplets is possible in an MHD simulation (see §4. 2;Smullen et al. in prep, Offner & Arce 2015). The two droplet-like structures examined in §4.2 appear to correspond to shock-induced features in the MHD simulation, and throughout the evolution of the simulated cube, material accumulates at the converging points of the shock features. Given the active star formation in the same simulated cube emerge in regions near the converging points of multiple shocks, we speculate that a droplet might evolve into a star forming core if accumulation continues and/or if the associated shock-induced feature converges/collides with other shocks.
More work is needed to understand the formation and evolution of droplets and coherent structures in general. With the GAS data, we hope to extend our analyses on coherent structures to other nearby molecular clouds. The GAS observations of NH 3 hyperfine line emission also allows an analysis of the internal velocity structures, which would shed light on the potential rotational and shear motions in the droplets/coherent cores. On the other hand, more targeted modeling and a statistical approach are needed to further understand the physical processes involved in the formation of droplets and the role the droplets might play in star formation.
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. AP acknowledges the support of the Russian Science Foundation project 18-12-00351. To compare the droplets identified in this work to previously known dense cores, we correct the physical properties summarized in Goodman et al. (1993) by more recent distance measurements. The updated distances are summarized below: 5. Regions associated with clouds and clumps in Oph N: L43/RNO90, L43, L260 (a.k.a. L255), L158, L234E, L234A, and L63. These regions are usually associated with the Ophiuchus complex or, on a larger scope, the Upper Sco-Oph-Cen complex. Goodman et al. (1993) adopted the same distance for these regions as for L1696A.
Here we use an updated distance measurement of 125 ± 18 pc to the Ophiuchus complex by Schlafly et al. (2014). This is in good agreement with the widely used 125 ± 45 pc, measured by de Geus et al. (1989).
6. Regions associated with Cepheus Flare: The Cepheus Flare spans more than 10 degrees from North to South on the plane of the sky, and is known to have a complicated structure with multiple concentrations of material at different distances. Here we adopt different distance measurements for different regions in Cepheus Flare, and note that these distances were used by Kauffmann et al. (2008) side-by-side. Note that Schlafly et al. (2014) measured 360 ± 35 pc for the southern part of Cepheus Flare and 900 ± 90 pc for the northern part of Cepheus Flare. See discussions in Schlafly et al. (2014).
8. Other regions of which the distances have not updated since the 1990s but are cited recently. Here we provide a list of the original references and the most recent year when each reference was cited.
The resulting change in distance, D, affects the measured radius, R, of each core listed in Table 1 in Goodman et al. (1993) according to a linear relation, R ∝ D. Since Goodman et al. (1993) calculated the mass based on volume density derived from NH 3 hyperfine line fitting, the change in distance affects the mass by M ∝ D 3 . See §2.3.1 for details.

B. A GALLERY OF CLOSE-UP VIEWS OF THE DROPLETS AND THE DROPLET CANDIDATES
In §3.1, we explain the steps we take to identify the droplets and the droplet candidates. The resulting droplets and droplet candidates are shown in Figs. 1, 2, and 5 for L1688, and Fig. 3, 4, and 6 for B18. Here we provide a gallery of close-up views of these droplets and droplet candidates. The gallery can be found at https://github.com/hopehhchen/ Droplets/tree/master/Droplets/plots/droplets.
The quantity shown in each panel of the figure is denoted in the top left corner, where N H2 is the Herschel column density, T dust is the Herschel dust temperature, T peak is the NH 3 brightness, σ NH3 is the observed NH 3 velocity dispersion, V LSR is the velocity centroid from fitting the NH 3 (1, 1) hyperfine line profile, and T kin is the kinetic temperature from fitting the NH 3 (1, 1) and (2,2) profiles (thus its smaller footprint due to the lack of detection of the NH 3 (2, 2) emission at some pixels).
The thick contour (black or white) in each panel marks the outline of the mask used to define the boundary of the droplet. The crosshair and the circle (red or blue) show the position centroid and the effective radius (R eff ) of the droplet. The red contour in the panel that shows the observed NH 3 velocity dispersion (σ NH3 ) corresponds to the outline of the regions where velocity dispersion is found to be subsonic.
Due to the varying contrast, we adjust the color scale used in each panel from droplet to droplet, but the span of the color scale between the two extreme colors remains the same for each quantity across different droplets. The grayscale used to plot N H2 ranges from lower column density in lighter gray to higher column density in darker gray and spans a total of one order of magnitude in column density (from white-the lowest column density, to black-the highest column density). The color scale used to plot T dust ranges from lower dust temperature in darker orange to higher dust temperature in lighter yellow and spans a total of six degrees in dust temperature (from dark red-the lowest dust temperature, to light yellow-the highest dust temperature). Similar to the grayscale used to plot N H2 , the grayscale used to plot T peak spans a total of an order of magnitude in NH 3 brightness (from white-the lowest T peak , to black-the T peak ). The color scale used to plot σ NH3 is fixed and shows observed NH 3 velocity dispersion between 0.05 km s −1 (light yellow) to 0.40 km s −1 (dark blue). The color scale used to plot V LSR ranges from more redshifted V LSR in red to more blueshifted V LSR in blue. Similar to the color scale used to plot T dust , the color scale used to plot T kin spans a total of six degrees in dust temperature (from dark red-the lowest kinetic temperature temperature, to light yellow-the highest kinetic temperature temperature).
The physical scale on the plane of the sky is noted by the horizontal line in the top right corner. The black circular area in the lower left corner corresponds to the GAS beam at 23 GHz.

C. DROPLETS AT POSITIONS OF DENSE CORES AND OTHER KNOWN STRUCTURES
Two of the 18 droplets defined in §3.1 are found near the positions of two dense cores observed and analyzed by Benson & Myers (1989), Goodman et al. (1993), andLadd et al. (1994). These are L1688-d11 and B18-d4, with centroid positions found within one GBT FWHM beam size (32 ) of the centers of L1696A and TMC-2A, respectively. Fig. 25 shows how the basic properties measured in this work using data from the Green Bank Ammonia Survey compare to properties measured by Benson & Myers (1989), Goodman et al. (1993), andLadd et al. (1994). We note that the observations done by Benson & Myers (1989) and Ladd et al. (1994) did not spatially resolve the "transition to coherence" , as was done by Pineda et al. (2010) for B5. For reference, the spatial resolution of the observations done by Benson & Myers (1989) and Ladd et al. (1994) is a factor of ∼ 2.5 coarser than that of modern GBT observations. The velocity resolution (at 23 GHz) of the observations done by Benson & Myers (1989) and Ladd et al. (1994) ranges from 0.07 to 0.20 km s −1 , compared to 0.07 km s −1 of the GBT observations done by the Green Bank Ammonia Survey (Friesen et al. 2017). See §2 for details.
Other than these two droplets that can be associated one-to-one with previous core structures observed in NH 3 emission, other droplets might be associated with previously known cores or density features. A full comparison with previous observations requires careful treatment of differences in observational setups and the methods used to define the structures, and such a comparison is beyond the scope of this paper. Here we list previously known cores and density features potentially associated with droplets in Table 7 and Table 8, based on a thorough search of the SIMBAD Astronomical Database 13 . Figure 25. (a) The mass-size distribution of the droplets identified in this work (blue circles) and the dense cores examined by Goodman et al. (1993) (green circles). The droplets found at positions of known dense cores and the corresponding dense cores are highlighted and connected by black lines. These are L1688-d11, found at the position of L1696A, and B18-d4, found at the position of TMC-2A. As in Fig. 10a, the black line shows a power-law relation between the mass and the effective radius, and randomly selected 10% of the accepted parameters in the MCMC chain used to find the power-law fit are plotted as transparent lines for reference. The solid gray line shows the empirical relation based on observations of larger-scale structures examined by Larson (1981). (b) The σtot-size distribution of the same structures shown in (a). As in Fig. 10b, the horizontal lines show σtot expected for structures where the non-thermal component is equal to the sonic speed (cs; thicker line) and half the sonic speed (thinner line) of a medium with a mean molecular weight of 2.37 u at a temperature of 10 K. (c) Gravitational potential energy, ΩG, plotted against internal kinetic energy, ΩK (Equation 6), for the same structures shown in (a). As in Fig. 11a, the red band from the lower left to the top right marks the equilibrium between ΩG and ΩK (solid red line) within an order of magnitude (pink band), according to the virial equation (Equation 6; omitting the pressure term). The black line marks where the conventional virial parameter, αvir, has a value of 2 (Equation 9). (d) The energy term representing the confinement provided by the ambient gas pressure, ΩP, plotted agains the internal kinetic energy, ΩK (Equation 6), for the same structures shown in (a). As in Fig. 11b, the red band from the lower left to the top right marks an equilibrium between ΩP and ΩK (solid red line) within an order of magnitude (pink red band), according to the virial equation (Equation 6; omitting the gravitational term). Structures in the parameter space above the red line (equilibrium) are expected to be dominated by the ambient gas pressure.

D. UNCERTAINTY IN THE RADIUS MEASUREMENT
The uncertainty in the radius measurement lies in two aspects. First, the radius measurement is limited by the intrinsic resolution of the observations. In the case of this paper, since the droplet boundary is defined by the change in linewidth based on GBT observations of NH 3 emission, the uncertainty in the radius measurement scales with the pixel size of the linewidth map. For the Nyquist-sampled linewidth map produced by Friesen et al. (2017), the pixel size equals to ∼ 0.007 pc at the distance of L1688 and B18.
Second, since the droplet boundary is not perfectly circular, assigning a single number to describe the size (radius) of the droplet boundary is subject to the uncertainty due to the non-circular shape of the boundary. In this paper, we estimate the lower and the upper bounds of the radius by measuring the radius of the largest circle that can be enclosed by the droplet boundary and the smallest circle that can enclose the droplet boundary, respectively. See Fig.  26. Figure 26. Droplet L1688-d6 as an example of how the lower and the upper bounds of the radius measurement are defined. The droplet is shown as an irregular hatched area. The red circle shows the effective radius, R eff , derived from the principal component analysis (PCA) weighted by the peak NH3 brightness, T peak (see §3.2 and Equation 1). The inner blue circle marks the largest circle that can be enclosed by the droplet boundary, which we use as the lower bound in the radius measurement. The outer blue circle marks the smallest circle that can enclose the droplet boundary, which we use as the upper bound in the radius measurement. Notice that the circles used in determining the lower and the upper bounds are required to center at the position centroid of the droplet (the positions listed in Table 1).
The difference between R eff and the lower or the upper bound of the radius is then required to be larger than the uncertainty due to the finite resolution of the GBT observations (i.e., 0.02 pc at the distances of L1688 and B18). The resulting uncertainty is listed in Table 1.
Besides using the principal component analysis (PCA; see §3.2) to find the radius, another common way to determine "effective radius" for a non-circular shape is to measure the projected area, A, and find the radius of the circle that has the same area (e.g., Rosolowsky & Leroy 2006). The effective radius found through the projected area is then: For each droplet, R eff,A lies within the range between the lower and the upper bounds determined using the enclosed circles (see above and Fig. 26), and deviates by less than 10% from R eff , determined from the PCA. For example, this translates to 10% of difference in the gravitational potential energy, Ω G . In the analyses presented in this paper, the uncertainty in R eff determined using the lower and the upper bounds (Fig. 26) is propagated to the uncertainties of other quantities, which are shown in corresponding plots and tables. Since R eff,A lies between the lower and the upper bounds determined for the radius measurement, using R eff,A , instead of R eff , in these analyses will only have an effect within the uncertainty reported throughout this paper.

E. BASELINE SUBTRACTION
In the analyses presented in this paper, the mass and related quantities such as the density are estimated after a baseline correction for the line-of-sight material outside the targeted volume. The method we apply in this paper is similar to the "clipping paradigm" examined by Rosolowsky et al. (2008b) to estimate the physical properties of a compact structure. Fig. 9 schematically demonstrates how the clipping paradigm can be a reasonable way to remove the contribution to column density measurements from the material along the same line of sight but outside the targeted structure. In the virial analysis presented in §3.3 and in the analyses of the radial density profiles presented in §4.1.1 and §4.1.2, we use the same method to estimate the density of the gas surrounding the volume under discussion.
In comparison, a simple sum of column densities measured within a certain projected area on the plane of the sky overestimates the mass by including the contribution from the material along the entire line of sight (see Fig. 27). Similarly, when estimating the average density within a shell-shaped volume surrounding the targeted structure (as done in §3. 3.3), summing the column densities measured within a ring-shaped area on the plane of the sky overestimates the mass and thus, the density (different shades of gray in Fig. 27). The typical difference between the mass estimated after applying the clipping and the mass estimated without any clipping (a simple sum) is ∼ 25%. In the virial analysis presented in §3.3, this amounts to a ∼ 50% of uncertainty in the estimate of the gravitational potential energy and a ∼ 25% of uncertainty in either of the kinetic energy and the energy term representing the ambient gas pressure confinement. These uncertainties are included in the uncertainties listed in Table 2 and do not qualitatively change the results presented in subsequent discussions. Figure 27. This cartoon shows the mass (integrated column density) on a column density distribution derived from a 2D column density map (left; with the vertical axis corresponding to the column density) and the corresponding volumes in the 3D space (right; with the line of sight along the vertical axis), if no baseline removal is applied. The solid shaded area (in black/dark gray/gray) corresponds to the mass (integrated column density) estimated from the 2D column density map (left) and the material occupying the corresponding volume in the 3D space (right).
More sophisticated ways to remove contributions from material in the foreground and background may involve removing contributions from column density structures larger than a certain size scale, for example, using a transform algorithm like the wavelet decomposition. While such algorithms perform well in analyses of compact structures, the uncertainty becomes unclear if we are interested in both the mass within the targeted structure (as we are in §3.2) and the density of the surrounding material (as we are in §3. 3.3). A single background removal can result in an overestimated mass of the structure at the center (the black areas in Fig. 28, compared to Fig. 9), and when estimating the density of the surrounding material, a single background removal would give an estimate for a hollow cylindrical volume instead of the shell-shaped volume (different shades of gray in Fig. 28, compared to Fig. 9). While theoretically, we can resolve the issue by optimizing the transform algorithm to perform differently for different purposes, we adopt the clipping method in this paper 1) to fully avoid double counting the contribution from the material in the same volume when estimating the mass in and outside a structure and 2) for its simplicity and ease of error estimation. As demonstrated in Fig. 9, if the structure is spherical, the clipping method would give the exact masses for the layers of materials at different radial distances. Figure 28. This cartoon shows the mass (integrated column density) on a column density distribution derived from a 2D column density map (left; with the vertical axis corresponding to the column density) and the corresponding volumes in the 3D space (right; with the line of sight along the vertical axis), if a single baseline removal is applied. The solid shaded area (in black/dark gray/gray) corresponds to the mass (integrated column density) estimated from the 2D column density map (left) and the material occupying the corresponding volume in the 3D space (right). Schematically, a single baseline removal corresponds to the "subtraction" of column density below the dashed line in the panel on the left. For a structure with a spherical shape, such subtraction corresponds to the removal of the contribution from material outside the dashed line in the panel on the right. Depending on the algorithm used for the baseline removal, the "subtraction" might not exactly correspond to the removal of a flat-top function as shown in this cartoon. The cartoon is used only to demonstrate how a single baseline removal does not fit the purpose of simultaneously estimating the masses within different layers at different radial distances.
The main uncertainty resulted from using the clipping method with circular annuli (as done in §3.3.3, §4.1.1, and §4.1.2) is then the deviation in the shape of the targeted structure from a sphere. As mentioned in §3.1, most of the droplets have aspect ratios between 1 and 2, with the exceptions of L1688-d1 with an aspect ratio of ∼ 2.50, L1688-d6 with an aspect ratio of ∼ 2.52, and B18-d5 with an aspect ratio of ∼ 2.03. Thus, we estimate that the uncertainty resulted from using circular annuli is no larger than a factor of 2. See discussions in Appendix D.

F. UNCERTAINTY IN MASS DUE TO THE POTENTIAL BIAS IN SED FITTING OF HERSCHEL OBSERVATIONS
Lastly, we examine the uncertainty in the potentially biased SED fitting of the Herschel observations. As presented above, we use the column density map obtained via SED fitting of Herschel observations to estimate the mass of the droplets ( §2.2). Using the Herschel column density to estimate the mass and the fits to the NH 3 line profiles to estimate the velocity dispersion allows mutually independent measurements of the mass-size and the mass-linewidth relations (Fig. 10). However, it is a known issue that SED fitting of emissions in Herschel bands might be biased, especially towards cold and dense regions (Shetty et al. 2009a,b;Kelly et al. 2012). In the cold and dense regions, there can be a certain degree of redundancy between a high dust temperature and a high column density. As a result, the SED fitting can overestimate the dust temperature and underestimate the column density, which would result in underestimated masses for the droplets in this case.
Consistently with what Friesen et al. (2017) pointed out, Fig. 29a shows that the dust temperature, T dust , is systematically 2 to 3 K higher than the kinetic temperature of the dense gas traced by NH 3 emission, T kin . Fig. 29a also shows that, for pixels within the boundaries of the droplets, the difference between T dust and T kin could be even larger, up to 6 K. Fig. 29b further shows that, for pixels within the boundaries of the droplets, not only is the difference between T dust and T kin larger than the median value of the entire cloud, the pixel-by-pixel NH 3 abundance obtained by dividing the N NH3 (from fitting the NH 3 hyperfine line profiles; see Friesen et al. 2017) by N H2 (from SED fitting of Herschel observations) is higher than the cloud median. The distribution is consistent with overestimated temperature and underestimated column density in the SED fitting of Herschel observations.
In this section, we try to estimate the effects of underestimated column density in the SED fitting of Herschel observations. In particular, we examine the effects of the underestimated column density on the virial analysis presented in §3.3. We compare N H2 , obtained via the SED fitting of Herschel observations, to N NH3 , obtained via fitting the NH 3 hyperfine line profiles (see Friesen et al. 2017, for details). Unfortunately, for the model used in the NH 3 hyperfine line fitting, we need detections of emission from both the NH 3 (1, 1) and (2,2) lines to determine the population ratios between the two states, in order to estimate N NH3 and T kin . While all pixels within the droplets are detected in NH 3 (1, 1) emission, not all pixels within the droplets are detected in NH 3 (2, 2) emission, so we can only obtain estimates of N NH3 in the densest regions within the droplets. Thus, estimating the mass solely from N NH3 is difficult, especially given that we expect the column density to decrease toward the outer edge of a droplet.
Thus, in order to assess the potential bias in the column density obtained via SED fitting of Herschel observations, we used the pixels within the droplet boundaries where we have measurements of both N NH3 and N H2 (i.e., the pixels where we have significant detection of NH 3 (2, 2) emission). We compare the abundance of the droplets (obtained by dividing N NH3 by N H2 ) to the median value of the cloud and assume that the difference in abundance between the droplet values and the cloud median is fully due to the underestimated N H2 in the droplets. Assuming the underlying, "real" NH 3 abundance, X real ≡ N NH3,droplet /N H2,real , is equal to the median NH 3 abundance of the cloud, X cloud = N NH3,cloud /N H2,cloud , we calculate a correction factor, : where N H2,real is the underlying, "real" column density, and N H2,droplet is the column density measured from the SED fitting for pixels within the droplet boundaries. We can then estimate the "real" mass using this correction factor: where M real is the underlying, "real" mass, and M droplet is the measured mass of the droplet (from Herschel column density; §2.2). After applying the correction factor to the mass, Ω G and Ω K in the virial analysis ( §3.3) are changed by ratios ∝ 2 and ∝ , respectively. The lefthand panels of Fig. 30 show the change as arrows, on top of the original mass-size relation (Fig. 10a) and the original comparisons between various terms in the virial analysis presented in §3.3 (cf. Fig. 11 and Fig. 12). The righthand panels of Fig. 30 show the resulting plots after applying the correction. Fig.  30a-2 shows that the mass-size relation after the correction is very slightly less steep and closer to what is found for cloud-scale structures M ∝ R 2 . Fig. 30b-2 shows that, after the correction, a total of 6 droplets (out of 18 droplets identified in §3.1) are now gravitationally "bound" based on the conventional criterion of α vir = 2, as opposed to only 2 droplets that were bound before applying the correction. The correction on the gravitational potential energy also makes the gravitational term, Ω G , in the virial analysis appear more comparable to the ambient pressure term, Ω P (Fig. 30d-2).
The correction also affects the normalized radial profiles of density. The characteristic radius, r c , which is used to normalize the radial distance from the center is dependent on the density measured at the center of a droplet and Figure 29. A comparison between properties based on fitting NH3 hyperfine line profiles and from the SED fitting of the Herschel observations, using the L1688 region as an example. (a) Pixel-by-pixel distributions of temperatures measured from fitting the NH3 hyperfine line profiles (kinetic temperature, T kin ; left) and from SED fitting of Herschel observations (the dust temperature, T dust ; right). Only pixels with significant detection of NH3 (1, 1) emission are included in this plot (i.e., the total number of pixels included on the left half of the plot is the same as that on the right). The distributions are normalized by the total number of pixels in each group (the cloud or the droplets), so the height of each bin in the histograms correspond to the frequency of a certain range of values occurring in each group (shown along the horizontal axis in percentage). The gray histograms show the distributions of all the pixels outside the boundaries of the droplets, and the blue histograms show the distributions of all the pixels inside the droplet boundaries. The median of each distribution is shown as a horizontal line. (b) 2D histogram showing the distribution between the difference between the dust temperature and the kinetic temperature (T dust − T kin ), as a function of the NH3 abundance (XNH 3 = NNH 3 /NH 2 , where NH 2 is derived from the SED fitting of the Herschel observations). The 2D histogram in each panel shows the distribution of pixels with significant detection of NH3 (1, 1) emission in the entire map, with the pixel frequency defined as the percentage of pixels on the map falling in each 2D bin in the 2D histogram. The blue dots show the distribution of individual pixels within the droplet boundaries. The horizontal line shows where T dust = T kin , and the white cross marks the median values of the abundance and the difference in temperature of the entire cloud.
is changed by a ratio of ∝ 1/ √ , making r c smaller and consequently x larger. On the other hand, y ≡ ρ/ρ cen is dimensionless and is thus not affected by the correction on the mass (or equivalently, on the density). The resulting change makes the normalized density profiles look even shallower and closer to ρ ∝ r −1 at larger distances (Fig. 31).
Overall, the correction does not qualitatively alter the results of the analyses presented in §3.2 and §3.3. With the corrected mass, the droplets still appear to follow the same power-law mass-size relation found for dense cores (Fig. 30a-2). Most of the droplets are still gravitationally unbound (Fig. 30b-2), and the ambient pressure remains important in confining the droplets (Fig. 30c-2). And, the density profiles of the droplets appear even shallower and seem to remain continuous from the typical density profile found for cloud-scale structures (Fig. 31a-2).
Again, since we did not detect NH 3 (2, 2) emission everywhere within the droplet boundaries with the GAS observations, and since we hope to examine the mass-size relation with each term independently measured, we base the analyses presented in this paper on the mass estimated from the Herschel column density. The column density obtained from the SED fitting of Herschel observations also gives better estimates of the radial density profiles outside regions where we find the dense gas (as traced by NH 3 emission). The "correction" examined in this section serves only as an estimate of the uncertainty in the SED fitting of Herschel observations, and we emphasize that a more sophisticated approach is needed to further determine the effects of varying NH 3 abundances due to changes in the astrochemical environments. Figure 30. The left four plots show the change in the physical properties corresponding to the correction using NNH 3 from the fits to the NH3 hyperfine line profiles, plotted on top of the original scatter plots. The right four plots show the data points after applying the correction, along the same axes. (a-1) The mass-size distribution of the original data points before the correction, as shown in Fig. 10a, with the change in mass due to the correction shown as arrows. (a-2) The mass-size distribution after applying the correction, plotted along the same axes and with the same data points for dense cores (not affected by the correction). (b-1) The gravitational potential energy plotted against the internal kinetic energy, as shown in Fig. 11a. The change in both quantities is plotted as arrows. (b-2) The gravitational potential energy plotted against the internal kinetic energy, after applying the correction, plotted along the same axes and with the same data points for dense cores (not affected by the correction). (c-1) The sum of the gravitational potential energy and the energy due to the pressure exerted on the cores by the thermal and non-thermal motions of the ambient gas, plotted against the internal kinetic energy, as shown in Fig. 12a. The change due to the correction in both quantities are plotted as arrows. (c-2) The sum of gravitational potential energy and the ambient pressure energy term, plotted against the internal kinetic energy, after applying the correction, plotted along the same axes and with the data points for dense cores (not affected by the correction). (d-1) The gravitational potential energy, plotted against the energy due to the pressure exerted on the core surfaces by the thermal and non-thermal motions of the ambient gas, as shown in Fig. 12. The change in the gravitational potential energy due to the correction is plotted as arrows. (d-2) The gravitational potential energy, plotted against the ambient pressure energy, after applying the correction, plotted along the same axes and with the data points for the dense cores (not affected by the correction).  (Equation 16), which is dependent on ρcen. No correction is applied on y ≡ ρ/ρcen, assuming that the same correction factor is applicable on the volume density measured at different distances from the centroid position of a droplet.

G. RADIAL PROFILES IN PHYSICAL UNITS
In §4.1.1, we examine the radial profiles and compare them to the Bonnor-Ebert sphere (Ebert 1955;Bonnor 1956;Spitzer 1968). Following the dimensionless analysis in Spitzer (1968), we show that the radial density profiles of the droplets are generally shallower than the Bonnor-Ebert sphere (Fig. 13). The analysis in dimensionless units allows comparisons between droplets of different sizes ( §4.1.1). Fig. 32 shows the radial profiles of the volume density and the pressure in physical units. Again, it demonstrates that the comparison between droplets of different sizes is difficult in the physical units. See also Fig. 8 for the radial profile of linewidths in physical units. The radial pressure profiles of the droplets, similarly color coded by the ratio between the gravitational potential energy, ΩG, and the internal kinetic energy, ΩK. Here the pressure measurement is expressed in the unit of K cm −3 , as a ratio between the measured pressure and the Boltzmann constant. L1688-d2, L1688-d5, and L1688-d6 are specifically marked, either because the droplets are elongated or because they sit near the edge of the regions where NH3 (1, 1) is detected, both of which render the profiles potentially biased. Since the dumbbell shape of L1688-d1 affects this analysis which assumes spherical geometry, L1688-d1 is not included in this plot.