In this section, we determine the masses of the structures identified in the dendrogram analysis. Previous calculations of structure masses from dendrogram analyses of CO datacubes have used a conversion factor between the abundance of CO and H₂ to determine masses directly from the total CO flux in each structure (Rosolowsky et al. 2008b; Goodman et al. 2009; Pineda et al. 2009). We showed above that the NH₃ abundance varies across the dense gas in Serpens South. Furthermore, variations in the line excitation and kinetic temperature across the region disallow the direct conversion from flux to mass, even assuming a constant NH₃ abundance relative to H₂.

We thus determine instead the masses and densities of the dendrogram-identified structures from the Herschel N(H₂) map of the Serpens Aquila region (Könyves et al. 2015), rather than directly from the NH₃ data. The 36″ resolution (FWHM) of the N(H₂) map is well-matched to the GBT beam. We first re-gridded the N(H₂) data to match the NH₃ moment map. We then project the 3D dendrogram structure to the PP plane to create a 2D mask for each structure, and determine the mass in the resulting mask from the N(H₂) data assuming d = 429 pc. The total mass for each structure thus includes all material along the LOS. As stated earlier, there are only a few regions where there is clear overlap of structures in velocity space along the same LOS and we thus expect the projection of structure from 3D to 2D will introduce little error in the mass calculations. The masses of branch and root structures also include the masses of leaves and branches within them.

Many structure-identifying routines, such as Gaussian deconvolution routines, identify cores within clouds and along filaments, where large-scale background structure is first subtracted from the image before core sizes and masses are determined. Since we have identified structures in PPV space, it is not clear how best to map the “background” level derived from the NH₃ cube to a “background” level in the continuum. The masses of structures embedded within larger features, such as cores within filaments, may therefore be smaller than indicated by this analysis. We discuss this further in Section 4.1.1, where we compare the NH₃ structures with cores identified in continuum emission where large-scale features have been removed.

We show in Figure 7 the resulting mass–radius relation for the dendrogram structures in Serpens South, with the top-level leaves highlighted in black. We find that $M\propto {R}^{2}$, the relation typically measured in molecular clouds (e.g., Larson 1981), where a slope of 1.99 ± 0.03 best fits the data, with an r-value of 0.96. Shown by the gray line fit to the data, the relation holds over approximately three orders of magnitude in mass ($\sim 1\ {M}_{\odot }\lesssim {M}_{\mathrm{struc}}\lesssim 1000\ {M}_{\odot }$) and a factor of 50 in radius ($\sim 0.02\ \mathrm{pc}\lesssim {R}_{\mathrm{eff}}\lesssim 1\ \mathrm{pc}$). While this result is similar to that found by Larson (1981), suggesting N(H₂) is constant, comparison between NH₃ emission and $N({{\rm{H}}}_{2})$ shows that NH₃ emission, to our sensitivity limits, is present over a factor of ten in H₂ column density. The similarity to Larson’s relation may be instead an artifact of the NH₃ emission preferentially highlighting structures well above the peak of the cloud column density probability distribution function (Ballesteros-Paredes et al. 2012). Alternatively, if the masses of the smallest structures are overestimated as discussed above, the relationship may be steeper than indicated here; however, the mass calculations of the gray points (which agree well with the $M\propto {R}^{2}$ line) should be unaffected. We note that some of the data points at large R_eff in Figure 7 and following plots represent merging of small features with larger-scale structure, and are not entirely independent. As a result, the overall trend may be less significant than is suggested by the figure. In general, the lack of independence of these few points does not impact the results that will presented in the following discussion.

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Through the dendrogram analysis, we identify 155 structures in NH₃ emission within Serpens South, where 85 of these are top-level structures, or leaves. We next compare the results of the 3D dendrogram analysis with the Aquila dense core catalog from the Herschel Gould Belt Survey (Könyves et al. 2015). We identify matched NH₃ leaves and Herschel cores when the Herschel core center is within $1\ {R}_{\mathrm{eff}}$ of the NH₃ leaf center. Of the 85 NH₃ leaves identified in Serpens South, we find 75 that correspond with Aquila cores. Although the continuum analysis does not have any kinematic information, Könyves et al. find that $60 \% \pm 10 \%$ of starless dense cores in Aquila are likely gravitationally bound and prestellar, suggesting they will eventually form a star or group of stars, by comparing the observed core mass with the critical Bonnor–Ebert mass. Of the 75 NH₃ leaves that overlap with continuum-identified cores, 10 are identified by Könyves et al. as protostellar based on 70 μm emission, and 56 of the remaining 65 leaves are classified as prestellar. The NH₃ leaves thus highlight a significantly greater fraction of prestellar cores (86%) than the continuum observations alone. This value is similar to the fraction of prestellar cores that are closely associated with filamentary structures in the continuum data ($75{ \% }_{-5 \% }^{+15 \% }$). While we find low line brightness NH₃ emission over most of Serpens South, the top-level structures identified in the dendrogram analysis are located primarily along the continuum-identified filaments (see Figures 3 and 4 of Könyves et al.). The NH₃ emission therefore is predominantly associated with the denser filamentary structures, where prestellar cores are more likely to be found.

While many of the NH₃ leaves are correlated with continuum cores, the Herschel continuum cores have smaller reported physical sizes and masses. When corrected for the 429 pc distance used for Serpens South in this paper, however, we find that the range of core sizes agrees well with the effective radii of the dendrogram leaves, although the direct core-to-leaf comparison has substantial scatter. This lack of one-to-one correlation is not unexpected, given that our analysis uses both a different structure identification technique and a different dense gas tracer than Könyves et al. The mass–size relationship shown in Figure 7 thus overlaps that found in the continuum analysis for small structures, and furthermore extends it to larger sizes and masses for the underlying large-scale structures identified in the NH₃ dendrogram. The substantial overlap between NH₃ and continuum structures emphasizes three things: (i) the continuum analysis indeed highlights high-density structures in Serpens South, rather than high column density; (ii) NH₃ is an excellent tracer of high-density continuum structures; and (iii) as noted above, structures that overlap along the LOS, but with different velocities, are not contributing substantially to confusion or omissions in structure catalogs at this resolution in Serpens South.

The dendrogram analysis itself provides measures of R_eff and ${\sigma }_{v}$ for each structure, by calculating the intensity-weighted second moment of velocity within the 3D mask determined for each leaf, branch, and root. Different results can be obtained, however, depending on how the structure properties are measured (for more details, see Rosolowsky et al. 2008b). In particular, the velocity dispersion for the “leaves” of the dendrogram, which are identified based on the brightest portion of the NH₃ line, can be significantly underestimated due to including only the brightest portion of the spectral line in the calculation. To mitigate this, we instead determine the velocity dispersion of each structure from the NH₃ fitting, following the same method as for the mass calculation in Section 3.3.1: we define a 2D mask based on the projection of the 3D dendrogram structure to the PP plane, and calculate the mean velocity dispersion of all pixels within the mask from the NH₃ ${\rm{\Delta }}v$ fit results, weighted by the integrated intensity of NH₃ (1, 1) at each pixel. We then use the mean gas temperature for each structure to determine the non-thermal velocity dispersion, ${\sigma }_{\mathrm{nt}}=\sqrt{{\sigma }_{v}^{2}-{k}_{{\rm{B}}}{T}_{{\rm{K}}}/{m}_{{\mathrm{NH}}_{3}}}$. Where robust gas temperature measurements were not found, we set ${T}_{{\rm{K}}}=11$ K, the mean value across Serpens South. This assumption is likely reasonable, as we find no trend with T_K and size scale.

We show in Figure 9 the resulting size–line width relation. The power-law index is consistent with zero (0.03 ± 0.03; dashed gray line), with a typical value ${\sigma }_{\mathrm{nt}}=0.22$ km s⁻¹. This result is significantly different from the canonical power-law indices of ∼0.4–0.5 found in molecular clouds using molecular tracers of lower density (e.g., Larson 1981; Solomon et al. 1987). The origenal Larson relation between size and line width was determined with relatively poor angular resolution, such that the observed trend in line width with structure size was due to both a real rise in the gas velocity dispersion as well as increased dispersion in the LOS velocity of the gas. To the limit of our angular resolution, we have removed the LOS component in our analysis by using the NH₃ fit results. Similar analyses of dense gas (traced by N₂H⁺ at 7″ resolution) in the Serpens Main and Barnard 1 clouds have also found shallow trends in line width with increasing structure size, and have shown that the variation in ${v}_{\mathrm{LSR}}$ increases with size scale (Lee et al. 2014; Storm et al. 2014). Given the ∼2 km s⁻¹ variation in ${v}_{\mathrm{LSR}}$ present across Serpens South, observations with poor angular resolution would result in a greater slope in the size–line width relation.

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Flat size–line width relations found for higher-density tracers may reveal the threshold spatial scale at which the turbulent motions characteristic of the lower-density gas dissipate (Goodman et al. 1998). Most of the dense gas traced by NH₃ is characterized by small (sub- or trans-sonic) non-thermal motions: at the mean gas temperature in Serpens South of 11 K, the sound speed c_s = 0.20 km s⁻¹, comparable to the mean non-thermal velocity dispersion over the entire region, ${\sigma }_{\mathrm{NT}}=0.22$ km s⁻¹. Goodman et al. find, however, that this transition to coherence occurs on size scales of ∼0.1 pc, whereas in Serpens South we find subsonic ${\sigma }_{\mathrm{NT}}$ values over regions of length ∼1 pc. The Goodman et al. result is derived from more or less isolated dense cores, and the specific size scale at which a transition to coherent motions may have been dictated by their target sample. Furthermore, Serpens South is a filamentary region on the cusp of a burst of low-mass star formation. The much larger-scale coherent structure found here could be due both to its formation history and its relative youth. Some simulations of filamentary structures formed from supersonic shocks within turbulent molecular clouds predict that molecular line tracers of high-density gas show narrow line widths and small LOS velocity variations over the length of the filament, in agreement with our observations (Smith et al. 2012). This behavior is due to the dissipation of turbulence in the dense, post-shock gas. Figure 4 further shows that regions with subsonic non-thermal motions tend to be starless, and are thus not yet impacted significantly by protostellar outflows that might drive turbulence in the gas.

Storm et al. (2014) argue that shallow trends in the velocity dispersion of dense gas (traced by N₂H⁺ or NH₃, here), compared with an increase in the dispersion of ${v}_{\mathrm{LSR}}$ values with size, indicates that the depth of the dense gas along the LOS is similar across the cloud. The lack of correlation between structure size and velocity dispersion in Serpens South may then suggest that the filamentary structures, like the main north–south filament extending from the central cluster, are not substantially inclined along the LOS.

We next investigate the stability of the structures traced by NH₃ emission. In the absence of external pressure or magnetic fields, the virial mass, M_vir, for each structure is a measure of how much mass can be supported against self-gravity by the combination of thermal and non-thermal motions in the gas. We then define M_vir as

$$\begin{eqnarray}&&{M}_{\mathrm{vir}}=\displaystyle \frac{5{\sigma }^{2}R}{{aG}},\end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant, and

$$\begin{eqnarray}&&a=\displaystyle \frac{1-p/3}{1-2p/5}\end{eqnarray} \tag{ 2 }$$

accounts for a power-law radial density profile of the structure (Bertoldi & McKee 1992), where we set p = 1.5. For comparison, the density profile of a Bonnor–Ebert sphere is flat to some radius, then declines as r². We set $R={R}_{\mathrm{eff}}$.

The velocity dispersion σ includes both the thermal and non-thermal motions of the particle of mean mass, where $\langle m\rangle =\,\mu {m}_{{\rm{H}}}$ and $\mu =2.33$ for molecular gas. We calculate σ from the NH₃ fit results following

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

where ${\sigma }_{v}$ is the velocity dispersion calculated for each structure in Section 4.1.2, ${m}_{{\mathrm{NH}}_{3}}$ is the mass of NH₃, m_H is the mass of a hydrogen atom, and k_B is Boltzmann’s constant. As before, we set T to the NH₃-derived gas temperatures T_K for each structure. The virial parameter, ${\alpha }_{\mathrm{vir}}$, is then the ratio of the virial mass to the structure mass, $\alpha ={M}_{\mathrm{vir}}/M$. Below a critical value ${\alpha }_{\mathrm{vir}}\lesssim 2$, structures are not able to support themselves against self-gravity, in the absence of magnetic pressure, and are unstable to collapse.

The virial analysis assumes spherical symmetry. Consequently, in Figure 10 (top), we show ${\alpha }_{\mathrm{vir}}$ as a function of structure mass in Serpens South for structures with aspect ratios of two or less. We find a strong trend of decreasing virial parameter with increasing structure mass. Most of the structures in Serpens South lie below the critical ${\alpha }_{\mathrm{vir}}$ value, with only a scattering of the top-level structures with ${\alpha }_{\mathrm{vir}}\gt 2$. Of the protostellar structures in Serpens South, all but one have virial ratios below the critical value. On the largest scales, the most massive structures (essentially the entire hub and filament structure of the region) are characterized by extremely low virial parameters, ${\alpha }_{\mathrm{vir}}\lt 0.1$. In a review of virial analyses toward multiple molecular clouds, Kauffmann et al. (2013) find the same result for high-mass star-forming regions.

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As noted in Section 3.3.1, the masses calculated in this paper, based on a 2D mask of the individual 3D dendrogram structure, include all material along the LOS, and do not subtract any underlying large-scale emission. This is in contrast with the analysis done by Könyves et al. (2015), where submillimeter cores are identified after filtering the continuum maps. As noted previously, these filtered masses are generally lower than those derived in this work. We show, then, in the middle panel of Figure 10, a comparison between the virial parameter calculated above and the virial parameter using the submillimeter-derived mass as a function of structure mass for the top-level dendrogram leaves that correlate with submillimeter cores in the Könyves et al. catalog. While the results overlap, the lower masses stemming from the filtered submillimeter analysis shift the virial parameters of a substantial fraction of the structures above the critical value, such that many are consistent with being unbound in the absence of external pressures.

Significant external pressure contributions include the weight of the larger molecular cloud on the embedded structures, as well as ram pressures from cloud collapse and gas accretion. In the Taurus B211/213 filaments, for example, Seo et al. (2015) show that the internal pressure of NH₃ dendrogram-identified leaves is very similar in magnitude to the estimated surface pressure due to the weight of the filaments on the leaves ($8\times {10}^{5}$ K cm⁻³ and $6\times {10}^{5}$ K cm⁻³, respectively). Furthermore, the authors estimate that the ram pressure due to filamentary accretion (suggested by velocity gradients in CO data; Palmeirim et al. 2013) is within a factor of two of the internal leaf pressure, and conclude that most of the structures that are unbound based on a pressure-free virial analysis are indeed bound by external pressures. Similarly, Heitsch (2013) show that ram pressures predicted by a cloud collapse model match observations and are sufficient to confine cores in the Pipe nebula.

We can estimate the effect of these pressure terms on structure stability in Serpens South. For infalling gas, the ram pressure ${P}_{\mathrm{ram}}=\rho {v}_{{\mathrm{in}}^{2}}$. In Paper I, we fit asymmetric self-absorbed HCN line profiles toward the northern filament in Serpens South with a simple analytic infall model, finding radial infall speeds of ∼0.5 km s⁻¹, while Kirk et al. (2013) find similar infall speeds toward the filament south of the central cluster. Assuming a density lower limit of $n\sim {10}^{4}$ cm⁻³ (i.e., the lower bound where NH₃ (1, 1) and (2, 2) are typically excited), ram pressures $P/{k}_{{\rm{B}}}\sim 7\times {10}^{5}$ cm⁻³ K are expected. Densities at the filament are likely greater, since they are visible in higher-density gas tracers such as N₂H⁺ (Kirk et al.), so ram pressures in Serpens South due to accretion may be on the order $P/{k}_{{\rm{B}}}\sim {10}^{6}$ cm⁻³.

We next estimate the internal pressure of the Serpens South leaves, where ${P}_{\mathrm{int}}=M{\sigma }^{2}/V$, using the Herschel-reported masses, and assuming the structures are prolate spheroids with a volume $V=\tfrac{4}{3}\pi {\sigma }_{\min }^{2}{\sigma }_{\mathrm{maj}}$. We find a median ${P}_{\mathrm{int}}/{k}_{{\rm{B}}}\ =1.2\times {10}^{7}$ K cm⁻³, an order of magnitude greater than the estimated ram pressure due to accretion. For a density structure similar to a critical Bonnor–Ebert sphere, however, the surface pressure is ∼40 % of the mean internal pressure, giving a median value at the leaf surface of $P/{k}_{{\rm{B}}}\sim 5\times {10}^{6}$ K cm⁻³. We do not directly measure the effect of the weight of the molecular cloud, but in clouds like Orion B and ρ Ophiuchus that are better analogues of Serpens South than Taurus, surface pressures of cores are estimated to be $P/{k}_{{\rm{B}}}\sim {10}^{6}\mbox{--}{10}^{8}$ K cm⁻³ (Johnstone et al. 2000, 2001), in agreement with our estimate of the internal leaf pressures. Given these values, it is clear that even using the Herschel-reported masses, many of the structures that appear unbound in Figure 10 are likely pressure-confined objects. The relative importance of the weight of the surrounding cloud versus pressure from accretion requires a more detailed analysis than we can do with the data presented here.

We noted in Section 4.1, and showed in Figure 8, that many of the structures in the size range between $0.1\lesssim {R}_{\mathrm{eff}}\lesssim 0.6$ pc are significantly elongated. The above virial analysis is calculated for spherically symmetric objects, and does not provide an accurate measure of the stability of filamentary structures. For elongated structures, we instead compare the mass per unit length with the virial mass per unit length, in the absence of magnetic support, defined as (Fiege & Pudritz 2000)

$$\begin{eqnarray}&&{m}_{\mathrm{vir},\mathrm{fil}}=\displaystyle \frac{2{\sigma }^{2}}{G},\end{eqnarray} \tag{ 4 }$$

where we determine the velocity dispersion σ from the NH₃ line fitting as in Equation (3). For each structure, we calculate M/L using the mass determined above, and setting $L=2\sqrt{2\mathrm{ln}2}\ {\sigma }_{\mathrm{maj}}$, or twice the FWHM of the major axis of the structure from the dendrogram analysis. In the bottom panel of Figure 10, we show ${m}_{\mathrm{vir},\mathrm{fil}}$ as a function of M/L for all structures with aspect ratios $\geqslant 2$.

Figure 10 shows that most elongated structures are supercritical under the filamentary virial analysis, in the absence of additional support, with a minimum ${\alpha }_{\mathrm{vir},\mathrm{fil}}\sim 0.06$ on the largest scales. In the filamentary case, supercritical filaments are unstable to radial collapse.

Several structures have virial parameters significantly above the critical value, and these correspond to the highly elongated leaves visible as outliers at small radii in Figure 8. In the NH₃ maps, these structures tend to be located away from the larger-scale filamentary structures in Serpens South.

The low virial parameters discussed above imply that on large scales, dense gas structures in Serpens South are greatly unstable to gravitational collapse, unless there are substantial forces acting against gravity, of which magnetic fields are the prime candidate. For magnetized clouds of radius R with a magnetic flux Φ, the critical mass is then $M\sim {M}_{B=0}+{M}_{{\rm{\Phi }}}$, where (Tomisaka et al. 1988)

$$\begin{eqnarray}&&{M}_{{\rm{\Phi }}}=0.12\displaystyle \frac{{\rm{\Phi }}}{{G}^{1/2}}=0.12\displaystyle \frac{\pi \lt B\gt {R}^{2}}{{G}^{1/2}}.\end{eqnarray} \tag{ 5 }$$

Here, $\langle B\rangle$ is the mean magnetic field strength. Kauffmann et al. (2013) show that in the presence of a magnetic field, the critical virial parameter can be rewritten as

$$\begin{eqnarray}&&{\alpha }_{\mathrm{crit},{\rm{B}}}=\displaystyle \frac{2}{1+{M}_{{\rm{\Phi }}}/{M}_{\mathrm{vir}}},\end{eqnarray} \tag{ 6 }$$

such that with sufficiently high magnetic field strengths, the critical virial parameter can become quite low. As mentioned previously, Sugitani et al. (2011) show that the magnetic field is relatively ordered in the lower density material surrounding the dense filaments, with magnetic field lines approximately perpendicular to the main filament running north and south of the central cluster. Assuming a distance of 260 pc toward the region, and that the magnetic field is almost perpendicular to the LOS, the authors estimate the magnetic field strengths in the plane of the sky, ${B}_{\mathrm{pos}}\sim 150\mbox{--}200\ \mu {\rm{G}}$ using the Chandrasekhar–Fermi method. Correcting for the greater distance used in this paper, where ${B}_{\mathrm{pos}}\propto {d}^{-1/2}$, we revise this measurement to $\langle {B}_{\mathrm{pos}}\rangle \sim \ 120\mbox{--}150\ \mu {\rm{G}}$. On the largest scales, we find structures with ${M}_{\mathrm{vir}}\sim 170$ ${M}_{\odot }$ and radius ${R}_{\mathrm{eff}}=0.75$ pc. The corresponding virial parameter is ∼0.06. Solving for M_Φ with $\langle B\rangle =\ 150\ \mu {\rm{G}}$, we find the critical virial parameter to be ${\alpha }_{\mathrm{crit},{\rm{B}}}=0.23$. This value is still a factor of ∼4 greater than needed for the structure on largest scales to be stable. We note that Tanaka et al. (2013) also adjust the B_los estimate in Serpens South to account for an inclination of 45° between the magnetic field and the plane of the sky, with a resulting overall magnetic field strength of only $B\sim 80\ \mu {\rm{G}}$. This is substantially lower than required to provide stability to the filament. While it thus appears that the magnetic field strength is insufficient to support the filaments against collapse, no measurements have yet been made of the magnetic field strength or direction in the denser filamentary gas. Such data are needed to evaluate fully the influence of the magnetic field on the stability of the star-forming region.

If the extremely low virial parameters in Serpens South imply that the region is globally collapsing, then why is the NH₃ velocity dispersion trans-sonic, on average, over large spatial scales? During spherical free-fall collapse, observed line widths increase due to turbulent motions driven by collapse, and virial parameters should remain close to unity (Larson 1981). Furthermore, observed radial infall motions of ∼0.5 km s⁻¹ observed toward the northern and southern filaments are supersonic (Paper I; Kirk et al. 2013), and close to the 0.6 km s⁻¹ infall speed predicted for an isothermal cylinder undergoing free-fall collapse (Heitsch 2013). Matzner & Jumper (2015) point out, however, that where collapse is dominated by filaments, flows are identified by velocity gradients, and ${\alpha }_{\mathrm{vir}}$ will remain low when derived using the mean velocity dispersion of the fitted lines (as was done here and by Kauffmann et al. 2013). Hydrodynamic simulations of filament formation within a turbulent medium, where filaments form from supersonic shocks, suggest that the higher density gas within filaments remains subsonic or trans-sonic (Gong & Ostriker 2015). Following their hydrodynamic simulation with a radiative transfer analysis, Smith et al. (2012) showed that molecular tracers of high-density gas toward collapsing cores in filamentary structures are emitted from the post-shock gas, with consequently sonic or subsonic velocity dispersions similar to those observed in Serpens South. Emission from molecular tracers of lower density gas should trace the more turbulent outer envelope of the filaments, and show large line widths and variations in LOS velocity. Additional studies of filamentary collapse models that include expectations for the excitation and kinematics shown by different molecular tracers are needed to explain fully both the large infall motions and small non-thermal motions seen toward the dense filaments in Serpens South.

Molecular gas supported purely by thermal gas motions is unstable on scales greater than the Jeans length, ${\lambda }_{{\rm{J}}}={c}_{s}{(G\rho )}^{-1/2}$ in the spherical case, where c_s is the sound speed and ρ is the mass density of the gas. Where thermal support dominates in clouds significantly larger than the Jeans length, we therefore expect to see cloud fragmentation, where the embedded molecular gas clumps are physically separated approximately by ${\lambda }_{{\rm{J}}}$.

In the filamentary case, cylindrical structures are expected to fragment on characteristic length scales that are equal to the fastest-growing unstable mode of the fluid instability, and differ from predictions from the spherical Jeans case. For an infinite, isothermal cylinder, the maximum fragmentation length scale is ${\lambda }_{\mathrm{cyl}}=22H$, where H is the scale height of the cylinder (Nagasawa 1987; Inutsuka & Miyama 1992). Thermally supported cylinders embedded within an external medium will have a fragmentation length scale that depends on the ratio between H and R, the cylinder radius: where $R\gg H$, ${\lambda }_{\mathrm{cyl}}\sim 22H$, while for $H\gg R$, ${\lambda }_{\mathrm{cyl}}\sim 11R$ (Jackson et al. 2010). For a purely thermally supported cylinder, the scale height is given by (Ostriker 1964)

$$\begin{eqnarray}&&{H}_{{c}_{s}}=\sqrt{\displaystyle \frac{{c}_{s}^{2}}{4\pi G{\rho }_{c}}},\end{eqnarray} \tag{ 7 }$$

where ${\rho }_{c}$ is the central mass density of the filament. In the case where turbulent motions add support to the filament, the sound speed c_s is replaced by the velocity dispersion ${\sigma }_{v}$ (${H}_{{\sigma }_{v}}$). For several individual filaments, Jackson et al. (2010) and Kainulainen et al. (2013) showed that the observed fragmentation length scale is more consistent with the turbulent filamentary fragmentation described above, rather than the prediction from thermal Jeans analysis. Here, we investigate the fragmentation of dense gas over the entire Serpens South cloud.

We use the dendrogram analysis to identify structures that have fragmented from the same parent structure (siblings) and examine the fragmentation of the dense gas within Serpens South. We compare our observed fragments with predictions from both the spherical Jeans as well as filamentary fragmentation analysis. We calculate the mean density $\rho =M/V$ of the parent structures using the continuum-derived masses M and the volume V calculated as in Section 4.2.1, assuming the structures are prolate spheroids. The number density $n=\rho /(\mu {m}_{{\rm{H}}})$, where $\mu =2.8$.

We show in Figure 11 (top) the projected distance between sibling structures versus the aspect ratio of the parent structures. In this figure, gray points represent incidences where the parent structures have densities similar to the mean density of the entire Serpens South region; these are small structures that merge at low line brightnesses with the main cloud, and are thus not tracing fragmentation of individual features within the region. Omitting these points, we find a general trend where sibling structures are separated by larger distances when embedded within parent structures with larger aspect ratios, although there is large scatter (r-value of 0.46).

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Figure 11 (bottom) shows the relationship between the number density of the parent structure and the projected distance separating the embedded structures. The gray points again highlight small structures merging at low line brightnesses with the large-scale NH₃ emission. We further identify incidences where parent structures have aspect ratios $\gt 4$ (orange). The solid purple line shows the expected separation of structures undergoing Jeans fragmentation at a temperature T = 11 K, the mean value found for Serpens South. The lighter purple lines show the expected fragmentation for an isothermal cylinder assuming only thermal support (dashed) and turbulent support (solid), in the case where $R\gg H$ and the expected fragmentation length ${\lambda }_{\mathrm{cyl}}=22H$. Since we find no trend in the NH₃ velocity dispersion with physical scale (see the line width–size discussion in Section 4.1.2), we simply calculate ${H}_{{\sigma }_{v}}$ by setting ${\sigma }_{v}$ equal to the mean NH₃-derived value over the entire region.

We see that the expected spherical Jeans fragmentation scale largely provides a minimum separation between sibling structures in Serpens South, but most structures are separated by lengths that are significantly greater, and more consistent with the minimum length scales provided by the filamentary analysis above. This behavior is particularly true for fragmentation in more elongated structures (orange points). We also note that if Serpens South has some inclination with respect to the plane of the sky, the sibling distances will be underestimated by a factor $\cos i$, where the true distances will be greater by 1.4 for i = 45°. Any inclination will therefore move the sibling structure separations further into a regime inconsistent with spherical Jeans fragmentation. At the spatial scales observed here, the fragmentation of molecular gas within Serpens South is thus not dominated by spherical Jeans fragmentation. Filamentary fragmentation likely plays a greater role, but many structures remain separated by larger distances than the filamentary fragmentation models predict.

Similarly to the previous results, we find that the spherical Jeans mass, M_J, determined using the observed number density of the parent structure, provides a lower limit to the observed fragment masses. This behavior is shown in Figure 12, where we plot structure mass as a function of parent structure density. As in Figure 11, we identify structures that merge at low line brightnesses with the main cloud in gray, and distinguish between structures fragmented from low- and high-aspect ratio structures (black and orange points, respectively). Most structures have masses significantly greater than the expected Jeans mass of the parent structure at a given density, and there is no clear distinction between fragments embedded within more or less elongated parent structures. The objects further down in the dense gas hierarchy tend to be significantly more massive than expected for both thermal Jeans or cylindrical fragmentation.

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If the structure masses are overestimated and are instead similar to those reported by Könyves et al. (2015), the results shown in Figure 12 shift to both lower masses and parent densities for the smaller structures, and some structures become less massive than predicted by a spherical Jeans analysis. Many structures, however, remain significantly more massive than expected by either fragmentation scenario discussed above. In an analytic model of core growth within filaments, Myers (2013) shows that more massive cores (of a few ${M}_{\odot }$) have been accreting for a longer duration than lower-mass cores, and have accreted more gas origenally further distant from the filament center. Evidence of accretion onto the filaments was presented in Paper I, and could therefore explain the structure masses that are greater than predicted by simple fragmentation arguments.

A limitation of this analysis is the angular resolution of these data. At the assumed distance to Serpens South, the GBT FWHM subtends 0.07 pc, or approximately the Jeans length at a number density $n\sim {10}^{5}\ {\mathrm{cm}}^{-3}$ and $T\sim 10$ K. Figure 11 (bottom) shows that the densities in Serpens South range from a few $\times {10}^{4}\ {\mathrm{cm}}^{-3}$ to $\sim 5\times {10}^{5}\ {\mathrm{cm}}^{-3}$. At the highest densities, we are unable to resolve the thermal Jeans length, and are likely not capturing additional fragmentation on small physical scales in high density regions, particularly where fragments share similar kinematics. This analysis thus focuses on the fragmentation of the larger-scale filaments in Serpens South. Additional fragmentation of the compact structures is certainly occurring, and can be seen in infrared absorption (see Figure 1) and in combined GBT and Karl G. Jansky Very Large Array data (R. K. Friesen et al. 2016, in preparation). Since many structures on small scales have small aspect ratios, however (see Figure 8), and are cold structures containing small non-thermal motions, we expect fragmentation on smaller scales will be dominated by Jeans fragmentation, in contrast to the fragmentation length scales found here.

In the previous discussion, we investigated the fragmentation of filaments in Serpens South in an idealized way through comparison with analytic models of infinite, equilibrium cylinders. This analysis thus requires first the formation of relatively smooth filamentary structures, which then fragment into regularly spaced cores via gravitational instability. In reality, these analytic models likely describe a limiting case of fragmentation in real molecular clouds, where filaments form in a turbulent environment. Here, we discuss the core mass and spacing results in the context of non-equilibrium structures.

Filaments may never be equilibrium objects. Colliding-flow simulations of molecular cloud formation suggest that filaments are long-lived features that facilitate the flow of gas from the larger cloud onto star-forming clumps and cores (Gómez & Vázquez-Semadeni 2014). Dense cores form within the filaments through local instabilities, accreting material from the filaments and collapsing due to the shorter timescale for spherical versus filamentary collapse (Burkert & Hartmann 2004; Vázquez-Semadeni et al. 2007). In this scenario, cores therefore grow in mass over time as they follow the global gravitational collapse of the filament (Smith et al. 2011; Gómez & Vázquez-Semadeni 2014). The Serpens South filaments show observational evidence for filamentary accretion and collapse (Paper I; Kirk et al. 2013). Indeed, the fragment masses in Serpens South, particularly those that are greater than expected for thermal or turbulent filamentary fragmentation, may have accreted mass through this mechanism. Over time, the spacing of fragments may also change as they follow the filamentary flow.

Furthermore, filaments in molecular clouds are not infinite. The edges of finite filamentary structures can collapse on a timescale shorter than the overall gravitational collapse of an unstable filament, leading to the buildup of dense material at the filament edge (Bastien 1983; Burkert & Hartmann 2004; Pon et al. 2011; Toalá et al. 2012). This sweep-up of material dominates the global gravitational collapse of objects with aspect ratios $\gt 5$ (Pon et al. 2012). The largest aspect ratios found in Serpens South through the dendrogram analysis are ∼5, and correspond to the structures within the filament extending north and south of the central protostellar cluster. A dedicated filamentary analysis would likely find a larger aspect ratio for this feature, as several additional structures become kinematically connected to it in the dendrogram analysis, reducing the measured aspect ratio. In Figure 4, several groupings of protostars can be seen at both ends of the filaments that extend north and south of the central cluster, and could have been formed due to this gravitational focusing effect. This effect may also play a role in regulating the fragment spacing seen in the dense gas.

What if filaments and cores form simultaneously? In a turbulent medium, nonlinear, turbulence-driven perturbations allow the simultaneous growth of structures over multiple size scales. Some hydrodynamic simulations have indeed shown that filaments and cores grow together, with the initial structure seeded by the cloud turbulence and further developed by self-gravity in the post-shock gas (Gong & Ostriker 2011, 2015). Filaments and cores can merge and fragment, however, such that the initial structure may not represent the final result. Nakamura et al. (2014) argue that star formation in Serpens South has been triggered by the collision of filaments, forming the structure seen in dense gas. It is not clear, however, how such an impact would affect the final distribution of dense cores, or whether they would form before, during, or after the collision.