\subsubsection{FellWalker} \label{FWP}
[HK moved & edited material here to appendix]
\subsection{Radii}
For our analysis, estimates of core radial extent are needed. Both detection algorithms deliver non-circular morphologies, and therefore need to be approximated by circles to have radii estimated. Since Getsources outputs the semi-major and semi-minor axes of an ellipse representing each core, an effective radial size can be found by determining the radius of a circle of the same area, i.e., the square root of the product of the axes. Since these algorithms report sizes based on the FWHM, a larger size should be taken to incorporate the full extent of the core. Therefore, we will take two times the FWHM size to be the core's full extent, which is equivalent to taking the FWHM to be the radius of the core.
The 14.1" beam size of the James Clerk Maxwell Telescope at 850 $\mu$m \citep{Dempsey13} has the effect of broadening the apparent size of the cores recorded in the Getsources extraction. A process of deconvolution was applied to the cores, removing the estimated broadening effects of resolution and leaving only the core size. This deconvolution is achieved by simply subtracting the beam size for JCMT at 850 $\mu$m from the given core size in quadrature. Some cores identified in the aforementioned catalogs, however, have effective radii of less than the beam size, and in this case, the standard deconvolution procedure does not work. To remedy this problem, a cut was made. If the deconvolved radius would have worked out to a size smaller than half that of the beam, it was taken to equal half of the beam size (7.05"). The result was then converted from an angular size to true size (in parsecs) using the small angle formula and distances to each object, which we take to be 139 pc for L1688 \citep{Mamajek08}, 140 pc for B18 \citep{Loinard08}, and 235 pc for NGC 1333 \citep{Hirota08}. That true size was used as the effective radius in all future analysis.
The process of getting effective radii from the FellWalker catalogs is similar. Like the other methods, a size limit of half the beam size was imposed, but the methods used to get the radius in the first place differed. FellWalker assigns significant pixels to a parent core, and writes those pixel assignments to an output file. This file can be read, and the number of pixels associated with a core can be determined. This amount is converted to an area, and, assuming that the core can be approximated as a circle, an effective radius can be determined. Once that number is found, the deconvolution and conversion to size in parsecs can finally be applied as described for the CuTEx and Getsources results.
To establish effective sizes in the GAS property maps, the data were reconvolved to account for the beam size of the Green Bank Telescope (32"), since its large beam would spread out emission more dramatically than seen in the JCMT data. The same idea was applied to the Herschel and FCRAO data, but using their larger respective beam sizes (36" for Herschel, 45" for FCRAO) \citep{Narayanan12}. For all reconvolution calculations, the relevant beam size was added and the JCMT beam size was subtracted from the effective radius, all in quadrature. The result was taken to represent the apparent radius of the core in that data set. Using these apparent dimensions, we managed to accurately identify the pixels associated with each core.
\subsection{Masses}
Estimates of the mass of each core seen in the SCUBA-2 data can be made using flux-based estimates in 850 $\mu$m continuum. For source detections in the submillimeter regime of JCMT, the following equation can be applied, which was originally presented by \citet{Hildebrand83}:
\begin{equation}
M={} \frac{{S_{\nu}}D^2}{{\kappa_{\nu}}{B_{\nu}}(T)}\\
\end{equation}
where at a given wavelength $\nu$, $S_{\nu}$ is the total flux, D is the distance to the cloud, ${\kappa}_{\nu}$ is the opacity, and $B_{\nu}(T)$ is the blackbody function at a given temperature T. \citet{Kirk16} derives an simplified version of this equation for their work on Orion B, which can easily be applied to our data:
\begin{equation}
\begin{aligned}
M ={} & 1.06 * {S_{850{\mu}m}} * (e^{17\x{K}/T}-1) \\
& * ({\frac{\kappa_{850{\mu}m}}{0.0125 {\x{cm}^2}{\x{g}^{-1}}}})^{-1} * ({\frac{D}{415\x{pc}}})^2 \\
\end{aligned}
\end{equation}
$S_{850{\mu}m}$ is the total flux at 850 $\mu$m, as established by source extraction on the 850$\mu$m JCMT SCUBA-2 data, while the temperature (T) is derived from the GAS data for the region within the boundary of the core as defined in the section 2.6. The opacity at 850 $\mu$m ($\kappa_{850{\mu}m}$) is assumed to be $0.0125{\x{cm}^2}{\x{g}^{-1}}$ for all clouds, following publications on the Orion Molecular Cloud by \citet{Pattle15}, \citet{Salji15} and \citet{Kirk16}. Similar values can be assumed for our sample of Gould Belt clouds.
\subsection{Data Selection}
While the reconvolved extents of each core would provide a logical search radius for data on any given core, in many cases this wide of an area can create problems. In portions of each cloud, very stark contrasts exist in linewidth between adjacent pixels, consistent with the transition to coherence described in \citet{Pineda10}. Subsequently, there are multiple cases where the largely quiescent gas of an identified core is surrounded by gas exhibiting much higher line widths \citep[as can be seen from the linewidth maps in][]{Friesen16}. In the interest of collecting data from the core itself and not the surrounding cloud, selecting the inner region of the core rather than the entire extent reduces the effects of contamination from outside gas and other nearby cores.
For the Getsources catalog, the reconvolved elliptical footprints made for logical regions to collect data over in the GAS property maps. To reduce the contamination of the core's data by its surroundings, however, only half of the reconvolved extents will be use rather than the full sizes mentioned earlier. Pixels that fell inside the ellipse defined by the given dimensions and rotation angle for each core were selected, and a weighted average was taken of the points using the relevant error map to determine weights.
Since the $^{13}$CO and column density data by nature includes information beyond the scale of cores, the full extents of cores were used for pixel selection. In this case, data across a core's entire ``surface" is desirable, not just the innermost portion. For column density, all points inside the full extent of a core were averaged and this value was taken to be the local column density. The same selection technique was employed on the $^{13}$CO data, though additional processing was required to find line widths, as described in Section \ref{COLW}.
Since FellWalker does not fit ellipses to sources, the aforementioned procedure is not relevant the algorithm's outputs. Here, the GAS data is re-gridded onto the 850$\mu$m JCMT SCUBA-2 detection map, and for each source ID in that map, the corresponding pixels are drawn from the GAS data map. The same procedure is employed for the column density and $^{13}$CO computations.
\subsection{Column Density}
Column density maps for each region are drawn from \citet{Singh17}, who used a combination of Herschel and Planck data to generate their maps. For examining the effects of pressure from cloud weight, however, the value of interest is the weight of the cloud surrounding the cores, not of the cores themselves. Therefore, when acquiring column density data, the small-scale structure inherent to the core and the large-scale structure of the cloud must be separated. The \citet{Singh17} maps were therefore decomposed using an \textit{\'{a} Trous} wavelet decomposition, which breaks the data down by scale in a manner similar to that of \citet{Kainulainen14}. As performed by \citet{Kirk17}, all frames with a scale significantly larger than the level of cores were summed, producing a composite frame that captures all features comfortably above the scale of the cores. In our source extraction, cores are generally less than 0.1 parsecs across, \citep[comparable to the values of 0.03-0.2 pc asserted in][]{Bergin07}, so at a scales of 0.5 pc and above, we should be including the majority of background material with very minimal contribution from the cores, as desired. Arguably, 0.25 parsecs may make for a better scale, but as a conservative estimate, 0.5 pc will be used for the main analysis. The effects of choosing a different scale will be explored in Appendix \ref{ApB}.
Due to differences in distance, the angular scale in pixels varies throughout the three clouds explored. Figures 4, 5, and 6 show the Herschel maps before and after decomposition. For Ophiuchus and Taurus, the smallest scale included in the large-scale column density maps was 64 pixels, while for the more distant Perseus cloud, 32 pixels was selected as the smallest scale. This corresponds to scales of 0.60 pc for Ophiuchus and Taurus, and 0.51 pc for Perseus.
\subsection{$^{13}$CO Line widths} \label{COLW}
$^{13}$CO data from FCRAO \citep{Ridge06,Narayanan12} were sampled to determine line widths of the gas surrounding the cores, which is an indicator of turbulent binding. This sampling was done by summing the spectra in all selected pixels, and fitting a Gaussian to the result. The returned value for Gaussian sigma was taken to be the observed velocity dispersion in the gas, $\sigma_{obs, ^{13}CO}$. As described in \citet{Pattle15}, $\sigma_{obs, ^{13}CO}$ has contributions from both $T_{kin}$ and non-thermal components $\sigma_{NT}$, expressed by the following equation:
\begin{equation} \label{EQ3}
\sigma_{obs, ^{13}CO}^2 = {} \sigma_{NT}^2 + c_{s, ^{13}CO}^2\\
\end{equation}
where $c_{s, ^{13}CO}$ is the sound speed of $^{13}$CO at the relevant temperature. To compute the total line width $\sigma_{tot, ^{13}CO}$, we subtract the thermal contribution from $^{13}$CO and add the expected thermal contribution from the gas as a whole (assuming that all such gas shares the same temperature found in the NH$_3$ property maps) which results in the following equation for $\sigma_{tot, ^{13}CO}$:
\begin{equation} \label{EQ4}
\sigma_{tot, ^{13}CO} = {} \sqrt{{\sigma_{obs, ^{13}CO}^2}-\frac{k_BT_{kin}}{\mu_{^{13}CO}m_H}+\frac{k_BT_{kin}}{\mu_{mean}m_H}}\\
\end{equation}
This equation is employed to convert observed $\sigma_{obs, ^{13}CO}^2$ into $\sigma_{tot, ^{13}CO}^2$, which is the value needed for a turbulent pressure analysis.
Analysis
\subsection{Kinetic Support versus Gravity}
We begin our analysis by studying the balance between gravity and internal pressure for cores in B18, NGC 1333, and L1688, which will expose whether gravity alone is sufficient to prompt the collapse of a typical dense core. A simple Jeans analysis was used as a first look at this balance, using the core sizes and masses identified by Getsources. The thermal Jeans criterion \citep[e.g., see][]{Bertoldi92}, which is displayed in Equation \ref{EQ5}, defines the mass at which a cloud of radius $R_J$ and kinetic temperature $T_{kin}$ will begin to experience gravitational collapse, barring other factors.
\begin{equation} \label{EQ5}
M_J = {} \frac{5R_J}{2G}\frac{k_BT_{kin}}{\mu_{mean}m_H}\\
\end{equation}
In the above equation, G is the gravitational constant, $k_B$ is the Boltzmann Constant, $m_H$ is the mass of a hydrogen atom, and $\mu_{mean}$ is the mean molecular weight. The value for $\mu_{mean}$ used for the analysis is 2.8, following \citet{Kauffmann08}. With the inclusion of non-thermal internal motion, Equation \ref{EQ5} can be expressed in terms of the total line width $\sigma_{tot}$:
\begin{equation}
M_J = {} \frac{{5R_J}{\sigma_{tot}^2}}{{2G}}\\
\end{equation}
If the above equality is satisfied, the cloud is in equilibrium. Therefore, for given values of $M_J$ and $R_J$, a certain $\sigma_{tot}$ will put the system in equilibrium. This value marks the velocity dispersion required to resist gravitational collapse, and we define that value as $\sigma_{grav}$.
Using the same technique for conversion employed in Equations \ref{EQ3} and \ref{EQ4}, the total velocity dispersion in a cloud with an observed line width in ammonia emission $\sigma_{obs}$ and a given $T_{kin}$ can be expressed as:
\begin{equation}
\sigma_{tot} = {} \sqrt{{\sigma_{obs}^2}-\frac{k_BT_{kin}}{\mu_{NH_3}m_H}+\frac{k_BT_{kin}}{\mu_{mean}m_H}}\\
\end{equation}
Larger-scale motion in the cloud can also contribute to $\sigma_{tot}$. To determine the effect of these motions on our value for $\sigma_{tot}$, we calculated the standard deviation of the centroid velocity $V_{LSR}$ across each core in every cloud and compared it to the original values for $\sigma_{tot}$. The values found for $\sigma{V_{LSR}}$ lie between 4 m/s and 220 m/s with a mean of 50 m/s, with NGC 1333 cores at the top of that range and B18 cores at the bottom of that range. This is around an order of magnitude less than typical line widths, which generally range between 120 m/s and 900 m/s, with a mean of 384 m/s across all cores. Although those ranges do overlap, all cores with a high $\sigma{V_{LSR}}$ are accompanied by a wide line width, which seems to indicate that relative to the effects of thermal and non-thermal pressure, $\sigma{V_{LSR}}$ is not particularly significant. Therefore, contributions from large-scale cloud motion will not be considered in calculations of total line width.
In all three regions, it is evident that internal turbulent motions in the gas dominate over thermal contributions. Figure 7 shows protostars and starless cores in all three regions plotted against the size-line width relation from \citet{Larson81}. Most cores lie above the values from Larson's relationship and have a line width trend that is generally independent of radius. Although this opposes Larson findings, it should be noted that we are working at lower scales than those explored by Larson. The lower end of \citet{Larson81}'s size scale range corresponds to the upper end of the core-level scale we are exploring, and at those scales, cores appear to fall around the Larson line. This indicates that Larson's size-line width relation may be valid, however it clearly breaks down at the scale of cores.
For each region, Figures 8-10 show the balance between gravity and internal pressure using the formulas derived from the Jeans criterion. We will also make use of the virial parameter $\alpha$ \citep{Bertoldi92}, defined as
\begin{equation}
\alpha = {} \frac{5\sigma_{tot}^2R}{GM}
\end{equation}
Note that although the $\sigma_{grav}$/$\sigma_{tot}$ and $\alpha$ plots use the same set of formulas and values to determine boundedness, the Jeans (mass vs radius) plot does not. Unlike in the other plots, the Jeans plot uses estimated global values for T$_{kin}$, and decides on the mass needed for binding given the reported global average kinetic temperature. This mass will be smaller than the mass required to bind a core when including internal turbulence, therefore making the Jeans line an under-approximation of this mass. This is reflected in the plots, where a much higher fraction of cores are considered bound in the Jeans plot compared to the $\sigma_{grav}$/$\sigma_{tot}$ and $\alpha$ plots. Also note that in the $\sigma_{grav}$/$\sigma_{tot}$ and $\alpha$ plots, the same set of cores are bound and unbound, a reassuring result given quantitative equivalence of the two measures. In the Jeans plot, all cores bound in the other plots are also bound, with the addition of a set of others made bound by the lower requirement for binding.
Although there is variation in overall morphology between the three clouds, all show a few common trends. For example, with respect to boundedness, all clouds have more than half of cores below their respective Jeans masses, and L1688 and NGC 1333 both have over 85\% of their cores above a value of two for $\alpha$, and with $\sigma_{tot}$\textgreater$\sigma_{grav}$. All of these factors suggest that typical cores are not gravitationally bound. B18 appears to have the most gravitationally bound cores of the sample, with almost half of cores having $\sigma_{grav}$\textgreater$\sigma_{tot}$. However, with only nine cores, this result has limited significance. Regardless, for a majority of cores in any of these regions to be identified as bound, other factors beyond self-gravity are required. Note that these statements do not consider class II, III, or flat-spectrum protostars, which will not be included in the analysis of the virial state of dense cores due to their different properties.
Unsurprisingly, across all regions, more massive cores are more likely to be bound. In NGC 1333, Class 0/I protostellar cores, which were usually both small and massive, represented the largest set of gravitationally bound cores. This is not surprising, as these are stars still in their protostellar envelope, having only recently seen sufficient density in their centers to begin nuclear fusion. However, interestingly, smaller masses for such objects in L1688 and B18 caused this trend to not carry over into the other regions. However, across protostars in all regions, smaller radii are consistently reported, with many lying at or around the resolution limit for the SCUBA-2 data. This is also to be expected, as protostellar cores, by definition, must have collapsed to a high enough density to form a star.
\subsubsection{Cloud Weight}
Given that this analysis depends on the large-scale structure surrounding the cores as opposed to the cores themselves, the scale-filtered density map shown earlier for structures larger than 0.5 parsecs will be used. Operating under an assumption that the clouds are spherical in nature, column density at a certain location can be used to estimate the depth at which a core is embedded in the cloud. At the large scales considered in the 0.5 pc-filtered column density maps, the assumption that the cloud is spherical is a reasonable approximation. Under this assumption, pressure from the surrounding cloud due to cloud weight can be expressed as follows \citep{McKee89,Kirk06}:
\begin{equation}
P_w={}{\pi}G{\Sigma}{\bar{\Sigma}}\\
\end{equation}
where P$_w$ is the pressure from cloud weight, $\Sigma$ is the column density along the line of sight of the core, and $\bar{\Sigma}$ is the mean column density across the entire cloud. $\bar{\Sigma}$ is calculated by averaging the values for the 0.5-pc-filtered column density map over the entire GAS field, which resulted in values for average column density of $\bar{N}=4.6*10^{21} \x{cm}^{-2}$ for L1688, $\bar{N}=2.0*10^{21} \x{cm}^{-2}$ for B18, and $\bar{N}=2.7*10^{21} \x{cm}^{-2}$ for NGC 1333.
For a full virial analysis, the effects of the forces involved in binding must be compared directly. The value for each of these terms can be expressed through the formalism presented by \citet{Pattle15}:
\begin{equation}
\Omega_k={}\frac{3}{2}M{\sigma}_{tot}^2\\
\end{equation}
\begin{equation}
\Omega_g={}\frac{-1}{2\sqrt{\pi}}\frac{GM^2}{R}\\
\end{equation}
\begin{equation} \label{PT}
\Omega_{p,w}={}-4\pi{P_{w}R^3}\\
\end{equation}
$\Omega_k$ is the internal pressure (kinetic) term, $\Omega_g$ is the gravitational term, and $\Omega_{p,w}$ is the pressure term from cloud weight. Cores that satisfy -$(\Omega_g + \Omega_{p,w})/2\Omega_k=1$ are in virial equilibrium, and the left hand side of that equation can be used as a measure for boundedness of the core, including cloud weight, gravity, and internal kinetic factors. This ratio is important to determining the virial state of the system, creating a clear way to differentiate between bound and unbound cores for each of the three clouds investigated. Figures 11-13 show this parameter against $\Omega_g/\Omega_p$ and $-\Omega_g/\Omega_k$, which display the gravity-only virial state and gravity-pressure balance respectively. If $-\Omega_g/\Omega_k$\textgreater2, then gravity is sufficient to bind the core, while $\Omega_g/\Omega_p$\textgreater1 simply indicates that gravity is dominant over cloud weight pressure. These two indicators provide different methods of determining what forces are dominant in each core.
Clearly, all clouds still appear mostly super-virial and therefore unbound. However, across all three regions, about 10\% of cores in each cloud have been pushed over the boundedness line by the cloud weight estimate to become sub-virial. On average, B18 appears to be closest to having mostly bound cores, but with only eleven cores that have complete data, this statement is not particularly robust. For these cores to be mostly sub-virial, we still need to explore other factors.
\subsubsection{External Turbulent Pressure}
A separate form of pressure is exerted on each core by turbulent motions of the surrounding cloud. These motions can be traced by transitions from molecules that are common at low densities, such as $^{13}$CO, which will be used in this paper. As described in Section 2.8, line widths are found using Gaussian fitting, and the result is put into the following equation, which follows from the ideal gas law:
\begin{equation}
{P_t}={}\rho_{^{13}CO}\times{\mu_{mean}}\times{m_{H}}\times{\sigma_{tot, ^{13}CO}^2}\\
\end{equation}
where $\rho_{^{13}CO}$ is the density of the gas that $^{13}$CO is probing. Since $^{13}$CO freezes out at densities above $10^5$ $\x{cm}^{-3}$ \citep{DiFrancesco07}, we select a slightly less dense value of $10^4$ $\x{cm}^{-3}$ to represent a more standard density of the ambient cloud. The resulting value for $P_t$ can be entered into Equation \ref{PT} in place of $P_w$ to generate a value for an external turbulent pressure term, $\Omega_{p,t}$. When added to the cloud weight pressure term, a total pressure term is generated, which can then be used in virial plots as in \S 3.2.1. Figures 14-16 show virial plots with turbulent pressure included.
After adding both pressure terms, around half of cores in L1688 and NGC 1333 and all but two of B18's cores qualify as sub-virial. Furthermore, over 80\% of cores across all regions have pressure dominating gravity. Evidently external pressure, and most notably turbulent pressure, is the force with the most significant effect on the binding of the cores in NGC 1333, L1688, and B18.
Discussion
\subsection{Overview of Data}
[HK notes - table included here is now a separate file that needs to be re-uploaded. Reference in Ronan's tex will be 'MDT'. New filename is tab\_cloudsummary\_GS.tex]
Most of the results found in this paper is consistent with past works. Most cores identified in the Getsources catalog for L1688 have radii reported that are of the same order as the ones in the \citet{Pattle15} CuTEx catalog, but with a higher upper limit. This small difference is not surprising given that CuTEx focuses on sharp peaks, and will therefore likely miss broad peaks that tend to accompany the most spatially extended structures in SCUBA-2 data. Masses also compare well, lying between 0.03 M$_\odot$ and 2 M$_{\odot}$ in the \citet{Pattle15} data and between 0.01 M$_\odot$and 3 M$_{\odot}$ in the Getsources results. Similar values are found in NGC 1333 and B18, although NGC 1333, on average, is host to higher-mass and more extended core structures compared to the very small and dense cores in L1688. Trend comparisons are less meaningful for B18 due to the very small number of cores. The B18 cores, however, all fall well within the range of masses and radii found for cores in NGC 1333. All of these factors suggest a gravitational term consistent with values reported in literature.
The values determined for external pressure also seem well within the previously determined ranges for each given cloud. L1688, for example, has median turbulent pressures of roughly $P/k_B=2.6\times10^6$ $cm^{-3}K$, matching up nicely with the estimates from \citet{Maruta10} with an estimate of $P/k_B=3\times10^6$ $cm^{-3}K$, but well below the estimates of \citet{Johnstone00} and \citet{Pattle15}, which are both around $P/k_B=2\times10^7$ $cm^{-3}K$. The range of turbulent pressures we find overlaps significantly with the \citet{Maruta10} estimate, but less so with the \citet{Johnstone00} and \citet{Pattle15} estimates. Therefore, using different techniques of determining binding pressure, significantly larger values may be found, making the cores appear more bound. This possibility will be investigated in depth in Appendix B.
Results for local properties in L1688 also match closely with those found in \citet{Friesen09}, which investigated the properties of Ophiuchus B, C, and F using NH$_3$ data from the GBT, Australia Telescope Compact Array, and Very Large Array. The properties found matched up closely with established values for $\sigma_{tot}$ and T$_{kin}$, showing higher kinetic temperatures of around 15 K in Ophiuchus B and F, compared to lower values of around 12 K for Ophiuchus C. Also like in \citet{Friesen09}, we found high line widths generally around 0.3-0.4 kms$^{-1}$ in Ophiuchus B, compared to values near 0.2 kms$^{-1}$ for most of Ophiuchus F and lower values of around 0.1 kms$^{-1}$ in central Ophiuchus C. Although we could only test correspondence with this paper in a few select regions, the close correspondence is comforting given that the same region is being observed using the same molecular tracer.
\citet{Friesen09} also identified cores in Ophiuchus B, C, and F using the CLUMPFIND algorithm. Most cores found fell between 0.013 and 0.025 pc as effective radii, sizes the correspond closely with most cores in those regions found in this paper. Also like in this paper, the cores identified in by \citet{Friesen09} in Ophiuchus C have radii closer to the upper edge of the aforementioned size range. This indicates not only that the \citet{Friesen16} results for core identification match up with this paper, but that the correspondence remains strong when breaking it down by region as well.
To investigate the virial state of starless cores, it is important to consider not only the overall results for boundedness, but the distribution of sub-virial and super-virial cores as well. Any apparent patterns may provide important insight into the nature of bound and unbound cores in these clouds. Figures 17-19 show the distribution of both sub- and super-virial cores in the clouds investigated. Although the trends are less clear in Perseus, it seems evident that the sub-virial cores are overwhelmingly isolated cores, many of which are Ammonia-bright. Meanwhile, cores in highly crowded regions such as Ophiuchus B and central Ophiuchus A are over 90\% unbound. This same pattern can be seen in NGC 1333, as most crowded filament to the southeast the cloud's centre is populated primarily by super-virial cores. B18, which has no such crowded regions, also has no super-virial starless cores, with the only super-virial structures being protostars. This is an interesting and counterintuitive result, as these regions we are identifying as unbound appear to be associated with the most active star formation.
If this is not accurate to reality, there are two main possibilities that may explain why: source detection and artificial linewidth broadening. With respect to source detection, the difference in the number of cores detected in the centre of Ophiuchus A varies greatly by detection technique, with 10 found in Getsources results and only 3 found using FellWalker. While the Getsources results align more closely with past catalogs such as the one from \citet{Pattle15}, such a discrepancy suggests that the number of cores in these dense regions is not well constrained, and therefore, so are the areas. The more the larger structures are sub-divided, the smaller the estimated radii become. Since a factor of two reduction in radius causes a factor of eight decrease in the turbulent pressure term, such effects could easily make up for the boundedness of many of these cores.
Constraining line widths is also more challenging in these more clustered regions. Since NH$_3$ exists down to densities of roughly $10^4$ cm$^{-2}$, some gas outside of cores will emit radiation. In isolated environments, the core dominates, and we reports its generally low line widths. However, in crowded regions, much of the gas surrounding the cores will be dense enough to house NH$_3$, which may allow the surrounding gas, which we expect to be more turbulent, to dominate in our measurement of line width. If this is true, this higher-density gas surrounding cores in these dense clumps may contribute to external pressure, rather than internal pressure. Precisely establishing that point of transition between the core and surrounding gas is often challenging with dense gas tracers. The effects of the surrounding gas broadening line widths may explain why isolated high-NH$_3$ integrated intensity features almost always have low line widths, while clustered features are usually accompanied by much higher line widths - higher than what has been found in past studies using different tracers \citep[e.g.][]{Maruta10}.
\subsection{Differences Between Clouds}
Although the three regions have often been discussed monolithically in this paper, they are in fact quite different, which is exemplified quite well by the values in Table \ref{MDT}. All have different morphologies, and different levels of dominance for each force considered. Figure 20 shows all cores found across all regions to illustrate the differing distributions for the three clouds.
\subsubsection{L1688}
L1688 represents the most densely populated and compact environment of the set, with numerous small, low mass protostellar cores. The median core mass was only a mere 0.10 M$_\odot$, compared to NGC 1333 and B18, which have median mass values almost 4 and 6 times higher, respectively. This makes L1688 the region in which gravity is least significant. Its cores sizes were also the smallest of set, with a median value of 0.014 pc, compared to NGC 1333 at 0.018 pc, and Taurus almost double that at 0.031 pc. It also sports the highest kinetic temperatures, total line widths, and cloud weight pressures, with median values of 16.3 K, 0.4 kms$^{-1}$, and P$_w$/k$_B=9.2\times10^5$, respectively. The notable situation in which L1688 does not have the most extreme pressure-related terms is in external turbulent pressure, where NGC 1333 has much higher pressures. Since this term dominates binding across all three regions, L1688's weaker pressure binding combined with its very high internal turbulent pressures makes it the only region studied in which the median starless core is super-virial, despite the higher cloud weight.
\subsubsection{B18}
Across all kinetic measures, B18 is the quietest of the three regions, with the lowest median values for line width, kinetic temperature, and pressure (of all kinds). However, due to the cores all having low line widths, even B18's low pressure environment is sufficient to bind all identified starless cores. This is helped by the fact that the cores in B18 have the highest masses of the clouds considered, making gravity more significant than in any other region. Although both values are much lower than in any other region, Taurus is also tied with Perseus for having the highest ratio of external turbulent pressure to cloud weight pressure, with external turbulent pressure making up 88\% of external binding pressure total in both regions. The larger sizes of B18 cores are also key to the pressure term's binding of cores, in spite of the lower line widths in the external turbulent pressure estimate.
\subsubsection{NGC 1333}
While NGC 1333 is between the extremes of B18 and L1688 on most properties, it is most notable in the very high turbulence estimates. Figure 21 shows the line widths across the three regions. B18 is shows two have all but a couple of cores with line widths outside of a main cluster centered at around 0.6 kms$^{-1}$, while L1688 has a distribution centered on 0.6 kms$^{-1}$ and NGC 1333 sports line widths that are often closer to 1.2 kms$^{-1}$. Evidently, NGC 1333 is the most turbulent environment, which aids in binding a larger fraction of cores than are found to be sub-virial in L1688. However, unlike L1688, it does not have insignificant gravity binding, with a median gravity/pressure ratio of 0.1.
\subsection{Comparison to Orion Results from \citet{Kirk17}}
A similar analysis was performed on the northern section of Orion A in \citet{Kirk17}. It used much of the same data, drawing from Getsources results on JCMT SCUBA-2 data for core identification, using GAS DR1 data for internal kinetic properties, and getting information on cloud weight pressure from the column density maps from \citet{Lombardi14}. The conclusion reached, which differs somewhat from our results, is that cloud weight pressure alone is sufficient to bind most cores. The picture painted by GAS data for Orion A is not unlike this paper's results in L1688 and NGC 1333, as cores are found with line widths mainly between 0.2 kms$^{-1}$ and 0.5 kms$^{-1}$, compared to this paper, which also reveals many cores in that same range (although some more quiescent cores are also shown to have results down to 0.1 kms$^{-1}$). This indicates that Orion A does not have significantly different kinetic properties. Instead, the main difference appears to come in the cloud's structure.
The mean column density reported in the \citet{Kirk17} paper is reported to be $\bar{\x{N}}=3.9\times10^{22}$ cm$^{-2}$. This is dramatically higher than the mean values found in any of our regions, with the highest $\bar{\x{N}}$ being $4.6\times10^{21}$ cm$^{-2}$ found in L1688 - almost an order of magnitude smaller. Since the equation for cloud weight pressure involves multiplying the mean value by the local value, this inflation of the pressure term result is effectively squared, producing almost a factor of 100 difference in cloud weight pressure values. That said, Orion A is a structurally different cloud. Rather than the low mass star forming regions of Perseus, Taurus, and Ophiuchus, Orion is notable for being the nearest site hosting the formation of high mass stars, with a few newly-formed O-stars present \citep{Hillenbrand97}. It is also much more massive, with $10^5$ M$_\odot$ of material in Orion A \citep{Maddalena86} compared to around 3000 M$_\odot$ in the Ophiuchus Complex \citep{Loren89}, for example. Orion A's massive size allows for much higher cloud weight pressures, which seem to dominate the binding of its cores. This may suggest that while cores in smaller, lower mass star forming regions are dominated by turbulence, the centers of Giant Molecular Clouds (GMCs) like Orion A have much more significant contributions from cloud weight.
\subsection{Implications for Star Formation in Low-Mass Star Forming Regions}
Consistent with past work, gravity alone does not appear to provide sufficient binding to overcome turbulent motion inside each core. Rather, external pressure is the dominant factor. The importance of external pressure binding in molecular clouds is not a new consideration \citep{Bertoldi92}, with cloud weight and turbulent pressure being a couple of the most frequently considered forms of it. In this paper, external turbulent pressure clearly dominated the forces supporting collapse in all regions surveyed. This turbulence-dominated picture suggests that these cores evolve primarily according to the dissipation of turbulence \citep[e.g.][]{Goodman98}.
\citet{Goodman98} show that in small, dense structures, line widths are consistently subsonic and nearly constant with size. This transition to coherence in dense structures is observed directly in \citet{Pineda10}, with the conclusion that cores generally represent regions of low line width, with more turbulent, higher linewidth gas contaminating at larger scales. This suggests that as cores form and their densities increase, the transition to coherence described in \citet{Goodman98} occurs, and turbulence within the core dissipates. This decrease in internal turbulence pressure without a drop in external pressure would then fuel the collapse of the core. Given the lack of force provided by cloud weight and gravity in these clouds, this dynamic model of core evolution seems likely, in which the dissipation of turbulence prompts core collapse.
Conclusions
The Green Bank Ammonia survey provided us with high-quality kinetic data at densities common to dense cores, providing us with some of the most accurate information on internal pressure support to date. When supplemented with 850 $\mu$m continuum data, column density maps, and FCRAO $^{13}$CO data, a comparison can be drawn between the new kinetic data and existing data that offers insight into pressure and gravitational forces that support the collapse of cores. With this information, we were able to perform a complete virial analysis of cores in three nearby Gould Belt Clouds. We find that around half of all cores in both L1688 and NGC 1333 are sub-virial, while all pre-stellar cores in B18 are sub-virial. The majority of this binding comes from turbulent pressure from the surrounding cloud, which was traced using $^{13}$CO data. Gravity and cloud weight also provide significant contributions, with gravity being most notable in protostellar cores, especially those in NGC 1333. Due to the dominance of turbulent pressure, we conclude that the dissipation of turbulence on the scale of cores is likely to be a major factor in prompting core collapse in these low mass star forming regions.
[Appendix writing not currently included here]