# A Parsec-Scale Catalog of Molecular Clouds in the Solar Neighborhood Based
on 3D Dust Mapping: Implications for the Mass-Size Relation

Shlomo Cahlon Center for Astrophysics $\mid$ Harvard & Smithsonian, 60 Garden
St., Cambridge, MA, USA 02138  Catherine Zucker Center for Astrophysics $\mid$
Harvard & Smithsonian, 60 Garden St., Cambridge, MA, USA 02138  Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
Alyssa Goodman Center for Astrophysics $\mid$ Harvard & Smithsonian, 60 Garden
St., Cambridge, MA, USA 02138  Charles Lada Center for Astrophysics $\mid$
Harvard & Smithsonian, 60 Garden St., Cambridge, MA, USA 02138  João Alves
University of Vienna, Department of Astrophysics, Türkenschanzstraße 17, 1180
Vienna, Austria Shlomo T. Z. Cahlon<EMAIL_ADDRESS>
###### Abstract

We dendrogram the Leike et al. (2020) 3D dust map, leveraging its $\sim 1$ pc
spatial resolution to produce a uniform catalog of molecular clouds in the
solar neighborhood. Using accurate distances, we measure the properties of 65
clouds in true 3D space, eliminating much of the uncertainty in mass, size,
and density. Clouds in the catalog contain a total of $1.1\times 10^{5}\;\rm
M_{\sun}$, span distances of $116-440$ pc, and include a dozen well-studied
clouds in the literature. In addition to deriving cloud properties in 3D
volume density space, we create 2D dust extinction maps from the 3D data by
projecting the 3D clouds onto a 2D “Sky” view. We measure the properties of
the 2D clouds separately from the 3D clouds. We compare the scaling relation
between the masses and sizes of clouds following Larson (1981). We find that
our 2D projected mass-size relation, $M\propto r^{2.1}$, agrees with Larson’s
Third Relation, but our 3D derived properties lead to a scaling relation of
about one order larger: $M\propto r^{2.9}$. Validating predictions from theory
and numerical simulations, our results indicate that the mass-size relation is
sensitive to whether column or volume density is used to define clouds, since
mass scales with area in 2D ($M\propto r^{2}$) and with volume in 3D
($M\propto r^{3}$). Our results imply a roughly constant column and volume
density in 2D and 3D, respectively, for molecular clouds, as would be expected
for clouds where the lower density, larger volume-filling gas dominates the
cloud mass budget.

Molecular clouds, Solar neighborhood, Star forming regions, Scaling relations

††software: Astropy (Astropy Collaboration et al., 2013, 2018), glue (Beaumont
et al., 2015), Unity 3D (Technologies, 2015), Astrodendro (Robitaille et al.,
2019)

##  1 Introduction

Star formation takes place in molecular clouds, which are associated with the
densest and coldest phase of the interstellar medium (ISM). Studying the
properties of molecular clouds has thus long been the focus of star formation
research, as the structure of these clouds has a direct impact on the
location, number, size, and mass of newly formed stars (Rosolowsky et al.,
2008).

Maps of the extinction or emission from dust trace out the interstellar medium
(ISM) in “position-position” or “p-p” space, on the 2D plane of the sky
(Lombardi, 2009; Lada et al., 2009). Spectral-line observations of the ISM can
add a third dimension, owing to the Doppler effect, which allows for
conversion of wavelength or frequency to velocity. The resulting so-called
“postion-position-velocity” or “p-p-v” cubes, can be analyzed as 2D maps
(integrating over velocity) or as pseudo-3D maps, where velocity is treated as
a non-spatial third dimension.

Catalogs of molecular clouds have previously been derived using both p-p and
p-p-v data. Rice et al. (2016) use the dendrogram technique (Rosolowsky et
al., 2008) to extract and analyze molecular clouds from the CO p-p-v survey of
Dame et al. (2001), identifying over a thousand clouds across the full
Galactic plane. Miville-Deschênes et al. (2016) apply a hierarchical cluster
identification method to a Gaussian decomposition of Dame et al. (2001) and
produce a catalog of 8,107 clouds covering the entire Galactic plane. Using 2D
extinction maps derived from the NICEST color excess method (Lombardi, 2009),
Dobashi (2011) identify over 7,000 dark clouds in the Galactic plane using a
fixed extinction threshold.

Numerical simulations show that projection effects intrinsic to p-p and p-p-v
space impact the study of cloud structures (e.g. Shetty et al., 2010;
Ballesteros-Paredes & Mac Low, 2002a). Comparing simulated p-p-p- and p-p-v-
derived clouds’ overlap, Beaumont et al. (2013) find that studying clouds in
p-p-v space (rather than in true physical 3D space) can induce approximately
40% scatter in their masses, sizes, and velocity dispersions. Moreover,
Beaumont et al. (2013) demonstrate that many p-p-v structures can be
fictitious, especially in “crowded” regions. Thus, accurate estimates of cloud
properties depend critically on studying clouds in position-position-position
space, which requires knowledge of clouds’ distances.

In the past few years, distance estimates to molecular clouds have improved
dramatically. Using so-called 3D dust mapping, Schlafly et al. (2014) produce
one of the first uniform catalogs of accurate distances to nearby molecular
clouds, with typical distance uncertainties of $\approx 10\%$. Specifically,
Schlafly et al. (2014) use multi-band photometry from Pan-STARRS1 (Chambers et
al., 2019) to infer self-consistent distances and extinctions for a large
number of stars across the solar neighborhood, the key ingredients necessary
for constructing a 3D dust map (see also Green et al., 2015). The advent of
the Gaia mission (Gaia Collaboration et al., 2016), and especially the results
from its second and third data releases, Gaia DR2 and DR3 (Brown et al., 2018;
Lindegren et al., 2021), has made it possible to construct ever-more-accurate
3D-dust-based distances to clouds, owing to stellar parallax measurements for
millions of stars in the solar vicinity. Zucker & Speagle et al. (2019)
utilize the Gaia DR2 data release to produce an accurate catalog of distance
estimates to molecular clouds, with uncertainties on the order of $5\%-6\%$
(see also Zucker et al., 2020; Yan et al., 2019)

Building on the accurate distances enabled in the Gaia era, there have only
been two molecular cloud catalogs based on true three-dimensional “p-p-p” data
obtained from 3D dust mapping, as presented in Chen et al. (2020) and
Dharmawardena et al. (2023). Chen et al. (2020) obtain a catalog of 567
molecular clouds using the 3D dust map of Chen et al. (2018). However, the
molecular clouds are typically resolved in distance on $\approx$ hundreds of
parsec scales, so key cloud properties, including the sizes of clouds, are
still derived using 2D projections. Dharmawardena et al. (2023) also derive a
catalog of molecular cloud properties towards sixteen complexes within $\rm
1-2\;kpc$ from the Sun using their 3D dust mapping algorithm DUSTRIBUTION,
which leverages stellar distance and extinction estimates from Fouesneau et
al. (2022). Applying the astrodendro package (Robitaille et al., 2019) to 3D
dust cutouts around each complex, Dharmawardena et al. (2023) obtain estimates
of e.g. the volume, mass, and density for each cloud and its myriad of
substructure in “p-p-p” space (see also Dharmawardena et al., 2022).

However, one of the highest resolution 3D dust maps over appreciable volumes
of the solar neighborhood is the Leike et al. (2020) map, which traces the
structure of the local interstellar medium at $\sim 1$ pc distance resolution.
Leveraging distance and extinction estimates from the StarHorse catalog
(Anders et al., 2019), Leike et al. (2020) utilize a combination of Gaussian
Processes and Information Field Theory to produce a highly resolved 3D dust
map that charts molecular clouds out to a distance of $d\approx 400$ pc with
distance uncertainties lower than 1%. Such accurate distance uncertainties
enable the extraction and characterization of molecular clouds in true 3D
p-p-p space.

In this work, we dendrogram the Leike et al. (2020) 3D dust map and uniformly
analyze the properties of resolved molecular clouds derived in p-p-p space. We
produce a catalog of 65 distinct local molecular clouds, including a dozen
well-studied clouds in the literature, and compare our results to extant
literature derived primarily from p-p-v and p-p space. In §2 we present the
Leike et al. (2020) data used to create the catalog. In §3 we present the
dendrogram technique applied to the data to derive the properties of our
molecular clouds in real 3D space. We then describe how we project our data
into 2D space following Zucker et al. (2021), in order to measure the 2D
properties of clouds. In §4 we summarize our cloud property results and
characterize Larson’s mass-size relation in both 2D and 3D space. In §5 we
hypothesize what could be driving differences in the 3D- and 2D- derived mass-
size relations, and discuss our mass-size results in the context of existing
literature, including previous exploration of the mass-size relation using
analytic theory and mathematical modeling (c.f. Ballesteros-Paredes et al.,
2012). Finally, we conclude in §6

##  2 Data

Leike et al. (2020) reconstruct the 3D dust distribution in a Heliocentric
Galactic Cartesian reference frame out to a distance of $\approx 400$ pc
($-370\;{\rm pc}<xy<370\;{\rm pc},-270\;{\rm pc}<z<270\;{\rm pc}$). This
distance range includes about a dozen well-studied star-forming regions,
including Taurus, Perseus, and Orion. We convert from the native units of the
Leike et al. (2020) map (optical depth in the $Gaia$ $G$ band per 1 pc) to
volume density of hydrogen nuclei ($n_{\rm H}$) following Zucker et al. (2021)
and Bialy et al. (2021). We derive all results in this work using the total
volume density of hydrogen nuclei, including contributions from both atomic
and molecular hydrogen gas.

Figure 1:  3D spatial map of the local interstellar medium, showing all gas
with density $n>5\;\rm cm^{-3}$ (gray) and clouds identified in this work
(blue). The Sun is shown at center in yellow. An interactive version of this
figure is available here.

##  3 Methods

###  3.1 Generating 3D Dendrogram

After converting the 3D dust map of Leike et al. (2020) to total volume
density of hydrogen nuclei, we segment the Leike et al. (2020) 3D dust map
into a set of molecular cloud features and measure their properties using the
dendrogram algorithm. To do so, we build upon the existing functionality for
dendrogramming 3D p-p-v data in the astrodendro package. Abstractly, the
dendrogram algorithm constructs a tree starting from the highest density point
in N-dimensional density (volume density in this work) data, moving to the
next largest value and connecting along isosurfaces of constant density. A
leaf is defined to be a feature without any descendants. Each time a local
maximum point is found (i.e. a leaf), the algorithm determines, based on
neighboring maxima and the behavior of the contour levels between maxima,
whether to join the pixel to an existing structure, or to create a new
structure. Once a local minimum point between the two structures is found, it
is classified as a branch that connects the two structures. Iterations of this
procedure will eventually either merge all values into a single tree or create
multiple trees. Moreover, once the data are contoured with levels, the
algorithm searches through every contour level, starting from the top, and
records how many local maxima are above each contour level. When the surface
around two local maxima merge together, that density level is recorded as a
branch. If more than two local maxima merge together between two successive
contour levels, the algorithm will continue to search with better tuned
contour levels such that every merger includes up to two leaves. The
dendrogram algorithm depends on three user-defined parameters set in the
astrodendro package, $n_{min}$, $\Delta_{n}$, and $\\#_{voxels}$:

  1. 1.

$n_{min}$: the minimum absolute volume density threshold for a structure to be
included as part of the dendrogram.

  2. 2.

$\Delta_{n}$: how significant a leaf must be in order to be considered an
independent entity. The significance is measured from the difference between
its peak density and the density value at which it is being merged into the
tree.

  3. 3.

$\\#_{voxels}$: the minimum number of voxels needed for a leaf to be
considered an independent entity. If a leaf is about to be joined onto a
branch or another leaf, the algorithm checks the leaf’s number of voxels. If
the leaf’s number of voxels is lower than $\\#_{voxels}$, the algorithm
combines it with the branch or leaf it is being merged with, so that it is no
longer considered a separate entity.

The dendrogram is most sensitive to changes in $n_{min}$, as this value
determines the minimum volume density value required to classify local maxima
as meaningful. Setting a high $n_{min}$ can lead to multiple isolated single-
leaf trees, especially when dealing with 3D dust maps. To determine what
values to adopt for these parameters, we created a set of different
combinations of values and compared the subsequent masses computed using these
values to a benchmark sample of about a dozen well-studied clouds in p-p space
from Lada et al. (2010) and p-p-p space from Zucker et al. (2021). We tailored
the parameters to reproduce similar results to the bench-marked cloud samples.
After testing multiple values for the parameters, we ultimately settled on a
fixed threshold of: $n_{min}=25\;\rm cm^{-3}$, $\Delta_{n}=25\;\rm cm^{-3}$,
$\\#_{voxels}=150$ voxels. The result of this procedure is a hierarchy of
cloud emission, where each structure in the dendrogram corresponds to a
contiguous, resolved feature in 3D Heliocentric Galactic Cartesian $xyz$
coordinates, bounded by a surface of constant volume density. In the next
section, we filter the dendrogram to extract clouds and their properties.

Figure 2: Dendrogram of the Leike et al. (2020) 3D dust map. Structures that
survive the cloud filtering scheme summarized in §3.1.1 are color-coded by
their mean density ($n_{average}$ in the context of Table 1), with filtered
out structures shown in black (totaling $3.7\times 10^{4}\;\rm M_{\sun}$). A
constant isosurface density of $n_{min}=25\;\rm cm^{-3}$ (dashed horizontal
line) is used to define cloud boundaries and will pre-ordain the narrow range
of mean cloud volume densities ($33\;{\rm cm^{-3}}\leq n_{average}\leq
92\;{\rm cm^{-3}}$) seen in our sample.

####  3.1.1 Filtering 3D Features

After generating a hierarchy of cloud structure using the dendrogram approach,
we need to convert the dendrogram tree into a meaningful set of molecular
cloud features for analysis. In order to avoid both spurious features (with
relatively low mass) and double counting of nested clouds (i.e. counting a
branch and its leaves as separate structures), we introduce a filtering
scheme.

Our filtering scheme is based on the adoption of a minimum cloud mass,
$M_{min}$, and cloud radius, $r_{min}$, required for a feature to be included
in the catalog. The mass is computed for all features as described in §3.1.2
below. We define $M_{min}=500\;\rm M_{\odot}$ and $r_{min}=2\;\rm pc$. By
using this definition, the algorithm only includes structures that correspond
to trunks in the dendrogram that are above $M_{min}$ and $r_{min}$. All other
structures are removed to avoid counting nested clouds, which by definition
are not trunks.

Given that most dendrograms computed from 3D dust maps are composed of an
ensemble of single structure trees, retaining only the trunk feature means
retaining the structures defined by isosurfaces near $n_{min}=25$ cm-3. The
filtered molecular cloud features extracted using this approach are
highlighted and overlaid on the underlying Leike et al. (2020) 3D dust map in
Figure 1. The implication of this filtering method is that we are removing
most of the cloud hierarchy and limiting the extracted clouds to a narrow
range in mean cloud density. In theory, we could achieve a similar catalog by
thresholding the Leike et al. (2020) volume above a density of $n_{min}=25$
cm-3. However, we choose to dendrogram both to utilize the existing
functionality of the astrodendro package (Robitaille et al., 2019) and to
enable further follow-up studies of the full hierarchy.

In Figure 2, we show the full dendrogram, and highlight the cloud features
that survive filtering. The combined mass of the filtered out structures is
$3.7\times 10^{4}\;\rm M_{\sun}$ and are shown in black in Figure 2. The
remaining, surviving clouds are color-coded by the mean density, with the
clouds possessing mean densities between $n_{average}=33-92\;\rm cm^{-3}$. As
we will discuss further in §5, the narrow range in density of extracted clouds
will pre-ordain the mass-size results we obtain in §4 due to the large filling
factor of low density gas near the chosen $n_{min}$ (see e.g. further
discussion in Beaumont et al., 2012).

Finally, we also recognize that a relatively low volume density isosurface is
defining the 3D cloud boundaries. However, it is not possible to define clouds
using significantly higher volume density thresholds because the Leike et al.
(2020) 3D dust map is not sensitive to the highest volume density regions
within molecular clouds, due to the map’s reliance on optical stellar
photometry and astrometry from Gaia (Brown et al., 2018). Despite not being
sensitive to very high volume densities, Zucker et al. (2021) show that the
Leike et al. (2020) is still recovering cloud properties based on 2D
integrated approaches, as determined for a benchmark sample of well-studied
clouds. Thus, the low isosurface levels defining cloud boundaries should not
have a significant effect on our mass results.

####  3.1.2 Measuring Cloud Properties in 3D

To calculate the properties of clouds in p-p-p space, we again build on
existing infrastructure for computing cloud properties in p-p and p-p-v space
using the astrodendro package (Robitaille et al., 2019). In order to calculate
cloud properties, we take as input the dendrogrammed cloud structure, where
each cloud structure consists of a set of contiguous volume density voxels
bounded by a surface of constant volume density. We first calculate physical
properties for all structures, and subsequently define the final cloud catalog
through filtering as explained in §3.1.1. Our catalog includes the following
properties for every cloud (see Table 1). 111We adapted and extended a version
of the astrodendro package written by Dario Colombo and Ana-Duarte Cabral. The
software is available here:
https://github.com/dendrograms/astrodendro/pull/147/files

  1. 1.

$V$ (pc3): Exact volume of the structure in p-p-p space

  2. 2.

$r$ (pc): Equivalent radius of the sphere occupying the same volume as the
volume exact $V$

  3. 3.

$x,y,z$ (pc): Central $x$, $y$, and $z$ position of the cloud in Heliocentric
Galactic Cartesian coordinates

  4. 4.

$l$ (∘), $b$ (∘), and $d$ (pc). The cloud’s Galactic coordinates (longitude,
latitude, and distance), computed from the mean $x$, $y$, and $z$ position of
the cloud in Heliocentric Galactic Cartesian coordinates

  5. 5.

$M$ (M⊙): the mass of the cloud, calculated as follows:

$M=\Sigma\;dM_{i}=\;\Sigma\;1.37\;m_{p}\times n_{i}\times dV_{i}$ (1)

The total mass is the sum of the mass in each individual voxel $dM_{i}$,
computed by multiplying the volume density in the $i$th voxel ($n_{i}$) by the
mean molecular weight of hydrogen ($1.37\times m_{p}$, correcting for the
helium abundance) and its volume (1 pc3)

  6. 6.

$n_{peak}$ (cm-3): maximum volume density within the cloud

  7. 7.

$A$ (pc2): surface area of the cloud, calculated by assuming a spherical
geometry with a radius of $r$ and determining the cross-sectional area ($\pi
r^{2}$)

  8. 8.

$\Sigma$ ($\rm\frac{M_{\odot}}{pc^{2}}$): the mass surface density of the
cloud, given as the cloud’s mass divided by its surface area

  9. 9.

$n_{average}$ (cm-3): average volume density of the cloud

###  3.2 2D Methods

####  3.2.1 Converting 3D Dust Data into 2D Extinction Maps

Once a catalog of cloud features is defined and characterized in 3D, we create
a 2D projected map of each 3D cloud following the methodology of Zucker et al.
(2021) (see their §3.3 for full details). Briefly, that work uses the yt
package (Turk et al., 2011) to integrate 3D volume density cubes (containing
the 3D cloud of interest) along the line of sight and produce 2D maps of the
total hydrogen column density. For each cloud, we obtain a 3D volume density
sub-region suitable for projection by extracting a cutout of the Leike et al.
(2020) 3D dust map using the minimum and maximum extent of the cloud
boundaries along $x$, $y$, and $z$. Once we obtain the projected 2D column
density maps, we convert from total hydrogen column density to visual
extinction in the $K$ band using a relation from Lada et al. (2009) of
$\frac{N({\rm HI})+2N({\rm H_{\rm 2}})}{A_{K}}=1.67\times
10^{22}\rm\;cm^{-2}\;mag^{-1}$ to produce a map of $K$-band extinction,
$A_{K}$. Converting to $A_{K}$ allows us to compare to previous 2D cloud
catalogs built on similar maps of integrated dust extinction from Lada et al.
(2009).

To analyze the 2D maps, we extract clouds on the plane of the sky using the
existing p-p dendramming functionality of the astrodendro package. In order to
understand how 3D clouds map to 2D projected space we use Zucker et al. (2021)
as a guide, who analyze the 3D cartesian space ($x,y,z$) and Galactic
($l,b,d$) coordinates of a sample of famous nearby clouds. Identifying the
relevant plane of the sky features from the 3D projected famous cloud data, we
then use Lada et al. (2009) as a benchmark to determine the optimal
$A_{K_{{min}}}$, $\Delta_{A_{K}}$, and $\\#_{pixels}$ parameters (where
$A_{K_{{min}}}$, $\Delta_{A_{K}}$, and $\\#_{pixels}$ are the 2D analogs of
the $n_{min}$, $\Delta_{n}$, and $\\#_{voxels}$ 3D input parameters described
in §3.1) for computing the dendrogram, with the goal of obtaining similar
cloud sizes as derived in Lada et al. (2009). With the intention of also
having a single 2D structure representing each 3D cloud feature, we settle on
$A_{K_{{min}}}=0.05$ mag, $\Delta_{A_{K}}=0.05$ mag, and $\\#_{pixels}=300$
pixels as the dendrogram input parameters.

In Figure 3, we show an example of the 3D to 2D projection for a single
feature in the catalog (the Perseus Molecular Cloud). We emphasize that due to
the imperfect mapping from 3D to 2D space — stemming from the complex
geometries of individual clouds (Zucker et al., 2021) — a few 3D features do
not have a 2D counterpart, largely because they were sub-divided into multiple
components and failed to produce a cloud feature with the same mass and/or
size minima adopted for the 3D catalog. Our goal in this work is not to
measure the most accurate 2D-based properties of molecular clouds. Rather, we
seek to understand how defining features in 3D versus 2D projected space
(given the same underlying 3D data) affects cloud properties in aggregate.

Nevertheless, for clouds that have a 2D counterpart meeting these criteria
(61/65 clouds, or $\approx 94\%$ of the 3D sample), the morphological matching
between 3D and 2D cloud shapes is clear. We show and discuss the
correspondence between 3D and 2D further in Appendix §B

Figure 3: Comparison between the 3D and projected 2D dust data for the Perseus
Molecular Cloud. The feature corresponding to the Perseus Molecular Cloud in
3D volume density space is shown in blue. The 3D dust has been projected onto
the plane of the sky and shown via an $A_{K}$ extinction map in the
background, where the 2D feature corresponding to the Perseus Molecular Cloud
is shown in red.

####  3.2.2 Calculating Properties for 2D Projected Structures

After creating the dendrogram based on the $A_{K}$ maps, we filter the
features in our 2D catalog by implementing the same minimum mass threshold
($M_{min}=500\;\rm M_{\odot}$) and radius threshold ($r_{min}=2\;\rm pc$) as
our 3D data. In 2D space we measure the following properties:

  1. 1.

$l$ (∘): Central longitude of the cloud

  2. 2.

$b$ (∘): Central latitude of the cloud

  3. 3.

$d$ (pc): The distance of the cloud222Retrieved from the 3D cloud structure
before projecting.

  4. 4.

$A$ (pc2): Exact surface area of the structure in p-p space

  5. 5.

$M$ (M⊙): Mass of the cloud, derived using the following mass surface density
relation from Zari et al. (2016) and the distance of the cloud originally
detected in 3D:

$\frac{\Sigma_{\rm gas}}{A_{K}}=\mu m_{p}\beta_{K}=183\;{\rm M_{\odot}}\;\rm
pc^{-2}\;mag^{-1}$ (2)

  6. 6.

$r$ (pc): Radius calculated using the exact area of the structure in p-p space
$\bigl{(}\sqrt{\frac{A}{\pi}}\bigr{)}$ assuming a spherical geometry

  7. 7.

$\Sigma$ ($\rm\frac{M_{\odot}}{pc^{2}}$): the mass surface density of the
cloud, given as the cloud’s mass divided by its surface area

##  4 Results

###  4.1 Summary of 3D and 2D Cloud Properties

In Table 1, we present a summary of the properties of molecular clouds derived
in 3D space following §3.1.2. A machine readable version of Table 1 and its
associated astrodendro dendrogram file is available online at the Harvard
Dataverse (DOI:10.7910/DVN/BFYDG8). The 3D catalog contains a total mass of
$1.1\times 10^{5}\;\rm M_{\odot}$ across the 65 cloud features identified in
3D volume density space, with an average cloud mass of $M=1.7\times
10^{3}\;\rm M_{\odot}$. The distance range of the clouds spans $d=116-440$ pc.
The typical average density of clouds in the 3D catalog is $n_{\rm
average}=47\;\rm cm^{-3}$, while the typical peak density is about an order of
magnitude higher ($n_{\rm peak}=414\;\rm cm^{-3}$). We find an average cloud
volume of $V=1220\;\rm pc^{3}$ and an average equivalent radius assuming a
spherical geometry of $r=6$ pc, though we emphasize that some of the clouds
show more complex, extended geometries. While every 3D cloud feature is
assigned a unique identifier, the catalog includes a number of well-studied
clouds in the literature, including Perseus, Taurus, Lupus, Chamaeleon,
Cepheus, and the Orion complex (Orion A, Orion B, and $\lambda$ Orion) which
have been denoted as such in a separate column in Table 1 to aid comparison
with existing studies.

In Table 2 we present the corresponding catalog of 2D cloud properties derived
from the projected 3D data following §3.2.2. A machine readable version of
Table 2 is likewise available online at the Harvard Dataverse
(DOI:10.7910/DVN/BFYDG8). The 2D catalog contains a total mass of $2\times
10^{5}\;\rm M_{\odot}$ across the 61 cloud features derived from the
projection of the 3D cloud data. One of these 61 clouds, feature 7, is broken
into two components, leading to a total of 62 clouds in Table 2. Moreover, in
Table 2 we use the same cloud identifiers as Table 1, which specifies how each
2D cloud feature maps to its 3D counterpart. In the case a cloud in Table 1
has no corresponding cloud identifier in Table 2, the cloud was filtered for
not meeting our minimum mass threshold of 500 $\rm M_{\odot}$. The typical
radius of clouds in the 2D catalog is marginally larger than the 3D catalog,
averaging $r=7$ pc, and the average cloud mass is about a factor of two
higher, at $M=3.2\times 10^{3}\;\rm M_{\odot}$.

Considering the ensemble of clouds, the total mass of the entire 2D-derived
catalog is approximately $\approx 1.9\times$ higher than the total mass of the
3D-derived catalog. The discrepancy suggests that projecting 3D gas density
into 2D can alter the perceived shape of molecular clouds enough to bear
significantly on their derived mass. This effect likely stems from the
complications of projecting a non-spherical 3D cloud geometry onto the plane
of the sky, resulting in a different cloud boundary definition. However,
diffuse emission in the vicinity of the cloud also likely plays a major role.

As a testament to the impact of diffuse intervening gas, recall that we create
2D dust maps by projecting 3D dust cutouts which were extracted using a
bounding box corresponding to the the minimum and maximum extents of the 3D
cloud features in $xyz$ space. Over the sample of 3D cutouts, we compute the
ratio of the mass inside the 3D features to the mass outside the feature but
within the bounding box used for 2D projections. We find that, over the full
sample, there is $2\times$ as much mass outside the 3D dendrogrammed features
as within them. Thus, if even half of this excess mass in the vicinity of each
cloud is incorporated into the 2D cloud definitions, this contamination would
be enough to account for the discrepancy in total mass between the 3D- and
2D-derived catalogs that we observe.

###  4.2 Fitting the Mass-Size Relation

We use the masses and sizes in Table 1 and 2 to explore the mass-size
relation, first proposed by Larson (1981). Larson (1981) conclude that the
mass $M$ contained within a cloud of radius $r$ obeys a power-law of the form:

$M(r)=a\times\left(\frac{r}{\rm pc}\right)^{b}$ (3)

Larson (1981) obtain the relation that $M(r)=460\;{\rm
M_{\odot}}\times(\frac{r}{\rm pc})^{1.9}$ or more generally, that the mass of
a cloud is proportional to its area, implying constant column density. This
law of constant column density has come to be known as one of the fundamental
properties of molecular cloud structure (McKee & Ostriker, 2007). Recently, a
similar relationship between the masses and lengths $L$ of filaments
($M\propto L^{2}$) has also been found (Hacar et al., 2022) and attributed to
turbulent fragmentation. With the goal of testing whether the dimensionality
used to define clouds affects our results, we fit the mass-size relation in
both 3D and 2D space. Fitting the relation in log-log space, we use a linear-
least-squares fitter to obtain $a$, $b$ and their associated uncertainties.
For the 3D results we obtain $\log M=2.9\times\log r(\pm 0.1)+0.86(\pm 0.3)$
such that:

$M(r)=7\;\rm M_{\odot}\times(\frac{\it r}{\rm pc})^{2.9}$ (4)

And for the 2D results, we obtain $\log M=2.1\times\log r(\pm 0.2)+1.59(\pm
0.4)$, such that:

$M(r)=39\;\rm M_{\odot}\times(\frac{\it r}{\rm pc})^{2.1}$ (5)

Moreover, we have included in the Appendix (see §A) another version of the 2D
catalog with a minimum extinction threshold when defining the dendrogram of
$A_{K_{min}}=0.1$ mag. This test yielded fewer features, as expected, but
maintained a similar mass-size relation of $M(r)=83\;\rm
M_{\odot}\times(\frac{\it r}{\rm pc})^{1.9}$, confirming that the scaling of
the mass-size relation does not depend on the threshold used to define cloud
boundaries. We also repeat both fits using only the subset of the clouds which
are well-studied in the literature (e.g. Perseus, Taurus, Lupus, Chamaeleon,
Cepheus, and the Orion complex), finding that the results agree with the full
catalog fits within our reported uncertainties.

In Figure 4, we plot mass versus size for our 3D and 2D catalogs with the
best-fits overlaid and four lines of constant volume density
($n=15,30,100,300\;\rm cm^{-3}$), assuming purely spherical geometries. Figure
4 shows that the clouds in the 3D catalog lay between $n=30\;\rm cm^{-3}$ and
$n=100\;\rm cm^{-3}$ lines of constant volume density, which we will argue in
§5 is a consequence of our dendrogramming procedure and the narrow volume
density range probed by the Leike et al. (2020) 3D dust map.

Table 1: 3D Cloud Catalog

Cloud | Complex | $x$ | $y$ | $z$ | $l$ | $b$ | $d$ | $n_{average}$ | $n_{peak}$ | $M$ | $r$ | $A$ | $V$ | $\Sigma$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | pc | pc | pc | ∘ | ∘ | pc | cm-3 | cm-3 | M⊙ | pc | pc2 | pc3 | $\frac{\rm M_{\odot}}{\rm pc^{2}}$
0 | - | -218 | 82 | -157 | 159.2 | -33.9 | 282 | 52 | 314 | 1352 | 5.8 | 106 | 826 | 12
1 | Orion A* | -340 | -236 | -148 | 214.8 | -19.8 | 440 | 38 | 120 | 774 | 5.4 | 90 | 647 | 8
2 | Orion A* | -348 | -184 | -143 | 207.9 | -20.0 | 419 | 45 | 221 | 1624 | 6.5 | 133 | 1163 | 12
3 | Orion A* | -320 | -204 | -135 | 212.6 | -19.6 | 403 | 44 | 1074 | 3996 | 8.9 | 247 | 2923 | 16
4 | - | -288 | -162 | -116 | 209.4 | -19.3 | 350 | 92 | 1349 | 2947 | 6.3 | 123 | 1029 | 23
5 | - | -356 | -158 | -117 | 204.0 | -16.8 | 406 | 53 | 915 | 590 | 4.4 | 61 | 359 | 9
6 | Perseus | -257 | 97 | -102 | 159.3 | -20.4 | 294 | 40 | 154 | 6425 | 10.7 | 361 | 5159 | 17
7 | Orion B* | -351 | -172 | -100 | 206.1 | -14.4 | 404 | 45 | 716 | 10878 | 12.3 | 472 | 7736 | 23
8 | - | 140 | -147 | -113 | 313.5 | -29.1 | 232 | 47 | 427 | 590 | 4.6 | 66 | 406 | 8
9 | Orion $\lambda$* | -363 | -101 | -108 | 195.6 | -16.0 | 392 | 38 | 375 | 2019 | 7.4 | 173 | 1718 | 11
10 | - | -280 | -136 | -102 | 206.0 | -18.2 | 328 | 85 | 1436 | 1867 | 5.5 | 95 | 704 | 19
11 | - | -277 | -114 | -101 | 202.4 | -18.8 | 316 | 90 | 3505 | 2207 | 5.7 | 103 | 790 | 21
12 | Orion $\lambda$* | -360 | -76 | -86 | 192.0 | -13.2 | 378 | 39 | 838 | 1526 | 6.7 | 140 | 1248 | 10
13 | - | 142 | -145 | -84 | 314.6 | -22.5 | 220 | 66 | 620 | 758 | 4.4 | 62 | 368 | 12
14 | Orion $\lambda$* | -359 | -129 | -79 | 199.7 | -11.7 | 390 | 39 | 146 | 1031 | 5.9 | 107 | 843 | 9
15 | Orion $\lambda$* | -350 | -94 | -78 | 195.0 | -12.2 | 371 | 72 | 1321 | 1335 | 5.2 | 85 | 594 | 15
16 | - | -254 | 118 | -76 | 155.0 | -15.3 | 291 | 42 | 364 | 925 | 5.5 | 95 | 697 | 9
17 | - | -61 | -363 | -69 | 260.4 | -10.7 | 374 | 55 | 838 | 1649 | 6.1 | 117 | 953 | 14
18 | - | -129 | -310 | -74 | 247.4 | -12.4 | 344 | 66 | 500 | 595 | 4.1 | 52 | 289 | 11
19 | - | -347 | -140 | -69 | 202.0 | -10.6 | 381 | 36 | 325 | 626 | 5.1 | 82 | 560 | 7
20 | Musca and Chamaeleon | 88 | -153 | -48 | 300.0 | -15.3 | 183 | 53 | 483 | 6162 | 9.6 | 291 | 3734 | 21
21 | - | -212 | -13 | -43 | 183.8 | -11.5 | 216 | 38 | 284 | 660 | 5.1 | 81 | 554 | 8
22 | Taurus | -141 | 20 | -37 | 171.6 | -14.8 | 147 | 47 | 562 | 6352 | 10.1 | 320 | 4312 | 19
23 | - | -41 | -313 | -41 | 262.4 | -7.5 | 318 | 63 | 799 | 1244 | 5.3 | 89 | 632 | 14
24 | - | 191 | 111 | -24 | 30.1 | -6.3 | 223 | 43 | 209 | 548 | 4.6 | 66 | 411 | 8
25 | - | 281 | 101 | -17 | 19.8 | -3.4 | 299 | 37 | 126 | 761 | 5.4 | 91 | 657 | 8
26 | - | -197 | 44 | -15 | 167.3 | -4.4 | 202 | 39 | 214 | 590 | 4.9 | 74 | 488 | 7
27 | - | 272 | 282 | -12 | 46.0 | -1.8 | 392 | 40 | 197 | 4501 | 9.5 | 285 | 3625 | 15
28 | - | 74 | -171 | -13 | 293.4 | -4.0 | 187 | 41 | 275 | 639 | 4.9 | 75 | 495 | 8
29 | - | 215 | 120 | -8 | 29.2 | -2.0 | 247 | 36 | 104 | 718 | 5.3 | 89 | 639 | 8
30 | - | 96 | -159 | -7 | 301.1 | -2.4 | 186 | 43 | 267 | 613 | 4.8 | 72 | 461 | 8
31 | Lupus | 193 | -35 | 12 | 349.6 | 3.5 | 197 | 40 | 263 | 2267 | 7.6 | 180 | 1816 | 12
32 | - | -245 | 112 | 8 | 155.3 | 1.9 | 269 | 44 | 348 | 896 | 5.4 | 90 | 645 | 9
33 | Lupus | 149 | -59 | 16 | 338.5 | 5.9 | 161 | 44 | 398 | 4210 | 9.0 | 253 | 3029 | 16
34 | Pipe and Ophiuchus | 134 | -5 | 29 | 357.8 | 12.3 | 138 | 47 | 432 | 8177 | 11.0 | 378 | 5529 | 21
35 | - | 228 | 114 | 21 | 26.5 | 4.7 | 256 | 35 | 157 | 2188 | 7.8 | 189 | 1969 | 11
36 | - | 201 | 77 | 21 | 21.1 | 5.6 | 217 | 33 | 90 | 601 | 5.2 | 84 | 586 | 7
37 | - | -225 | 263 | 24 | 130.6 | 4.0 | 347 | 39 | 155 | 868 | 5.5 | 95 | 707 | 9
38 | - | -205 | 283 | 24 | 125.9 | 4.0 | 350 | 42 | 162 | 633 | 4.9 | 74 | 485 | 8
39 | - | -148 | 318 | 31 | 115.0 | 5.2 | 352 | 43 | 250 | 1043 | 5.7 | 101 | 772 | 10
40 | - | -175 | 267 | 28 | 123.2 | 5.0 | 321 | 46 | 331 | 511 | 4.4 | 61 | 359 | 8
41 | - | 219 | 113 | 31 | 27.2 | 7.3 | 249 | 39 | 145 | 552 | 4.8 | 71 | 453 | 7
42 | Ophiuchus | 141 | -24 | 38 | 350.0 | 15.1 | 148 | 43 | 180 | 1054 | 5.7 | 102 | 777 | 10
43 | - | -199 | 207 | 39 | 133.9 | 7.9 | 290 | 44 | 420 | 1388 | 6.2 | 121 | 1009 | 11
44 | - | -106 | 218 | 37 | 116.1 | 8.7 | 245 | 42 | 239 | 522 | 4.6 | 65 | 397 | 8
45 | - | -164 | 271 | 39 | 121.2 | 7.1 | 319 | 47 | 584 | 621 | 4.6 | 67 | 420 | 9
46 | Ophiuchus | 107 | 18 | 39 | 9.9 | 19.9 | 116 | 46 | 376 | 640 | 4.7 | 70 | 443 | 9
47 | - | -245 | 221 | 42 | 138.0 | 7.3 | 333 | 44 | 265 | 858 | 5.3 | 88 | 626 | 9
48 | Ophiuchus | 141 | 10 | 43 | 4.1 | 17.2 | 148 | 37 | 298 | 660 | 5.1 | 83 | 570 | 7
49 | Lupus | 138 | -55 | 45 | 338.2 | 16.7 | 156 | 63 | 460 | 663 | 4.3 | 58 | 340 | 11
50 | Lupus | 168 | -71 | 48 | 337.0 | 14.9 | 189 | 43 | 269 | 595 | 4.7 | 70 | 442 | 8
51 | - | -223 | 166 | 48 | 143.3 | 9.9 | 282 | 44 | 185 | 609 | 4.7 | 69 | 439 | 8
52 | - | -132 | 179 | 54 | 126.5 | 13.8 | 230 | 46 | 280 | 1589 | 6.4 | 129 | 1106 | 12
53 | - | -144 | 275 | 56 | 117.7 | 10.4 | 315 | 42 | 434 | 1193 | 6.0 | 112 | 903 | 10
54 | - | -195 | 208 | 64 | 133.2 | 12.6 | 293 | 46 | 340 | 964 | 5.4 | 93 | 675 | 10
55 | - | -191 | 227 | 73 | 130.0 | 13.9 | 306 | 50 | 436 | 904 | 5.2 | 84 | 583 | 10
56 | Cepheus | -78 | 320 | 80 | 103.7 | 13.8 | 339 | 53 | 316 | 854 | 5.0 | 77 | 515 | 11
57 | Cepheus | -133 | 303 | 99 | 113.8 | 16.6 | 346 | 44 | 627 | 4535 | 9.2 | 267 | 3283 | 17
58 | Cepheus | -69 | 318 | 89 | 102.3 | 15.4 | 338 | 68 | 724 | 988 | 4.8 | 73 | 469 | 13
59 | Cepheus | -79 | 328 | 108 | 103.6 | 17.8 | 354 | 35 | 132 | 568 | 5.0 | 78 | 518 | 7
60 | Cepheus | -143 | 293 | 116 | 116.0 | 19.6 | 346 | 44 | 265 | 1065 | 5.7 | 102 | 780 | 10
61 | Cepheus | -128 | 320 | 126 | 111.8 | 20.2 | 367 | 50 | 382 | 1261 | 5.8 | 104 | 802 | 12
62 | - | -165 | 268 | 145 | 121.6 | 24.7 | 347 | 40 | 321 | 960 | 5.7 | 100 | 760 | 9
63 | - | -179 | 265 | 150 | 124.0 | 25.2 | 353 | 43 | 465 | 705 | 5.0 | 78 | 519 | 9
64 | - | -175 | 256 | 181 | 124.4 | 30.2 | 360 | 37 | 115 | 675 | 5.2 | 84 | 580 | 8

**footnotetext: The Orion Clouds lie at the very edge of the Leike et al.
(2020) 3D dust grid, thus adding more uncertainty to their derived properties
and should be treated with caution.

Note. — Properties of local molecular clouds derived in 3D p-p-p space. (1)
Cloud Index (2) Association with well-studied nearby cloud complexes (3-5)
Heliocentric Galactic Cartesian coordinates $x$,$y$,$z$ (6-7) Central Galactic
Longitude $l$ and Latitude $b$ (8) Distance (9) Average volume density (10)
Peak volume density (11) Mass (12) Radius (13) Surface area (14) Volume (15)
Surface mass density.

Note. — A machine readable version of this table is available at
https://doi.org/10.7910/DVN/BFYDG8.

Table 2: 2D Projected Catalog

Cloud | Complex | $l$ | $b$ | $d$ | $M$ | $r$ | $A$ | $\Sigma$
---|---|---|---|---|---|---|---|---
|  | ∘ | ∘ | pc | $\rm M_{\odot}$ | pc | pc2 | $\rm\frac{\rm M_{\odot}}{\rm pc^{2}}$
0 | - | 159.1 | -33.8 | 282 | 1609 | 5.2 | 84 | 19
1 | Orion A* | 214.9 | -19.6 | 440 | 1184 | 5.2 | 85 | 14
2 | Orion A* | 207.8 | -20.0 | 419 | 2179 | 6.6 | 137 | 16
3 | Orion A* | 212.7 | -19.5 | 403 | 6159 | 10.4 | 341 | 18
4 | - | 209.2 | -19.3 | 350 | 2758 | 6.2 | 121 | 23
6 | Perseus | 159.0 | -20.3 | 294 | 13083 | 14.7 | 679 | 19
$7_{0}$ | Orion B* | 206.1 | -13.9 | 404 | 20143 | 18.0 | 1013 | 20
$7_{1}$ | Orion B* | 204.0 | -16.6 | 404 | 595 | 3.7 | 43 | 14
8 | - | 313.5 | -28.9 | 232 | 887 | 4.4 | 59 | 15
9 | Orion $\lambda$* | 195.1 | -16.0 | 392 | 2917 | 8.2 | 209 | 14
12 | Orion $\lambda$* | 192.0 | -13.3 | 378 | 3055 | 8.7 | 238 | 13
13 | - | 313.9 | -22.4 | 220 | 508 | 3.3 | 34 | 15
14 | Orion $\lambda$* | 199.7 | -11.6 | 390 | 1537 | 5.6 | 98 | 16
15 | Orion $\lambda$* | 194.8 | -12.0 | 371 | 1436 | 5.1 | 81 | 18
16 | - | 154.9 | -15.3 | 291 | 1507 | 5.5 | 95 | 16
17 | - | 260.2 | -10.5 | 374 | 1458 | 5.5 | 96 | 15
18 | - | 247.5 | -12.4 | 344 | 590 | 3.0 | 28 | 21
19 | - | 201.9 | -10.4 | 381 | 1635 | 5.9 | 109 | 15
20 | Musca and Chamaeleon | 299.5 | -14.7 | 183 | 10500 | 14.6 | 668 | 16
21 | - | 183.5 | -11.7 | 216 | 1267 | 5.7 | 102 | 12
22 | Taurus | 171.2 | -14.2 | 147 | 16652 | 17.3 | 936 | 18
23 | - | 262.8 | -7.6 | 318 | 615 | 3.5 | 38 | 16
24 | - | 30.4 | -6.0 | 223 | 618 | 3.8 | 46 | 13
25 | - | 20.1 | -3.4 | 299 | 1576 | 6.3 | 124 | 13
26 | - | 167.0 | -4.1 | 202 | 620 | 4.0 | 49 | 13
27 | - | 46.4 | -1.5 | 392 | 10262 | 14.1 | 625 | 16
28 | - | 293.4 | -3.9 | 187 | 975 | 4.6 | 67 | 15
29 | - | 29.3 | -1.8 | 247 | 1206 | 5.5 | 93 | 13
30 | - | 301.3 | -2.3 | 186 | 721 | 4.0 | 49 | 15
31 | Lupus | 349.6 | 3.8 | 197 | 4179 | 9.5 | 285 | 15
32 | - | 155.3 | 1.9 | 269 | 1247 | 5.3 | 88 | 14
33 | Lupus | 338.3 | 6.1 | 161 | 8462 | 13.0 | 530 | 16
34 | Pipe and Ophiuchus | 0.0 | 13.7 | 138 | 30146 | 22.0 | 1516 | 20
35 | - | 26.4 | 5.3 | 256 | 8225 | 12.1 | 463 | 18
36 | - | 21.4 | 5.3 | 217 | 1915 | 7.2 | 161 | 12
37 | - | 131.2 | 4.2 | 347 | 1792 | 6.4 | 130 | 14
38 | - | 126.0 | 4.0 | 350 | 876 | 4.5 | 62 | 14
39 | - | 115.0 | 5.1 | 352 | 1743 | 6.1 | 115 | 15
40 | - | 123.2 | 5.1 | 321 | 577 | 3.7 | 41 | 14
41 | - | 27.3 | 7.3 | 249 | 871 | 4.6 | 66 | 13
42 | Ophiuchus | 350.3 | 15.2 | 148 | 1454 | 5.5 | 95 | 15
43 | - | 133.7 | 7.9 | 290 | 1998 | 6.7 | 142 | 14
44 | - | 115.9 | 8.7 | 245 | 677 | 4.1 | 52 | 13
45 | - | 121.4 | 7.0 | 319 | 616 | 3.7 | 42 | 15
46 | Ophiuchus | 9.5 | 19.9 | 116 | 1018 | 4.8 | 73 | 14
47 | - | 137.9 | 7.5 | 333 | 1158 | 5.1 | 81 | 14
48 | Ophiuchus | 4.5 | 17.3 | 148 | 1172 | 5.4 | 92 | 13
49 | Lupus | 338.4 | 16.9 | 156 | 703 | 3.6 | 39 | 18
51 | - | 143.1 | 10.0 | 282 | 702 | 3.9 | 48 | 15
52 | - | 126.3 | 13.9 | 230 | 2273 | 6.8 | 144 | 16
53 | - | 117.4 | 10.8 | 315 | 1256 | 5.6 | 98 | 13
54 | - | 133.1 | 12.7 | 293 | 1224 | 5.2 | 85 | 14
55 | - | 129.8 | 14.0 | 306 | 1177 | 4.9 | 76 | 16
56 | Cepheus | 103.6 | 13.9 | 339 | 998 | 4.2 | 54 | 18
57 | Cepheus | 113.7 | 16.7 | 346 | 6944 | 10.6 | 353 | 20
58 | Cepheus | 102.2 | 15.4 | 338 | 1055 | 4.1 | 53 | 20
59 | Cepheus | 103.4 | 17.8 | 354 | 1019 | 5.2 | 85 | 12
60 | Cepheus | 116.0 | 19.7 | 346 | 1447 | 5.4 | 92 | 16
61 | Cepheus | 111.7 | 20.2 | 367 | 1556 | 5.5 | 94 | 17
62 | - | 121.4 | 24.8 | 347 | 1431 | 5.7 | 102 | 14
63 | - | 124.0 | 25.3 | 353 | 976 | 4.6 | 65 | 15
64 | - | 124.3 | 30.2 | 360 | 998 | 4.9 | 74 | 14

**footnotetext: The Orion Clouds lie at the very edge of the Leike et al.
(2020) 3D dust grid, thus adding more uncertainty to their derived properties
and should be treated with caution.

Note. — Properties of the 2D projected local molecular clouds (1) Cloud index
(2) Association with well-studied nearby cloud complexes (3-4) Central
Galactic longitude $l$ and latitude $b$ (5) Distance (6) Mass (7) Radius (8)
Exact area (9) Surface mass density. A small fraction of the 3D clouds did not
have a 2D counterpart that met our cloud definition, which accounts for the
fact that there are 62 structures identified in 2D after projecting our 65 3D
structures from Table 1 on to plane of the sky.

Note. — A machine readable version of this table is available at
https://doi.org/10.7910/DVN/BFYDG8.

Note. — Projecting feature 7 from 3D into 2D with $A_{K_{min}}=0.05$ mag
yields two components, $7_{0}$ and $7_{1}$.

Note. — A version of this table with a higher minimum threshold for cloud
boundary definition, $A_{K,min}=0.1$ mag, is available in Appendix §A.

Figure 4: Comparison of the mass-size relation derived from our 3D catalog
(blue points) and 2D catalog (red points) of molecular cloud properties. In
3D, $M\propto r^{2.9}$, while in 2D, $M\propto r^{2.1}$. We demarcate the 95%
confidence interval with the thin semi-transparent band around each best-fit
line. We include four lines of constant volume density
($n=15,30,100,300\rm\;cm^{-3}$) in dashed semi-transparent black lines. The
gray triangles represent the inaccessible volume and column densities, either
due to the minimum volume and extinction thresholds required for inclusion in
the 3D and 2D dendrograms ($n<25\;\rm cm^{-3}$ and $A_{K}<0.05\;\rm mag$) or
the inability of the 3D dust map to probe higher densities due to the map’s
reliance on optical stellar photometry ($n\gtrsim 100\;\rm cm^{-3}$ or
$A_{K}\gtrsim 0.2-0.3\;\rm mag$).

##  5 Discussion

Figure 5: 3D and 2D mass-size relations for the 3D dust (blue) and projected
2D dust (red) cloud catalogs, in comparison to five catalogs from the
literature, including (a) Miville-Deschênes et al. (2016), (b) Chen et al.
(2020), (c) Larson (1981), (d) Rice et al. (2016) and (e) Dharmawardena et al.
(2023).

Here we compare our results for the 3D and 2D mass-size relation with extant
results from the literature. We base our comparison on the work of Lada & Dame
(2020), which analyzes both dust-based cloud catalogs and CO-based cloud
catalogs to investigate the nature of the mass-size relation in the Milky Way.
Specifically, we consider the original cloud sample of Larson (1981),
alongside more recent cloud catalogs re-analyzed in Lada & Dame (2020),
including: the Rice et al. (2016) catalog based on a dendrogram decomposition
of the Dame et al. (2001) ${}^{\rm 12}{\rm CO}$ survey; the Miville-Deschênes
et al. (2016) catalog based on a hierarchical clustering algorithm applied to
Gaussian fits of the same Dame et al. (2001) ${}^{\rm 12}\rm CO$ survey; and
the Chen et al. (2020) catalog based on the 3D dust map of Chen et al. (2018).
We also include a comparison to Dharmawardena et al. (2023), which is the only
other study to extract cloud properties in p-p-p space. For consistency, we
compare to the Dharmawardena et al. (2023) primary trunks only catalog (see
Table 2 in their work).

In Figure 5, we plot our mass-size results in context. Following Lada & Dame
(2020), the catalogs from Rice et al. (2016), Miville-Deschênes et al. (2016)
and Chen et al. (2020) are all consistent with $b=2$ ($M\sim r^{2}$), implying
constant surface (column) density across molecular clouds. While some of the
clouds in these catalogs may have originally been identified using 3D
position-position-velocity data (Rice et al., 2016; Miville-Deschênes et al.,
2016) or even 3D position-position-position data (Chen et al., 2020), the
masses and/or sizes of these clouds have all been measured in 2D by projecting
the 3D data on to the plane of the sky (Chen et al., 2020; Rice et al., 2016;
Miville-Deschênes et al., 2016).

By measuring the masses and sizes of fully-resolved molecular clouds directly
in 3D volume density space, we find a larger power law slope of approximately
three, or $b=2.9$. This discrepancy suggests that the power law slope is
directly dependent on the number of dimensions used to measure mass and area,
with the power law slope consistent with $b=2$ when measuring cloud properties
in column density space and $b=3$ when measuring cloud properties in volume
density space. We directly test this hypothesis by projecting the 3D volume
density data into 2D, re-defining the cloud boundaries in column density space
and re-measuring their masses and areas. We find that despite stemming from
the same underlying 3D dust data, the projected results are consistent with a
shallower power law slope of $b=2.1$.

While Dharmawardena et al. (2023) argue that their p-p-p-based mass-size
results are consistent with Larson’s $b=1.9$ relationship, Figure 4 suggests
that the Dharmawardena et al. (2023) results may be closer to $b=3$, albeit
with an order of magnitude higher cloud masses. We attribute the difference in
3D-derived cloud masses to the existence of more extended cloud substructure
along the line of sight in Dharmawardena et al. (2023), which is not observed
in Leike et al. (2020). This claim is supported by Figure 6, which shows that
the Taurus molecular cloud — one of the sixteen cloud complexes in the
Dharmawardena et al. (2023) sample — spans distances of $d=93-342$ pc, about a
factor of five more extended than we find in this work ($d=127-170$ pc; based
on Leike et al., 2020).

The higher power law slope for the 3D catalogs observationally validates
previous predictions for the scaling of the mass-size relation based on a
combination of extant 2D observational results, numerical simulations, and
analytic theory. For example, Shetty et al. (2010) measure masses and sizes of
clouds in both volume and column density space based on hydrodynamical
simulations, finding $b=3.03\pm 0.02$ in 3D and $b=1.95\pm 0.03$ in 2D.
Likewise, Ballesteros-Paredes et al. (2012) argue that the mass-size relation
depends on cloud boundary definition, with column density definitions yielding
a power-law slope $b=2$ and volume density definitions yielding $b=3$. The
Ballesteros-Paredes et al. (2012) argument stems from the fact that for clouds
with similar boundary definitions, the filling factor of dense structures is
small while the filling factor of fluffier structures used to define the cloud
boundaries is high, implying that mass should scale with the area in 2D and
with volume in 3D (see also e.g. Ballesteros-Paredes & Mac Low, 2002b;
Ballesteros-Paredes et al., 2019). When clouds are defined as isocontours or
isosurfaces above a particular threshold, the average column or volume density
of the cloud is similar to the adopted threshold, because a large fraction of
pixels or voxels in the cloud lie close to the threshold value.

As an observational counterpart to the investigations of Shetty et al. (2010)
and Ballesteros-Paredes et al. (2012), Beaumont et al. (2012) examines the
mass-size relationship in terms of the column density PDF and its possible
variation within and between clouds. Leveraging 2D dust extinction maps from
Lombardi et al. (2010), Beaumont et al. (2012) find that for structures
defined with a constant extinction threshold, the mean of the column density
PDF within each structure varies less than the region-to-region dispersion in
area, naturally yielding $M\propto A\propto r^{2}$

In our work, the thresholds used to define cloud boundaries lie close to the
minimum volume density ($n_{min}$) or extinction ($A_{K_{min}}$) threshold
required for a feature to be included in the dendrogram, and are roughly
constant across the sample. As seen in Figure 4, on the lower mass end, the
clouds all lie above the $n=25\;\rm cm^{-3}$ line of constant volume density
and $N=8\times 10^{20}\;\rm cm^{-2}$ line of constant column density, which is
pre-determined by the minimum volume density ($n_{min}=25\;\rm cm^{-3}$) and
extinction threshold ($A_{K_{min}}=0.05\;\rm mag$) required for inclusion in
the 3D and 2D catalogs, respectively. On the higher mass end, the clouds lie
below $n\approx 100\;\rm cm^{-3}$ and $N\approx 4\times 10^{21}\;\rm cm^{-2}$
which is also pre-determined by the fact that the Leike et al. (2020) 3D dust
map is not sensitive to the densest, most extinguished regions in molecular
clouds ($A_{K}\gtrsim 0.3\;\rm mag$) due to their reliance on optical
photometry (see e.g. discussion in §4.4 of Zucker et al., 2021). Thus,
following Ballesteros-Paredes et al. (2019), Beaumont et al. (2012), and
Shetty et al. (2010), mass should scale with area in 2D and volume in 3D given
the narrow range of column and volume density probed, which we validate here
for the first time using the same underlying observational data.

Figure 6: A bird’s eye comparison between the Taurus molecular cloud complex
as analyzed in Dharmawardena et al. (2023) (orange) and in this work (blue),
based on the 3D dust map from Leike et al. (2020). Both 3D dust cutouts have
been integrated over the same range in $z$. Dharmawardena et al. (2023) find
roughly an order of magnitude higher cloud masses across their sample of
sixteen local clouds compared to this work, likely due to the presence of
cloud substructure at farther distances along the line of sight, which is
largely not detected in Leike et al. (2020).

##  6 Conclusions

Using the Leike et al. (2020) 3D dust map with a distance resolution of 1 pc,
we extend the dendrogram technique to position-position-position space to
extract and measure the properties of clouds in 3D physical space, including
their 3D positions, masses, sizes, and volume densities. To compare with
extant results, we also create synthetic 2D dust extinction maps from the 3D
dust distributions and derive similar properties for the same clouds defined
on the plane of the sky. Given the masses and sizes of clouds obtained in 2D
and 3D space, we fit the mass-size relation following Larson (1981).
Consistent with predictions from extant observational studies and numerical
simulations (see e.g. Beaumont et al., 2012; Ballesteros-Paredes et al.,
2019), we find that our 2D projected mass-size relation, $M\propto r^{2.1}$,
agrees with the original Larson (1981) results ($M\propto r^{2}$), where mass
scales according to the cloud’s area. However, we obtain a steeper power-law
slope for the 3D results, $M\propto r^{2.9}$, where the mass scales according
to the cloud’s volume. This difference in scaling is a natural consequence of
the roughly constant thresholds used to define cloud boundaries, in
combination with the fact that the PDF of column and volume density do not
systematically scale with structure size. Future work connecting these high-
resolution 3D cloud results to complementary tracers of a cloud’s CO
kinematics should enable further constraints not only on Larson’s other
relations (see also e.g. Kainulainen et al., 2021, for insight into the
Kennicutt-Schmidt relation) but also on the physical conditions of star
formation within molecular clouds.

##  7 Acknowledgements

The visualization, exploration, and interpretation of data presented in this
work were made possible using the glue visualization software, supported under
NSF grant numbers OAC-1739657 and CDS&E:AAG-1908419. AG and CZ acknowledge
support by NASA ADAP grant 80NSSC21K0634 “Knitting Together the Milky Way: An
Integrated Model of the Galaxy’s Stars, Gas, and Dust.” CZ acknowledges that
support for this work was provided by NASA through the NASA Hubble Fellowship
grant #HST-HF2-51498.001 awarded by the Space Telescope Science Institute,
which is operated by the Association of Universities for Research in
Astronomy, Inc., for NASA, under contract NAS5-26555. The authors acknowledge
Thomas Dame and Andreas Burkert for helpful discussions that contributed to
this work. We also thank Thavisha Dharmawardena for help with Figure 6.

##  Appendix A Dependence of 2D Catalog Properties on Minimum Extinction
Threshold

To confirm that the 2D mass-size relation is robust to the choice of boundary
definition we repeat the cloud extraction procedure described in §3.2.2 but
using a higher minimum extinction threshold of $A_{K_{min}}=0.1$ mag (in
comparison to $A_{K_{min}}=0.05$ mag whose results are described in §4). As
expected, this higher-extinction-threshold version yields fewer features, as
well as divided several of the clouds into multiple components333If a single
3D feature is broken up into multiple 2D features, the same distance is
assigned for each 2D feature.. Nevertheless, the catalog maintains a similar
mass-size relation of $\log M=1.9\times\log{r}(\pm 0.2)+1.92(\pm 0.4)$, such
that $M(r)=83\;{\rm M_{\odot}}\times(\frac{r}{\rm pc})^{1.9}$, confirming that
the scaling of the mass-size relation does not depend on the threshold used to
define cloud boundaries.

Table 3: 2D Catalog Properties with $A_{K_{min}}=0.1$ mag

Cloud | Component ID | Complex | $l$ | $b$ | $d$ | $M$ | r | $A$ | $\Sigma$
---|---|---|---|---|---|---|---|---|---
|  |  | ∘ | ∘ | pc | $\rm M_{\odot}$ | pc | $\text{pc}^{2}$ | $\frac{M_{\odot}}{\rm pc^{2}}$
2 | 0 | Orion A* | 207.8 | -20.0 | 419 | 2179 | 6.6 | 137 | 16
4 | 0 | - | 209.2 | -19.3 | 350 | 1879 | 4.4 | 61 | 31
4 | 1 | - | 209.4 | -19.4 | 350 | 1559 | 4.0 | 49 | 32
4 | 2 | - | 209.4 | -19.4 | 350 | 1458 | 3.8 | 46 | 32
4 | 3 | - | 209.4 | -19.4 | 350 | 1356 | 3.7 | 43 | 32
4 | 4 | - | 209.5 | -19.5 | 350 | 1043 | 3.2 | 31 | 34
6 | 0 | Perseus | 159.1 | -20.4 | 294 | 8638 | 10.5 | 347 | 25
7 | 0 | Orion B* | 206.2 | -14.3 | 404 | 13038 | 12.6 | 498 | 26
7 | 1 | Orion B* | 203.9 | -11.8 | 404 | 595 | 2.9 | 26 | 23
11 | 0 | - | 192.0 | -13.3 | 378 | 2868 | 8.5 | 225 | 13
11 | 1 | - | 191.9 | -14.1 | 378 | 1480 | 5.9 | 110 | 14
11 | 2 | - | 192.0 | -11.7 | 378 | 849 | 4.4 | 60 | 14
13 | 0 | - | 199.7 | -11.6 | 390 | 1537 | 5.6 | 98 | 16
15 | 0 | Orion $\lambda$* | 155.2 | -14.7 | 291 | 503 | 2.6 | 21 | 24
16 | 0 | - | 260.2 | -10.5 | 374 | 1458 | 5.5 | 96 | 15
18 | 0 | - | 297.0 | -15.8 | 183 | 1124 | 3.5 | 38 | 30
20 | 0 | Musca and Chamaeleon | 171.2 | -15.1 | 147 | 7596 | 9.4 | 278 | 27
25 | 0 | - | 46.3 | -1.5 | 392 | 9037 | 13.3 | 554 | 16
28 | 0 | - | 301.3 | -2.3 | 186 | 721 | 4.0 | 49 | 15
31 | 0 | Lupus | 338.3 | 6.1 | 161 | 8462 | 13.0 | 530 | 16
32 | 0 | - | 358.4 | 17.5 | 138 | 14493 | 12.9 | 523 | 28
32 | 1 | - | 354.4 | 16.3 | 138 | 4437 | 6.1 | 116 | 38
32 | 2 | - | 357.0 | 19.3 | 138 | 1574 | 3.8 | 44 | 36
33 | 0 | Lupus | 26.4 | 5.3 | 256 | 8225 | 12.1 | 463 | 18
44 | 0 | - | 9.5 | 19.9 | 116 | 1018 | 4.8 | 73 | 14
47 | 0 | - | 338.4 | 16.9 | 156 | 703 | 3.6 | 39 | 18
50 | 0 | Lupus | 126.3 | 13.9 | 230 | 2273 | 6.8 | 144 | 16
53 | 0 | - | 129.8 | 14.0 | 306 | 1177 | 4.9 | 76 | 16
54 | 0 | - | 103.6 | 13.9 | 339 | 998 | 4.2 | 54 | 18
55 | 0 | - | 113.6 | 16.7 | 346 | 4582 | 7.5 | 175 | 26
56 | 0 | Cepheus | 102.2 | 15.4 | 338 | 677 | 2.8 | 24 | 28
59 | 0 | Cepheus | 111.6 | 20.1 | 367 | 657 | 2.8 | 25 | 26
60 | 0 | Cepheus | 124.0 | 25.3 | 347 | 943 | 4.5 | 63 | 15
61 | 0 | Cepheus | 124.0 | 25.3 | 353 | 976 | 4.6 | 65 | 15

**footnotetext: The Orion Clouds lie at the very edge of the Leike et al.
(2020) 3D dust grid, thus adding more uncertainty to their derived properties
and should be treated with caution.

Note. — Properties of the 2D projected local molecular clouds (1) Cloud index
(2) Component ID (3) Association with well-studied nearby cloud complexes
(4-5) Central Galactic longitude $l$ and latitude $b$ (6) Distance (7) Mass
(8) Radius (9) Exact area (10) Surface mass density.

Note. — A machine readable version of this table is available at
https://doi.org/10.7910/DVN/BFYDG8.

##  Appendix B Morphological Matching Between 3D Projected and 2D Cloud
Features

After projecting the 3D data on to the plane of the sky to derive the 2D Cloud
features, we compare the morphological agreement between each 3D projected
feature and its corresponding 2D cloud counterpart. As seen in Figure 7 for a
subset of the sample, we overall find good morphological agreement between the
3D and 2D clouds. The remainder of the morphological maps can be accessed at
https://doi.org/10.7910/DVN/BFYDG8.

3D 2D Maps Comparison

Figure 7: Morphological matching between 3D projected and 2D cloud features
for a subset of six clouds in the catalog (Clouds 0, 3, 4, 9, 12, 15). The
semi-transparent red on the 2D cloud panels denote 2D projected cloud
components that are filtered out from the catalog, since they did not meet our
minimum mass or radius requirements.

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