# Magnetic fields in the Southern Coalsack and beyond

M.J.F. Versteeg Department of Astrophysics/IMAPP, Radboud University, PO Box
9010, 6500 GL Nijmegen, The Netherlands  Y. Angarita Department of
Astrophysics/IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, The
Netherlands  A.M. Magalhães Depto. de Astronomia, IAG, Universidade de São
Paulo, Brazil  M. Haverkorn Department of Astrophysics/IMAPP, Radboud
University, PO Box 9010, 6500 GL Nijmegen, The Netherlands  C.V. Rodrigues
Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais (INPE/MCTI),
Av. dos Astronautas, 1758, São José dos Campos, SP, Brazil  R. Santos-Lima
Depto. de Astronomia, IAG, Universidade de São Paulo, Brazil  Koji S. Kawabata
Hiroshima Astrophysical Science Center, Hiroshima University, Kagamiyama,
Higashi-Hiroshima, Hiroshima, 739-8526, Japan

###### Abstract

Starlight polarimetry, when combined with accurate distance measurements,
allows for exploration of the three-dimensional structure of local magnetic
fields in great detail. We present optical polarimetric observations of stars
in and close to the Southern Coalsack, taken from the Interstellar
Polarization Survey (IPS). Located in five fields of view approximately 0.3°
by 0.3° in size, these data represent the highest density of optical
polarimetric observations in the Southern Coalsack to date. Using these data,
combined with accurate distances and extinctions based on Gaia data, we are
able to characterize the magnetic field of the Coalsack and disentangle
contributions to the polarization caused by the Southern Coalsack and a
background structure. For the Southern Coalsack, we find an average magnetic
field orientation of $\theta\sim 75\arcdeg$ with respect to the Galactic north
pole and an average plane-of-sky magnetic field strength of approximately
$B_{POS}=10$ $\mu G$, using the Davis-Chandrasekhar-Fermi (DCF) method. These
values are in agreement with some earlier estimates of the Coalsack’s magnetic
field. In order to study the distant structure, we introduce a simple method
to separate and isolate the polarization of distant stars from foreground
contribution. For the distant structure, which we estimate to be located at a
distance of approximately 1.3-1.5 kpc, we find an average magnetic field
orientation of $\theta\sim 100\arcdeg$ and we estimate a field strength of
$B_{POS}\sim 10\ \mu G$, although this will remain highly uncertain until the
precise nature of the distant structure can be uncovered.

††facilities: LNA:BC0.6m

##  1 Introduction

Optical starlight polarization is a powerful tracer of interstellar material
and magnetic fields. (Hall, 1949; Hiltner, 1949). Asymmetric dust grains in
the interstellar medium (ISM) align themselves with the Galactic Magnetic
Field (GMF) (see e.g. Lazarian, 2007) and subsequently preferentially absorb a
fraction of the radiation polarized perpendicularly to the magnetic field
lines, see e.g. Clarke (2010); Whittet (2022). Thus, stars that emit
unpolarized light may show partial polarization when shining through the dusty
ISM. The polarization angle corresponds to the local orientation of the plane-
of-sky component of the magnetic field and the degree of polarization may be
indicative of the dust conditions along the line of sight (LOS), as well as
the magnetic field orientation. In addition, the dispersion of the
polarization angles can be used to determine the strength of the GMF, using
for example the Davis-Chandrasekhar-Fermi method (Davis, 1951; Chandrasekhar &
Fermi, 1953), hereafter DCF. For a dominating regular magnetic field with an
additional, but relatively small turbulent component, the polarization angles
would follow a roughly Gaussian distribution, centered around the local
orientation of the plane-of-sky component of the magnetic field. Large-scale
deviations from a simple Gaussian distribution may be indicative of, for
example, changes in a large-scale magnetic field direction along the LOS or
the existence of intervening structures such as clouds.

The Southern Coalsack (see also Nyman, 2008, for an overview) is a nearby
large dark nebula located in the Galactic plane, centered at approximately
l=303°, b=0° with an on-sky size of about 10 degrees, see Figure 1. It is
located at a distance of $d\sim 200\textrm{ pc}$ (see e.g. Franco 1989, who
finds a distance of $d=180\pm 26\textrm{ pc}$, Seidensticker & Schmidt-Kaler
1989 who identify two overlapping clouds with the closest at a distance of
$d=188\pm 4.1\textrm{ pc}$ and Dharmawardena et al. 2023, who place the
Coalsack at a distance of approximately $d\sim 150\textrm{ pc}$). The Coalsack
has been studied intensively at various wavelengths, in CO by Nyman et al.
(1989) and Kato et al. (1999), revealing a large and complex fragmented
structure; in X-rays and FUV by Andersson et al. (2004) showing a large hot
envelope surrounding the Coalsack; and in $uvby$ and $H_{\beta}$ by Franco
(1989), who studied the dark cloud complex as well as the regions immediately
in front of and behind the Coalsack. McClure-Griffiths et al. (2001) have
observed HI in the direction of the Coalsack, revealing a large HI shell
located behind the Southern Coalsack. Recent dust maps (Lallement et al.,
2022; Vergely et al., 2022; Edenhofer et al., 2023) show that the Coalsack is
likely attached to the wall of the Local Bubble (LB). Furthermore, the
Coalsack has been studied polarimetrically as well by Jones et al. (1984) in
the near-infrared (NIR), Andersson & Potter (2005)]in the optical, and by Lada
et al. (2004), who base their calculations on observations from Jones et al.
(1984). The latter two studies lead to estimates of the magnetic field
strength ranging from 23 to 93 $\mu$G for the component perpendicular to the
LOS. The plane-of-sky orientation of the magnetic field was found to be
between $\theta\sim 60\arcdeg$ and $\theta\sim 80\arcdeg$ (see Andersson &
Potter, 2005, Table 2) and Jones et al. (1984), their Figure 4. Note that, in
light of recent evidence that the Coalsack and LB wall are attached, it is
likely that unless explicitly accounted for, these estimates of magnetic field
properties for the Southern Coalsack also contain contributions from the LB
wall. The angles mentioned above and throughout the rest of the paper will be
given in the Galactic coordinate system, increasing from North through East.

Using high spatial density observations from the Interstellar Polarization
Survey, General ISM (IPS-GI) catalog (Magalhães et al. in prep, Versteeg et
al. 2023) in combination with reliable distance measurements from Anders et
al. (2022) based on Gaia EDR3 (Gaia Collaboration et al., 2021), we are able
to analyze the magnetic field properties of the Southern Coalsack in great
detail.

The high density of stars allows us to distinguish and separate different
polarizing components or layers along the LOS and determine their individual
magnetic field orientations and strengths. However, because the polarized
signal comprises contributions from all structures along the LOS, the
influence of foreground structures must be removed to be able to study any
distant structures in isolation.

In this paper we present the highest density of stellar optical polarimetric
observations in the Southern Coalsack yet, in five fields of view toward the
Southern Coalsack, showing in great detail the orientation as well as the
strength of the magnetic fields in the Southern Coalsack cloud complex.
Furthermore, we present a simple method which can be used to isolate the
background signal from foreground contamination, allowing for studies of the
magnetic fields behind the Coalsack. The paper is organized as follows. We
describe the polarimetric observations in Section 2. We outline our methods in
Section 3. We then apply these methods and present our results on a field-by-
field basis in Section 4. We discuss our results in Section 5 and summarize
and conclude our findings in Section 6.

##  2 Observations

In this paper, we have used polarimetric observations from IPS-GI (Magalhães
et al. 2005; Versteeg et al. 2023; Magalhães et al. in prep). This catalog
contains linear polarimetric measurements of starlight polarization in the
V-band of over 41000 stars, located in discrete fields in or close to the
Galactic plane. Each field is typically 0.3° by 0.3° on the sky and contains
on average 1000 stars. We have identified five fields located close to or in
the Southern Coalsack. These are fields C11, C15, C39, C40, and C54, following
Versteeg et al. (2023) nomenclature. The positions of the fields relative to
the CO emission from the Coalsack (Nyman et al., 1989) are shown in the top
right of Figure 1.

In order to add distance information, we have cross-matched the IPS data to
two auxiliary catalogs. Firstly, we match IPS data to the GAIA EDR3 (Gaia
Collaboration et al., 2021) catalog using criteria based on position and
magnitude. We find matches within a 3″ position (corresponding to $1.5\sigma$
in the IPS position uncertainty) and 2 mag brightness (to account for the
differences in IPS’ V-band and GAIA’s G-band magnitudes). After matching, we
use GAIA’s source ID to find matches with the StarHorse2021 catalog (Anders et
al., 2022) which provides distances and other stellar parameters. To ensure
that the analysis is applied only to the highest quality data, we have applied
six quality filters, which are explained in more detail in Versteeg et al.
(2023).

  1. 1.

$p/dp>5$, where _p_ is the degree of linear polarization
($p=\sqrt{q^{2}+u^{2}}$) and _dp_ is its associated error; q and u are the
normalized linear Stokes parameters, which ensures that only high-quality
polarimetric observations are taken into consideration,

  2. 2.

$dp<0.8\%$, which removes sources with spurious polarimetric errors,

  3. 3.

$fidelity>0.5$, which is a quality filter defined by Rybizki et al. (2022) and
removes spurious results from GAIA fitting routines,

  4. 4.

$sh_{outflag}=0$, which removes spurious results from the output of the
StarHorse tool used by Anders et al. (2022),

  5. 5.

$C^{*}/\sigma_{C^{*}}<5$, where $C^{*}$ is the corrected color excess and
$\sigma_{C^{*}}$ the 1$\sigma$ scatter. This filter excludes sources with
spurious color excess (see also Riello et al., 2021, equation 18 therein),

  6. 6.

Only high-certainty matches, removing results for which the cross-matching
between the IPS and GAIA EDR3 catalogs did not result in a unique match.

Having applied the six quality filters, we retain 1543 stars divided over five
fields, listed in Table 1.

Figure 1: Top right: locations of the five fields analyzed in this work. Gray
rectangles represent the data, black markers indicate mean position of all
stars in the field. Background contours show the total intensity in CO (Nyman
et al., 1989) at the $2\textrm{ K km s}^{-1}$ and $6\textrm{ K km s}^{-1}$
levels. The surrounding histograms show the polarization angle distributions
for each of the fields including a Gaussian fit to the distribution (see also
Section 3.1). Field C11, indicated with a circular marker, lies outside the
area observed by Nyman et al. (1989).  Table 1: Coalsack Observations and Data

Field | Comp. | N. * |  $\langle$Dist.$\rangle$ | $\langle\textrm{A}_{V}\rangle$ |  $\langle$p$\rangle$ | $\langle\theta\rangle$ |  $\sigma_{\theta}$aaThe polarization angle dispersion has been corrected with the mean error in the polarization angle as in Equation 2. |  $\sigma_{v}$ddfootnotemark:  |  $\rho$ddfootnotemark:  |  $\textrm{B}_{\perp}$ (DCF) |  $\textrm{B}_{\perp}$ (ST)
---|---|---|---|---|---|---|---|---|---|---|---
|  |  | kpc | mag | % | deg | deg | $\textrm{km s}^{-1}$ | $10^{-22}\textrm{ g cm}^{-3}$ | $\mu\textrm{G}$ | $\mu\textrm{G}$
| N | 88 | $0.9^{+0.004}_{-0.004}$ | $0.7^{+0.01}_{-0.01}$ | $1.4\pm 0.1$ | $64.2\pm 1.4$ | $4.3\pm 0.4$ |  |  |  |
C11 | D | 193 | $2.0^{+0.01}_{-0.01}$ | $1.4^{+0.02}_{-0.02}$ | $1.8\pm 0.1$ | $83.8\pm 1.6$ |  |  |  |  |
| Diso. | 193 | $2.0^{+0.01}_{-0.01}$ | $1.4^{+0.02}_{-0.02}$ | $1.1\pm 0.1$ | $110.8\pm 2.3$ |  $6.8\pm 0.7$bbIsolated distant $\sigma_{\theta}$ was corrected for the contribution of the nearby component angle dispersion using Equation 3. | 7.5 | 3.68E-2 | $\sim 20$ | $\sim 10$
| Neither | 91 |  |  |  |  |  |  |  |  |
| N | 194 | $1.3^{+0.01}_{-0.01}$ | $1.5^{+0.02}_{-0.02}$ | $1.6\pm 0.1$ | $70.2\pm 1.5$ | $12.1\pm 0.8$ | $0.8\pm 0.03$ | $3.1\pm 0.9$ |  $12.5\pm 2.1$ccUsing a structure function of polarization angles, see also Section 5.6, we find a magnetic field strength $B=31.2\pm 4.7\ \mu$G. | $8.1\pm 2.0$
C15 | D | 267 | $2.4^{+0.02}_{-0.02}$ | $3.2^{+0.02}_{-0.02}$ | $3.6\pm 0.1$ | $84.8\pm 1.2$ |  |  |  |  |
| Diso. | 267 | $2.4^{+0.02}_{-0.02}$ | $3.2^{+0.02}_{-0.02}$ | $1.7\pm 0.1$ | $98.2\pm 2.2$ |  $15.0\pm 2.5$bbIsolated distant $\sigma_{\theta}$ was corrected for the contribution of the nearby component angle dispersion using Equation 3. | 7.5 | 3.68E-2 | $\sim 10$ | $\sim 5$
C39 | Full | 309 | $1.7^{+0.01}_{-0.01}$ | $2.0^{+0.02}_{-0.02}$ | $2.1\pm 0.1$ | $83.2\pm 2.0$ | $9.5\pm 0.3$ | $0.7\pm 0.07$ | $1.2\pm 0.4$ | $8.1\pm 1.5$ | $4.6\pm 1.5$
C40 | Full | 264 | $2.0^{+0.02}_{-0.02}$ | $2.3^{+0.02}_{-0.02}$ | $2.0\pm 0.2$ | $64.5\pm 2.4$ | $10.8\pm 0.6$ | $0.8\pm 0.05$ | $2.1\pm 0.6$ | $10.8\pm 1.9$ | $6.6\pm 1.8$
C54 | Full | 137 | $1.4^{+0.01}_{-0.01}$ | $1.3^{+0.02}_{-0.02}$ | $1.5\pm 0.1$ | $75.5\pm 2.0$ | $12.0\pm 0.6$ |  |  $<0.9\pm 0.3$eefootnotemark:  |  |

ddfootnotemark:

Based on CO data from Nyman et al. (1989) eefootnotemark: The density is an
upper limit, see Section 4.5.

Note. — For each field: its field number (1); description of components along
the LOS (2), where N is Nearby, D is Distant, Diso is the isolated Distant
component and Neither is the data not assigned to a data cluster (Section
4.1); the number of stars in each component (3); the mean distance (4) and
V-band extinction (5); the weighted average degree of polarization (6); the
weighted average polarization angle of that component (7); the polarization
angle dispersion (8); the LOS velocity dispersion (9); the local density (10)
and the resulting magnetic field strength from the DCF and ST methods, (11)
and (12), respectively.

##  3 Methods

###  3.1 Nearby and Distant component partitioning

Figure 1 shows the distributions of Galactic polarization angles in each of
the five fields under consideration. All fields show a clear limited range of
angles, indicating a dominating regular field orientation along the LOS with
small deviations. However, fields C11 and C15 appear to show a bimodality in
their distributions.

A LOS can probe different interstellar structures located at different
distances, which can produce the bimodality seen in fields C11 and C15. In
this Section, we describe our procedure to disentangle possible different
interstellar structures along the LOS probed by our five fields in and near
the Coalsack.

In order to determine the number of discrete polarimetric components for each
field, we fit different Gaussian distributions (single Gaussian curves, as
well as the sum of two, three or four Gaussians) to the polarization angle
histograms. The number of bins was determined using the Knuth rule (Knuth,
2006, 2019), which is a data-based method that can be used to find the optimum
number of histogram bins, based on Bayesian probability theory. The bin width
of the histograms is thus different for each field. Using the Chi-squared
value as a measure of goodness-of-fit, we are able to determine the number of
components in each field, shown in Table 1. For fields C39, C40 and C54 a
single Gaussian distribution provides a good fit. For fields C11 and C15, a
model consisting of two Gaussians is more appropriate. We also perform an
F-test (Lupton, 1993) and find that for fields C11, C15 and C40, it is
warranted to include second Gaussian component to the model at the 99 per cent
confidence level. The case of field C40 will be discussed in more detail in
Section 5.1.. Further analysis of the data, including partitioning, is done in
the q, u, distance parameter space.

In the case of fields C11 and C15, stars belonging to either component are
uniformly distributed across the field-of-view. It therefore seems unlikely
that a small separate cloud or other dusty structure exists nearby in
projection on the sky, but rather that one component is located at a different
distance from the other. Indeed, this is confirmed by comparing the linear
Stokes parameters q and u and the distances, see also Figures 3 and 6.
Therefore, partitioning of components based on q, u and the distance $d$ is
justified. However, because of the unique characteristics of the three-
dimensional q, u, d distributions for each field, we had to use different
methods to partition the data into its discrete components, which will be
explained in more detail in Sections 4.1 and 4.2.

After partitioning, components are either assigned “nearby” or “distant” (N
and D in Table 1, respectively). The nearby component consists of those stars
at smaller distances where we can reasonably expect the Southern Coalsack to
be the dominant polarizing feature. The light of these nearby stars is
polarized by both the Southern Coalsack and, to a lesser extent, the LB wall.
The distant stars are further away – typically far behind the Coalsack cloud
complex – and show polarimetric signatures of at least two dominant polarizing
structures along the LOS.

In order to isolate the distant signal from the contributions of the nearby
structure, characteristic values for the Stokes q and u parameters are
determined for the nearby stars. These are the median values for q and u of
all stars in the 80th distance percentile. Choosing the 80th percentile most
distant stars of the nearby population ensures that we have a sizable
representative sample. Using the most distant stars in the nearby population
guarantees that we probe the entirety of the nearby ISM, including possible
minor contributions from less dense dust behind the Coalsack. These proxy
numbers are then subtracted from the distant population polarimetric
measurements, resulting in foreground-corrected distant measurement (Diso. in
table 1). We also correct for nearby contributions to the polarization angle
dispersion, see Section 3.2 and Equation 3. We can then determine the
characteristics of the magnetic field for each component individually. Because
the LB wall and Coalsack are likely attached (see e.g. Edenhofer et al., 2023,
and Figure 6 below), we are unable to disentangle contributions from the LB
wall to the polarized signal of the nearby population of stars.

All average polarimetric values presented in Table 1 are calculated from the
weighted Stokes parameters, where the weights are set equal to the inverse of
the associated errors. The distances and V-band extinctions are taken from
Anders et al. (2022) and the averages of these values, taking their asymmetric
errors into account, are calculated following the method described by Barlow
(2003), which was developed into a numerical method described in Laursen et
al. (2019).

###  3.2 DCF parameters

The DCF method (Davis, 1951; Chandrasekhar & Fermi, 1953) relates the magnetic
field strength of the plane-of-sky magnetic field component (${B}_{\perp}$,
hereafter B) to the polarization angle dispersion, gas velocity dispersion and
gas mass density as follows:

$\textrm{B}_{\perp}=f\sqrt{4\pi\rho}\frac{\sigma_{v}}{\sigma_{\theta}},$ (1)

where $\rho$ is the local density of the gas (in $\textrm{g cm}^{-3}$) and
$\sigma_{v}$ is the velocity dispersion of the gas (in $\textrm{cm s}^{-1}$).
$\sigma_{\theta}$ refers to the dispersion in the polarization angles (in
radians) and $f$ is a correction factor, meant to compensate for the fact that
the DCF method generally overestimates magnetic field strengths. Numerical
simulations of molecular clouds (Ostriker et al., 2001; Heitsch et al., 2001)
and protostellar cores (Padoan et al., 2001) show that $f=0.5$ gives
reasonable estimates for the field strength..

We use CO maps to find DCF input parameters related to the specifics of the
local ISM, $\sigma_{v}$ and $\rho$. CO maps in the general area of the
Southern Coalsack are available as part of the Dame et al. (2001) composite CO
map. Specifically, the surveys of Dame et al. (1987) (DHT01 in the composite
map), Nyman et al. (1989) (DHT35) and Bronfman et al. (1989) (DHT36) cover
(part of) the region in which the observed fields can be found. The CO
component associated with the Southern Coalsack has a velocity of
$\sim-4\textrm{ km s}^{-1}$ (Nyman et al., 1989; Beuther et al., 2011). For
fields C15, C39 and C40, the typical Coalsack CO signature line at $-4\textrm{
km s}^{-1}$ is well visible above the noise in all three possible maps. Of
these three options, the map by Nyman et al. (1989) has the highest spatial
and spectral resolution and is thus used. Although the ${}^{12}\textrm{CO}$
may be optically thick especially in high-density regions, Nyman et al. (1989)
have also observed ${}^{13}\textrm{CO}$ in the Southern Coalsack and find that
the line profiles of both species show great similarity (see Figure 3 in Nyman
et al., 1989) except in the most dense regions. Because all our fields are
located away from the most dense regions, we are confident using
${}^{12}\textrm{CO}$ from Nyman et al. (1989) as a tracer.

Field C11 lies outside the area observed by Nyman et al. (1989), see also
Figure 1. Field C54 lies on the edge of the area observed by Nyman et al.
(1989) and is only partially covered. In this field, no feature at $-4\textrm{
km s}^{-1}$ is detected. Therefore, we cannot reliably determine the magnetic
field strengths in the foreground of field C11 and the entire LOS of field
C54.

For the fields where we do have reliable CO observations, a Gaussian fit to
the spectrum at a velocity $-4\textrm{ km s}^{-1}$ will provide us with the
velocity dispersion $\sigma_{v}$, which will be used in the DCF method.

The gas density $\rho$ is determined using the total CO intensity in the
observed field. The intensity is converted to an $\textrm{H}_{2}$ column
density using the Bolatto et al. (2013) conversion factor $X_{CO}=2\times
10^{20}\textrm{ cm}^{-2}\textrm{(K km s}^{-1}\textrm{)}^{-1}\pm 30\%$.
Finally, we convert this value to a spatial number density by dividing the
column density by the path length through the cloud.

The path length is the thickness of the Southern Coalsack along the LOS. This
path length was determined using the differential extinction ($A_{V}/d$) as a
function of distance in GAIA data in the field. We downloaded all available
data at the location of the Southern Coalsack in the Anders et al. (2022)
catalog (i.e. all available stars in the region $298\arcdeg>l>308\arcdeg$,
$-4.5\arcdeg>b>2.5\arcdeg$ for Galactic longitude and latitude, respectively)
and plotted $A_{V}/d$ versus the distance $d$, see Figure 2. This shows an
increase in differential extinction starting at $\sim 185\textrm{ pc}$, which
peaks a distance of $\sim 210\textrm{ pc}$. The increase inDistances
normalized extinction coincides with the near edge of the cloud (i.e. the
distance to its near edge) and is in agreement with other measures of the
distance to the Coalsack ($d\sim 180\textrm{ pc}$, see e.g. Franco, 1989;
Seidensticker & Schmidt-Kaler, 1989). The far edge of the cloud is indicated
by the peak in normalized extinction located at $\sim 210\textrm{ pc}$. Beyond
this distance, the differential extinction decreases because there is little
to no increase in $A_{V}$ per unit distance. It is thus expected that the
cloud does not extend farther than the distance at which the peak of the
excess $A_{V}/d$ is located. Fitting a Gaussian curve to the excess
differential extinction, we can use the half-width at half maximum (HWHM) as
an indicator of cloud thickness. We find that $l_{Coalsack}=HWHM=\sqrt{2ln2}\
\sigma=25.7\pm 0.6\textrm{ pc}$. This is in agreement with other observations
of 3D cloud structure, most recently by Dharmawardena et al. (2023), who find
a thickness of $29\textrm{ pc}$.

Finally, to convert the number density to a mass density, we assume the gas to
be 90 % molecular hydrogen ($\textrm{H}_{2}$) with a mass of
$m_{H2}=3.348\times 10^{-24}\textrm{ g}$ and 10% helium ($m_{He}=6.646\times
10^{-24}\textrm{ g}$) which will give us the local gas density $\rho$.

Figure 2: Distance-normalized extinction ($A_{V}/dist$) versus distance for
$\sim 2.5$ million stars in the general area of the Southern Coalsack. Data
taken from Anders et al. (2022). Red data points are the individual stars, the
grey background indicates the 25%-75% quantile range and the black solid line
indicates the median. The increase in $A_{V}/dist$ is caused by the presence
of the Coalsack at a distance of $\sim 185\textrm{ pc}$. The inset shows a
Gaussian fit to the $A_{V}/dist$ excess. We estimate a line-of-sight thickness
of $25.7\pm 0.6\textrm{ pc}$.

The final input parameter for the DCF method, the polarization angle
dispersion $\sigma_{\theta}$, is corrected by the contribution to the observed
dispersion originating from the observational errors using

$\sigma_{\theta}=\sqrt{{\sigma_{\theta,G}}^{2}-\langle{\sigma_{\theta,err}}\rangle^{2}},$
(2)

where $\sigma_{\theta}$ is the polarization angle dispersion,
$\sigma_{\theta,G}$ is the standard deviation of the Gaussian fit to the
polarization angle distribution (or some isolated component of the
distribution) and $\langle\sigma_{\theta,err}\rangle$ is the mean error,
following the method from Pereyra & Magalhães (2007).
$\langle\sigma_{\theta,err}\rangle$ is typically small (using
$\sigma_{\theta}=0.5(\sigma_{p}/p)$ rad and an SNR of 5, the rms uncertainty
is $\sigma_{\theta}<5.8$ deg. The typical average error per field is around
$\langle\sigma_{\theta,err}\rangle\sim 2\arcdeg$), which constitutes a small
correction to the Gaussian fit.

For distant components, it is assumed that the ISM conditions reflect the
canonical diffuse ISM values. Therefore, we adopt a value of
$n(H_{2})=1\textrm{ cm}^{-3}$ for the gas density (see e.g. Ferrière, 2001).
For the velocity dispersion, we take the average value of the range
$\sigma_{v}\sim 5-10\textrm{ km s}^{-1}$ (see e.g. Burkert, 2006; McKee &
Ostriker, 2007; Dobbs et al., 2011) such that $\sigma_{v}=7.5\textrm{ km
s}^{-1}$. Although it is possible, even likely, that the polarization of the
distant stellar population is caused by a structure more dense than the
diffuse ISM, such as dust clouds, filaments, or spiral arms, we currently do
not know details about this secondary structure and therefore assume canonical
diffuse ISM values. The validity of these assumptions will be discussed in
more detail in Section 5.7.

In order to correct for contribution to the polarization angle dispersion of
the far component by the nearby polarizing structure, we subtract in
quadrature the nearby population’s angle dispersion from the far population
dispersion:

$\sigma_{\theta,D,corr}=\sqrt{\sigma_{\theta,D}^{2}-\sigma_{\theta,N}^{2}}.$
(3)

Equation 3 is a coarse approximation of correction for the contribution to the
angle dispersion from a foreground polarizing structure. The resulting field
strengths of the distant components will therefore be reported as order-of-
magnitude estimates rounded to the nearest five $\mu$G because of the large
uncertainties regarding the characteristics of the distant structures.

The error on the plane-of-sky magnetic field strength (see equation 1) will be
calculated as:

$\frac{\delta
B}{B}=\sqrt{\left(\frac{1}{2}\frac{\delta\rho}{\rho}\right)^{2}+\left(\frac{\delta\sigma_{v}}{\sigma_{v}}\right)^{2}+\left(\frac{\delta\sigma_{\theta}}{\sigma_{\theta}}\right)^{2}},$
(4)

derived from conventional error propagation. For the fields where CO maps are
used to determine the gas density,

$\frac{\delta\rho}{\rho}=\sqrt{\left(\frac{\delta
W_{CO}}{W_{CO}}\right)^{2}+\left(\frac{\delta
X_{CO}}{X_{CO}}\right)^{2}+\left(\frac{\delta
l_{cloud}}{l_{cloud}}\right)^{2}},$ (5)

where $W_{CO}$ is the total CO intensity, $X_{CO}$ is the CO-
to-$\textrm{H}{{}_{2}}$ conversion factor (Bolatto et al., 2013) and
$l_{cloud}$ is the thickness of the Coalsack. The input parameters for the DCF
method and their associated errors (Eqs. 1 and 4) can be found in Table 1.

Alternative methods for determining the magnetic field strength will be
discussed in Sections 5.8 and 5.6.

##  4 Results

Because each field has different and unique properties, the analysis of the
data differs from field to field. Therefore, we will discuss the results of
each field separately.

###  4.1 C11

Field C11 ($l=307.6\arcdeg,b=1.3\arcdeg$) contains 372 stars. This field is
located outside the lowest CO contours at $2\textrm{ km s}^{-1}$, see the
circular marker in Figure 1. Using multiple Gaussian fits, we find the
distribution of polarization angles comprises two components, see Figure 1,
top left histogram, indicating at least two structures along the LOS. We first
create two partitions to separate the data with polarization dominated by the
nearby structure from that dominated by both structures, see also Section 3.1.
This particular field is very suited for partitioning using the SciKit-learn
(Pedregosa et al., 2011) DBSCAN algorithm (Ester et al., 1996). The algorithm
finds clusters of overdensities in data separated by regions that are less
densely populated. We applied this method to three parameters: the two linear
polarization Stokes parameters q and u and the distance d, all normalized to
the range 0-1. We set the DBSCAN parameters $\epsilon$ and $min_{samples}$ to
$\epsilon=0.12$ and $min_{samples}=81$, respectively, based on silhouette
score optimization. $\epsilon$ specifies the maximum distance between two data
points beyond which they will not be considered part of the same cluster and
$min_{samples}$ defines the minimum population of a dense region. The
algorithm will use these two parameters to create the appropriate number of
clusters in the data. In the case of field C11, this creates two components of
clustered data (the nearby and distant components) and a component consisting
of outlier stars that belong to neither cluster, see Figure 3, top, called
“noise” in the method (Ester et al., 1996). We emphasize that these data are
high-quality, valid observations of stars, they just do not belong to the most
dense clusters in the normalized q, u, d-space. We therefore do not apply
further analysis of the magnetic field on the outlier points.

Having created two distinct partitions in the data (the red and blue
components in Figure 3, top), we use the method outlined in Section 3.1 to
compute the polarization properties of the distant component. The polarimetric
characteristics for each of the two components are presented in Table 1. The
components are indicative of two distinct polarizing components along the LOS
with very different magnetic field orientations. The nearby population, with
an average distance of 900 pc, is dominated by a structure with a magnetic
field orientation of $\theta_{N}=64\arcdeg$, which we determine to be the
Southern Coalsack. The average distance of the nearby partition extends far
beyond the far edge of the Coalsack (i.e. $\sim 215$ pc, see also Section 3
and Figure 2), but the region immediately behind the Coalsack is mostly devoid
of any dust (McClure-Griffiths et al., 2001; Nyman, 2008, see also Figure 6).
The distant component, located at a much larger average distance of 2 kpc,
shows a preferential magnetic field orientation of $\theta_{D}=84\arcdeg$.
After isolating the distant polarization from nearby contamination as outlined
in Section 3.1, the difference in orientation between the two components
becomes even larger with $\theta_{D,iso.}=110\arcdeg$. The turnover point
between the two partition occurs at a distance of approximately $\sim
1.3\textrm{ kpc}$, which is where we expect the secondary structure to begin.
This dominant polarizing feature in the distant component will be discussed
further in Section 5.7.

Determination of the strength of the magnetic field requires certain
information about the local ISM conditions. For field C11, no reliable
estimates for the gas velocity dispersion and gas density are available in the
Nyman et al. (1989) CO map because the field lies outside the area observed.
Therefore, we cannot determine the magnetic field strength of the nearby
polarizing structure.

The magnetic field strength of the distant structure, $B_{D,iso.}$, can be
calculated using the characteristic diffuse ISM velocity dispersion and gas
density values from Section 3.2, i.e. $\sigma_{v}=7.5\textrm{ km s}^{-1}$ and
$n(H_{2})=1\textrm{ cm}^{-3}$, respectively. Using these values, the
polarization angle dispersion as determined using Equation 2 and the DCF
method (Equation 1), we find a magnetic field with a strength of around
$B_{D,iso.}\sim 20\ \mu\textrm{G}$.

Figure 3: Distributions of Stokes parameters q and u, and distance d in kpc.
Top: field C11. Red and blue indicate the nearby and distant partitions,
respectively, created using DBSCAN. Gray indicated data point that are not
part of either component, see also Section 4.1. Bottom: field C15. Red and
blue indicate the nearby and distant partitions, respectively created using
K-means clustering; see also Section 4.2.

###  4.2 C15

Field C15, located at $l=304.7\arcdeg$, $b=-0.2\arcdeg$, is a moderately-
densely populated field (461 stars) located near the edge of the Coalsack in
CO. The distribution of polarization angles for field C15 shows two distinct
components, see Figure 1, bottom left histogram. As described in Section 3.1,
we want to create two partitions in the data. The three-dimensional
distribution of q, u and distance show that these data are well-suited to be
divided into two partitions using the SciKit-learn (Pedregosa et al., 2011)
K-means algorithm due to the overall low density and uniform distribution
across the parameter space, and the similar size of the two potential
components. We applied this algorithm to Stokes parameters q and u and the
distance d. The K-means algorithm iteratively minimizes the Euclidean distance
between each data point and an assumed centroid, updating the position of the
centroid at each step until an optimal position is found. All parameters were
normalized to the range 0-1 prior to partitioning to ensure an accurate
result. The only input parameter for the K-means algorithm is the number $n$
of clusters that will be created. Having found that the best fit to the
distribution of polarization angles has two Gaussian components, this is set
to $n=2$.

Having created a nearby and distant component (the red and blue components in
Figure 3, bottom, respectively), we applied the method described in Section 3
to isolate the distant stars from any polarizing contamination originating
from the nearby cloud. The mean polarimetric characteristics for each of the
components are presented in Table 1. The magnetic field in the nearby
structure has a mean orientation of around $\theta_{N}=70\arcdeg$, whereas the
distant and isolated distant magnetic fields are oriented along
$\theta_{D}=85\arcdeg$ and $\theta_{D,iso.}=100\arcdeg$, respectively. The
boundary between the two polarizing regimes occurs at a distance of
approximately $\sim 1.6\textrm{ kpc}$. The nature of this secondary structure
will be discussed in Section 5.7.

We determine the input parameters for the DCF method (Equation 1) using the
methods described in Section 3.2. The parameters used to determine the
magnetic field strength can be found in Table 1. We find magnetic field
strengths for the nearby and (isolated) distant structures of $B_{N}=12.5\pm
2.1\ \mu\textrm{G}$ and $B_{D,iso.}\sim 10\ \mu\textrm{G}$, respectively.

###  4.3 C39

C39 is a field located near the Northern edge of the Coalsack at
$l=303.1\arcdeg,b=1.6\arcdeg$, see Figure 1, and comprises 309 stars. The fit
to the polarization angle distribution for field C39 agrees best with a
single-component Gaussian, see Figure 1, bottom middle histogram. The mean
polarimetric parameters of this single component are described in Table 1. We
believe that this LOS is dominated by the Southern Coalsack nearby. The mean
orientation of the magnetic field in this LOS is $\theta=83\arcdeg$.

Next, we determined the parameters necessary to apply the DCF method following
the methods described in Section 3.2. The parameters are summarized in Table
1. We use those parameters and Equation 1 to determine the magnetic field
strength and find $B=8.1\pm 1.5\ \mu\textrm{G}$ for field C39.

###  4.4 C40

Field C40, located at $l=302.2\arcdeg,b=-1.2\arcdeg$, probes a more central
region of the Southern Coalsack. This field contains 264 stars. The
distribution of polarization angles for field C40 has been fit by a single
Gaussian curve, see Figure 1, bottom right histogram. We find that the chi-
squared of the fit becomes slightly lower by adding a second Gaussian
component centered around $\theta\sim 96\arcdeg$, but we are unable to
attribute this component to an overdensity in either the q, u, distance
parameter space, or the q, u, sky-coordinate parameter space. This will be
explored further in the discussion in Section 5.1. The single polarizing
component along the LOS is oriented along an angle of $\theta=65\arcdeg$.

The magnetic field strength along this LOS can be calculated using the
polarization angle dispersion $\sigma_{\theta}$, gas velocity dispersion
$\sigma_{v}$ and gas density $\rho$ as in Section 3.2 and are presented in
Table 1. Using these values as input for Equation 1, we find a magnetic field
strength of $B=10.8\pm 1.9\ \mu\textrm{G}$.

###  4.5 C54

Field C54 is located near the Northeastern edge of the Southern Coalsack
($l=305.2\arcdeg,b=1.3\arcdeg$, see also Figure 1) and comprises 137 stars.
The distribution of polarization angles is best characterized by a single
Gaussian, see Figure 1, center left histogram. The mean magnetic field
orientation is $\theta=76\arcdeg$.

Determination of the magnetic field strength is made difficult by the fact
that this field lies partially outside of the region observed by Nyman et al.
(1989). Only half of the field is covered in this CO map, see also Section
3.2. Because no suffiecient CO maps are available with which to determine the
local gas density and gas velocity dispersion, we are unable to calculate the
magnetic field strength in this field.

##  5 Discussion

###  5.1 Possible secondary components

The method of identification and isolation of different polarized stellar
populations using clustering algorithms depends on being able to find those
populations in q, u, distance parameter space. We describe some limitations of
the method in two cases below.

A small change in angle may be visible in Figure 4, field C39, between 1 and 2
kpc. This angle does not appear in the corresponding histogram (Figure 1,
bottom center), nor does it correspond to a cluster in q, u, distance-space
(Figure 6). Therefore, we cannot identify this cluster in the data, and thus
cannot quantify the effect is may or may not have on our results. However, if
the angle change were real, we here interpret is as dispersion and the actual
magnetic field strength would likely be higher.

As mentioned in Section 4.4, we find a possible secondary component in the
polarization angle distribution for field C40. Adding a secondary component to
the Gaussian fit lowers the $\chi^{2}$ from $\chi^{2}_{1\textrm{ comp}}=8.4$
to $\chi^{2}_{2\textrm{ comps}}=5.0$. The secondary Gaussian component has an
average angle $\mu=95.8\pm 1.9\arcdeg$ and a standard deviation of
$\sigma=6.0\pm 1.9\arcdeg$. An F-test (Lupton, 1993) confirms that the
addition of this second Gaussian component to the model is statistically
warranted. However, we are unable to isolate this component for further
analysis because we cannot correlate this component to a spatial parameter,
neither the on-sky Galactic coordinates $l$, $b$ nor the distance $d$.
Therefore, it is likely that the excess exists all throughout the observed
field. Indeed, we find that the Planck polarization angle observed at the
location of field C40 is $\theta_{Gal,Planck}=94.4\arcdeg$, which is in
agreement with the possible secondary Gaussian component. We therefore
consider the possibility that a structure with a magnetic field orientation
$\theta\sim 95\arcdeg$ exists farther along the LOS. This structure could
either be a single polarizing dust structure (such as a cloud or arm) located
at a distance beyond what is probed by optical starlight or a diffuse large-
scale component that dominates the Planck signal (which probes the ISM along
the entire LOS). Instead, the observed signal from starlight polarimetry
probes only the local ISM and its magnetic field, which is dominated by the
Southern Coalsack.

###  5.2 Magnetic field orientations

We now discuss the orientations of the various magnetic field components. As
mentioned above, for the nearby population of stars, the polarization is
induced by both the Southern Coalsack and the LB wall. However, because these
two structures are attached (Lallement et al., 2022; Vergely et al., 2022;
Edenhofer et al., 2023), it is not possible for us to disentangle the two
contributions. Although field C11 does not show a CO feature that may be
associated to the Coalsack, these dust models indicate that the LOS of field
C11 has a Coalsack-like dust feature and thus may be considered part of the
Coalsack.

Firstly, the components that are dominated by the nearby structure (i.e. C11,
nearby component, C15, nearby component, and fields C39, C40 and C54) are
generally oriented along an angle $\theta\sim 60-80\arcdeg$. This is in
agreement with other measurements of the magnetic field orientation for the
Southern Coalsack (see Andersson & Potter, 2005, table 2 therein).

In addition, the fact that the nearby component of field C11 (located outside
the main CO emitting region) shows a similar orientation to the fields located
within the CO contours and is in agreement with other findings (see Andersson
& Potter, 2005, Table 2) provides evidence that the Southern Coalsack extends
beyond the edges suggested by the CO emission (Nyman et al., 1989), see also
Figure 1. This is corroborated by the dust models of Lallement et al. (2022)
and Vergely et al. (2022), who see Coalsack-like dust features in all five
fields, see also the extinction profiles in Figure 6, as well as Andersson et
al. (2004), who observe a large hot envelope surrounding the Coalsack.

Secondly, the distant structure-dominated components (i.e. C11, distant
component, and C15, distant component) are oriented along angles that are
higher compared to the foreground: $\theta\sim 100-120\arcdeg$. The
polarization angle of the distant component could be caused by the large-
scale, regular GMF component in the Milky Way. A toy model GMF in the Galactic
plane, following the spiral arms, would show a polarization angle
$\theta=90\arcdeg$. However, more complex models of the GMF show out-of-plane
components (Jansson & Farrar, 2012; Rodrigues et al., 2019; Shukurov et al.,
2019), and so do infrared polarimetric observations in the plane in the first
quadrant (Clemens et al., 2020). Indeed, the observations presented above show
a departure from $90\arcdeg$ toward higher polarization angles. This distant
component will be discussed in more detail in Section 5.7.

###  5.3 Magnetic field strengths

For two fields (C11 nearby and C54), we are unable to accurately determine the
magnetic field strength because these fields lie (partially) outside of the
observed area by Nyman et al. (1989). To get an order-of-magnitude idea of the
field strength in those regions, we use the average gas density and gas
velocity dispersion from the fields where we do have a CO detection (C15
nearby, C39 and C40). For the foreground of field C11, we find then a magnetic
field strength for the nearby component of $B_{N}=27.1\pm 5.2\textrm{
}\mu\textrm{G}$. For field C54, we extrapolate the available data to arrive at
an upper limit for the gas density $\rho<0.9\pm 0.3\times 10^{-22}\textrm{ g
}\textrm{cm}^{-3}$. Because no Coalsack signature is detected, we use the
average value from fields C15 (nearby), C39 and C40 for the gas velocity
dispersion to arrive at a magnetic field strength of $B<6.1\pm 1.2\
\mu\textrm{G}$. These values are speculative but in rough agreement with the
findings in the other fields.

Firstly, we find that components dominated by the Southern Coalsack (i.e. C11
(nearby component), C15 (nearby component), C39, C40 and C54) have plane-of-
sky magnetic fields with a strength ranging between $B\sim 8\ \mu\textrm{G}$
and $B\sim 25\ \mu\textrm{G}$, with an average nearby field strength of $B\sim
13\ \mu\textrm{G}$. As the magnetic field component parallel to the LOS is
unknown, this constitutes a lower limit to the total magnetic field strength.
Even though Zeeman measurements probe the parallel component of the magnetic
field, large statistical samples of Zeeman measurements can give an idea of
typical field strengths in clouds. Myers et al. (1995) find parallel magnetic
field strengths in diffuse clouds from 4 to 19 $\mu$G, while Crutcher (1999)
discusses field strengths in 27 molecular clouds and reports values from a few
$\mu$G to 3.1 mG. The observed perpendicular field strength in the Southern
Coalsack is consistent with these values.

Furthermore, we find that our magnetic field strengths $B\sim 10\
\mu\textrm{G}$ are broadly in agreement with other measurements of the
strength of the perpendicular component of the magnetic field in the Southern
Coalsack– differences between our findings and those from literature can
largely be attributed to differences in methodical choices. Lada et al.
(2004), who base their calculations on observations by Jones et al. (1984),
find a magnetic field strength of $B=23\ \mu\textrm{G}$. A re-analysis of Lada
et al. (2004) by Bhat & Andersson (2011), using the same observations from
Jones et al. (1984), finds a magnetic field strength of $B\sim 32\
\mu\textrm{G}$. Andersson & Potter (2005) calculate a magnetic field strength
of $B=93\ \mu\textrm{G}$ and have corrected the magnetic field strength of the
Coalsack for foreground contributions to the polarization angle dispersion
using polarimetric radiative transfer. Without this correction, the magnetic
field strength is found to be $B=53\ \mu\textrm{G}$ These results are in rough
agreement with our findings, taking into consideration that we have used a
correction factor f=0.5 in our application of the DCF method.

Secondly, we find that distant components, after isolation from nearby
contributions, have magnetic field strengths of around $B\sim 10-20\
\mu\textrm{G}$. These values are found using canonical values for the diffuse
ISM for the gas density $\rho$ and velocity dispersion $\sigma_{v}$ and will
be discussed in more detail in Section 5.7.

We note that Andersson & Potter (2005) see signatures of the LB wall in their
analysis of the polarimetric data which has a significant effect on their
determination of the magnetic field strength of the Coalsack. However, they do
not see LB Wall contributions in all of their fields. It is likely that the
contribution of the LB wall to the polarized signal is not uniform across the
sky. Due to varying dust contents and changing magnetic field geometries
(Bailey et al., 2016; Medan & Andersson, 2019; Pelgrims et al., 2020), the LB
wall contribution could be significant in some areas and negligible in others.
Because we cannot distinguish any contribution from the LB wall, we are unable
to accurately assess the contribution of the LB wall to the polarimetry of the
observed stars. It is also possible that the edge of the Southern Coalsack
coincides with the LB wall. This is strengthened by the fact that in
extinction, we do not see separate components for the LB wall and the
Coalsack, see e.g. Figure 6, bottom line plots. This means that we are unable
to disentangle contributions from the LB wall and the Coalsack.

We emphasize that IPS-GI fields are located away from the most dense regions
of the Coalsack (see Figure 1). As such, the magnetic field orientations and
strengths may be accurate for those specific locations, but might not be
representative of the entire Southern Coalsack.

We correct polarization angle dispersion of the distant populations by
subtracting in quadrature the nearby angle dispersion (Equation 3). This
approximation is necessary because we lack precise details about the local ISM
conditions of the distant polarizing structure. If those details were known,
it would be valuable to apply a more detailed method, such as a full polarized
radiative transfer analysis (see e.g. Andersson & Potter, 2005)

Finally, we note that the errors in the magnetic field strength are dominated
by the uncertainties in the gas density, especially due to the error in the
CO-to-$\textrm{H}_{2}$ conversion factor (Bolatto et al., 2013). The error in
density makes up between 80-90% of the total error in magnetic field strength.
However, because of the location of the fields (in the Galactic plane, in the
Southern sky) very few other tracers of a similar scale and resolution to
DHT35 are available. There are some observations of stars near the Coalsack.
Andersson et al. (2002) observe CH in three stars close to the Southern
Coalsack and Corradi et al. (2004) observe a further four stars in Na I. These
observations show velocity dispersions a bit larger than our findings using
CO, but they do not cover the same area that is observed in CO and
specifically not the regions of the Coalsack where our observed fields are
located.

Summarizing, various estimates for magnetic field strengths in the Coalsack
are quoted in the literature. However, a detailed look at the distinct methods
of analysis shows that there are a number of valid reasons why differing field
strengths are derived, such as the inability to separate the LB wall
contribution, analyses that only look at certain parts of the Coalsack and
inclusion of correction factors in the DCF method or not.

###  5.4 Comparison with the Heiles catalog

The agglomeration of polarimetric catalogs by Heiles (2000) also contains
stars located in or close to the Southern Coalsack. We have identified 84
stars (taken from van Smith, 1956; Mathewson & Ford, 1970; Klare & Neckel,
1977; Reiz & Franco, 1998, see also Heiles (2000) for more details and a
quantitative comparison between these data sources) that are located within
the lowest boundaries in the CO maps of Nyman et al. (1989). Using these
polarimetric observations, we find average polarimetric values of $p=1.2\pm
0.07\%$ and $\theta_{gal}=86.3\pm 1.6\arcdeg$. We use the mean values for
$\rho$ and $\sigma_{v}$ to compute a magnetic field strength of $B=12.2\pm
2.3\ \mu\textrm{G}$ using the DCF method and $B=7.0\pm 1.2\ \mu\textrm{G}$
using the ST method. The magnetic field orientation and strength are in
agreement with field strengths determined based on IPS data (Table 1). We have
identified one star in common between the two catalogs: HD 113754, located in
IPS field C15 at a position $l=304.71\arcdeg,b=-0.17\arcdeg$. We consider this
star to be a background star at a distance $d=1.9\textrm{ kpc}$. The
polarimetric parameters taken from both Heiles (2000) and the IPS catalog are
in rough agreement: the degree of polarization is $p=4.1\pm 0.2\%$ and
$p=5.2\pm 0.05\%$ in the former and latter catalogs, respectively. The
polarization angle is $\theta_{Gal}=76\pm 1.1\arcdeg$ and
$\theta_{Gal}=76.4\pm 0.3\arcdeg$ for Heiles (2000) and IPS, respectively.

###  5.5 Comparison with Planck observations

Observations of polarized dust emission by Planck provide another auxiliary
measurement to compare our findings to. We use the thermal dust maps at
variable resolution to determine a single orientation for the magnetic field
as measured by Planck. In Figure 4 we plot the polarization angle from the
IPS-GI observations with their distance. The dotted lines indicate the
magnetic field orientation as measured by Planck. Figure 4 shows that Planck
sometimes measures a field parallel to the nearby structure (i.e. the Southern
Coalsack) as in field C11, it sometimes is parallel to the distant structure
(as in field C15) and it sometimes finds a different orientation altogether
(as in fields C39, C40 and C54).

Figure 4: Per field, for each star its polarization angle and distance.
Distances taken from Anders et al. (2022). The dotted lines indicate the
average magnetic field orientation at the location of the center of the field
taken from the Planck 353 GHz HFI maps. The shaded area shows the uncertainty
of the Planck polarization angle.

Despite this, we do find some consistencies. Firstly, in fields where IPS-GI
find more than one polarimetric component (fields C11 and C15), Planck finds a
magnetic field orientation $\theta_{Gal}<90\arcdeg$, indicating that the
nearby component is a significant contribution to the Planck polarization.
Conversely, in fields with only one component, Planck generally finds
orientations $\theta_{Gal}>90\arcdeg$. Planck appears to pick up on a dominant
background that lies beyond the distances probed by starlight polarization in
certain fields, see also Section 5.1.

For fields C15, C39 and possibly C40, we see that with increasing distance,
there is a closer agreement between the optical and submillimeter polarization
angles. This is likely a sign that in those fields, the most distant stars
trace the same dust as Planck does and that there is little dust contributing
to the polarized submillimeter signal beyond that distance.

###  5.6 Structure functions of polarization angles

The DCF method derives the field strength from the dispersion of the
polarization angles, the displacement of which is assumed to be caused by the
propagation of incompressible transverse magnetohydrodynamic (MHD) waves known
as Alfvén waves. The DCF method assumes that the polarization angle dispersion
measured across the field of view is representative of the total turbulent
angle dispersion. Therefore, if the largest turbulent scale is larger than the
field of view, the measured angle dispersion is a lower limit and the derived
magnetic field is an upper limit to the true value. On the other hand, if the
maximum turbulent scale is smaller than the resolution, the angle dispersion
is solely caused by large-scale variations in the regular magnetic field
component. In such a case, the magnetic field strength can be obtained using
the method described by Hildebrand et al. (2009), who propose using a more
accurate way to calculate the plane-of-sky turbulent dispersion of local
structured fields (e.g. in clouds) via the structure function of the
polarization angles, the Angular Dispersion Function (ADF):

$\langle\Delta\theta^{2}(\ell)\rangle^{1/2}=\left(\frac{1}{N(\ell)}\sum^{N(\ell)}_{i=1}[\mathbf{\theta(x)-\theta(x+\boldsymbol{\ell})}]^{2}\right)^{1/2},$
(6)

where $\theta(\mathbf{x})$ is the polarization angle at the position
$\mathbf{x}$ and $\ell$ is the angular separation between two sources in the
sky. This method does not need to assume a model of the magnetic field and
provides an independent measurement of the turbulent to large-scale magnetic
field strength ratio, $\langle B_{t}^{2}\rangle^{1/2}/B$ (see Hildebrand et
al., 2009, for more details).

We use Equation 6 to find the ADF in the nearby components of the Coalsack’s
IPS-GI regions, i.e. C11 and C15 (see Figure 5). For these two nearby
components, we can apply the structure function method because we can
approximate that part of the LOS as a single polarizing dust sheet with a
population of stars, all with similar polarimetric properties, located
immediately behind the dusty screen. This simplification is necessary because
we use the plane-of-sky position to determine the scale length in the
structure function, disregarding the distance between the stars. Because all
nearby stars are polarized by the Southern Coalsack only, and there are no
other dusty structures that may add to the polarization of the starlight, we
can safely make this simplification. This also explains why we cannot readily
apply this method to the other LOSs or the distant components of fields C11
and C15– we cannot simplify those LOS to a single polarizing dust screen with
a single stellar population with similar polarimetric properties, because
there are multiple polarizing structures along the LOS.

Figure 5 shows the ADF for the nearby populations of fields C11 and C15.
Starting with field C15, the ADF increases with the observed scale up to
$\ell\sim 0.1\arcdeg$ and then flattens. In field C11, on the other hand, the
ADF is basically constant throughout the observed scales. Due to the angular
size of the field-of-view, which is 0.3° by 0.3°, we only trust scales up to
$\sim 0.2\arcdeg$, above which the statistics may be inaccurate. This is also
true for the first scale bin.

There are different ways we can interpret the ADFs of Figure 5. Firstly, if
the turbulent scale $\delta$ is larger than the smallest scale we can probe,
i.e. $\delta>\ell_{min}$ with $\ell_{min}\sim 0.01\arcdeg$, and smaller than
the large-scale variations of the magnetic field, $d$, the ADF will increase
linearly (in log-log space) with the smallest scales until it reaches the
outer scale of the turbulence. In this case, we would be observing a turbulent
scale of 0.35 pc at the assumed cloud distance ($\sim 200$ pc) in field C15,
but we can not calculate the magnetic field strength.

Secondly, if the turbulent scale is smaller than the smallest scale we can
probe, i.e. $\delta<\ell_{min}$ such that $\delta<\ell\ll d$, then
$\ell_{min}$ is the turbulence scale upper limit and the ADF can be fitted at
the smallest scales as described in Hildebrand et al. (2009):

$\langle\Delta\theta^{2}(\ell)\rangle_{tot}=b^{2}+m^{2}\ell^{2}+\sigma^{2}_{M}(\ell),$
(7)

where $\langle\Delta\theta^{2}(\ell)\rangle_{tot}$ is the total function
calculated from the observations, $\sigma^{2}_{M}(\ell)$ are the measurement
uncertainties that must be subtracted to obtain the values of Equation 6, $b$
is the turbulent contribution, and $m\ell$ is the contribution of the large-
scale fluctuations. According to Hildebrand et al. (2009), $b^{2}$ is
proportional to $\sigma_{\theta}^{2}$, which is also assumed to be
approximately $\delta B/B$, the variation in the magnetic field about its
large-scale component. Therefore, we can calculate the the regular magnetic
field strength as

$B=\sqrt{8\pi\rho}\frac{\sigma_{v}}{b}~{}~{}~{}~{}\mathrm{for}~{}~{}~{}~{}B_{t}\ll
B,$ (8)

where $B_{t}$ is the turbulent magnetic field strength and the turbulent
velocity dispersion, $\sigma_{v}$, and density, $\rho$, of the cloud must be
very well known. A correction factor f as in Equation 1 is not required. Then,
if this is the case for field C15, the structure function fit gives
$b=14.59\arcdeg\pm 0.13\arcdeg$ (the red line in Figure 5, top row), and the
magnetic field strength would be $31.2\pm 4.7\ \mu$G considering the
parameters of Table 1.

Finally, we find that the structure function of C11 (Figure 5, bottom left) is
constant (i.e. fitting the structure function with a parabola as in Field C15
versus a straight horizontal line results in similar $\chi^{2}$-values). This
suggests that the large-scale variations are not significant and the turbulent
outer scale is below the minimum scale we can probe, $\ell_{min}$. This means
that the magnetic field is significantly regular as it can also be observed in
the polarization vector map (Figure 5, bottom right) and the polarization
angle histogram for this field, Figure 1, top left.

Figure 5: Top left: the square root of the structure function of the
polarization angles (the ADF, see Eq. 6) of the nearby population of field
C15. The red line indicates the best fit to the first five bins. Top right:
polarization vector plot of the nearby stars of field C15. The red marked star
indicates an observation removed from further analysis. Bottom left: the ADF
(Eq. 6 of the polarization angles of the nearby population of field C11.
Bottom right: polarization vector plot of the stars in the nearby population
of field C11.

###  5.7 Distant structure

The observations of fields C11 and C15 (see Sections 4.1 and 4.2,
respectively) indicate lines-of-sight with at least two polarizing structures.
This is most clearly evident when we plot the Stokes parameters q and u as a
function of distance as in Figure 6, top left and right. Figure 6, top left
and right shows a somewhat constant value for q and u up to a certain
distance, beyond which the Stokes parameters assume a different value. The
distance at which this happens is around $d\sim 1.3\textrm{ kpc}$ for field
C11, and $d\sim 1.5\textrm{ kpc}$ for C15. We interpret this as follows:
starting from Earth, the LOS first penetrates the Southern Coalsack. This is a
dense, dusty structure which causes a polarized signal. This affects the stars
within the Coalsack, but also those behind the cloud complex. The region
immediately behind the Coalsack appears to be mostly devoid of (dense) dust
(Seidensticker & Schmidt-Kaler, 1989), see also Figure 6. This means that
stars located in this low-density region will exhibit a polarized signal
dominated by the Southern Coalsack nearby. This is what is represented in
Figure 6 at distances $d<1.5\textrm{ kpc}$.

Figure 6: Scatter plots of Stokes parameters q (top panel) and u (middle
panel) versus distance for all fields. The bottom profiles show the
differential extinction (or extinction density) in mag/kpc taken from
Lallement et al. (2022); Vergely et al. (2022). The dotted and dashed vertical
lines in the scatter plots of fields C11 and C15 indicate the average distance
of the nearby and distant populations, respectively. The turnover point at
which the polarized signal changes occurs at a distance of approximately 1.3
kpc.

Around $d\sim 1.5\textrm{ kpc}$, a change in polarimetry is observed. This
must be caused by a second polarizing component along the LOS, indicating
another dusty, dense structure capable of dominating the polarimetry of the
stars. There are various possibilities of what precisely constitutes this
structure.

Firstly, the CO observations of Beuther et al. (2011) show not only the well-
known Coalsack signal at $-4\textrm{ km s}^{-1}$, but also a second component
at a velocity of $-30\textrm{ km s}^{-1}$. Beuther et al. (2011) put this
component at a distance of $d=3.1\textrm{ kpc}$, which is too great a distance
to account for the changes in polarimetry seen in Figure 6. Therefore, we
dismiss this possibility.

Secondly, there is evidence of a very large HI shell (GSH 304-00-12) located
at a distance of about $d\sim 1.2\textrm{ kpc}$ behind the Southern Coalsack
(McClure-Griffiths et al., 2001). This structure is located at the exact
distance at which we see a change in polarimetry in fields C11 and C15.
However, this HI shell is very large (approximately 20° by 30°) which means we
should be able to see this shell in the distant components of all the IPS
fields, assuming that the dust in the HI shell is distributed uniformly enough
to be detectable throughout the shell. However, we do not see similar changes
in the polarimetry of fields C39, C40 and C54.

The final possibility is that the starlight from the distant stars is
polarized by dust in the Carina spiral arm (see also Seidensticker & Schmidt-
Kaler, 1989) as suggested by Andersson & Potter (2005), who see the spiral arm
in the polarimetric signature of one of their components (their component 2).
Their magnetic field orientation of this component, described as the Carina
arm, is $\theta=88.5\arcdeg$ on average, which is a bit lower than our
measurements for the orientation (i.e. $\theta\sim 100-120\arcdeg$). However,
they, as well as e.g. Seidensticker & Schmidt-Kaler (1989) estimate a distance
to the Carina arm of $d=1.3\textrm{ kpc}$ which does coincide with our
estimates of the distance of the second polarizing structure. However, because
of the large size of the spiral arm, we would expect to see signatures of the
Carina arm in the polarimetry of most, if not all, observed fields-of-view.
Therefore, this distant component is believed to be a structure at roughly the
Carina arm’s distance, likely inside the arm.

Our detections of the distant structure rely on a change in polarization
observed along the LOS. If the secondary structure were to have similar
magnetic field properties as the nearby structure, the method described above
would not be sensitive to the distant polarizing screen. In that case, a more
detailed analysis of dust models would be warranted.

We note that the two fields that show a strong secondary polarizing component
along the LOS, i.e. fields C11 and C15, are both located toward the Northeast
of the Southern Coalsack. Therefore, it is possible that the second structure
only exists in that region, that its dust content is limited to that region or
that outside of that region its magnetic field properties are more similar to
that of the Coalsack in front. However, we note that these fields are located
near the outskirts of the Coalsack. This means that polarimetric signal of
these specific lines-of-sight may be less dominated by the Coalsack. For
example, the dust content along the edges of the Coalsack may be less than in
more dense regions toward the center. This would allow a secondary structure
to be more visible in the observations. More observations are necessary to
fully confirm the nature of this secondary structure.

###  5.8 Other methods of field strength determination

In the analysis presented above, we have chosen to use the DCF method (Davis,
1951; Chandrasekhar & Fermi, 1953), see also Equation 1, to determine the
strength of the magnetic field components. Although this method is very well-
established and many adaptations and corrections to the method have been
suggested, it is known that the DCF method overestimates the magnetic field
strength if a correction factor is not applied. Therefore, it is desirable to
compare different methods of field strength determination.

An alternative to the DCF method based was presented by Skalidis & Tassis
(2021), hereafter ST, who have relaxed the assumption of incompressibility
necessary for the DCF method and instead derived:

$B=\sqrt{2\pi\rho}\frac{\sigma_{v}}{\sqrt{\sigma_{\theta}}}$ (9)

where $\rho$ is the gas density, $\sigma_{v}$ is the velocity dispersion and
$\sigma_{\theta}$ is the polarization angle dispersion. We have also used the
ST method to calculate the strengths of the various magnetic field components.
The results can be found in Table 1. The ST method finds lower magnetic field
strengths than the DCF method. The differences are largest for the strongest
magnetic fields (i.e. C15 (nearby) and C40). Overall, we find that using the
ST method, magnetic fields are found to be around 50% weaker compared to the
DCF method.

Another method that can be used to determine magnetic field strength is the
Differential Measure Analysis (DMA), see Lazarian et al. (2022). This method
uses gradients (in the form of structure functions) of the gas velocity and
polarization angle dispersion and is based on MHD turbulence theory. Lazarian
et al. (2022) estimate the value of the correction factor f for different
turbulent regimes. However, the required properties, such as Mach numbers and
plasma $\beta$, are not well known in the Coalsack. However, if we assume
low-$\beta$ conditions in the Coalsack (as in their Table 1) and sub-alfvenic
turbulence (Li et al., 2013), Figure 13 in Lazarian et al. (2022) shows that
the factor f can range from 0.2 to 0.9 depending on Alfvenic mach number and
line-of-sight angle of the local magnetic field. In addition, we do not have
detailed enough information to construct a structure function of gas
velocities across the fields-of-view. As these quantities are either unknown
or very uncertain, we do not have sufficient information to further constrain
the magnetic field in the Coalsack with this method.

##  6 Summary and Conclusions

We have presented optical V-band observations of starlight polarization from
IPS-GI in five discrete fields-of-view in or close to the Southern Coalsack.
Combining these observations with accurate distance measurements, we are able,
in two of our fields, to disentangle starlight polarization dominated by the
Coalsack and a polarimetric signature of a dusty structure behind the
Coalsack, at a distance of roughly $\sim 1.3-1.5\textrm{ kpc}$. Using the
K-means clustering and DBSCAN algorithms and by subtracting the nearby
polarimetric contribution from the distant polarization, we are able to study
both structures independently. For the Coalsack-dominated nearby population,
we find magnetic field orientations in the range $\theta_{N}=65-85\arcdeg$.
For the polarizing distant structure, we find a plane-of-sky magnetic field
orientation of $\theta_{D}=100-110\arcdeg$. Furthermore, using CO observations
of Nyman et al. (1989), we can use the DCF method to find an estimate of the
strength of the plane-of-sky component of the magnetic fields of the nearby
and distant structures. The strengths we find are in the range $B=8-12\
\mu\textrm{G}$ and $B\sim 10\ \mu\textrm{G}$ for the nearby and distant
structures, respectively. The magnetic field strength of the distant structure
is an order-of-magnitude estimation due to uncertainties in its local gas
density and gas velocity dispersion. Further observations may help uncover its
precise nature and aid in improving the field strength estimation.

The authors are very grateful to the reviewers, whose helpful suggestions and
insights helped us improve the manuscript. MJFV acknowledges Wouter Veltkamp
for his contributions to Figure 1. Over the years, IPS data has been gathered
by a number of dedicated observers, to whom the authors are very grateful:
Flaviane C. F. Benedito, Alex Carciofi, Cassia Fernandez, Tibério Ferrari,
Livia S. C. A. Ferreira, Viviana S. Gabriel, Aiara Lobo-Gomes, Luciana de
Matos, Rocio Melgarejo, Antonio Pereyra, Nadili Ribeiro, Marcelo Rubinho,
Daiane B. Seriacopi, Fernando Silva, Rodolfo Valentim and Aline Vidotto. MJFV,
MH and YAA acknowledge funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No 772663). The authors acknowledge Interstellar Institute’s program
”With Two Eyes” and the Paris-Saclay University’s Institut Pascal for hosting
discussions that nourished the development of the ideas behind this work. This
research has used data, tools or materials developed as part of the EXPLORE
project that has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No 101004214. AMM’s
work and Optical/NIR Polarimetry at IAG has been supported over the years by
several grants from São Paulo state funding agency FAPESP, especially
01/12589-1 and 10/19694-4. AMM has also been partially supported by Brazilian
agency CNPq (grant 310506/2015-8). AMM graduate students have been provided
grants over the years from Brazilian agency CAPES. CVR thanks Conselho
Nacional de Desenvolvimento Científico e Tecnológico - CNPq (Proc:
310930/2021-9).

## References

  * Anders et al. (2022) Anders, F., Khalatyan, A., Queiroz, A. B. A., et al. 2022, A&A, 658, A91, doi: 10.1051/0004-6361/202142369
  * Andersson et al. (2004) Andersson, B. G., Knauth, D. C., Snowden, S. L., Shelton, R. L., & Wannier, P. G. 2004, ApJ, 606, 341, doi: 10.1086/382861
  * Andersson & Potter (2005) Andersson, B. G., & Potter, S. B. 2005, MNRAS, 356, 1088, doi: 10.1111/j.1365-2966.2004.08538.x
  * Andersson et al. (2002) Andersson, B. G., Wannier, P. G., & Crawford, I. A. 2002, MNRAS, 334, 327, doi: 10.1046/j.1365-8711.2002.t01-1-05465.x
  * Bailey et al. (2016) Bailey, M., van Loon, J. T., Farhang, A., et al. 2016, A&A, 585, A12, doi: 10.1051/0004-6361/201526656
  * Barlow (2003) Barlow, R. 2003, arXiv e-prints, physics/0306138.  https://arxiv.org/abs/physics/0306138
  * Beuther et al. (2011) Beuther, H., Kainulainen, J., Henning, T., Plume, R., & Heitsch, F. 2011, A&A, 533, A17, doi: 10.1051/0004-6361/201116746
  * Bhat & Andersson (2011) Bhat, N. D. R., & Andersson, B. G. 2011, ApJ, 729, 38, doi: 10.1088/0004-637X/729/1/38
  * Bolatto et al. (2013) Bolatto, A. D., Wolfire, M., & Leroy, A. K. 2013, ARA&A, 51, 207, doi: 10.1146/annurev-astro-082812-140944
  * Bronfman et al. (1989) Bronfman, L., Alvarez, H., Cohen, R. S., & Thaddeus, P. 1989, ApJS, 71, 481, doi: 10.1086/191384
  * Burkert (2006) Burkert, A. 2006, Comptes Rendus Physique, 7, 433, doi: 10.1016/j.crhy.2006.05.001
  * Chandrasekhar & Fermi (1953) Chandrasekhar, S., & Fermi, E. 1953, ApJ, 118, 113, doi: 10.1086/145731
  * Clarke (2010) Clarke, D. 2010, Stellar Polarimetry (John Wiley & Sons)
  * Clemens et al. (2020) Clemens, D. P., Cashman, L. R., Cerny, C., et al. 2020, ApJS, 249, 23, doi: 10.3847/1538-4365/ab9f30
  * Corradi et al. (2004) Corradi, W. J. B., Franco, G. A. P., & Knude, J. 2004, MNRAS, 347, 1065, doi: 10.1111/j.1365-2966.2004.07219.x
  * Crutcher (1999) Crutcher, R. M. 1999, ApJ, 520, 706, doi: 10.1086/307483
  * Dame et al. (2001) Dame, T. M., Hartmann, D., & Thaddeus, P. 2001, ApJ, 547, 792, doi: 10.1086/318388
  * Dame et al. (1987) Dame, T. M., Ungerechts, H., Cohen, R. S., et al. 1987, ApJ, 322, 706, doi: 10.1086/165766
  * Davis (1951) Davis, L. 1951, Physical Review, 81, 890, doi: 10.1103/PhysRev.81.890.2
  * Dharmawardena et al. (2023) Dharmawardena, T. E., Bailer-Jones, C. A. L., Fouesneau, M., et al. 2023, MNRAS, 519, 228, doi: 10.1093/mnras/stac2790
  * Dobbs et al. (2011) Dobbs, C. L., Burkert, A., & Pringle, J. E. 2011, MNRAS, 417, 1318, doi: 10.1111/j.1365-2966.2011.19346.x
  * Edenhofer et al. (2023) Edenhofer, G., Zucker, C., Frank, P., et al. 2023, arXiv e-prints, arXiv:2308.01295, doi: 10.48550/arXiv.2308.01295
  * Ester et al. (1996) Ester, M., Kriegel, H.-P., Sander, J., & Xu, X. 1996, in Proceedings of the Second International Conference on Knowledge Discovery and Data Mining, KDD’96 (AAAI Press), 226–231
  * Ferrière (2001) Ferrière, K. M. 2001, Reviews of Modern Physics, 73, 1031, doi: 10.1103/RevModPhys.73.1031
  * Franco (1989) Franco, G. A. P. 1989, A&A, 215, 119
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1, doi: 10.1051/0004-6361/202039657
  * Hall (1949) Hall, J. S. 1949, Science, 109, 166, doi: 10.1126/science.109.2825.166
  * Heiles (2000) Heiles, C. 2000, The Astronomical Journal, 119, 923
  * Heitsch et al. (2001) Heitsch, F., Zweibel, E. G., Mac Low, M.-M., Li, P., & Norman, M. L. 2001, ApJ, 561, 800, doi: 10.1086/323489
  * Hildebrand et al. (2009) Hildebrand, R. H., Kirby, L., Dotson, J. L., Houde, M., & Vaillancourt, J. E. 2009, ApJ, 696, 567, doi: 10.1088/0004-637X/696/1/567
  * Hiltner (1949) Hiltner, W. A. 1949, Science, 109, 165, doi: 10.1126/science.109.2825.165
  * Jansson & Farrar (2012) Jansson, R., & Farrar, G. R. 2012, ApJ, 757, 14, doi: 10.1088/0004-637X/757/1/14
  * Jones et al. (1984) Jones, T. J., Hyland, A. R., & Bailey, J. 1984, ApJ, 282, 675, doi: 10.1086/162247
  * Kato et al. (1999) Kato, S., Mizuno, N., Asayama, S.-i., et al. 1999, PASJ, 51, 883, doi: 10.1093/pasj/51.6.883
  * Klare & Neckel (1977) Klare, G., & Neckel, T. 1977, A&AS, 27, 215
  * Knuth (2006) Knuth, K. H. 2006, arXiv preprint physics/0605197
  * Knuth (2019) —. 2019, Digital Signal Processing, 95, 102581
  * Lada et al. (2004) Lada, C. J., Huard, T. L., Crews, L. J., & Alves, J. F. 2004, ApJ, 610, 303, doi: 10.1086/421517
  * Lallement et al. (2022) Lallement, R., Vergely, J. L., Babusiaux, C., & Cox, N. L. J. 2022, A&A, 661, A147, doi: 10.1051/0004-6361/202142846
  * Laursen et al. (2019) Laursen, P., Sommer-Larsen, J., Milvang-Jensen, B., Fynbo, J. P. U., & Razoumov, A. O. 2019, A&A, 627, A84, doi: 10.1051/0004-6361/201833645
  * Lazarian (2007) Lazarian, A. 2007, J. Quant. Spec. Radiat. Transf., 106, 225, doi: 10.1016/j.jqsrt.2007.01.038
  * Lazarian et al. (2022) Lazarian, A., Yuen, K. H., & Pogosyan, D. 2022, ApJ, 935, 77, doi: 10.3847/1538-4357/ac6877
  * Li et al. (2013) Li, H.-b., Fang, M., Henning, T., & Kainulainen, J. 2013, MNRAS, 436, 3707, doi: 10.1093/mnras/stt1849
  * Lupton (1993) Lupton, R. 1993, Statistics in theory and practice (Princeton University Press)
  * Magalhães et al. (2005) Magalhães, A. M., Pereyra, A., Melgarejo, R., et al. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 343, Astronomical Polarimetry: Current Status and Future Directions, ed. A. Adamson, C. Aspin, C. Davis, & T. Fujiyoshi, 305
  * Mathewson & Ford (1970) Mathewson, D. S., & Ford, V. L. 1970, MmRAS, 74, 139
  * McClure-Griffiths et al. (2001) McClure-Griffiths, N. M., Dickey, J. M., Gaensler, B. M., & Green, A. J. 2001, ApJ, 562, 424, doi: 10.1086/323859
  * McKee & Ostriker (2007) McKee, C. F., & Ostriker, E. C. 2007, ARA&A, 45, 565, doi: 10.1146/annurev.astro.45.051806.110602
  * Medan & Andersson (2019) Medan, I., & Andersson, B. G. 2019, ApJ, 873, 87, doi: 10.3847/1538-4357/ab063c
  * Myers et al. (1995) Myers, P. C., Goodman, A. A., Gusten, R., & Heiles, C. 1995, ApJ, 442, 177, doi: 10.1086/175433
  * Nyman (2008) Nyman, L. Å. 2008, in Handbook of Star Forming Regions, Volume II, ed. B. Reipurth, Vol. 5, 222
  * Nyman et al. (1989) Nyman, L. A., Bronfman, L., & Thaddeus, P. 1989, A&A, 216, 185
  * Ostriker et al. (2001) Ostriker, E. C., Stone, J. M., & Gammie, C. F. 2001, ApJ, 546, 980, doi: 10.1086/318290
  * Padoan et al. (2001) Padoan, P., Goodman, A., Draine, B. T., et al. 2001, ApJ, 559, 1005, doi: 10.1086/322504
  * Pedregosa et al. (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., et al. 2011, Journal of Machine Learning Research, 12, 2825
  * Pelgrims et al. (2020) Pelgrims, V., Ferrière, K., Boulanger, F., Lallement, R., & Montier, L. 2020, A&A, 636, A17, doi: 10.1051/0004-6361/201937157
  * Pereyra & Magalhães (2007) Pereyra, A., & Magalhães, A. M. 2007, ApJ, 662, 1014, doi: 10.1086/517906
  * Reiz & Franco (1998) Reiz, A., & Franco, G. A. P. 1998, A&AS, 130, 133, doi: 10.1051/aas:1998216
  * Riello et al. (2021) Riello, M., De Angeli, F., Evans, D. W., et al. 2021, A&A, 649, A3, doi: 10.1051/0004-6361/202039587
  * Rodrigues et al. (2019) Rodrigues, L. F. S., Chamandy, L., Shukurov, A., Baugh, C. M., & Taylor, A. R. 2019, MNRAS, 483, 2424, doi: 10.1093/mnras/sty3270
  * Rybizki et al. (2022) Rybizki, J., Green, G. M., Rix, H.-W., et al. 2022, MNRAS, 510, 2597, doi: 10.1093/mnras/stab3588
  * Seidensticker & Schmidt-Kaler (1989) Seidensticker, K. J., & Schmidt-Kaler, T. 1989, A&A, 225, 192
  * Shukurov et al. (2019) Shukurov, A., Rodrigues, L. F. S., Bushby, P. J., Hollins, J., & Rachen, J. P. 2019, A&A, 623, A113, doi: 10.1051/0004-6361/201834642
  * Skalidis & Tassis (2021) Skalidis, R., & Tassis, K. 2021, A&A, 647, A186, doi: 10.1051/0004-6361/202039779
  * van Smith (1956) van Smith, E. P. 1956, ApJ, 124, 43, doi: 10.1086/146202
  * Vergely et al. (2022) Vergely, J. L., Lallement, R., & Cox, N. L. J. 2022, A&A, 664, A174, doi: 10.1051/0004-6361/202243319
  * Versteeg et al. (2023) Versteeg, M. J. F., Magalhães, A. M., Haverkorn, M., et al. 2023, AJ, 165, 87, doi: 10.3847/1538-3881/aca8fd
  * Whittet (2022) Whittet, D. C. B. 2022, Dust in the Galactic Environment (Third Edition), doi: 10.1088/2514-3433/ac7204