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Astronomy
&
Astrophysics
A&A, 684, A162 (2024)
https://doi.org/10.1051/0004-6361/202349015
© The Authors 2024
The first degree-scale starlight-polarization-based tomography
map of the magnetized interstellar medium⋆,⋆⋆
V. Pelgrims1,2,3, N. Mandarakas2,3, R. Skalidis4, K. Tassis2,3, G. V. Panopoulou5, V. Pavlidou2,3,
D. Blinov2,3, S. Kiehlmann2,3, S. E. Clark6,7, B. S. Hensley8, S. Romanopoulos2,3, A. Basyrov9,
H. K. Eriksen9, M. Falalaki2,3, T. Ghosh10 ,11 , E. Gjerløw9, J. A. Kypriotakis2,3, S. Maharana12 , A. Papadaki2,3,13 ,
T. J. Pearson4, S. B. Potter12,14 , A. N. Ramaprakash1,4,11, A. C. S. Readhead4, and I. K. Wehus9
1Université Libre de Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium
e-mail: vincent.pelgrims@ulb.be
2Institute of Astrophysics, Foundation for Research and Technology-Hellas, N. Plastira 100, Vassilika Vouton, 71110 Heraklion,
Greece
3Department of Physics, and Institute for Theoretical and Computational Physics, University of Crete, Voutes University campus,
70013 Heraklion, Greece
4Owens Valley Radio Observatory, California Institute of Technology, MC 249-17, Pasadena, CA 91125, USA
5Department of Space, Earth & Environment, Chalmers University of Technology, 412 93 Gothenburg, Sweden
6Department of Physics, Stanford University, Stanford, CA 94305, USA
7Kavli Institute for Particle Astrophysics & Cosmology, PO Box 2450, Stanford University, Stanford, CA 94305, USA
8Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA, USA
9Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway
10 National Institute of Science Education and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar 752050, Odisha,
India
11 Inter-University Centre for Astronomy and Astrophysics, Post bag 4, Ganeshkhind, Pune, 411007, India
12 South African Astronomical Observatory, PO Box 9, Observatory, 7935 Cape Town, South Africa
13 Institute of Computer Science, Foundation for Research and Technology-Hellas, 71110 Heraklion, Greece
14 Department of Physics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa
Received 19 December 2023 / Accepted 4 February 2024
ABSTRACT
We present the first degree-scale tomography map of the dusty magnetized interstellar medium (ISM) from stellar polarimetry and
distance measurements. We used the RoboPol polarimeter at Skinakas Observatory to conduct a survey of the polarization of starlight
in a region of the sky of about four square degrees. We propose a Bayesian method to decompose the stellar-polarization source field
along the distance to invert the three-dimensional (3D) volume occupied by the observed stars. We used this method to obtain the first
3D map of the dusty magnetized ISM. Specifically, we produced a tomography map of the orientation of the plane-of-sky component
of the magnetic field threading the diffuse, dusty regions responsible for the stellar polarization. For the targeted region centered on
Galactic coordinates (l,b)≈(103.3◦,22.3◦), we identified several ISM clouds. Most of the lines of sight intersect more than one cloud.
A very nearby component was detected in the foreground of a dominant component from which most of the polarization signal comes
and which we identified as being an intersection of the wall of the Local Bubble and the Cepheus Flare. Farther clouds, with a distance
of up to 2 kpc, were similarly detected. Some of them likely correspond to intermediate-velocity clouds seen in H Ispectra in this
region of the sky. We found that the orientation of the plane-of-sky component of the magnetic field changes along distance for most
of the lines of sight. Our study demonstrates that starlight polarization data coupled to distance measures have the power to reveal the
great complexity of the dusty magnetized ISM in 3D and, in particular, to provide local measurements of the plane-of-sky component
of the magnetic field in dusty regions. This demonstrates that the inversion of large data volumes, as expected from the PASIPHAE
survey, will provide the necessary means to move forward in the modeling of the Galactic magnetic field and of the dusty magnetized
ISM as a contaminant in observations of the cosmic microwave background polarization.
Key words. polarization – methods: statistical – dust, extinction – ISM: magnetic fields – ISM: structure
1. Introduction
The polarization of starlight is a powerful probe of the magne-
tized interstellar medium (ISM). Starlight acquires a polarization
⋆Table 1 is available at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https:
//cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/684/A162
⋆⋆ The 3D map obtained in this paper can be visualized at https:
//pasiphae.science/visualization
due to dichroic absorption by aspherical interstellar dust grains,
which align their minor axis with the magnetic field (e.g., Davis
& Greenstein 1951;Andersson et al. 2015). The polarization
position angle of starlight is parallel to the plane-of-sky (POS)
component of the magnetic field, and the maximum degree of
polarization is proportional to the column density of the polariz-
ing dust through which the light beam passes. Since its discovery
(Hiltner 1949;Hall 1949), the polarization of starlight has con-
tributed significantly to the study of the magnetic field in our
A162, page 1 of 25
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Pelgrims, V., et al.: A&A, 684, A162 (2024)
Galaxy and to our understanding of its role as an agent of the
Galactic ecosystem, from the smallest to the largest scales (e.g.,
Spoelstra 1972;Ellis & Axon 1978;Goodman et al. 1990;Heiles
1996;Heyer et al. 2008;Nishiyama et al. 2010;Li et al. 2013;
Berdyugin et al. 2014;Doi et al. 2023). Once stellar distances
are known, starlight polarization could provide information on
the properties of the magnetized ISM directly in three dimension
(3D; Panopoulou et al. 2019), and with good spatial resolution
given the high stellar density.
In recent years, the Gaia satellite (Gaia Collaboration
2016) has provided the data necessary for the precise local-
ization in 3D space of more than a billion stars in our Galaxy
(e.g., Bailer-Jones et al. 2021;Gaia Collaboration 2021,2023;
Lindegren et al. 2021). By combining stellar parallax and
reddening data, several teams have been successful in recon-
structing 3D tomography maps of the dust density distribution
in large volumes centered on the Sun, up to around 3 kpc in
the Galactic disk and up to around 1.2 kpc in the halo (Green
et al. 2019;Lallement et al. 2019,2022;Leike & Enßlin 2019;
Leike et al. 2020;Vergely et al. 2022;Edenhofer et al. 2024),
or in more focused areas. This formidable community effort has
revolutionized our view of the 3D structure of dust distribution
in the ISM, and it has already enabled the modeling and better
understanding of some of the main structures in our cosmic
neighborhood and their history (e.g., Pelgrims et al. 2020;
Alves et al. 2020;Das et al. 2020;Bialy et al. 2021;Zucker
et al. 2022;Großschedl et al. 2018;Marchal & Martin 2023;
Ivanova et al. 2021;Tahani et al. 2022). Such a 3D mapping,
with the additional knowledge of magnetic field properties,
would be certain to enable breakthroughs and discoveries
in several research topics, as discussed below and in Pelgrims
et al. (2023).
For example, a tomographic view of the magnetized ISM
would offer new avenues to address open questions such as
the role of the magnetic field in star formation (see, e.g.,
Mouschovias et al. 2006;McKee & Ostriker 2007), and the
search for the sources of ultra-high energy cosmic rays (see,
e.g., Boulanger et al. 2018;Magkos & Pavlidou 2019;Tsouros
et al. 2024). Likewise, by providing measurements of the polar-
ization properties for each individual dust cloud in 3D space,
such a tomography map would enable significant progress in the
modeling and characterization of the dusty magnetized ISM as
a contaminating foreground to observations of the polarization
of the cosmic microwave background (Tassis & Pavlidou 2015;
Martínez-Solaeche et al. 2018;Pelgrims et al. 2021). Hence,
it would help clear the path for an unbiased study of one of
the first moments of the Universe’s history. This holds true,
although less directly, for the characterization of the Galactic
synchrotron emission as demonstrated by Panopoulou et al.
(2021). Combined with multiwavelength observations of the
polarized emission of dust in the submillimeter, knowledge of
the polarization properties of each individual dust cloud would
also enable us to advance our modeling of astrophysical dust
(e.g., Hensley & Draine 2021), to better determine its composi-
tion and understand its interaction with its cosmic environment,
and to better assess its role as a building block of life.
Mapping the dust distribution in 3D over large volumes
(kiloparsecs) required millions of stellar extinction and distance
measurements. A similar amount of data is likely required for
magnetic field mapping in such a volume. Currently, the avail-
ability of stellar polarization data is limited (Panopoulou et al.
2023), with only the inner Galaxy having millions of stars
with measured near-IR polarization (GPIPS survey, Clemens
et al. 2020). In the future, planned optical polarimetric surveys
will deliver millions of measurements throughout the entire sky
(Magalhães et al. 2005,2012;Tassis et al. 2018;Covino et al.
2020). It is therefore timely to develop techniques to analyze
these forthcoming datasets in such a way that would make it
possible to extract from the data most of the information on the
properties of the dusty magnetized ISM in 3D.
The observed polarization of each single star is the inte-
grated effect of the dichroic absorption from all ISM clouds lying
between us and the star. This line-of-sight (LOS) integration
needs to be inverted in order to derive the complex 3D structure
of the magnetized ISM from starlight polarization and distance
measurements. Recently, we have developed the first standalone
method to perform this LOS inversion through Bayesian model-
ing (Pelgrims et al. 2023). This method, which takes into account
all sources of uncertainties in polarization and parallax measure-
ments, works on a per LOS basis. It has the advantage that it
can be easily automated to run on a large set of sightlines. We
have extensively tested the method and its performance based
on simulated data. We further applied it to existing data for
two sightlines and demonstrated that this new method leads to
results that are fully consistent with previously obtained results
but within a robust Bayesian framework and, therefore, put on a
more solid footing.
In this paper, we continue our preparation for the large
datasets to come. We aim to develop a pipeline capable of invert-
ing measurements of parallax and starlight polarization for an
actual, extended volume of 3D space in order to derive the prop-
erties of the dusty magnetized ISM in 3D. We develop a first
pipeline based on our LOS-inversion method and apply it to
observations taken for an extended region of the sky. Our goal
is to obtain, with as few assumptions as possible, a first tomog-
raphy map of the POS component of the magnetic field in dusty
regions from which 3D properties of the magnetized ISM can be
accessed.
For the purposes of this work, we carried out a survey of
starlight polarization for a continuous region of the sky covering
about four square degrees. We present our survey and the result-
ing dataset in Sect. 2. We devote Sect. 3to the description of our
data analysis pipeline and its application to our data to obtain
the first degree-scale tomography map of the magnetized ISM
from starlight polarization and parallax measurements. Section 4
presents our main results and how we produce and visualize our
3D map of the POS component of the magnetic field in dusty
region from the posterior distributions output by our Bayesian
analysis. Our 3D map extends up to 3 kpc distance and covers a
sky region of about four square degrees. We discuss our results in
comparison with other probes of the (magnetized) ISM in Sect. 5
and conclude in Sect. 6.
2. Dataset
To demonstrate the feasibility of starlight-polarization-based
tomography of the diffuse magnetized ISM, Panopoulou et al.
(2019) obtained dense optical polarization measurements of stars
for two circular beams of 9.6 arcmin radius in a sky region that
they identified as being likely to exhibit complexity along the
distance axis based on inspection of H Ivelocity data and polar-
ized dust emission maps. These data indeed suggest possible
variations of the polarization signal both in the POS and along
distance as several components can be identified. As they showed
through the analysis of their polarization data along distance, and
as we recently confirmed (Pelgrims et al. 2023), one of the two
beams likely intersects at least two dust clouds while the other
A162, page 2 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
l[±]
b[±]
l[±]
b[±]
Fig. 1. Sky location of the surveyed area of about four square degrees. Left: full-sky map of the dust emission as seen by Planck at 353 GHz (Planck
Collaboration XII 2020). The color represents the intensity of dust emission on logarithmic scale. The line-integral-convolution texture shows the
polarization angle of the dust emission rotated by 90◦. Middle: a zoom-in of the map toward the surveyed regions, which includes part of the North
Celestial Pole Loop on the East of the map. Right: a closer view of the surveyed region. Black segments indicate the polarization orientation from
the stars in our survey and from Panopoulou et al. (2019). The segments are scaled according to the polarization fraction. Unpolarized stars appear
as dots. The blue, horizontal segment in the bottom right corner shows the scale for a 1% polarized star. Outlier candidates (see Sect. 3.3) are not
shown.
one likely intersects only one. Showing complexity both along
the LOS and in the POS, this region therefore seems to be well
suited to develop and test tomography methods to reconstruct the
magnetized ISM in 3D. Hence, we carried out a survey to expand
that region of the sky with optical starlight polarization data. The
final surveyed region, comprising about four square degrees, and
its location in the sky is shown in Fig. 1.
2.1. Survey strategy and observation plan
We aimed at obtaining polarization measurements for a large,
continuous region centered on Galactic coordinates (l,b)≈
(103.5◦,22.25◦)with complete star samples limited in magni-
tude. Due to limited observing time, it was infeasible to perform
a deep, photometrically uniform survey for the entire area.
Hence, we relied on H Iobservations to gauge a priori which
part of the targeted region likely intersects dust clouds at large
distances, in order to increase the number of stellar polarization
data accordingly.
As discussed by Panopoulou et al. (2019), and also shown
in Fig. 2, the averaged H I-velocity spectrum (measured in the
Local Standard of Rest, LSR) in this region of the sky shows
the existence of two very distinct components. The dominant
component is centered on vLSR ≈ −2.5 km s−1and we refer to it
as the low velocity cloud (LVC). The second component, with
a lower amplitude, is centered on vLSR ≈ −50 km s−1and we
refer to it as the intermediate velocity cloud (IVC). This velocity
component may be a southern extension of the IV Arch as iden-
tified by Kuntz & Danly (1996) who determined that it is located
at large distance (≳1 kpc). We used the HI4PI survey (HI4PI
Collaboration 2016) to look at the spatial distribution of the
intensity maps obtained from the integration of the velocity
spectra in channels corresponding to the peaks observed in the
averaged velocity spectrum.
These data are shown in Fig. 2where we also show the
outline of the region for which we obtain a tomography recon-
struction in this work. As can be seen clearly in the right-hand
bottom panel of Fig. 2, the H Ivelocity spectra show power in
the IVC range in the eastern part of the region (with l≳103.5◦).
Thus, we concluded that this region is more likely to contain
dust clouds at larger distances than the other (western) part of
the region. Hence, as the stellar magnitude generally increases
with stellar distance, we conduct a deeper survey in that part of
the sky (with a limiting magnitude R≲15.5mag, compared to
R≲14 mag in the remaining area); to increase the number den-
sity of data points, in particular at large distances. Despite this
choice that resulted in more sparse stellar polarization measure-
ments on the right part of the region, a hint of distant clouds
is still found there (see Sects. 4and 5). This indicates that even
with a shallower polarization survey distant clouds could still be
recovered – though see Sect. 5where we discuss the reliability
of cloud detection.
We have conducted our survey so that the entire region is
photometrically complete in the Rband up to 14 mag (shallow
survey), and up to 15.5 mag for the part of the region located
at l≳103.5◦(deep survey). The upper panel of Fig. 3shows
the locations of stars on the sky for both the shallow and the
deep survey. We used the R-band magnitude and sky coordinates
from the USNO-B1.0 catalog (Monet et al. 2003) to plan our
observations.
2.2. Observations and data reduction
Observations of optical polarization were conducted using the
1.3-m telescope situated at Skinakas Observatory1in Crete
(1750 meters above the sea level, 24◦53′57′′E, 35◦12′43′′N).
The telescope is equipped with the RoboPol polarimeter, which
1https://skinakas.physics.uoc.gr
A162, page 3 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
106±105±104±103±102±101±
24±
23±
22±
21±
20±
Galactic Longitude
Galactic Latitude
20
40
60
80
IHI (K kms°1)
106±105±104±103±102±101±
24±
23±
22±
21±
20±
Galactic Longitude
Galactic Latitude
60
80
100
120
140
IHI (K kms°1)
Tb[K]
Tb[K]
vLSR[km/s]
vLSR[km/s]
[Kkm/s]
[Kkm/s]
IHI[Kkm/s]
IHI[Kkm/s]
Fig. 2. HIvelocity data from the HI4PI survey in the sky area toward the region surveyed in starlight polarization. The top panels show the
brightness temperature as a function of the gas velocity measured in the LSR. The blue spectrum corresponds to the velocity spectrum averaged
over the full region. Two dominant peaks with velocities in the LVC (≈ − 1km s−1) and IVC (≈ − 50 km s−1) ranges are seen. The bottom panels
show the column density maps resulting from the integration of the velocity spectra in the ranges depicted by shaded regions marked in the top
panels in the LVC (left) and IVC (right) ranges. The maps reveal a high degree of complexity in morphological structures. The magenta outline
indicates the sky region of 3.8 sq. deg. for which starlight-polarization-based tomography is obtained in this work (see Sect. 3.4).
consists of two adjacent half-wave retarders, with their fast-
axes rotated by 67.5◦, relative to each other, followed by two
Wollaston prisms with orthogonal fast-axes (Ramaprakash et al.
2019). This setup splits each incident ray into four rays with
different polarization states on a single CCD, which provide
information on the q=Q/Iand u=U/Inormalized Stokes
parameters in the instrument’s reference frame (see Eq. (1) in
King et al. 2014) with only one exposure. In this way, the use of a
fixed instrument configuration eliminates random and systematic
errors resulting from changes in the sky, imperfect alignment,
and non-uniformity of rotating optical elements, as the instru-
ment has no moving parts aside from the filter wheel. To enhance
the signal-to-noise ratio (S/N), a special mask was placed in the
center of the telescope focal plane where systematic uncertain-
ties have been estimated to be lower than 0.1% in the degree of
polarization (Skalidis et al. 2018;Ramaprakash et al. 2019).
The observations were carried out star-by-star over three
observation seasons, from May 2019 to November 2022. Each
star was measured in the mask. Optical polarization measure-
ments were obtained for each star in the Johnsons–Cousins
Rband. For each star, the observation exposure time was
estimated on-the-fly so as to guarantee that photon-noise-driven
uncertainties fall below the estimated uncertainties from instru-
mental calibration (which turned out not always to be the case
as discussed below). Zero- and highly polarized polarimetric
standards were observed every observing night to monitor the
instrumental polarization and polarization angle zero-point
through time, and to estimate the corresponding uncertainties
as described in Blinov et al. (2021,2023). We obtained
instrumental uncertainties in both qand uat the level of 0.1%.
Data reduction was performed with the standard RoboPol
pipeline (King et al. 2014;Panopoulou et al. 2015;Blinov et al.
2021). For any given source, we produced the stacked image
of all observations and deduced the linear Stokes qand u
parameters and their photon-noise-driven uncertainties through
differential aperture photometry as outlined in Ramaprakash
et al. (2019). Then we removed the contribution from instru-
mental polarization and added in quadrature observational and
instrumental uncertainties as:
q=qmeasured −qinstr,(1)
σq=rσmeasured
q2+σinstr
q2,(2)
and similarly for Stokes u, where the superscripts “measured”
and “instr” refer to the values obtained from differential pho-
tometry and from estimation of the instrumental polarization,
respectively. The polarization data are given in the equato-
rial coordinate system and follow the IAU convention for the
polarization position angle (zero at north, increasing toward
east). In total, we obtained reliable optical polarization data for
1530 stars, spending approximately 153 telescope hours.
2.3. Cross-match with Gaia and quality cuts
For the purpose of this work, we complemented our new polar-
ization measurements with data from Panopoulou et al. (2019)
A162, page 4 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
102.0102.5103.0103.5104.0104.5105.0
Galactic Longitudes [◦]
21.5
22.0
22.5
23.0
Galactic Latitudes [◦]
R
< 14 [mag]
R
14 [mag]
R
unknown
2 4 6 8 10 12 14 16
Distance Modulus µ= 5 log10(d)−5
0
25
50
75
100
125
Occurence
R
< 14 [mag]
R
14 [mag]
R
unknown
Fig. 3. Sky distribution (top) and distance (modulus) distribution
obtained by inverting the parallax (bottom) of our stellar sample with
reliable parallax estimates. The sample is divided according the USNO-
BR-band magnitude. Blue, red and black correspond to stars with
R<14 (shallow survey), R≥14 (deep survey) and unknown values
due to identification mismatch. The histograms on the bottom panel are
stacked.
for 192 stars. As we need estimates of stellar parallaxes and
their corresponding uncertainties to perform the tomography
decomposition along distance, we cross-matched our sample
with polarization measurements with the Gaia DR3 catalog
(Gaia Collaboration 2023).
We used a cross-match radius of 5 arcsec around the USNO-
B1.0 coordinates of the targets. We chose to use the USNO-B1.0
coordinates as the astrometric accuracy of the RoboPol pipeline
output varies with position on the field of view and can be
limited to a few arcseconds in some cases as a result of dis-
tortions and the 4-spot pattern of the stars which are modeled
within the software2. For most sources, there was a unique Gaia
source found within the search radius, to which we assigned the
match. Some sources had multiple matches with the Gaia cata-
log, as a result of the proximity of the target star with another
star. The cross-match was not always straightforward mainly due
to the lack of precise astrometry in the USNO-B1.0 catalog.
This problem mostly happened when, in the USNO-B survey,
adjacent stars (approximately less than 2′′ spatial separation)
2The limited astrometric accuracy of RoboPol is primarily due to
uncertainties in the joint modeling of the 4-spot pattern and field dis-
tortions. New generation wide-field polarimeters such as the Wide Area
Linear Optical Polarimeters (e.g., Maharana et al. 2020) will enable
improved astrometric accuracy with wider coverage and the absence of
the 4-spot pattern.
Fig. 4. 2D distribution of the number of stars in our catalog in the par-
allax – S/N plane. For better visualization, a dozen stars with large
parallax values (ϖ) and or large parallax S/N are not included in this
plot.
were blended3. We visually inspected all cases where multiple
matches were found by comparing the raw RoboPol images with
the USNO-B1.0 and Gaia catalogs. We also checked for possi-
ble misidentifications by comparing the R- and G-magnitudes of
the sources after the cross-match, correcting for a couple cases
where a fainter Gaia star had been mistakenly associated with a
bright USNO-B1.0 star. Finally, we note that we observed several
faint stars, which exceed the photometric magnitude limits of our
survey (R>15.5mag), because they were close (around 1′′ dis-
tance) to the target stars. These stars happened to lie within the
mask of RoboPol during the observations. The resulting obtained
S/N in their degree of polarization is usually low. In some cases,
this proximity led to photometrically blended measurements,
which we disregarded in our analysis.
The final cross-match was successful for 1698 stars for which
we retrieved their Gaia identifier, their G-band photometric mag-
nitude, their parallax and corresponding uncertainty, and their
renormalized unit weight error (RUWE). The latter measures
the quality of the single-star model to account for astromet-
ric Gaia observations, and must be used as a quality criterion
to guarantee the reliability of the solution, and thus of the
parallax estimates. We used stars with RUWE <1.4(as rec-
ommended) to avoid unreliable measurements that could occur
because of blended sources for example (Lindegren et al. 2018,
2021). Among the successfully cross-matched stars, 24 stars do
not have parallax information and 226 stars have RUWE ≥1.4or
are unknown. Consequently, after applying the quality criterion
we have 1448 stars with both reliable optical polarization mea-
surements and reliable parallax estimates. This is the star sample
that we use for our tomography of the magnetized ISM.
Some properties of our final dataset of 1448 stars are given
in Figs. 3–5, which illustrate the non-homogeneous character of
our sample.
The different cuts in R-band magnitude used to design our
survey based on H Icomplexity are clearly seen in the sky dis-
tribution shown in the top panel of Fig. 3. The histogram in the
bottom panel of the same figure clearly shows that fainter stars
(R≥14 mag) generally have larger distances (obtained sim-
ply by taking the inverse of the parallax) than brighter ones.
3In the USNO-B1.0 catalog, blended objects have a single identifier
and the magnitude of the star represents the total photometry of both
stars.
A162, page 5 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
4 2 0 2
q[%]
4
2
0
2
u[%]
0
100
200
0 100 200
1.0 0.5 0.0 0.5 1.0
log10(σq)[%]
1.0
0.5
0.0
0.5
1.0
log10(σu)[%]
0
100
200
0 200
Fig. 5. Distribution of the polarization properties for the stellar sample
with a reliable parallax estimate. Distributions of the Stokes parameters
(q’s and u’s) and of the logarithm of their uncertainties (log10 (σq)and
log10(σu)) are shown in the top and bottom panels, respectively. The
horizontal and vertical dashed lines are used for visual reference. They
indicate the q=0and u=0loci in the top panel and indicate the values
of 0.1% and 1% in polarization uncertainties in the bottom panel.
Consequently, the density of stars, in particular at large dis-
tances, is much larger in the eastern half of the observed region
than in the western one. We also discuss this in Sect. 3.4.
Figure 4shows the distribution of parallax and parallax S/N
in our sample. Most of our targets (with RUWE <1.4) have
parallax S/N higher than 10, demonstrating the reliability of
our distance markers through the ISM. The distributions of the
Stokes parameters and the (total) uncertainties of the entire sam-
ple with reliable estimate of the parallax are shown in Fig. 5. It
is seen from the 2D distribution of the Stokes parameters that a
certain number of stars have low polarization while the majority
have polarization degree at the per cent level or higher4. The
uncertainty on the Stokes parameters is at the level of 0.19%
4The polarization as a function of distance is shown in Sect. 3.3 and
we study it starting from Sect. 3.5.
in both qand ufor a large fraction of our sample, and there-
fore dominated by systematic uncertainty from the instrument
calibration, while the photon noise contributes significantly for
a subset of measurements. We decided to keep all measurements
since we are confident in their uncertainty estimates.
2.4. Coordinate system conversion
The Stokes parameters are measured in the equatorial celestial
coordinate system. We construct our 3D map of the POS com-
ponent of the magnetic field in the Galactic coordinate system,
as this seems natural in the context of Galactic tomography.
Although we could perform the tomographic decomposition in
the equatorial coordinate system and then convert the result-
ing 3D map in the Galactic coordinate system, we prefer to
first convert the starlight polarization data and then perform our
analysis to obtain the tomography result directly in the Galactic
coordinate system. Because of the change of coordinate sys-
tem, the values of the polarization position angles, and therefore
of the Stokes parameters, change to account for the change of
the orientation of the meridian from one reference frame to the
other at the location of the stars. The change of coordinate sys-
tem thus involves a rotation of the polarization plane which
is implemented using the rotation matrix (e.g., Tegmark & de
Oliveira-Costa 2001)
R= cos(2ψR) sin(2ψR)
−sin(2ψR) cos(2ψR)!,(3)
where the rotation angle (ψR) is defined locally. It depends on the
celestial coordinates (right ascension and declination) of a star
(α⋆, δ⋆)and on the celestial coordinates of the North Galactic
pole (αNGP, δNGP ), and is given by (e.g., Hutsemékers 1998)
ψR=arctan2{cos(δNGP) sin(αNGP −α⋆),
sin(δNGP) cos(δ⋆)
−sin(δ⋆) cos(δNGP) cos(αNGP −α⋆)},(4)
where we use the two arguments arctangent function to place the
resulting angle in the correct trigonometric quadrant. The Stokes
parameters of a star in the Galactic coordinate system are thus
obtained from the Stokes parameters in equatorial coordinate
system through the following:
qgal
ugal!=R qeq
ueq!.(5)
Similarly, it can be shown (e.g., see Appendix A of Planck
Collaboration Int. XIX 2015) that the noise covariance matrix of
the Stokes parameters in the Galactic reference frame is obtained
from the rotation matrix and the noise covariance matrix in the
equatorial coordinate system through
Cgal =R CeqRT,(6)
where RTis the transpose of the rotation matrix. Introducing a=
cos(2ψR)and b=sin(2ψR), the elements of the noise covariance
matrix are thus obtained as follows:
Cgal
qq =a2Ceq
qq +2a b Ceq
qu +b2Ceq
uu
Cgal
qu =a2−b2Ceq
qu +a b Ceq
uu −Ceq
qq(7)
Cgal
uu =b2Ceq
qq −2a b Ceq
qu +a2Ceq
uu.
A162, page 6 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
Table 1. Polarization catalog (abbreviated).
Column name Unit Description
Gaia_source_ID −Source identifier in Gaia DR3
RA degrees Right ascension (2016)
Dec degrees Declination (2016)
q% Relative Stokes q
e_q % Uncertainty in q
u% Relative Stokes u
e_u % Uncertainty in u
p% Degree of polarization
e_p % Uncertainties in p
EVPA degrees Polarization angle
e_EVPA degrees Uncertainties in EVPA
date −Observation date
Gmag G-band magnitude
plx mas Parallax
e_plx mas Uncertainty in parallax
d_Maha −Post tomography outlier flag
usage_flag −
0: missing or unreliable parallax
1: used in tomography
2: outlier (sigma-clipping)
The noise covariance matrix of the Stokes parameters remains
unchanged from one coordinate system to the other only when
there is no noise covariance between q’s and u’s and when the
noise uncertainties in both q’s and u’s are equal. In any other
situation, the noise covariance matrix changes and, most notably,
a non-zero off-diagonal term arises simply due to the change
of coordinates. The use of the exact analytical formalism given
above allows us to avoid issues related to the polarization bias in
low S/N regime and to avoid the need for estimating the noise
covariance matrix through Monte Carlo treatment.
2.5. Stellar polarization catalog
The polarization catalog is made publicly available through
CDS. The columns contained in the catalog are described in
Table 1. The coordinates given are from Gaia DR3 at epoch
2016. We include all stellar polarization measurements, includ-
ing those that were identified as outliers and those that did not
return a reliable distance (due to no match with Gaia or high
RUWE values). We distinguish between sources used in the
tomographic reconstruction in the catalog with a usage flag.
The usage flag is assigned a value of 0 for stars that did not
have a reliable distance estimate, 1 for stars that are used in
the reconstruction, and 2 for stars identified as outliers (see
Sect. 3.3).
3. Method and data analysis
In this paper we aim to obtain a 3D tomographic decomposition
of the dusty magnetized ISM for an extended region of the sky,
from stellar measurements of optical polarization and distance.
As there is no currently available method to perform a 3D inver-
sion of an actual volume of stellar polarization data, we design
an analysis pipeline based on the LOS-inversion method that we
presented in Pelgrims et al. (2023) and which is implemented in
the BISP-1 (Bayesian Inference of Starlight Polarization in 1D)
Python code5.
5https://github.com/vpelgrims/Bisp_1
3.1. BISP-1
Our maximum-likelihood method, implemented in BISP-1,
makes it possible to reconstruct the dusty magnetized ISM along
a single LOS using starlight polarization and distance (paral-
lax) only. It assumes that the effect of multiple clouds along
the LOS is well approximated by the vector addition of the lin-
ear Stokes parameters induced by each cloud separately – which
holds for the typically low polarization levels in the diffuse ISM
(Martin 1974;Patat et al. 2010;Panopoulou et al. 2019). Relying
on the nested sampling method (Skilling 2004) implemented in
the code dynesty (Speagle 2020), our algorithm is able to deter-
mine the number of components (dust clouds) along the LOS
and to determine the distance and polarization properties of each
component using six parameters per component: cloud distance
(dC), cloud mean polarization (qC,uC), and three parameters to
characterize the covariance matrix Cint encoding the intrinsic
scatter of stellar polarizations arising as a result of ISM tur-
bulence. As such, the method makes it possible to recover the
stellar-polarization source field which directly informs on the
local orientation of the POS component of the magnetic field
(the position angle) and on the local degree of polarization. The
latter is related to the dust grain density, the dust grain polariza-
tion efficiency and on the angle made by the magnetic field lines
with the POS, as explained in (e.g., Pelgrims et al. 2023). In this
picture, it is implicitly assumed that the aspherical dust grains
always align their shortest axis with the local orientation of the
magnetic field, at least statistically. This assumption is expected
to be true for the diffuse ISM and agrees with current state of
dust alignment theory (e.g., Draine & Hensley 2021;Hensley &
Draine 2021).
The model that we use to reconstruct the dusty magnetized
ISM assumes that the dust clouds can be represented as thin lay-
ers distributed along the sightline. That is, we consider that the
typical extent of dust clouds along the LOS is smaller than the
typical separation of stars or, in practice, smaller than the dis-
tance range spanned by the number of stars needed to allow for
the detection of a cloud given the amplitude of the polarization
it induces and all sources of scatter in the polarization data. We
expect this assumption to hold at high and intermediate Galactic
latitudes and for clouds of the cold neutral medium and molec-
ular clouds (Heiles 1976;Zucker et al. 2021;Marchal & Martin
2023). It is worth noting that the thin-layer approximation also
seems to hold in the denser regions of the ISM, at least for some
sightlines (Doi et al. 2023).
Our method (BISP-1) further assumes that all stellar mea-
surements trace the dusty magnetized ISM only (as opposed to
having intrinsic polarization), and it does not implement spa-
tial variation in the POS. To circumvent these limitations while
benefiting from the strengths of our LOS-inversion method,
we embed it in a multistep process to form our 3D inversion
pipeline, as described below.
3.2. Design of the 3D inversion pipeline
The broad outline for the workflow of our 3D inversion pipeline
is illustrated in the flowchart given in Fig. 6. From the origi-
nal sample of stars in the extended region, we first identify stars
that likely trace the magnetized ISM (as opposed to the intrin-
sically polarized candidates). Then, for a set of sightlines that
span the observed region, we create two sets of overlapping sub-
samples centered on each LOS, as detailed in Sect. 3.4. The first
is meant to capture ISM properties at small distances despite the
low stellar density; the second corresponds to the highest angular
A162, page 7 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
Stellar catalog
Purity
Outliers
ISM probes
Sky sampling and
Beam definition
Beam samples
(hybrid, conical)
3D-map making
Differential quantities in 3D
Validation and
by-products
LOS-based inversion in
hybrid beams
(uninformed priors)
Best model per LOS
LOS-based inversion in
conical beams
(informed priors)
Converting posteriors
to priors
Bayesian inference
Fig. 6. Flowchart for the 3D inversion pipeline, from the stellar cata-
log with reliable parallax estimates (top) to the 3D maps of differential
quantities (bottom).
resolution we can achieve. We then perform the LOS inversion
for each LOS; first on the subsamples which connect different
sightlines at small distances, and then at higher resolution tak-
ing into account the information obtained in the previous step.
Finally, we produce the 3D map of the magnetized ISM from the
posterior distributions obtained for each LOS individually. Each
step is detailed in this section and in Sect. 4, and illustrated by
applying the method to our dataset.
3.3. Identification of outliers
Some stars may exhibit intrinsic polarization, possibly due to
the existence of a circumstellar disk or other asymmetries in
the object (e.g., Fadeyev 2007;Cotton et al. 2016;Gontcharov
& Mosenkov 2019). These stars usually show either a higher
degree of polarization or an unrelated polarization position angle
as compared to their neighboring stars, and sometimes they dis-
play both. As BISP-1 assumes that all starlight-polarization
data points trace the dusty magnetized ISM, the first step is to
remove from the original sample all stars for which their polar-
ization is unlikely to be of only interstellar origin. This includes
intrinsically polarized stars and also any other outliers.
To identify these stars, we adopt recursive sigma-clipping in
groups of neighboring stars. Based on the sky coordinates and
distance estimate of every star, we identify the group of Nneigh-
bors of each star in 3D space using their heliocentric Cartesian
coordinates, ignoring the distance uncertainties from parallax
measurements. We estimate the weighted-average Stokes param-
eters (ˆ
¯
s) and associated covariance ( ˆ
C) in the (q,u)plane from
the neighbors, and then compute the Mahalanobis distance of
the polarization of the central star (s⋆) as compared to the 2D
bivariate distribution from the neighbors as
d⋆
Maha =q(s⋆−ˆ
¯
s)†Σ⋆−1(s⋆−ˆ
¯
s),(8)
where the noise covariance matrix of the polarization of the cen-
tral star (C⋆
obs) is added to the covariance matrix from the bivari-
ate distribution (Σ⋆=C⋆
obs +ˆ
C) to compute the Mahalanobis
distance. If the Mahalanobis distance exceeds some threshold,
the probability that the polarization of the central star is drawn
from the same parent distribution as the neighbors (assumed to
be representative of the magnetized ISM) is small and, therefore,
the central star must be identified as an outlier. Repeating this
process for all the stars in the original sample and running the
whole process until no additional outliers are identified yields a
catalog of outliers and a “clean” sample of stars whose polar-
ization likely traces the dusty magnetized ISM. Only the clean
sample of stars is then considered to infer the dusty magnetized
ISM. The exact list of outliers depends on our specific choice
for the size of the neighbor groups and the adopted threshold
in significance level. In addition, the sensitivity of the “clean
sample” membership to these parameters should also ideally be
tested against distance uncertainties. However, if we can recover
the 3D dusty magnetized ISM from the clean sample of stars,
we can test a posteriori the hypothesis that the polarization of
a given star is given by the magnetized ISM only, as we do in
Sect. 5.5. This has the potential to lead to a somewhat more
robust list of candidate targets for being intrinsically polarized
stars, or at least outliers, and can trigger follow-up observations.
However, we notice that caution has to be made for the choice
of the significance-level threshold. A too strong selection crite-
rion would discard stars that merely pick up fluctuations of the
magnetized ISM. This would subsequently lead to an underesti-
mation of the turbulence-induced intrinsic scatter which we want
to avoid. This point is further discussed in Sect. 5.5.
We apply the recursive sigma-clipping approach to the full
sample of stars discussed in Sect. 2.3. We adopt a size of N=30
to build the groups of neighbors and we choose to flag every
star as an outlier if its polarization shows a probability of less
than 1% for it to be drawn from the same parent distribution
as the neighbors. In our case, the number of outliers remained
constant after three iterations. Using these parameters, 18 stars
out of 1448 are identified as outlier candidates. This represents
a fraction of 1.2% of our full sample. In Fig. 7, we show the
stellar polarization data plotted against the distance modulus
(µ=5 log10(d)−5) and highlight the outliers. These outliers
are not considered in the analysis discussed below.
3.4. Definition of subsamples
BISP-1 assumes that all the stars in a sample lie along a nar-
row, one dimensional beam. It consequently returns the structure
of the dusty magnetized ISM averaged in the POS over the
sky region spanned by the input sample, which we refer to our
“beam”. The method also relies on the dust-layer model that we
A162, page 8 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
Fig. 7. Stellar relative Stokes parameters (q,u) versus the distance mod-
ulus for the full sample. The qVand uV(in the Equatorial coordinate
system) are shown by green circles and blue diamonds, respectively.
The vertical error bars indicate the uncertainties (noise and systematic)
on the Stokes parameters and the horizontal error bars represent the
uncertainties on the star distance modulus converted from the 1σpar-
allax uncertainties. Outliers identified from the iterative sigma-clipping
approach in groups of nearest neighbors are highlighted with red-filled
circles for qand purple-filled squares for u. To facilitate the visual clar-
ity of the plot we restrict the range of (q,u) and µvalues. Ten stars have
µ < 5and 30 stars have µ > 14.
have introduced (Pelgrims et al. 2023) and which we expect to
hold true as long as the magnetic field and dust density do not
vary appreciably in our beam. The validity of these assumptions
is tested by the data in the following. Depending on the geometry
of the volume filled by the star sample, the averaging scale in the
POS may depend on distance. For example, if a conical geome-
try is chosen to define the star subsamples, the averaging scale at
small distances is much smaller than the one at large distances.
A cylindrical geometry would keep the averaging scale constant.
The number density of constraints (i.e., of stars) as a function
of distance also depends on the chosen geometry of the beam,
the specific size of the volume encompassed by the data, and the
actual 3D spatial distribution of stars. Hence, there is a trade-off
between having a sufficient number of data points to constrain
the model and the achieved resolution. The resolution also needs
to be sufficiently good to minimize the POS variations of ISM
properties in the beam and thus ensure that the intrinsic scatter,
which we fit for, is not dominated by POS variations of the mean
ISM properties.
We find that running the BISP-1 decomposition solely on
subsamples defined according to conical beams is not appropri-
ate. The reason is that the spatial distribution of nearby stars
is sparse and that, unless a very large opening angle is chosen,
nearby clouds may be missed or their parameters loosely deter-
mined due to the absence of a sufficient number of constraints
in the required range of distances. If a large opening angle is
chosen in order to be able to capture those nearby clouds, then
the signal of any faraway structure would be averaged out and
therefore likely missed. Alternatively, a cylindrical geometry for
our beam would lead to a constant resolution in the transverse
direction to the distance axis. However, at high and intermediate
Galactic latitudes the number density of stars decreases substan-
tially after approximately 1 kpc. Faraway clouds would then be
missed, due to the sparsity of data points, unless a large cylinder
radius is chosen. A cylindrical geometry would also imply that
any detail at small distances would be averaged out over large
0.0
0.5
1.0
1.5
2.0
Angular aperture [◦]
hyb. cone cyl.
0 500 1000 1500 2000 2500
Distance to observer [pc]
0.0
2.5
5.0
7.5
10.0
Transverse distance [pc]
hyb. cone cyl.
Fig. 8. Beam sizes as a function of distance from the observer. Top:
angular aperture radius of our hybrid beam (continuous gray) resulting
from the combination of a conical beam (red dashed) and cylindri-
cal beam (dot-dashed). Bottom: extent of the beams in the transverse
direction to the LOS. Same line convention as in the top panel. In this
example the cylinder radius is fixed at 2.5 pc and the angular radius of
the cone at 13.74 arcmin. For the hybrid beams, the cylindrical geometry
prevails at distances smaller than ≈630 pc, while the conical geometry
prevails at larger distances.
angular scales. Thus, a cylindrical geometry alone is also not
appropriate to define our beam samples.
To guarantee a good angular resolution at all distances, and
to avoid missing clouds or only placing loose constraints on their
distance and polarization properties, we adopt a two-step hier-
archical decomposition process. The first step is performed on
samples defined according to a hybrid geometry for our beam:
the beam follows a cylindrical geometry at low distances and
a conical geometry at large distances. For the second step, the
star samples are defined in a beam with conical geometry only.
The idea is to perform the BISP-1 decomposition in the sec-
ond step using priors defined from the posterior distributions
obtained in the first step. In this way, constraints on distances
and polarization properties of nearby clouds are obtained from
the decomposition on hybrid-beam samples (with larger num-
ber of stars at lower distances) while good angular resolution is
achieved, even at low distances, from the conical-beam samples.
We use the same opening angle for the conical part of the hybrid
beam and the purely conical beam so that they match at large
distances.
In Fig. 8, we show the distance-dependent angular radius of
our beams along with the corresponding physical scale in the
POS for the specific choice of opening angle and cylinder radius
that we use. Our hybrid beam centered on a given LOS has an
angular size that depends on distance. Any star is considered as
part of a given subsample if its angular separation to the LOS is
lower than the maximum between the opening angle of the cone
and the distance-dependent angular size of the cylinder evaluated
at the star distance. For the conical beam, the angular separation
cutoff is constant.
A162, page 9 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
To cover an extended region of the sky, such as the one we
observed, we adopt a moving-window strategy. That is, we sam-
ple the observed region with a large number of sightlines. For
each LOS, we define a subsample of stars according to our choice
for the beam geometries. The star samples of neighboring sight-
lines are overlapping and, in the first step, the overlap extends
to larger angular scale for nearby stars than that for distant stars.
This strategy ensures a continuous scan of the observed region
and also implies that the results of the LOS decomposition of
neighboring sightlines will not be independent.
For our stellar polarization measurements spanning a region
of about four square degrees, we sample the sky according to a
HEALPix tessellation (Górski et al. 2005;Zonca et al. 2019) with
the resolution parameter Nside =512. Each pixel center defines
a LOS which we take as the symmetry axis of our beam. The
angular separation of neighboring sightlines is about 6.9 arcmin.
To define our hybrid beam, we adopt a value of 2.5 pc for the
cylinder radius and an angular radius of 13.74 arcmin for the
cone. The conical beam has the same value of 13.74 arcmin for
the angular radius. Each conical beam spans a sky area of about
0.16 square degrees. We choose the angular radius of the conical
beam so that (i) at least 20 stars are contained within each LOS
and (ii) every pixel has at least two neighbors to ensure conti-
nuity at the edges of the map. The sky region is covered by an
ensemble of 287 sightlines. Panels a–c of Fig. 9show the num-
ber of stars per bins of distance modulus for all the 287 sightlines
covering the regions of the hybrid beams, of the conical beams,
and of the cylindrical beams, respectively. We see that the den-
sity of stars is generally very low at distances smaller than 300 pc
for the conical beams and at distances larger than 1 kpc for the
cylindrical beams. This justifies the use of the hybrid beams in
extracting information of nearby and distant ISM clouds. With
our choice of beam parameters, the geometry of the hybrid beam
transitions from cylindrical to conical at a distance of about
630 pc (µ≈9). Figure 10 shows the number of stars per beam
samples for the conical beam geometry. The higher density for
points at l≳103.5◦is due to the aforementioned survey strategy
(Sect. 2.1).
3.5. LOS decomposition of the dusty magnetized ISM
Once the star subsamples are defined for the entire observed
region, we independently apply the BISP-1 code to each sub-
sample, first on the hybrid-beam samples and then on the
conical-beam samples.
3.5.1. Step 1 – LOS decomposition in hybrid beams
For each LOS, we test the layer-model for one to four clouds
along the distance with uniform priors on all the model param-
eters. The limits of the priors on cloud parallaxes are set so
that the minimum allowed distance is 20 pc and the maximum
allowed distance of the farthest cloud is the minimum between
3.5 kpc and the maximum distance of the stars in the analyzed
sample. The upper distance limit may thus vary from one LOS
to another. The value of 3.5 kpc (corresponding to a distance
modulus of µ≈12.7) originates from the fact that beyond this
distance our data sample generally become very scarce as also
seen in Fig. 3. In addition, we make sure that there are at least
five stars between clouds. This is required by the BISP-1 code
(Pelgrims et al. 2023).
The uniform priors for the cloud polarization parameters are
set as follows. For the mean polarization parameters (qCand
uC), we compute the maximum of the absolute values of both
(a)
30 50 100 200 300 500 1000 2000
Distance [pc]
2 4 6 8 10 12
µDistance modulus
0
5
10
15
20
Number of stars
50
100
150
200
250
# of LOS
(b)
30 50 100 200 300 500 1000 2000
Distance [pc]
2 4 6 8 10 12
µDistance modulus
0
5
10
15
20
Number of stars
50
100
150
200
250
# of LOS
(c)
30 50 100 200 300 500 1000 2000
Distance [pc]
2 4 6 8 10 12
µDistance modulus
0
5
10
15
20
Number of stars
50
100
150
200
250
# of LOS
Fig. 9. Distribution of the number of stars in beam samples. (a) Number
of stars per bin of distance modulus for the 287 sightlines sampling the
observed regions when the beam geometry is hybrid. The colors indicate
the number of sightlines for which a specific number of stars in a given
distance bin is observed. The red-continuous line indicates the median
number of stars per bin of distance for the entire set of sightlines. (b)
Same as for (a) but for the conical-beam geometry. (c) Same as for (a)
but for the cylindrical-beam geometry.
the stellar Stokes parameters in the star samples. This value is
used to define the limits of the top-hat priors on both qCand
uC. This definition of the prior limits, while not fully general,
is valid in our case as we know that the extinction is domi-
nated by nearby components as shown for example by comparing
Planck dust column-density map and 3D star extinction maps
(e.g., O’Callaghan et al. 2023). The diagonal elements of the
intrinsic-scatter covariance matrix are positive definite and we
require Cint,qq,Cint,uu ∈[0,10−4]. This is a very loose range for
possible values as it allows for a spread in stellar Stokes param-
eter due to turbulence as high as 1% in degree of polarization.
The off-diagonal element Cint,qu is initially bounded by ±10−4
A162, page 10 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
102.0102.5103.0103.5104.0104.5105.0
Galactic Longitude [◦]
21.5
22.0
22.5
23.0
Galactic Latitude [◦]
20
40
60
80
100
120
140
Nstar
Fig. 10. Number of stars per conical beam. The blue circle in the top
right corner indicates the size of our conical beam with an angular
radius of 13.74 arcmin. The magenta contour surrounds all HEALPix
pixels whose center coincides with the center of the beams defining our
star subsamples.
but is further constrained to verify |Cint,qu|<(Cint,qq Cint,uu)1/2
inside BISP-1 to ensure that the covariance matrix is invertible.
We use BISP-1 to run the nested sampling experiment using
1000 live points and sample the parameter space until an uncer-
tainty of around 0.01 is achieved on the log of the model
evidence. Typically, this requires 15 000–80000 nested sampling
iterations, corresponding to several hundred million calls of the
log-likelihood function of each model.
For each of the tested models, BISP-1 leads to estimates of
the log of the evidence, the maximum log-likelihood value and to
the estimated posterior distributions of all model parameters. As
already demonstrated and pointed out in Pelgrims et al. (2023),
the posterior distributions of the cloud parallaxes have generally
complex shapes, mainly due to the sparse and uneven distribu-
tion of stars along distance, and might also be piled up on the far
edge of the prior domains. The latter case happens whenever the
data are not enough to determine the cloud properties or if there
is no cloud to be found.
To deal with this peculiarity, we follow the same idea as
in Pelgrims et al. (2023) and rely on the following automated
analysis of the marginalized posterior distribution on the cloud
parallax. The idea is that the value at maximum-likelihood must
belong to (one of) the main mode(s) of the marginalized poste-
rior distribution on the cloud parallax and that this mode must
not be piled-up on the lower limit of the prior. If both criteria are
verified, then we qualify the fit as valid. We then select from the
6×NCdimensional posterior distribution (where NCis the num-
ber of layer in the model) all samples that belong to this mode
for the remainder of the analysis. In practice, we analyze the
marginalized posterior distribution using the peak-finder algo-
rithm find_peaks of the SciPy Python library (Virtanen et al.
2020) which identifies all local maxima through simple com-
parison of neighboring values. We thus find local maxima of
the marginalized posterior distribution and the range of parallax
values corresponding to the extent of the corresponding peaks.
We compute the fraction of the posterior distribution which cor-
responds to each peak and we consider that it is (one of) the
dominant peak(s) if this fraction exceeds the threshold of 30%.
Our results do not depend strongly on this choice.
We consider all solutions for which the posterior distribution
on the cloud parallax of the farthest cloud passes this selection
criterion to be valid. Any solution which does not satisfy this
criterion is discarded in this first step. Finally, for each tested
model j, we compute the Akaike Information Criterion (AIC) as
AICj=2M−2 log( ˆ
Lj),(9)
102.0102.5103.0103.5104.0104.5105.0
Galactic Longitude [◦]
21.5
22.0
22.5
23.0
Galactic Latitude [◦]
1
2
3
4
Number of clouds
Fig. 11. Map of the number of dust clouds identified at the end of Step 1
from the analysis of the hybrid-beam samples obtained for each beam
centered on the pixel centers.
based on the estimated maximum likelihood ( ˆ
Lj) and the num-
ber of parameters in the model (M). We then compare the model
performances by computing the probability
Pj|{m}=exp (min
m{AICm} − AICj)/2,(10)
that, among the tested models {m}, each model jis actually the
one that minimizes the loss of information (Boisbunon et al.
2014), as in Pelgrims et al. (2023). The best model is the one
with Pj|{m}=1. We note that, according to the selection proce-
dure explained above, we are giving some chances to solutions
with large number of clouds (up to four) to be considered as the
best model while the data itself might not be enough to place
strong constraints on the farthest cloud. However, we checked all
the solutions and found no evidence of spurious detections that
might have resulted from bad fits.
A map of the number of clouds per LOS determined in this
first step from the analysis of the hybrid-beam samples is shown
in Fig. 11. For most of the surveyed area, we find evidence of two
clouds per LOS. The maximum number of clouds is three, and a
few sightlines show only one cloud.
3.5.2. From posteriors to priors
We now aim at informing the maximum-likelihood analysis of
data in conical beams from the results of the analysis of hybrid
beam samples. To do so, we would ideally want to resample the
posterior distributions obtained in Step 1, at least for the cloud-
parallax parameters. This is however currently not possible using
BISP-1 which does not make it possible to input prior samples as
it relies on the Python nested sampling software dynesty which
does not have this feature. For the purpose of the this work, we
thus work around this limitation as explained below and leave
the development of improved solutions for the future. We choose
to ignore the possible correlations between model parameters
and only extract (and propagate) information on cloud parallax
and cloud mean-polarization properties from the marginalized
posterior distributions of these parameters. By ignoring possi-
ble correlations, we know we are losing part of the information
gained in Step 1.
For each of the valid models selected in Step 1 and their cor-
responding estimated (marginalized) posteriors, we define priors
on the parameters of the models to be constrained by the data
in Step 2. For the cases discussed above where the marginalized
posterior distribution on the cloud parallax of the farthest cloud
is multimodal, only the “valid” subset of the posterior samples is
used to define the priors.
A162, page 11 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
Neither a Gaussian nor a top-hat distribution generally rep-
resent the posterior distributions of the cloud parallax well;
however, we find that in our case, they are sufficiently effective
to impose constraints on the cloud’s parallax. We choose to con-
struct Gaussian or top-hat priors on cloud parallaxes from the
estimated posteriors as follows. We denote {ϖC}the posterior
sample of parallax for a given cloud. For a given cloud, the esti-
mated mean ( ˆ
¯ϖC)and standard deviation ( ˆσϖC) of the Gaussian
prior, and the estimated minimum ( ˆϖmin
C) and maximum ( ˆϖmax
C)
limits of the top-hat prior are obtained from percentiles of the
posterior sample of the cloud parallax as
ˆ
¯ϖC={ϖC}50
ˆσϖC=({ϖC}84 − {ϖC}16 )/2
ˆϖmin
C={ϖC}0.01
ˆϖmax
C={ϖC}99.99,(11)
where {ϖC}Xdenotes the X-th percentile of the sample distri-
bution. The prior distributions on the cloud parallax are then
defined as follows:
PG(ϖC)=1
√2πˆσϖC
exp (−(ϖC−ˆ
¯ϖC)2
2 ˆσ2
ϖC)(12)
PH(ϖC)=
1
ˆϖmax
C−ˆϖmin
C
if ϖC∈[ ˆϖmin
C,ˆϖmax
C]
0otherwise,(13)
for the Gaussian and the top-hat respectively.
For our data, we find that, even for the cases where the three-
layer model is favored, only the two nearest clouds are located
in the distance range where the beam geometry is cylindrical.
Therefore, we do not need to update the priors of the most dis-
tant cloud parameters and simply left it as is for Step 1. That is,
we inform the modeling of the data in conical beams (Step 2)
from the fit in hybrid beams only for clouds that are found in
the distance range where the hybrid and conical beams differ. If
ˆ
¯ϖC/ˆσϖC>2and if there are at least five stars with parallaxes
lower than ˆϖC−ˆσϖC, then we adopt the Gaussian prior. In all
other cases, we adopt a top-hat prior for this parameter. This is
to avoid unphysical negative parallax values and to allow for the
possibility of a farther away cloud.
Defining the priors on the cloud mean polarization from
the corresponding posteriors is easier than for the parallax as
the marginalized posteriors are generally close to Gaussian. To
build the Gaussian priors on the mean Stokes parameters of the
cloud, we consider the means of the posterior distribution sam-
ples and both the standard deviations of those samples (taken
individually) and the mean of the posterior distribution of the
corresponding element of the intrinsic-scatter covariance matrix.
If ˆ
¯qCand ˆσqCare the estimated mean and standard deviation of
the posterior distributions on qC(the mean qStokes parameter
of the cloud), and if ˆ
Cint,qq is the estimated mean of the posterior
on the qq element of the covariance matrix of the intrinsic scat-
ter, then the Gaussian prior on qCis defined with a mean of ˆ
¯qC
and a standard deviation of ( ˆσ2
qC
+ˆ
Cint,qq)1/2. The same applies
for uC. We include the terms from the intrinsic scatter in the def-
inition of the priors on the mean Stokes parameters to account
for the fact that, in the conical beams (higher angular resolu-
tion at lower distance), the “mean” polarization may, in general,
pick a local fluctuation of the turbulent magnetized ISM defined
at larger angular scales in the hybrid beams. Accordingly, we
do not modify the priors for the elements of the intrinsic-scatter
covariance matrix because, without further assumptions, we do
102.0102.5103.0103.5104.0104.5105.0
Galactic Longitude [◦]
21.5
22.0
22.5
23.0
Galactic Latitude [◦]
1
2
3
4
Number of clouds
Fig. 12. Map of the number of dust clouds along the sightlines identified
at the end of Step 2 in the conical beam centered on the pixel centers.
The blue circle in the top-right corner indicates the extent of the conical
beam.
not know how the intrinsic scatter evolves as a function of physi-
cal and angular scales. Consequently, we keep the same uniform
priors defined above throughout the analysis.
3.5.3. Step 2 – LOS decomposition in conical beams
For each LOS, we have obtained a LOS decomposition in the
hybrid beam, we have selected the best model, and we have
defined priors from the estimated posterior distributions of this
model. In this second step, we use BISP-1 to decompose the
starlight polarization along distance for the star samples defined
in the conical beams using the information gained above. For
each LOS, we test the models with two, three, and four layers and
use the informed priors for the two nearest clouds only. For the
cases where the one-layer model was favored in Step 1, we test
this model and a higher number of layers using the informed pri-
ors only on the nearest layer. In all cases we force the additional
cloud(s) to be located at larger distances.
We use BISP-1 to run the nested sampling experiment using
1000 live points and sample the parameter space until an uncer-
tainty of around 0.01 is achieved on the log of the model
evidence. The required number of nested-sampling iterations
was similar or higher than in Step 1.
When all models have been evaluated, we proceed as in
Step 1 to inspect the solutions based on the marginalized
posterior on the cloud parallax and keep only the valid recon-
structions, and to select the best-model which minimizes the
loss of information based on the AIC criterion. Figure 12 shows
the number of clouds along each of the sightlines sampling the
observed regions, as in Fig. 11. Again, the number of clouds
ranges from one to three. A large fraction of the sightlines in
this sky area intersects two clouds along the LOS. This map is
further discussed in the Sect. 4.
The solutions of neighboring sightlines are not independent.
First, the samples of stars from neighboring sightlines overlap,
even for the conical beams, because we explicitly decided to
oversample the observed region with a large number of sight-
lines and the beam size is more than twice as large as the
angular separation between adjacent sightlines. Second, the solu-
tions obtained at the end of the second step are constrained by
the larger angular scales at short distances since the priors are
defined from the results obtained from the hybrid beam samples.
As a result, we obtain the posterior distributions for our decom-
position of the POS component of the magnetic field in dusty
regions for a simply connected 3D volume without gaps.
A162, page 12 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
3.6. Validation
To validate the result of our tomographic reconstruction, we rely
on inspection of the significance of the polarization residuals of
all the stars on which the reconstruction is based. We consider
that we can be confident in our reconstruction if only a few data
points show significant deviations from our model prediction,
and if these data points are not clustered in the 3D volume. If, on
the contrary, too large a fraction of stars show significant residu-
als, or if they are clustered in 3D space, this would indicate flaws
or limitations in our tomographic reconstruction. In particular,
the clustering of significant residuals would indicate the pres-
ence of features in the data that the model is not able to account
for within uncertainties.
To estimate the significance of the residuals for all the stars
taken individually, we compare the individual stellar observa-
tional data to the modeled data from the tomography results. We
proceed as follows. Since we do not have reconstruction for each
LOS toward each star individually but only for each LOS toward
the center of HEALPix pixels at the center of our sample beams,
we first identify the pixel in which the star falls in. We then read
the posterior samples of the best-model decomposition obtained
earlier for the corresponding beam.
By resampling the posterior distribution, we can then eval-
uate the expected polarization of the star at its given distance.
According to our layer model (Pelgrims et al. 2023), the ISM
contribution to a star polarization at distance d⋆toward a given
LOS is thus described by the stochastic model:
m⋆=G2(¯
m,Cint),(14)
with m⋆=( ˆq⋆ˆu⋆)†and where G2(¯
m,Cint)denotes a bivariate
normal distribution with mean ¯
mand covariance matrix Cint . The
values of the cloud’s mean polarization and covariance matrix
are obtained from the posterior samples and added, cloud wise,
up to the distance of the star. The noise covariance matrix of
the polarization measurements of the star can then be added
to the intrinsic-scatter covariance matrix (Σ⋆=C⋆
obs +Cint) and
the observation (s⋆=(q⋆u⋆)†) compared to the value predicted
from our 3D reconstruction using the Mahalanobis distance:
d⋆
Maha =q(s⋆−¯
m)†Σ⋆−1(s⋆−¯
m).(15)
For each individual star, a single value of d⋆
Maha is obtained for
each sample of the posteriors. By resampling the posteriors of
the model parameters and resampling the parallax distribution
of the star, we obtain a distribution of d⋆
Maha. This distribution
informs us on the likelihood that the observed polarization is due
to the dusty magnetized ISM given our 3D reconstruction. This
estimate accounts for both the turbulence-induced scatter and
observational noise in polarization and parallax. Large values of
d⋆
Maha indicate significant residuals. In practice, a star is identified
as an outlier (i.e., a star whose polarization is poorly predicted
by our LOS model and thus showing significant residuals) if the
median of its Mahalanobis-distance distribution is larger than a
given threshold value. In 2D, the square of the Mahalanobis dis-
tances is expected to follow a χ2distribution with two degrees of
freedom. Accordingly, we can compute the threshold value cor-
responding to a given probability (Pth) to observe by chance a
Mahalanobis distance greater than that. In 2D the threshold value
is obtained as dth
Maha =p−2 log(Pth), where log is the natural
logarithm.
Having obtained the Mahalanobis-distance distributions for
all the stars in our sample, we look at the locations of stars with
102.0102.5103.0103.5104.0104.5105.0
Galactic Longitude [ ]
21.5
22.0
22.5
23.0
Galactic Latitude [ ]
pv
0.05
pv
0.01
Fig. 13. Sky map of residual significance. All the stars in the sample
from which we performed our tomographic inversion are represented.
Transparent gray dots (gray crosses) show stars that do (do not) fall
in an HEALPix pixel for which we have tomography data. Blue and
green circles show the stars for which the median of their distribution
of Mahalanobis distances exceed a threshold values corresponding to
the p-value indicated in the legend (see text). The lower the p-value, the
more significant the residuals.
significant residuals in the 3D space. We show in Fig. 13 the sky
locations of the stars for which the median of their Mahalanobis-
distance distributions exceed threshold values corresponding to
the probabilities of 5% and 1% of observing by chance greater
values than that. The stars with significant residuals correspond-
ing to lower probability of being compatible with the model are
highlighted with blue and green circles, respectively.
The fractions of our star sample which show p-values lower
than the 5% and 1% threshold are 3% and 0.3%, respectively,
and no star shows a p-value lower than 0.02%. These values are
further discussed in Sect. 5.5. We see from Fig. 13 that the stars
with significant residuals do not cluster in particular places of the
sky. Visually it seems that there is an excess of significant resid-
uals in the eastern half of the observed region but this is due to
the larger stellar density in that region. However, we observed
(not shown on the figure) mild preference for the stars with sig-
nificant residuals to be located at large distance. This is likely an
effect of the low number density of stars at large distance and of
the limited angular size of our sample beam as, for example, two
of the four stars with their p-value lower than 1% are the farthest
stars in their beam. Though, as the residuals are not very signif-
icant and that this trend is mild, we consider that this effect has
no substantial effect on the present results.
This validation test makes it possible to verify the reliabil-
ity of the assumptions underlying our modeling by looking at
the spatial distribution of the polarization residuals. On the one
hand, the violation of the thin-layer assumption would lead to
systematic increases of the polarization residuals close to the
reconstructed cloud distances. And, on the other hand, any sig-
nificant variation of the polarization signal in the POS within the
beam would lead to gradients in the residuals. We searched for
such possible trends and could not find any. This suggests that
our working assumptions are appropriate to model our dataset.
4. Results
The output of the 3D-inversion pipeline developed in the pre-
vious section consists of an ensemble of LOS decomposition of
stellar polarization for non-independent samples. For each beam,
we identified the number of components (shown in Fig. 12) and
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Pelgrims, V., et al.: A&A, 684, A162 (2024)
0 500 1000 1500 2000 2500
Cloud distance [pc]
100
101
102
Occurence
ˆ
¯
d
Fig. 14. Histogram of estimated mean cloud distances ( ˆ
¯
dC). Distribution
of mean posterior cloud distances ( ˆ
¯
dC) for the best fit models determined
for all sightlines. The vertical lines indicate the limits to define the dis-
tance bins used in Sect. 4.1.
obtained the posterior distributions on their distances (cloud par-
allaxes, ϖC) and polarization properties (qC,uC,Cint). From
these, we can now infer the properties of the POS component
of the magnetic field in dusty regions. We examine the results
at the mean of the posterior distribution in Sect. 4.1 to infer the
main features of the dusty magnetized ISM in the observed 3D
volume. Then we build 3D maps of the posterior distributions in
Sect. 4.2 and visualize the main output in Sect. 4.3. The discus-
sion, interpretation, and validation of the results are provided in
Sect. 5along with caveats of the method.
4.1. Basic exploration of the output
From the posterior distributions, we extract the estimated mean
values for the cloud parallax ( ˆ
¯ϖC) and mean polarization ( ˆ
¯qC
and ˆ
¯uC) for each LOS. In Fig. 14 we show the histogram of
the (estimated) mean cloud distance ( ˆ
¯
dC=1/ˆ
¯ϖC) for all clouds
and all analyzed sightlines. This histogram shows two main
separated peaks centered on 62 pc and 380 pc. A number of
clouds are also found at distances larger than 1 kpc. Relying on
this observation, we divide the distance axis in three bins for
nearby (dC≤265 pc), intermediate (dC∈[265,650] pc), and
distant (dC>650 pc) clouds. For each distance bin, we gener-
ate maps of the cloud mean distance and mean polarization as
shown in Figs. 15 and 16, respectively. The polarization maps
are obtained by introducing the mean degree of polarization and
mean polarization angle from the mean Stokes parameters as
ˆ
¯pC=(ˆ
¯qC+ˆ
¯uC)1/2and ˆ
¯
ψC=0.5 arctan2( ˆ
¯uC,ˆ
¯qC)where we make
use of the two-argument arctangent function to handle the π-
ambiguity of the arctangent. In these maps, empty pixels of the
observed regions mark the absence of detected clouds with mean
distance in the specific range of distances. We find that all sight-
lines intersect a cloud in the intermediate range of distances. A
large fraction of the sightlines intersect both the nearby- and
intermediate-distance components. As seen in Figs. 15 and 16,
these components show a high degree of regularity in their dis-
tances and mean polarization properties, suggesting that each of
these two components is related to physical entities (real inter-
stellar clouds). The distances of detected clouds in the large
distance bin are more scattered but we notice several clusterings
of cloud detections in the 3D space for components with similar
polarization properties.
Over the observed region of the sky, the mean estimated dis-
tance to the nearby cloud ranges from 30 pc to 170 pc and has
a median of 58 pc. Its mean estimated degree of polarization
102.0102.5103.0103.5104.0104.5105.0
Galactic Longitude [◦]
21.5
22.0
22.5
23.0
Galactic Latitude [◦]
800
1000
1200
1400
1600
1800
2000
2200
dCdistant [pc]
102.0102.5103.0103.5104.0104.5105.0
Galactic Longitude [◦]
21.5
22.0
22.5
23.0
Galactic Latitude [◦]
300
350
400
450
500
550
600
650
dCintermediate [pc]
102.0102.5103.0103.5104.0104.5105.0
Galactic Longitude [◦]
21.5
22.0
22.5
23.0
Galactic Latitude [◦]
0
50
100
150
200
250
dCnearby [pc]
Fig. 15. Maps of the estimated mean cloud distances in the three bins
of distances identified from Fig. 14, from far away (top) to nearby (bot-
tom). The color scales span the full range of distance in each bin. Empty
pixels of the observed regions mark the absence of clouds in the specific
distance range. The magenta contour and the blue circle are as in Fig. 10.
ranges from 0.06% to 0.33% with a median value of 0.19%. The
estimated POS orientation of the magnetic field of this compo-
nent varies smoothly across the region. It ranges from 49.4◦to
126.9◦with a mean of about 80.5◦.
The mean estimated distance to the intermediate cloud
ranges from 354 pc to 448 pc with a median of 374 pc. Its mean
estimated degree of polarization shows spatial variation over the
observed region. It is also much higher than the nearby compo-
nent, spanning the range from 0.68% to 2.03% with a median
value of 1.45%. This is the dominant polarizing “screen” in the
region, the effect of which was already clearly seen in Fig. 7. The
mean estimated polarization angle of this component is nearly
uniform throughout the observed region. It ranges from 28.4◦
to 51.9◦and has a mean value of 39.5◦. The standard devia-
tion of all the polarization angles is 4.7◦. This small dispersion
indicates that the POS projection of the ordered component of
the magnetic field in this cloud is nearly uniform despite the
inhomogeneities seen in the mean degree of polarization.
At larger distances, we find several distinct clouds intersected
by different sightlines. The distances of the distant clouds in
neighboring pixels agree within uncertainties. Most noticeably,
we find a cloud toward (l,b)≈(103.8◦,22.4◦) with mean dis-
tance ranging from about 1600 pc to 1850 pc and which has a
A162, page 14 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
Galactic Longitude [
]
Galactic Latitude [
]
pCdistant [%]
Galactic Longitude [
]
Galactic Latitude [
]
pCintermediate [%]
Galactic Longitude [
]
Galactic Latitude [
]
pCnearby [%]
Fig. 16. Same as for Fig. 15 but for the cloud mean polarization. The
color scales is identical for the three maps and informs on the mean
degree of polarization ( ˆ
¯pC). The red segments indicate the orientation
of the mean polarization in the clouds ( ˆ
¯
ψC) which directly traces the
orientation of the mean POS-component of the magnetic field.
mean degree of polarization of about 0.26% and a mean polar-
ization angle of about 136◦. This is nearly perpendicular (≈85◦)
to the mean polarization orientation of the dominant polariza-
tion screen for the same sky pixels. This small region of the
sky is slightly to the northwest of the “2-cloud LOS” studied
in Panopoulou et al. (2019), Clark & Hensley (2019), Pelgrims
et al. (2023) at (l,b)=(104.1◦,22.3◦). In the east-southeast of
this “2-cloud LOS”, toward (l,b)≈(104.3◦,22.2◦), we detect
a cloud with mean distances ranging from 1700 pc to 2300 pc
which has a mean degree of polarization of about 0.28% and
a mean polarization angle of 66.2◦, about 23◦away from the
mean polarization angle of the dominant polarization screen in
the same sky pixels. In the southwestern part of the surveyed
region, toward (l,b)≈(102.4◦,21.9◦), we detect a cloud at dis-
tance between 1270 pc and 1500 pc. It has a mean degree of
polarization of about 0.58% and a mean polarization angle of
16.5◦, about 20◦away from the mean polarization angle of the
dominant polarization screen in the same sky pixels.
We also detect other groups of pixels with cloud detec-
tion close to the edges of the observed region. One at (l,b)≈
(104.4◦,22.8◦) with a somewhat large scatter in mean distance,
ranging from 1760 pc to 2400 pc, and another one in the upper
nearby intermediate distant
0.00
0.05
0.10
0.15
0.20
0.25
0.30
det(
C
int) [%2]
Fig. 17. Violin diagrams of the distributions of the square root of
ˆ
¯
det(Cint)obtained from the posterior distributions and for the three
distance ranges defined from Fig. 14. In each diagram, the hori-
zontal segments indicate the 16th, 50th and 84th percentiles of the
distribution.
right corner of the observed region, with a mean distance of
about 1170 pc. Other distant clouds are detected sparsely in the
region. We have checked the tomographic decompositions of
all these LOS individually, and none of these cloud detections
appear to be the result of a statistical flaw in our analysis. We
checked the posterior distributions of the different tested mod-
els to make sure that the automated model selection worked
as expected. We additionally checked the residuals and visually
compared the data and models in the (q,u)−µspace. In all cases,
the best model was correctly identified.
We provide more discussion on our results and their reliabil-
ity in Sect. 5. There we also relate them to known structures of
the ISM that have already been studied in the literature, based on
stellar extinction and H Idata, in particular.
As a last step in the basic exploration of the output, we
look at the covariance matrices that encode the intrinsic scatter
of polarization properties within clouds (Cint). We evaluate the
determinant of Cint for each sample of the posterior distributions,
for each dust layer of the best model, and for each LOS individu-
ally. This quantity is related to the level of turbulence-induced
intrinsic scatter within a cloud (see Appendix B of Pelgrims
et al. 2023). We then estimate the median of the distributions
of the determinants ( ˆ
¯
det(Cint)) for each layer and each LOS. We
finally build the distributions of the medians splitting the sample
in terms of the cloud distances, as before. We show these distri-
butions in Fig. 17 in the form of violin diagrams. We note that,
while all the sightlines likely trace the same clouds in the nearby
and intermediate distance ranges, this is unlikely to be the case
for the distant distance range, as argued before. This is likely
the reason for the apparent wider distribution of ˆ
¯
det(Cint)in this
distance range than for the others. From Fig. 17, it is seen that,
at least for the clouds at nearby and intermediate distances, the
intrinsic scatter is detected above observational noise. This indi-
cates that, in principle, subsequent analyses of the covariance
matrices could lead to a detailed characterization of fluctuations
in the magnetized ISM. We will explore such an avenue in future
work.
4.2. 3D-map making
In this subsection, we build 3D maps of the dusty magnetized
ISM from the posterior distributions of all model parameters
obtained for each LOS of the observed region. For each LOS,
we construct the probability density function (PDF) of having a
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Pelgrims, V., et al.: A&A, 684, A162 (2024)
cloud at a given distance as follows. We stack the marginalized
posterior distributions of the cloud parallax of each component,
and obtain the distance distribution P(dC)by inverting every par-
allax value. The PDF is estimated from this distribution using a
moving window of length ∆das:
PDF(d)=1
∆dZd+∆d/2
d−∆d/2
P(dC) ddC.(16)
By construction PDF(d)is normalized to the number of compo-
nents intersected by the LOS. Then, using the estimated PDF(d),
we construct the distance profiles of the different polarization
properties by marginalizing the posterior distributions of each
polarization parameter in distance bins. We thus estimate at any
distance (d) the differential of any of the Stokes parameters or of
the elements of the intrinsic-scatter covariance matrix as:
δsC(d)="ZsCP(sC|d) dsC#PDF(d),(17)
where sCis any of qC,uC,Cint,qq,Cint,uu or Cint,qu and dCis the
distance given by the inverse of the parallax. P(sC|d)is the con-
ditional probability of having a value sCgiven a cloud distance
dCat value d:
P(sC|d)=P(d,sC)
PDF(d),(18)
where P(d,sC)is the 2D-marginalized posterior distribution
between distance and the chosen polarization parameter sC,
obtained by mapping the posterior P(ϖC,sC)in distance space.
The units of the differentials of the Stokes parameters, δqand δu,
are in polarization fraction per parsec, and the units of the differ-
ential of the intrinsic scatter covariance matrix, (δCint,xy, where
the subscripts xand ydenote either qor u) are polarization frac-
tion per parsec squared. In the remainder of this paper, we focus
on the mean properties of the dusty magnetized ISM that we can
infer from stellar polarization and refer to future work for the
study and characterization of its fluctuations.
Due to the limited number of samples used in estimating
the posterior distributions, spurious noise is observed in the dis-
tance profiles. To reduce this, we smooth the estimated distance
profiles along LOS using a Gaussian kernel. The posteriors are
not sampled uniformly along LOS. This is because the sam-
pling happens in the parallax space and also because the density
of stars varies significantly with distance. It is not possible to
choose a constant kernel value that would both smooth the noise
at large distances and not severely dilute the signal at small
distances. Therefore, we choose to smooth the profiles with a
Gaussian kernel that varies with distance as
σk(d)=30 +20/(π/2) arctan((d−500)/50),(19)
where σk(d)and dare given in parsec. This choice is rather arbi-
trary, but it effectively smooths the noise at all distances. The
choice for σkis such that it is close to 10 pc at a distance of
50 pc, it increases smoothly around 500 pc, and it reaches a value
of about 50 pc for all distances larger than 1 kpc. The radial pro-
files and subsequent visualization are not strongly dependent on
this choice. However, we recommend using the posterior dis-
tributions directly rather than the profiles for any subsequent
quantitative analysis.
We show a set of such radial profiles of the Stokes parame-
ters for 15 sightlines in Fig. 18 (middle and bottom panels). Nine
30 50 100 200 300 500 1000 2000
Distance [pc]
0.00
0.01
0.02
0.03
PDF(dC)
0.005
0.000
0.005
δqC(d)[%/pc]
2 4 6 8 10 12
Distance Modulus: µ= 5 log10(d)−5
0.00
0.01
0.02
0.03
0.04
δuC(d)[%/pc]
Fig. 18. Variation along the distance of the normalized probability den-
sity distribution of the cloud distances (top) and of the differentials of
the Stokes parameters δqand δu(middle and bottom) obtained from
the marginalization of the full posterior distributions in distance bins
for 15 sightlines as explained in the text. The moving-window length of
10 pc is adopted. The curves are smoothed with a Gaussian kernel that
varies with distance as explained in the text. The same color corre-
sponds to the same LOS across panels.
sightlines are randomly chosen in the observed regions to which
we add six sightlines that intersect three components as identi-
fied in Fig. 11 (those with b≤22.5◦and l≥103.5◦). Similar
profiles can be constructed for the three parameters δCint,xy. The
top panel of Fig. 18 shows the PDF of cloud distances for the
same set of sightlines. To construct these profiles, we used a
moving-window width of ∆d=10 pc and sample the distance
axis every parsec up to a distance of 3 kpc.
By repeating this process for all sightlines, we finally
obtained the values of the differentials at any grid point sam-
pling the 3D space corresponding to the observed sky region.
These are the 3D maps that describe the dusty magnetized ISM
in the observed sky region. Obtaining these maps is the main
result of this paper. We present some visualizations of the maps
in the next subsection. The 3D maps come naturally in a spher-
ical coordinate system centered on the observer. From the 3D
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30 50 100 200 300 500 1000 2000
Distance [pc]
0.4
0.2
0.0
0.2
Rd
0δq(dC) ddC[%]
0
1
2
Rd
0δu(dC) ddC[%]
0
1
2
p?(d)[%]
2 4 6 8 10 12
Distance Modulus: µ= 5 log10(d)−5
50
75
100
ψ?(d)[◦]
Fig. 19. Polarization of a test star as a function of its distance for the
same set of sightlines as shown in Fig. 18. The first and second rows
show the cumulatives of the radial profiles of the differentials δqand δu
as a function of distance. The third and fourth rows show the degree of
polarization and polarization angle observed for the test star, as derived
from the cumulatives of derived δqand δu. For p⋆<0.05%,ψ⋆is
masked (shown by thin gray line).
maps of the differential of the Stokes parameters, 3D maps of
the polarization degree (δp) and of the polarization angle (δψ)
can be derived. The 3D map of the polarization degree can
be interpreted as a map of the density of polarizing material
through space, weighted by the geometrical factor from the local
inclination of the magnetic field on the LOS and by the local
polarization efficiency. The 3D map of the polarization angle
informs us on the orientation of the POS component (as seen
from us, the observer) of the magnetic field locally. However,
because the degree of polarization and the polarization angle are
not additive quantities, as opposed to the Stokes parameters (and
so of their differentials), the maps of δpand δψ must not be inte-
grated along distance. In the first and second rows of Fig. 19, we
show the cumulatives of the differentials δqand δuas a func-
tion of distance for the same sightlines as presented in Fig. 18.
The cumulatives at a distance dcorrespond to the Stokes param-
eters that would be observed for a test star at that distance if we
neglect the effects from turbulence. In the third and bottom rows
we show the derived (mean) degree of polarization (p⋆(d)) and
(mean) polarization angle (ψ⋆(d)) that would be observed for a
star at the specific distance. These quantities are obtained at any
distance dfrom the cumulatives of δqand δu, not from the cumu-
lative of δpand δψ. The depolarization effect (decrease in p⋆)
due to the superposition of clouds with misaligned POS com-
ponent of the magnetic field is clearly observed for some of the
sightlines.
4.3. Visualization of the 3D maps
The portion of the Galactic space covered by our observations
extends over ≈3.8square degrees across the sky and extends up
to 3.5 kpc. As a result, the geometry of the 3D maps that we
construct is very elongated in radial distance and resembles a
“pencil beam” having the geometry of a rectangular pyramid.
This makes it difficult to visualize the results in 3D. To cir-
cumvent this limitation, we transformed the spherical coordinate
system using the distance modulus as the radial coordinate as
X=µ(d) cos l′cos b′
Y=µ(d) sin l′cos b′
Z=µ(d) sin b′,
(20)
where µ(d)=5 (log10(d)−1) is the distance modulus with d
given in parsec and (l′,b′)are the angular coordinates (longi-
tude and latitude) defined such that (l′,b′)=(0,0) points toward
the center of the observed region. In our case, we thus have
X≈µ(d),Y≈µ(d)l′, and Z≈µ(d)b′. This coordinate system
is ill-defined at small distances. However, we are not affected
by this issue since all stars have a distance larger than 20 pc and
that our 3D maps do not extend at distance modulus smaller than
µ=2.
We thus project the 3D maps of the differentials constructed
above in this coordinate system. The sampling on angular coor-
dinates is fixed by the centers of the HEALPix pixels used to
define our samples from which the LOS decompositions were
performed. The sampling on radial distance is such that we
have values for the differentials at every parsec for the range
µ∈[2,12.7]. As defined, the X-axis runs through the center of
the observed region at (l,b)=(103.3◦,22.3◦),Yincreases with
increasing longitude and Zwith latitude.
Despite the use of this coordinate system, the volume is still
more extended toward the X-axis. For visualization purposes, we
thus shrink that axis by a factor of 10 as compared to the others.
We then construct a regular Cartesian grid made of 2563voxels,
with limits such that the volume X/10 ×Y×Zfits in. Due to
the rectangular pyramid geometry of the inverted dataset, most
of the volume is empty (no data). We use linear interpolation
to obtain the values of the differentials at every voxel position
from the 3D maps of the differentials in the modified spherical
coordinate system. The data cubes can now be visualized.
4.3.1. Plane projections
We start by producing plane projections of the data cubes. The
data cubes are integrated along the Y-axis and the Z-axis of
the Cartesian grid to produce the vertical and horizontal plane
projections, respectively. Figure 20 shows the results of such pro-
jection for the differentials of the (mean) Stokes parameters. In
this figure, and as explained in its caption, we tweak the color
scales in order to visualize the very faint features at large X
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Pelgrims, V., et al.: A&A, 684, A162 (2024)
0.1
0.0
0.1
Z
Plane Projection of δq
2 4 6 8 10 12
X
0.3
0.2
0.1
0.0
0.1
0.2
0.3
Y
0.01
0.00
0.01
[%/
pc
]
0.1
0.0
0.1
Z
Plane Projection of δu
2 4 6 8 10 12
X
0.3
0.2
0.1
0.0
0.1
0.2
0.3
Y
0.02
0.00
0.02
[%/
pc
]
Fig. 20. Plane projections of δq(left) and δu(right) in the vertical (top) and horizontal (bottom) planes. The vertical and horizontal planes are
defined by averaging the data cubes (rectangular pyramid) in the Cartesian grid along the Y-axis and the Z-axis, respectively. The observer (us) is
on the left and distance increases to the right. In order to visualize faint features, the color values (c) are obtained from the projected values (v) as
c=sign(v)√|v|. The units of the color scale are therefore p%/pc. The color scales are symmetrical about zero and the range of δuis twice as large
as that of δq, reflecting the difference in magnitude of the two quantities that is observed in Fig. 18.
values. The faint signal of the distant components seen in Fig. 18
at µ≈10 are indeed dimmed because of their small extent in
space and by the integration over the full length of the data
cube axes. The nearby and dominant components already seen
in Fig. 18 are striking on these plane projections. The dominant
component appears nearly planar at constant X(distance) and
with very coherent polarization properties. This is reminiscent
of what we already discussed in Sect. 4.1. The change of colors
along distance indicates that the POS component of the mag-
netic field in the nearby, intermediate, and large distance ranges
are misaligned.
Plane-projection maps reveal features that are somewhat
elongated along the distance axis. This is clearly visible at small
distances. Most of it is due to the use of the moving window of
10 pc and the smoothing that we use to construct the distance
profiles of the differentials and due to the choice of the loga-
rithmic scale (distance modulus) for visualization. However, it is
worth emphasizing that the constructed 3D maps result from the
marginalization of the posterior distributions along the distance
axis. Therefore, some real extensions along the LOS exist in the
maps and are related to the level of constraints we can impose
on cloud distances given the stellar data. They do not inform the
actual extension of a dust cloud along the LOS but rather reflect
our uncertainties on cloud distances. These features are similar to
the “finger-of-god effect” commonly seen in 3D dust map recon-
structions. We recall that the model that we use to reconstruct the
dusty magnetized ISM from stellar data in polarization and dis-
tance assumes that the clouds are thin – they have no dimension
along the LOS.
4.3.2. Visualization in 3D
Visualizing a pseudo-vector field in 3D presents more chal-
lenges than visualizing a scalar field, such as the dust density
distribution for example. While preparing this paper, we
have started addressing these challenges building up on the
framework underlying ASTER ION , the tool developed in
Konstantinou et al. (2022) to simulate magnetized dust clouds in
3D. ASTER ION relies on real-time 3D visualization techniques
with virtual reality capabilities. It is implemented in the real-
time-engine called Unreal Engine 4 (UE4)6and allows for the
rendering of details of the magnetized ISM and enables the user
to fly through the simulated environment as is done in video
games.
Extending the capabilities of this software, we are now able
to visualize the main properties of our tomography map. Yet, the
current version of the software does not render the contribution
from the intrinsic scatter nor does it visualize the uncertainties
on our 3D reconstruction. Only the solution corresponding to the
means of our posterior distributions (or at the maximum likeli-
hood values) on the cloud distances and their mean polarization
can be visualized. Specifically, the software makes it possible
to place the magnetized dust cloud in 3D and to visualize the
“differentials” of δpCand δψCusing colors, transparency and a
3D version of the line integral convolution technique (Cabral &
Leedom 1993). The visualization of the 3D map obtained in this
paper can be accessed online7.
5. Discussion
In this section, we compare the results of our 3D reconstruction
of the POS component of the magnetized ISM from starlight
polarization and distance data to other datasets that inform on
the complexity of the ISM. We also discuss the limitations of
our results and caveats of our method.
6UE4 is a complete suite of development tools that allows for the
visualization and immersive virtual worlds, multiplatform deployment,
asset and plugin marketplace, among other features (https://www.
unrealengine.com/en-US/unreal).
7https://pasiphae.science/visualization
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Pelgrims, V., et al.: A&A, 684, A162 (2024)
5.1. Comparison with 3D dust extinction data
We start with a comparison of our results with a 3D dust extinc-
tion map that was obtained from Gaia parallax and from both
spectroscopic and photometric extinction measurements, includ-
ing from Gaia and 2MASS, and which informs on the dust
density distribution in 3D space in a Cartesian box of 3×3×
0.8kpc3centered on the Sun (Vergely et al. 2022). Our starlight-
polarization-based tomography map is independent from the 3D
extinction map as it relies on different observables (aside from
the parallax) and as the stellar polarization sample is (much)
smaller than the stellar extinction sample. Both 3D maps inform
us on different properties of the ISM (extinction versus polariza-
tion). In addition, the 3D map of dust extinction and our map of
stellar polarization have different spatial resolutions both along
the LOS and in the POS. The spatial resolution of the 3D dust
map that we consider is 10 pc (and sampled by 3D voxels with
side length of 5 pc). Our map has a varying spatial resolution,
first because the LOS inversion is not bound to a particular sam-
pling (or gridding) of the distance, and second because the signal
along each LOS has been smoothed with a distance-dependent
Gaussian kernel. Despite these differences, we wish to verify
whether our approach identifies clouds at distances similar to
the cloud locations in the 3D dust map. We thus construct 1D
profiles of extinction as a function of distance to compare the
locations of peaks in the dust distribution along the LOS with
the locations of clouds in our tomography map.
For this first qualitative comparison between dust extinction
and dust polarization tomography data, we sample both 3D maps
with a set of sightlines that go through the surveyed sky area
and set by the HEALPix tessellation with Nside =128. For each
LOS we extract the distance profiles of differential extinction
(A′
v) and of the “differential” of the degree of polarization (δpC),
computed locally from the differentials of the Stokes parame-
ters (δqCand δuC). We present the profiles in Fig. 21 where we
also indicate with a vertical strip the estimate of the distance
to the inner surface of the Local Bubble as derived in Pelgrims
et al. (2020)8from the 3D extinction map of Lallement et al.
(2019). Because of the finite size of the 3D map of Vergely et al.
(2022), the A′
vprofiles do not extend over the whole distance
range for which we recover information from stellar polarization.
Comparing the profiles from dust extinction and dust polariza-
tion tomography data, we notice that both tracers indicate the
presence of a very nearby component at µ≲5and a dominant
component at µ≈7.9.
It is remarkable that the stellar polarization data makes it
possible to recover the very nearby component at µ≲5although
it appears to be very shallow in the extinction profiles. The
difference between extinction and polarization data in relative
amplitudes between the nearby component and the dominant one
could point to differences in magnetic field inclination or dust
polarization properties. However, differences in 3D-map reso-
lutions could artificially dilute more the signal of the nearby
component in 3D dust density map than in polarization. Thus,
this comparison will require confirmation from a dedicated anal-
ysis which goes beyond the scope of this paper. Meanwhile, it
should be noted that our polarization tomography data indepen-
dently confirms the presence of a very close dust cloud identified
in the 3D dust extinction map allowing us to argue that these
small features in the 3D dust map are real (at least in the sur-
veyed region). Evidence for nearby ISM clouds within the Local
Bubble are also provided by pulsar-scintillation studies (Ocker
8https://doi.org/10.7910/DVN/RHPVNC
30 50 100 200 300 500 1000 2000
Distance [pc]
10 3
10 2
δpC[%/pc]
2 4 6 8 10 12
Distance Modulus: µ= 5 log10(d)−5
10 4
10 3
10 2
A0
v[mag/pc]
Fig. 21. Qualitative comparison of distance profiles between dust polar-
ization and dust extinction tomography results. Top: profiles of the
“differential” of the degree of polarization (δpC, computed locally from
δqCand δuC) as a function of distance modulus as inferred from our
polarization 3D map. Bottom: differential extinction (A′
v) as a function
of distance modulus as inferred from the 3D extinction map of Vergely
et al. (2022). No extinction data is available at distance larger than about
1 kpc. The surveyed sky area is sampled according to an HEALPix map
with Nside =128. Each LOS is represented with a different color, the
same color is used across panels. The green vertical strip indicates the
range of distances to the inner surface of the Local Bubble in this sky
area (Pelgrims et al. 2020).
et al. 2024 and references therein) and stellar line absorption
(Peek et al. 2011).
The agreement between distances to the dominant peak seen
in the A′
vand δpCprofiles at µ≈7.9is remarkable. However,
while this component appears as an isolated an narrow peak in
the δpCprofiles, it appears broader toward lower distances, pos-
sibly featuring a second peak, in the A′
vprofiles. According to
Pelgrims et al. (2020), the nearby peak would correspond to the
wall of the Local Bubble. The main peak (centered in µ≈7.9)
could correspond to the diffuse Cepheus Flare as several molec-
ular clouds in this part of the sky show similar distances (see
Schlafly et al. 2014).
However, it is unclear whether the broadening of the peak in
the A′
vprofiles actually indicates the presence of two close dust
components, or if this shape simply stems from the fact that there
is too little stellar extinction data right in front of the dominant
component to effectively constrain the lower distance limit of a
single cloud centered at the main peak. We recall that A′
vpro-
files, like our distance profiles in the cloud parameters, reflect
the shape of the posterior distribution, which is closely related
to the uneven distribution of stars along the distance axis. Addi-
tionally, we note that we have identified the presence of a cloud
with distance µ≈7, several degrees away in the southeastern
direction, outside of the surveyed region, and that the outskirt of
this cloud is intersected by our region. This is observed both in
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Pelgrims, V., et al.: A&A, 684, A162 (2024)
3D maps from Vergely et al. (2022) and Edenhofer et al. (2024).
It is therefore possible that the peak broadening in the A′
vprofiles
is due to the limited resolution of the 3D dust density maps, or
that the broadening indicates a real dust overdensity. In the latter
scenario, we understand why we are unable to find evidence for
the existence of this cloud using starlight polarization by the fol-
lowing. If there are two dust components in the range µ∈[6,9],
they are relatively close in distance. Thus the number of stars
with polarization measurements that would be useful to identify
a cloud and constrain its polarization properties in between the
two peaks is very low. We have checked that, indeed, we have few
polarization measurements in that distance range in our sample.
Furthermore, the amplitude of the jump in degree of polariza-
tion that is induced by the dominant screen is very large. This
hampers the possible identification of a counterpart of the wall
of the Local Bubble in its vicinity as it would require a large
number of data points. This illustrates one of the limitations in
our reconstruction which likely comes from the limited depth of
our survey. An alternative explanation would be that the near-
est of these two clouds simply does not induce polarization. This
could happen if this cloud is devoid of polarizing dust grains or if
both its column density is very low and its permeating magnetic
field lines are nearly perpendicular to the POS.
One major difference between our approach and the approach
of Lallement et al. (2019) and Vergely et al. (2022) is that they
use a fixed spatial kernel in the 3D space to infer the dust den-
sity in a grid while the distance to the cloud is a free parameter
in our model. Together with the thin-layer assumption in our
model, this might explain the tighter constraints on cloud dis-
tances that we obtained (378 pc ±14 pc), compared to the fixed
maximum resolution of 10 pc in the 3D dust density map of
(Vergely et al. 2022).
Overall, the agreement between our tomography map from
stellar polarization with the tomography map of dust density
from stellar extinction is remarkable given the very different
nature of the two datasets. We find good agreement in both the
number of components and their distances to the Sun. This indi-
cates that a systematic and dedicated analysis will help study
spatial variations of the ISM properties such as the polarization
efficiency of the dust grains and the inclination of the magnetic
field lines with respect to the LOS. However, our qualitative
comparison taught us that comparing tomography data in polar-
ization and density obtained from different approaches (as it is
the case here) and with different resolutions could lead to con-
fusion. To benefit from the comparison of dust extinction and
polarization data, it is essential to perform detailed analysis treat-
ing the different datasets in a self-consistent manner. We leave
the task of devising such a framework for future work.
5.2. Detection of distant clouds and HIdata
In Fig. 21, we observe farther away clouds in the δpCprofiles
where no information exists in A′
vprofiles, although indications
for faraway clouds might be guessed in the far edge of the extinc-
tion profiles (at µ≈10). To corroborate our findings of faraway
clouds (≳1kpc), we first turn to the inspection of H Ivelocity
spectra. As discussed in Sect. 2, we expect to observe complex
HIspectra with features at intermediate velocities if faraway dust
clouds are present along these sightlines.
For this purpose, we extract the H Ivelocity spectra from
the HI4PI data cube (HI4PI Collaboration 2016) in pixels cor-
responding to the sightlines for which we detected a cloud with
mean distance in the large distance bin (dC>650 pc) in Sect. 4.1.
These sightlines can be seen in the top panel of Fig. 16, for
80 60 40 20 0 20 40
vLSR [km/s]
0
5
10
15
20
Tb[K]
Fig. 22. HIvelocity spectra for sightlines with far away clouds. The
spectra are sorted according to the Galactic longitudes of their sightlines
and shown with color ranging from blue to red. For visual perception,
the velocity spectra are smoothed with a Gaussian kernel with width of
1.9 km s−1.
example. We sorted the sightlines according to their Galactic
longitude and present the spectra in Fig. 22 from blue (low
longitude) to red (high longitude). We exclude the range of
high-velocity clouds (vLSR ≲−90 km s−1) and velocities above
40 km s−1where there is no signal.
The H Ivelocity spectra clearly show complexity with sev-
eral components. All sightlines at large Galactic longitude show
strong IVC components (|vLSR|≳35 km s−1). These IVC compo-
nents do not vanish at lower longitude but are shallower. The fact
that there is power in the velocity spectra for |vLSR|>35 km s−1
for all these sightlines makes us confident in our detection of
faraway clouds. However, the evidence for IVC components at
low Galactic longitudes is milder than for high longitudes; the
column density of these clouds is much lower in this part of
the surveyed region, as already inferred from Fig. 2. The spec-
tra corresponding to the upper right part of the region, with
l<103◦and b>22.1◦, are the only ones not showing local max-
ima at vLSR <−20 km s−1. At this stage we do not know if this
is an indication for spurious detection of faraway clouds in our
starlight-polarization based tomography. We discuss further the
reliability of our cloud detections in Sect. 5.3.
In their analysis of a small subset of the polarization data
studied in this paper, Panopoulou et al. (2019) studied starlight
polarization within a beam of about 9.6 arcmin radius toward
(l,b)=(104.1◦,22.3◦)and made the identification of a faraway
dust cloud with an IVC with velocity of about −50 km s−1. Hav-
ing enlarged the surveyed area with stellar polarization, we see
that IVCs are detected in this part of the sky using starlight polar-
ization, although not specifically in the pixel closest to the LOS
studied by Panopoulou et al. (2019), but in neighboring pixels
(eastward and westward). It is not clear from our polarization
tomography data whether these faraway clouds form a contin-
uous entity in 3D space or if they are independent. Assuming
a common distance of 1700 pc, the detected clouds are at least
15 pc apart. The orientations of the POS component of the mag-
netic field that we find are also different in the three main clusters
of pixels with faraway-cloud detection. A different POS mag-
netic field orientation is however compatible with the finding of
Clark & Hensley (2019) who used the orientation of H Ifibers to
infer this quantity. This is illustrated in Fig. 23 where we overlaid
our results from starlight-polarization-based tomography for the
clouds in the large distance bin (as in the top panel of Fig. 16)
to the H I-orientation data from Clark & Hensley (2019). We
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102.0103.0104.0105.0
Galactic Longitude [◦]
21.0
21.5
22.0
22.5
23.0
23.5
Galactic Latitude [◦]
0.5
1.0
1.5
2.0
NHI [cm 2]
1e20
Fig. 23. HI-orientation data integrated in the velocity range vLSR ∈
[−60.45,−44.99] km s−1toward our surveyed area. The background
color shows the H Iintensity on a linear stretch, and the texture rep-
resents the POS magnetic field orientation inferred from H Ifibers. The
red segments indicate the mean POS magnetic field orientation obtained
from starlight polarization tomography for clouds with distances larger
than 1 kpc, as in Fig. 16. The white-dotted circle shows the “2-cloud
region” studied in Panopoulou et al. (2019). The magenta contour and
the blue circle are as in Fig. 10.
also obtain consistent results if we instead use the H Iorienta-
tion maps from Halal et al. (2023). The H Idata is integrated
over the velocity range vLSR ∈[−60.45,−44.99] km s−1. The
POS component of the magnetic field implied by the orienta-
tion of H Ifibers is shown using line-integral-convolution (LIC)
texture. The alignment between H Istructures and the magnetic
field is driven by the orientation of anisotropic, cold gas struc-
tures (Clark et al. 2019;Peek & Clark 2019;Murray et al. 2020;
Kalberla et al. 2020;Kalberla 2023), so the LIC pattern may not
trace the magnetic field well away from regions of prominent H I
emission. An IVC component, with varying POS magnetic field
component, is prominent in the upper left corner of the surveyed
area. Visual comparison of the orientation of the POS compo-
nent of the magnetic field obtained from starlight-polarization
tomography and H Idata reveals a good qualitative agreement
where faraway clouds are detected, in particular toward the IVC.
However, we notice that we do not detect faraway components
toward all sightlines sampling the prominent IVC, especially in
the closest pixel toward the “2-cloud” region of Panopoulou et al.
(2019), and that our cloud detections in the western part of the
region do not have counterparts in this velocity range. We further
discuss these differences in the next subsection.
Overall, our comparison between starlight-polarization and
HIdata points toward the great potential of combining the dif-
ferent tracers of the magnetized ISM to study and characterize its
properties in detail. Going far beyond the scope of this paper, we
will explore the potential of such a synergy in future work. Mean-
while, all of the above indicates that our inversion method for
obtaining, from stellar polarization and parallax alone, a tomo-
graphic view of the POS component of the magnetic field in
dusty regions leads to reliable results.
5.3. Reliability of cloud detection and limitations
To present our results, in Sect. 4, and to compare them with other
probes of the ISM, in the two subsections above, we focus on the
tomographic solutions corresponding to the best model (number
of clouds) obtained per LOS. However, it is interesting to look
102.0102.5103.0103.5104.0104.5105.0
Galactic Longitude [◦]
21.5
22.0
22.5
23.0
Galactic Latitude [◦]
1
2
3
4
Number of clouds
102.0102.5103.0103.5104.0104.5105.0
Galactic Longitude [◦]
21.5
22.0
22.5
23.0
Galactic Latitude [◦]
Conical beam
0.0
0.16
0.33
0.5
0.66
0.83
1.0
Probability of second-best model
102.0102.5103.0103.5104.0104.5105.0
Galactic Longitude [◦]
21.5
22.0
22.5
23.0
Galactic Latitude [◦]
Conical beam
1
2
3
4
Number of clouds
Fig. 24. Maps of the second-best model: (top) number of clouds and
(middle) probability for the second-best model to be the model that
minimizes the loss of information against the best model. The bottom
panel combines the information from the two upper panels. It indicates
the number of clouds (color) whose opacity is given by the probabil-
ity. Transparent pixels correspond to low probabilities for second-best
models.
at the second-best model (defined below) and to compare the
performance between the best and second-best models that we
obtained. In fact this helps quantify the robustness of a solution
(a given model) and, also, the reliability of a cloud detection.
In Sect. 3.5.3, at the end of Step 2 in the inversion process,
we obtained for each LOS, the estimated maximum-likelihood
values for all tested models (different number of layers). Among
the tested models, the best model was identified comparing the
model performances based on their AIC values (see Eqs. (9)
and (10)). By ranking the AIC values, we can also identify the
second-best model. The probability that this model is actually
the model that minimizes the loss of information as compared to
the best model (i.e., its Pj|{m}value) quantifies by how much it is
outperformed by the best model.
In the top panel of Fig. 24, we show the map of the number of
clouds per LOS corresponding to the second-best model. In the
bottom panel of the same figure, we show the map of the second-
best model probabilities. In these maps, white pixels (enclosed in
the magenta outline) indicate that no second-best model can be
identified. That is, there is no “valid” reconstruction with a dif-
ferent number of clouds along the LOS. In practice, this happens
for several sightlines where the best model is a 2-layer model,
A162, page 21 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
and for which the 3-layer and 4-layer models led to bad posterior
distributions (see Sect. 3.5.1). This means that either there is no
additional cloud farther away than the second cloud, or the data
is not good enough to make it possible to detect it solely based
on stellar data. We recall that when a 2-layer model is selected
as the best model in Step 1, the 1-layer model is not considered
in Step 2.
The comparison of the number of clouds per LOS from the
best model (Fig. 12) and from the second-best model (top panel
of Fig. 24), taking into account its probability, suggests that
for most of the sightlines where the 3-layer (2-layer) model is
favored by the data, we cannot safely ignore the 2-layer (3-layer)
model. Assuming the presence of faraway clouds in the eastern
(left) part of the region, as motivated by H Idata, this indicates
that the data is not enough to allow for a strong and robust cloud
detection at large distances, suggesting that we reach the limit
of the cloud-detection capability given the data. In the western
(right) part of the region, the probability for the 2-layer model is
generally high for the 3-layer sightlines, suggesting either unre-
liable (spurious) detection of faraway cloud or, again, that we
reach the limit of the cloud-detection capability.
The cloud-detection capability is primarily limited by the
number of measurements and their uncertainties as compared
to the strength of the polarization signal induced by a cloud to
its background stars (see Sect. 4 in Pelgrims et al. 2023). This
shows that the reliability of our tomographic results at large dis-
tances is limited by the current depth of our survey and that more
polarization data is needed to enable a stronger model separation.
However, given the complementary nature of stellar extinction,
HIand starlight polarization data, which we also highlighted
above, there is the possibility to incorporate such external data in
the inversion process. In principle this would help differentiate
between competing models, even in the absence of additional
stellar polarization data. We will undertake this endeavor in
future work.
The depth of the survey, and therefore the number density of
the measurements, also sets a limit on the maximum angular res-
olution at which we can achieve the LOS inversion of starlight
polarization data. The angular resolution is directly linked to
one of the main limitations of using BISP-1. The method does
not explicitly take into account POS variations of the polar-
ization signal within the beam other than through the intrinsic
scatter term. We use a fixed geometry to define our beam and
consider a top-hat acceptance window to include stars in our
sample. As a result, while modeling the ISM along distance,
stars at the edge of the beam (on the POS) contribute the same
as stars at the center of the beam. This is fine as long as the
polarization signal does not vary much on the POS and can be
described by our model. However, artifacts (such as overestima-
tion of the intrinsic-scatter covariance matrix and biases in the
mean polarization properties) may be expected as soon as the
polarization data cannot be well described by a bivariate normal
distribution. This may happen in the presence of substantial POS
variations at the angular scale comparable to the size of our beam
or if only part of the beam intersects a cloud. Such shortcom-
ings in polarization data modeling can naturally hamper cloud
detection.
This limitation likely explains differences between the
starlight-polarization and H Itomographic data at large distances
seen in Fig. 24, in particular toward the prominent IVC. Specif-
ically, H Idata suggests substantial variation of the signal in our
beam of 13.76 arcmin radius, and in particular, an abrupt change
of the POS component of the magnetic field toward the “2-cloud
region” of Panopoulou et al. (2019). We notice that part of these
limitations could be overcome by introducing weights on the
polarization data while computing the log-likelihood to account
for the angular distance of each star with respect to the center
of the beam, that is, of the LOS. We will test this idea in future
work.
5.4. Caveats of the inversion method
The main caveats of our approach to obtain a 3D map of the
POS component of the magnetic field in dusty regions come
from the use of our Bayesian method (BISP-1) which works
along the LOS. This method requires the definition of beams
within which the starlight polarization data is “averaged” on the
POS and modeled as a bivariate normal distribution with a mean
and covariance matrix according to the dust-layer model we
rely on.
The first limitation of the designed approach is that it uses a
fixed beam geometry and does not explicitly correlate solutions
for different beams. We address this problem by choosing to
oversample the sky with non-independent beam samples and by
applying our decomposition method to each of them. However,
while the solutions for overlapping beam samples are not inde-
pendent, the correlation is not quantified nor controlled through,
for example, the use of density power spectrum as it is the
case for 3D dust density mapping (e.g., Green et al. 2019;
Lallement et al. 2019,2022;Leike & Enßlin 2019;Vergely et al.
2022;Edenhofer et al. 2024). Introducing such a correlation for
polarization tomography is not a trivial task and would require
assumptions that we wish to avoid at this stage, in particular
because of the intertwined nature of matter and magnetic field,
and of the inherent degeneracy between the density of polariz-
ing dust and the inclination of magnetic field with respect to the
LOS. A careful analysis of our tomography results in combina-
tion with simulation-based studies will help shed light on how
to implement such correlations for the case of dust polarization.
We will address this question in future work.
A second, important limitation of BISP-1 is that it does not
explicitly account for POS variations of the polarization sig-
nal. For the diffuse sightlines targeted in this work, significant
POS variations within the beam do not seem to be present, as
our modeling provides a good description of the data. As men-
tioned in Sect. 3.6, we searched for possible systematic variations
of the polarization residuals within our beam and could not
find any. This suggests that any POS variation of the polariza-
tion signal in our beam are successfully characterized by the
intrinsic-scatter covariance matrix. However, we expect more
significant POS variations to arise toward denser regions, for
example toward nearby molecular clouds, where the column
density varies by an order of magnitude within tens of arcsec-
onds (e.g., filaments measured by Herschel,André et al. 2010).
Some nearby molecular clouds have been targeted with deep
polarimetric surveys (e.g., Pereyra & Magalhães 2004;Santos
et al. 2017), approaching a density of 1000 measurements per
square degree. In such regions, a careful examination of the
choice of beam size and the POS variations of column density
would be necessary when using BISP-1 to decompose polariza-
tion along the LOS, as envisioned by e.g. Soler et al. (2016).
Given the existing stellar polarization data in the literature, it
appears that the dataset presented here is particularly favorable
for applying BISP-1 in a moving-window scan scheme due to its
combination of high number density of stars and the fact that it
probes diffuse sightlines, conditions that will be encountered by
the PASIPHAE survey targeting high and intermediate Galactic
latitudes.
A162, page 22 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
5.5. Astrophysical use of the output
Mapping continuously the stellar-polarization source field in 3D
opens the way to tackle several science objectives that were thus
far out of reach or left to the study of specific clouds or LOS. A
direct use of our results, which does not require postprocessing
of the tomography map, is the production of a list of intrinsi-
cally polarized-star candidates. Other possible uses, which will
however require further postprocessing and specific analysis, are
mentioned in Sect. 6.
Using the same procedure as in Sect. 3.6, we proceed to the
estimation of the significance of the residuals in polarization
for all the stars making our polarization samples, with success-
ful Gaia cross-match and which satisfy the quality criterion on
parallax estimate (RUWE ≤1.4, see Sect. 2.3). This sample is
made of 1448 stars among which 18 were identified as possi-
ble outliers according to the recursive sigma-clipping approach
employed in Sect. 3.3. Among the 1448 stars, 1392 fall in an
HEALPix pixel for which we have a model for the magnetized
ISM along distance. For each of them we obtain a distribution
of the Mahalanobis distance values (d⋆
Maha) informing us on the
likelihood that the measured polarization is compatible with our
picture of the dusty magnetized ISM, taking into account all
sources of uncertainties and scatter in model and observations.
We show the histogram of the median of the d⋆
Maha distributions
for the full sample in Fig. 25 where we separate the stars flagged
as outliers in Sect. 3.3 from the others. This figure confirms
that most of the outliers discarded from the tomography analysis
indeed show polarization properties that are not compatible with
the picture of the dusty magnetized ISM that we reconstructed.
In some sense, this also further validates our 3D reconstruction.
Using our tomography map to estimate the likelihood that
the polarization of any star falling in the reconstructed 3D vol-
ume is solely due to the dusty magnetized ISM provides us with
a robust way to identify outlier candidates and therefore intrin-
sically polarized-star candidates. We thus add the values of the
median of the d⋆
Maha distribution that we obtained for each star
to our published catalog (see Table 1). The higher this value,
the more likely the target is to be an outlier. This may be used
to plan follow-up observations to study these sources in more
detail. Only additional study will confirm whether these targets
are real outlying data points or if they merely pick up fluctu-
ations of the magnetized ISM that are unaccounted for in our
reconstruction. Our analysis also confirms that the fraction of
intrinsically polarized stars in the ISM is rather low, at least at
these Galactic latitudes. Only 14 stars out of 1448 have a p-value
lower than 0.2% for their polarization to be induced by the ISM
only. That is, only about 1% of our sample may be made of intrin-
sically polarized stars whereas we did not try to minimize this
fraction while planning our observation, for example based on
stellar type.
We must note a possible intrinsic degeneracy in our pro-
cedure. As mentioned in Sect. 3.3, during the sigma-clipping
procedure, it is possible to inadvertently exclude from the analy-
sis data points that merely capture fluctuations in the magnetized
ISM. In this case, a too low level of intrinsic scatter would
be obtained from the fit and, accordingly, any discarded points
would show a large d⋆
Maha value in the a posteriori test described
above. The list of outlier candidates thus depends on the choice
of the hyper parameters of the sigma-clipping procedure, and
more specifically on the choice of the used significance thresh-
old. This illustrates the relevance of follow-up observations and
analysis of the outlier candidates. However, if the turbulence-
induced intrinsic scatter had to be systematically underestimated
02468
¯
d?
Maha
100
101
102
Occurence
outliers
ISM probes
Fig. 25. Histogram of the median of the per star d⋆
Maha distributions. The
histograms for the “ISM probes” and “outliers” identified in Sect. 3.3
are separated (in blue and red) and stacked on top of each other. Notice
the logarithmic scale of the vertical axis. The dashed vertical lines cor-
respond to the p-value thresholds of 5%, 1%, and 0.2%. Most of the
outliers show significant residuals.
in our reconstruction, we would obtain a distribution of d⋆
Maha
values that would be statistically shifted toward large values and
that would not correspond to a bivariate normal distribution of
the polarization residuals. This is not what we obtain and what
is also shown in Fig. 25. On the contrary, we obtain a distribu-
tion of d⋆
Maha values that is slightly shifted to lower values than
expected and which, therefore, suggests a small overestimation
of the covariance matrices. We understand the latter as coming
from unaccounted for variation of the polarization signal in the
POS within our beam.
6. Summary and concluding remarks
In this work, we performed a survey of optical starlight polariza-
tion for a continuous region covering about four square degrees
centered on (l,b)=(122◦,33◦), and designed a pipeline to
obtain the first 3D map of the dusty magnetized ISM based on
a Bayesian analysis of the measurements in stellar polarization
and distances only. Obtaining this map, which corresponds to an
extended volume in 3D space, is the main result of this paper.
Our reconstruction corresponds to a sky area of about 3.8 square
degrees and extends up to 3 kpc from the Sun. We found that
the 3D volume covered by our tomography data is populated by
several clouds and that a large fraction of the sightlines in the sur-
veyed regions intersect at least two clouds, one being very close
to the Sun with a distance of about 62 pc and a dominant polar-
izing screen at about 375 pc. Distant clouds are also detected up
to a distance of about 2 kpc. We were able to corroborate our
findings using a 3D dust extinction map and H I-velocity spec-
tra. We are thus confident that our inversion pipeline works and
that stellar data in polarization and distance alone are a powerful
probe of the magnetized ISM. Specifically, for the diffuse ISM
where dust grains are expected to align their shortest axes with
the ambient magnetic field lines, starlight polarization allows us
to determine locally, in 3D space, the orientation of the POS
component of the magnetic field (the position angle) and the
amplitude of the starlight-polarization source field (the degree of
polarization) which depends on the local dust density, dust grain
polarization efficiency, and on the inclination of the magnetic
field lines with respect to the sightlines.
We obtained our polarization tomography map by adopt-
ing a moving-window strategy to scan the surveyed region with
A162, page 23 of 25
Pelgrims, V., et al.: A&A, 684, A162 (2024)
non-independent beams and making use of the LOS-inversion
method implemented in BISP-1. This allowed us to invert the
data and reconstruct the dusty magnetized ISM in 3D while
keeping the number of assumptions to its minimum. Namely, we
relied on the thin-dust layer model developed in Pelgrims et al.
(2023) and assumed that its assumptions are valid for the adopted
beam size of 13.74 arcmin radius. We expect that the analysis of
the obtained 3D map will enable the unbiased study and charac-
terization of the properties of the dusty magnetized ISM in 3D,
such as through 3D correlation functions. This will enable us in
the future to develop 3D inversion methods capable of taking
into account variations and correlations in space, both along the
distance and in the POS.
Finally, we expect that polarization tomography maps, as the
one we obtained in this work and that provides local measure-
ments of the POS component of the magnetic field and individ-
ual cloud polarization properties, will enable breakthroughs in
the modeling of the Galactic magnetic field, in the modeling and
characterization of the dusty magnetized ISM as a contaminant
foreground in observations of the cosmic microwave background
polarization, and in the modeling of astrophysical dust. These
research goals, along with the estimation of the strength of the
magnetic field through the quantification of the variation of the
POS component of the magnetic field (e.g., Skalidis & Tassis
2021;Skalidis et al. 2021), will necessitate dedicated analyses
and postprocessing of our 3D maps, such as to obtain proper
boundaries of dust clouds. In future works we will explore these
research directions based on the 3D map that we have pre-
sented here. This will set the stage for future analyses that will
greatly benefit from the most awaited polarization data from the
PASIPHAE survey (Tassis et al. 2018).
Acknowledgements. We thank an anonymous referee for a thorough and con-
structive report that helped us improve the clarity of the paper. The PASIPHAE
program is supported by grants from the European Research Council (ERC)
under grant agreements No. 7712821 and No. 772253; by the National Science
Foundation (NSF) award AST-2109127; by the National Research Foundation
of South Africa under the National Equipment Programme; by the Stavros
Niarchos Foundation under grant PASIPHAE; and by the Infosys Foundation.
This work was also partly supported by the ERC grant agreements No. 819478.
V.P. acknowledges funding from a Marie Curie Action of the European Union
(grant agreement No. 101107047). V.P. also thanks Philipp Frank and Sebastian
Hutschenreuter for the fruitful and inspiring discussions related to this work
during the program "Toward a Comprehensive Model of the Galactic Mag-
netic Field" at Nordita in April 2023, which was partly supported by NordForsk
and Royal Astronomical Society. V.Pa. acknowledges support by the Hellenic
Foundation for Research and Innovation (H.F.R.I.) under the “First Call for
H.F.R.I. Research Projects to support Faculty members and Researchers and
the procurement of high-cost research equipment grant” (Project 1552 CIRCE).
V.Pa. also acknowledges support from the Foundation of Research and Tech-
nology – Hellas Synergy Grants Program through project MagMASim, jointly
implemented by the Institute of Astrophysics and the Institute of Applied and
Computational Mathematics. K.T. and A.P. acknowledge support from the Foun-
dation of Research and Technology – Hellas Synergy Grants Program through
project POLAR, jointly implemented by the Institute of Astrophysics and the
Institute of Computer Science. T.G. is grateful to the Inter-University Centre
for Astronomy and Astrophysics (IUCAA), Pune, India for providing the Asso-
ciateship programme under which a part of this work was carried out. This
work was supported by NSF grant AST-2109127 and was carried out in part at
the Jet Propulsion Laboratory, California Institute of Technology, under a con-
tract with the National Aeronautics and Space Administration. 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.
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