# Sub-mm Wavelength Polarimetry of IRC+10 216

B-G Andersson Institute for Scientific Research, Boston College, 140
Commonwealth Avenue, Chestnut Hill, MA 02467, USA  SOFIA Science Center,
Universities Space Research Association, NASA Ames Research Center, M.S.
N232-12, Moffett Field, CA 94035, USA  Janik Karoly Jeremiah Horrocks
Institute, University of Central Lancashire, Preston PR1 2HE, UK  Pierre
Bastien Institut de Recherche sur les Exoplanètes (iREx) & Centre de Recherche
en Astrophysique du Québec (CRAQ),
Université de Montréal, Département de Physique, 1375, Avenue Thérèse-Lavoie-
Roux, Montréal, QC, H2V 0B3, Canada  Archana Soam Indian Institute of
Astrophysics, II Block, Koramangala, Bengaluru 560034, India  SOFIA Science
Center, Universities Space Research Association, NASA Ames Research Center,
M.S. N232-12, Moffett Field, CA 94035, USA  Simon Coudé Department of Earth,
Environment, and Physics, Worcester State University, Worcester, MA 01602, USA
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA  SOFIA Science Center, Universities Space Research
Association, NASA Ames Research Center, M.S. N232-12, Moffett Field, CA 94035,
USA  Mehrnoosh Tahani Banting and KIPAC Fellow: Kavli Institute for Particle
Astrophysics & Cosmology (KIPAC), Stanford University, Stanford, CA 94305, USA
Covington Fellow: Dominion Radio Astrophysical Observatory, Herzberg Astronomy
and Astrophysics Research Centre, National Research Council Canada, P. O. Box
248, Penticton, BC V2A 6J9 Canada  Michael S. Gordon Ball Aerospace, 1600
Commerce St., Boulder, CO 80301  Sydney Fox-Middleton  Physics Department,
Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053, USA  SOFIA
Science Center, Universities Space Research Association, NASA Ames Research
Center, M.S. N232-12, Moffett Field, CA 94035, USA

###### Abstract

We present SCUBA-2/POL-2 850 $\mu$m polarimetric observations of the
circumstellar envelope (CSE) of the carbon-rich asymptotic giant branch (AGB)
star IRC+10 216. Both FIR and optical polarization data indicate grains
aligned with their long axis in the radial direction relative to the central
star. The 850 $\mu$m polarization does not show this simple structure. The 850
$\mu$m data are indicative, albeit not conclusive, of a magnetic dipole
geometry. Assuming such a simple dipole geometry, the resulting 850 $\mu$m
polarization geometry is consistent with both Zeeman observations and small-
scale structure in the CSE. While there is significant spectral line
polarization contained within the SCUBA-2 850 $\mu$m pass-band for the source,
it is unlikely that our broadband polarization results are dominated by line
polarization. To explain the required grain alignment, grain mineralogy
effects, due to either fossil silicate grains from the earlier oxygen-rich AGB
phase of the star, or due to the incorporation of ferromagnetic inclusions in
the largest grains, may play a role. We argue that the most likely explanation
is due to a new alignment mechanism (Lazarian, 2020) wherein a charged grain,
moving relative to the magnetic field, precesses around the induced electric
field and therefore aligns with the magnetic field. This mechanism is
particularly attractive as the optical, FIR, and sub-mm wave polarization of
the carbon dust can then be explained in a consistent way, differing simply
due to the charge state of the grains.

††software: Astropy (Astropy Collaboration et al., 2013), Matplotlib (Hunter,
2007), SciPy (Virtanen et al., 2020), NumPy (Harris et al., 2020), Starlink
(Currie et al., 2014), SMURF (Chapin et al., 2011)

##  1 Introduction

Visual light interstellar medium (ISM) polarization was discovered in 1949
(Hiltner, 1949a, b; Hall, 1949), and from the start hypothesized to be
associated with dichroic extinction by asymmetric dust grains aligned with the
Galactic magnetic field. The corresponding emission polarization was first
detected by Cudlip et al. (1982). Observations of dust-induced polarization
covering the UV-to-sub-mm111For the purpose of the present paper, we will
distinguish between Far Infrared (FIR) and sub-mm wave, as the former covering
the wavelength range where the atmosphere is fully opaque from the ground,
i.e. $\lambda\approx 30-300\ \mu$m, and the latter being $\lambda\approx 350\
\mu$m – 1 mm. In practice, here, the distinction is between SOFIA/HAWC+, and
JCMT/SCUBA2/POL-2 or ALMA observations. wavelength range serve as efficient
probes of both the geometry and strength of the interstellar magnetic fields
(e.g. Davis, 1951; Chandrasekhar & Fermi, 1953; Houde et al., 2009; Lazarian
et al., 2022), being widely observable with modest telescope size and
observing time and straightforward to calibrate. The general association of
dust polarization with magnetic fields has been securely established, and the
general mechanism of the grain alignment is now understood as being radiation
driven (cf Andersson et al., 2015, henceforth ALV15), but with many details
still needing to be explored in detail.

A quantitative theory of grain alignment based on radiative torques has been
refined and tested over the last few decades (e.g. Dolginov & Mitrofanov,
1976; Draine & Weingartner, 1996; Lazarian & Hoang, 2007, ALV15). This
radiative alignment torque (RAT) theory predicts that the transfer of angular
momentum from photons with wavelengths less than grain diameter ($\lambda<2a$,
where a is the effective grain radius) will spin up an irregular grain222The
coupling to the radiation field does not fully cease at $\lambda>2a$, but
falls with a steep dependence; (cf Hoang et al., 2021). For a paramagnetic
grain, some of the rotational energy is exchanged for spin-flips in the solid
(the “Barnett effect”; Purcell, 1979). This both causes the angular momentum
of the grain to align with the grain’s axis of maximum inertia, so-called
“internal alignment”, through Barnett dissipation (cf. Purcell, 1979; Lazarian
& Draine, 1999) and, in steady state, magnetizes the grain. The magnetized
grain then Larmor precesses around the external magnetic field ($B$-field),
and continued radiative torques aligns the grains with the $B$-field (so
called “B-RAT”). This paradigm has now been well-tested, including
observations of a correlation with the radiation intensity (Medan & Andersson,
2019; Santos et al., 2019; Soam et al., 2021), a correlation of aligned dust
grain size with opacity into a cloud (Andersson & Potter, 2007; Vaillancourt
et al., 2020), a dependence on the angle between the radiation and the
$B$-fields (Andersson et al., 2011; Vaillancourt & Andersson, 2015), and a
loss of the grain alignment in the starless cores (Jones et al., 2015; Alves
et al., 2014, ALV15). However, several detailed aspects of the RAT paradigm
are still in need of further clarification

In particular, the role of grain mineralogy, and the influence of gas-grain
relative velocities, remain to be fully understood. With “mineralogy” we here
refer to the composition and structure of the solid, as it influences
particularly the magnetic properties (dia-, para- or ferromagnetism) of the
grains. One observational effect, related to the grain mineralogy, still
needing to be fully understood is why carbonaceous grains appear not to be
aligned in the interstellar medium (ISM).

While the silicate feature at $\sim$9.7 $\mu$m shows significant polarization
(Smith et al., 2000), the aliphatic CH feature at 3.4 $\mu$m does not (Chiar
et al., 2006). As shown by Chiar et al. (2013), carbon grains will, when
exposed to atomic hydrogen and UV radiation, develop CH bonds on their
surfaces. The lack of polarization in the 3.4 $\mu$m feature therefore implies
that carbonaceous grains are not aligned with any external reference direction
in the ISM.

Chiar et al. (2006) also addressed the possibility, and behavior, of composite
grains by comparing the fractional polarization ($p/\tau$), where p is the
polarization in the line, and $\tau$ is the opacity in the (Stokes I)
absorption line, in the silicate and CH features towards the Quintuplet
cluster members GCS 3-II and 3-IV. They find that the upper limits on
($p/\tau)_{\mathrm{CH}}$ compared with the measured values for
($p/\tau)_{\mathrm{Sil}}$ argues against the model of composite grains with a
silicate core and organic refractory mantles (ORM) and conclude that “the
agent responsible for the hydrocarbon feature in the diffuse ISM is located in
a grain population that is both physically separate from the silicates and far
less efficient as a producer of polarization”.

Recent theoretical work (Draine & Hensley, 2021) argues, however, that the
observational constraints on the CH feature polarization are just above the
expected level of polarization from composite grains consisting of a mix of
amorphous silicates and other materials - so called “astrodust” \- provided
that the 3.4 $\mu$m absorption is preferentially located in grain surface
layers. Constraining the alignment characteristics of pure carbon dust
therefore would be a valuable constraint on ISM modeling.

The non-detection of polarization in the 3.4 $\mu$m line is consistent with
the FIR/sub-mm polarization spectrum models of Draine & Fraisse (2009), which
requires two dust components - a cooler aligned one and a warmer unaligned
one. Aligned silicate and unaligned carbon grains would explain the FIR/sub-mm
wave polarization spectrum measured by Vaillancourt et al. (2008). We note,
however, that recent observations (e.g., Ashton et al., 2018) do not see the
distinct concave spectrum discussed by Vaillancourt et al. (2008).

As noted above, two kinds of alignment in dust grains are required to cause
polarization. “Internal alignment” is accomplished when the grain spins around
one of its principal axes (usually the axis of maximal inertia = smallest
axis; the lowest energy state for a given total angular momentum), yielding a
constant projected shape of that grain. Under angular momentum conservation,
internal alignment requires efficient energy dissipation in the grain bulk,
which Purcell (1979) showed was most efficiently accomplished through Barnett
relaxation in paramagnetic grains (cf Lazarian & Draine, 1999). “External
alignment” signifies the alignment of the individual grains’ spin axes with an
external reference direction, usually the $B$-field. Both types of alignment
assume paramagnetic solids, which in the ISM applies to silicate grains.
Carbon grains are, however, diamagnetic and expected to show neither fast
internal alignment nor to respond to magnetic fields.

Variants of RAT alignment are also possible, dependent on the radiation field
strength and the grain mineralogy. For grains with efficient internal
alignment exposed to strong, highly anisotropic radiation fields, the external
alignment direction is expected to change to the radiation field propagation
direction (its “k-vector”; so called “k-RATs”). This phenomenon may have been
detected in the Orion region (Chuss et al., 2019; Pattle et al., 2021) and
possibly in protostellar disks (Kataoka et al., 2017).

RAT theory does predict that also carbonaceous grains are spun up by an
anisotropic radiation field (as seen in the laboratory by Abbas et al.
(2006)). Hoang & Lazarian (2009) have shown that for strong and anisotropic
enough radiation fields, grains without internal alignment can experience a
second-order k-RAT alignment. Such alignment is likely to be weak and bi-
modal, with some grains oriented with their minor axis along the radiation
field direction and some with the major axis in that direction. In regions
with supersonic gas-grain drift, this drift can preferentially randomize the
former grain-orientation, because of the larger cross-section towards the
flow, leaving a net alignment of grains with their long axis in the gas-dust
flow direction, thus causing polarization.

As discussed recently by Lazarian (2020), a new alignment mechanism should
exist for 1) charged carbon grains with 2) a net velocity perpendicular to the
magnetic field. Such grains get aligned via precession of their electric
dipole moment around the induction electric field (E $\propto$
v${}_{grain}\times$ B). Both of the required conditions are likely fulfilled
in the outer envelope of AGB star winds, due to the radiative driving of the
dust in the AGB wind, and the photoelectric grain charging by the UV light in
the interstellar radiation field. As noted by Lazarian (2020), for inherently
magnetically active grains (paramagnetic or stronger) B-RATs would, in most
situations, dominate this mechanism. Hence, such “E-RAT” alignment is
observationally expected to be associated with carbon grains. This
electrically induced radiative alignment can cause the grains to align with
either their long or short axis along the induced electric field, depending on
the relative precession rate around the electrical and magnetic fields:
($\Omega_{E}$) and ($\Omega_{B}$). This ratio is given by Lazarian (2020,
equation 9)

$\frac{\Omega_{E}}{\Omega_{B}}=\frac{p_{el,J}V_{grain\perp}}{\mu c}$ (1)

where pel,J is the grain’s electric dipole moment parallel to its angular
momentum, $V_{grain\perp}$ is the grain’s velocity perpendicular to the
direction of the magnetic field and $\mu$ is the grain’s magnetic moment

As shown by Lazarian (2020, their Table 1, and Sec. 6.2), if the precession
rate around the E- or B-fields ($\Omega_{E}$) are faster than around the
radiation field direction ($\Omega_{k}$), then, if $\Omega_{E}>\Omega_{B}$,
theory predicts that the observed emission polarization should be parallel to
the $B$-field (opposite to the situation in interstellar - B-RAT - alignment).
If the precession rate around the B-field ($\Omega_{B}$) is faster, the
polarization will be perpendicular to the $B$-field (see Lazarian (2020) for
details). Therefore, in addition to probing for the magnetic field geometry,
such alignment would provide a measure of the micro-physics (including
charging) of the grains.

In the general ISM, silicate and carbonaceous dust are generally well mixed,
making it very difficult to decisively probe this effect, even in
environments, such as shock fronts, where systematic supersonic gas-dust drift
may exist (Hoang & Lee, 2020). We have therefore initiated a program of
observations targeting grain alignment in the chemically segregated
environments of AGB star envelopes using FIR and sub-mm wave emission from the
CSE, and optical polarimetry observations of background stars.

As medium-mass stars first ascend the AGB their CSEs reflect the cosmic
carbon-to-oxygen abundance ratio [C]/[O] of $<$1\. Because of the chemical
stability of the CO molecule, all the carbon in these oxygen-rich CSEs is then
tied up in that gas-phase molecule and the dust formed consists of silicates
and metal oxide grains (Gail & Sedlmayr, 2013). As the thermal pulses of the
later AGB evolution progress, (Lattanzio & Wood, 2004) newly synthesized
carbon is dredged up to the stellar surface and ejected into the CSE,
increasing the [C]/[O] ratio to unity (so called S-type stars) and eventually
to $>$1 when the star becomes C-rich (Whittet, 2003; Olofsson, 2004). The gas
is now depleted of oxygen (which is tied up in CO), and the dust formed
consists of various forms of carbon solids, including amorphous carbon and
silicon carbide (e.g. Ivezić & Elitzur, 1996a).

Observations of the planetary nebula BD +30∘3639 show that the size of the
object at wavelengths dominated by silicate features is larger than that at
wavelengths dominated by carbonaceous spectral features (Guzman-Ramirez et
al., 2015), indicating that the O-to-C rich history of the shell may, in some
sources, be traced through high-resolution observations.

IRC+10 216 is a well known infrared object discovered in 1969 by Eric Becklin
(Becklin et al., 1969) and identified as a dust enshrouded carbon star the
following year (Miller, 1970). At a distance of only $d=123\pm$14 pc;
(Groenewegen et al., 2012) it can be resolved by many telescopes, and it has
been extensively studied both in continuum and line radiation (e.g. Cernicharo
et al., 2010; Decin et al., 2011, and refs. therein). The extensive dust and
gas circumstellar shell ends in a termination shock, where the CSE runs into
the ISM, seen in both the ultraviolet (Sahai & Chronopoulos, 2010) and the FIR
(Ladjal et al., 2010). The effective extent of the CSE of IRC+10 216 depends
on the wavelength of observation and its depth. For reference, the CSE can be
detected in visible light (azimuthally averaged to $\sim$200″(Mauron &
Huggins, 2000), the 70$\mu$m extended emission can be detected to at least
285″, (Dharmawardena et al., 2018) and the asymmetric astrosphere is located
at $\sim$ 500-600″(Sahai & Chronopoulos, 2010) from the central star.

With a high C/O ratio (C/O=1.4 Winters et al. (1994), Milam et al. (2009, and
ref.s therein)), the dust is fully carbonaceous, consisting of 95% amorphous
carbon and 5% SiC by mass; Ivezić & Elitzur (1996a). While iron is found to be
significantly depleted in the gas phase of the CSE (Mauron & Huggins, 2010),
no direct observational evidence for iron grains or iron inclusions in the
carbon grains, has been found.

Evidence for magnetic fields in the IRC+10 216 CSE come from observations of
the Goldreich-Kylafis (G-K) effect in several molecules (Girart et al., 2012)
with the Submillimeter Array (SMA), and from Zeeman observation in the J=1-0
line of CN (Duthu et al., 2017) with the IRAM 30 m telescope. The G-K
observations on small scales indicate a “global radial pattern” (Girart et
al., 2012). However, the CN Zeeman observation shows significant spatial
variations (including field direction reversals) in the line of sight field
strength around the star, indicating deviations from spherical symmetry.
Hence, while there are strong indications of a magnetic field in the CSE, the
geometry of the field is not fully clear. Evidence for simple dipole
geometries in AGB star CSEs come, for instance, from Szymczak et al. (2001),
using OH maser emission and Vlemmings et al. (2005), using H2O masers. Both
find that the magnetic field on hundreds-to-thousands of AUs scales in the AGB
star VX Sgr (Tabernero et al., 2021) can be modeled as resulting from a
dipole.

For sources such as AGB stars, observed with unresolved aperture or centered
slit polarimetry, asymmetries in the dust distribution and grain alignment
effects are very difficult to separate. Kahane et al. (1997); Bastien (2003)
acquired spatially unresolved aperture polarimetry of a sample of 68 AGB stars
in a band centered at 0.883$\mu$m. They divided their sample into two groups,
spherical and aspherical envelopes, based on their CO line profiles. Both
polarization and CO line profiles can reveal the presence of non-spherical
envelopes. No statistical difference was found in polarization histograms of
stars with “normal” (spherical) and “abnormal” (aspherical) CO line profiles
(Kahane et al., 1997). However, the polarization data showed a significant
difference between C-rich and O-rich envelopes, with C-rich CSE presenting a
higher polarization than their O-rich counterparts (Kahane et al., 1997;
Bastien, 2003), indicating a more asymmetric dust distribution in the former.
Bieging et al. (2006) performed optical (0.42 - 0.84$\mu$m) spectro-
polarimetry of 21 AGB stars, 13 proto-planetary nebulae, and two R CrB-type
stars. They, also, found a higher fraction of polarization for the carbon rich
AGB stars (five of six) than for the oxygen rich stars (eight of fourteen),
which they attributed to the later developmental stage of the carbon-rich
stars. All of these observations are, however, likely dominated by scattering
polarization. High-resolution, Active Optics (AO) supported, observations in
the optical and near infrared can resolve the inner part of the CSEs (e.g.
Kastner & Weintraub, 1994, 1996; Jeffers et al., 2014; Khouri et al., 2020;
Montargès et al., 2023) and provide important information about the spatial
dust distribution and grain size distribution. However, since these
observations are dominated by [Rayleigh] scattering, they do not provide
information about the grain alignment. In most radiative transfer modeling of
these kinds of data, (e.g. Montargès et al., 2023) the grains are either
assumed to be symmetrical or have a random spatial orientation. However,
imaging FIR and sub-mm wave observations can resolve the CSEs and can
differentiate between the two mechanisms of polarization (scattering and
dichroic emission), as the emission wavelengths are much larger than the grain
sizes. Similarly, optical polarimetry of background stars with careful
subtraction of the diffuse, scattered, light from the CSE can show how the
grains are aligned.

IRC+10 216 has been previously observed at multiple wavelengths in scattering,
dichroic extinction, and emission polarization. Optical through near-infrared
polarization observation of the central part of the CSE has been reported by
e.g. Dyck et al. (1971); Cohen & Schmidt (1982); Trammell et al. (1994)
showing significant polarization rising to the blue (40% at 0.67 $\mu$m (Cohen
& Schmidt, 1982)), consistent with Rayleigh (dust) scattering. The small scale
dust distribution (2-3″ scales, and hence much smaller than the beams in our
studies) has also been extensively studied in recent years. Kastner &
Weintraub (1994, 1996) mapped the large-scale structure of the CSE at J, H and
K-bands, while Murakawa et al. (2005), using 2 $\mu$m polarization, found a
NW-SE structure that they identified as a possible dust torus. Optical broad-
band polarimetry (Jeffers et al., 2014) reveal a similar structure, which is
also consistent with multi-epoch HST imaging (Kim et al., 2021). In an
accompanying paper (Andersson et al., 2023b, in preparation, Andersson et al.,
2018)333See also presentation no. 5, e-proceedings of “From Stars to Galaxies
II”: http://cosmicorigins.space/fstgii we discuss polarimetry of background
stars shining through the CSE. While we detect strong scattered light
polarization, we can, with careful sky-subtraction, also determine the
dichroic extinction polarization towards these stars.

Our SOFIA/HAWC+ observations at 53 $\mu$m (at 5″ resolution Andersson et al.,
2022) show a uniform radial polarization for IRC+10 216. The FIR polarization
maps at 154 and 214 $\mu$m also show predominantly radial polarization, albeit
with lower S/N and at worse resolution. A tight correlation between the FIR
polarization and dust temperature indicates radiative grain alignment, via the
second-order k-RAT process discussed above. The low polarization efficiency
seen in the inner parts of the 850 $\mu$m data presented here (Figure 10) is
consistent with k-RAT alignment of grains with no (or slow) internal alignment
(Hoang & Lazarian, 2009).

Here we present new JCMT/SCUBA-2/POL-2 observations of the IRC+10 216 CSE and
analyze these in the context of SOFIA/HAWC+ and optical polarimetry (as well
as other archival data sets) and the general RAT grain alignment paradigm.
Along with Andersson et al. (2022), this paper is the start of a comprehensive
study we’ve named the Survey of Polarization in AGB Circumstellar Envelopes
(SPACE). We are working to complement the studies of IRC+10216 with
observations of O-rich and S-type AGB stars IK Tau and W Aql.

The paper is structured as follows: We present the observations and data
reduction method in Section 2. Section 3 introduces the analysis of our
results and Section 4 summarizes these before Section 5 where we discuss our
results and possible alignment mechanisms. Finally, we summarize our findings
in Section 6.

##  2 Observations and Data Reduction

IRC+10 216 was observed with the SCUBA-2/POL-2 instrument combination (Holland
et al., 2013; Bastien et al., 2011) on the James Clark Maxwell Telescope,
during 2018 January 11 and 12. Six observations lasting 41 minutes each of
IRC+10 216 were carried out. The regular POL-CV daisy pattern for
SCUBA-2/POL-2 covers the star and all of its circumstellar envelope well. We
acquired 12 such patterns to reach a sensitivity of about 3 mJy/beam in
polarized intensity in 8 arcsec2 pixels. Six additional observations were
acquired in flexible mode on January 19. We have also included 8 repeats that
are $\sim$31.5 minutes long from archival observations on CADC444Canadian
Astronomy Data Centre: https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/
(Project ID: M19BP001; PI: Peter Scicluna).

During one observation on January 11, the telescope reached an elevation of
83°. However, upon inspection of the data from that observation, no
distortions of the data due to the JCMT alt-az mount could be detected. The
membrane in front of the telescope, in regular observations, was removed
during the period 2017 December 5 to 2018 January 10 to carry out
commissioning tests targeted at better characterizing the instrumental
polarization due to the telescope and membrane. The data for this project
were, therefore, acquired just after the reinstallation of the Gore-Tex
membrane in front of the telescope. Comparison of the instrumental
polarization after the membrane was reinstalled with those before it was
removed showed no significant differences.

IRC+10 216 was previously observed with the earlier JCMT sub-mm wave
polarimeter SCUPOL. These observations were reprocessed, combined and
presented by Matthews et al. (2009). Because these data were of lower signal-
to-noise, we have not included them in the present analysis.

We reduced the JCMT/POL-2 850 $\mu$m observations using the pol2map command of
the SMURF package (Chapin et al., 2011) in the Starlink software (Currie et
al., 2014). In the first step, using pol2map, the Stokes I, Q and U time
streams are separated from the raw observations using the calcqu command. Then
the Stokes I time streams are processed using the command makemap to produce
an initial Stokes I map. This Stokes I map is used as a fixed-SNR mask for the
second step of the reduction where final Stokes I, Q and U maps are made. The
second step consists of running pol2map again, however we use
skyloop555http://starlink.eao.hawaii.edu/docs/sc22.pdf to make the final maps
rather than makemap. Skyloop is an iterative map-making command that reduces
the growth of large-scale structures that may form due to the map-making
routine by combining all the observations at each iteration, rather than each
observation being created individually and then combined at the end. A
polarization vector catalog is then created using the Stokes I, Q and U maps.
The data were reduced using a pixel size of 8$\arcsec$ with a 14$\arcsec$
beam. The final Stokes I, Q, and U maps were flux calibrated, in units of $\rm
mJy\,beam^{-1}$, using a flux calibration factor (FCF) for 850 $\mu$m of 748
$\rm Jy\,beam^{-1}\,pW^{-1}$ and in units of $\rm mJy\,arcsec^{-2}$ using a
FCF of 2.93 $\rm Jy\,arcsec^{-2}\,pW^{-1}$ 666These conversions were done
using the CALIBRATE-SCUBA-2-DATA recipe under the PICARD package in STARLINK..

The final polarization values in the vector catalog are debiased using the
Stokes Q and U variances to remove the statistical bias in regions of low
signal-to-noise ratio (SNR; Wardle & Kronberg, 1974).

The values for the debiased polarization $P$ were calculated from

$P=\frac{1}{I}\sqrt{Q^{2}+U^{2}-\frac{1}{2}(\delta Q^{2}+\delta U^{2})}\,\,,$
(2)

where I, Q, and U are the Stokes parameters, and $\delta Q$, and $\delta U$
are the uncertainties for Stokes Q and U. The uncertainty $\delta P$ of
polarization was obtained using

$\delta P=\sqrt{\frac{(Q^{2}\delta Q^{2}+U^{2}\delta
U^{2})}{I^{2}(Q^{2}+U^{2})}+\frac{\delta I^{2}(Q^{2}+U^{2})}{I^{4}}}\,\,,$ (3)

with $\delta I$ being the uncertainty for the Stokes I total intensity.

The polarization position angles $\theta$, increasing from north to east in
the sky projection, were measured using the relation

${\theta=\frac{1}{2}\tan^{-1}\frac{U}{Q}}\,.$ (4)

The corresponding uncertainties in $\theta$ were calculated using

$\delta\theta=\frac{1}{2}\frac{\sqrt{Q^{2}\delta U^{2}+U^{2}\delta
Q^{2}}}{(Q^{2}+U^{2})}\times\frac{180^{\circ}}{\pi}\,\,$ (5)

Vectors were then selected using SNR cuts of $I>$ 20 $\delta I$ and $P>$ 2.5
$\delta P$. The selection criteria were chosen as a compromise between high
SNR constraints and maintaining a significant number of polarization vectors.
The cut in polarization is roughly equal to an uncertainty in the position
angle of $\delta\theta=11.5^{\circ}$ (Naghizadeh-Khouei & Clarke, 1993) which
is quite reasonable. The SNR constraints are common throughout studies using
SCUBA-2/POL-2 data (e.g. Kwon et al., 2018; Liu et al., 2019).

The earlier observations of IRC+10 216 with SCUPOL (Matthews et al., 2009) are
of lower SNR with cuts of $I>$ 0 and $P>$ 2 $\delta P$. We show the SCUPOL
polarization vectors plotted with our POL-2 vectors in Figure 1. When applying
a similar SNR cut to the SCUPOL vectors as our own, we have $\sim$50% more
vectors with POL-2, increasing from 15 to 22 vectors. We can also compare the
vectors that spatially match. There are only seven vector pairs which
spatially match within 8$\arcsec$ which we have chosen due to our vector
catalog being binned to 8$\arcsec$. While the polarization amount is generally
mutually consistent, the polarization angles vary significantly, with only
three vectors agreeing well within the position angle uncertainties for
moderate SNR data (Naghizadeh-Khouei & Clarke, 1993). We note that SCUPOL was
a different polarimeter and observed in a different mode than POL-2 which is a
significant upgrade in sensitivity and ability to deal with atmospheric
variations. So we do not expect perfect agreement, but the POL-2 data are of
higher quality.

Figure 1: The SCUPOL polarization vectors (Matthews et al., 2009) with
$I/\delta_{I}>0$ and $P/\delta_{P}>2$ are shown as black dotted vectors
(binned to a 10$\arcsec$ grid), with our POL-2 polarization vectors in solid
black (on an 8$\arcsec$ grid) with $I/\delta_{I}>20$ and $P/\delta_{P}>2.5$.
The POL-2 data has better sensitivity than SCUPOL data, with approximately 50%
more vectors at the same SNR cut. The color background shows the POL-2 Stokes
I component with the SCUBA-2/POL-2 850 $\mu$m beam size shown in the bottom
left. All beam sizes quoted in this paper are in terms of full-width at half
maximum (FWHM)

Figure 2 shows the azimuthally averaged variation of intensities of different
wavelength emissions with the offset from IRC+10 216. The beam-sizes
corresponding to each observation are also indicated in the plots. The
variation of 53 $\mu$m intensity with radius is clearly different from that of
850 $\mu$m, while both are decreasing with offset. However, a similarity in
the intensity variations of 154 and 850 $\mu$m can be noticed in the right
panel of the figure. The data for the 53 and 154 $\mu$m traces are taken from
Andersson et al. (2022).

Figure 2: (Left panel) The variation of Stokes $I$ intensity with the
projected radius of IRC+10 216 in 53 and 850 $\mu$m wavelengths is shown.
(These are azimuthally averaged fluxes around the star. The shells of
2.5$\arcsec$ widths were drawn around the central star and average flux was
extracted in those shells.) The right panel shows the same plot, but for 154
and 850 $\mu$m wavelengths. Vertical dashed lines indicate the beam sizes in
respective wavelengths.

##  3 Analysis

Both at small (e.g. De Beck et al., 2012) and large (e.g. Mauron & Huggins,
1999) scales, the over-all density of the shell is well-described by a
$n\propto r^{-2}$ density, out to the astrosphere (Sahai & Chronopoulos, 2010;
Ladjal et al., 2010). Modeling using both DUSTY (Ivezic et al., 1999) e.g.
Ivezić & Elitzur (1996a); Andersson et al. (2022) and MCMax (Min et al.,
2009), e.g. De Beck et al. (2012) yield good agreement between the emitted
spectra and a monotonously decreasing temperature distribution in the envelope
(e.g. Ivezić & Elitzur, 1996b; De Beck et al., 2012; Andersson et al., 2022).
The emission at both FIR and sub-mm wavelengths are optically thin (Ivezić &
Elitzur, 1996a; Andersson et al., 2022, Table 4 of the latter), and for the
radial offsets from the star considered here, the dust temperature is low
enough ($\lesssim$200 K) that all our observed wavelengths are well into the
Rayleigh-Jean limit. Therefore, the Stokes $I$ emission from the shell, for
all wavelengths considered, is dominated by the density profile in the CSE.

###  3.1 Radial coordinate polarization

To better study the radial polarization in IRC+10 216, we use “radial Stokes
parameters” $Q_{r}$ and $U_{r}$, as defined in equation 6, where the reference
frame is rotated by the polar angle at each point with respect to the star
location:

$Q_{r}=+Q\cos{2\xi}+U\sin{2\xi},\leavevmode\nobreak\ \leavevmode\nobreak\
U_{r}=-Q\sin 2\xi+U\cos 2\xi,$ (6)

where $\xi=\arctan{\frac{x-x_{0}}{y-y_{0}}}$ is the polar angle of a given
pixel coordinate position $(x,y)$ (corresponding to equatorial coordinates)
and $(x_{0},y_{0})$ is the location of the star. Because of the moderate
Declination of IRC+10 216 ($\sim$13.25∘) we have omitted the cos($\delta$)
factor for the Right Ascension distance in the numerator.

Positive $Q_{r}$ shows radial polarization, while negative $Q_{r}$ indicates
tangential polarization. $U_{r}$ represents polarization with an angle of $\pm
45^{\circ}$ from the radial direction. Therefore, zero $U_{r}$ indicates
purely radial or tangential polarization.

We take thin annuli ($\sim 2.2\arcsec$ in HAWC$+$ and $\sim 14\arcsec$ in JCMT
observations) around the star and find the averaged $Q_{r}$ and $U_{r}$ in
each annulus to identify the radial profile of $Q_{r}$ and $U_{r}$. Following
that we determine

$\theta_{r}=0.5\times\arctan\big{(}\frac{<U_{r}>}{<Q_{r}>}\big{)},$ (7)

as defined by Tahani et al. (2023) and shown in Figure 3 for the HAWC$+$ 53
$\mu$m (upper left panel), HAWC$+$ 154 $\mu$m (upper right panel) and the JCMT
POL-2 850 $\mu$m observations (lower panel). The data have the same
constraints of SNR($I$)$>20$ and SNR($P$)$>2.5$. Allowing for 5% uncertainty,
a point with $\cos{\theta_{r}}>0.95$ (including their error bar) represents
radial polarization at that radius. This choice is clearly somewhat arbitrary,
but as can be seen from Figure 3 provides a reasonable separation between
systematically radial and non-radial polarization. The HAWC$+$ 53 $\mu$m
observations show clear radial polarization between 6″and 45″, while the JCMT
observations do not indicate a clear radial polarization. The HAWC$+$ 154
$\mu$m observations show radial polarization only from 5″to 28″.

The blue x markers in Figure 3 show the number of pixels with polarization
measurements included for each radial bin. Comparing the uncertainties
indicated for the cos($\theta_{r}$) values with the variation in the number of
points per bin shows that the dominant source for the larger uncertainties is
dispersion within each sample, rather than localized higher measurement
errors.

Figure 3: Radial polarization with SNR($I$)$>20$ and SNR($P$)$>2.5$. The $x$
and $y$ axes represent the radial profile of $\cos{\theta_{r}}$ as described
in equation 7 and the radius from the center of the star. The dashed
horizontal line shows $\cos{\theta_{r}}=0.95$, with points above this line
(including their error bars) representing radial polarization (allowing for 5%
uncertainty; see Tahani et al. (2023) for more detail). The blue dots
illustrate the number of pixels (satisfying the SNR cut) in each annulus.
Upper left panel: HAWC$+$ 53 $\mu$m observations. Upper right panel: HAWC$+$
154 $\mu$m observations. Lower left panel: JCMT POL-2 observations. Lower
right panel: A cartoon illustrating the radial polarization frame (see Eq. 6).

###  3.2 CN Zeeman

Duthu et al. (2017) observed the Zeeman effect in the IRC+10 216 CSE, using
the CN (J=1–0) line at 113 GHz with the IRAM 30m telescope. They detected the
line-of-sight magnetic field in five locations around the central star (plus
one upper limit) with radial offsets ranging from 18 to 27$\arcsec$. Because
of the varying distances from the star, they - and we - scale the observations
to a common distance from the star to allow comparisons with models. They
normalized their measurements to a common distance using a $r^{-1}$
dependence, based on Vlemmings (2012). As shown by Figure 4 a coherent
variation of the line of sight $B$-field is indicated by these data. We used
this $r^{-1}$ scaling, as well as a $r^{-3}$ scaling, appropriate for a dipole
field, and fit the resulting values to an azimuthal cosine function.

$B_{LOS}=a\cdot cos(\Theta-\Theta_{0})$ (8)

With the limited number of data points, and marginal signal-to-noise these
fits are not conclusive, but provide an indication of the magnetic field
structure in the CSE, especially the orientation of a possible dipole field
orientation. Figure 4 shows the best fit cosine functions (weighted, in solid
lines, and unweighted, in dashed lines) of the Duthu et al. (2017) data for
both scalings ($r^{-3}$ in blue and $r^{-1}$ in black). We show both types of
fits, since we scaled also the uncertainties in the $B$-field by the two
radial scaling. In the fits, we have set the value at (-18, -10) - an upper
limit in Duthu et al. (2017) \- to zero and assigned the upper limit as the
uncertainty (indicated in Figure 4 by wedges showing the upper limit and an
open symbol for the value used in the fits). Consistent results are found if
the point is simply omitted. The various fits give mutually consistent
results. In all cases, a pole orientation around 90-125∘ East of North is
found, albeit with significant uncertainties. For the $r^{-1}$ scaling, the
weighted and unweighted fits, we find pole-orientations of 113$\pm$17∘ and
119$\pm$21∘, respectively. For the $r^{-3}$ scaling, the weighted and
unweighted fits yield pole-orientations of 91$\pm$16∘ and 124$\pm$21∘,
respectively.

Figure 4: The line-of-sight magnetic field strength, based on the CN (J=1-0)
Zeeman observations of Duthu et al. (2017), scaled to a common radial distance
from the central star using an r-3 scaling (blue open symbols and curves,
right axis) and using an r-1 scaling (black, filled, symbols and curves, left
axis). The weighted (W; solid lines) and unweighted (UW; dashed lines) fits
yield consistent results, both in amplitude and phase (see text for details).

###  3.3 Dipole Modeling

Given the clearly non-radial polarization pattern in the 850 $\mu$m data,
contrasting with the FIR HAWC+ 53 and 154 $\mu$m results, an alternative
polarization geometry is required - unless the geometry is inherently random.
However, a truly random orientation of the grains and/or geometry, would
produce a zero polarization. Hence, we need an ordered structure in the CSE
which could allow grain alignment and produce the non-radial polarization
pattern observed. Based on the results of Szymczak et al. (2001) and Vlemmings
et al. (2005), the fits to the Zeeman data discussed above, and the simplicity
of geometry (lowest order terms of a spherical harmonic expansion), we chose
this to be a projected dipole field. This implies a magnetic dipole, but does
not at this point require it.

We therefore compared the observed 850 $\mu$m polarization pattern to a simple
model of a projected magnetic dipole of the form described in Appendix A.
Because of the simplifying assumptions in our model, we only use the position
angles, but not the amount of polarization, in our fitting. We generated a
family of models with varying rotation angles $\Theta$, where $\Theta$ is the
rotation of the dipole model (E-vectors) in the plane-of-sky (POS). We took
the CSE to be the central area where $I/\delta_{I}>10$ (though we maintained
our $I/\delta_{I}>20$ cut on polarization vectors in this area) which was a
region with diameter of $\approx$104$\arcsec$. We generated the dipole on a
104$\times$104 grid and set the spacing to 1$\arcsec$. We sought a maximum
likelihood estimate (MLE) (Bevington & Robinson, 1992) of the model rotation
angle, by performing a $\chi^{2}_{\rm red}$ minimization of the model fits to
the polarization position angle data. We used the $\chi^{2}_{\rm red}$
normalization in the least-squares MLE since this provides an indication of
the absolute goodness of the model fits. Because the uncertainties for
polarization data (amount, $p$ and position angle $\theta$) for marginal
signal-to-noise are not Gaussian, care has to be taken in such minimization.
We used the analysis of such non-Gaussian position angle uncertainty
distributions by Naghizadeh-Khouei & Clarke 1993, specifically their eq. 3 & 4
to assign the weights in the $\chi^{2}_{\rm red}$ modeling of the position
angles. With our limited selection of only vectors with p $>$ 2.5$\delta_{p}$
this should be a minor concern but we include the factor of 1.2 in Eq. 9.

In the fitting, we fix the center of the dipole model at the center of the
observed map. We calculated models on a fixed rectangular grid, over a POS
rotation $\Theta$ of 0∘ to 180∘, in spacings of 5 degrees. For each case, we
find the closest model vector to each observed polarization vector. We then
calculate a goodness of fit parameter (reduced $\chi^{2}_{\rm red}$; Bevington
& Robinson, 1992, but see also (Andrae et al., 2010)) for that rotation
$\Theta$:

$\chi^{2}_{\rm
red}=\frac{\sum\limits_{i=1}^{n}\frac{(\theta_{obs}-\theta_{model})^{2}}{(1.2\,\delta_{\theta})^{2}}}{N-N_{c}}$
(9)

where $N$ is the number of data points fitted (25 POL-2 vectors passed our S/N
threshold), $N_{c}$ is the number of constraints in the model (here just 1:
$\Theta$), $\delta_{\theta}$ is the uncertainty in polarization position angle
(see eq. 5) and $\theta_{obs}$ and $\theta_{model}$ are observed polarization
position angle (see eq. 4) and polarization position angle from the model,
respectively. The best fit model is overlaid on the observational data in the
left-hand panel of Figure 5. The resulting reduced $\chi^{2}_{\rm red}$ plot
is shown in the right panel of Figure 5. We find a clear minimum
$\chi^{2}_{\rm red}$ of $\sim$25.6 when the dipole is rotated to
$\Theta_{E}$=125∘ East of North (or -55∘ with the normal POL-2 notation). The
average measurement uncertainty for the position angles is 11.5∘ with a
standard deviation of 6.5∘. The rotation angle is well constrained, with a
fitting uncertainty of $\sim$15∘ .

Figure 5: Left: The 850 $\mu$m dust emission map is shown with the POL-2 850
$\mu$m polarization vectors ($I>$ 20$\delta_{I}$ and $p>$ 2.5$\delta_{p}$)
plotted in black over the best fit dipole morphology shown by the cyan stream-
plot (the orientation of the field-lines represent the polarization
“E-vectors”). The image is zoomed in to show just the envelope. Right: A plot
of the reduced $\chi^{2}_{\rm red}$ value which is found using Eq. 9 as it
varies with the different field rotations $\Theta$. A minimum can be seen
around $\Theta$ = 125∘ (E-vectors) with an uncertainty of $\sim$15∘

The minimum value of reduced $\chi^{2}_{\rm red}\sim 26$ in our analysis
indicates that this is either, over-all not a good fit or, that the model is
not a complete description of the data. Given the clear minimum in the
$\chi^{2}_{\rm red}$ we explore its significance with two simple tests (cf
Andrae et al., 2010). Figure 6 (left) shows the distribution of residuals
between the observed and modeled position angles at the $\chi^{2}_{\rm red}$
minimum. While broad, the residuals show a peaked distribution centered on
zero (within the uncertainty). The normalized residuals also show a clear
Gaussian distribution, centered at zero, albeit with $\sigma^{2}>>1$.
Additionally, we performed a jack-knife test (Lupton, 1993; Andrae et al.,
2010), where we omitted each data point, in turn, and repeated the
$\chi^{2}_{\rm red}$ minimization. The resulting variation in $\chi^{2}_{\rm
red}(\Theta)$ is shown in the right-hand panel of Figure 6. The general shape
of the fit-results, and location of the minimum, are consistent for all data-
point exclusions. These results point to a dipole as being a meaningful part
of the true, underlying, magnetic field geometry. More and higher signal-to-
noise data, as well as a full radiative transfer model treatment, are required
to conclusively address this question.

Figure 6: To test the validity of the $\chi^{2}$ minimization, noting that the
reduced $\chi^{2}$ never approaches zero, we follow Andrae et al. (2010).
(Left) The residuals for the best fit solution yields a peaked distribution.
(Right) Recalculating the $\chi^{2}$ minimization while eliminating each of
the data points, in turn, shows a consistent $\chi^{2}$ minimum. The ordinate
shows the eliminated data point (arbitrarily numbered).

With the reasonable fit to the 850 $\mu$m data, and since the 154 $\mu$m
SOFIA/HAWC+ data at offsets beyond $\sim$30″ from the central star show
significant deviations from a fully radial polarization pattern, we applied
the model also to the 154 $\mu$m SOFIA/HAWC+ observations. If we combine a
‘mono-pole’ (radial) geometry inside 30″ with a dipole model at larger radii
for the 154 $\mu$m observations, the vectors in the north-eastern quadrant can
be better fit than by a radial geometry alone. Even so, the over-all reduced
$\chi^{2}$ for the fit with only a mono-pole component is lower than for the
combined fit.

###  3.4 Comparison of polarization properties in different wavelengths

Figure 7: Left panel shows the SOFIA/HAWC+ 53 $\mu$m polarization vectors in
cyan in the envelope of IRC+10 216 with JCMT/POL-2 850 $\mu$m polarization
vectors overlaid in black. Background image is the 53 $\mu$m emission. The
right-hand panel shows the JCMT/POL-2 850 $\mu$m data, in black, overlaid on
SOFIA/HAWC+ 154 $\mu$m emission with polarization vectors (in cyan). Because
of the large beam sizes, relative to the offset from the CSE center, the 850
$\mu$m data may be prone to significant beam-averaging and not show the true
polarization. In this figure, we show the over-sampled maps of the HAWC+ data
to emphasize the differences in polarization geometry. Beam sizes are shown in
the lower left corner.

The clear differences in the polarization geometry between the 53 and 850
$\mu$m data, makes it is worthwhile to directly compare the polarization at
several wavelengths. Figure 7 compares the polarization geometry for the 53,
154 and 850 $\mu$m data. The 154 $\mu$m does show a somewhat intermediary
geometry, possibly indicating a transition from the radial polarization seen
at 53 $\mu$m to the “dipole-like” structure seen at 850 $\mu$m. We note,
however, that while the beam sizes at 154 and 850 $\mu$m are comparable, they
are significantly larger than that for 53 $\mu$m, so the comparison is not
fully straight-forward.

Figure 8 shows the polarization fractions at 53, 154 and 850 $\mu$m, as
functions of the total intensities. A similar drop from high polarization and
low intensities at larger radii to low polarization and high intensities near
the star can clearly be seen for all wavelengths. Fitting single component
power-laws, the drop is steeper for 850 $\mu$m, with a slope (exponent of a
power-law) of -0.89$\pm$0.04. Shallower, single-slope relations are found for
154 $\mu$m (slope= -0.41$\pm$0.09) and for 53 $\mu$m (slope= -0.31). As is
clear from Figure 8, a single component power-law may, however, not be
adequate, especially for the 53 $\mu$m data. A two power-law fit to the 53
$\mu$m data yields an initial slope consistent with that for the 850 $\mu$m
data (slope= -0.85$\pm$0.06), with a slope of -0.45$\pm$0.05 at higher
intensities, slightly offset to higher polarization fractions. The 154 $\mu$m
polarization shows an indication of a similar offset/second-slope at the
highest intensities in that band (I$\geq$0.2 Jy/arcsec2). This potential
second distribution is most strongly indicated by the fact that if the 154
$\mu$m data were fitted to only intensities less than $\sim$0.1 Jy/arcsec2 a
significantly steeper power-law would have resulted (slope= -0.71$\pm$0.02).
The “jump” in the 53 $\mu$m polarization occurs at $\sim$3 Jy/arcsec2. As
Figure 2 shows, the two intensities occur at similar radial distance from the
central star of $r\approx 12.5\arcsec$. We note that at this radius, the
Stokes $I$ intensity for the 850 $\mu$m emission is $\sim$6.5 mJy/arcsec2 and
therefore well beyond the current polarization sensitivity. Hence, the
polarization fractions in each band vary in similar manners with a steeper
slope at large distances in the CSE and shallower slopes inside about 12.5″.
This may indicate a change in alignment mechanism within the shell, or a
reduction in the alignment efficiency (including disalignment effects) closer
to the star. While the higher gas density and dust temperature closer to the
central star would both, nominally, imply larger collisional and radiative
disalignment (cf Draine & Lazarian, 1998), we would not expect them, by
themselves, to be able to explain the discrete change in slope of the
polarization. Both the radiation field and gas density in CSEs are expected to
be continuous and vary smoothly with radius. Although significant, small
scale, structure is known to exist in the CSE (eg. Mauron & Huggins, 1999;
Leão et al., 2006; Ladjal et al., 2010), the general r-2 radial density
dependence will still dominate the radial density profile. Hence, the observed
discrete changes in polarization efficiency, likely require a more qualitative
change in the grain alignment, such as a different mechanism.

Figure 8: The polarization fraction for the 53, 154, and 850 $\mu$m
polarization is plotted against the Stokes $I$ intensities at each wavelength.
For the 850 $\mu$m data, a single well-defined power-law is apparent. The 53
$\mu$m is well fit by a two-component distribution with a steeper slope at
small fluxes (similar to the 850 $\mu$m one) followed by a shallower slope,
offset to somewhat higher values at the intersection. The 154 $\mu$m falls in
between the two, with a hint of an offset, seen in the 53 $\mu$m data, at the
high end of the 154 $\mu$m intensity distribution. If the 154 $\mu$m data are
fit to only intensities less than 0.1 Jy/arcsec2 an exponent consistent with
that for the steep part of the 53 $\mu$m and the 850 $\mu$m data is found. See
text for details. All data have been debiased.

###  3.5 Line Polarization

As noted above, Goldreich-Kylafis polarization (Goldreich & Kylafis, 1981,
1982) has been observed in several sub-mm wave lines in the envelope of IRC+10
216 (Girart et al., 2012). They mapped the linear polarization from the lines
of CO (J=3–2) at $\nu$=345.796 GHz, SiS (J=19–18) at $\nu$=344.779 GHz, and CS
7–6 at $\nu$=342.883 GHz. These lines are all located at wavelengths with
transmissions above 95% of the peak transmission of the SCUBA-2 850 $\mu$m
filter (Cookson et al., 2018). The SMA primary beam (6m antennae) at 850
$\mu$m is about 36″and Girart et al. (2012) achieved a synthesized beam of
2.6″$\times$1.6″. Also, as shown by Girart et al. (2006), the SMA filters out
emission that arises from structures larger than about 10″, so the two data
sets are not trivial to compare in a quantitative way. However, if we use the
CO data from Girart et al. (2012) and the conversion factors given in Drabek
et al. (2012), we can estimate the possible Goldreich-Kylafis contamination in
our observations. Using the Girart et al. (2012) measurement of the Stokes $Q$
parameter, we find a peak line brightness temperature of 2 K and a width of 20
km/s. If the JCMT conversion factor is valid also for the Stokes $Q$
parameter, we find a polarized intensity on the order of 30 mJy/beam which
would contribute a significant signal in our polarized intensity, assuming the
spatial filtering does not strongly affect the comparison.

Figure 9: Comparison between the (upper left) JCMT/POL-2 850 $\mu$m (black
vectors) and SOFIA/HAWC+ 53 $\mu$m (cyan vectors) polarization with the line
polarization observation by Girart et al. (2012) (reproduced with permission).
The red square in the upper-left map shows the approximate extent of the three
sub-figures from Girart et al. (2012). In the three panels from Girart et al.
(2012) the linearly polarized intensity is shown as color plots, while the
contours represent the Stokes $I$ emission (see Girart et al. (2012) for
further details). While the JCMT/POL-2 data at this scale is prone to some
beam averaging, the polarization geometries between the continuum and line
polarization are not consistent with a common origin. Note that the caption of
Figure 2 in Girart et al. (2012) mislabels the SiS and CS images.

However, if we compare the polarization geometry between our observations
(SOFIA/HAWC+ and JCMT/POL-2) with those of Girart et al. (2012) the picture
becomes more complicated. As Figure 9 shows, the latter only detect emission
in the very inner part of the CSE and our field of view, where we might expect
beam averaging effects in the JCMT/POL-2 data. Nonetheless, none of the CO,
SiS or CS line emission data show polarization geometries that are consistent
with those seen in the continuum, either at 53 or 850 $\mu$m. In addition, the
position angle of the CS (J=7-6) emission (PA$\sim 48^{\circ}$) is close to
perpendicular to those for CO (J=3-2) (PA$\sim 155-170^{\circ}$) and SiS
(J=19-18) (PA$\sim 139^{\circ}$), and the different line polarization
components would therefore offset each other within the broad SCUBA-2 band-
pass filter (and larger beam).

###  3.6 Grain Charging

Charged grains can interact with a surrounding magnetic field, even for
diamagnetic grains. As we discuss below (Sec. 5.3), a recently formulated
grain alignment mechanism (“E-RAT” Lazarian, 2020), predicts that grains not
susceptible to paramagnetic interactions can achieve efficient alignment with
the magnetic field, if they are charged and posses a relative motion to the
field.

For dense AGB star CSEs the photo-ionization of the gas, and charging of the
dust, is expected to be strongest in the outer envelope and originate from the
UV light in the diffuse interstellar field (because of the cool effective
temperature of the central star very little UV flux is expected from its
photosphere). Since we do not have any direct observational tracer of the
grain charging, we used the gas electron density calculated in the detailed
chemical modeling of IRC+10 216 by Cordiner & Millar (2009) and Li et al.
(2014) as a proxy for the grain charging as a function of radius. Figure 10
shows the comparison of polarization fraction to the electron density.

For interstellar conditions, where grain-electron collisions are an important
aspect of the grain charge balance (Weingartner & Draine, 2001), we would
expect the grain charge to be proportional to 1/n(e). However, the conditions
in the CSE (low fractional ionization, fully molecular hydrogen, high flow-
speed and steep density gradient, etc.) may mean that the collisional effects
are negligible, and the grain charge will be determined by photo-electric
emission balanced by the electrostatic potential of the grain, and hence be
dependent on the opacity to the interstellar radiation field, but not involve
the 1/n(e) factor, as implied by the semi-empirical correlation seen in Figure
10. Detailed modeling, applicable to the conditions of the CSE (beyond the
scope of this paper), will be needed to fully address this issue.

While for small grains the electric moment can originate as part of grain
structure (Draine & Lazarian, 1998), for the large grains indicated in the
IRC+10 216 CSE (e.g. Groenewegen, 1997) grain charging is likely required (A.
Lazarian, 2022, private communications). Since the electric dipole moment of a
grain depends on the total charge on the grain (cf. Weingartner, 2006;
Lazarian, 2020), we use the space density of electrons as our proxy for grain
charging and electric alignment. The blue dashed line in the outer envelope
corresponds to electrons released by photo-ionization, while the red dashed
curve corresponds to electrons due to cosmic ray ionization. Ivlev et al.
(2015) have argued that grain-charging by cosmic rays in the inner envelope,
which is similar in gas parameters to their region “$\mathcal{I}$”, should be
inefficient.

Figure 10: (Left) The amount of 850 $\mu$m polarization is well correlated
with the electron density by photo-ionization due to the external diffuse
radiation (UV) field (blue dashed line) calculated by Li et al. (2014) in the
IRC+10 216 envelope (the scaling of the two vertical axes have been
arbitrarily set to show the correlation). The red dashed curve represents
modeled electron density generated by cosmic rays in the denser inner shell.
As shown by Ivlev et al. (2015) grain-charging by cosmic rays in the inner
envelope (similar to their region “$\mathcal{I}$”) is inefficient. (Right) A
direct comparison between polarization and electron density for distances
beyond 20″ (where the latter is dominated by photo-ionization in the Li et al.
(2014) models) yields a formally significant correlation.

##  4 Results

Our FIR, especially the 53 $\mu$m, and optical polarization data (Andersson et
al., 2022, Andersson et al., 2023b, in preparation) indicate grain alignment
with the long axis in the radial direction away from the central star,
supporting a second-order radiative grain alignment mechanism (Hoang &
Lazarian, 2009). We stress that, as we argued in the former paper (Andersson
et al., 2022), the radial dependence of the FIR polarization does not support
a mechanical (gas-dust drift) alignment mechanism (e.g. Gold, 1952) as the
primary driver. The JCMT/POL-2 850 $\mu$m data, presented here, do, however,
not show this simple radial pattern, even in the inner part of the CSE.

The analysis utilizing the radial coordinate polarization (Sec. 3.1) shows a
distinct difference between the FIR (SOFIA/HAWC+) and sub-mm wave
(SCUBA-2/POL-2) data. Again, for the FIR, the $\cos{\theta_{r}}>0.95$ (for all
53 $\mu$m data and inside $r\approx$ 32″ for the 154 $\mu$m data) indicate a
clear radial polarization pattern (Figure 3, upper panels). While of lower
signal-to-noise, and at somewhat different scales, the plots indicate neither
a radial nor a tangential pattern in the sub-mm wave data in Figure 3, lower
panel.

Using the Zeeman measurement results of Duthu et al. (2017), we find a
periodic variation in the line-of-sight magnetic field around the star, with
the pole of the possible underlying dipole field oriented at around 90-125∘,
albeit with relatively large uncertainties. We therefore assume a multipole
expansion of a potential magnetic field and apply a simple dipole model fit to
the 850 $\mu$m data.

As Figure 5 shows, there is a clear $\chi^{2}$ minimum found for a rotation
angle of $\Theta_{E}$=125∘ The modeling is done for E-vectors. The
relationship to a possible magnetic dipole field axis is ambiguous, depending
on the alignment mechanism, even if involving magnetic fields (see Sec. 5.3).
The polarization symmetry axis is close to that of the Zeeman results and of
the torus seen by Jeffers et al. (2014).

While our simple model uses only a projected dipole term for a possible
magnetic field and does not fully account for all measured orientations of the
polarization field, it is clear that the FIR and sub-mm wave data do not show
the same (dominant) polarization patterns.

Comparing to the line polarimetry by Girart et al. (2012) shows that, while
significant, Goldreich-Kylafis polarization of spectral lines within the
SCUBA-2 pass-band is unlikely to explain the observed 850 $\mu$m continuum
polarization (Sec. 3.5).

As shown by Figure 8, the slope (power-law exponent) of the fractional
polarization as a function of total intensity is nominally steeper for the 850
$\mu$m polarization, than for the 53 $\mu$m or 154 $\mu$m ones. Allowing a
two-power-law fit to the 53 (and 154) $\mu$m data, however, shows that at
small respective intensities all three share a common slope of approximately
-0.85. The second power-law parameters are only measurable for the 53 $\mu$m
data, yielding a slope of -0.45, with an offset to higher polarization
fractions for a given intensity, with the transition between the two at
$\sim$3 Jy/arcsec2. The 154 $\mu$m polarization shows an indication of a
similar change at $I\approx$ 0.2 Jy/arcsec2. These intensities both occur at a
radial offset from the star of $\sim$12.5″ (Figure 2). While this is
comparable to the beam size at 154 $\mu$m, it is significantly bigger than the
53 $\mu$m beam.

Figure 10 shows a correlation between grain charging and alignment in the
outer envelope of IRC+10 216 beyond an offset radius of $\sim$20″.

##  5 Discussion

As shown by our polarization maps and the azimuthally averaged radial
polarization plots, the 850 $\mu$m polarization is dominated by neither a
radial nor an azimuthal pattern. The small number, and asymmetric
distribution, of the JCMT polarization detections make the difference between
53 $\mu$m and 850 $\mu$m data difficult to interpret conclusively. However,
the non-radial polarization in the latter data set does require a different
polarization mechanism than we have proposed for the FIR data.

For the inner part of the CSE, the large beam size (14″), relative to the
offset from the center of the CSE, means that the JCMT data is prone to,
possibly severe, beam-averaging effects. However, as can be seen from Figure 7
the beam-averaged polarization even inside $\sim$20″of the central star is
often azimuthal (or, at least, non-radial) in orientation. By symmetry, a
beam-averaged radial pattern would remain radial, and therefore it is unlikely
that the beam size can explain this dichotomy between the FIR and sub-mm wave
polarization for dust at a (3D) radial distance from the central star
comparable to the projected distance of these measurements.

If the more efficient 850 $\mu$m polarization is due to the dust, then it
indicates that some of the grains have acquired efficient internal
dissipation, to be aligned by some version of the RAT mechanism, or are
aligned by the RAT-like mechanical alignment (MET) mechanism discussed by
Lazarian & Hoang (2021), where gas particles reflect off of a helical grain
surface and therefore align the grain’s angular momentum vector with the gas-
dust drift velocity. MET alignment would therefore, as would first-order
k-RAT, produce azimuthal polarization in the AGB star envelope. However, given
that only a small number of points show close to azimuthal polarization,
neither first order k-RAT nor MET dominated alignment seem likely. As noted by
Hoang & Lazarian (2009) weak “direct RAT alignment” can be achieved along the
$B$-field also without internal alignment which for low-J attractor points can
be with the long axis along the magnetic field.

Some level of internal alignment may be achieved from inelastic dissipation in
porous grains (Purcell, 1979, Tram, 2022, private communications). This seems
somewhat unlikely in the IRC+10 216 CSE, given the need to preserve large
grains (Groenewegen, 1997) under strong radiative disruptive driving (Hoang et
al., 2019) which, for porous grains with small tensile strength, would result
in the large aligned grains being destroyed. We note that the grain disruption
is limited to grains at high-J attractor points (Lazarian & Hoang, 2021), and
a significant fraction of the dust may therefore not be affected by the
disruption.

We are therefore left with magnetically aligned grains, possibly due to
mineralogy changes or driven by grain charging and induced electric fields
(Lazarian, 2020) or some as yet unidentified grain alignment mechanism.
Contamination by polarized line emission (Cernicharo 2021, private
communications) also cannot be fully discounted, at this stage, as discussed
above.

Given the differences in polarization geometry between FIR and sub-mm wave
data, we used the Zeeman observations from Duthu et al. (2017) to search for
periodic variability in the line-of-sight magnetic field around the central
star. While there are only five data points (plus an upper limit) we find a
good fit to a cosine function of the azimuthal angle, $\Theta$ around the star
with the extremum located at $\sim$90-125∘. We interpret this as indicating a
dipole pattern (being the second order component of a spherical harmonics
expansion). A dipole with its symmetry axis in the plane of the sky will not
show any net line-of-sight polarization, since in the equatorial plane the
$B$-field is in the plane of the sky, while towards the poles there will be
equal contributions of positive and negative fields along the line of sight
(assuming optically thin emission). However, if the symmetry axis is tilted
out of the plane of the sky, a net line-of-sight magnetic flux will result (as
seen by e.g. Vlemmings et al., 2005).

As our modeling (Figure 5) shows, a simple projected dipole model rotated by
125∘ east of north (E-vectors) yields a distinct nominal $\chi^{2}$ minimum in
the fit of the 850 $\mu$m data, with a symmetry axis consistent with that
implied by the Zeeman data. This provides some support for B-RAT alignment of
the grains, or some other magnetically defined alignment mechanism giving rise
to the 850 $\mu$m polarization. As noted above, the large absolute minimum
value of the reduced $\chi^{2}$, means that a projected dipole is not, by
itself, a complete or accurate model of the data. However, the localization of
the symmetry axis seems to indicate that it likely forms part of the true
polarization geometry.

It is noteworthy that Jeffers et al. (2014) have modeled the scattered light
in the inner $\sim$3″of the IRC+10 216 CSE as due to a torus centered on the
star, with a symmetry axis of $\sim$75∘ east of north. In our optical data,
the only stars whose polarization diverges significantly from being azimuthal
with respect to the central star IRC+10 216 intercept this symmetry axis. If
this represents the angular momentum/rotation axis of the CSE, it would imply
a close alignment with the indicated magnetic axis.

###  5.1 Polarized Line Emission Contamination

While the nominal polarized line flux in the G-K polarization is similar to
the SCUBA-2 polarized flux, the geometry of the two types of polarization do
not seem consistent. Because the spatial sampling of the SMA and the JCMT are
significantly different, we cannot conclusively rule out a measurable line
polarization contamination in our data, but a dominant contribution from G-K
polarization seems unlikely, because of the geometry. Simultaneous line and
continuum polarization of IRC+10 216 with ALMA are planned for Cycle 9 to
better address this issue.

###  5.2 Magnetic Dust Polarization Mechanisms

For dust polarization to be tracing a magnetic field in the CSE, the grains
dominating the 850 $\mu$m polarization must have acquired at least the
equivalent of paramagnetic characteristics. This might be caused by variations
in the grain chemistry with radius from the star, or with grain size (or some
combination of these).

First, when considering the grain chemistry, mineralogy of AGB envelopes
naturally show a radial variation due to the evolution on the AGB. As noted
above, when a star first enters the AGB it is oxygen rich. As the thermal-
pulse driven dredge-up of newly synthesized material is mixed up to the
stellar surface and ejected into the CSE it eventually becomes carbon rich.
This may leave silicate grains in the outer part of the CSE, with carbonaceous
dust in the inner part of the envelopes, as reported for the planetary nebula
BD+30∘3639 (Guzman-Ramirez et al., 2015). The JCMT/POL-2 850 $\mu$m
observations may therefore be tracing this outer, colder, fossil silicate dust
seeing only dust at large distances from the star, even when projected close
to the central star. However, because the emission is optically thin, the
dominant contribution also to the 850 $\mu$m Stokes $I$ emission originates in
the denser gas at relatively small radii away from the central star. Given the
expansion velocity of 14 km/s and a distance of 123 pc, the presence of such
fossil silicates would also require that the transition from oxygen to carbon
rich chemistry would have been very recent, or that the mixing of the phases
be very efficient (cf Leão et al., 2006; Sahai & Chronopoulos, 2010).

The possibility for explaining efficient alignment due to grain size, for the
grains dominating at 850 $\mu$m, parallels the proposed explanation of the
shape of the optical polarization - “Serkowski” \- curve under Davis-
Greenstein (henceforth “DG”) alignment by Mathis (1986), using super-
paramagnetic grain inclusions. The empirical Serkowski curve for diffuse ISM
lines of sight requires that only grains with radii larger than
$\sim$0.045$\mu$m are aligned (Kim & Martin, 1995). RAT alignment explains
this as due to the lack of radiation below the Lyman limit in neutral gas (the
condition for efficient RAT alignment being $\lambda<$ 2a, where a is the
effective grain radius). Because DG alignment becomes more efficient for
smaller grains (e.g. Draine, 2011), a mechanism was needed to deactivate it
for grains smaller than this size. Because DG alignment further requires
significantly higher magnetic susceptibility in the grains than for ordinary
silicates (Jones & Spitzer, 1967), Mathis proposed that the lack of small
grain alignment was due to the fact that grains get aligned only if they
contain at least one “super-paramagnetic inclusion” (of e.g. metallic iron,
FeO, FeS, etc., (cf Goodman et al., 1995)). While mid-infrared observations
show that the dust in the IRC+10 216 CSE is fully carbonaceous (95% amorphous
carbon and 5% SiC; Ivezić & Elitzur (1996a)), iron is found to be depleted in
the CSE gas (Mauron & Huggins, 2010). It is therefore possible that the
difference between FIR and sub-mm wave polarization originates in differences
in the alignment properties of smaller (and hotter) grains seen in the
MIR/FIR, and larger (cooler) ones seen in the sub-mm, where the latter have
acquired inclusions of e.g. ferromagnetic metallic iron or iron carbide (Fe3C;
Wang et al., 2017). Since this is only a change in the alignment efficiency,
it would not affect the Stokes $I$ profile.

The fractional polarization, as a function of Stokes $I$ intensity, shows a
bimodal dependence in the 53 $\mu$m data, with a hint of a similar behavior at
154 $\mu$m, indicating a change in alignment behavior at about 12.5″ from the
star. Since the slope at higher intensities (closer to the star) is shallower
and the absolute level is higher for this distribution than the steeper slope
extending further out, this is unlikely to be due to radiative dust disruption
(Hoang et al., 2019, 2021). Quantitative modeling will be needed to explore
the origin of this bimodality, but does support a change in grain alignment
physics with radius in the CSE.

###  5.3 Electrically Induced Alignment

Because of the cool photosphere of CW Leo777For clarity, we refer to the
central star by its name when specifically addressing stellar parameters, and
“IRC+10 216” for the extended object, including the star and the high column
density of the CSE, the inner part of the envelope is expected to be close to
neutral (modulo cosmic ray ionization; Cordiner & Millar, 2009), as supported
by the strong absorption in K I (ionization potential of 4.3 eV) seen in “Star
6” of Mauron & Huggins (2010), even at an offset of 37″ from the center of
IRC+10 216.

There is growing evidence for some kind of UV source at or close to the center
of the CSE. Ionized carbon (through the 158 $\mu$m line of [C II]) has been
observed with Herschel/HIFI toward the center of the CSE (Reach et al., 2022),
as has high-J line of HC3N with ALMA (Siebert et al., 2022), which both
require UV photons for their production. Based on imaging observations by Kim
et al. (2015, 2021) and variability studies by e.g. Cernicharo et al. (2015);
Decin et al. (2015); Guélin et al. (2018), Siebert et al. (2022) added a
central UV source - most likely a “Solar-like companion” \- to their chemical
modeling of the HC3N abundance. The inner density peak of HC3N resulting from
this assumption is, however, located within a small enough radius ($\lesssim$
5″ ) that our observations would be significantly beam averaged in this
region.

For the outer parts of the CSE, UV photons from the diffuse interstellar field
can penetrate the medium and ionize the gas, as well as charge the grains.
While the work function for interstellar grains is not well known, the value
of $W\approx$4.4 eV, appropriate for graphite (Weingartner & Draine, 2001), is
lower than the ionization potential of both hydrogen and carbon. Since both
the grain charging and ionization fraction in the gas are due to the external
UV field, we will therefore, here, use the gas electron density as a proxy for
grain charge. As discussed above, the validity of this assumption depends on
grain-electron collisions to be negligible. Figure 10 shows a clear
correlation between the polarization fraction and electron density at offsets
$\gtrsim 20\arcsec\ $from the center of IRC+10 216.

As discussed by Ivlev et al. (2015) and Ibáñez-Mejía et al. (2019), while the
radiative grain charging at low extinctions for carbon grains is efficient,
due to their low work function, the cosmic ray grain charging (mostly through
secondary H2 UV emission in the Lyman and Werner bands) is less so, by at
least an order of magnitude (see Figure 3 of Ibáñez-Mejía et al. (2019)).
This, possibly together with the beam averaging at the very center, likely
explains the lack of correlation between $n$(e-) and $P$ inside $r\approx
20\arcsec$.

For emission polarimetry the polarization fraction $P$ inherently has the
appropriate line of sight averaging needed to address alignment effects (as
noted by Hildebrand et al. (2000) for extinction and emission polarization,
generally: $P_{\mathrm{em}}=-P_{\mathrm{ext}}/\tau$, where the minus sign
denotes the fact that the emission and extinction polarization position angles
are orthogonal.) No significant correlation is seen between $n$(e-) and the
polarized intensity; $IP$. Because of the non-uniform density structure (and
spherical symmetry) of the CSE, our comparison is, of course, still only
approximate, but points to a likely correlation between grain alignment and
grain charging, in the outer envelope.

As discussed above, Lazarian (2020) predicts that charged carbon grains with a
net velocity perpendicular to the magnetic field should align with the
magnetic and electrical fields (since the direction of the induced E-field is
given by the direction of the $B$-field). If the precession rate around the
E-field ($\Omega_{E}$) is faster than around the radiation field direction
($\Omega_{k}$) and the B-field ($\Omega_{B}$), then theory predicts that the
grain should align with their long axis parallel with the B-field and thence
the observed emission polarization should be parallel to the $B$-field. Since
the polarization field (E-vector) symmetry, from our modeling, agrees with
that from Zeeman data, our results indicate that this is the case
($\Omega_{E}>\Omega_{B}$). If confirmed, these results can therefore provide
constraints on the charging, drift velocity, and geometry of the grains.

To understand the dichotomy between FIR and optical polarization on the one
hand and sub-mm wave continuum polarization on the other in the envelope of
IRC+10 216, sub-mm observations at higher spatial resolution and better
signal-to-noise are needed. Also, line and continuum polarimetry in the sub-mm
wave range acquired with the same instrument would be highly valuable to
understand the relative contributions of dust and Goldreich-Kylafis
polarization. The 7-m array of ALMA (and eventually the Morita array) can
reach 4″ resolution for polarimetry at 870$\mu$m - closely matching that of
the SOFIA/HAWC+ 53 $\mu$m observations - and the high signal-to-noise needed
to sensitively probe the cold dust in the inner [projected] part of the CSE.

The 3.4 $\mu$m aliphatic CH line is likely too inherently weak to be
detectable in polarization in CSEs (Chiar et al. (2000) reports line opacities
of $\tau\sim$0.2 towards the Galactic center for visual extinctions exceeding
30 magnitudes). However, the solid-state SiC line at 11.3$\mu$m has been
reliably detected in several carbon rich CSEs (e.g. Sloan et al., 1998).
Unfortunately, very little polarization data are available to specifically
probe the alignment of such grains. The polarization data of Smith et al.
(2000) may show a hint of polarization in this line for the carbon-rich post-
AGB star AFGL 618 (showing a shape consistent with the flux spectra of carbon
stars from Sloan et al. (1998). While the amplitude of this polarization line
in AFGL 618 is only marginal with respect to its uncertainties, the possible
detection of SiC polarization suggests that a systematic study of polarization
in the line would be very valuable to further probe carbonaceous grain
alignment.

##  6 Conclusions

We present 850 $\mu$m JCMT/SCUBA-2/POL-2 polarimetry of the circumstellar
envelope of the carbon rich AGB star IRC+10 216. In contrast to FIR 53 $\mu$m
and optical polarimetry, which show an ordered radial (azimuthal) polarization
pattern in the FIR (optical), implying dust grains aligned with their long
axis in the radial direction away from the star, the sub-mm wave polarization
is more complex. Our data are not sufficiently extensive to allow a complete
modeling of the 850 $\mu$m polarization data. However, based on an analysis of
the Zeeman effect measurements in the CN (J=1-0) line by Duthu et al. (2017)
and an assumption of the validity of a multipole decomposition of the magnetic
field, we fitted a rotated, projected, dipole to the 850 $\mu$m polarization
data. Under this model assumption, a distinct preference for a symmetry axis
is found, consistent with both the symmetry axis of the Zeeman results and the
dust torus reported by Jeffers et al. (2014). These results imply that, if the
polarization is due to aligned dust grains, the dust traced by sub-mm wave
observations is aligned with a magnetic field. The optical and FIR
polarization, in contrast, indicate magnetically inactive dust aligned via a
second-order pure radiative mechanism. To reconcile this dichotomy we consider
several possible explanations:

  * •

The 850 $\mu$m continuum polarization may be dominated by Goldreich-Kylafis
line polarization e.g. in the lines of CO (J=3-2), SiS (J=19-18) and CS
(J=7-6), contained within the SCUBA-2 pass band.

  * •

The cold dust probed by our 850 $\mu$m observations may have a different dust
mineralogy than the dust probed by the optical and FIR observations. We
hypothesize that this difference can arise from one of two processes:

    * –

The sub-mm wave data may be probing the fossil silicate dust from the oxygen-
rich phase of the AGB star, located in the outer parts of the CSE, or

    * –

The largest carbon grains may have acquired magnetically active inclusions
from the depleted metals in the CSE (e.g. metallic iron or Fe3C).

  * •

A new alignment mechanism (Lazarian, 2020) applicable to charged grains with
small magnetic susceptibilities (i.e. carbon grains) may explain our results,
as the polarization fraction in the 850 $\mu$m data are well correlated with
the calculated electron densities (Li et al., 2014) in the envelope.

Based on an Occam’s razor argument, the latter possibility is particularly
attractive, as the FIR and sub-mm wave polarization difference then is only a
matter of differences in the grain charging state.

If the magnetic alignment of the grains seen at sub-mm wavelengths, and the
dipole geometry of the resulting polarization, can be confirmed, the large
scale structure of the magnetic field around IRC+10 216 will be known. The
connection to the dust structures seen at small scales (Jeffers et al., 2014)
are particularly intriguing.

Further continuum and line data, at higher resolution and sensitivity,
preferably also at intermediate wavelengths are needed to confirm and further
explore these results. Our results indicate that pure carbon grains are likely
not aligned in the ISM, and thereby provide observational constraints on the
properties of composite grains (e.g. Draine & Hensley, 2021).

B-G A., A.S. and S. F-M gratefully acknowledge the support from the National
Science Foundation under grant AST-1715876 and the SOFIA project under grant
05_0048. J.K. acknowledges funding from the Moses Holden Studentship in
support of his PhD. Financial support for SC was provided by NASA through
award 08_0186 issued by USRA. M.Tahani is supported by the Banting Fellowship
(Natural Sciences and Engineering Research Council Canada) hosted at Stanford
University. We thank professor T. Miller for providing the data tables of
electron density in the IRC+10 216 shell and several helpful discussions. We
thank Prof. J. Cernicharo for pointing out the possibility that our data are
dominated by line polarization. We thank Prof. A. Lazarian and Dr. R. Sankrit
for several insight- and help-full suggestions. The James Clerk Maxwell
Telescope is operated by the East Asian Observatory on behalf of The National
Astronomical Observatory of Japan; Academia Sinica Institute of Astronomy and
Astrophysics; the Korea Astronomy and Space Science Institute; Center for
Astronomical Mega-Science (as well as the National Key R&D Program of China
with No. 2017YFA0402700). Additional funding support is provided by the
Science and Technology Facilities Council of the United Kingdom and
participating universities and organizations in the United Kingdom and Canada.
The authors wish to recognize and acknowledge the very significant cultural
role and reverence that the summit of Maunakea has always had within the
indigenous Hawaiian community. We are most fortunate to have the opportunity
to conduct observations from this mountain. Additional funds for the
construction of SCUBA-2 and POL-2 were provided by the Canada Foundation for
Innovation. This research used the facilities of the Canadian Astronomy Data
Centre operated by the National Research Council of Canada with the support of
the Canadian Space Agency. This work is based in part on observations made
with the NASA/DLR Stratospheric Observatory for Infrared Astronomy (SOFIA).
SOFIA is jointly operated by the Universities Space Research Association, Inc.
(USRA), under NASA contract NNA17BF53C, and the Deutsches SOFIA Institut (DSI)
under DLR contract 50 OK 0901 to the University of Stuttgart.  JCMT/POL-2,
SOFIA/HAWC+

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##  Appendix A Dipole model

The magnetic field model is based on a functional form described by King et
al. (2018) to predict the polarization in synthetic magnetohydronamical (MHD)
simulations. This model was then adapted to estimate the polarization from
magnetically-aligned dust grains in a torus by Lopez-Rodriguez et al. (2020).
The implementation here has been simplified to model a rotationally-symmetric
dipole field.

For the magnetic field configuration, we assume a rotationally-symmetric
dipole of the form:

$\displaystyle B_{r}$ $\displaystyle=$
$\displaystyle-2\,B_{0}\,\cos{\left(\phi+\psi\right)}$ (A1) $\displaystyle
B_{\theta}$ $\displaystyle=$ $\displaystyle-
B_{0}\,\sin{\left(\phi+\psi\right)},$ (A2)

where $B_{0}$ is proportional to $r^{-2}$, $\phi$ is the azimuthal angle and
$\psi$ is the pitch angle of the dipole. This represents a somewhat simplified
dipole for two-dimensional computation. In Cartesian coordinates, the field
becomes:

$\displaystyle B_{x}$ $\displaystyle=$
$\displaystyle\left(-B_{\theta}\,\sin{\phi}\right)+\left(B_{r}\,\cos{\phi}\right)$
(A3) $\displaystyle B_{y}$ $\displaystyle=$
$\displaystyle\left(B_{\theta}\,\cos{\phi}\right)+\left(B_{r}\,\sin{\phi}\right)$
(A4) $\displaystyle B_{z}$ $\displaystyle=$ $\displaystyle B_{0}(r).$ (A5)

When this dipole is viewed at an inclination angle $i$ and tilt $\Theta$
w.r.t. the plane of the sky, the observer’s frame becomes:

$\displaystyle B_{x^{s}}$ $\displaystyle=$ $\displaystyle
B_{x}\,\cos{\Theta}+\left(B_{y}\,\cos{i}-B_{z}\,\sin{i}\right)\,\sin{\Theta}$
(A6) $\displaystyle B_{y^{s}}$ $\displaystyle=$ $\displaystyle-
B_{x}\,\sin{\Theta}+\left(B_{y}\,\cos{i}-B_{z}\,\sin{i}\right)\,\cos{\Theta}$
(A7) $\displaystyle B_{z^{s}}$ $\displaystyle=$ $\displaystyle
B_{y}\,\sin{i}+B_{z}\,\cos{i}.$ (A8)

Following Lopez-Rodriguez et al. (2020), we then express the Stokes parameters
in terms of the local magnetic field $B=(B_{x},B_{y},B_{z})$ in the optically-
thin case $\tau_{\lambda}\ll 1$, for each point in a map on the sky of RA, Dec
($\alpha,\delta$):

$\displaystyle I(\alpha,\delta)$ $\displaystyle=$ $\displaystyle\int
n\left(1-p_{0}\left(\frac{B_{x}^{2}+B_{y}^{2}}{B^{2}}-\frac{2}{3}\right)\right)\,ds$
(A9) $\displaystyle Q(\alpha,\delta)$ $\displaystyle=$ $\displaystyle
p_{0}\int n\left(\frac{B_{y}^{2}-B_{x}^{2}}{B^{2}}\right)\,ds$ (A10)
$\displaystyle U(\alpha,\delta)$ $\displaystyle=$ $\displaystyle p_{0}\int
n\left(\frac{2B_{x}B_{y}}{B^{2}}\right)\,ds$ (A11)

where the $x$ and $y$ axes are in the plane of the sky, $z$ is along the line-
of-sight, and $n$ describes the volume density. In our simulations, ds is
along the z-axis. We implement $n$ as a two-dimensional Gaussian function to
generate an arbitrary “gas” volume density. $p_{0}$ relates dust grain cross-
sections with magnetic alignment properties. Because we are not, here,
attempting to fit the polarization fraction, but only fit the position angles
between data and model, and because dipoles are rotationally symmetric, the
density distribution in the model is not of primary importance. We note that,
because the model does not include quantitative polarized radiative transfer
calculations, we have limited our model fitting to the symmetric case with a
zero inclination angle of the dipole axis, relative to the plane of the sky,
e.g. a face-on view.