# Evolution of the disk in the Be binary $\delta$ Scorpii probed during three
periastron passages

R.G. Rast Department of Physics and Astronomy, The University of Western
Ontario, London, ON N6A 3K7, Canada  C.E. Jones Department of Physics and
Astronomy, The University of Western Ontario, London, ON N6A 3K7, Canada  A.C.
Carciofi Instituto de Astronomia, Geofíca e Ciéncias Atmosféricas,
Universidade de São Paulo, Brazil  M.W. Suffak Department of Physics and
Astronomy, The University of Western Ontario, London, ON N6A 3K7, Canada
A.C.F. Silva Instituto de Astronomia, Geofíca e Ciéncias Atmosféricas,
Universidade de São Paulo, Brazil  G.W. Henry Center of Excellence in
Information Systems, Tennessee State University, Nashville, TN 37209, USA  C.
Tycner  Department of Physics, Central Michigan University, Mt. Pleasant, MI
48859, USA

###### Abstract

We examine the evolution of the disk surrounding the Be star in the highly
eccentric binary system $\delta$ Scorpii over its three most recent periastron
passages. $V$-band and $B-V$ photometry, along with H$\alpha$ spectroscopy are
combined with a new set of extensive multi-band polarimetry data to produce a
detailed comparison of the disk’s physical conditions during the time periods
surrounding each closest approach of the secondary star. We use the three-
dimensional Monte Carlo radiative transfer code HDUST and smoothed particle
hydrodynamics (SPH) code to support our observations with models of disk
evolution, discussing the behaviour of the H$\alpha$ and He i 6678 lines,
$V$-band magnitude, and polarization degree. We compare the characteristics of
the disk immediately before each periastron passage to create a baseline for
the unperturbed disk. We find that the extent of the H$\alpha$ emitting region
increased between each periastron passage, and that transient asymmetries in
the disk become more pronounced with each successive encounter. Asymmetries of
the H$\alpha$ and He i 6678 lines in 2011 indicate that perturbations
propagate inward through the disk near periastron. When the disk’s direction
of orbit is opposite to that of the secondary, the parameters used in our
models do not produce spiral density enhancements in the H$\alpha$ emitting
region because the tidal interaction time is short due to the relative
velocities of the disk particles with the secondary. The effects of the
secondary star on the disk are short-lived and the disk shows independent
evolution between each periastron event.

Be stars (142) — Binary stars (154) — Circumstellar disks (235) — Johnson
photometry (871) — Spectroscopy (1558) — Starlight polarization (1571)

##  1 Introduction

Be stars are non-supergiant stars of spectral class B that have exhibited
Balmer lines in emission at some point in their lifetime (Jaschek et al.,
1981; Collins, 1987). Classical Be stars are more narrowly defined as main
sequence (or slightly evolved) B-type stars whose Balmer emission lines
originate from circumstellar disks of gas that are rotating in Keplerian
fashion and fed by the central star (Bjorkman, 2012). These stars rotate at
rates greater than any other non-degenerate stars (Townsend et al., 2004).
Their rotation rates are usually thought to be more than 75% of the critical
velocity, where the gravitational force is balanced by the centrifugal force
at the equator (Rivinius et al., 2013; Townsend et al., 2004; Frémat et al.,
2005). In addition, the stars exhibit multi-mode, non-radial pulsations that
may play a role in the star-disk mass transfer (Baade et al., 2018; Baade &
Rivinius, 2020; Labadie-Bartz et al., 2022).

The disks surrounding classical Be stars have been successfully described
using the viscous decretion disk model (Lee et al., 1991). Dynamically, this
model is similar to the one for viscous accretion disks, only instead of gas
flowing inward onto the star as is the case with accretion, the gas flows
outward. The central star contributes mass and angular momentum via parcels of
gas to the inner disk. The gas is then launched into Keplerian orbit and
drifts outward facilitated by viscosity. This model has been used in the
radiative transfer code HDUST to predict observables of Be stars at specific
times (Carciofi & Bjorkman, 2006). For example, de Almeida et al. (2020) used
HDUST to successfully predict the spectral energy distribution (SED), $V$-band
polarization angle, as well as the H$\alpha$ and Br$\gamma$ line profiles of
the star $o$ Aqr. Similarly, HDUST was used by Ghoreyshi et al. (2021) to
reproduce observational trends in photometry, spectra and broad-band
polarimetry for $\omega$ CMa.

Among Be stars, the H$\alpha$ line (6562 Å) may be singly-peaked (usually
observed more pole-on) or doubly-peaked (at higher inclinations when viewing
Doppler shifted material) (Hanuschik, 1989; Hummel, 1994; Sigut et al., 2020;
Sigut & Ghafourian, 2023). Edge-on stars can also exhibit shell
characteristics, with deep absorption cores that may be flanked by emission
peaks. Similar to the other H i lines, the H$\alpha$ emission line is formed
predominantly through recombination in the disk (Porter & Rivinius, 2003;
Bjorkman, 2012). For an optically thick disk, the strength of the H$\alpha$
flux is directly proportional to the area of the emitting region (Tycner et
al., 2005), which constitutes a lower limit on the extent of the circumstellar
disk. Information about the inner regions of the disk may be carried in the He
i 6678 Å line, which is produced near the Be star.

In many Be stars, the equivalent width (EW) and full width at half maximum
(FWHM) of the H$\alpha$ emission line have been observed to vary over time,
indicating changes in the size of the H$\alpha$ emitting region or changes in
the density of the emitting volume. The strength of the H$\alpha$ line can be
used to track the evolution of a star as it acquires a circumstellar disk and
enters its Be phase, or as it loses the disk to appear as a typical B star
(Wisniewski et al., 2010). For example, $\omega$ Ori was observed to build a
disk over a period of a few years (Hayes & Guinan, 1984). By contrast, the
disks of 60 Cyg and Aqr dissipated over periods of roughly 1000 days and 2440
days, respectively (Wisniewski et al., 2010), while the disk of 66 Oph has
been dissipating for more than 20 years (Marr et al., 2021). Over a few
decades, the Be star $\gamma$ Cas famously underwent two shell phases
(Baldwin, 1940; Borre et al., 2020) where the disk may have been dynamically
destroyed (Baade et al., 2023) before regaining the disk and remaining
relatively stable in recent years (Borre et al., 2020).

The shape of the H$\alpha$ profile also encodes information about the density
distribution (and velocites) of the disk. The ratio of the relative flux of
the violet and red peaks (V/R) of double-peaked lines may vary cyclically over
5 to 10 years (Okazaki, 1997; Štefl et al., 2009). The cycle lengths and
amplitudes of the variations may also change over time (Ruždjak et al., 2009).
The most successful and widely accepted model to address this involves one-
armed spiral density waves in the inner disk (Okazaki, 1997; Carciofi et al.,
2009). Density enhancements have been directly observed (Vakili et al., 1998),
supporting this model.

Photometric observations of Be stars can also reveal important information
about disk size and mass-loss rates. $V$-band photometry is of particular
interest in Be stars. The majority of the $V$-band flux originates in the
inner regions of the disk, typically within two stellar radii
($\mathrm{R_{\star}}$) with the exact emitting regions depending on disk
density (Carciofi, 2011; Rivinius et al., 2013). Therefore, the observations
in the $V$-band are the fastest indicators to signal changes in the disk’s
size or density due to photospheric activity such as outbursts (Carciofi,
2011). For example, Wisniewski et al. (2010) found that changes in the
H$\alpha$ profile, which forms in a more extended volume of the disk, lagged
behind the $V$ band by hundreds of days.

Together with the polarization angle, changes in the polarization degree have
been attributed to mass ejections and resulting density alterations of the
disk (Carciofi et al., 2007). The polarization signature originates in the
disk, where unpolarized light undergoes Thompson scattering (Wood et al.,
1997). The relative strength of the polarization, polarization position angle,
and wavelength-dependence of the polarization give information about the disk
geometry, number of scatterers in the disk, and the density of the disk,
respectively (Haubois et al., 2014). During disk growth, the polarization in
the Balmer jump rises rapidly while the $V$-band polarization degree increases
at a slower rate (Halonen & Jones, 2013). Conversely, during dissipation
events, the polarization in the Balmer jump drops rapidly while the $V$-band
polarization degree takes longer to decrease (Halonen & Jones, 2013). Since
the Balmer jump forms at smaller disk radii than the $V$-band, this suggests
disk growth is caused by mass loss near the stellar surface, while the inner
regions of the disk are the first to disappear during dissipation events. This
pattern has been observed in $\omega$ Ori (Sonneborn et al., 1988).

Be stars, like other high-mass stars, are commonly found in binary systems. In
a study of 340 Be stars, Miroshnichenko (2016) concluded that more than 50% of
Be stars brighter than $V$ $\sim$ 4 mag were in binary systems. Klement et al.
(2019) studied the infrared and radio emissions from 57 classical Be stars and
found indirect evidence of binarity in 26 of these objects. Dodd et al. (2024)
found binary fractions of 28% for B and 17% for Be stars at separations
between 0.02 and 0.2 arcsec. They attribute the relative lack of binary Be
stars at these separations to the presence of stripped companions, which
constitute an increasing fraction of detected secondary stars in Be binary
systems (Marchant & Bodensteiner, 2023) and would not have been detected by
their methods (Dodd et al., 2024).

Tidal interactions between the secondary star and the disk may result in
structural distortions or truncations of the disk. Two-armed waves have been
proposed in connection to such tidal perturbations (Okazaki et al., 2002).
They are associated with much smaller V/R amplitudes that are phase-locked
with the binary orbital period (Panoglou et al., 2018). For example, the
H$\alpha$ profile of $\pi$ Aqr has shown radial velocity and V/R variations
that are phase-locked to the orbital period of the system (Bjorkman et al.,
2002; Zharikov et al., 2013). Miroshnichenko et al. (2023) suggest that these
V/R variations can be used to detect binarity in Be stars if other methods are
inconclusive. Okazaki et al. (2002) studied a subgroup of Be/X-ray systems
where the disk was aligned to the orbital plane, and found that the density
within the truncation radius (which is located where the tidal torque and
viscous torque are balanced) was larger than in the untruncated disks of
single Be stars. In eccentric orbits, the truncation radius is smaller at
periastron, and a tightly-wound spiral structure may be generated at closest
approach (Okazaki et al., 2002). Cyr et al. (2020) showed that for large
misalignment angles and higher viscosities, the spiral arms are folded more
closely. If the eccentricity of the system is close to one or if the disk and
orbital plane are misaligned by more than 60∘, these truncation effects are
inefficient (Martin et al., 2011; Okazaki, 2007).

$\delta$ Scorpii ($\delta$ Sco) is a bright spectroscopic binary system
consisting of a 15 M⊙ Be star and an 9 M⊙ secondary. While the presence of the
secondary star has been confirmed through interferometry, its spectral lines
have yet to be directly detected as its brightness is roughly one-fifth that
of the primary (Bedding, 1993; Tango et al., 2009; Hutter et al., 2021). As a
result, its spectral class has not been determined to a high degree of
certainty (Bodensteiner et al., 2020), although observations are compatible
with a main sequence star (Tango et al., 2009; Carciofi et al., 2006;
Miroshnichenko et al., 2013). This fact makes $\delta$ Sco of special
interest, as a literature study by Bodensteiner et al. (2020) found a lack of
verified main sequence companions among Be stars earlier than B1.5. The
inclinations of the primary star and its disk have been measured through
interferometry to be approximately 26∘ (Meilland et al., 2013) and 27∘ (Che et
al., 2012), respectively. Similarly, Carciofi et al. (2006) found a stellar
inclination of 38∘ $\pm$ 5∘ using polarimetry models that assumed the disk was
equatorial.

This system is highly eccentric (e $\approx$ 0.94) with a period of
approximately 10.8 years (Tango et al., 2009; Tycner et al., 2011; Meilland et
al., 2011). At periastron, Tycner et al. (2011) have found the minimum
separation between the two stars to be 6.14 mas. Adopting a distance of 150
pc, as determined by astrometric parallax (van Leeuwen, 2007), this
corresponds to a separation of 0.92 AU or 29 $R_{\star}$, where $R_{\star}$
represents the radius of the primary star.

The primary star in this system represents one of few examples where a Be star
has been consistently observed as its disk grew and evolved over three
decades. Sco was first reported to display faint H$\alpha$ emission in 1990
(Cote & van Kerkwijk, 1993) and showed strong H$\alpha$ emission by the summer
of 2000 (Miroshnichenko et al., 2001). Since then, the binary system has
undergone three periastron passages: on UT 2000 September 10 (Tycner et al.,
2011), UT 2011 July 3 (Che et al., 2012), and UT 2022 April 24 (based on the
orbital period of Tycner et al. 2011). The behaviour of the system during its
periastron passages in 2000 and 2011 has been studied by Miroshnichenko et al.
(2001), Tycner et al. (2011), Miroshnichenko et al. (2013), Che et al. (2012)
and others. Suffak et al. (2020) studied long-term varations of the system
over nearly two decades from 2000 to 2018, finding significant disk growth
during 2009 to 2011 as well as evidence of partial disk dissipation at
apastron. The behaviour of the system during its most recent periastron in
April of 2022 has yet to be studied.

In this paper, we present for the first time a systematic comparison of the
behaviour of the disk surrounding the Be component of $\delta$ Sco during the
binary system’s three most recent periastra. We discuss how the evolution of
the disk between each periastron passage influences the interactions between
the secondary star and the disk at closest approach. We utilize H$\alpha$
spectroscopy, $V$-band photometry, and multiband polarimetry to probe disk
structure, size and density, supporting these observations with 3-dimensional
simulations and radiative transfer calculations. Section 2 details our
methodology, including a description of the codes used in our analysis.
Section 3 provides our observations and their sources, along with any
reduction methods applied. Our findings are presented in Section 4, while
Section 5 contains our discussion and Section 6 provides our conclusions.

##  2 Methodology

Our model predictions were accomplished using two three-dimensional codes:
HDUST, used to investigate the behaviour of the $V$-band polarimetry in the
2010s, and a smoothed particle hydrodynamics (SPH) code, used to trace the
structure and extent of the disk through the three close encounters between
the secondary and primary.

HDUST is a non-local thermodynamic equilibrium (NLTE) radiative transfer code
that uses the Monte Carlo method to predict observables for the models of
circumstellar disks (Carciofi & Bjorkman, 2006). Given particular input
parameters of a Be star-disk system, such as stellar mass, radius and rotation
rate, disk radius, and disk density, HDUST simulates the random propagation of
photon packets through the disk medium. It calculates the temperature
structure of the disk, along with the ionization fraction and level
populations in the NLTE regime. HDUST can predict images, spectral lines,
photometry, and polarization in wavelengths of interest. It has been used for
studies on individual Be stars (for example, see Suffak et al. 2020,
Richardson et al. 2021, Ghoreyshi et al. 2021, Marr et al. 2021, Marr et al.
2022, Rubio et al. 2023) and for large statistical studies (for example, see
Rímulo et al. 2018 and Suffak et al. 2023).

In our HDUST models, we used a power law for volume density of the disk

$\rho(r,z)=\rho_{0}\left(\frac{R_{\star}}{r}\right)^{n}\exp\left(-\frac{z^{2}}{2H^{2}}\right)\,,$
(1)

where $r$ is the radial position in the disk, $z$ is the vertical position in
the disk, $R_{\star}$ is the equatorial radius of the Be star, $\rho_{0}$ is
the density where $r$ = $R_{\star}$ and $z=0$, $n$ is the parameter that
governs how quickly the density drops off with increasing $r$, and $H$
represents the scale height.

The SPH code was first developed by Benz et al. (1990) and Benz (1990). Bate
et al. (1995) modified the code to increase efficiency by adding a Runge-Kutta
integrator, and Okazaki et al. (2002) further adapted it for the viscous
decretion disks of Be binary systems. It has been used extensively to follow
Be star disk structure over time. For recent work, see Panoglou et al. (2016),
Cyr et al. (2017) and Suffak et al. (2022).

The SPH simulations begin with a diskless system, given the values for the
mass and radii of both stellar components and the orbital parameters of the
system. We then inject material into the disk at a prescribed rate near the
primary star. The disk material flows slowly outward facilitated by viscosity
following the prescription of Shakura & Sunyaev (1973):

$\nu=\alpha c_{s}H\,,$ (2)

where $\nu$ is the viscosity, $\alpha$ is a dimensionless parameter that
scales the viscosity (hereafter called the viscosity parameter), $c_{s}$ is
the sound speed in the disk, and $H$ is the scale height.

##  3 Observations and Data

###  3.1 Photometry

Photometric data for 2009 to 2023 were collected at the Fairborn Observatory
in the Patagonia Mountains of southern Arizona, USA. These observations were
acquired using the T3 0.4 m automatic photoelectric telescope (APT) equipped
with a photometer that measures photometric count rates with an EMI 9924B
photomultiplier tube. These observations each have an accuracy of
approximately 0.003 to 0.005 mag on a good night, with seasonal brightness
means of 0.0001 to 0.0002 mag (Jones et al., 2011; Henry, 1999). Observations
were recorded in the Johnson $B$ and $V$ bandpasses. The data spanning 2009 to
2019 were first published in Suffak et al. (2020).

Additional data for 2000 to 2023 were obtained from the American Association
of Variable Star Observers (AAVSO)111https://www.aavso.org/data-access. To
ensure high-quality measurements, we selected $V$-band values that were
flagged as “validated,” meaning they were peer reviewed by AAVSO members.
Together with the observations from Fairborn, these data created excellent
coverage for the periastron years of interest as well as the time period
between.

###  3.2 Spectroscopy

Spectral data for 2000 were obtained from the Ritter Observatory Public
Archive. They were observed at the Ritter Observatory in Toledo, Ohio, USA,
using a 1.06 m telescope equipped with a fiber-fed echelle spectrograph and a
CCD camera manufactured by Wright Instruments Ltd. These spectra have a
resolving power of $R$ = 26,000 over a wavelength range of 5285 to 6527 Å,
split across 9 orders. We focused on the order containing the H$\alpha$ line,
ranging from 6527 to 6594 Å.

Spectra for the 2005 to 2018 period were also obtained from the Lowell
Observatory in Flagstaff, Arizona, USA. These were taken with the 1.1 m John
S. Hall telescope equipped with an echelle spectrograph with a resolving power
of $R$ = 10,000. These observations from the Lowell Observatory were first
published in Suffak et al. (2020). Three additional spectra for 2022 used in
this study have been acquired at the 0.4 m telescope of the Brooks Observatory
at Central Michigan University (CMU). The telescope was equipped with LHIRES
III spectrograph from Shelyak Instruments with 2400 lines/mm grating, used in
conjunction with a Atik 460EX monochrome CCD camera and 50 $\mu$m slit. This
resulted in H$\alpha$ spectra with a resolving power of $R$ = 11,000.

For the period spanning 2011 to 2023, additional observations were taken from
the BeSS database222http://basebe.obspm.fr/basebe/ (Neiner et al., 2011),
observed with a 0.4 m telescope at the observatory of the Vereinigung der
Sternfreunde Köln, near Cologne, Germany. For 2011-2017, these were taken with
a Shelyak LHIRES III spectrograph with a resolving power of $R$ = 11,000. For
2017-2023, the spectra were taken with a Shelyak LHIRES III spectrograph with
a resolving power of $R$ = 20,000.

All spectral files for $\delta$ Sco containing the He i 6678 Å line were also
downloaded from BeSS. Spectra with low SNR or bad wavelength calibration were
deleted to remove outliers, and selected observations were not restricted to a
specific observing site. The available data provided coverage for the
periastron in 2011, but not in 2000 or 2022.

###  3.3 Polarimetry

Table 1: $\delta$ Sco’s observed field stars.  Object | Spectral ClassaaSpectral Class taken from SIMBAD database. | Angular Separation (∘) | DistancebbDistance for $\delta$ Sco from Hipparcos (van Leeuwen, 2007); distances for the field stars based on Gaia DR3 (Babusiaux et al., 2023) (pc)
---|---|---|---
$\delta$ Sco | B0.3IV | 0 | $150\begin{subarray}{c}+0.4\\\ -0.5\end{subarray}$
HD 142902 | A9V | 0.54 | 149.5 $\pm$ 0.4
HD 143567 | B9V | 0.74 | 134.7 $\pm$ 0.5
HD 143600 | B9.5V | 0.44 | 142.6 $\pm$ 0.6
HD 144548 | F7V | 1.71 | 150.9 $\pm$ 1.3
Table 2: Parameters of the Serkowski function for the field stars of $\delta$ Sco and the mean polarization angles.  Object |  $P_{max}$ (%) | $\lambda_{\rm max}$ | K |  $\left\langle\theta\right\rangle$ (°)
---|---|---|---|---
HD 142902 | 0.411 $\begin{subarray}{c}+0.084\\\ -0.004\end{subarray}$ | 0.568 $\begin{subarray}{c}+0.018\\\ -0.475\end{subarray}$ | 0.95 $\pm$ 0.07 | 148.0 $\pm$ 0.3
HD 143567 | 0.553 $\begin{subarray}{c}+0.008\\\ -0.007\end{subarray}$ | 0.595 $\begin{subarray}{c}+0.016\\\ -0.535\end{subarray}$ | 1.00 $\pm$ 0.08 | 142.3 $\pm$ 0.3
HD 143600 | 0.628 $\begin{subarray}{c}+0.008\\\ -0.099\end{subarray}$ | 0.614 $\begin{subarray}{c}+0.013\\\ -0.569\end{subarray}$ | 1.03 $\pm$ 0.08 | 144.6 $\pm$ 0.4
HD 144548 | 0.602 $\begin{subarray}{c}+0.008\\\ -0.008\end{subarray}$ | 0.703 $\begin{subarray}{c}+0.010\\\ -0.011\end{subarray}$ | 1.18 $\pm$ 0.08 | 144.4 $\pm$ 0.6

The polarimetric observations of $\delta$ Sco were carried out beginning in
2010 at Pico dos Dias Observatory (OPD), located in Minas Gerais, Brazil, and
operated by the National Laboratory of Astrophysics (LNA). These observations
were conducted using the 1.60m Perkin-Elmer telescope and the 0.60m Boller &
Chivens telescope, jointly with the IAGPOL polarimeter (described by Magalhaes
et al., 1996) to obtain linear polarization in the optical range with BVRI
filters. The polarimeter consists of a half-wave retarder and a calcite prism
analyzer, with the light beams ultimately being recorded by a CCD camera. In a
typical observation, a sequence of images at eight half-wave plate positions
(WPP) separated by 22.5∘ is acquired. For each observing night, at least one
polarized standard star was observed to calibrate the polarization angle in
the equatorial system. Data reduction was performed using software packages
described by Magalhaes et al. (1984, 1996) and Carciofi et al. (2007). Data
for $\delta$ Sco prior to 2010 were obtained from the MAST
archive333https://archive.stsci.edu/hpol/index.html. These data were collected
using the University of Winsconsin’s Half-wave Spectropolarimeter (HPOL) at
the Pine Bluff Observatory near Madison, Winsconsin (Nordsieck & Harris,
1996).

In parallel with the polarimetric observations, additional observations were
conducted at OPD to determine the interstellar polarization (ISP) along the
line of sight of $\delta$ Sco, using the same observational procedures and
data reduction methods previously mentioned. Knowing the value of the ISP is
crucial for the study of Be stars because it allows us to ascertain the
intrinsic polarization of the star, which arises due to the presence of the
disk. The method of field stars is the simplest way to determine the value of
ISP. It involves selecting and observing stars that are physically close to
the target both in radial and angular distance. If these field stars do not
exhibit intrinsic polarization, the observed polarization will serve as a
proxy of the ISP. Four field stars of $\delta$ Sco were observed: HD 144548,
HD 143567, HD 142902, and HD 143600. Additional data for these stars and
$\delta$ Sco are detailed in Table 1.

The spectral dependence of linear polarization in the interstellar medium with
wavelength is described by the Serkowski Law (Serkowski et al., 1975)

$\frac{P(\lambda)}{P_{\rm max}}=\exp\left[-K\ln^{2}\left(\frac{\lambda_{\rm
max}}{\lambda}\right)\right],$ (3)

where $P_{\rm max}$ is the maximum polarization that occurs at a wavelength
$\lambda_{\rm max}$, $K$ is a dimensionless constant that describes the
inverse width of the polarization curve peaking around $\lambda_{\rm max}$
(Cotton et al., 2019). If $K$ is treated as a third free parameter, it can be
shown that $\lambda_{\rm max}$ and $K$ are linearly related following the
relationship (Martin et al. 1992)

$K=c_{1}\lambda_{\rm max}+c_{2}.$ (4)

Whittet et al. (1992) define the values of $c_{1}=1.66\pm 0.09$ and
$c_{2}=0.01\pm 0.05$, with $\lambda_{\rm max}$ in micrometers ($\mu\rm m$).
Equation 3, with $K$ given by Eq. 4, is also known as the Serkowski-Wilking
Law (Wilking et al., 1980).

For each dataset obtained for $\delta$ Sco’s field stars, the Serkowski
function (Eq. 3) is fitted using the Monte Carlo method via Markov Chains. The
fits return the values of $P_{\rm max}$ and $\lambda_{\rm max}$ for each curve
(Table 2). In addition to the parameters above, an additional parameter is
necessary to fully described the ISP, namely the polarization position angle,
$\theta_{IS}$.

Figure 1: Left: Polarization map for $\delta$ Sco’s field. The direction of
each bar represents the polarization position angle, while the length of the
bars represents the magnitude of the maximum polarization. Right: Maximum
polarization and distance of each field star. The dashed vertical line
represents the distance of $\delta$ Sco, and the dotted lines represent the
distance uncertainties.

Fig. 1 displays the positions of the stars in the sky (declination,
$\Delta_{DEC}$, and right ascension, $\Delta_{RA}$) along with their
respective polarization pseudovectors, representing the maximum polarization
degree and polarization angle in each observed filter. In this map, there is
consistency in the direction of the polarization pseudovectors, which may
indicate that all the stars sample the interstellar medium behind which
$\delta$ Sco is located. Fig. 1 (right) presents the maximum polarization of
each field star along with its respective distance.

To find the maximum polarization of the interstellar medium ($P_{\rm max}^{\rm
IS}$) and the interstellar polarization angle ($\theta_{IS}$), an average was
calculated from the respective values of each field star, resulting in $P_{\rm
max}^{\rm IS}=0.549\%\pm 0.003\%$ and $\theta_{IS}=144.8^{\circ}\pm
0.2^{\circ}$, respectively. In order to find the ISP in each filter, we used
the Serkowski-Wilking Law, reaching the results $P_{\rm IS}^{\rm B}=0.490\%\pm
0.006\%$, $P_{\rm IS}^{\rm V}=0.539\%\pm 0.003\%$, $P_{\rm IS}^{\rm
R}=0.549\%\pm 0.003\%$ and $P_{\rm IS}^{\rm I}=0.505\%\pm 0.005\%$, for the
BVRI filters, respectively. Hall (1958) found the polarization degree of
$\delta$ Sco in $V$-band filter to be $P_{\rm V}=0.60\%$, during the diskless
phase of the star. Therefore, this value refers to the level of ISP along the
line of sight. This value is very close to what we obtain from observations of
$\delta$ Sco’s field stars in the $V$-band ($P_{\rm IS}^{\rm V}=0.539\%\pm
0.003\%$).

##  4 Results

Figure 2: $V$-band photometry and H$\alpha$ spectroscopy for $\delta$ Sco from
2000 to 2023. The time scale on the bottom x-axis is the Modified Julian Date
(MJD), while the top x-axis provides the year. Panel a) shows $V$-band
photometry, while panel b) shows the $B-V$ colour index and panels c), d), e),
and f) show H$\alpha$ equivalent width (EW), full width at half maximum
(FWHM), peak separation ($\Delta$vp), and emission peak ratio (V/R),
respectively. The legends indicate the data sources for each panel. The
periastron times are indicated by vertical lines.

###  4.1 Photometry

$V$-band photometry for the years spanning 2000 to 2023 is shown in panel (a)
of Fig. 2. The mean magnitude has brightened between each consecutive
periastron, with the changes between 2000-2011 being especially large. The
$B-V$ colour index, plotted in panel (b) of Fig. 2, shows variations that are
concurrent with the $V$-band fluctuations, and they follow a similar trend.
Figure 3 shows the $V$ magnitude versus colour index ($B-V$) for 2009 to 2022.
The system reddened during the time leading up to the 2011 periastron, but
remained approximately the same colour between 2011 and 2022.

Panel a) of Fig. 4 shows the $V$-band data for the three most recent
periastra. The system dimmed by a small amount around the 2000 periastron, but
this may have been related to the large-amplitude variability observed in the
following years. Jones et al. (2011) found cyclical brightness variations in
the data spanning 2009 to 2012, although these cycles have not been detected
in the photometry from 2013 onwards. The data do not indicate significant
perturbations of the disk in 2011 or 2022.

Figure 3: $V$ magnitude vs. the $B-V$ colour index for $\delta$ Sco from 2009
to 2022. The legend indicates the data for each year. Figure 4: Comparison of
the $V$-band, H$\alpha$ and polarization data for $\delta$ Sco during the time
surrounding the 2000, 2011 and 2022 periastra. Panel a) shows the $V$-band
magnitude. Panels b), c) d), e) and f) show the equivalent width (EW), full
width at half maximum (FWHM), peak separation ($\Delta$vp), emission peak
ratio (V/R), and EWV/EWR for the H$\alpha$ line, respectively. Panel (g) shows
the $V$-band intrinsic polarization, while panel (h) shows the PA. The x-axis
measures time until the periastron for each year. The vertical lines represent
the periastra.

###  4.2 Spectroscopy

As mentioned in Section 1, previous studies on $\delta$ Sco have been unable
to detect the spectral lines of the secondary star. If the radial velocities
between the primary and secondary stars at periastron are sufficiently large,
it could be possible to differentiate the lines originating from the primary
and the secondary. However, the maximum radial velocity of the primary star at
periastron is $\approx$ 50 km s-1, corresponding to a shift of roughly 0.1 nm
(Miroshnichenko et al., 2013; Tango et al., 2009). These shifts are small
compared to the width of the H$\alpha$ line, and therefore the secondary star
remains undetected through spectroscopy.

Panels (c), (d), (e) and (f) of Fig. 2 show the EW, FWHM, peak separation
$\Delta$vp and V/R of the H$\alpha$ emission line from 2000 to 2023. Overall,
the data during the periastra do not show abnormalities in comparison to the
rest of the observations. The V/R and FWHM variations in 2011 are the main
exceptions to this trend.

Figure 5: Emission peak ratio (V/R) for the He i 6678 line during 2011. The
vertical line represents the periastron time.

Fig. 5 shows the He i 6678 Å line for the time period surrounding the 2011
periastron. Similar to the H$\alpha$ line, this spectral line shows enhanced
blue emission near periastron, but delayed by approximately one month. Since
the He i 6678 line forms in the inner disk while the H$\alpha$ line forms over
a larger emitting volume, this lag may indicate that the outer disk was
perturbed first, then the helium emitting volume was perturbed later.

Panels (b), (c), (d) and (e) of Fig. 4 plot the H$\alpha$ EW, FWHM,
$\Delta$vp, and V/R for only the periastra. Panel (f) of Fig. 4 plots the
ratio of the EW of the violet side of the H$\alpha$ line to the EW of the red
side (EWV/EWR). The radial velocities for the H$\alpha$ line were found using
the mirrored profile method, fitting the wings of the lines to compare the
direct and reversed line profiles. The radial velocity measurement for each
observation was then used as the line center, and the EW of the line on either
side of this value was calculated. This ratio can be used to trace line
asymmetries when the profile is singly-peaked or shows complex peak structure,
in which cases the V/R ratio cannot be used.

Fig. 4 shows a general lack of repetitive patterns throughout the periastra.
With the exception of the V/R ratio, the measured values leading up to each
periastron are different. The unifying feature between periastron years is the
EWV/EWR, which consistently peaks near periastron.

###  4.3 Polarimetry

By vector subtraction, we can extract the intrinsic data of $\delta$ Sco by
separating the interstellar component from the observed measurements. Fig. 6
exhibits the variation of intrinsic polarization degree and angle over time
for the $V$-band. We computed an average of the intrinsic polarization angles
in all four filters, resulting in $\theta=73.42^{\circ}\pm 0.14^{\circ}$.

Figure 6: Top: $V$-band photometry for $\delta$ Sco during the period
2001-2023. Center: Intrinsic polarization degree. Bottom: Polarization angle.
The time scale on the bottom x-axis is Modified Julian Date, while the scale
on the top x-axis is the decimal year.

An alternative method for acquiring the intrinsic polarization angle of a star
is through the observed $Q$ and $U$ Stokes parameters (Draper et al., 2014).
The positioning of $\delta$ Sco’s data points on the $QU$ diagram for the
$V$-band is illustrated in Fig. 7. The $QU$ diagrams for the remaining filters
are found in the Appendix. The data in all filters adhere to a consistent
linear pattern, signifying a steady intrinsic angle of $\theta=74.4^{\circ}\pm
0.1^{\circ}$. This angle is in agreement with the calculated intrinsic
measurement obtained using the field stars method, and offers strong
supporting evidence for the ISP determination.

Figure 7: (a) QU Diagram of $\delta$ Sco in the $B$-band. The dashed lines
indicate the most likely solution of the damped least-squares method
(Levenberg–Marquardt algorithm), with a reduced chi-squared ($\chi^{2}$) of
31.8678. (b) Same as in (a), but for the $V$-band, with reduced
$\chi^{2}=24.9883$. (c) Same as in (a), but for the $R$-band, with reduced
$\chi^{2}=22.0285$. (d) Same as in (a), but for the $I$-band, with reduced
$\chi^{2}=56.4171$

.

Fig. 6 compares the long-term trends in polarization vs. $V$-band photometry.
The polarization degree, shown in panel (b), shows a gradual increase leading
up to 2011, followed by a decline over the next four years. By 2015, the
polarization degree stabilized at its lowest recorded values. Curiously, this
minimum in the average value of the polarization corresponds to when the
V-band photometric data is close to its maximum value. The position angle (PA)
is shown in panel (c). This shows the greatest range of variation in 2011, but
remains roughly stable in every other year. The scatter observed in the data
is not a result of observational errors; instead, it represents intrinsic
short-term variations.

##  5 Discussion

The highly eccentric binary orbit of $\delta$ Sco has important implications
for the tidal interactions between the secondary star and the primary’s disk.
The periastron passage is a brief event; if we assume the orbital parameters
found by Tycner et al. (2011), the secondary is within 50 $R_{\star}$ of the
primary for no more than 40 days. Here, we choose the astrometric solution for
parallax rather than the dynamical parallax estimates by Miroshnichenko et al.
(2013), Tycner et al. (2011) and Tango et al. (2009), due to the large flux
ratios of the system. Although the minimum separation of the two stars is
small (approximately 29 $R_{\star}$, assuming a stellar radius of 6.8 R⊙), the
secondary star is acting on the disk at this distance for only a matter of
hours. Since the secondary star is near the primary for a small portion of its
orbit, the interaction time is much smaller than for systems with smaller
eccentricities (Panoglou et al., 2016). For the remainder of the system’s
nearly 11-year period, the disk is not influenced by the secondary star and
can evolve independently. At each encounter, the secondary star interacts with
a disk with different characteristics.

The observed trends in photometry, spectroscopy and polarimetry give useful
insight into how the disk characteristics differ leading up to each closest
approach, resulting in distinct patterns in the observations for each passage.
We will examine the photometric evidence first, then turn to the H$\alpha$
spectra and polarimetry. We then conclude this section with a summary
comparing the physical characteristics of the disk at each periastron passage.

###  5.1 Disk evolution

After the periastron in 2000, the $V$-band photometry shown in panel (a) of
Fig. 2 shows a significant drop in brightness beginning in 2005 which reached
a minimum between 2006 and 2007. The H$\alpha$ EW, shown in panel (c) of Fig.
2, exhibits a similar pattern during these years, although its response lagged
behind the $V$-band because the H$\alpha$ emission line is formed in a larger
volume of the disk (Carciofi, 2011). Suffak et al. (2020) discussed these
variations in detail, attributing the drop in brightness and H$\alpha$
emission strength to partial dissipation of the young disk. Such behaviour has
also been observed in 66 Oph (Marr et al., 2021), as well as $\omega$ CMa
(Ghoreyshi et al., 2018).

The trend in $V$ vs. $B-V$ shown in Fig. 3, and the long-term trends shown in
panels (a) and (b) in Fig. 2, indicate that $\delta$ Sco has been both
brightening and reddening since 2009. For (near) pole-on viewing, as is the
case for this star, this reddening is expected during periods of disk growth
(Haubois et al., 2012). In simple terms, reddening occurs because the disk’s
contribution to the total flux builds as the disk grows (Vieira & Carciofi,
2017), and the disk has a lower temperature than the star (Carciofi &
Bjorkman, 2006; Vieira et al., 2015). Most of the growth appears to have
occurred between 2009 and 2012, as those years correspond to the largest
changes in both $V$ and $B-V$. While the system brightened by roughly one-
tenth of a magnitude in the $V$-band between the two most recent periastron
passages, the system did not redden significantly during this time. Fig. 3 is
a standard example of the variability resulting from the growth and
dissipation of a disk for a pole-on Be star, as predicted by Haubois et al.
(2012) and observed in numerous Be stars (for example, see de Wit et al.
2006).

The decrease in FWHM values between 2000 and 2011, as shown in panel (c) of
Fig. 4, also points to a period of disk growth during the same period. This is
consistent with the findings of Suffak et al. (2020), who concluded on the
basis of H$\alpha$ EW values and $V$-band photometry that the disk underwent
disk building and dissipating periods between 2000 to 2011, yielding an
overall increase in the spatial extent of the disk during this period. The
leveling off of FWHM values after 2011 suggests that disk growth was less
significant between 2011 and 2022. However, the disk must have grown somewhat,
evidenced by the decrease in peak separation between 2011 and 2022 and the
slow secular increase in $V$-band flux in the past decade.

The radius of the disk, $R_{d}$, can be related to the peak separation
$\Delta$vp and rotational velocity of the star $v\sin{i}$ through

$\frac{R_{d}}{R_{\star}}=\left(\frac{2v\sin{i}}{\Delta v_{p}}\right)^{2}\,,$
(5)

assuming Keplerian rotation (Huang, 1972; Hummel & Vrancken, 1995; Meilland et
al., 2007). Adopting a v $\sin{i}$ of 175 $\kmps$ (Bernacca & Perinotto,
1970), then, we can use H$\alpha$ peak separation values to roughly estimate
the radius of the H$\alpha$ emitting region (ER) during each periastron year.

In 2000, peak separation values began to change roughly 20 days before
periastron. They rose from 150 $\kmps$, corresponding to an H$\alpha$ ER of
roughly 5 $R_{\star}$, to approximately 170 $\kmps$, corresponding to an
H$\alpha$ ER of 4.2 $R_{\star}$ at closest approach. These values for peak
separation at periastron are slightly larger than those reported by
Miroshnichenko et al. (2001), but yield values for the H$\alpha$ ER radius
that are consistent with the disk radius of 5.3 $R_{\star}$ reported by
Miroshnichenko et al. (2003) for 2003, and somewhat lower than the estimate of
7 $R_{\star}$ by Carciofi et al. (2006). All estimates agree that the
H$\alpha$ ER radius in 2000 fell well within the first Lagrange point at
$\approx$ 20 $R_{\star}$, (Miroshnichenko et al., 2013; Eggleton, 1983; Kopal,
1959), so mass transfer between the primary and the secondary star during
periastron during the first periastron, in 2000, is unlikely. The secondary
star would have passed no closer than 19 to 22 $R_{\star}$ from the H$\alpha$
ER of the disk.

In 2011, the peak separation had lowered to approximately 70 $\kmps$,
corresponding to an H$\alpha$ ER radius of about 25 $R_{\star}$. This value is
similar to the model-based estimate for the size of the H$\alpha$ ER by Suffak
et al. (2020) for that year, but somewhat larger than the interferometry
measurements of 14.9 $R_{\star}$ in 2007 and 9 $\pm$ 3 $R_{\star}$ in 2010 by
Millan-Gabet et al. (2010) and Meilland et al. (2011), respectively. If we
ignore any truncation effects on the disk, the secondary star could have
passed within a few $R_{\star}$ of the H$\alpha$ ER. However, the peak
separation reached a maximum of roughly 90 $\kmps$, which indicates the
H$\alpha$ ER was truncated to approximately 15 $R_{\star}$ at closest
approach.

In 2022, the double-peaked structure of the H$\alpha$ line, as seen in panel
(a) of Fig. 8, was not discernible until the week before minimum separation
between the primary and the secondary. Peak separation at closest approach was
approximately 50 $\kmps$, corresponding to a disk radius of 49 $R_{\star}$.
This value is consistent with the estimate by Suffak et al. (2020) for 2018.
Interestingly, inspection of the changes in the line profile in 2022 reveals
that the upper limit for the disk radius may be smaller than indicated by the
peak separation. The line profile remains relatively unchanging and singly
peaked up to 30 days before closest approach. After this point, the line shows
a more complicated profile with significant asymmetries and increased peak
separation indicating partial disk destruction. If these asymmetries are
caused by interactions with the secondary star, this may suggest that the
upper limit for the disk’s H$\alpha$ ER in early 2022 is roughly 30
$R_{\star}$, since line profile changes were not apparent until the secondary
star was within this distance.

Figure 8: a) The H$\alpha$ line profile as observed approximately one week
before closest approach in 2000, 2011 and 2022. The 2000 spectrum was taken
nine days prior, while the 2011 and 2022 spectra were captured seven days
before closest approach. The narrow absorption features in the continuum are
telluric lines. (b) A top-down view of the position of the secondary star
relative to the primary star and the H$\alpha$ emitting region (ER) at the
time each spectrum in (a) was observed. Sizes of the primary and secondary
stars are not to scale. (c) Same as in (a), except the selected spectra were
observed on the dates nearest closest approach. The spectrum in 2000 was
observed three days prior to closest approach, the spectrum for 2011 was
observed on the day of closest approach, and the spectrum for 2022 was taken
four days before closest approach. (d) Same as in (b), but for (c). (e) Same
as in (a), except the selected spectra were observed approximately one week
after closest approach. The spectrum in 2000 was observed ten days after,
while the spectrum for 2011 was observed eight days after and the spectrum for
2022 was taken nine days after closest approach. (f) Same as in (b), but for
(e).

Fig. 9 shows the binary orbit of $\delta$ Sco on the orbital x-y plane, where
the origin is centered on the primary star. The orbital path of the secondary
star was calculated using the orbital parameters published in Tycner et al.
(2011), which were produced from high angular resolution interferometric
observations. The extent of the H$\alpha$ ER for each periastron year, based
on the peak separation values discussed above, is also plotted. The secondary
star, whose position is plotted at 5-day intervals for 50 days prior and
following periastron, interacts with the disk for different lengths of time
during each consecutive periastra.

Figure 9: The binary orbit of $\delta$ Sco, in the orbital x-y plane, showing
the orbital solution (blue line) relative to the primary star (red star) along
with the estimated sizes of the disk’s H$\alpha$ emitting region (ER) for each
periastron year (concentric blue circles). The position of the secondary star
(gray circles) is plotted at 5 day intervals for 50 days before and after
minimum separation. The motion of the secondary star is counter-clockwise.

Panel (a) of Fig. 8 compares the observed H$\alpha$ line profile roughly 1
week before closest approach in each periastron year. Since these spectra were
taken at approximately the same orbital phase (within 2 days of each other),
the radial velocities of the system for each observation are comparable. The
calculated position of the secondary star on its orbital path at the moment
each spectrum was observed is shown in panel (b). The intensity of the line
has become progressively larger for each periastron, indicative of disk
evolution during the intervening years. In 2000, the disk was the smallest and
therefore the least perturbed of the three periastra, evidenced both by the
larger peak separation and roughly equal peak heights. The profiles for 2011
and 2022, where the emission is stronger at smaller wavelengths, may represent
increasingly perturbed disks. In addition to showing higher relative flux at
the central wavelengths, the relative flux of the violet side of the line
increased significantly more than the red side for the two most recent
periastron years.

These asymmetries of the line profiles are still more apparent in the spectra
observed nearest each periastron event, shown in panel (c) of Fig. 8. The 2011
spectrum shows a red peak enhancement that is not as pronounced in either of
the other two years. The 2022 spectrum does not show much variation from panel
(a), however this is not unexpected as the two spectra were observed only
three days apart. Ratios of the EW of the violet side of the line to the EW of
the red side (EWV/EWR) should demonstrate these asymmetries more clearly than
the V/R ratios, due to the complex nature of the H$\alpha$ peak structure.
Indeed, panel (f) of Fig. 4 shows the expected peaks in EWV/EWR around closest
approach. In the spectra observed roughly one week after periastron, presented
in panel (e), a less complex double-peaked structure is more apparent in both
2011 and 2022.

We note that the H$\alpha$ line and the He i 6678 line both showed enhanced
blue emission near the 2011 periastron, with the peak in the He I 6678 V/R
observed approximately one month after the peak in the H$\alpha$ EWV/EWR. This
excess blue emission could be caused by density and velocity perturbations
propagating inward through the disk in response to the secondary star’s
gravitational influence. As the secondary star approaches the primary during
the time leading up to periastron, it interacts with the blue-shifted side of
the disk first (see Figs. 2 and 3 in Che et al. 2012). Another contributing
factor could be the secondary star illuminating the disk, heating the portion
of it nearest the secondary which results in stronger line emission. This
phenomenon has been observed before in the Be star 59 Cyg, where the secondary
is a hot subdwarf star (Peters et al., 2013). In the future, a more dedicated
modelling effort is needed to test these hypotheses.

Overall, the general lack of repetitive features in the data shown in Fig. 4
can be explained by different disk characteristics at each passage of the
secondary star. Due to the high eccentricity of the system, the Be disk can
evolve independently of the secondary star for the majority of the orbit. This
explanation is supported by radius estimates of the H$\alpha$ peak separations
presented here, in addition to the long-term modelling of the system by Suffak
et al. (2020). The asymmetric H$\alpha$ profiles during the close encounters
in 2011 and 2022 indicate more significant disk perturbations during those
events than during the 2000 passage when the disk was smaller.

###  5.2 Polarization

Panel (b) of Fig. 6 shows distinct drops in the $V$-band polarization degree
in 2008 and 2010. These abrupt decreases may be explained by mass ejection
events. When the mass injection rate increases suddenly, enhanced densities in
the inner disk lead to increased continuum emission, which in turn causes the
$V$-band polarization to drop (Haubois et al., 2014). This downturn is
temporary because, as the disk grows, the number of scatterers increases so
the polarization must also rise (Haubois et al., 2014). This would be
consistent with Sco’s behaviour in 2011, where the polarization rapidly
increases to its highest levels in our observation set.

Following the 2011 periastron, the drop in $V$-band polarization degree, rise
in $V$-band photometry and decrease in H$\alpha$ EW shown in Fig. 2 indicate
sustained disk growth. The decrease in polarization degree could be explained
through a combination of multiple scattering and post-scattering absorption.
In a tenuous disk, polarization is initially linearly proportional to density.
However, for large density values, the polarization begins to flatten out and
eventually, it should begin to decrease. This is expected because photons
scattered in dense disks may be subsequently reabsorbed rather than escaping
the disk, causing an overall reduction in polarization degree.

Figure 10: Mean polarization degree ($P_{int}$) vs. the density at the
innermost regions of the disk ($\rho_{0}$) for the $V$-band, produced by HDUST
for an inclination angle of 35∘. The best-fit $\rho_{0}$ values for $\delta$
Sco from Suffak et al. (2020) for 2010 to 2018 are shaded in light green. The
light blue shaded region represents the observed intrinsic polarization in the
$V$ band from 2010 to 2018.

To investigate this explanation, a set of 26 HDUST models were run with
varying density values and with the stellar parameters listed in Table 3. The
number density $n_{0}$ of the disk was increased incrementally from 5$\times
10^{12}$ cm-3 to 5$\times 10^{14}$ cm-3. For these values of $n_{0}$, the
density at the innermost regions of the disk $\rho_{0}$ has values of 5$\times
10^{-12}$ g cm-3 to 5$\times 10^{-10}$ g cm-3. Suffak et al. (2020) found that
the disk of $\delta$ Sco had $\rho_{0}$ $\approx$ 5$\times 10^{-11}$ g cm-3 to
2.5$\times 10^{-10}$ g cm-3 for 2010 to 2018. Fig 10 shows the mean
polarization level vs. $\rho_{0}$ for these models. The $V$-band polarization
degree was estimated by calculating the mean polarization degree across the
$V$-band (500 to 650 nm).

Fig. 10 shows that as $\rho_{0}$ increases, the intrinsic polarization degree
also increases as expected. However, the intrinsic polarization degree reaches
a maximum and then drops off for larger values of $\rho_{0}$ when the disk is
very dense and optically thick. After $\rho_{0}\approx$ 3$\times 10^{-10}$ g
cm-3, the polarization degree stays approximately constant. Fig. 10 was
constructed to help us understand the observed trends in polarization degree.
We note that we held $n$ constant at $n=3.2$ (see Eq. 1), consistent with the
values from Suffak et al. (2020), and we did not attempt to match the observed
polarization degree in detail.

###  5.3 SPH models

Based on near-infrared (1.6 to 2.4 μm) spectrointerferomic data, Che et al.
(2012) suggested that the direction of the secondary star’s orbital motion
vector is opposite to that of the disk’s rotation. In such a retrograde orbit,
as the secondary is travelling toward the observer, the disk is rotating away.
Using near-infrared interferometry and radio mm observations, Štefl et al.
(2012) confirmed this finding. The direction of the secondary star’s motion
relative to the rotating disk has important implications on the expected
variations in all observables. A retrograde orbit reduces the interaction
times between the secondary star and individual particles in the outer regions
of the disk, therefore reducing the strength of the tidal effects (Overton et
al., 2024).

To investigate the effects of the secondary’s direction on disk geometry,
size, and any potential disruptions at closest approach, an SPH model was run
for both the prograde and the retrograde cases, where the disk and the
secondary’s orbit were coplanar. The parameters used in our SPH models are
shown in Table 3. These values represent the most up to date stellar and
orbital parameters, primarily from Suffak et al. (2020) and Tycner et al.
(2011). The mass and radius of the secondary star are taken from the
interpolated values for a B2 star in Silaj et al. (2014), based on the values
given by Cox (2000). The mass loss rate utilized here is consistent with the
rate from Suffak et al. (2022). The value adopted for $\alpha$ has been
commonly used in previous studies using this code and falls within the lower
end of estimates made by Rímulo et al. (2018) and Ghoreyshi et al. (2018). Our
simulations followed the disk evolution for three periastron passages,
beginning at apastron half a period before the first periastron. While an
extensive grid of SPH models was beyond the scope of this study, these models
aid in the interpretation of the observational data presented earlier.

Table 3: Parameters of the SPH models.  Parameter | Value
---|---
Primary Mass | 15.3 M⊙
Secondary Mass | 9.11 M⊙
Primary Radius | 6.8 R⊙
Secondary Radius | 5.33 R⊙
Orbital Period | 3951 days
Eccentricity | 0.938
$\alpha$ | 0.1
Mass loss rate |  $1\times 10^{-7}~{}$M⊙ yr-1

The SPH models show changes in disk structure and spatial extent at each
periastron passage. Fig. 11 plots the surface density vs. radius for various
values of the azimuthal angle $\phi$ during the three periastra for the
retrograde case. The angle $\phi$ is measured with respect to the binary’s
semi-major axis, where $\phi=180^{\circ}$ is in the direction of apastron.
Therefore, the $\phi=0^{\circ}$ line probes the density along the direction of
the periastron, when the secondary star is at its closest approach. For both
the prograde and the retrograde models, the outer limit of the disk has grown
between each periastron. In addition, the slope of surface density vs. radius
becomes shallower at each consecutive periastron. These two facts together
indicate that the disk has a larger radius at each successive periastron
event, consistent with the observational findings based on H$\alpha$
spectroscopy and polarimetry discussed above.

Both the prograde and retrograde models also indicate that the disk becomes
more asymmetric with each passing, although this effect is less pronounced in
the retrograde case. In both cases, the surface density shows perturbations at
periastron once the outer limit of the disk approaches $\approx$ 20
$R_{\star}$. This can be taken as an approximate value for the radius where
the disk becomes perturbed by the secondary star. The values for the radius of
the disk in the second and third periastra are lower than the estimated values
for 2011 and 2022 discussed above; this may be because the estimate is
approximate, as it does not take into consideration the diffuse emission. In
the future, it would be interesting to run a more extensive set of SPH models
with different values for the scaling parameter $\alpha$. As discussed in
Brown et al. (2019) and Panoglou et al. (2016), higher values of $\alpha$
allow the disk material to flow outwards more quickly, enabling faster disk
growth for the same mass loss rate. However, the overall trend of disk growth
and increased perturbations at each closest approach supports the observations
discussed previously.

Fig. 12 shows a snapshot of the prograde simulation, immediately after the
second periastron passage. A two-armed ($m=2$) spiral density enhancement is
clearly visible in the outer regions of the disk. This density wave is similar
to those produced in SPH simulations of binary Be stars by Cyr et al. (2020).
These arms dissipate within 0.02 to 0.03 orbital periods (80 to 120 days) of
closest approach, once the distance between the secondary star and the disk is
sufficiently large for the tidal torques to become negligible. Fig. 13 shows a
snapshot at the same point in time as in Fig. 12, but for the retrograde case.
The secondary star does not produce a clear spiral arm structure in this case,
as the interaction time between the secondary star and individual disk
particles is shorter than in the prograde case due to their relative
velocities. Overall, our models for the retrograde case are consistent with
the observed photometric, spectral and polarimetric data near periastron for
the disk parameters we used (see Table 3).

Figure 11: Plots of surface density vs. radius, from the coplanar retrograde
SPH simulation for the three periastron years using the paramters from Table
3. (a) is for the first periastron passage, while (b) and (c) are for the
second and third, respectively. The azimuthal angle $\phi$ is measured with
respect to the binary’s semi-major axis, where $\phi=0^{\circ}$ is in the
direction of periastron. Figure 12: Top-down snapshot of an SPH simulation
based on the parameters of $\delta$ Sco shown in Table 3, in which the disk is
prograde with respect to the binary orbit. This image is taken immediately
after the second periastron passage. The colour denotes the column density as
shown in the colourbar. The primary and secondary stars are denoted by small
black points, and the motion of the secondary star is counterclockwise. The
brown dots represent rogue medium-density particles that have been launched
from the disk. Figure 13: Top-down snapshot of an SPH simulation based on the
parameters of $\delta$ Sco shown in Table 3, in which the disk is retrograde
with respect to the binary orbit. This image is taken immediately after the
second periastron passage. The format is the same as Fig. 12.

##  6 Conclusions

Using photometric, spectral, and polarimetric data to trace the behaviour of
the Be binary $\delta$ Sco through its three most recent periastron events, we
can create a cohesive description of the disk’s physical characteristics at
each encounter with the secondary star. While we focused on the behaviour of
the disk during each close encounter, the system’s high eccentricity and long
period necessitated the consideration of the disk’s evolution during the
intervals between each event. The disk evolved independently of the secondary
star between each passage, leading to different responses to the presence of
the secondary at each closest approach.

During the 2000 periastron, the disk was young and relatively small. It
experienced truncation from the passage of the secondary star, indicated by a
drop in brightness and increase in H$\alpha$ EW and FWHM around closest
approach as seen in Fig. 4, but otherwise was relatively unperturbed. The
polarization and $V$-band photometry shown in Fig. 6 indicate increasing disk
density leading up to 2011, consistent with a building disk. This is further
evidenced by the overall increase in system brightness and stronger H$\alpha$
emission shown in Fig. 4. In 2011, the H$\alpha$ line profile in Fig. 8 and He
i 6678 line (see Fig. 5) exhibit significant asymmetries near minimum
separation, indicating perturbations that propagated to the inner regions of
the disk, but these damped out shortly after periastron. Polarization data are
consistent with disk growth in the years following 2011. By 2022, the disk had
grown again in extent, as shown by continued increase in system brightness and
H$\alpha$ emission in addition to a further drop in peak separation, shown in
Fig. 4. Once again, the H$\alpha$ line showed significant asymmetries around
closest approach, as demonstrated by the EW ratios in Fig. 4 and the profiles
shown in Fig. 8.

Overall, the passage of the secondary star produces increasingly larger
perturbations in the disk with each passage, due to the growth of the disk
between periastra. Our model with a retrograde coplanar disk, shown in Fig. 13
indicates that these perturbations are short-lived, as the tidal torques
quickly become negligible due to the large eccentricity of the system. One of
the features of our data that remain unexplained is the drop in polarizaiton
after 2013. In the future, it would be interesting to vary the viscosity
parameter $\alpha$ in an extensive suite of SPH models.

The authors would like to thank the referee, Dietrich Baade, for his comments
which greatly improved the quality of this paper. This work has made use of
the BeSS database, operated at LESIA, Observatoire de Meudon, France:
http://basebe.obspm.fr. Special thanks to observer Ernst Pollmann for
contributions of H$\alpha$ spectra. We thank Jonathan Labadie-Bartz for
providing processed spectra containing helium lines obtained from BeSS. Ritter
Observatory Public Archive is supported by the National Science Foundation
Program for Research and Education with Small Telescopes (PREST). We
acknowledge with thanks the variable star observations from the AAVSO
International Database contributed by observers worldwide and used in this
research. C.E.J. acknowledges support through the National Science and
Engineering Research Council of Canada. A.C.C. acknowledges support from CNPq
(grant 311446/2019-1) and FAPESP (grants 2018/04055-8 and 2019/13354-1). This
work was made possible through the use of the Shared Hierarchical Academic
Research Computing Network (SHARCNET). G.W.H. thanks Lou Boyd, Director of
Fairborn Observatory, for his decades of support for our robotic telescopes.
C.T. would like to acknowledge, with thanks, FRCE grant support from Central
Michigan University. We acknowledge the use of SPLASH (Price, 2007) for
rendering and visualization of our figures.

## References

  * Baade et al. (2023) Baade, D., Labadie-Bartz, J., Rivinius, T., & Carciofi, A. C. 2023, A&A, 678, A47, doi: 10.1051/0004-6361/202244149
  * Baade & Rivinius (2020) Baade, D., & Rivinius, T. 2020, in Stars and their Variability Observed from Space, ed. C. Neiner, W. W. Weiss, D. Baade, R. E. Griffin, C. C. Lovekin, & A. F. J. Moffat, 35–38
  * Baade et al. (2018) Baade, D., Rivinius, T., Pigulski, A., et al. 2018, in 3rd BRITE Science Conference, ed. G. A. Wade, D. Baade, J. A. Guzik, & R. Smolec, Vol. 8, 69–76, doi: 10.48550/arXiv.1708.08413
  * Babusiaux et al. (2023) Babusiaux, C., Fabricius, C., Khanna, S., et al. 2023, A&A, 674, A32, doi: 10.1051/0004-6361/202243790
  * Baldwin (1940) Baldwin, R. B. 1940, ApJ, 92, 82, doi: 10.1086/144203
  * Bate et al. (1995) Bate, M. R., Bonnell, I. A., & Price, N. M. 1995, MNRAS, 277, 362, doi: 10.1093/mnras/277.2.362
  * Bedding (1993) Bedding, T. R. 1993, AJ, 106, 768, doi: 10.1086/116684
  * Benz (1990) Benz, W. 1990, in Numerical Modelling of Nonlinear Stellar Pulsations Problems and Prospects, ed. J. R. Buchler, 269
  * Benz et al. (1990) Benz, W., Bowers, R. L., Cameron, A. G. W., & Press, W. H. . 1990, ApJ, 348, 647, doi: 10.1086/168273
  * Bernacca & Perinotto (1970) Bernacca, P. L., & Perinotto, M. 1970, Contributi dell’Osservatorio Astrofisica dell’Universita di Padova in Asiago, 239, 1
  * Bjorkman (2012) Bjorkman, J. E. 2012, in Astronomical Society of the Pacific Conference Series, Vol. 464, Circumstellar Dynamics at High Resolution, ed. A. C. Carciofi & T. Rivinius, 85
  * Bjorkman et al. (2002) Bjorkman, K. S., Miroshnichenko, A. S., McDavid, D., & Pogrosheva, T. M. 2002, ApJ, 573, 812, doi: 10.1086/340751
  * Bodensteiner et al. (2020) Bodensteiner, J., Shenar, T., & Sana, H. 2020, A&A, 641, A42, doi: 10.1051/0004-6361/202037640
  * Borre et al. (2020) Borre, C. C., Baade, D., Pigulski, A., et al. 2020, A&A, 635, A140, doi: 10.1051/0004-6361/201937062
  * Brown et al. (2019) Brown, R. O., Coe, M. J., Ho, W. C. G., & Okazaki, A. T. 2019, MNRAS, 488, 387, doi: 10.1093/mnras/stz1757
  * Carciofi (2011) Carciofi, A. C. 2011, in Active OB Stars: Structure, Evolution, Mass Loss, and Critical Limits, ed. C. Neiner, G. Wade, G. Meynet, & G. Peters, Vol. 272, 325–336, doi: 10.1017/S1743921311010738
  * Carciofi & Bjorkman (2006) Carciofi, A. C., & Bjorkman, J. E. 2006, ApJ, 639, 1081, doi: 10.1086/499483
  * Carciofi et al. (2007) Carciofi, A. C., Magalhães, A. M., Leister, N. V., Bjorkman, J. E., & Levenhagen, R. S. 2007, ApJ, 671, L49, doi: 10.1086/524772
  * Carciofi et al. (2009) Carciofi, A. C., Okazaki, A. T., Le Bouquin, J. B., et al. 2009, A&A, 504, 915, doi: 10.1051/0004-6361/200810962
  * Carciofi et al. (2006) Carciofi, A. C., Miroshnichenko, A. S., Kusakin, A. V., et al. 2006, ApJ, 652, 1617, doi: 10.1086/507935
  * Che et al. (2012) Che, X., Monnier, J. D., Tycner, C., et al. 2012, ApJ, 757, 29, doi: 10.1088/0004-637X/757/1/29
  * Collins (1987) Collins, George W., I. 1987, in IAU Colloq. 92: Physics of Be Stars, ed. A. Slettebak & T. P. Snow, 3
  * Cote & van Kerkwijk (1993) Cote, J., & van Kerkwijk, M. H. 1993, A&A, 274, 870
  * Cotton et al. (2019) Cotton, D. V., Marshall, J. P., Frisch, P. C., et al. 2019, MNRAS, 483, 3636, doi: 10.1093/mnras/sty3318
  * Cox (2000) Cox, A. N. 2000, Allen’s astrophysical quantities (New York: AIP Press; Springer)
  * Cyr et al. (2020) Cyr, I. H., Jones, C. E., Carciofi, A. C., et al. 2020, MNRAS, 497, 3525, doi: 10.1093/mnras/staa2176
  * Cyr et al. (2017) Cyr, I. H., Jones, C. E., Panoglou, D., Carciofi, A. C., & Okazaki, A. T. 2017, MNRAS, 471, 596, doi: 10.1093/mnras/stx1427
  * de Almeida et al. (2020) de Almeida, E. S. G., Meilland, A., Domiciano de Souza, A., et al. 2020, A&A, 636, A110, doi: 10.1051/0004-6361/201936039
  * de Wit et al. (2006) de Wit, W. J., Lamers, H. J. G. L. M., Marquette, J. B., & Beaulieu, J. P. 2006, A&A, 456, 1027, doi: 10.1051/0004-6361:20065137
  * Dodd et al. (2024) Dodd, J. M., Oudmaijer, R. D., Radley, I. C., Vioque, M., & Frost, A. J. 2024, MNRAS, 527, 3076, doi: 10.1093/mnras/stad3105
  * Draper et al. (2014) Draper, Z. H., Wisniewski, J. P., Bjorkman, K. S., et al. 2014, ApJ, 786, 120, doi: 10.1088/0004-637X/786/2/120
  * Eggleton (1983) Eggleton, P. P. 1983, ApJ, 268, 368, doi: 10.1086/160960
  * Frémat et al. (2005) Frémat, Y., Zorec, J., Hubert, A. M., & Floquet, M. 2005, A&A, 440, 305, doi: 10.1051/0004-6361:20042229
  * Ghoreyshi et al. (2021) Ghoreyshi, M. R., Carciofi, A. C., Jones, C. E., et al. 2021, ApJ, 909, 149, doi: 10.3847/1538-4357/abdd1e
  * Ghoreyshi et al. (2018) Ghoreyshi, M. R., Carciofi, A. C., Rímulo, L. R., et al. 2018, MNRAS, 479, 2214, doi: 10.1093/mnras/sty1577
  * Hall (1958) Hall, J. S. 1958, Publications of the U.S. Naval Observatory Second Series, 17, 275
  * Halonen & Jones (2013) Halonen, R. J., & Jones, C. E. 2013, ApJ, 765, 17, doi: 10.1088/0004-637X/765/1/17
  * Hanuschik (1989) Hanuschik, R. W. 1989, Ap&SS, 161, 61, doi: 10.1007/BF00653238
  * Haubois et al. (2012) Haubois, X., Carciofi, A. C., Rivinius, T., Okazaki, A. T., & Bjorkman, J. E. 2012, ApJ, 756, 156, doi: 10.1088/0004-637X/756/2/156
  * Haubois et al. (2014) Haubois, X., Mota, B. C., Carciofi, A. C., et al. 2014, ApJ, 785, 12, doi: 10.1088/0004-637X/785/1/12
  * Hayes & Guinan (1984) Hayes, D. P., & Guinan, E. F. 1984, ApJ, 279, 721, doi: 10.1086/161938
  * Henry (1999) Henry, G. W. 1999, PASP, 111, 845, doi: 10.1086/316388
  * Huang (1972) Huang, S.-S. 1972, ApJ, 171, 549, doi: 10.1086/151309
  * Hummel (1994) Hummel, W. 1994, A&A, 289, 458
  * Hummel & Vrancken (1995) Hummel, W., & Vrancken, M. 1995, A&A, 302, 751
  * Hutter et al. (2021) Hutter, D. J., Tycner, C., Zavala, R. T., et al. 2021, ApJS, 257, 69, doi: 10.3847/1538-4365/ac23cb
  * Jaschek et al. (1981) Jaschek, M., Slettebak, A., & Jaschek, C. 1981, Be Star Newsletter, 4, 9
  * Jones et al. (2011) Jones, C. E., Tycner, C., & Smith, A. D. 2011, AJ, 141, 150, doi: 10.1088/0004-6256/141/5/150
  * Klement et al. (2019) Klement, R., Carciofi, A. C., Rivinius, T., et al. 2019, ApJ, 885, 147, doi: 10.3847/1538-4357/ab48e7
  * Kopal (1959) Kopal, Z. 1959, Close binary systems (The International Astrophysics Series, London: Chapman & Hall)
  * Labadie-Bartz et al. (2022) Labadie-Bartz, J., Carciofi, A. C., Henrique de Amorim, T., et al. 2022, AJ, 163, 226, doi: 10.3847/1538-3881/ac5abd
  * Lee et al. (1991) Lee, U., Osaki, Y., & Saio, H. 1991, MNRAS, 250, 432, doi: 10.1093/mnras/250.2.432
  * Magalhaes et al. (1984) Magalhaes, A. M., Benedetti, E., & Roland, E. H. 1984, PASP, 96, 383, doi: 10.1086/131351
  * Magalhaes et al. (1996) Magalhaes, A. M., Rodrigues, C. V., Margoniner, V. E., Pereyra, A., & Heathcote, S. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 97, Polarimetry of the Interstellar Medium, ed. W. G. Roberge & D. C. B. Whittet, 118
  * Marchant & Bodensteiner (2023) Marchant, P., & Bodensteiner, J. 2023, arXiv e-prints, arXiv:2311.01865, doi: 10.48550/arXiv.2311.01865
  * Marr et al. (2021) Marr, K. C., Jones, C. E., Carciofi, A. C., et al. 2021, ApJ, 912, 76, doi: 10.3847/1538-4357/abed4c
  * Marr et al. (2022) Marr, K. C., Jones, C. E., Tycner, C., Carciofi, A. C., & Silva, A. C. F. 2022, ApJ, 928, 145, doi: 10.3847/1538-4357/ac551b
  * Martin et al. (1992) Martin, P. G., Adamson, A. J., Whittet, D. C. B., et al. 1992, ApJ, 392, 691
  * Martin et al. (2011) Martin, R. G., Pringle, J. E., Tout, C. A., & Lubow, S. H. 2011, MNRAS, 416, 2827, doi: 10.1111/j.1365-2966.2011.19231.x
  * Meilland et al. (2013) Meilland, A., Stee, P., Spang, A., et al. 2013, A&A, 550, L5, doi: 10.1051/0004-6361/201220712
  * Meilland et al. (2007) Meilland, A., Stee, P., Vannier, M., et al. 2007, A&A, 464, 59, doi: 10.1051/0004-6361:20064848
  * Meilland et al. (2011) Meilland, A., Delaa, O., Stee, P., et al. 2011, A&A, 532, A80, doi: 10.1051/0004-6361/201116798
  * Millan-Gabet et al. (2010) Millan-Gabet, R., Monnier, J. D., Touhami, Y., et al. 2010, ApJ, 723, 544, doi: 10.1088/0004-637X/723/1/544
  * Miroshnichenko (2016) Miroshnichenko, A. S. 2016, in Astronomical Society of the Pacific Conference Series, Vol. 506, Bright Emissaries: Be Stars as Messengers of Star-Disk Physics, ed. T. A. A. Sigut & C. E. Jones, 71
  * Miroshnichenko et al. (2001) Miroshnichenko, A. S., Fabregat, J., Bjorkman, K. S., et al. 2001, A&A, 377, 485, doi: 10.1051/0004-6361:20010911
  * Miroshnichenko et al. (2003) Miroshnichenko, A. S., Bjorkman, K. S., Morrison, N. D., et al. 2003, A&A, 408, 305, doi: 10.1051/0004-6361:20030965
  * Miroshnichenko et al. (2013) Miroshnichenko, A. S., Pasechnik, A. V., Manset, N., et al. 2013, ApJ, 766, 119, doi: 10.1088/0004-637X/766/2/119
  * Miroshnichenko et al. (2023) Miroshnichenko, A. S., Chari, R., Danford, S., et al. 2023, Galaxies, 11, 83, doi: 10.3390/galaxies11040083
  * Neiner et al. (2011) Neiner, C., de Batz, B., Cochard, F., et al. 2011, AJ, 142, 149, doi: 10.1088/0004-6256/142/5/149
  * Nordsieck & Harris (1996) Nordsieck, K. H., & Harris, W. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 97, Polarimetry of the Interstellar Medium, ed. W. G. Roberge & D. C. B. Whittet, 100
  * Okazaki (1997) Okazaki, A. T. 1997, A&A, 318, 548
  * Okazaki (2007) Okazaki, A. T. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 367, Massive Stars in Interactive Binaries, ed. N. St. -Louis & A. F. J. Moffat, 485
  * Okazaki et al. (2002) Okazaki, A. T., Bate, M. R., Ogilvie, G. I., & Pringle, J. E. 2002, MNRAS, 337, 967, doi: 10.1046/j.1365-8711.2002.05960.x
  * Overton et al. (2024) Overton, M., Martin, R. G., Lubow, S. H., & Lepp, S. 2024, MNRAS, 528, L106, doi: 10.1093/mnrasl/slad172
  * Panoglou et al. (2016) Panoglou, D., Carciofi, A. C., Vieira, R. G., et al. 2016, MNRAS, 461, 2616, doi: 10.1093/mnras/stw1508
  * Panoglou et al. (2018) Panoglou, D., Faes, D. M., Carciofi, A. C., et al. 2018, MNRAS, 473, 3039, doi: 10.1093/mnras/stx2497
  * Peters et al. (2013) Peters, G. J., Pewett, T. D., Gies, D. R., Touhami, Y. N., & Grundstrom, E. D. 2013, ApJ, 765, 2, doi: 10.1088/0004-637X/765/1/2
  * Porter & Rivinius (2003) Porter, J. M., & Rivinius, T. 2003, PASP, 115, 1153, doi: 10.1086/378307
  * Price (2007) Price, D. J. 2007, PASA, 24, 159, doi: 10.1071/AS07022
  * Richardson et al. (2021) Richardson, N. D., Thizy, O., Bjorkman, J. E., et al. 2021, MNRAS, 508, 2002, doi: 10.1093/mnras/stab2759
  * Rímulo et al. (2018) Rímulo, L. R., Carciofi, A. C., Vieira, R. G., et al. 2018, MNRAS, 476, 3555, doi: 10.1093/mnras/sty431
  * Rivinius et al. (2013) Rivinius, T., Carciofi, A. C., & Martayan, C. 2013, A&A Rev., 21, 69, doi: 10.1007/s00159-013-0069-0
  * Rubio et al. (2023) Rubio, A. C., Carciofi, A. C., Ticiani, P., et al. 2023, MNRAS, 526, 3007, doi: 10.1093/mnras/stad2652
  * Ruždjak et al. (2009) Ruždjak, D., Božić, H., Harmanec, P., et al. 2009, A&A, 506, 1319, doi: 10.1051/0004-6361/200810526
  * Serkowski et al. (1975) Serkowski, K., Mathewson, D. S., & Ford, V. L. 1975, ApJ, 196, 261, doi: 10.1086/153410
  * Shakura & Sunyaev (1973) Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
  * Sigut & Ghafourian (2023) Sigut, T. A. A., & Ghafourian, N. R. 2023, ApJ, 948, 34, doi: 10.3847/1538-4357/ac940c
  * Sigut et al. (2020) Sigut, T. A. A., Mahjour, A. K., & Tycner, C. 2020, ApJ, 894, 18, doi: 10.3847/1538-4357/ab8386
  * Silaj et al. (2014) Silaj, J., Jones, C. E., Sigut, T. A. A., & Tycner, C. 2014, ApJ, 795, 82, doi: 10.1088/0004-637X/795/1/82
  * Sonneborn et al. (1988) Sonneborn, G., Grady, C. A., Wu, C.-C., et al. 1988, ApJ, 325, 784, doi: 10.1086/166049
  * Suffak et al. (2022) Suffak, M., Jones, C. E., & Carciofi, A. C. 2022, MNRAS, 509, 931, doi: 10.1093/mnras/stab3024
  * Suffak et al. (2023) Suffak, M. W., Jones, C. E., Carciofi, A. C., & de Amorim, T. H. 2023, MNRAS, 526, 782, doi: 10.1093/mnras/stad2781
  * Suffak et al. (2020) Suffak, M. W., Jones, C. E., Tycner, C., et al. 2020, ApJ, 890, 86, doi: 10.3847/1538-4357/ab68dc
  * Tango et al. (2009) Tango, W. J., Davis, J., Jacob, A. P., et al. 2009, MNRAS, 396, 842, doi: 10.1111/j.1365-2966.2009.14272.x
  * Townsend et al. (2004) Townsend, R. H. D., Owocki, S. P., & Howarth, I. D. 2004, MNRAS, 350, 189, doi: 10.1111/j.1365-2966.2004.07627.x
  * Tycner et al. (2011) Tycner, C., Ames, A., Zavala, R. T., et al. 2011, ApJ, 729, L5, doi: 10.1088/2041-8205/729/1/L5
  * Tycner et al. (2005) Tycner, C., Lester, J. B., Hajian, A. R., et al. 2005, ApJ, 624, 359, doi: 10.1086/429126
  * Vakili et al. (1998) Vakili, F., Mourard, D., Stee, P., et al. 1998, A&A, 335, 261
  * van Leeuwen (2007) van Leeuwen, F. 2007, A&A, 474, 653, doi: 10.1051/0004-6361:20078357
  * Vieira & Carciofi (2017) Vieira, R. G., & Carciofi, A. C. 2017, in Astronomical Society of the Pacific Conference Series, Vol. 508, The B[e] Phenomenon: Forty Years of Studies, ed. A. Miroshnichenko, S. Zharikov, D. Korčáková, & M. Wolf, 33, doi: 10.48550/arXiv.1707.03105
  * Vieira et al. (2015) Vieira, R. G., Carciofi, A. C., & Bjorkman, J. E. 2015, MNRAS, 454, 2107, doi: 10.1093/mnras/stv2074
  * Štefl et al. (2009) Štefl, S., Rivinius, T., Carciofi, A. C., et al. 2009, A&A, 504, 929, doi: 10.1051/0004-6361/200811573
  * Štefl et al. (2012) Štefl, S., LeBouquin, J. B., Rivinius, T., et al. 2012, in Astronomical Society of the Pacific Conference Series, Vol. 464, Circumstellar Dynamics at High Resolution, ed. A. C. Carciofi & T. Rivinius, 197
  * Whittet et al. (1992) Whittet, D. C. B., Martin, P. G., Hough, J. H., et al. 1992, ApJ, 386, 562
  * Wilking et al. (1980) Wilking, B. A., Lebofsky, M. J., Martin, P. G., Rieke, G. H., & Kemp, J. C. 1980, ApJ, 235, 905
  * Wisniewski et al. (2010) Wisniewski, J. P., Draper, Z. H., Bjorkman, K. S., et al. 2010, ApJ, 709, 1306, doi: 10.1088/0004-637X/709/2/1306
  * Wood et al. (1997) Wood, K., Bjorkman, K. S., & Bjorkman, J. E. 1997, ApJ, 477, 926, doi: 10.1086/303747
  * Zharikov et al. (2013) Zharikov, S. V., Miroshnichenko, A. S., Pollmann, E., et al. 2013, A&A, 560, A30, doi: 10.1051/0004-6361/201322114