#  $\epsilon$ Sagittarii: An Extreme Rapid Rotator with a Decretion Disk

Jeremy Bailey School of Physics, University of New South Wales, Sydney, NSW
2052, Australia  Fiona Lewis  School of Physics, University of New South
Wales, Sydney, NSW 2052, Australia  Ian D. Howarth University College London,
Gower Street, London WC1E 6BT, U.K.  Daniel V. Cotton Monterey Institute for
Research in Astronomy, 200 Eighth Street, Marina, CA, 93933, USA.  Western
Sydney University, Locked Bag 1797, Penrith-South DC, NSW 1797, Australia
Jonathan P. Marshall Institute of Astronomy and Astrophysics, Academia Sinica,
11F of AS/NTU
Astronomy-Mathematics Building, No.1, Sec. 4, Roosevelt Rd, Taipei 106216,
Taiwan  Lucyna Kedziora-Chudczer University of Southern Queensland, Centre for
Astrophysics, Toowoomba, QLD 4350, Australia

###### Abstract

We report high-precision multi-wavelength linear-polarization observations of
the bright B9 (or A0) star $\epsilon$ Sagittarii. The polarization shows the
distinctive wavelength dependence expected for a rapidly rotating star.
Analysis of the polarization data reveals an angular rotation rate $\omega$ (=
$\Omega/\Omega_{\rm crit})$ of 0.995 or greater, the highest yet measured for
a star in our galaxy. An additional wavelength-independent polarization
component is attributed to electron scattering in a low-density edge-on gas
disk that also produces the narrow absorption components seen in the spectrum.
Several properties of the star (polarization due to a disk, occasional weak
H$\alpha$ emission, and multiple periodicities seen in space photometry)
resemble those of Be stars, but the level of activity in all cases is much
lower than that of typical Be stars. The stellar properties are inconsistent
with single rotating-star evolutionary tracks, indicating that it is most
likely a product of binary interaction. The star is an excellent candidate for
observation by interferometry, optical spectropolarimetry to detect the Öhman
effect, and UV polarimetry; any of which would allow its extreme rotation to
be tested and its stellar properties to be refined.

Polarimetry (1278), Stellar rotation (1629), Circumstellar disks (235)

††facilities: AAT (HIPPI, HIPPI-2, IRIS2), CFHT (ESPaDOnS), VLT:Kueyen (UVES),
IUE, TESS

##  1 Introduction

The non-spherical shape of a rapidly rotating star results in a net linear
polarization of the integrated light of the star. This effect was first
predicted by Harrington & Collins (1968), but the resulting polarizations are
small, and have only recently been observed (Cotton et al., 2017a; Bailey et
al., 2020a; Lewis et al., 2022; Howarth et al., 2023). The polarization has a
distinctive wavelength dependence, and is strongly dependent on the rotation
rate $\omega$, the axial inclination and the gravity of the star. This allows
polarization data, combined with other observations, to set significant
constraints on the stellar properties.

$\epsilon$ Sagittarii (HD 169022, hereafter $\epsilon$ Sgr) is a bright (V =
1.85) rapidly rotating star at a distance of 44 pc (van Leeuwen, 2007) and
with a number of anomalous properties. The spectral type is variously reported
as B9 IVp (Slettebak, 1975), B9.5 III (Houk, 1982), A0 II (Gray & Garrison,
1987) and A0 III (Paunzen et al., 2001). Narrow, “shell” absorption components
in the Balmer and other lines are seen (Slettebak, 1975), and Gray & Garrison
(1987) call it a “proto-shell star”. It has been suggested to be a $\lambda$
Boo star (Slettebak, 1975). However, Baschek & Slettebak (1988) conclude, from
a detailed abundance study, that $\epsilon$ Sgr, while “peculiar in both
spectrum and colors, is definitely not a $\lambda$ Boo star”.

A companion star of magnitude V $\sim$ 7.7 at a distance of $\sim$ 2.1 arc
seconds was discovered by Golimowski et al. (1993). Infrared observations by
Hubrig et al. (2001) give the K magnitude of the companion as 6.5, and are
consistent with it being a main-sequence star of 0.95 $\mbox{M}_{\odot}$with
$T_{\rm eff}$= 5808 K.

$\epsilon$ Sgr was found to have an infrared (IR) excess in 60 $\mu$m Infra
Red Astronomical Satellite (IRAS) data by Cote (1987). The IRAS excess was
interpreted by Rhee et al. (2007) as originating from a dust disk with a
temperature of 100 K at a radius of 3.5 arc seconds from the star (equivalent
to 150 au) and a fractional luminosity ($L_{\rm dust}/L_{\star}=4.46\times
10^{-6}$). The IR excess was also detected by Spitzer at 13 and 31 $\mu$m
(Chen et al., 2014).

Polarization observations of $\epsilon$ Sgr were reported by Cotton et al.
(2016a) and Bailey et al. (2017), and were larger than expected at its
distance, implying the presence of intrinsic polarization. In this paper we
report a more extensive set of polarization observations and compare these
with models of rapidly rotating stars.

##  2 Observations

###  2.1 Polarization Measurements

Table 1: Summary of Observing Runs and telescope-polarization (TP)
Calibrations

| Telescope and Instrument Set-Upa | Observationsb | Calibrationc
---|---|---|---
Run | Date Ranged | Instr. | Tel. | f/ | Ap. | Mod. | Filt. | Det.e | $n$ | $\lambda_{\rm eff}$ | Eff. | $q_{\text{TP}}$ | $u_{\rm TP}$
| (UT) |  |  |  | ($\arcsec$) |  |  |  |  | (nm) | ($\%$) | (ppm) | (ppm)
2014AUG | 2014-09-01 | HIPPI | AAT | 8 | 6.6 | BNS-E1 | $g^{\prime}$ | B | 1 | 463.1 | 89.0 |  $-$39.9 $\pm$ 0.9 |  $-$38.2 $\pm$ 0.9
2016FEB/JUNf | 2016-02-27 to 06-25 | HIPPI | AAT | 8 | 6.6 | BNS-E2 | $g^{\prime}$ | B | 2 | 463.8 | 86.9 |  $-$20.5 $\pm$ 1.7 | 4.5 $\pm$ 1.8
|  |  |  |  |  |  | $r^{\prime}$ | R | 1 | 619.9 | 80.6 |  $-$8.2 $\pm$ 1.7 | 1.3 $\pm$ 1.7
m2016JUN | 2016-06-10 | M-HIPPI | UNSW | 11 | 58.9 | MT | Clear | B | 1 | 467.9 | 75.5 |  $-$64.5 $\pm$ 4.3 |  $-$7.6 $\pm$ 4.4
m2016JUL | 2016-08-31 | M-HIPPI | UNSW | 11 | 58.9 | MT | Clear | B | 1 | 464.0 | 74.3 |  $-$74.5 $\pm$ 3.1 |  $-$11.7 $\pm$ 3.1
2017JUN | 2017-06-26 to 07-04 | HIPPI | AAT | 8 | 6.6 | BNS-E2 | 425SP | B | 1 | 400.1 | 49.4 |  $-$7.6 $\pm$ 3.8 | 8.8 $\pm$ 3.8
|  |  |  |  |  |  | $g^{\prime}$ | B | 1 | 463.3 | 86.8 |  $-$9.1 $\pm$ 1.5 |  $-$2.6 $\pm$ 1.4
|  |  |  |  |  |  | 650LP | R | 1 | 717.6 | 62.7 |  $-$8.6 $\pm$ 2.5 |  $-$5.3 $\pm$ 2.5
2017AUG | 2017-08-08 to 08-14 | HIPPI | AAT | 8 | 6.6 | BNS-E2 | 425SP | B | 1 | 400.2 | 49.5 |  $-$7.6 $\pm$ 3.8 | 8.8 $\pm$ 3.8
|  |  |  |  |  |  | 500SP | B | 1 | 435.3 | 74.0 |  $-$10.0 $\pm$ 1.7 |  $-$0.4 $\pm$ 1.6
|  |  |  |  |  |  | $r^{\prime}$ | R | 2 | 619.2 | 80.7 |  $-$10.6 $\pm$ 1.3 |  $-$7.1 $\pm$ 1.3
2018MAR | 2018-03-18 to 04-06 | HIPPI-2 | AAT | 8* | 25.5 | BNS-E3 | $g^{\prime}$ | B | 2 | 461.7 | 81.5 | 128.1 $\pm$ 0.9 | 3.8 $\pm$ 0.9
|  |  |  |  |  |  | $r^{\prime}$ | R | 2 | 620.9 | 84.5 | 109.1 $\pm$ 1.4 | 6.9 $\pm$ 1.4
2018JULg | 2018-07-12 to 07-25 | HIPPI-2 | AAT | 8* | 11.9 | BNS-E4 | $V$ | B | 1 | 532.3 | 96.6 |  $-$20.1 $\pm$ 1.5 | 2.3 $\pm$ 1.5
|  |  |  |  |  |  | $r^{\prime}$ | B | 1 | 602.0 | 88.9 |  $-$10.3 $\pm$ 2.2 | 3.6 $\pm$ 2.2
2018AUGg | 2018-08-21 to 09-01 | HIPPI-2 | AAT | 8* | 11.9 | BNS-E5 | 500SP | B | 1 | 438.2 | 55.1 | 1.9 $\pm$ 1.4 | 18.0 $\pm$ 1.4
|  |  |  |  |  |  | $V$ | B | 1 | 532.4 | 95.7 |  $-$20.1 $\pm$ 1.5 | 2.3 $\pm$ 1.5
|  |  |  |  |  |  | 650LP | R | 1 | 720.9 | 76.8 |  $-$6.5 $\pm$ 1.8 | 3.8 $\pm$ 1.8
2023APR/MAYh | 2023-06-03 | HIPPI-2 | AAT | 15 | 12.7 | ML-E1 | 425SP | B | 1 | 398.8 | 79.3 |  $-$1.0 $\pm$ 2.4 | 7.8 $\pm$ 2.1
|  |  |  |  |  |  | 500SP | B | 1 | 438.2 | 86.9 |  $-$20.1 $\pm$ 1.7 |  $-$5.1 $\pm$ 1.5
|  |  |  |  |  |  | $g^{\prime}$ | B | 1 | 460.2 | 89.2 | 5.9 $\pm$ 1.0 | 0.0 $\pm$ 1.0
|  |  |  |  |  |  | $r^{\prime}$ | B | 1 | 601.8 | 64.2 | 4.9 $\pm$ 4.5 |  $-$19.7 $\pm$ 3.9
|  |  |  |  |  |  | $r^{\prime}$ | R | 1 | 620.8 | 60.5 |  $-$2.7 $\pm$ 3.2 |  $-$18.8 $\pm$ 3.3
|  |  |  |  |  |  | 650LP | R | 1 | 719.1 | 44.6 |  $-$36.6 $\pm$ 5.8 |  $-$35.4 $\pm$ 6.2

Notes:

* Indicates use of a 2$\times$ negative achromatic lens, effectively making the foci f/16.

a A full description, along with transmission curves for all the components
and modulation characterisation of each modulator (‘Mod.’) in the specified
performance era, can be found in Bailey et al. (2020b).

b Mean values are given as representative of the observations made of
$\epsilon$ Sgr. Individual values are given in Table 2 for each observation;
$n$ is the number of observations of $\epsilon$ Sgr.

c Most of the observations used to determine the TP and the high-polarization
standards observed to calibrate position angle (PA), are described in Bailey
et al. (2017) (m2016JUN, m2016JUL), Bailey et al. (2015) (2014MAY), Cotton et
al. (2017b) (2016FEB, 2016JUN) Cotton et al. (2019) (2017JUN, 2017AUG), and
Bailey et al. (2020b) (other runs). This is the first data presented from the
2023MAY run; 2-6 total observations per band of HD 2151, HD 61421, HD 102647,
HD 102870 or HD 140573 were used to determine TP; PA was calibrated with
reference to HD 84810, HD 161471, HD 111613 and HD 183143.

d Dates given are for observations of $\epsilon$ Sgr and/or control stars.

e B, R indicate blue- and red-sensitive H10720-210 and H10720-20
photomultiplier-tube detectors, respectively.

f TP calibration was carried out using observations combined from 2016FEB and
2016JUN runs.

g TP calibration was carried out using observations combined from 2018JUL and
2018AUG runs.

h TP calibration was carried out using observations combined from 2023APR A/B
and 2023MAY runs.

Table 2: Polarization Observations of $\epsilon$ Sgr (Sorted by Effective
Wavelength).

Run | UT (mid-point) | Dwell | Exp. | Filt. | Det.a | $\lambda_{\rm eff}$ | Eff. | $q$ | $u$ | $p$ | $\theta$
---|---|---|---|---|---|---|---|---|---|---|---
|  | (s) | (s) |  |  | (nm) | (%) | (ppm) | (ppm) | (ppm) | (∘)
2023MAY | 2023-06-03 16:39:20 | 1615 | 1280 | 425SP | B | 398.8 | 79.3 |  $-$5.8 $\pm$ 12.1 |  $-$13.6 $\pm$ 12.0 | 14.8 $\pm$ 12.0 | 123.6 $\pm$ 27.9
2017JUN | 2017-06-26 16:29:55 | 1557 | 800 | 425SP | B | 400.1 | 49.4 | 6.7 $\pm$ 15.6 |  $-$50.3 $\pm$ 15.6 | 50.7 $\pm$ 15.6 | 138.8 $\pm$ 09.2
2017AUG | 2017-08-10 14:14:07 | 2124 | 800 | 425SP | B | 400.2 | 49.5 | 28.1 $\pm$ 14.6 |  $-$19.4 $\pm$ 14.5 | 34.1 $\pm$ 14.6 | 162.7 $\pm$ 14.2
2023MAY | 2023-06-03 17:03:20 | 996 | 640 | 500SP | B | 433.4 | 86.9 | 36.0 $\pm$ 06.7 | 60.4 $\pm$ 06.7 | 70.3 $\pm$ 06.7 | 29.6 $\pm$ 02.7
2017AUG | 2017-08-11 13:53:51 | 1506 | 640 | 500SP | B | 435.3 | 74.0 | 21.8 $\pm$ 08.9 | 81.0 $\pm$ 08.9 | 83.9 $\pm$ 08.9 | 37.5 $\pm$ 03.0
2018AUG | 2018-08-21 14:16:31 | 973 | 640 | 500SP | B | 438.2 | 54.9 | 25.1 $\pm$ 06.8 | 83.2 $\pm$ 06.8 | 86.9 $\pm$ 06.8 | 36.6 $\pm$ 02.2
2023MAY | 2023-06-03 16:17:39 | 983 | 640 | $g^{\prime}$ | B | 460.2 | 89.2 | 19.3 $\pm$ 02.5 | 95.1 $\pm$ 02.6 | 97.0 $\pm$ 02.6 | 39.3 $\pm$ 00.8
2018MARs | 2018-04-06 17:39:10 | 1032 | 640 | $g^{\prime}$ | B | 461.7 | 81.5 | 71.3 $\pm$ 02.5 | 99.9 $\pm$ 02.5 | 122.7 $\pm$ 02.6 | 27.2 $\pm$ 00.6
2018MARl | 2018-04-06 17:19:21 | 1189 | 640 | $g^{\prime}$ | B | 461.8 | 81.6 | 16.2 $\pm$ 02.9 | 95.5 $\pm$ 02.7 | 96.9 $\pm$ 02.8 | 40.2 $\pm$ 00.8
2014AUG | 2014-09-01 10:11:12 | 1843 | 640 | $g^{\prime}$ | B | 463.1 | 89.0 | 41.0 $\pm$ 03.6 | 164.0 $\pm$ 03.6 | 169.0 $\pm$ 03.6 | 38.0 $\pm$ 00.6
2017JUN | 2017-06-26 16:02:50 | 1588 | 800 | $g^{\prime}$ | B | 463.3 | 86.8 | 13.4 $\pm$ 06.0 | 102.0 $\pm$ 04.2 | 102.9 $\pm$ 05.1 | 41.3 $\pm$ 01.7
2016FEB | 2016-03-01 18:37:33 | 973 | 480 | $g^{\prime}$ | B | 463.8 | 86.9 | 27.6 $\pm$ 03.7 | 88.8 $\pm$ 03.7 | 93.0 $\pm$ 03.7 | 36.4 $\pm$ 01.2
2016FEB | 2016-02-29 18:34:56 | 1012 | 480 | $g^{\prime}$ | B | 463.9 | 87.0 | 36.2 $\pm$ 04.0 | 85.1 $\pm$ 04.1 | 92.5 $\pm$ 04.0 | 33.5 $\pm$ 01.2
m2016JUL | 2016-08-31 10:37:32 | 1609 | 800 | Clear | B | 464.0 | 74.3 | 43.5 $\pm$ 19.9 | 123.2 $\pm$ 20.6 | 130.7 $\pm$ 20.3 | 35.3 $\pm$ 04.4
m2016JUN | 2016-06-10 10:47:32 | 1850 | 800 | Clear | B | 467.9 | 75.5 | 29.5 $\pm$ 21.9 | 98.45 $\pm$ 21.8 | 102.7 $\pm$ 21.8 | 36.7 $\pm$ 06.2
2018JUL | 2018-07-25 16:08:13 | 1025 | 640 | $V$ | B | 532.3 | 96.6 | 39.8 $\pm$ 04.1 | 217.1 $\pm$ 04.1 | 220.7 $\pm$ 04.1 | 39.8 $\pm$ 00.5
2018AUG | 2018-08-21 14:36:42 | 1345 | 960 | $V$ | B | 532.4 | 95.7 | 45.2 $\pm$ 03.1 | 178.5 $\pm$ 03.0 | 184.1 $\pm$ 03.1 | 37.9 $\pm$ 00.5
2023MAY | 2023-06-03 17:23:21 | 1405 | 960 | $r^{\prime}$ | B | 601.8 | 64.2 | 31.5 $\pm$ 07.5 | 261.8 $\pm$ 06.7 | 263.7 $\pm$ 07.1 | 41.6 $\pm$ 00.8
2018JUL | 2018-07-25 16:26:18 | 1045 | 710 | $r^{\prime}$ | B | 602.0 | 88.9 | 51.0 $\pm$ 07.3 | 262.8 $\pm$ 06.4 | 267.7 $\pm$ 06.8 | 39.5 $\pm$ 00.8
2017AUG | 2017-08-09 13:12:24 | 1880 | 640 | $r^{\prime}$ | R | 619.2 | 80.7 | 61.1 $\pm$ 03.9 | 281.3 $\pm$ 03.7 | 287.9 $\pm$ 03.8 | 38.9 $\pm$ 00.4
2017AUG | 2017-08-08 14:18:24 | 1339 | 640 | $r^{\prime}$ | R | 619.3 | 80.7 | 63.8 $\pm$ 03.7 | 283.9 $\pm$ 03.7 | 291.0 $\pm$ 03.7 | 38.7 $\pm$ 00.4
2016FEB | 2016-02-27 16:41:49 | 1237 | 640 | $r^{\prime}$ | R | 619.9 | 80.6 | 30.3 $\pm$ 05.2 | 227.5 $\pm$ 05.4 | 229.5 $\pm$ 05.3 | 41.2 $\pm$ 00.7
2023MAY | 2023-06-03 18:24:03 | 958 | 640 | $r^{\prime}$ | R | 620.8 | 60.5 | 64.5 $\pm$ 05.1 | 284.8 $\pm$ 05.2 | 292.0 $\pm$ 05.2 | 38.6 $\pm$ 00.5
2018MARl | 2018-03-18 18:49:50 | 980 | 640 | $r^{\prime}$ | R | 620.9 | 84.5 | 78.4 $\pm$ 03.2 | 276.1 $\pm$ 03.2 | 287.0 $\pm$ 03.2 | 37.1 $\pm$ 00.3
2018MARs | 2018-03-18 19:11:58 | 1300 | 640 | $r^{\prime}$ | R | 620.9 | 84.5 | 71.9 $\pm$ 04.6 | 278.7 $\pm$ 04.4 | 287.8 $\pm$ 04.4 | 37.8 $\pm$ 00.5
2017JUN | 2017-07-04 13:08:09 | 2165 | 1440 | 650LP | R | 717.6 | 62.7 | 81.9 $\pm$ 04.8 | 328.5 $\pm$ 04.8 | 338.6 $\pm$ 04.8 | 38.0 $\pm$ 00.4
2023MAY | 2023-06-03 18:45:19 | 1634 | 1280 | 650LP | R | 719.1 | 44.6 | 101.3 $\pm$ 08.1 | 341.2 $\pm$ 08.5 | 355.9 $\pm$ 08.3 | 36.7 $\pm$ 00.7
2018AUG | 2018-08-23 11:47:56 | 1232 | 960 | 650LP | R | 720.9 | 76.8 | 52.4 $\pm$ 04.1 | 318.2 $\pm$ 04.1 | 322.5 $\pm$ 04.1 | 40.3 $\pm$ 00.4

a B, R indicate blue- and red-sensitive H10720-210 and H10720-20
photomultiplier-tube detectors, respectively. s/l Aperture sizes are as given
in Table 1 except where indicated by these superscripts; $l$: 25.5 arc
seconds, $s$: 5.3 arc seconds – small enough to exclude the reported debris
disk but not the binary companion (Rodriguez & Zuckerman, 2012).

We report here 28 linear polarization observations of $\epsilon$ Sgr. Of
these, 26 were made on the 3.9-m Anglo-Australian Telescope (AAT) at Siding
Spring Observatory using either the HIPPI (high precision polarimetric
instrument, Bailey et al., 2015) or HIPPI-2 (Bailey et al., 2020b)
polarimeters. Two observations were made with the Mini-HIPPI (Bailey et al.,
2017) polarimeter on the 35-cm telescope at the University of New South Wales
(UNSW) observatory in Sydney. These 3 polarimeters all use similar techniques,
making use of ferro-electric liquid-crystal modulators operating at 500 Hz,
and compact photomultiplier tubes (PMTs) as detectors. Some of the
observations were previously reported by Cotton et al. (2016a) and Bailey et
al. (2017), but all the data presented here have been re-analysed using the
methods described by Bailey et al. (2020b). We adopt a new calibration of
modulator performance that has been carried out for an upcoming paper (Cotton
et al., in prep.). The resulting final polarization values are similar to past
work.

Details of the observing runs, made between 2014 and 2023, as well as
instrumental details are listed in Table 1. The polarization observations of
$\epsilon$ Sgr are listed in Table 2. New observations of interstellar control
stars are described in Appendix A. The AAT observations were obtained with a
range of broad-band filters. The Sloan Digital Sky Survey (SDSS) $g^{\prime}$
and $r^{\prime}$ filters used with HIPPI were made by Omega Optics. For
HIPPI-2 the corresponding filters were generation-2 filters from Astrodon
Photometrics. The two Mini-HIPPI observations were made with no filter
(Clear). A bandpass model as described in Bailey et al. (2020b) was used to
determine the effective wavelength and modulation efficiency for each
observation. This model takes account of all the optical components as well as
the source spectrum, and atmospheric transmission. These values are listed in
Table 2 and the effective wavelengths for these observations range from
$\sim$400 nm to $\sim$720 nm.

###  2.2 Imaging

Infrared images of $\epsilon$ Sgr were obtained on 2016 Sep 10 using the IRIS2
instrument (Tinney et al., 2004) on the 3.9m Anglo-Australian Telescope in a
narrow band filter at a wavelength of 2.3 $\mu$m. The separation ($\rho$) and
position angle of the companion were measured from the images with the Python
Photutils package (Bradley et al., 2022) using profile fitting methods based
on the DAOPHOT algorithms (Stetson, 1987). The results, together with those
from past observations, are listed in Table 3. The separation of the companion
has been slowly increasing, with little change to the position angle.

The companion was included in the aperture for all our polarization
observations. However, it contributes less than 1% of the optical-region
light, and as a main-sequence G star is not expected to have significant
intrinsic polarization (Cotton et al., 2017b).

Table 3: Measurements of the $\epsilon$ Sgr Binary System.

UT Date |  $\rho$ (″) | pa (°) |  $\Delta$m (mag) | Ref
---|---|---|---|---
1992-06-18 | 2.06$\pm$0.01 | 142$\pm$1 | 5.85 (V) | 1
1999-03a | 2.39 | 142 | 4.6 (K) | 2
2016-09-10 | 2.58$\pm$0.04 | 140$\pm$1 | 4.2 (2.3$\mu$m) | 3

Notes:

References: 1 Golimowski et al. (1993), 2 Hubrig et al. (2001), 3. This paper
(Section 2.2)

a The exact date of the observation is not reported.

###  2.3 Archival Spectroscopy

Spectra of $\epsilon$ Sgr have been obtained from the archive of the Canada
France Hawaii Telescope (CFHT). These were taken with the Echelle
Spectropolarimetric Device for Observation of Stars (ESPaDOnS) instrument
(Donati, 2003) on 2014 Sep 8. The spectra cover 370 to 1000 nm at R $\sim$
70000\. The individual integrations have been combined to form a single
spectrum which was used for the analysis in Section 3.3 and 3.4.

We also used three spectra covering the H$\alpha$ region taken with the
ultraviolet and visual echelle spectrograph (UVES, Dekker et al., 2000) on the
European Southern Observatory (ESO) Very Large Telescope (VLT)111Based on data
obtained from the ESO Science Archive Facility with DOI: 10.18727/archive/50
(catalog 10.18727/archive/50). These cover 472.6 to 683.5 nm and have R =
74450. The dates of observation were 2016 Apr 22, May 17 and Jun 1.

##  3 Modelling

###  3.1 Polarization of Rotating Stars

We model the polarization of rotating stars using the methods described in
detail by Bailey et al. (2020a) and also used by Cotton et al. (2017a), Lewis
et al. (2022) and Howarth et al. (2023). We use a Roche model for the rotating
star and assume gravity darkening according to Espinosa Lara & Rieutord
(2011). This allows us to determine the distribution of local temperature and
gravity as a function of latitude. We use a set of custom ATLAS9 (Castelli &
Kurucz, 2003) models to represent the local stellar atmosphere at 46
colatitudes from 0 to 90° at 2° steps. For each of these model atmospheres we
then calculate the specific intensity and polarization of the emergent
radiation. These calculations use a version of the SYNSPEC spectral synthesis
code (Hubeny et al., 1985; Hubeny, 2012) which we have modified to do full
polarized radiative transfer using the vector linearized discrete ordinate
radiative transfer (VLIDORT) code of Spurr (2006).

We overlay a rectangular grid of pixels over the projected view of the star.
For each pixel we determine the local temperature, gravity, and the surface
normal viewing angle. We then use these values to interpolate in our set of
radiative transfer calculations to determine the local specific intensity and
polarization. Summing over all the pixels then gives the integrated flux and
polarization as a function of wavelength.

Figure 1: Polarization observations (rotated to put all the polarization into
the Q stokes parameter) compared with a model of polarization for a rapidly
rotating star. The observations show similar wavelength dependence to the
model but the polarization is offset by $\sim$300 ppm in the positive
direction.

###  3.2 Polarization Components

Figure 1 shows the polarization observations rotated through $\sim$40°, which
puts the polarization primarily in the Q Stokes parameter. The polarization
shows a large change with wavelength from values of about -50 ppm at $\sim$400
nm, to more than 300 ppm at $\sim$720 nm. This large wavelength change matches
that expected for a star with a large rotation rate. However, when we compare
the observations with a rapidly-rotating-star model, as shown in Figure 1, we
find that the observed polarization is offset in a positive direction from the
modelled value by $\sim$300 ppm. In this plot negative polarization is
perpendicular to the star’s rotation axis, positive polarization is parallel
to the axis.

For the polarization to be consistent with rotational distortion, there must
be an additional source of polarization of $\sim$300 ppm with a position angle
parallel to the star’s rotation axis. In our previous studies of rotating
stars we have been able to model the observations using a rotating star model
and a small contribution of interstellar polarization. For $\epsilon$ Sgr at a
distance of 44 pc (van Leeuwen, 2007) the expected level of interstellar
polarization is $\sim$60 ppm according to the models of Cotton et al. (2017b)
and a value as high as 300 ppm would be extremely unlikely (see Appendix A).
Furthermore the interstellar polarization would have to be aligned with the
rotation axis of the star to provide the required offset. This would not be
consistent with what is measured for nearby stars and seems intrinsically
unlikely.

For $\epsilon$ Sgr there is, however, an additional likely source of
polarization with the required properties. The narrow shell absorption
features in the spectrum, as described in Section 1, indicates the presence of
circumstellar gas. In such a rapidly rotating star this is most likely in the
form of a narrow equatorial disk. Such disks are common in rapidly rotating B
stars as seen in the classical Be stars (Rivinius et al., 2013). One of the
important observable features of Be stars is polarization due to starlight
scattered from the disk. The typical polarization level in nearby Be stars
(where interstellar polarization is unimportant) is $\sim$3000–6000 ppm
(Cotton et al., 2016a).

The much lower polarization ($\sim$300 ppm) we infer for the disk in
$\epsilon$ Sgr, as well as the lack of strong H$\alpha$ emission, indicates a
disk of much lower density than those usually found in classical Be stars. At
low densities we expect electron scattering to be the dominant opacity source
leading to a flat, wavelength-independent polarization (Halonen & Jones, 2013;
Rivinius et al., 2013). This differs from the wavelength-dependent
polarization structure seen in the higher-density disks of classical Be stars
(Poeckert, Bastien, & Landstreet, 1979; Wood, Bjorkman, & Bjorkman, 1997)
where bound–free absorption is important. Comparison with the models of
Halonen & Jones (2013) suggests a gas density of $\sim 10^{-13}$ g cm-3
whereas densities up to $\sim 10^{-10}$ g cm-3 are seen in classical Be stars.

The polarization position angle of a Be star disk is perpendicular to the disk
(Quirrenbach et al., 1997). Hence the polarization is parallel to the star’s
rotation axis, as required for our offsetting polarization component.

###  3.3 Polarization Variability

The polarization observations of $\epsilon$ Sgr (Table 2) show considerable
scatter between observations in the same filter on different dates. The total
range in $p$ values is 77 ppm at $g^{\prime}$ and 62 ppm at $r^{\prime}$. This
is much larger than the statistical errors, and much larger than the scatter
seen in our similar observations of other rapidly rotating stars using the
same instruments (e.g. Bailey et al., 2020a; Howarth et al., 2023). The most
extreme values are seen in some of the earliest observations in 2014 and 2016,
whereas observations closely spaced in time generally show good agreement.
This suggests variability of the polarization on long timescales. Such
variability is not unexpected for a system with a disk component as described
in the previous section. Polarization in Be stars is often variable (Coyne,
1976).

Independent evidence for a variable Be-type disk in $\epsilon$ Sgr comes from
the archival spectroscopy (Section 2.3). Figure 2 shows two spectra of
$\epsilon$ Sgr. One of these, the CFHT spectrum from 2014 Sep 8, shows weak
double-peaked emission in H$\alpha$. The other spectrum, taken on 2016 Apr 22
with VLT UVES shows a broad photospheric absorption and narrow shell-
absorption core. Similar behavior is seen in the other UVES spectra from 2016.
Since the earlier spectroscopy discussed in Section 1 makes no mention of
emission we assume the 2016 spectra represent the more usual state.
Interestingly the highest $g^{\prime}$ polarization was observed on 2014 Sep
1, only a few days before the spectrum showing the emission, whereas
polarization observations in 2016 show some of the lowest values.

Figure 2: Spectra of $\epsilon$ Sgr in the H$\alpha$ region. The UVES (ESO
VLT) spectrum is from 2016 Apr 22 and shows broad absorption and the narrow
shell absorption component. The ESPaDOnS (CFHT) spectrum on 2014 Sep 8 shows
additionally double-peaked core emission. The very narrow absorption features
are telluric lines.

###  3.4 Model Grid

Figure 3: Results of Fourier-Transform modelling of CFHT spectra of the MgII
448.1nm line to determine $v_{\rm e}\sin{i}$. The value plotted is the rms
difference between observed and modelled Fourier Transforms.

The polarization of a rotating-star model (as described in Section 3.1)
depends on four main parameters: the polar temperature and gravity ($T_{p}$,
$g_{p}$), the rotation rate ($\omega$ = $\Omega/\Omega_{\rm crit}$)222Where
the subscript ‘crit’ is used to denote critical rotation throughout this
paper., and the inclination of the star’s rotation axis ($i$). The
polarization observations alone are not sufficient to determine these four
parameters so, as in our previous studies (Bailey et al., 2020a; Lewis et al.,
2022; Howarth et al., 2023) we use additional information to provide
relationships between these parameters.

The first of these is the spectral-energy distribution, which principally
constrains the temperature and angular diameter (i.e., radius). We use
photometry from Johnson et al. (1966) (optical) and archival spectroscopy
obtained with the International Ultraviolet Explorer (IUE; UV). IUE spectra
taken in low-resolution mode ($\Delta\lambda\simeq 0.6$ nm) through the
spectrographs’ large aperture provide the most reliable spectrophotometry.
Only one spectrum is available for each of the short- and long-wavelength
ranges ($\lambda\lambda\simeq 115$–198, 185–335 nm), but the data are well
exposed (SWP 16443, LWR 12689). We adjusted the fluxes to match the current
HST calibration by using results from Massa & Fitzpatrick (2000), including
the minor update described by Fitzpatrick et al. (2019). This gives an
$\sim$8% increase in integrated UV flux over the standard pipeline product
(and a $\sim$1.5% increase in the inferred $T_{\rm eff}$). The resulting
integrated observed flux in the adopted 123.5–321 nm region is 1.40 $\times$
10-6 erg cm-2 s-1. We then applied an extinction correction for $E(B-V)=0\fm
008$, estimated using the g-tomo tool333https://explore-platform.eu/sdas
(based on results from Lallement et al. 2022), with a Seaton (1979) extinction
curve (adopting $R\equiv A(V)/E(B-V)=3.1$).

The second constraint is the value of $v_{\rm e}\sin{i}$. We determined new
values of $v_{\rm e}\sin{i}$ using a Fourier Transform analysis of archival
CFHT spectra. We used the MgII 448.1 nm line, chosen as the rotational line
shape is less affected by shell absorption than alternatives such as SiII
634.7 nm. The analysis was done by comparing model spectra, calculated using
Roche geometry, to the observations in Fourier-Transform space (see Howarth et
al., 2023, for a more detailed description of this approach). Because of the
rotational distortion of the star and the strong gravity darkening the
inferred $v_{\rm e}\sin{i}$ values depend on the rotation rate $\omega$, as
shown in Figure 3. For slow rotations (alone) the inferred $v_{\rm e}\sin{i}$
is in reasonable agreement with the value of 236 km s-1 given by Royer et al.
(2002).

We adopt the $v_{\rm e}\sin{i}\text{--}\omega$ relation of Figure 3, together
with the the Hipparcos parallax of 22.76$\pm$0.24 mas (van Leeuwen 2007; the
star is too bright for Gaia). We can then construct a two-dimensional grid of
stellar parameters where all other values can be established from the two
parameters $\omega$ and $i$. Each model in this grid corresponds to a star
that fits the spectral-energy distribution and $v_{\rm e}\sin{i}$. We can then
calculate the polarization for each of these grid models and compare with the
observations.

Figure 4: Fit to $\epsilon$ Sgr grid using the rotational polarization models
plus a fixed polarization offset. In this and similar figures each point on
the grid corresponds to a set of stellar parameters that fits the spectral
energy distribution and $v_{\rm e}\sin{i}$ of $\epsilon$ Sgr as described in
Section 3.4. The plot shows how well each of these grid models fits the
polarization observations. Figure 5: As Figure 4 but using the rotational
polarization models plus a disk and up to 30ppm interstellar component Figure
6: As Figure 4 but using the rotational polarization models plus a disk and up
to 100 ppm interstellar component. The solid lines show the 2$\sigma$ contours
on the rotational parameters as fitted to these models (see Section 3.6). The
dash-dot lines shows the corresponding extended range of parameters when the
uncertainties on the input parameters to the model grid (in particular $v_{\rm
e}\sin{i}$) are taken into account. Table 4: Parameters of Best Fit Region in
Grid Model Variants (Figures 4 to 6)

Variant | QU Offset | Disk + Interstellar
---|---|---
$p_{\rm max}$ (ppm) | 0.0 | 30.0 | 60.0 | 100.0
$\theta_{\rm ism}$ (deg)a |  | 144.3 | 137.1 | 134.2
$p_{\rm disk}$ (ppm)a | 276 | 299 | 330 | 367
$\theta_{\rm rot}$ (deg) | 40.1 | 39.9 | 39.9 | 39.9
$\omega(i=90)$ | 0.9977(7) | 0.9975(7) | 0.9972(7) | 0.9968(9)
$\omega(i=88)$ | 0.9977(7) | 0.9974(7) | 0.9971(7) | 0.9967(9)
$\omega(i=86)$ | 0.9981(6) | 0.9978(7) | 0.9975(7) | 0.9971(8)
$\omega(i=84)$ | 0.9986(6) | 0.9983(7) | 0.9979(6) | 0.9974(7)
$\omega(i=82)$ | 0.999 | 0.999 | 0.9987(5) | 0.9982(6)
$\omega(i=80)$ |  |  | 0.999 | 0.999

Notes:

Figures in parentheses are the errors in the last digit of the value. When the
best fit for an inclination is in the $\omega$=0.999 bin the value give is the
bin centre and no error can be determined.

a Varies slightly with inclination, average value is given.

Table 5: Best-fitting Stellar Parameters.

Parameter |  $i$=90a | Besta | Rangeb (2$\sigma$)
---|---|---|---
$\omega$ | 0.997 | 0.999 | 0.995 – 1.0
$v_{\rm e}/v_{\rm crit}$ | 0.95 | 0.97 | 0.94 – 1.0
Inclination, $i$ [°] | 90 | 80 | 76 – 90
$T_{\rm eff}$ [K] | 10091 | 9950 | 9845 – 10096
$T_{p}$ [K] | 11791 | 11721 | 11548 – 11813
$T_{e}$ [K] | 7884 | 7433 | 7324 – 7492
Log($L$/$\mbox{L}_{\odot}$) | 2.709 | 2.696 | 2.669 – 2.726
Rp [$\mbox{R}_{\odot}$] | 5.98 | 6.01 | 5.90 – 6.08
Re [$\mbox{R}_{\odot}$] | 8.59 | 8.80 | 8.47 – 8.89
Mass [$\mbox{M}_{\odot}$] | 3.69 | 3.80 | 3.69 – 3.90
$\log{g_{p}}$ [dex cgs] | 3.45 | 3.46 | 3.45 – 3.48
$\log{g_{e}}$ [dex cgs] | 2.24 | 2.00 | 1.99 – 2.36
$P_{\rm rot}$ [days] | 1.63 | 1.61 | 1.57 – 1.64

Notes:

a The parameters listed are those of the grid model (see Section 3.4) giving
the lowest $\chi^{2}$ among all models (Best), and the best model at
inclination of 90 degrees.

b Range is the full range of stellar parameters corresponding to the region
enclosed by the dot-dash line in Figure 6.

###  3.5 Comparison with Observations

In comparing the model grid calculations with the observations we have to
contend with the variability of the system (Section 3.3) and the presence of
additional polarization components due to the disk and interstellar
polarization (Section 3.2). There is insufficient data to attempt to correct
for the variability. However, examination of the polarization measurements
shows that the earliest observations (those in 2014 to 2016) seem to show the
largest scatter, and variability at these times is confirmed by the spectra
shown in Figure 2. We therefore chose to exclude these observations from the
analysis, which was restricted to the 22 observations covering 2017 – 2023.
There is still some evidence of variability, which shows up as excess noise
compared with that expected from the measurement errors.

The additional polarization components were handled by fitting the
observations with several variants of the model. The same grid of calculations
was used for the polarization of the distorted star, but different models were
used for the polarization of the additional components. Initial tests over a
wider and coarser grid of models showed that only the models with high
$\omega$ were compatible with the polarization. The final grid covered the
range 0.980 $\leq\omega\leq$ 0.999 and 70 $\leq i\leq$ 90 and the relevant
section of this region is shown in Figures 4 to 6.

In the first variant (Figure 4) we represent the additional polarization as a
fixed (i.e. wavelength independent) offset in Q and U. Thus at each point in
the grid we fit three values, the Q-offset, U-offset and the position angle of
the star’s rotation axis ($\theta_{\rm rot}$). With this model we find that
for the best-fitting (lowest-$\chi^{2}$) points in Figure 4, the additional
QU-offset corresponds to a position angle of 38.5$\arcdeg$, and the star’s
rotation axis is at 40.1$\arcdeg$. The polarization is within 1.6$\arcdeg$ of
that expected for an equatorial disk (Section 3.2). The small difference in
angles can be understood as due to an additional contribution from
interstellar polarization.

In the remaining variants we represent the additional polarization as a
combination of a wavelength-independent disk contribution, and an
interstellar-polarization component following the ‘Serkowski law’ (Serkowski,
1973; Serkowski et al., 1975; Wilking et al., 1982; Whittet et al., 1992). We
set the $\lambda_{\rm max}$ of the interstellar polarization to 470 nm, the
typical value found within the Local Hot Bubble (Marshall et al., 2020 and
refs. within). In these models there are four fitted parameters at each grid
point: $p_{\rm max}$ and the position angle $\theta_{\rm ism}$ of the
interstellar polarization, the position angle $\theta_{\rm rot}$ of the star’s
rotation axis, and the polarization of the disk $p_{\rm disk}$. The disk is
assumed to be in the equatorial plane so its polarization position angle is
$\theta_{\rm rot}$.

There is some degeneracy in these models between $p_{\rm max}$ and $p_{\rm
disk}$. We therefore ran three versions of the models where we applied the
restricted ranges $0<p_{\rm max}\leq 30$ ppm, $0<p_{\rm max}\leq 60$ ppm, and
$0<p_{\rm max}\leq 100$ ppm, based on our analysis of the likely value of
interstellar polarization as described in Appendix A. Although we constrain
$p_{\rm max}$ to a range, the best-fitting models in all these cases have
$p_{\rm max}$ at the largest allowed value. The results for two of these cases
are plotted in Figures 5 and 6, and all the variants are described in Table 4.

All these variants (Figures 4–6) produce quite similar results, with a
“valley” of low $\chi^{2}$ values running from $\omega\sim 0.997$ at
$i=90\arcdeg$ to $\omega=0.999$ at $i\sim 80\arcdeg$. Thus despite different
assumptions about the interstellar polarization, almost the same stellar
parameters are being selected in all these cases.

In Table 5 we list the stellar parameters corresponding to the two grid models
at the ends of this $\chi^{2}$ valley. The model labelled “Best” (in the ‘Disk
+ 100 ppm’ set) is the one that has the lowest $\chi^{2}$ and is at $\omega$ =
0.999, $i$ = 80$\arcdeg$ (in some of the other model variants the best fit is
in the adjacent grid point at $i$ = 82$\arcdeg$). However, the very strong
shell absorption seen in $\epsilon$ Sgr, despite the low density of the disk,
might favour a higher-inclination model like the $\omega$ = 0.997, $i$ =
90$\arcdeg$ case also listed in Table 5.

Figure 7 shows images of the stellar models for the two cases listed in Table
5. The models show the distribution of specific intensity over the star,
showing the strong effect of gravity darkening at the equator, and is overlaid
with polarization vectors. The polarization is dominated by light from the
bright regions at top and bottom of the image resulting in the negative
(perpendicular to the rotation axis) integrated polarization at this
wavelength.

Figure 7: Polarization and intensity images for the two models of $\epsilon$
Sgr listed in Table 5. Specific intensity is overlaid with polarization
vectors for a wavelength of 400 nm.

###  3.6 Uncertainties

To determine the uncertainties on the fitted values of $\omega$ and $i$ we use
the bootstrap method (Press et al., 2007). For each of the model grid variants
we ran the modelling on 1000 trials using random selections of the 22
observations with replacement (so that each observations can be selected more
than once). For each trial we determined the best $\omega$ value at each
inclination by fitting the $\chi^{2}$ values with a cubic function. We then
determined the standard deviation of these values over the 1000 trials. These
results give the errors on $\omega$ listed in Table 4 and plotted as error
bars on Figure 4 to 6. This method of error determination does not depend on
the errors on the individual polarization measurements, but on the actual
scatter on the data, which is largely due to the polarization variability of
the star.

Using these bootstrap results we can determine the $\Delta\chi^{2}$ ranges
corresponding to 1 or 2$\sigma$, and thus determine the 2-$\sigma$ contours on
$\omega$ and $i$ which are shown as the solid lines in Figure 6. We use Figure
6 (the ‘Disk + 100 ppm’ variant) as this gives the lowest $\chi^{2}$ values,
and has a somewhat broader distribution in $\omega$, but largely overlays the
best-fit regions covered by the other model variants.

Additional uncertainties are introduced by the uncertainties in the input
parameters to the model grid (Section 3.4). These were assessed by running the
modelling for auxiliary grids calculated for adjusted values of $v_{\rm
e}\sin{i}$, parallax and UV flux in turn. The most significant effect is that
due to an estimated 10 km s-1 uncertainty in $v_{\rm e}\sin{i}$ which has the
effect of moving the $\chi^{2}$ valley up and down in $\omega$. This leads to
the wider 2-$\sigma$ contours shown by the dash-dot lines in Figure 6.
Uncertainties in parallax and UV flux do not significantly change the fitted
values of $\omega$ and $i$, but do have a small effect on the relationship to
other stellar parameters. The wider contours (dash-dot line) were then used to
determine the allowed range of each stellar parameter, as listed in Table 5.

##  4 Discussion

###  4.1 There is No Polarizing Debris Disk

Previously, HIPPI observations of $\epsilon$ Sgr were reported by Cotton et
al. (2016b). They interpreted the observed polarization in the context of the
system’s reported infrared excess (Rodriguez & Zuckerman, 2012). Cotton et al.
present a model comprising two dust components (to reproduce the continuum IR
excess) in addition to the stars. A cold dust belt (responsible for the bulk
of the IR excess) was assumed to be circumbinary, with a warm dust belt
orbiting one of the two stars. A residual polarization of 30 ppm remained
after accounting for polarization induced by the interstellar medium. To
reproduce this a fractional polarization of around 50% of the total IR excess
would be required. Such a value is much higher than those measured for dust
grains in other debris disks, which peak at around 20 to 30% (e.g. Krivova et
al., 2000; Graham et al., 2007; Anche et al., 2023). Since the Cotton et al.
(2016b) model was proposed, work by Vandeportal et al. (2019) has shown that
warm (mid-IR) excesses represent substantially lower dust masses and
contribute less to polarization compared to cool (far-IR) excesses with lower
fractional luminosities but representing greater dust masses. This further
reduces the plausibility of the proposed disk model.

The large beam size of IRAS and the low galactic latitude of the system
($|b|<10\arcdeg$) also lead to the possibility that the flux density
measurement at 60 $\mu$m is contaminated by diffuse background emission.
Indeed, the IRAS point source catalog (Joint Iras Science, 1994) notes a
quality flag of “2” for the 60 $\mu$m band. Applying a downward revision of
the flux, as implied, would only increase the tension between the level of IR
excess and the polarization efficiency of the disk dust required.

In the light of observations presented here we can eliminate the disk model as
a plausible explanation for the polarimetric signal arising from the system.
Specifically, the invariance of the position angle of the polarimetric signal
between epochs, despite the secondary’s proper motion, is inconsistent with
illumination of a circumbinary dust disk. The agreement between the
$r^{\prime}$ observations with small and large aperture sizes in the 2018MAR
run is also inconsistent with this hypothesis. Furthermore, the level of
fractional polarization required would be a factor of two higher than other
systems for which such measurements are available (see above, and also
Marshall et al., 2023).

Given that Rodriguez & Zuckerman (2012) find the system to be unstable with
the accepted debris disk architecture and that IR excess may also be produced
by free-free emission in the (previously unidentified) Be disk (Kastner &
Mazzali, 1989), it is likely no debris disk exists at all. The nature of
$\epsilon$ Sgr as a rapidly rotating B star with a gas disk is therefore the
origin of its polarization.

###  4.2 Rapid Rotation

The rotation as a fraction of criticality that we measure for $\epsilon$ Sgr
of $\omega$ $\sim$ 0.997, which corresponds to $v_{\rm e}/v_{\rm crit}$
$\sim$0.95, is the largest yet measured for a star in our galaxy444VFTS285, an
O star in the Large Magellanic Cloud, has been reported as having $\omega$ =
1.00 (Shepard et al., 2022). The previous highest was obtained by Domiciano de
Souza et al. (2014) who measured the rotation of Achernar using interferometry
as $\omega$ = 0.980, $v_{\rm e}/v_{\rm crit}$ = 0.883. Achernar is a Be star,
but the measurements were made during a quiescent phase where the Be disk was
absent.

$\epsilon$ Sgr stood out in our polarization study of bright, rapidly-rotating
stars because of the large amplitude of the wavelength-dependent polarization
variations – $>$300 ppm, compared with $<$100 ppm for the other rapidly-
rotating stars (Cotton et al., 2017a; Bailey et al., 2020a; Howarth et al.,
2023). The only other star with a comparable polarization amplitude was
$\theta$ Sco (Lewis et al., 2022). In that case the rotation was slower than
in $\epsilon$ Sgr, but the low gravity of this evolved star enhanced the
polarization.

Multiwavelength polarimetry should therefore be a good way of finding other
similar extreme rotators, and establishing how common such objects are. In the
19 stars we have observed for our AAT polarimetry program on rotating stars we
have found no other candidates for such extreme rotation. The program is based
on stars down to V $\sim$ 3.5 from the list of van Belle (2012).

Figure 8: Comparison of observations and models for the polarization of
$\epsilon$ Sgr as a function of wavelength. The models show the predictions
due to stellar rotation. The disk and interstellar polarization have been
applied as a correction to the observations. The black solid line is the model
for $\omega$ = 0.997, $i$ = 90$\arcdeg$. Blue diamonds are the predicted
polarization values for each observed filter (they are not necessarily exactly
on the line, because they are averages over the filter bandpass). Grey lines
show the model predictions for grid models with lower values of $\omega$.
Figure 9: Periodogram of TESS light curve of $\epsilon$ Sgr (TESS sector 13).
In the upper panel it is shown at full scale. In the lower panel it is
compared with the periodogram of the bright classical Be star Achernar
($\alpha$ Eri) from TESS sector 2. The presence of two frequency groups with a
2-to-1 frequency ratio is common in Be stars, but the amplitude seen in
$\epsilon$ Sgr is much lower than is commonly seen.

###  4.3 Relationship to Be Stars

The value of $v_{\rm e}/v_{\rm crit}$ $\sim$0.95 that we find for $\epsilon$
Sgr is interesting because this is the value that was suggested (Townsend et
al., 2004; Owocki, 2005) as required for “weak” processes such as pulsation to
be able to launch material from the star into orbit. If all Be stars rotated
this fast, the existence of Be disks would be easier to explain. However,
statistical analysis of $v_{\rm e}\sin{i}$ data for Be stars (Cranmer 2005
[but see Howarth 2007]; Zorec et al. 2016) suggests that most Be stars rotate
much slower than this, with a broad distribution of $v_{\rm e}/v_{\rm crit}$
values from $\sim$0.5 to $\sim$0.95, with the mode of the distribution at 0.66
(Zorec et al., 2016). There are also rapidly-rotating B stars (e.g. Regulus,
Cotton et al., 2017a) that are not Be stars.

While $\epsilon$ Sgr has a rotation rate at the top end of the range inferred
for Be stars it is certainly not a typical classical Be star. It does appear
to have an equatorial disk, but it is very weak compared with typical Be
stars. Weak H$\alpha$ emission has been observed on one occasion but is
normally absent, as discussed in section 3.3. Thus, these results confirm that
rotation alone cannot be the key requirement for formation of Be disks.
Rather, there must be some additional mechanism operating in Be stars that can
launch material from the surface into orbit. In $\epsilon$ Sgr this mechanism
is either not operating or is operating with much reduced efficiency, enabling
the formation of only a weak disk despite the very rapid rotation. We note
that Classical Be stars are more common at early B spectral types and less
common at the B9/A0 spectral type of $\epsilon$ Sgr. Huang et al. (2010)
suggest that the threshold rotation rate for forming a Be disk is much higher
for later type stars.

Most classical Be stars show photometric variability on short timescales.
Observations with the Transiting Exoplanet Survey Satellite (TESS, Ricker et
al., 2015) typically show multiple periodic signals at frequencies between 0.5
and 4 day-1 (Labadie-Bartz et al., 2022). These are normally interpreted as
non-radial pulsations, which may be related to the mass ejection process that
forms the disk (Rivinius et al., 2013).

In Figure 9 we show the periodogram of the TESS two-minute cadence light curve
of $\epsilon$ Sgr for sector 13 (2019 Jun 19 — Jul 17)555The TESS data used
here are available at MAST: 10.17909/t1ae-rd45 (catalog 10.17909/t1ae-rd45).
The pattern seen of two groups of peaks (here at $\sim$0.65 and 1.3 days-1),
with a frequency ratio of roughly 1 to 2 is one that is very commonly seen in
Be stars (e.g. Walker et al., 2005; Semaan et al., 2018; Baade et al., 2018;
Labadie-Bartz et al., 2022). However the amplitude (around 140 ppm for the
highest peak) is low compared with those more typically seen in classical Be
stars, which are in the parts per thousands (Labadie-Bartz et al., 2022).

As an example we show in the lower panel for Figure 9 the periodogram of
$\epsilon$ Sgr compared with that of the bright Be star Achernar ($\alpha$
Eri, HD 10144), based on observations obtained from TESS sector 2 (2018 Aug 23
— Sep 20). At that time Achernar was in an active state. It has been quiescent
without a disk since late 2019666According to our polarization monitoring of
Achernar over the last 10 years.. The frequency structure seen in Achernar is
quite similar to that of $\epsilon$ Sgr, but the peak amplitude is 25 times
higher in Achernar.

###  4.4 Evolutionary Status

Figure 10: Location of $\epsilon$ Sgr on the HR diagram compared with
evolutionary models for rotating stars from Georgy et al. (2013). Isochrones
are shown at 10 Myr intervals from 150 to 190 Myr, and then at 196 and 197
Myr.  Figure 11: Plot of $\omega$ against $\log(g_{\rm p})$ for $\epsilon$ Sgr
and the same models as in Figure 10. The rotation rate ($\omega$) reaches a
maximum at a point corresponding to the “hook” in the HR diagram, and then
declines rapidly as the star evolves across the Hertzprung gap.

In Figure 10 we show the position of $\epsilon$ Sgr on an HR diagram compared
with evolutionary models for rotating stars of solar metallicity (Z = 0.014)
from Georgy et al. (2013). The models are for $M/\mbox{M}_{\odot}$ of 3.0 and
4.0 (our measured value is $M/\mbox{M}_{\odot}$ = 3.69), and initial rotation
rates of $\omega$ = 0.7, 0.8 and 0.9. The location of $\epsilon$ Sgr is that
of a star just beginning to evolve away from the main sequence. Its position
corresponds to the hydrogen shell burning phase where the star is crossing the
Hertzprung gap. This is a very rapid phase of stellar evolution and such stars
are not commonly observed.

Figure 11 shows the comparison with the same set of models in a plot of
$\omega$ against $\log(g_{\rm p})$. The $\log(g_{\rm p})$ we measure for
$\epsilon$ Sgr is again consistent with a star beginning to evolve away from
the main sequence. The maximum rotation rate for these model stars corresponds
to the “hook” in the HR diagram where the star contracts after the end of core
hydrogen burning. At this point a star born with $\omega$ = 0.9 or greater
reaches near critical rotation. However, Figures 10 and 11 show that our
measured parameters for $\epsilon$ Sgr place the star at a point beyond the
“hook” where the star is expanding and the rotation rate is slowing. The
rotation we measure for $\epsilon$ Sgr is therefore too high to match any of
these single rotating star models. Its rapid rotation may therefore be the
result of close binary evolution (Pols et al., 1991).

The visible companion (Section 2.2) seems to have colours consistent with a
main sequence star making it unlikely to have been involved in the interaction
that caused the rapid rotation. $\epsilon$ Sgr is known to be an X-ray source
(Berghoefer et al., 1996; Hubrig et al., 2001), which might indicate an unseen
companion.

##  5 Future Observations

Our results on $\epsilon$ Sgr could be further tested by a number of other
observations.

###  5.1 Interferometry

Interferometry has been successful in imaging the rotational distortion of a
number of stars of similar brightness and spectral type to $\epsilon$ Sgr such
as Regulus (Che et al., 2011), Altair (Monnier et al., 2007) and $\alpha$ Oph
(Zhao et al., 2009). Such observations should be able to confirm the extreme
rotational oblateness and detect the variation of surface brightness due to
gravity darkening. Interferometry is better than polarimetry at measuring the
inclination and hence distinguishing between the range of models that are
consistent with the polarimetry.

Figure 12: Modelled polarization for $\epsilon$ Sgr at high spectral
resolution. The pure Öhman effect (due to rotational line broadening) is seen
most clearly on the CaII K line at 393 nm, the strongest metal line in this
region. On the strongly Stark broadened hydrogen lines (the five strongest
lines in this region) the Öhman effect in Q/I is swamped by other effects, but
the Öhman effect is still present in U/I, and is a strong function of
inclination

###  5.2 Spectropolarimetry and the Öhman Effect

Current observations of the polarization due to the rotational distortion of
stars, such as those presented in this paper, have been made with broad-band
polarimeters because these instruments can achieve the high precision needed
to measure the small rotational signals. However, further information on
stellar rotation could be obtained from spectropolarimetric observations with
sufficient resolution to resolve the lines.

These effects include the Öhman effect first predicted in 1946 (Öhman, 1946).
The Öhman effect arises because at different points in the line profile of a
rotationally broadened line the absorption arises in different parts of the
star and therefore changes the integrated polarization we observe. Models of
the effect have been given by Collins & Cranmer (1991). The effect is small
and has not yet been observed. Since $\epsilon$ Sgr has the largest rotational
signal yet seen in continuum polarization it is interesting to consider
whether observations of the Öhman effect for this star are feasible and
useful.

Because our polarization modelling using SYNSPEC/VLIDORT (Section 3.1) is
carried out at high spectral resolution, our models predict what should be
seen in spectropolarimetry. As in the modelling done by Collins & Cranmer
(1991) the spectral lines are treated as pure absorbers. No line polarization
processes are involved. The polarization is due to the same processes that
produce the continuum polarization (primarily electron scattering).

Models for the polarization wavelength dependence of $\epsilon$ Sgr are shown
in Figure 12 for the 380 – 450 nm region. This shows the problem with
detecting the Öhman effect in a star of this spectral type. The spectrum is
dominated by hydrogen lines, which are strongly Stark broadened by an amount
that exceeds the rotational broadening. The pure Öhman effect is only seen on
the metal lines, of which the CaII K line at 393 nm is the strongest in this
region. The large polarization features seen on the hydrogen lines in Q/I are
not due to the Öhman effect, but are due to the fact that in these strong
lines we are seeing different layers of the star in line and continuum.

The Öhman effect features seen in U/I are more interesting because they show a
strong dependence on inclination. The features seen on the hydrogen lines are
smeared out by the Stark broadening, but may still be observable and would
provide another way of refining the inclination of the star. Such an
observation would require spectropolarimetry with R $>$ 5000 and precision
$\sim$ 5 ppm or better.

###  5.3 Ultraviolet (UV) Polarimetry

Models of rotating stars predict that polarization values are expected to
increase at shorter wavelengths (Collins et al., 1991; Jones et al., 2022)
reaching $\sim$1% (10000 ppm) at 120 nm for large $\omega$ and high
inclination. Since the polarization due to a disk is not strongly wavelength
dependent, and interstellar polarization decreases at shorter wavelengths,
observations in the UV would remove any possible confusion between these
different sources. Such observations may be possible with future space
missions similar to the proposed Polstar satellite (Scowen et al., 2022).

##  6 Conclusions

Multi-wavelength high-precision polarization observations of $\epsilon$ Sgr
show the distinctive wavelength dependence expected for a rotationally
distorted star, but with a much higher amplitude ($\sim$ 300 ppm) than seen in
other rapidly rotating stars. In order to fit a rotating-star model, an
additional positive polarization component is needed. We attribute this to
polarization in a low-density ($\sim$10-13 g cm-3) equatorial gas disk which
also produces the narrow shell absorption features seen in the spectrum.

Detailed modelling of the polarization together with additional data including
the spectral-energy distribution and $v_{\rm e}\sin{i}$ reveals a very high
rotation rate of $\omega$ $\geq$ 0.995, the highest yet measured for a star in
our galaxy. The modelling also results in a detailed set of stellar
parameters. The star is rotating too rapidly to be consistent with single
rotating star evolutionary tracks, suggesting that it is likely to be a
product of binary evolution.

$\epsilon$ Sgr shares some features with Be stars in having a low density gas
disk, weak H$\alpha$ emission seen on one occasion, and multiple periodicities
in its TESS light curve. However, the disk is of very low density and the TESS
periodicites are of very low amplitude compared with typical Be stars. The
results confirm that rapid rotation alone cannot be the cause of the Be
phenomenon, and support the idea that an additional mechanism is needed to
eject material into a Be disk.

$\epsilon$ Sgr is an excellent target for future observations by
interferometry, spectropolarimetry and UV polarimetry which would allow the
extreme rotation to be confirmed and the stellar parameters refined.

##  7 Acknowledgments

.

Based in part on data obtained at Siding Spring Observatory. We acknowledge
the traditional owners of the land on which the AAT stands, the Gamilaraay
people, and pay our respects to elders past and present. JPM acknowledges
research support by the National Science and Technology Council of Taiwan
under grant NSTC 112-2112-M-001-032-MY3. DVC thanks the Friends of MIRA for
their support.

##  Appendix A Interstellar Polarization

Figure 13: Plot of the polarization of stars near to $\epsilon$ Sgr,
representative of interstellar polarization, as projected onto the sky (left)
– vectors represent the PA (clockwise N over E) – and debiased according to
$\hat{p}=\sqrt{p^{2}-p_{e}^{2}}$ as a function of distance (right). Where
multiple observations exist, the error-weighted average is displayed. The grey
vector and dot represent the interstellar polarization of $\epsilon$ Sgr as
predicted by the model in Cotton et al. (2017b), whereas their black
counterparts average the observations, and correct them for wavelength as if
they are interstellar measurements. All observations are scaled to 450 nm as
if they are purely interstellar with $\lambda_{\rm max}$=470 nm. The controls
are numbered in order of angular separation as follows, 1: HD 169586, 2: HD
165135, 3: HD 177474, 4: HD 166916, 6: HD 176687, 7: HD 165189, 8: HD 157919,
8 HD 156384, 9: HD 174309, 10: HD 160915, 11: HD 181240, 12: HD 160691, 13: HD
188114, 14: HD 151680, 15: HD 157172, 16: HD 191408, 17: HD 153580, 18: HD
192310, 19: HD 155125, 20: HD 151504, 21: HD 194640, 22: HD 180409, 23: HD
151192, 24: HD 165499, 25: HD 153631, 26: HD 146070, 27: HD 167425, 28: HD
197157, 29: HD 143114, 30: HD 173168, 31: HD 199288, 32: HD 164259, 33: HD
168723, 34: HD 162521, 35: HD 181391, 36: HD 197692, 37: HD 145518, 38: HD
192947, 39: HD 144766, 40: HD 185124, 41: HD 177389, 42: HD 164651. Note that
for stars with $p<\sigma_{p}$ the PA’s in the left hand panels are not well
defined, since $\sigma_{\rm PA}>28\arcdeg$. Table 6: Observations of
Interstellar Control Stars

Control | SpT | Run | UT | Dwell | Exp. | $\lambda_{\rm eff}$ | Eff. | $q$ | $u$ | $p$ | $\theta$
---|---|---|---|---|---|---|---|---|---|---|---
HD |  |  |  | (s) | (s) | (nm) | (%) | (ppm) | (ppm) | (ppm) | (∘)
157919 | F5III-IV | 2018MAR | 2018-03-27 19:22:38 | 1114 | 640 | 468.3 | 84.2 | 18.5 $\pm$ 05.8 |  $-$20.4 $\pm$ 05.6 | 27.5 $\pm$ 05.7 | 156.1 $\pm$ 06.0
160691 | G3IV-V | 2018MAR | 2018-04-04 17:15:17 | 1780 | 1280 | 472.1 | 85.7 | 14.5 $\pm$ 06.0 | 8.1 $\pm$ 05.9 | 16.6 $\pm$ 05.9 | 14.6 $\pm$ 11.2
165189 | A6V | 2018AUG | 2018-09-01 10:51:27 | 971 | 640 | 464.4 | 58.7 |  $-$56.6 $\pm$ 11.7 | 0.1 $\pm$ 11.2 | 56.6 $\pm$ 11.5 | 89.9 $\pm$ 05.9
168723 | K0III-IV | 2018MAR | 2018-03-29 19:00:51 | 1001 | 640 | 475.0 | 86.9 | 10.2 $\pm$ 04.6 | 22.7 $\pm$ 04.4 | 24.9 $\pm$ 04.5 | 32.9 $\pm$ 05.2
|  | 2018JUL | 2018-07-12 10:54:07 | 1100 | 640 | 475.3 | 86.1 | 1.7 $\pm$ 04.6 | 17.0 $\pm$ 04.5 | 17.1 $\pm$ 04.6 | 42.1 $\pm$ 07.8
177474 | F8V | 2017AUG | 2017-08-14 13:49:16 | 1663 | 960 | 471.3 | 89.2 | 3.4 $\pm$ 05.8 |  $-$17.6 $\pm$ 05.9 | 17.9 $\pm$ 05.8 | 140.5 $\pm$ 09.9
181240 | A8III | 2018MAR | 2018-04-05 18:40:37 | 1716 | 1280 | 465.3 | 83.0 |  $-$88.4 $\pm$ 06.9 |  $-$47.5 $\pm$ 06.9 | 100.4 $\pm$ 06.9 | 104.1 $\pm$ 02.0
|  | 2018MAR | 2018-04-06 18:25:17 | 1726 | 1280 | 465.4 | 83.0 |  $-$104.2 $\pm$ 06.7 |  $-$54.2 $\pm$ 06.6 | 117.5 $\pm$ 06.7 | 103.7 $\pm$ 01.6
191408 | K2.5V | 2016JUN | 2016-06-25 12:21:58 | 3029 | 1680 | 477.2 | 90.9 |  $-$8.6 $\pm$ 09.3 |  $-$32.0 $\pm$ 08.7 | 33.1 $\pm$ 09.0 | 127.5 $\pm$ 08.0
192310 | K2+V | 2016JUN | 2016-06-25 15:41:28 | 3464 | 2560 | 476.1 | 90.7 |  $-$9.8 $\pm$ 07.2 |  $-$5.9 $\pm$ 07.1 | 11.4 $\pm$ 07.2 | 105.5 $\pm$ 22.3
|  | 2018JUL | 2018-07-20 13:21:57 | 2203 | 1600 | 475.3 | 86.0 |  $-$17.7 $\pm$ 06.9 |  $-$3.5 $\pm$ 08.3 | 18.0 $\pm$ 07.6 | 95.6 $\pm$ 13.9
R |  | 2018JUL | 2018-07-23 11:07:49 | 1772 | 1280 | 627.0 | 84.4 |  $-$14.8 $\pm$ 08.9 |  $-$18.4 $\pm$ 09.8 | 23.6 $\pm$ 09.3 | 115.6 $\pm$ 12.7
R |  | 2018JUL | 2018-07-23 11:33:17 | 1051 | 640 | 627.0 | 84.4 | 14.0 $\pm$ 12.0 |  $-$16.7 $\pm$ 12.3 | 21.8 $\pm$ 12.1 | 155.0 $\pm$ 19.7
192947 | G9III | 2018JUL | 2018-07-18 12:25:25 | 1068 | 640 | 474.7 | 85.8 |  $-$16.1 $\pm$ 04.7 | 15.7 $\pm$ 04.7 | 22.5 $\pm$ 04.7 | 67.9 $\pm$ 06.1

Notes: The same aperture as used for the $\epsilon$ Sgr observations specified
in Table 1 in the same run was used. Spectral types (SpT) are from SIMBAD. R
Indicates SDSS $r^{\prime}$ filter paired with R PMT detectors; all other
tabulated observations used the SDSS $g^{\prime}$ filter paired with the B PMT
detectors.

In Figure 13 we present a map of the polarization of nearby stars. The stars
have been selected such that their polarization is likely to be interstellar
dominated. We interrogated observations made with high precision polarimeters,
reported in Bailey et al. (2010); Cotton et al. (2016a); Marshall et al.
(2016); Cotton et al. (2017a, b); Bailey et al. (2017); Bott et al. (2018);
Cotton et al. (2019, 2020); Bailey et al. (2020a); Marshall et al. (2020);
Piirola et al. (2020); Lewis et al. (2022); Marshall et al. (2023), and
selected stars with spectral types between A0 and K2.5 within 35∘. From this
list we removed rapidly rotating stars, those in close binary systems,
magnetically active stars, systems hosting significant debris disks, and any
other star known to display intrinsic polarization. To this we added a small
number of interstellar controls newly observed for this work (Table 6). The
observations were not all made in the same bandpass so, as we have done in
past work, we assumed $\lambda_{\rm max}=470$ nm and used the Serkowski Law to
scale the polarization magnitude to correspond to 450 nm – representative of
the $g^{\prime}$ band.

The right panel of Figure 13 illustrates that most stars near $\epsilon$ Sgr
have polarizations corresponding to $\leq$ 2 ppm/pc, though the closest star,
HD 169586, is a little greater. This is in line with the trends previously
reported in Cotton et al. (2017b), which if extrapolated predict 60.0 $\pm$
10.5 ppm for $\epsilon$ Sgr (grey dot).

The position angles of the interstellar controls are shown in the left panel
of Figure 13. Cotton et al. (2017b) presents a formula for predicting position
angle of interstellar polarization for a star based on the angular separation
weighted mean of nearby stars that gives 163.1∘ $\pm$ 29.1 (grey bar).
However, this was only meant to apply to stars within 30 pc of the Sun, where
potential controls are sparse. The paper also found better agreement than the
nominal error between stars within 10∘ of each other on the sky; the median
position angle of the nearest 5 stars to $\epsilon$ Sgr is 145.6∘; it is
likely that the interstellar position angle is within 15-20∘ of this.

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