11institutetext: Department of Physics, Faculty of Science, University of
Zagreb, 10 000 Zagreb, Croatia
11email<EMAIL_ADDRESS>22institutetext: Astrophysics Group, Keele
University, Staffordshire, ST5 5BG, UK 33institutetext: Institute of
Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
44institutetext: University of Applied Sciences, 10 410 Velika Gorica, Croatia

# High-mass eclipsing binaries: a testbed for models of interior structure and
evolution††thanks: Based on observations made with the ESO 3.6 m Telescope and
the HARPS spectrograph, operated on La Silla, Chile by the European Southern
Observatory. ††thanks: Table A.1 is only available in electronic form at the
CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via
https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/

Accurate fundamental properties and surface chemical composition for V1034
Sco, GL Car, V573 Car and V346 Cen

K. Pavlovski 11 J. Southworth 22 A. Tkachenko 33 T. Van Reeth 33 E. Tamajo 44

###### Abstract

Aims. The surface chemical compositions of stars are affected by physical
processes which bring the products of thermonuclear burning to the surface.
Despite their potential in understanding the structure and evolution of stars,
elemental abundances are available for only a few high-mass binary stars. We
aim to enlarge this sample by determining the physical properties and
photospheric abundances for four eclipsing binary systems containing high-mass
stars: V1034 Sco, GL Car, V573 Car and V346 Cen. The components have masses
8–17$\,{\rm M}_{\odot}$ and effective temperatures from 22 500 to 32 200 K,
and are all on the main sequence.

Methods. We present new high-resolution and high signal-to-noise spectroscopy
from HARPS, and analyse them using spectral disentangling and NLTE spectral
synthesis. We model existing light curves and new photometry from the TESS
satellite.

Results. We measure the stellar masses to 0.6–2.0% precision, radii to
0.8–1.7% precision, effective temperatures to 1.1–1.6% precision, and
abundances of C, N, O, Mg and Si. The abundances are similar to those found in
our previous studies of high-mass eclipsing binaries; our sample now comprises
25 high-mass stars in 13 binary systems. We also find tidally-excited
pulsations in V346 Cen.

Conclusions. We reinforce our previous conclusions: interior chemical element
transport is not as efficient in binary star components as in their single-
star counterparts in the same mass regime and evolutionary stage, possibly due
to the effects of tidal forces. Our ultimate goal is to provide a larger
sample of OB-type stars in binaries which would enable a thorough comparison
to stellar evolutionary models, as well as to single high-mass stars.

###### Key Words.:

stars: fundamental parameters – stars: evolution – binaries: spectroscopic –
binaries: eclipsing – stars: abundances

##  1 Introduction

The interior structure and evolution of a star are largely determined by its
mass and chemical composition at formation. Precise and accurate observational
constraints on these fundamental physical quantities are required for the
validation, calibration and improvement of theoretical models of the interior
structure and evolution of stars. Despite being much more complex than single
stars, binary star systems are a treasure trove for testing stellar structure
and evolution models and understanding how these might be improved. In the
case of eclipsing binaries (EBs) where both components are detected
spectroscopically, it is possible to measure their masses and radii with high
precision and accuracy using only orbital mechanics and geometry. Detached
systems are particularly valuable as they are expected to evolve as single
stars without alteration of their evolution by mass transfer episodes.

Table 1: Basic characteristics of binary systems studied in this work.

Binary | Other | Orbital | $V_{\rm max}$ | Spectral | Age | Cluster | Apsidal
---|---|---|---|---|---|---|---
system | designation | period (d) | (mag) | types | (Myr) | membership | period (yr)
V1034 Sco | CPD$-$41o7742 | 2.44 | 8.80 | O9.5 V + B1-1.5 V | 3–8 | NGC 6231 | $23.4\pm 0.8$
GL Car | HD 306168 | 2.42 | 9.74 | B0.5 V + B1 V | 2.0 | NGC 3572 | $25.20\pm 0.02$
V573 Car | CPD$-$59o2628 | 1.47 | 9.47 | O9.5 V + B0.3 V | 2 | Trumpler 16 | -
V346 Cen | HD 101837 | 6.32 | 9.57 | B1/3II/III | 10 | Stock 14 | $306\pm 4$

111References to the quantities are given in Section 2.

The role of precise empirical mass measurements is difficult to overstate for
validating and calibrating modern and sophisticated stellar models. Herrero et
al. (1992) presented a study of 25 luminous galactic OB-type stars and
reported a discrepancy between the masses inferred from their spectra (via
wind theory) and those predicted by evolutionary models. The authors termed
the effect the “mass discrepancy” and emphasised the difficulty in attributing
it to either of the two theories (wind and stellar evolution) involved. Since
then, many attempts have been made to diagnose the cause of the mass
discrepancy in intermediate- to high-mass stars.

Given the high precision and accuracy that SB2 detached eclipsing binaries
(dEBs) allow us to achieve in measurements of mass and surface gravity (e.g.,
Torres et al. 2010), these objects are important in studying the mass
discrepancy. Burkholder et al. (1997) studied seven early-type spectroscopic
binaries with masses below 15$\,{\rm M}_{\odot}$ and reported a good agreement
between masses inferred from binary dynamics and those estimated with
evolutionary models in all cases where the stars are non-interacting. Guinan
et al. (2000) and Pavlovski et al. (2009) presented independent studies of the
high-mass SB2 dEB V380 Cyg and reported a substantial mass discrepancy for the
evolved primary component, in the sense that its dynamical mass is too low
compared to the predictions of standard stellar models. Whereas Guinan et al.
(2000) showed that the discrepancy could be resolved by introducing extra
near-core mixing in the form of convective core overshooting, Pavlovski et al.
(2009) found that rotationally induced mixing in models was insufficient to
explain the mass discrepancy. Indeed, Tkachenko et al. (2014b) demonstrated
that only the combined effects of rotation and convective core overshooting
can account for the mass discrepancy observed in V380 Cyg.

Recently, Massey et al. (2012), Morrell et al. (2014), Mahy et al. (2015), and
Pavlovski et al. (2018) reported systematic discrepancies between the
Keplerian and evolutionary masses of stars less massive than 30$\,{\rm
M}_{\odot}$. Mahy et al. (2020b) found a good agreement between the
spectroscopic and dynamical masses for 26 early-type binary components whereas
their evolutionary masses appear to be systematically overestimated. These
results hint towards models of interior structure and evolution being the
primary cause of the mass discrepancy. Tkachenko et al. (2020) and Johnston
(2021) demonstrated that the problem cannot be attributed to differences in
observation and analysis methods between research groups, and instead showed
that the mass discrepancy progressively increases with the evolutionary stage
of the star. In particular, the authors found that higher convective core
masses were required in models of stellar structure and evolution for stars
that are born with a convective core. The effects of excess core mass can be
efficiently mimicked with an enhanced mixing in the near-core regions,
irrespective of the true cause(s) of the mixing.

Connecting the treatment of interior mixing in stellar evolution models and
the mass discrepancy requires extra observational constraints. Surface
chemical composition measurements are ideal because chemical abundance
patterns are expected to be substantially altered by various mechanisms of
interior mixing and chemical element transport in stars. For example, Heap et
al. (2006) found surface nitrogen enrichment in 80% of their sample stars and
speculated on the role of rotation in causing this phenomenon. Overall, these
observational findings are in good agreement with predictions from rotating
stellar evolution models for high-mass stars (Meynet & Maeder 2000; Maeder &
Meynet 2000; Heger et al. 2000; Heger & Langer 2000; Langer 2012) with the
caveat that the observed nitrogen enrichments (Heap et al. 2006) are larger
than those predicted by the models.

Hunter et al. (2008, 2009) studied a large sample of intermediate- to high-
mass stars in the Magellanic Clouds. Some of their findings corroborate the
theory of rotationally-induced mixing while others contradict it (e.g.,
rapidly and slowly rotating stars without and with substantial surface
nitrogen enrichment, respectively). Slowly-rotating, nitrogen-enriched stars
were also found by Markova et al. (2018) and the authors suggested that
inadequacies of the models in these particular cases might be related to the
efficiency of rotational mixing. At the same time, Pavlovski et al. (2018)
presented a detailed study of the surface chemical compositions in several
high-mass SB2 dEBs and found no dependence of the abundances of carbon (C),
nitrogen (N) and oxygen (O) on either the projected rotational velocity
($v\sin i$) or surface gravity ($\log g$) of the star.

Whilst changes in the photospheric CNO abundances of high-mass B-type single
stars have been found (Przybilla et al. 2010; Nieva & Przybilla 2012; Maeder
et al. 2014; Cazorla et al. 2017a, b; Markova et al. 2018), the role of
rotationally-induced mixing in the formation of these chemical abundance
patterns remains poorly quantified. At the same time, Aerts et al. (2014)
demonstrated that neither $v\sin i$ nor the rotational frequency of a star has
significant predictive power for the surface N abundance. Instead, the latter
correlates strongly with the effective temperature ($T_{\rm eff}$) of the star
and the frequency of its dominant acoustic oscillation mode. Furthermore,
Rogers et al. (2013) showed that internal gravity waves (IGWs) excited at the
convective-radiative boundary near the core in high-mass stars are efficient
in transporting angular momentum and chemicals on short timescales and over
large distances. Pedersen et al. (2018) demonstrated that the IGW-driven
functional form of the interior mixing profile is a good candidate to
simultaneously explain the observed properties of gravity-mode oscillations
and surface abundances in B-type stars.

SB2 dEBs are at the forefront of efforts to resolve deficiencies in
theoretical stellar models. The masses and radii of the component stars can be
measured precisely and independently of models, and the requirement for the
stars to have the same age and initial chemical composition at formation
provides an additional stringent constraint on theoretical models. Moreover,
measured masses and radii give a precise surface gravity which can be used to
break the degeneracy between $T_{\rm eff}$ and $\log g$ in spectral analysis,
boosting the accuracy of measurements of the surface chemical compositions of
the stars. The DEBCat222http://www.astro.keele.ac.uk/jkt/debcat/ catalogue of
dEBs (Southworth 2015) currently lists approximately 300 examples with
precisely-measured masses and radii, but only a small fraction of high-mass
systems have useful constraints on their photospheric chemical abundances
(Serenelli et al. 2021). In this study, we aim to enlarge the sample of high-
mass SB2 dEBs with accurately determined surface chemical abundance patterns.
In Sections 2 and 3, we present the sample and high-quality spectroscopic data
used in this study. Section 4 covers the determination of the spectroscopic
orbits of the stars, Sections 5 and 6 the inference of their atmospheric
parameters and chemical abundances, and Section 7 the light curve analysis.
The chemical compositions, ages and distances to the binary stars analysed are
compared in Section 8 to the properties of their parent clusters. We finish
with a discussion (Section 9) and conclusions (Section 10). It is important to
state that the various analyses presented in this work were performed
iteratively to ensure internal consistency in the derived results.

##  2 Sample

We selected four main-sequence (MS) dEBs for study, based on the masses of the
components, their membership of open clusters or associations, and on their
visibility during the telescope time we were allocated. Basic information on
these targets is given in Table 1. All eight stars have a spectral type of
late-O or early-B and a surface gravity between 3.7 or 4.2 dex, so are in a
relatively early evolutionary phase. All systems except V573 Car reside in
eccentric orbits. All are confirmed members of open clusters, although we did
not impose any additional constraints on their ages and/or chemical
compositions from the cluster membership. All of our targets except GL Car
were included in the homogeneous sample of Tkachenko et al. (2020).

V1034 Sco is located in the core of the open cluster NGC 6231, which in turn
is near the centre of the Sco OB1 association. A detailed spectroscopic and
X-ray study was presented by Sana et al. (2003). Light curves have been
presented and analysed by Sana et al. (2005) and Bouzid et al. (2005). The
most recent analysis is that published by Rosu et al. (2022b), who determined
physical properties of the components and measured apsidal motion from our
spectra (retrieved from the ESO archive) and the light curves from Bouzid et
al. (2005).

Figure 1:  Light curves used in the current study from our reduction of data
from the TESS satellite. They have been normalised to zero magnitude for
display purposes.

GL Car is a dEB studied by the Copenhagen group (Giménez & Clausen 1986) using
$uvby$ photometry. The system shows a significant orbital eccentricity
($e=0.157$) and fast apsidal motion with $U=25.20\pm 0.02$ yr (Giménez &
Garcia-Pelayo 1983; Giménez & Clausen 1986; Wolf et al. 2008). We know of no
previous time-series spectroscopy of the system, so the current work provides
the first measurements of its physical properties.

V573 Car is one of the brightest stars in very young open cluster Trumpler 16,
although its membership is disputed (Kaltcheva & Georgiev 1993). Its
spectroscopic binary nature was found by Walborn (1982), and the discovery of
eclipses was made by Freyhammer et al. (2001) during a study of the nearby
massive binary system $\eta$ Carinae. Freyhammer et al. (2001) obtained
extensive $uvby$ photometry and, combined with radial velocities (RVs) from
Levato et al. (1991), determined the physical properties of the system.

V346 Cen contains early B-type components (Houk & Cowley 1975) in an orbit
with a significant eccentricity. Apsidal motion is present with a period of
$U=306\pm 4$ yr (Giménez et al. 1986a; Drobek et al. 2013). High-quality light
curves in the Strömgren $uvby$ system were obtained and analysed by the
Copenhagen group (Giménez et al. 1986a, b). The only full spectroscopic
dataset for this system is our own HARPS data, available through the ESO
archive, and which were already analysed by Mayer et al. (2016).

##  3 Observations

###  3.1 Spectroscopy

The spectra presented in this work were all taken in one observing run333ESO
proposal 083.D-0040(A), PI J. Southworth over the nights 2–7 April 2009 using
the High-Accuracy Radial-velocity Planet Searcher (HARPS) cross-dispersed
échelle spectrograph (Mayor et al. 2003) at the 3.6-m telescope at ESO La
Silla. HARPS achieves extreme RV precision due to a high mechanical stability,
being fed by two optical fibres, sited in a vacuum chamber, and calibrated by
a Th-Ar emission lamp. We operated HARPS in the high-efficiency EGGS mode,
which has a resolving power of $R=80\,000$, and used the second fibre to
obtain the sky background during each observation. Each spectrum consists of
72 orders incident on two CCDs, covering 3780–6900 Å with a gap at 5304–5337 Å
between the CCDs.

We reduced the spectra using semi-automatic IRAF444IRAF is distributed by the
National Optical Astronomy Observatory, which are operated by the Association
of the Universities for Research in Astronomy, Inc., under cooperative
agreement with the NSF. scripts. Reduction of the spectra included the
standard steps: bias subtraction, flat-field correction, spectral order
localisation, extraction, and wavelength calibration. Normalisation of
extracted spectral orders was performed by fitting ninth-order polynomial
functions to selected continuum points in the blaze function. Since the Balmer
lines cover up to three consecutive spectral orders, these were normalised by
interpolating the blaze functions from adjacent orders as described by Kolbas
et al. (2015). The HARPS blaze functions are very stable so the normalisation
and merging of even these difficult orders produced very satisfactory results.

###  3.2 Photometry

Our analysis below originally relied on published ground-based light curves,
as will be discussed in Section 7. In the course of this work, additional data
became available from the NASA Transiting Exoplanet Survey Satellite (TESS), a
space-based mission that has observed most of the celestial sphere in sectors
of 27.4 d duration (Ricker et al. 2015). The TESS datasets used in the current
study are shown in Fig. 1. Additional data are shown in Fig. 17, and may be
useful in future for period or apsidal motion studies.

|
---|---
|
Figure 2:  Visualisation of the spectroscopic orbits of our targets. The best-
fitting orbits are shown with black lines and the RVs of the stars at the
times of observation with red symbols for the primary component, and blue
symbols for the secondary component. Note that these are not measured RVs,
hence the uncertainties in RVs are not assigned to individual symbols, because
we calculate orbital parameters directly from all observed spectra for each
system (see Section 4).

TESS observed V1034 Sco in sectors 12 (1800 s cadence) and 39 (600 s cadence).
We extracted the light curves using custom aperture masks. V1034 Sco is in a
crowded field and the TESS pixels subtend a large angle (21′′) so the light
curves contain a significant amount of third light. Our analysis was based on
sector 39 due to the better temporal sampling.

V346 Cen was observed using TESS in sectors 10 and 11 (1800 s cadence), and 37
and 38 (600 s cadence). The light curves available for download from
MAST555https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html are very
affected by the field crowding but are nevertheless much better than the
ground-based data for this object. We based our analysis on the data from
sectors 37 and 38.

TESS observed V573 Car in sectors 10, 36 and 37. Because this object is very
close to the extremely bright $\eta$ Car binary system, the standard data
products from TESS (Jenkins et al. 2016) are unreliable. We therefore
extracted photometry from the halo of V573 Car by making a careful customised
pixel selection for the aperture mask, with the aim to maximise the collected
flux of V573 Car compared to the flux of $\eta$ Car. The resulting light
curves are of relatively low quality and suffer from a large and varying
amount of third light, so we did not use these in our analysis. We note that
V573 Car was just outside the field of view of TESS during sector 11 but we
were still able to extract a light curve using halo photometry.

GL Car was observed by TESS in sectors 10 and 11 (1800 s cadence), and 37 (600
s cadence). The light curves available on MAST have eclipses that are too deep
so we again extracted our own photometry from the TESS full-frame images using
custom aperture masks.

##  4 Spectroscopic orbits

Table 2:  Parameters of the spectroscopic orbits for the four targets determined by SPD. Binary | $P$ | $T_{\rm peri}$ | $e$ | $\omega$ | $K_{\rm A}$ | $K_{\rm B}$ | $q$
---|---|---|---|---|---|---|---
system | (d) | (BJD) |  | (deg) | (km s1) | (km s1) |
V1034 Sco | 2.440656 | 51934.356$\pm$0.032 | 0.029 $\pm$ 0.003 | 191 $\pm$ 12 | 168.3 $\pm$ 0.3 | 299.0 $\pm$ 1.1 | 0.563 $\pm$ 0.002
GL Car | 2.422238 | 54901.182$\pm$0.015 | 0.146 (fixed) | 32.2 $\pm$ 2.7 | 244.6 $\pm$ 1.8 | 259.3 $\pm$ 1.6 | 0.943 $\pm$ 0.009
V573 Car | 1.469332 | - | 0.0 | 90 | 250.61 $\pm$ 0.71 | 306.3 $\pm$ 1.1 | 0.818 $\pm$ 0.011
V346 Cen | 6.321835 | 50452.543$\pm$0.016 | 0.289 $\pm$ 0.006 | 22.2 $\pm$ 1.3 | 135.3 $\pm$ 0.6 | 190.1 $\pm$ 0.7 | 0.712 $\pm$ 0.038

The spectra of binary systems containing high-mass stars are difficult to
analyse for several reasons. First, the $v\sin i$ values are typically large,
smearing out the spectral lines and causing the lines from the two components
to blend together even around the phases of maximum RV difference. Second,
there are relatively few spectral lines that are strong enough to provide
useful RV information. We therefore determined the spectroscopic orbits of the
stars using the method of spectral disentangling (SPD). This method was
introduced by Simon & Sturm (1994) in wavelength space and by Hadrava (1995)
in Fourier space. It represents the observed composite spectra of a binary
system as a sum of the individual spectra of the two stars shifted in RV
according to their orbital motion. SPD makes it possible to quantitatively
analyse time-series spectra of SB2 systems even when line blending is strong
(Hensberge et al. 2000; Pavlovski & Hensberge 2005). Note that no template
spectra are needed for SPD, thus avoiding any biases due to template mismatch
(Hensberge & Pavlovski 2007).

We used the FDBinary666http://sail.zpf.fer.hr/fd3 code (Ilijic et al. 2004) to
perform SPD in Fourier space using Fast Fourier Transform (FFT). For each
object we analysed all spectra simultaneously to determine the disentangled
spectra of the two stars and their spectroscopic orbital parameters. We fitted
directly for the orbital parameters, without the intermediate step of
calculating RVs. The orbital parameters were the orbital period, $P$, time of
periastron pasage, $T_{\rm peri}$, eccentricity, $e$, argument of periastron,
$\omega$, and velocity semiamplitudes, $K_{\rm A}$ and $K_{\rm B}$. The
orbital periods were held fixed as they are well determined from previous
analyses. The orbital solutions are given in Table 2 in which the mass ratio
($q=K_{\rm A}/K_{\rm B}$) is also given.

We also disentangled individual short segments of spectra in order to
concentrate on spectral lines of interest, avoid interstellar lines, and
achieve reasonable computation times. The Balmer lines were not used in the
determination of the spectroscopic orbits because they are much wider than the
changes in RV of the stars over an orbital cycle. The best fits were obtained
using the downhill simplex algorithm (Press et al. 1992). We found 100 runs
with 1000 iterations each to be sufficient to ensure the global minimum was
found whilst keeping the required computation time manageable. Convergence was
achieved quickly because of the high quality of the HARPS spectra and the
availability of preliminary orbital parameters from the literature.
Uncertainties in the results were obtained using 10 000 bootstrapping
simulations (Pavlovski et al. 2018). Fig. 2 is a visualisation of the
spectroscopic orbits of the four targets and the phase distribution of our
spectra.

###  4.1 V1034 Sco

V1034 Sco has been found to have a small eccentricity (Hill et al. 1974;
Levato & Morrell 1983). We have been able to measure precise velocity
amplitudes for the components (Table 2) which highlight the low mass ratio of
the system.

Our results are in good agreement with those from Sana et al. (2003, 2005) and
agree within the errorbars with those from Rosu et al. (2022b). We conclude
that the RV semi-amplitudes of the components of V1034 Sco are now well-
determined since the the accuracy achieved is about 0.2% for the primary and
0.4% for the secondary. The argument of periastron is quite uncertain due to
the small eccentricity, and is much better determined from the photometric
analysis in Section 7.

###  4.2 GL Car

To the best of our knowledge, our spectroscopic orbit for GL Car is the first
one published. The mass ratio is in fairly good agreement with the photometric
value of $q=0.943\pm 0.009$ found by Giménez & Clausen (1986). We fixed the
eccentricity to a value of 0.146, which is precisely known from analyses of
its apsidal motion (Wolf et al. 2008). We fitted for the argument of
periastron, which is well-determined when the eccentricity is fixed.

###  4.3 V537 Car

This is the only binary system with a circular orbit in our sample. The HARPS
spectra densely cover both quadratures. We also obtained spectra during the
primary and secondary minimum, but did not use these in our analysis because
the eclipses are not total.

A spectroscopic orbit for V573 Car has previously been published by Freyhammer
et al. (2001) but based on only two newly obtained spectra and eight spectra
taken from Levato & Malaroda (1982). The velocity amplitudes measured by these
authors are quite uncertain but agree with ours to within the errorbars.

###  4.4 V346 Cen

V346 Cen has a significant eccentricity of $e=0.289\pm 0.006$. Our HARPS
spectra have good phase coverage and SPD quickly converged to a stable
solution (Table 2). Our results are in reasonable agreement with the only
previous spectroscopic analysis of this system (Mayer et al. 2016), as
expected because they used the same spectra.

However, the velocity amplitudes we measured are both 1.5 km s-1 lower than
those of Mayer et al. (2016). We attribute this to differences in the methods
employed in the two analyses. In particular, Mayer et al. (2016) employed
cross-correlation (Zucker & Mazeh 1994) to determine RVs, using as templates
the disentangled spectra themselves. This approach is mathematically incorrect
and extensive numerical experiments have shown that it is not reliable (Ilijic
et al. 2001); SPD has instead been shown to be the best approach to
determining spectroscopic orbits (Southworth & Clausen 2007). Our approach
yields masses that are smaller by 0.26$\,{\rm M}_{\odot}$ and 0.11$\,{\rm
M}_{\odot}$ than those found by Mayer et al. (2016) for the primary and
secondary component of the system, respectively, which is larger than the
quoted uncertainties.

Figure 3:  Fits to the He i 4388 Å and He ii 4541 Å lines. The ionisation
balance of He i and He ii was used in the determination of $T_{\rm eff}$ for
the stars. The blue data are the disentangled spectra and the red lines the
best fits. The upper row is for the primary stars and the lower row is for the
secondary stars. The lower S/N for the secondary components arises because
they are fainter than the primary components. The absence of He ii 4541 Å
absorption in V1034 Sco B and V346 Cen B is obvious and indicates that $T_{\rm
eff}<23\,000$ K.

##  5 Atmospheric parameters

For determination of the atmospheric parameters and individual abundances of
C, N, O, Mg and Si, we employed a hybrid NLTE approach as described in detail
in Nieva & Przybilla (2007, 2012). A hybrid NLTE approach means that the
modelling combines hydrostatic, plane-parallel, and line-blanketed model
atmospheres in local thermodynamic equilibrium (LTE) with line formation
calculated in NLTE. We used the Atlas9 code (Kurucz 1979; Castelli & Kurucz
2003) for the calculations of model atmospheres. Then emergent fluxes and line
profiles were calculated with the codes Detail and Surface (Giddings 1980;
Butler & Giddings 1985). In Detail the coupled radiative transfer and
statistical equilibrium equations are solved, while Surface was used for the
calculations of NLTE synthetic spectra. The following model atoms were used in
these calculations: H i (Przybilla & Butler 2004), He i/ii (Przybilla 2005), C
ii/iii (Nieva & Przybilla 2006), N ii (Przybilla & Butler 2001), O i/ii
(Becker & Butler 1988; Przybilla et al. 2000), Mg ii (Przybilla et al. 2001),
and Si ii/iii/iv (Becker & Butler 1990).

Table 3:  The atmospheric parameters derived from optimal fitting of disentangled spectra of the components to a grid of NLTE spectra. Star | $T_{\rm eff}$ | $v\sin i$ | $\xi_{\rm t}$
---|---|---|---
| (K) | (km s-1) | (km s-1)
V1034 Sco A | 32 200 $\pm$ 500 | 169.8 $\pm$ 2.6 | 5 $\pm$ 1
V1034 Sco B | 25 800 $\pm$ 300 |  94.5 $\pm$ 3.3 | 5 $\pm$ 1
GL Car A | 30 950 $\pm$ 500 | 180.1 $\pm$ 2.2 | 4 $\pm$ 1
GL Car B | 30 400 $\pm$ 500 | 134.6 $\pm$ 3.5 | 2 $\pm$ 1
V573 Car A | 31 900 $\pm$ 400 | 184.6 $\pm$ 2.7 | 5 $\pm$ 1
V573 Car B | 28 700 $\pm$ 350 | 155.4 $\pm$ 3.1 | 3 $\pm$ 1
V346 Cen A | 26 100 $\pm$ 300 | 165.2 $\pm$ 2.8 | 5 $\pm$ 1
V346 Cen B | 22 500 $\pm$ 300 |  89.1 $\pm$ 2.3 | 5 $\pm$ 1
Table 4:  Abundances determined for the stars in our sample of binary systems.

Star | $\log\epsilon({\rm C})$ | $\log\epsilon({\rm N})$ | $\log\epsilon({\rm O})$ | [N/C] | [N/O] | $\log\epsilon({\rm Mg})$ | $\log\epsilon({\rm Si})$
---|---|---|---|---|---|---|---
V1034 Sco A | 8.39 $\pm$ 0.12 | 7.71 $\pm$ 0.12 | 8.76 $\pm$ 0.07 | $-0.68\pm 0.17$ | $-1.05\pm 0.14$ | 7.67 $\pm$ 0.14 | 7.56 $\pm$ 0.01
V1034 Sco B | 8.27 $\pm$ 0.05 | 7.67 $\pm$ 0.08 | 8.69 $\pm$ 0.12 | $-0.57\pm 0.09$ | $-1.02\pm 0.14$ | 7.45 $\pm$ 0.07 | 7.46 $\pm$ 0.14
GL Car A | 8.18 $\pm$ 0.08 | 7.69 $\pm$ 0.14 | 8.74 $\pm$ 0.12 | $-0.67\pm 0.16$ | $-1.04\pm 0.18$ | 7.52 $\pm$ 0.12 | 7.50 $\pm$ 0.12
GL Car B | 8.21 $\pm$ 0.12 | 7.72 $\pm$ 0.11 | 8.76 $\pm$ 0.12 | $-0.52\pm 0.16$ | $-0.85\pm 0.16$ | 7.50 $\pm$ 0.13 | 7.44 $\pm$ 0.14
V573 Car A | 8.30 $\pm$ 0.08 | 7.63 $\pm$ 0.10 | 8.67 $\pm$ 0.05 | $-0.57\pm 0.12$ | $-0.52\pm 0.11$ | 7.58 $\pm$ 0.08 | 7.57 $\pm$ 0.12
V573 Car B | 8.28 $\pm$ 0.05 | 7.76 $\pm$ 0.07 | 8.61 $\pm$ 0.04 | $-0.55\pm 0.09$ | $-0.94\pm 0.08$ | 7.45 $\pm$ 0.05 | 7.54 $\pm$ 0.13
V346 Cen A | 8.13 $\pm$ 0.05 | 7.68 $\pm$ 0.05 | 8.70 $\pm$ 0.04 | $-0.45\pm 0.07$ | $-1.02\pm 0.06$ | 7.70 $\pm$ 0.13 | 7.45 $\pm$ 0.16
V346 Cen B | 8.33 $\pm$ 0.06 | 7.72 $\pm$ 0.09 | 8.80 $\pm$ 0.07 | $-0.61\pm 0.11$ | $-1.08\pm 0.11$ | 7.40 $\pm$ 0.14 | 7.35 $\pm$ 0.17
This work | 8.27 $\pm$ 0.08 | 7.69 $\pm$ 0.05 | 8.70 $\pm$ 0.06 | $-0.58\pm 0.07$ | $-1.01\pm 0.07$ | 7.59 $\pm$ 0.14 | 7.49 $\pm$ 0.08
OB binariesaa$a$The abundances found for OB binaries in our previous work (Pavlovski et al. 2018) | 8.25 $\pm$ 0.07 | 7.69 $\pm$ 0.06 | 8.71 $\pm$ 0.05 | $-0.56\pm 0.08$ | $-1.02\pm 0.07$ | 7.56 $\pm$ 0.12 | 7.45 $\pm$ 0.09
B starsbb$b$The ‘present-day cosmic abundances’ for B stars (Nieva & Przybilla 2012) | 8.33 $\pm$ 0.04 | 7.79 $\pm$ 0.04 | 8.76 $\pm$ 0.05 | $-0.54\pm 0.06$ | $-0.97\pm 0.06$ | 7.56 $\pm$ 0.05 | 7.50 $\pm$ 0.05

777 The $T_{\rm eff}$ and $\log g$ values used for the construction of the
model atmospheres are given in Tables 3 and 7, respectively.

We used the disentangled spectra generated in the previous section to
determine the $T_{\rm eff}$, $v\sin i$, and microturbulent velocity ($\xi_{\rm
t}$) for each of the eight stars in our sample. This process was greatly
helped by the availability of $\log g$ values from the measured masses and
radii (see Section 7) so our analysis was performed iteratively. The
disentangled spectra were still in the common continuum of the binary system
so needed to be renormalised to the continuum of the individual component
stars. This was done iteratively alongside the light curve analysis, to arrive
at light ratios that were consistent between the two types of the analysis
(Ilijic et al. 2004; Pavlovski & Hensberge 2005).

The exception to the process above was GL Car, for which the light curve
solutions suffered from a degeneracy which caused the light ratio to be highly
uncertain. We therefore fitted the disentangled spectra to obtain the $T_{\rm
eff}$, $v\sin i$ and $\log g$ values and the light ratio directly, using the
approach of Tamajo et al. (2011) and Kolbas et al. (2015). After iteration
with the light curve solution, $\log g$ was fixed for the final measurements
of the remaining parameters. We have found that such spectroscopically
determined light ratios can be competitive with those from light curve
analysis (Pavlovski et al. 2009, 2018, 2022).

Since we are dealing with late-O, and early-B type stars, the helium
ionisation balance (He i/He ii) is a sensitive indicator of $T_{\rm eff}$. Our
spectra cover a broad spectral range and thus allowed us to use a large number
of lines: 4009, 4026, 4388, 4437, 4471, 4713, 4921, 5015, 5047, 5875 and 6678
Å for He i and 4200, 4541, 4686, and 5411 Å for He ii. Once a first set of
parameters was obtained, we made the light ratio a free parameter to check its
reliability. As a further check we also fitted the H$\delta$, H$\gamma$ and
H$\beta$ lines, during which we excluded wavelengths affected by interstellar
absorption (specifically the red wing of H$\beta$). We did not base our
$T_{\rm eff}$ measurements on the Balmer lines because their large width makes
them susceptible to errors due to continuum normalisation.

The He line strengths also depend on $\xi_{\rm t}$, which for hot stars can be
obtained by minimising the scatter in the O abundances. We started with the
assumptions of solar He abundance and $\xi_{\rm t}=2$ km s-1, and subsequently
relaxed each of them before refitting. Convergence was fast, taking either one
or two iterations for all eight stars. Once this was achieved, we repeated our
optimal fitting of the disentangled spectra described above. The results of
this process are given in Table 3. Below we compare our results to published
determinations for each system, except for GL Car for which there is no other
analysis based on modern spectroscopy.

###  5.1 V1034 Sco

In the most recent study, Rosu et al. (2022b) analysed disentangled spectra of
the components obtained from the HARPS spectra obtained in our observing run,
and available at the ESO archive. The $T_{\rm eff}$s they derived are within
1$\sigma$ uncertainty of our results. This is encouraging, especially as Rosu
et al. (2022b) used a different NLTE spectrum synthesis code to us.

Rosu et al. (2022b) also fitted for surface gravity, using the wings of the
Balmer and some He i lines, whereas we prefer the surface gravities determined
with a high precision from the masses and radii. The two analyses agree to
within 2$\sigma$, but the uncertainties of the values from Rosu et al. (2022b)
are much larger ($\pm$0.10 dex) than our own ($\pm$0.01 dex).

###  5.2 V573 Car

Freyhammer et al. (2001) studied V573 Car using two high-resolution spectra
from the FEROS spectrograph, taken near opposite quadratures. They fitted the
spectra with NLTE synthetic spectra for the He i, He ii, H$\delta$ and
H$\gamma$ lines. The helium lines of the components are not completely
resolved at quadrature due to the high $v\sin i$, and the Balmer lines are not
resolved.

The agreement between their and our $T_{\rm eff}$ measurements is well within
the 1$\sigma$ errorbars for component A, but only within 2$\sigma$ for
component B. We attribute this to the very small number of spectra available
to Freyhammer et al. (2001) compared to our own extensive dataset. We also
find that fitting disentangled spectra is superior to fitting individual
observed spectra because it avoids problems with blending of lines from the
two stars. Moreover, disentangled spectra have a higher S/N than individual
observed spectra.

###  5.3 V346 Cen

The atmospheric parameters for both components of V346 Cen were determined by
Mayer et al. (2016) from the same set of the HARPS spectra as we used. In
their analysis, Mayer et al. (2016) used optimal fitting of disentangled
spectra in similar manner as we did, for fixed surface gravities and
microturbulent velocities (fixed to $\xi_{\rm t}=2$ km s-1) and using a grid
of synthetic spectra from Lanz & Hubeny (2007). They found a large discrepancy
for the secondary: their spectroscopic analysis gave $20\,991\pm 190$ K and
their light curve analysis gave $25\,376\pm 18$ K. The former value is in much
better agreement with our result (Table 3), and the complete absence of the He
ii lines demands $T_{\rm eff}<23\,000$ K (based on a detailed examination of
theoretical spectra for the He ii 4686 Å line). Mayer et al. (2016) did not
discuss the He ii lines in the secondary’s spectrum at all. They also gave
unrealistically small uncertainties for $T_{\rm eff}$: $\pm$25 K for the
primary, $\pm$190 K for the secondary from spectroscopy, and $\pm$18 K for the
secondary from the light curve analysis. Such low uncertainties are typical
for formal errors of the fitting algorithms, but are unrealistic.

##  6 Abundance analysis

Figure 4:  Example of fits to N lines for our target stars. In this case the N
ii 3995 Å line is shown for V1034 Sco (primary star on the left, secondary
star on the right). The blue lines show the renormalised disentangled spectra
of the stars. The red lines show synthetic spectra from our precalculated grid
for three different abundances (labelled on the bottom right corner in each
panel).

Figure 5:  Same as Fig. 4 but for O lines in the components of GL Car. The
complex blend of O ii lines at 4941 and 4943 and the O ii line at 4955 Å are
shown.

Figure 6:  Same as Fig. 4 but for lines in the components of V573 Car. The
upper two panels show O ii lines at 4661, 4673 and 4676 Å (shown with green
lines), which are blended with C ii lines at 4659, 4663, 4665 and 4673 Å
(shown with red lines). The calculated synthetic spectrum is shown using a
black line. The lower two panels show the N ii 3995 Å lines.

Figure 7:  Same as Fig. 4 but for V346 Cen. The upper two panels show the C ii
lines at 5133–5151 Å, and the bottom two panels show the N ii 4630.5 Å line
for the two components.

With the $T_{\rm eff}$, $\xi_{\rm t}$ and $v\sin i$ from Section 5, and $\log
g$ from the masses and radii of the stars (Section 7), we have all quantities
needed for determining surface abundances. We calculated model atmospheres for
the $T_{\rm eff}$ and $\log g$ values of the components with the atlas9 code.
Then a grid of synthetic spectra was calculated in NLTE with detail, and
surface. The following species were considered: C, N, O, Mg and Si. Spectra
for a broad range of elemental abundances were calculated, spanning $\pm 0.05$
dex in steps of 0.05 dex, around the ‘present-day cosmic abundances’
determined in Nieva & Przybilla (2012) ($\log\epsilon({\rm C})=8.25$,
$\log\epsilon({\rm N})=7.69$, $\log\epsilon({\rm O})=8.71$, $\log\epsilon({\rm
Mg})=7.56$, and $\log\epsilon({\rm Si})=8.45$). These were broadened by the
instrumental broadening, and a rotational kernel. The microturbulent velocity
was taken into account in the line profile calculations with surface as
determined from minimising the scatter in the O abundances and given in Table
3. Abundances were determined by minimising the residuals ($\chi^{2}$
criterion) between the renormalised disentangled spectrum and the synthetic
spectrum. In the renormalisation of the disentangled spectra to their
individual continuum, the light ratio obtained in the light curve analysis
(Table 6) was used, except for GL Car where the spectroscopically determined
light ratio was employed.

The number of lines available for the abundance determination of a particular
element varies due primarily to $T_{\rm eff}$. For the $T_{\rm eff}$ range
covered by our target stars, the spectral lines of CNO are quite varied. The
most numerous spectral lines are for O, which is why we used them to determine
$\xi_{\rm t}$. The spectral lines of C are the least numerous, and the broad
wavelength coverage of HARPS spectra is of vital importance. We show examples
of the disentangled and synthetic spectra in Figs. 4 to 7 for selected CNO
lines.

The results for all five elements are given in Table 4, as well as the indices
[N/C] and [N/O]. Uncertainties were calculated including the standard
deviation of the mean for available spectral lines, and the uncertainty due to
uncertainties in $T_{\rm eff}$ and $\xi_{\rm t}$. Uncertainty in the
abundances due to uncertainties in the surface gravity are negligible since
$\log g$ is determined to high precision from the masses and radii.

Figure 8:  Individual abundances of carbon (left), nitrogen (middle), and
oxygen (right) as a function of surface gravity. The stars in the present
sample are indicated with filled blue circles while stars taken from our
previous studies are shown with open blue circles. The surface gravity
(obtained from the binary solution) is used as a proxy for stellar evolution.
Solid red lines show theoretical evolutionary tracks for a 15$\,{\rm
M}_{\odot}$ star and three values of the initial rotational velocity
$\Omega/\Omega_{\rm crit}$ = 0.1, 0.3, and 0.5 (Georgy et al. 2013). The
cosmic standard abundance values of Nieva & Przybilla (2012) are indicated
with horizontal dashed lines.

Figure 9:  Same as in Fig. 8 but for the [N/C] and [N/O] abundance indices.
Table 5:  Comparison of abundances determined for targets in present work to
their parent clusters.

NGC 6231 |  |  |
---|---|---|---
Element | Kilian et al. | Mathys et al. | This work
| (1994) | (2002) | V1034 Sco
$\log\epsilon({\rm C})$ | 8.37$\pm$0.05 | 8.29$\pm$0.17 | 8.33$\pm$0.08
$\log\epsilon{\rm(N)}$ | 7.85$\pm$0.05 | 7.85$\pm$0.10 | 7.69$\pm$0.08
$\log\epsilon({\rm O})$ | 8.61$\pm$0.05 | 8.30$\pm$0.42 | 8.73$\pm$0.12
$\log\epsilon({\rm Mg})$ | 7.39$\pm$0.04 | – | 7.56$\pm$0.11
$\log\epsilon({\rm Si})$ | 7.15$\pm$0.06 | – | 7.51$\pm$0.10
$[{\rm N/C}]$ |  $-$0.52$\pm$0.07 |  $-$0.44$\pm$0.20 |  $-$0.64$\pm$0.11
$[{\rm N/O}]$ |  $-$0.76$\pm$0.07 |  $-$0.45$\pm$0.40 |  $-$1.04$\pm$0.14
NGC 3293 |  |  |
Element | Hunter et al. | Morel et al. | V573 Car
| (2009) | (2022) | GL Car
$\log\epsilon({\rm C})$ | 7.97$\pm$0.19 | 8.13$\pm$0.16 | 8.24$\pm$0.17
$\log\epsilon{\rm(N)}$ | 7.60$\pm$0.15 | 7.72$\pm$0.14 | 7.70$\pm$0.27
$\log\epsilon({\rm O})$ | 8.65$\pm$0.17 | – | 8.70$\pm$0.18
$\log\epsilon({\rm Mg})$ | 7.22$\pm$0.16 | 7.45$\pm$0.18 | 7.50$\pm$0.20
$\log\epsilon({\rm Si})$ | 7.42$\pm$0.09 | 7.56$\pm$0.25 | 7.51$\pm$0.26
$[{\rm N/C}]$ |  $-$0.37$\pm$0.21 |  $-$0.40$\pm$0.21 |  $-$0.54$\pm$0.27
$[{\rm N/O}]$ |  $-$1.05$\pm$0.26 | – |  $-$1.00$\pm$0.11
888 V1034 Sco is member of the open cluster NGC 6231 for which abundance
analyses were published by Kilian et al. (1994) and Mathys et al. (2002). For
the open clusters NGC 3572 and Trumpler 16, parent clusters of GL Car and V573
Car, respectively, no abundance studies are available. We used abundance
studies of the open cluster NGC 3293, since it is part of the large Car OB1
complex, as are NGC 3572 and Trumpler 16.

A fairly good agreement between the abundances in both components of the same
binary system is seen from examination of Table 4. The most notable difference
is for the abundance of magnesium (Mg) which in three cases (V1034 Sco, V573
Car and V346 Cen) is modestly larger than the uncertainties. The Mg abundances
are inferred from a single line, Mg ii 4481 Å, so their uncertainties are
larger than for other species. For other elements, the observed abundance
differences are mostly below 0.1 dex, well within the 1$\sigma$ uncertainty
interval. Apart from Mg, the largest deviations are for the C abundance in
V346 Cen ($\log\epsilon({\rm C})_{\rm A}-\log\epsilon({\rm C})_{\rm
B}=-0.20\pm 0.08$ dex), the N abundance in V573 Car ($-0.13\pm 0.12$), and the
O abundance in V346 Cen ($-0.10\pm 0.08$). V346 Cen A has the lowest C
abundance among the eight stars, with $\log\epsilon(C)=8.13\pm 0.05$, almost
0.20 dex less than the mean C abundance. Contrary to this, the N abundance for
the same star is normal. It is also worth noting that V346 Cen A is the most
evolved in our sample of eight OB stars.

Fig. 8 compares individual CNO abundances determined for the eight stars in
this work to our previous abundance measurements in high-mass binaries. The
new determinations are shown in solid blue circles, whilst our previous
results are represented with open blue circles. Our previous determinations
are for 17 high-mass stars in nine binary systems: V578 Mon (Pavlovski &
Hensberge 2005; Garcia et al. 2014; Pavlovski et al. 2018), V453 Cyg
(Pavlovski & Southworth 2009; Pavlovski et al. 2018), V380 Cyg (Pavlovski et
al. 2009; Tkachenko et al. 2014b), $\sigma$ Sco (Tkachenko et al. 2014a),
$\alpha$ Vir (Tkachenko et al. 2016), CW Cep (Johnston et al. 2019), AH Cep
(Pavlovski et al. 2018), V478 Cyg (Pavlovski et al. 2018) and the primary
component in V621 Per (Southworth et al. in prep.). There are no discernable
systematics between the new and previous sample. This is expected because we
are consistently using the same reduction and analysis tools, so the 25 stars
(in 13 binary systems) represent an homogeneous sample. In the last three rows
of Table 4 the mean values for abundances in the present sample, in OB
binaries studied previously by us, and the ‘present cosmic abundance standard’
– an abundance pattern evaluated for B-type stars by Nieva & Przybilla (2012)
– are given for comparison. As already mentioned, both our samples are in
perfect agreement and there are no outliers. However, it can be seen that the
CNO abundances for OB stars in binary systems are below the cosmic abundance
standard, with very few exceptions. Nieva & Przybilla (2012) determined
elemental abundances for sample of sharp-lined early B-type stars, enabling a
very high accuracy. This was not an option for our work because our sample
stars are all in short-period binary systems so are either moderate or fast
rotators. Thus the spectral lines are usually broad and overlapping, making
the choice of suitable spectral lines for abundance determination more
limited, and thus affecting the accuracy of the results.

Fig. 8 presents a comparison of the inferred CNO abundances with theoretical
evolutionary tracks of a 15$\,{\rm M}_{\odot}$ star computed for three values
of the initial rotational velocity $\Omega/\Omega_{\rm crit}=0.1$, 0.3 and 0.5
(Georgy et al. 2013). In their model calculations Georgy et al. (2013) used
the following initial abundances: $\log\epsilon({\rm C})=8.28$,
$\log\epsilon({\rm N})=8.67$, $\log\epsilon({\rm O})=8.55$. The values for the
abundances of C and N are in fair agreement with our present and preious
findings (c.f. Table 5, but differ by 0.15 dex for the O abundance. Therefore,
we empirically ‘corrected’ the initial O abundance in the theoretical models,
and shift the O abundance upwards in Fig. 8, and accordingly for [N/O] in Fig.
9.

One can see that the models predict significant depletion and enhancement of C
and N, respectively, as the rotation rate of the star increases, while only a
marginal depletion is predicted for O. Moreover, these abundance trends are
substantial at the start of the main-sequence evolution already. However, the
individual abundances of CNO elements measured by us do not follow the
relations predicted by the models: instead we observe a scatter of values
around or slightly below the cosmic standard abundance values of Nieva &
Przybilla (2012).

Furthermore, the N to C abundance ratio index [N/C] is a sensitive probe of
the stellar evolution model predictions, as can be seen in Fig. 9. The models
suggest a noticeable increase of the surface N abundance with respect to the
abundance of C (top panel) and O (bottom panel) as the rotation rate of the
star $\Omega/\Omega_{\rm crit}$ increases. Similar to the individual elemental
abundances discussed above, we do not observe the increase in the [N/C] and
[N/O] indices as the surface gravity of the star decreases. Moreover, the bulk
of our abundance measurements cluster around the mean [N/C] and [N/O] values
found by Nieva & Przybilla (2012) in the solar neighbourhood, with the spread
being significantly smaller than one would expect if the abundance ratios were
altered substantially by the effect of stellar rotation.

No previous abundance determinations are available for any of the binary
systems analysed in this work. However, we can check our results against
published abundances for the open clusters our sample are members of.
Photospheric abundances for B-type stars in the open cluster NGC 6231 were
determined by Kilian et al. (1994) and Mathys et al. (2002). Results from
these studies are compared to our results for V1034 Sco in Table 5. The parent
clusters for the other three systems have not (yet) been subject to a chemical
composition study, but the open cluster NGC 3293 (which is part of young
association Car OB1 Turner et al. 1980) is well studied. Hunter et al. (2009)
determined abundances from 50 B-type stars, while in a recent publication
Morel et al. (2022) examined a large sample of about 150 B-type stars in the
framework of the Gaia-ESO Survey. Since the dEBs V573 Car and GL Car belongs
to the Car OB1 association, we use NGC 3293 as a proxy for the abundance
pattern in Car OB1. It is interesting that the massive sample of B-type stars
analysed in Morel et al. (2022), with a spread in $v\sin i$ values, show a
pattern of under-abundances compared to the standard solar abundances (Asplund
et al. 2009), in accordance with our general abundance pattern.

##  7 Light curve analysis

We assembled the available light curves of the four targets in this work and
modelled them using a consistent approach in order to determine their physical
properties. The light curves were fitted using the Wilson-Devinney (WD) code
(Wilson & Devinney 1971; Wilson 1979), which implements Roche geometry to
determine the shapes of the stars and thus the brightness of binary systems as
a function of orbital phase. We used the 2004 version of the WD code, driven
with the jktwd wrapper (Southworth et al. 2011).

For each system we performed a series of tests to determine the best approach
to modelling it with jktwd. Once we had arrived at the preferred solution, we
performed further tests to determine the range of plausible solutions and thus
the uncertainties in the fitted parameters. This step was taken because we
have consistently found that the formal errorbars calculated by the WD code
underestimate the true uncertainty of the fitted parameters (Pavlovski &
Southworth 2009; Pavlovski et al. 2009, 2018; Southworth et al. 2020), as
indicated in the user guide to the code (Wilson & Van Hamme 2004).

Unless otherwise specified we used Mode 0 in the WD code, which is for
detached binary systems where the light contributions for each star are fitted
individually, simple reflection, and the logarithmic limb darkening (LD) law.
We fitted for the potentials and light contributions of the two stars, the
orbital inclination and a phase shift with respect to the adopted orbital
ephemeris. The mass ratio was fixed at the spectroscopic value, bolometric
albedos were set to 1.0, synchronous rotation was assumed, and the gravity
brightening exponents were set to 1.0. A circular orbit was assumed for V573
Car but the possibility of an eccentric orbit was checked. The input LD
coefficients were obtained by bilinear interpolation in the tables of van
Hamme (1993).

For the purposes of determining the uncertainties in the fitted parameters, we
ran a series of alternative solutions for differing choice of WD code mode of
operation, choice of numerical resolution, treatment of reflection, choice of
LD law, whether the LD coefficients were fixed or fitted, treatment of third
light, variation of the mass ratio within the uncertainties, and the
possibility of orbital eccentricity (for V573 Car). We also considered the
effects of albedo, rotational velocity and gravity brightening, by fixing them
at different values and also attempting to fit for them directly.

The net result of this process was a default solution for each system,
accompanied by a measurement of how much each fitted parameter changed between
this default solution and each of the alternative solutions. These changes
were then added in quadrature to arrive at a final robust uncertainty value
for each fitted parameter. The results for all four systems are summarised in
Table 6. The fractional radii are volume-equivalent values obtained from the
lc flavour of the WD code.

Table 6:  Summary of the parameters for the wd2004 solutions of the light
curves of the systems.

Parameter |  wd2004 name | V1034 Sco | GL Car | V573 Car | V346 Cen
---|---|---|---|---|---
Control and fixed parameters: |  |  |  |  |
wd2004 operation mode | mode | 0 | 0 | 0 | 0
Treatment of reflection | mref | 1 | 1 | 1 | 1
Number of reflections | nref | 1 | 1 | 1 | 1
Limb darkening law | ld | 2 (logarithmic) | 2 (logarithmic) | 1 (linear) | 2 (logarithmic)
Numerical grid size (normal) | n1, n2 | 60 | 50 | 50 | 50
Numerical grid size (coarse) | n1l, n2l | 50 | 40 | 40 | 40
Fixed parameters: |  |  |  |  |
Orbital period (d) | period | 2.440646 | 2.4222681 | 1.4693316 | 6.3220088
Primary eclipse time (HJD) | hjd0 | 2451931.2652 | 2459321.2994 | 2450456.8164 | 2459335.5607
Mass ratio | rm | 0.563 | 0.943 | 0.818 | 0.712
$T_{\rm eff}$ star A (K) | tavh | 32 200 | 30 960 | 31 900 | 26 100
$T_{\rm eff}$ star B (K) | tavh | 25 800 | 30 390 | 28 700 | 22 500
Rotation rates | f1, f2 | 1.0, 1.0 | 1.67, 1.29 | 1.0, 1.0 | 2.49, 2.70
Gravity darkening | gr1, gr2 | 1.0, 1.0 | 1.0, 1.0 | 1.0, 1.0 | 1.0, 1.0
Bolometric albedos | alb1, alb2 | 1.0, 1.0 | 1.0, 1.0 | 1.0, 1.0 | 1.0, 1.0
Fitted parameters: |  |  |  |  |
Phase shift | pshift |  $-$0.0035 | 0.0413 | 0.0008 | 0.0702
Star A potential | phsv | $3.670\pm 0.044$ | $5.736\pm 0.033$ | $3.913\pm 0.024$ | $5.933\pm 0.015$
Star B potential | phsv | $4.228\pm 0.031$ | $5.766\pm 0.064$ | $4.106\pm 0.027$ | $8.012\pm 0.032$
Orbital inclination (°) | xincl | $81.80\pm 0.32$ | $86.57\pm 0.17$ | $80.52\pm 0.14$ | $84.97\pm 0.12$
Orbital eccentricity | e | $0.027\pm 0.013$ | $0.1465\pm 0.0004$ | 0.0 (fixed) | $0.2750\pm 0.0006$
Argument of periastron (°) | perr0 | $57\pm 18$ | $22.47\pm 0.36$ |  | $27.53\pm 0.28$
Light from star A ($u$ band) | hlum |  |  | 8.006 |
Light from star B ($u$ band) | clum |  |  | 4.269 |
Light from star A ($v$ band) | hlum |  |  | 7.861 |
Light from star B ($v$ band) | clum |  |  | 4.449 |
Light from star A ($b$ band) | hlum |  |  | 7.893 |
Light from star B($b$ band) | clum |  |  | 4.531 |
Light from star A ($y$ band) | hlum |  |  | 7.888 |
Light from star B ($y$ band) | clum |  |  | 4.551 |
Light from star A (TESS band) | hlum | $8.37\pm 0.10$ | $6.096\pm 0.081$ |  | $8.374\pm 0.086$
Light from star B (TESS band) | clum | $1.73\pm 0.08$ | $5.204\pm 0.102$ |  | $1.967\pm 0.006$
Third light (TESS band) | el3 | $0.239\pm 0.012$ | $0.166\pm 0.006$ |  | $0.214\pm 0.007$
Fractional radius of star A |  | $0.3300\pm 0.0032$ | $0.2203\pm 0.0015$ | $0.3308\pm 0.0025$ | $0.2088\pm 0.0014$
Fractional radius of star B |  | $0.1901\pm 0.0022$ | $0.2088\pm 0.0017$ | $0.2759\pm 0.0029$ | $0.1106\pm 0.0008$

999 Detailed descriptions of the control parameters can be found in the WD
code user guide (Wilson & Van Hamme 2004). A and B refer to the primary and
secondary stars, respectively. Uncertainties are only quoted when they have
been robustly assessed by comparison with a full set of alternative solutions.

###  7.1 V1034 Sco

Figure 10:  The light curves and best WD models for V1034 Sco. The
differential magnitudes are plotted versus orbital phase and are colour-coded
according to the central wavelengths of the passbands. The source and passband
of each light curve is labelled. The residuals of the fit are shown at the
base of the figure. Offsets have been applied between the light curves and
residuals for clarity.

Two photometric studies of V1034 Sco have been published. Bouzid et al. (2005)
presented light curves taken in the Strömgren $uvby$ filters, with 409, 645,
1058 and 1036 datapoints, respectively. Sana et al. (2005) obtained light
curves in two narrow-band filters, designated $\lambda$4685 and $\lambda$6051,
containing 112 and 138 datapoints, respectively. For our exploratory solutions
we used the Sana et al. (2005) data as the Bouzid et al. (2005) data are not
available.

In the course of this work a new light curve became available from sector 39
of the TESS satellite (see Section 3). As the TESS data are of much higher
quality than the other photometry, we have based our final results for V1034
Sco on these data. Before doing so, we performed a preliminary fit with
jktebop (Southworth 2013) to obtain an orbital ephemeris then phase-binned
these data down to 500 bins to decrease the computation time. Our final
solution is for an eccentric orbit, including third light, and the logarithmic
LD law and the linear LD coefficient fitted for each star and passband. The
main contributors to the uncertainty in the fractional radii are the treatment
of albedo and gravity darkening. Uncertainties arising from the choice of
numerical resolution, WD program mode, rotation rate and LD were all
significantly smaller and therefore contributed negligibly when all
uncertainties for each parameter were added in quadrature. The parameters and
their uncertainties are given in Table 6, and the best fits are shown in Fig.
10.

###  7.2 GL Car

Figure 11:  The light curves and best WD models for GL Car. Other comments are
the same as for Fig. 10.

Light curves of GL Car in the Strömgren $uvby$ system were obtained by Giménez
et al. (1985) using the 0.5 m Strömgren Automated Telescope at ESO La Silla.
They comprise 526 observations through each filter, 234 in the 1982 observing
season and 308 in the 1983 season. These data were analysed by Giménez &
Clausen (1986) using the wink code (Wood 1971). The observations were obtained
in electronic form from the archive of J. V. Clausen and used in the current
work to obtain a preliminary solution. We found values and uncertainties for
the fitted parameters in good agreement with those from Giménez & Clausen
(1986).

Subsequent to our analysis of the $uvby$ data a new light curve of GL Car
became available from TESS. We phase-binned this and modelled it using wd,
fitting for an eccentric orbit and third light. The rotation rates (F1 and F2)
were set to the ratios of the measured rotational velocities (Table 3) and the
synchronous values, determined iteratively. Unlike the $uvby$ data, the TESS
light curve shows a strong correlation between the light ratio and the amount
of third light. We therefore applied the light ratio from our spectroscopic
analysis. Because there is no mechanism to explicitly apply a spectroscopic
light ratio in wd2004 we propagated the light ratio from our spectral interval
(which corresponds closely to the Johnson $B$ band) to the Strömgren $uvby$
bands (see Southworth 2010) using atlas9 theoretical spectra (Castelli et al.
1997) and passband response functions from Maíz Apellániz (2006). We forced
wd2004 to match them by fixing the hlum parameters at the appropriate values.
Including this constraint greatly improved the reliability of the results.

We find precise fractional radii for GL Car once our spectroscopic light ratio
is included (Table 6). The uncertainties are dominated by those from this
light ratio, but are still below 1% and a factor of three smaller than those
from the $uvby$ data alone. They also agree well with the less precise results
from Giménez & Clausen (1986). The fitted orbital eccentricity is in excellent
agreement with that from its apsidal motion ($e=0.1459\pm 0.0015$ from Wolf et
al. 2008). The orbital phase of secondary eclipse has changed a lot between
the $uvby$ and TESS datasets, and the different morphology of the light curve
is obvious (see Fig. 11).

###  7.3 V573 Car

Figure 12:  The light curves and best WD models for V573 Car. Other comments
are the same as for Fig. 10.

V573 Car was studied by Freyhammer et al. (2001) using the Dutch 0.9 m
telescope at ESO La Silla. A total of 1910 observations were obtained through
the Strömgren filters: 763 in $y$, 513 in $b$, 350 in $v$ and 284 in $u$. We
fitted all four light curves simultaneously, using the ephemeris from
Freyhammer et al. (2001). We assumed a circular orbit in most cases, but did
run a fit with $e$ and $\omega$ free to check if this led to a better fit to
the data (it didn’t). We also assumed no third light, after attempts to fit
for it had a negligible effect on the results and also led to a slightly
negative value for this parameter. The best fit is shown in Fig. 12.

The fitted parameters were the potentials of the two stars, the orbital
inclination, a phase shift, and the light contributions of the two stars in
each passband. To avoid very small values for the light contributions we
renormalised each light curve to be at approximately zero relative magnitude
at quadrature. We adopted the linear LD law as it gave results very similar to
those for the logarithmic and square-root laws; attempts to fit for the LD
coefficients led to unphysical solutions. The rotational velocities of the
stars were held to the synchronous values.

We found that the solution of the light curves is degenerate in that
significantly different values of the ratio of the radii or the light
contributions of the stars led to almost indistinguishable fits. This was also
found by Freyhammer et al. (2001), who constrained their solution using a
light ratio measured from their spectra. We took the same approach.

For our final result (Table 6) we give the solution for fitting all four light
curves simultanously, constrained by the spectroscopic light ratio. The
uncertainties in the parameters include contributions from the uncertainty in
the spectroscopic light ratio, the effect of a change of 5% in the rotation
velocities of the stars, and the treatment of albedo and gravity darkening.
Other sources of uncertainty (see above) were checked and found to be
negligible. We were able to measure the fractional radii of the stars to
precisions of 0.8% (star A) and 1.1% (star B); the main contribution to these
uncertainties is the spectroscopic light ratio (for star A) and the treatment
of gravity darkening (for star B). The values we find are in reasonable
agreement with those from Freyhammer et al. (2001), but our uncertainties are
slightly larger.

Table 7:  Physical properties measured for the four systems analysed in this
work.

Parameter | V1034 Sco | GL Car | V573 Car | V346 Cen
---|---|---|---|---
Mass ratio | 0.5628 $\pm$ | 0.0021 | 0.943 $\pm$ | 0.012 | 0.8182 $\pm$ | 0.0037 | 0.7119 $\pm$ | 0.0038
Mass of star A ($\mathcal{M}^{\rm N}_{\odot}$) | 17.01 $\pm$ | 0.14 | 15.86 $\pm$ | 0.31 | 15.11 $\pm$ | 0.13 | 11.74 $\pm$ | 0.12
Mass of star B ($\mathcal{M}^{\rm N}_{\odot}$) | 9.573 $\pm$ | 0.053 | 14.95 $\pm$ | 0.30 | 12.365 $\pm$ | 0.096 | 8.359 $\pm$ | 0.089
Semimajor axis ($\mathcal{R}^{\rm N}_{\odot}$) | 22.767 $\pm$ | 0.053 | 23.79 $\pm$ | 0.15 | 16.412 $\pm$ | 0.044 | 39.12 $\pm$ | 0.13
Radius of star A ($\mathcal{R}^{\rm N}_{\odot}$) | 7.513 $\pm$ | 0.075 | 5.242 $\pm$ | 0.048 | 5.429 $\pm$ | 0.043 | 8.278 $\pm$ | 0.079
Radius of star B ($\mathcal{R}^{\rm N}_{\odot}$) | 4.328 $\pm$ | 0.051 | 4.968 $\pm$ | 0.051 | 4.528 $\pm$ | 0.049 | 4.123 $\pm$ | 0.072
Surface gravity of star A ($\log$[cgs]) | 3.917 $\pm$ | 0.009 | 4.199 $\pm$ | 0.007 | 4.148 $\pm$ | 0.007 | 3.672 $\pm$ | 0.008
Surface gravity of star B ($\log$[cgs]) | 4.147 $\pm$ | 0.010 | 4.220 $\pm$ | 0.008 | 4.218 $\pm$ | 0.009 | 4.130 $\pm$ | 0.015
Synch. rotational velocity of star A ( km s-1) | 155.7 $\pm$ | 1.6 | 109.5 $\pm$ | 1.0 | 186.9 $\pm$ | 1.5 | 66.25 $\pm$ | 0.64
Synch. rotational velocity of star B ( km s-1) | 89.7 $\pm$ | 1.1 | 103.8 $\pm$ | 1.1 | 155.9 $\pm$ | 1.7 | 33.00 $\pm$ | 0.57
$T_{\rm eff}$ of star A (K) | 32200 $\pm$ | 500 | 30960 $\pm$ | 500 | 31900 $\pm$ | 400 | 26100 $\pm$ | 300
$T_{\rm eff}$ of star B (K) | 25800 $\pm$ | 300 | 30390 $\pm$ | 500 | 28700 $\pm$ | 350 | 22500 $\pm$ | 300
Luminosity of star A $\log(L/\hbox{$\mathcal{L}^{\rm N}_{\odot}$})$ | 4.738 $\pm$ | 0.028 | 4.357 $\pm$ | 0.029 | 4.439 $\pm$ | 0.023 | 4.457 $\pm$ | 0.022
Luminosity of star B $\log(L/\hbox{$\mathcal{L}^{\rm N}_{\odot}$})$ | 3.874 $\pm$ | 0.028 | 4.278 $\pm$ | 0.030 | 4.098 $\pm$ | 0.023 | 3.594 $\pm$ | 0.028
Absolute bolometric magnitude of star A |  $-$7.104 $\pm$ | 0.071 |  $-$6.15 $\pm$ | 0.073 |  $-$6.331 $\pm$ | 0.057 |  $-$6.403 $\pm$ | 0.054
Absolute bolometric magnitude of star B |  $-$4.944 $\pm$ | 0.057 |  $-$5.96 $\pm$ | 0.075 |  $-$5.505 $\pm$ | 0.058 |  $-$4.245 $\pm$ | 0.069
Interstellar extinction $E(B-V)$ (mag) | 0.75 $\pm$ | 0.05 | 0.55 $\pm$ | 0.05 | 0.40 $\pm$ | 0.05 | 0.56 $\pm$ | 0.03
Distance (pc) | 1460 $\pm$ | 50 | 2278 $\pm$ | 63 | 2466 $\pm$ | 78 | 2290 $\pm$ | 60
Gaia DR3 parallax (mas) | 0.6452 $\pm$ | 0.0231 | 0.4232 $\pm$ | 0.0130 | 0.4428 $\pm$ | 0.0200 | 0.4380 $\pm$ | 0.0261
Gaia DR3 distance (pc) | 1550 $\pm$ | 56 | 2363 $\pm$ | 73 | 2260 $\pm$ | 100 | 2280 $\pm$ | 140

101010 The units labelled with a superscripted ‘N’ are given in terms of the
nominal solar quantities defined in IAU 2015 Resolution B3 (Prša et al. 2016).

###  7.4 V346 Cen

Figure 13:  The light curves and best WD models for V346 Cen. Other comments
are the same as for Fig. 10.

Extensive photometry in the Strömgren $uvby$ system was obtained by Giménez et
al. (1986b), comprising 1056 observations made simultaneously through all four
filters using the Strömgren Automated Telescope (Grønbech et al. 1976). These
data have been analysed by Giménez et al. (1986a) using the wink model, and by
Mayer et al. (2016) using the phoebe code. The two studies agree on the values
of the fractional radii to within the uncertainties quoted by Giménez et al.
(1986a) but not the uncertainties quoted by Mayer et al. (2016). We therefore
performed our own analysis of these data in order to assess robust errorbars
and check the level of agreement with the previous studies.

Our WD code model for the $uvby$ data provided a good fit to the observations
(Fig. 13) but required $\omega$ to be fixed at a suitable value to avoid the
fit diverging to unphysical solutions. We set the rotation rates to 2.49 and
2.70 based on the rotational velocities of the stars measured from the
disentangled spectra. The logarithmic LD law was adopted, although the other
two laws gave almost identical results. Third light was fixed at zero because
attempts to fit for it returned a small negative value that was consistent
with zero. Our results were in excellent agreement with those of Giménez et
al. (1986b).

After this work had been performed, light curves from sectors 37 and 38 of the
TESS satellite became available. These are of much higher quality so we used
them for our final analysis. We performed a preliminary fit with jktebop to
obtain an orbital ephemeris then phase-binned them into 500 bins to make the
computations faster. Our approach was the same as for the $uvby$ data except
that we were able to fit for $\omega$ and also needed to fit for third light
due to significant contamination of the TESS light curve. We found the best
fit to the TESS data to be highly stable against changes in mass ratio,
rotation rate, treatment of LD, albedo, gravity darkening and numerical grid
size. We had to fix the LD coefficients as they diverged to unphysical values
when we attempted to fit for them.

The final parameters and uncertainties of the fit are given in Table 6. The
fits are shown in Fig. 13, and two things are worth highlighting. First, the
morphology of the light curve has changed between the $uvby$ and TESS epochs
due to apsidal motion. The phase of secondary eclipse has changed and it is no
longer annular – the primary eclipse has become a transit instead. Second, the
TESS data show a clear pulsation signature. This affected the quality of our
solution and was probably why we were unable to fit for LD coefficients. The
pulsation almost certainly arises from the EB itself and not from the
contaminating light, because they are commensurate with the orbital period
(see Fig. 18). V346 Cen is therefore another high-mass EB showing pulsations
(Southworth & Bowman 2022). Because our light curve solution did not account
for pulsations, we have conservatively doubled the uncertainties in the
measured fractional radii.

####  7.4.1 Pulsations

Following the binary analysis, an analysis of the residual light curve
(hereafter called the pulsation light curve) revealed the presence of tidally
excited pulsations, as illustrated in Fig. 14. To measure this tidally induced
variability, we fitted sine waves, corresponding to the 20 lowest-order
orbital harmonic frequencies, to the out-of-eclipse part of the pulsation
light curve. Fitted orbital harmonics were accepted when the signal-to-noise
ratio $S/N\geq 4.0$, where $S/N$ was calculated as the ratio of the amplitude
of the fitted sine wave, and the average signal amplitude of the Lomb-Scargle
periodogram (Scargle 1982) in a $1\,\rm d^{-1}$ window around the considered
frequency. Finally, the measured amplitudes, phases and orbital harmonic
frequencies were optimised simultaneously by nonlinearly fitting them to the
pulsation light curve. Their values are listed in Table 8.

From these results, we determined that the tidally induced pulsation
corresponds to the $9^{\rm th}$ orbital harmonic, in agreement with what is
shown in Fig. 14. The physical origin of the other measured orbital harmonics
is less clear. While they may also partially correspond to tidally induced
pulsations, this could not be confirmed. At least part of it is likely caused
by the non-sinusoidal nature and orbital-phase dependent amplitude modulations
of the $9^{\rm th}$ orbital harmonic signal. Moreover, as shown in the middle
panel of Fig. 14, this pulsation has a minimum during the primary eclipse and
a maximum during the secondary eclipse, which indicates that it belongs to the
primary component.

Finally, after the significant tidally excited variability was removed from
the pulsation light curve, we evaluated the residuals. As illustrated in Fig.
15, the remaining data exhibit signatures of stochastic low-frequency
variability, as has been reported in the literature for other high-mass stars
(e.g., Bowman et al. 2019, 2020).

Figure 14:  Tidally excited pulsations of V346 Cen. Top: observed light curve
of V346 Cen for sectors 37 and 38, phase-folded with the binary orbital
period. The eclipses are indicated by the grey bands. Middle: Pulsation light
curve of V346 Cen for sectors 37 and 38, phase-folded with the binary orbital
period. Data points taken during the eclipses again lie within the grey bands.
Bottom: Lomb-Scargle periodogram, calculated for the out-of-eclipse data
points of the pulsation light curve for sectors 37 and 38. The dashed vertical
lines indicate harmonics of the orbital frequency. Figure 15:  Stochastic low-
frequency variability of V346 Cen. Top: part of the (out-of-eclipse) residual
light curve of V346 Cen, after fitting the orbital harmonics. Bottom: Lomb-
Scargle periodogram, calculated for the out-of-eclipse data points of the
residual light curve for sectors 37 and 38. Table 8:  Values of the amplitudes
$A$, frequencies $\nu$, phases $\phi$ and signal-to-noise ratios $S/N$ of the
orbital harmonics, calculated for the out-of-eclipse data points in the
pulsation light curve of V346 Cen.

$n_{\rm orb}$ |  $A$ (mmag) |  $\nu$ ($\rm d^{-1}$) |  $\phi$ ($2\pi$ rad) | $S/N$
---|---|---|---|---
1 | $0.813\pm 0.023$ | 0.15818130 | $0.0811\pm 0.0005$ | 6.3
5 | $0.812\pm 0.026$ | 0.79090648 | $0.378\pm 0.005$ | 6.7
6 | $0.771\pm 0.025$ | 0.94908778 | $-0.240\pm 0.005$ | 6.5
8 | $0.483\pm 0.025$ | 1.26545037 | $0.237\pm 0.008$ | 4.0
9 | $3.938\pm 0.025$ | 1.42363166 | $-0.2993\pm 0.0010$ | 33.4
111111 The frequency values were fixed at the indicated integer multiples of
the measured orbital frequency $\nu_{\rm orb}$.

###  7.5 Physical properties

We have determined the physical properties of the systems using the results
from the spectroscopic and photometric analyses outlined above. For this we
used the velocity amplitudes, $T_{\rm eff}$ values, $e$ and $\omega$ from the
spectroscopic analysis, and the fractional radii and orbital inclination from
the photometric analysis. To perform the calculations we used the jktabsdim
code (Southworth et al. 2005), which propagates the errorbar from each input
parameter using a perturbation analysis. We used a version of jktabsdim
modified to use the IAU system of nominal solar values (Prša et al. 2016) plus
the NIST 2018 values for the Newtonian gravitational constant and the Stefan-
Boltzmann constant. The results of this analysis are given in Table 7.

Distances have been derived using the measured radii and $T_{\rm eff}$s of the
stars, apparent magnitudes of the system in the Johnson-Cousins $UBVRI$ and
2MASS $JHK_{s}$ bands, and the theoretical bolometric corrections tabulated by
Girardi et al. (2002). We adjusted the interstellar extinction $E(B-V)$ to
obtain consistent distances in the optical and infrared passbands. These
results are given in Table 7 alongside the Gaia EDR3 parallaxes (Gaia
Collaboration 2016, 2021) and the distance from simple inversion of the
parallax. We see agreement within the errorbars, the most discrepant
(1.6$\sigma$) being for V573 Car. Very similar conclusions are drawn if we use
the geometric or photogeometric distances from Bailer-Jones et al. (2021). We
conclude that our results for all four targets are independently verified by
the Gaia parallaxes.

##  8 The parent clusters

Knowledge of the properties of stars in dEBs allows the determination of their
distance. Moreover, a comparison of the properties of dEBs to stellar
evolutionary models constrains their age. The age of stars in our sample,
except for GL Car, were determined from isochrone fitting in Tkachenko et al.
(2020) for two cases: (i) as a single star; (ii) as a binary where the two
components have the same age. Three different interior structures were assumed
in these calculations, hence in Table 9 we give lower and upper limits for the
age. The distances to the binary systems in our sample are given in Table 7.

Table 9:  Distances and ages for the binary systems in the present sample
compared to the parent clusters.

Binary | Distance (pc) | Age (Myr) | Cluster | Distance (pc) | Ref. | Age (Myr) | Ref.
---|---|---|---|---|---|---|---
V1034 Sco | $1460\pm 50$ | 5.0–7.7 — 5.3–7.0 | NGC 6321 | $1538\pm 20$ | 1 | 6.3 | 2
GL Car | $2278\pm 63$ | $2.0\pm 0.5$ | NGC 3572a | $2444\pm 33$ | 3 | 1–4 | 4
V573 Car | $2466\pm 78$ | 1.5–3.1 — 2.2–2.7 | Trumpler 16 | $2360\pm 505$ | 5 | $2\pm 1$ | 6
V346 Cen | $2290\pm 60$ | 10.5–16.0 — 10.7-16.0 | Stock 14 | $2439\pm 326$ | 7 | $10\pm 2$ | 7

121212 The distances to the binary systems are from the present work (Table
7). The ages were calculated by Tkachenko et al. (2020) except GL Car for
which the age is adopted from Giménez & Clausen (1986). Tkachenko et al.
(2020) estimated the age for two options: assuming the components are
individual stars, and constraining the age to be the same for both components.
Both measurements are given, separated by a vertical line.

(1) Banyard et al. (2022); (2) Kuhn et al. (2017); (3) Clariá (1976); (4)
Garcia (1994); (5) Göppl & Preibisch (2022); (6) Hur et al. (2012); (7)
Paunzen & Netopil (2006).

###  8.1 V1034 Sco and NGC 6231

The distance to V1034 Sco was evaluated by Sana et al. (2005) who found
$d=1528^{+117}_{-109}$ pc, which is within 1$\sigma$ of the distance we
calculated. The light curve solution in Bouzid et al. (2005) suffers from an
ambiguity in setting the primary’s $T_{\rm eff}$ so the authors calculated the
distance for both cases. The larger one is exactly the same as those reported
by Sana et al. (2005), while the shorter one is $d=1399^{+20}_{-20}$ pc. Mayer
et al. (2008) determined the distance to another dEB in this cluster, V1007
Sco, as 1622 pc (no uncertainty given) which is somewhat larger than the other
distance estimates mentioned here.

The open cluster NGC 6231 belongs to the star-formation complex Sco OB1 (Perry
et al. 1991). The cluster is the oldest and most massive in Sco OB1 (Damiani
et al. 2016). The ages of the cluster members have been estimated to be
between 2 and 8 Myr (Sung et al. 2013; Damiani et al. 2016; Kuhn et al. 2017),
with OB stars being an older population in the cluster. The cluster is rich in
spectroscopic binaries: García & Mermilliod (2001) listed about 30 systems of
which 16 are certain. Mayer et al. (2008) did an exhaustive search of the
cluster members, and listed ten EBs. The most recent distance determinations
to NGC 6231 are based on Gaia parallaxes. Kuhn et al. (2019) quoted
$d=1710^{+13}_{-100}$ pc using Gaia DR2, while Banyard et al. (2022) found the
median geometric and photogeometric distances for their sample of about 60
stars in the cluster using Gaia EDR3 parallaxes to be 1579 and 1576 pc,
respectively.

Rosu et al. (2022b) determined the age of V1034 Sco to be $\tau=6.8\pm 1.4$
Myr, in perfect agreement with the result of Tkachenko et al. (2020). Three
other binary systems that are members of this cluster were studied: HD 152248
(Rosu et al. 2020), HD 152219 (Rosu et al. 2022b) and HD 152218 (Rosu et al.
2022a). Their ages were determined from the apsidal motion rate and range from
5 to 9.5 Myr.

###  8.2 GL Car and NGC 3572/Collinder 240

Giménez & Clausen (1986) found a distance to GL Car of $d=2100$ pc. They did
not give an uncertainty but quoted an error of 100 pc due to bolometric
corrections and interstellar reddening. This distance is smaller than our
result and that from the Gaia DR3 parallax. Giménez & Clausen (1986)
extensively discussed possible physical relationships to the open clusters in
the vicinity of GL Car, which is in a region crowded with young open clusters
and in the direction of the Sagittarius-Carina spiral arm.

Membership of GL Car in NGC 3572 was proposed by Sahade & Berón Dàvila (1963).
Giménez & Clausen (1986) rejected this association due to the shorter distance
to the dEB than the cluster, and because NGC 3572 is a compact cluster with a
radius of 5′ and GL Car is at an angular distance of 40′. It was recognised
that the open cluster NGC 3572 consists of two overlapping clusters, one at
2.3 kpc and one at 3.0 kpc (Clariá 1976). The nearer cluster is also
considered by Clariá (1976) to be the probable nucleus of a scattered group of
OB stars located in the vicinity, identified as Collinder 240 and an extension
of Car OB2. This is a region in which the line of sight is tangential to the
molecular cloud ridge in the Carina Arm, and is projected on a rather small
area in the sky. It shows as a region with a higher concentration of OB stars,
but with a radial extension of several kpc.

The age of GL Car was determined to be $\tau=2.0\pm 0.5$ Myr (Giménez &
Clausen 1986). This is compatible with age determinations for Collinder 240,
$\tau\sim 1$ Myr, and Car OB2, $\tau=4$ Myr (Garcia 1994).

###  8.3 V573 Car and Trumpler 16

Freyhammer et al. (2001) determined a distance to V573 Car of $d=2600\pm 120$
pc, and an age of $\tau=1.5\pm 1.0$ Myr. Their distance determination is
within 1$\sigma$ of ours. Also, the very young age is confirmed with extensive
isochrone fitting to different stellar interior structure models in Tkachenko
et al. (2020), as summarised in Table 9.

V573 Car is situated near the centre of the open cluster Trumpler 16, close to
$\eta$ Carinae, the brightest star in the cluster, and one of the most
intriguing objects in the Galaxy. The cluster itself, with its neighbouring
clusters, Trumpler 14, and Trumpler 15, forms a chain of rich clusters in the
prominent Carina star-forming complex, a conspicous part of the Carina-Vela
spiral arm. The whole region is recognised as the young association Carina
OB1, which also includes NGC 3293 and several small open clusters, the H ii
region and prominent nebula NGC 3372 powered by $\eta$ Car (Smith 2006; Wright
2020). Pre-Gaia distance estimates relied mostly on multicolour photometry,
and gave distances in the range 2.2–2.9 kpc and young ages in the range 1–3
Myr (Hur et al. 2012). Using Gaia EDR3 Shull et al. (2021), Maíz Apellániz et
al. (2022) and Göppl & Preibisch (2022) found distances to the cluster
Trumpler 16 at the lower end of the range: $2.32\pm 0.12$, $2.38\pm 0.20$ and
$2.36\pm 0.05$ kpc, respectively, all within 1$\sigma$ of our determination.

###  8.4 V346 Cen and Stock 14

The distance and age of V346 Cen were also determined by Giménez et al.
(1986b), which allows a direct comparison with our results. Giménez et al.
(1986b) determined the distance $d=2.38\pm 0.18$ kpc, which is within
1$\sigma$ of our determination. The age of the binary system they found,
$\tau=10.0^{+5.8}_{-3.6}$ Myr, also agrees well with our result,
$\tau=10.7$–16.0 Myr (Table 9).

Stock 14, the parent cluster of V346 Cen, is described as a loose but clearly
defined open cluster (Moffat & Vogt 1975; Eichendorf & Reipurth 1979). The
most recent deep $UBV$ photometry of Stock 14 was obtained by Drobek et al.
(2013) primarily in a search for new variable stars. Their photometry allowed
determination of the distance and an estimate of the age for Stock 14,
$d=2399^{+56}_{-55}$ pc, and $\tau=20\pm 10$ Myr. The authors confirmed the
cluster membership of V346 Cen. Re-evaluation of the photometric distance and
age of Stock 14 by Paunzen & Netopil (2006) also favoured a shorter distance
than previous determinations. They obtained a distance of $d=2439\pm 326$ pc,
and an age of $\tau=10\pm 2$ Myr, in fine agreement with the extensive
photometric study by Drobek et al. (2013) as well as our results for V346 Cen.

##  9 Discussion

The results of the analyses above are summarised in Table 4 for elemental
abundances and Table 7 for fundamental stellar quantities. The stars in the
present work cover a range of mass (8.4–17.1$\,{\rm M}_{\odot}$), radius
(4.1–8.3$\,{\rm R}_{\odot}$), $T_{\rm eff}$ (22 500 to 32 200 K), surface
gravity (3.7–4.2 dex) and $v\sin i$ (90–185 km s-1) and are all unevolved main
sequence stars from late-O to early-B spectral types. We have achieved a high
accuracy in the fundamental stellar properties, with uncertainties in mass of
0.6–2.0%, radius of 0.8–1.7%, and $\log g$ of 0.009–0.021 dex. Having a
precise $\log g$ allows us to avoid its degeneracy with $T_{\rm eff}$ in
spectral analysis, resulting in uncertainties of 1.7–2.5% in $T_{\rm eff}$.
Since $T_{\rm eff}$ and $\log g$ are the principal quantities for specifying a
model atmosphere, precise values are a prerequisite in measuring chemical
abundances to a high precision. We now discuss the implications of our results
for two subjects: evolutionary models for high-mass stars, and chemical
evolution in high-mass binaries.

Studies of chemical abundances in high-mass stars mostly concentrate on more
advanced evolutionary stages, so it is difficult to perform a quantitative
comparison between our results and those published elsewhere. Martins et al.
(2017) presented a study of six short-period binary systems. Of them, two are
contact or overcontact systems so will have abundances altered by mass
transfer, one (DH Cep) has component stars considerably more massive than our
sample (38$\,{\rm M}_{\odot}$ and 33$\,{\rm M}_{\odot}$), while the remaining
three (Y Cyg, AH Cep and V478 Cyg) are suitable for the comparison. Of these,
AH Cep and V478 Cyg were analysed in our previous work (Pavlovski et al. 2018)
so a direct comparison is possible. Two studies agreed to within 2$\sigma$
uncertainties in the [N/C] and [N/O], but only because of the large
uncertainties quoted by Martins et al. (2017). It is hard to trace the reason
for this, but it may be related to the large uncertainties in the atmospheric
parameters in their study.

Results of a comprehensive analysis of a large sample of binary and/or
multiple stars in the Tarantula Nebula have recently been published (Almeida
et al. 2017; Mahy et al. 2020b, a) based on medium-resolution ($R=6400$)
spectra from VLT/FLAMES/GIRAFFE covering 3964–4567 Å. A total of 51 SB1 and
SB2 systems were studied, of which 13 are eclipsing. The atmospheric
parameters were determined using NLTE methods, and He, C and N abundances
derived. The objects studied fall into five different groups: (1) long-period
systems ($P>20$ d) with well-detached components; (2) eccentric short-period
($P<10$ d) detached binaries; (3) circular-orbit short-period ($P<10$ d)
binaries with strong tidal effects; (4) semi-detached systems; and (5) contact
systems. No N enrichment was found for binaries in the first two groups,
despite the components having $v\sin i$ values of 50–250 km s-1. This finding
is in disagreement with evolutionary models with rotationally induced mixing
(Maeder & Meynet 2000; Heger & Langer 2000; Heger et al. 2000). Furthermore, a
large N abundance was found for apparently slowly rotating stars in binaries.
This agrees with initial findings by Hunter et al. (2007, 2008, 2009) who
detected three distinctive groups in a diagram of [N/H] versus $v\sin i$ for
single OB stars (sometimes dubbed the “Hunter diagram”): (1) stars showing N
enrichment with $v\sin i$; (2) rapidly rotating stars with no sign of N
enrichment; and (3) stars with low $v\sin i$ and excessive N abundance.

In the third group of binaries from Mahy et al. (2020b), N enrichment was
found for the fast rotators. This is a group of stars in which the strongest
influence of tidal forces on rotationally induced mixing is expected,
following theoretical calculations by de Mink et al. (2009). By far the
largest N enhancement was found for stars with almost the lowest $v\sin i$ in
this group ($\sim$50 km s-1), just as in the case of findings for stars in the
first two groups. Mahy et al. (2020b) concluded that stars in detached
binaries (groups 1 to 3) are evolving as single stars. A lack of a clear
relationship between N abundance and $v\sin i$ is in conflict with theoretical
models and makes it hard to understand the effect of rotationally induced
turbulent mixing in stellar interiors.

A very recent comprehensive spectroscopic analysis of a large set of B-type
stars in the young open cluster NGC 3293 (Morel et al. 2022) also corroborates
these results: in the sample of almost 150 B-type stars of which the majority
have high $v\sin i$, apparently no star with excess N abundance was detected.
Only two stars are found with mild N enhancement, and these stars have a low
$v\sin i$. A lack of N enhancement in fast-rotating B stars, and conversely,
further evidence for N enhancement in low-$v\sin i$ B stars, is in clear
contradiction with theoretical evolutionary models which incorporate
rotationally-induced mixing.

A state-of-the-art statistical analysis was carried out by Aerts et al. (2014)
to identify possible mechanism(s) that could explain the distribution of stars
in the Hunter diagram. The authors collected a statistically significant
sample of well-studied Galactic single B stars for which seven observables
were available (surface N abundance, rotational frequency, magnetic field
strength, and the amplitude and frequency of their dominant acoustic and
gravity modes of oscillation). A multivariate analysis indicated that the
$T_{\rm eff}$ and the frequency of the dominant acoustic oscillation mode have
the most predictive power of the surface N abundance, whereas the rotational
frequency of the star does not have any predictive power at all. Up to now, no
follow-up studies have been undertaken to investigate these unexpected
results.

Strong support for rotationally induced mixing has come from detailed
abundance study for early B-type stars by Przybilla et al. (2008) and Nieva &
Przybilla (2012). The authors selected 20 early-B stars with a low $v\sin i$
to allow a high precision in determination of the atmospheric parameters and
chemical abundances. Przybilla et al. (2010) confirmed an observationally
tight correlation in the plot of abundance ratios N/C versus N/O, with a slope
predicted via nuclear reactions in the CNO process. The targets had a broad
evolutionary range, from dwarfs to supergiants, and their CNO abundances
followed predictions of the nuclear reaction theory.

Figure 16:  Left panel: Abundances of the CNO elements for high-mass stars in
a diagram of [N/C] index versus [N/O] index. Stars in binary systems (Section
6) are represented by solid blue circles. For comparison, abundance
determinations for single early-B type stars (Nieva & Przybilla 2012) are
represented by green circles. Single stars obey a trend indicated by an
analytical approximation to the nuclear reactions path for the CNO cycle
derived in Przybilla et al. (2010) and Maeder et al. (2014). The slope in this
diagram indicates a gradual enhancement of N at the expense of C. A slight
decrease in O abundance is also predicted. Whilst the single and binary stars
span almost identical mass and $T_{\rm eff}$ ranges, and are all main-sequence
stars, it is clear they do not share the same distribution (see Section 9).
Right panel: The observed [N/C] index for 23 high-mass stars in binaries
(solid blue circles), compared to single B-type stars showing in solid green
circles (Nieva & Przybilla 2012), as a function of surface gravity. Solid red
lines in the right panel show theoretical evolutionary tracks for a 15$\,{\rm
M}_{\odot}$ star and five values of the initial rotational velocity
$\Omega/\Omega_{\rm crit}$ = 0.1, 0.3, 0.5, 0.7, and 0.9 (Georgy et al. 2013).
Striking differences between single stars, and stars in binary systems are
discussed in Section 9.

From our current and previous (see Section 6) studies, we have a sample of 13
dEBs of which 25 components have measured CNO abundances. We compared these to
a sample of high-mass stars published in Przybilla et al. (2010) and Nieva &
Przybilla (2012) in the logarithmic N/C versus N/O diagram (left panel in Fig.
16). This is a powerful diagnostic tool in which the slope between [N/C] and
[N/O] represents changes in CNO abundances due to nuclear reactions as derived
in Przybilla et al. (2010). It is striking that the two samples cover the same
mass range (8–20$\,{\rm M}_{\odot}$) but do not fully overlap in the diagram.
For binary components there is a cut-off at [N/C] $\sim-0.4$ dex and [N/O]
$\sim-0.8$ dex and they cluster around values close to solar
([N/C]${}_{\odot}=-0.52$ dex and [N/O]${}_{\odot}=-1.00$ dex), but a slope can
be seen. The targets in the current work strengthen our previous conclusion
that properties of interior mixing in binary stars are different from and
might be less efficient than in single high-mass stars (Pavlovski et al.
2018).

This striking effect is also clearly seen in the diagram of [N/C] versus $\log
g$ (Fig. 16). Theoretical evolutionary tracks are presented for a 15$\,{\rm
M}_{\odot}$ star and five values of the initial rotational velocity
$\Omega/\Omega_{\rm crit}=0.1$, 0.3, 0.5, 0.7, and 0.9 (Georgy et al. 2013).
The overall spread in [N/C] could be interpreted as due to evolutionary
changes or (very) high initial rotational velocities. However, only single
stars from the sample of (Nieva & Przybilla 2012) tend to be consistent with
the large [N/C] ratio predicted by the models for large initial rotational
velocity values of $\Omega/\Omega_{\rm crit}\gtrapprox 0.5$ and [N/C]
$\gtrapprox-0.4$ dex. The main issue with the interpretation of the observed
distribution in the context of the rotationally induced mixing alone is the
generally low projected rotational velocity values ($v\,\sin\,i<30$ km s-1)
found by Nieva & Przybilla (2012) for about half of their sample stars. For
the effect of rotational mixing being alone responsible for the observed [N/C]
and [N/O] abundance ratios, one would require the majority of apparently slow
rotators in the sample of Nieva & Przybilla (2012) to be stars that are seen
pole-on. This is a highly improbable scenario, so we conclude that the CNO
abundances and their ratios observed in single high-mass stars are altered by
multiple processes rather than just a single mechanism of rotational mixing.
For example, high-mass stars are know to possess magnetic fields, stellar
winds, and pulsations. To this (strong) tidal effects in close high-mass
binary systems should be added. All these mechanisms, in one way or another,
are expected to impact the efficiency of internal mixing, and hence the
surface chemical composition.

However, the comparison between these two sets of empirical data, one with
single high-mass stars and the other with high-mass stars in binary systems,
is not straightforward. First and foremost, the sets differ in their
distributions of $v\sin i$. The set of single stars were deliberately selected
to be sharp-lined stars, so contains a mix of intrinsically slowly-rotating
stars and ones with small inclinations and thus small $\sin i$ terms. The set
of binary stars, on the other hand, contains objects whose equatorial
rotational velocities are accurately known, assuming their rotational and
orbital axes have been aligned during formation or by tidal effects.
Furthermore, even though the $v\sin i$ distribution could be statistically
corrected to intrinsic rotational velocities for single stars, there is a
substantial difference in the rotational history between single and binary
stars that one cannot easily account for. Evolution of stellar rotational
velocity from its initial value at the zero-age main sequence, and its
subsequent changes in the course of stellar evolution due primarily to changes
in radius, is substantially different due to tidal effects. This is
particularly important for short-period systems whose rotation is synchronised
with and thus governed by their orbital period. Nevertheless, the non-
detection of substantial changes in the CNO abundances of stars in binaries
contradicts the predictions of single-star rotational evolutionary models. For
the sample of Przybilla et al. (2010) and Nieva & Przybilla (2012), i.e.
single high-mass stars with low observed $v\sin i$, the possibility remains
that they agree with the theoretical predictions.

Tidal forces in binary and/or multiple systems affect the geometry of the
orbits and the shape and spin of the components (Mazeh 2008). In order of
increasing timescale, the stellar spin axes are aligned first, then their
rotation is synchronised, and finally the orbit is circularised. Later
evolution is dominated by mass transfer due to the increase in the sizes of
the component stars. Our hypothesis that tidal effects suppress the efficiency
of rotational mixing is not supported by theoretical calculations (de Mink et
al. 2013), which predict precisely the opposite: that short-period
circularised binary systems should experience rotationally induced turbulent
mixing in stellar interiors.

In looking for possible mechanisms which diminish turbulent mixing in the
components of binary systems, Koenigsberger et al. (2021) examined
differential rotation in asynchronous binary systems. If the components in a
binary system are not yet in synchronous rotation, tidally-induced and time-
variable differential rotation could develop. The calculations by
Koenigsberger et al. (2021) clearly show the role of asynchronism: the induced
rotation structure and its temporal variability depend on the degree of
departure from synchronism. The authors further speculated that, in this
context, slowly-rotating asynchronous binaries could have more efficient
mixing than the more rapidly-rotating but tidally locked systems. This shows
that processes triggered by asynchronous rotation in binary systems cannot be
ignored, while a comparison between samples of single and binary stars should
be done with particular care, even when the latter are in a detached
configuration. We note that 12 of the 14 binaries in our sample have eccentric
orbits but that most of the component stars rotate synchronously.

##  10 Conclusion

Despite their astrophysical importance, high-precision fundamental stellar
quantities (mass, radius, $T_{\rm eff}$) have been determined for only a few
high-mass stars in binary systems in our galaxy (Southworth 2015). Even fewer
have measurements of their surface chemical composition (Serenelli et al.
2021). In the present work we have added four more binary systems to this
list: V1034 Sco, V346 Cen, GL Car and V573 Car, containing stars of masses
from 8.4 to 17.1$\,{\rm M}_{\odot}$. Most of these stars are young, with only
two in the second half of their MS evolution.

We have presented high-quality HARPS spectra and analysed them using spectral
disentangling to determine their spectroscopic orbits and the individual
spectra of the component stars. These were analysed using an NLTE approach. We
have modelled the available light curves for our systems, comprising $uvby$
photometry in all cases and TESS photometry in three cases, to determine their
photometric parameters. Combining these analyses, we have determined high-
precision masses, radii, surface gravities, $T_{\rm eff}$ values, $v\sin i$
values and C, N, O, Mg and Si abundances for all eight stars in the four
binary systems. Of particular interest are the CNO abundances since these
elements are involved in core hydrogen burning through the CNO cycle. During a
star’s evolution its N abundance increases and its C abundance decreases.
Rotationally induced mixing of stellar material, or some other mixing
processes, could bring nuclear-processed material from the stellar core to the
surface. Therefore, the [N/C] ratio is a sensitive probe of interior mixing
processes during the MS evolutionary stage.

The CNO abundances determined in this work corroborate our previous findings
(Pavlovski et al. 2018) that interior mixing is different in binary stars to
single stars. A tight correlation of [N/C] with [N/O] versus the predicted
evolutionary changes has been found for single early B-type stars (Przybilla
et al. 2010; Nieva & Przybilla 2012), whereas binary systems in our sample
show much less variation in both [N/C] and [N/O]. However, care is needed when
comparing them with single stars due to the differences in rotational velocity
between these types of object. It remains true that the binary sample does not
reproduce the results found for a sample of single low-$v\sin i$ B-type stars.

On other hand, recent spectroscopic analysis of large samples of high-mass
stars in binaries (Mahy et al. 2020a), and single B-type stars in the young
open cluster NGC 3293 (Morel et al. 2022) apparently confirmed the lack of
substantial changes in CNO abundances for high-$v\sin i$ stars, i.e. for
intrinsically fast-rotating stars.

We speculate that proximity effect in binary systems somehow suppress mixing
and/or transport of chemical elements from the interior to the surface.
However, firmer conclusions will need a substantial expansion of the binary
stars sample and an extension to more massive and hotter stars, and/or wider
long-period binary systems.

## Acknowledgements

Careful reading of the manuscript and useful suggestions provided by the
referee are acknowledged. We are indebted to Keith Butler and Norbert
Przybilla for kindly sharing their codes, and the model atoms used in the
present work. KP and ET were initially supported by the Croatian Science
Foundation through research grant IP-2014-09-8656. The research leading to
these results has (partially) received funding from the KU Leuven Research
Council (grant C16/18/005: PARADISE) and from the BELgian federal Science
Policy Office (BELSPO) through PRODEX grant PLATO. TVR gratefully acknowledges
support from the Research Foundation Flanders (FWO) under grant agreement
number 12ZB620N.

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##  Appendix A Additional plots

Figure 17:  Light curves of our target stars, from our own reduction of data
from the TESS satellite, that were not included in the work in this paper, but
could useful for studies of the period changes, and apsidal motion. The
reduced photometric data are given in Table A.1 only available in electronic
form at the CDS (see article front page). Figure 18:  Light curve of V346 Cen
from the TESS satellite, plotted versus orbital phase but with a small
magnitude offset linearly dependent on time to shift successive cycles upward
in the figure. The earliest points are coloured red and the latest points are
coloured blue. The repetition of the pulsation signature with orbital phase is
easy to see.

Figure 19: Portions of the disentangled spectra of the stars (labelled)
studied in this work.