11institutetext: Argelander-Institut für Astronomie, Universität Bonn, Auf dem
Hügel 71, 53121 Bonn, Germany
11email<EMAIL_ADDRESS>22institutetext: The School of Physics and
Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel 33institutetext: Max-
Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany
44institutetext: Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001
Leuven, Belgium 55institutetext: Max Planck Institute for Astrophysics, Karl-
Schwarzschild-Strasse 1, 85748 Garching, Germany

#  An absence of binary companions to Wolf-Rayet stars in the Small Magellanic
Cloud ††thanks: Based on observations collected at the European Organisation
for Astronomical Research in the Southern Hemisphere under ESO programme
108.22M1.001.

Implications for mass loss and black hole masses at low metallicity

A. Schootemeijer 11 T. Shenar 22 N. Langer 1133 N. Grin 11 H. Sana 44 G.
Gräfener 11 C. Schürmann 11 C. Wang 55 X.-T. Xu 11

In order to predict the black hole mass distributions at high redshift, we
need to understand whether very massive single stars ($M\mathrel{\hbox{\hbox
to0.0pt{\hbox{\lower 4.0pt\hbox{$\sim$}}\hss}\hbox{$>$}}}40$ M⊙) at low
metallicity $Z$ lose their hydrogen-rich envelopes, like their metal-rich
counterparts, or whether a binary companion is required to achieve this. To
test this, we undertake a deep spectroscopic search for binary companions of
the seven apparently single Wolf-Rayet (WR) stars in the Small Magellanic
Cloud (SMC; where $Z\simeq 1/5\,Z_{\odot}$). For each of them, we acquired six
high-quality VLT-UVES spectra spread over a time period of 1.5 years. By using
the narrow N v lines in these spectra, we monitor radial velocity (RV)
variations to search for binary motion. We find low RV variations between 6
and 23 km/s for the seven WR stars, with a median standard deviation of $5$
km/s. Our Monte Carlo simulations imply probabilities below $\sim$5% for any
of our target WR stars to have a binary companion more massive than $\sim$5 M⊙
at orbital periods of less than a year. We estimate that the probability that
all our target WR stars have companions with orbital periods shorter than 10
yr is below $\sim$10-5, and argue that the observed modest RV variations may
originate from intrinsic atmosphere or wind variability. Our findings imply
that metal-poor massive stars born with $M\gtrsim 40$ M⊙ can lose most of
their hydrogen-rich envelopes via stellar winds or eruptive mass loss, which
strongly constrains their initial mass – black hole mass relation. We also
identify two of our seven target stars (SMC AB1 and SMC AB11) as runaway stars
with a peculiar radial velocity of $\sim$80 km/s. Moreover, with all five
previously detected WR binaries in the SMC exhibiting orbital periods of below
20 d, a puzzling absence of intermediate-to-long-period WR binaries has
emerged, with strong implications for the outcome of massive binary
interaction at low metallicity.

###### Key Words.:

Stars: Wolf-Rayet – Stars: massive – Stars: binaries – Stars: evolution –
Stars: winds – Stars: black holes

##  1 Introduction

Two recent developments in observational astronomy have sparked a great
interest in understanding very massive stars ($M>40$ M⊙) in the Early
Universe. First, the gravitational wave (GW) observatories of the LIGO/Virgo
collaboration have detected black hole (BH) mergers of tens of Solar masses
(Abbott et al., 2016). These BHs were most likely formed at low metallicity
(Giacobbo et al., 2018). Second, the James Webb Space Telescope just started a
new era in observational astronomy, and has already delivered spectacular
discoveries of bright blue galaxies born briefly after the Big Bang
(Castellano et al., 2022; Adams et al., 2023).

Very massive stars are crucial for both of the above topics, as they are
progenitors of BH-BH mergers and drivers of galaxy evolution. One question is
of particular relevance here: could very massive stars in the early Universe
lose their hydrogen-rich envelopes via stellar winds? Current stellar
evolution models give discrepant results (Yusof et al., 2013; Szécsi et al.,
2015; Pauli et al., 2022). If winds were strong enough for this, the
consequences are, i), a limitation of the mass of BHs produced by isolated
stars, and, ii), the formation of isolated hot Wolf-Rayet (WR) stars, which
may chemically enrich and ionize their environment. If not, massive stars in
the early universe without a binary companion remained massive and left behind
massive black holes. Indeed, for stars born more massive than 30 M⊙, wind mass
loss is the major uncertainty regarding their final mass as BHs (Fryer et al.,
2012). Also, massive stars in the early universe would then probably needed
companions to become WR stars.

A one-of-a-kind environment to study massive stars with chemical compositions
similar to those in the Early Universe is the Small Magellanic Cloud (SMC),
where the metallicitity ($Z$) is about five times lower than Solar (Venn,
1999). As a satellite galaxy of the MW only 62 kpc away (Graczyk et al., 2020)
it is close enough to study individual stars. There are twelve WR stars in the
SMC. Five of them are known to have a hot and massive companion, and for the
seven others previous searches did not find any companion stars (Westerlund,
1964; Smith, 1968; Sanduleak, 1968, 1969; Breysacher & Westerlund, 1978;
Azzopardi & Breysacher, 1979; Moffat, 1982; Moffat et al., 1985; Moffat, 1988;
Morgan et al., 1991; Bartzakos et al., 2001; Massey & Duffy, 2001; Massey,
2003; Foellmi et al., 2003a; Foellmi, 2004; Hainich et al., 2015; Shenar et
al., 2016, 2018; Neugent et al., 2018).

Figure 1: Normalized 100 Å wide cutouts of the co-added VLT-UVES spectrum of
SMC AB9. For reference, the blue arrow indicates the wavelength shift
resulting from a 300 km/s velocity shift at the central wavelength of each
panel.

To search for binary companions, Foellmi et al. (2003a) – hereafter F03 – have
monitored the radial velocity (RV) of all SMC WR stars except SMC AB 12, which
was discovered later (Massey, 2003; Foellmi, 2004) and has not yet been
subjected to RV monitoring, and SMC AB8, which is the only WO-type SMC WR star
and part of a binary system. F03 presented orbital solutions for the binary
stars and investigated the completeness of their study. They showed that for
binaries with orbital periods up to 300 days that consist of a 10 M⊙ WR star
with a 30 M⊙ companion, the projected semi-amplitude of the RV of the WR star
is typically larger than 30 km/s, which is the minimum value for which they
claim to be able to detect orbital motion. This threshold of 30 km/s is
estimated as $\sqrt{2}\sigma_{\mathrm{RV}}$, where $\sigma_{\mathrm{RV}}$ is
the measurement error with a reported value of 20 km/s.

However, this does not rule out the presence of companion stars in general,
since different orbital configurations are possible that would decrease the
projected orbital velocity of the WR star. First, WR masses in the SMC are
typically at least 20 M⊙ (Hainich et al., 2015; Shenar et al., 2016;
Schootemeijer & Langer, 2018) instead of the 10 M⊙ assumed by F03. Second,
interaction can take place for wider binaries that have orbital periods up to
about 10 years (Sana et al., 2012). Third, the companion might be
significantly lighter than 30 M⊙.

Therefore, with the knowledge gathered up to now it is possible that most or
even all SMC WR stars are in binaries, but that at least a few have escaped
detection due to long orbital periods and/or relatively low-mass companions
(Schootemeijer & Langer, 2018). Here, we present a modern RV monitoring survey
of the seven apparently-single WR stars in the SMC using data acquired with
the 8-meter Very Large Telescope (VLT). These data allow us to reach
conclusive statements regarding the binary status of the apparently single SMC
WR stars. This provides important constraints on the ability of isolated
massive stars in low-Z environments to shed their outer H-rich layers.

Our paper is organized as follows. In Sect. 2, we present the VLT data used in
this study and our method to measure RVs; the results of these measurements
are shown in Sect. 3. We discuss the impact of line profile variability on our
results in Sect. 4, the sensitivity of our campaign to binary motion in Sect.
5, and other possible signatures of binary companions in Sect. 6. Finally, we
discuss our findings in Sect. 7 and present our conclusions in Sect. 8.

##  2 Methods

###  2.1 Observations

We made use of high-resolution spectra taken by the Ultraviolet and Visual
Echelle Spectrograph (UVES; Dekker et al., 2000) instrument of the VLT. For
our monitoring program (program ID: 108.22M1.001; PI: Schootemeijer) six
spectra in total were obtained in Service Mode for each of our seven targets.
This was done during three semesters, between October 2021 and December 2022.
We used the standard DIC 2 437+760 setting and a slit width of 1”. The ESO CPL
pipeline (version 7.1.2) was used for the reduction of the spectra. The
wavelength calibration was done with the standard ESO procedure that uses a
thorium-argon lamp. We corrected the spectra for barycentric motion by
subtracting the RV measured (as described in Sect 2.2) for the extremely
narrow Ca ii interstellar absorption feature around 3970 Å. This correction
also serves as wavelength calibration. As a test, we also corrected for
barycentric motion using the baryCorr function from the PyAstronomy package
pyasl (Piskunov & Valenti, 2002). Both approaches provided similar results. We
chose to use the interstellar Ca ii method because it yielded RVs that are
slightly more constant in time.

The UVES instrument splits the collected light into two arms: a blue arm and a
red arm. For the blue arm, the covered wavelength range was 3730–5000 Å, the
typical signal to noise ratio (S/N; per resolution element, and measured with
the method of Stoehr et al., 2008) of the spectra was 40–50. The blue spectra
have a resolving power of $R=\lambda/\Delta\lambda\approx 41000$, which
translates into a bin size of $\Delta\lambda=0.03$ Å around 4500 Å. The lines
that we use for the RV measurements are in the wavelength range of the blue
arm. The spectra taken with the red arm in the wavelength range 5660–9460 Å
had a typical S/N of 25–30 and a resolving power of $R\approx 42000$.

SMC AB11 has a red source with a similar $G$ magnitude (both have $G=15.8$) 1”
away from it, causing the point spread functions of the two objects to blend.
To enable a relatively clean extraction of the WR spectrum, the slit
orientation for SMC AB11 was set to include both sources. The spectrum of the
WR star was then extracted from the 2D spectrum image by identifying the
spatial pixel beyond which the contribution of the red sources becomes
negligible.

For three out of seven targeted stars – SMC AB1, 2, and 4 – archival UVES
spectra from August 2006 (PI: Foellmi; see Marchenko et al., 2007) exist on
the ESO archive. In their campaign, a total of 10–12 spectra was taken during
two successive nights for each of these three stars. These spectra are in the
wavelength range 3930–6030 Å, and they have an S/N of $\sim$50 at a resolving
power of $R\approx 30000$.

Figure 2: Top: The N v 4604, 4620 Å doublet that is used for cross-
correlation, shown for SMC AB9. The co-added spectrum as well as the spectrum
that was taken during observing night 0 are displayed. For reference, we also
show spectrum 0 shifted by 40 km/s. The spectra are normalized and then
vertically shifted such that the integrated flux in the shown wavelength range
is zero. Bottom: Cross-correlation functions (CCFs) of the N v 4604, 4620 Å
doublet for the co-added template spectrum and the six individual spectra that
were taken. For this figure, the spectra were not corrected for barycentric
motion. Figure 3: RV measurements based on UVES-spectra taken during different
nights. The first of the six spectra of our observing campaign is taken during
observation night 0. The time interval in days between different observation
nights is written at the bottom of the subplots. For SMC AB1, 2 and, 4, we
also show measurements based on the pre-existing UVES-spectra of Marchenko et
al. (2007) (these are the data of observation nights -2 and -1). In this work
we opt to use the N v lines that are show with thicker lines.

###  2.2 Radial velocity measurements

We used a linear fit to the continuum to normalize our spectra (e.g., those
shown in Fig. 1). First we selected a wavelength range around an absorption
line of interest (e.g., 4596–4630 Å for the N v doublet shown in Fig. 2). We
measured the average flux in the leftmost and rightmost 7.5% of that
wavelength range. Then, we divided the observed flux by a linear fit to the
average flux in the central points of these two wavelength ranges.

Our method to measure the RV shifts of the normalized spectra is the cross-
correlation method from Zucker (2003). The same method has been successfully
applied to precisely measure the RVs of Galactic WR stars (Dsilva et al.,
2020, 2022, 2023). We briefly describe the method below. First, the normalized
spectra were shifted vertically such that the integrated flux is zero (Fig. 2,
top). After that, for each of our WR stars we created a template spectrum by
co-adding all available UVES spectra, starting with unshifted spectra. Then we
iteratively shifted each individual spectrum by its measured RV (as described
below) compared to the template spectrum, until the RVs no longer changed.
After that, we calculated the cross-correlation function (CCF) between the
individual spectra and the template spectrum (see bottom panel of Fig. 2). We
fitted a parabola to the CCF above 0.9 times its maximum value to obtain the
best-fit RV and the statistical error $\sigma_{\mathrm{CCF}}$; the peak of the
function fitted to the CCF is the best-fit RV value and the height and width
of the fit to the CCF determine the value of the statistical error.

##  3 Results

Figure 3 shows our RV measurements for the apparently single SMC WR stars. The
time interval in days between two successive observations is written at the
bottom of each panel.

###  3.1 Best lines for radial velocity measurements

To decide on which line to use for our analysis, we show RVs from different
lines in Fig. 3. The N v 4944 Å line is not observed in the relatively cold
SMC WR stars AB 2 and AB 4, for which Hainich et al. (2015) found
$T_{*}\approx 45$ kK, but it is detected in the spectra of the five other
targeted SMC WR stars (where $T_{*}\gtrsim 80$ kK).

Observation night -2 and -1 in Fig. 3 refer to observations of SMC AB1, 2, and
4 of Marchenko et al. (2007), which are densely spaced during two successive
nights. We notice that for these observations, the RVs measured with N v lines
(both the 4604 + 4620 Å doublet and the line around 4944 Å) remain relatively
stable. The RVs measured with He ii lines are not as constant. Also for the
spectra taken in our recent campaign, the RVs measured from N v lines are more
stable than those from He ii lines. For our seven target stars, we calculate
$\Delta$RV (the difference between the highest and the lowest RV measurement)
and $\sigma_{\mathrm{RV}}$ (the standard deviation of the RV measurements,
where we subtract 1.5 degree of freedom in the divisor). We show the median
values in Table 1. Compared to the measurements that use N v lines, the median
He ii 4339 Å and He ii 4542 Å $\Delta$RV measurements are $\sim$1.5 times
higher; the measurements with the He ii 4686 Å line yield a $\sim$2.5 times
higher median $\Delta$RV. Our interpretation for this is that the measurements
based on He ii are less accurate – for example because these lines are more
variable, weaker, or broader (Fig. 1). This is in agreement with findings of
Dsilva et al. (2022). A possible explanation for lower variability is that N v
lines form in deeper layers than He ii lines.

The cross-correlation method can also be applied to multiple line regions
simultaneously. We did so for the N v 4604 + 4620 Å doublet combined with N v
4944Å, as well as seven narrow He ii and N features combined (see bottom two
rows and caption of Table 1). We find slightly more constant RVs for the
combination of N v lines than for the N v 4604 + 4620 Å doublet individually.
For the seven narrow He ii and N v features combined, we find the most
constant RVs.

For the RV measurements presented later on, we elect to use the RVs measured
with the combination of the N v 4604 + 4620 Å doublet and N v 4944Å line,
because these are formed in the same relatively hot layers of the star that
are close to the stellar surface. We consider this the conservative approach,
since with the combined seven narrow He ii and N v lines we find the lowest RV
variations.

Table 1:  Median $\Delta$RV and $\sigma_{\mathrm{RV}}$ measured in our campaign for our seven target stars, using different lines. The narrow He ii and N lines from the last row are He ii 4100, 4200, 4339, 4542 Å in combination with N iv 4058 Å and N v 4604+4620, 4944 Å.  |  $\Delta$RVmedian | $\sigma_{\mathrm{RV,\,median}}$
---|---|---
Line (rest wavelength) | [km/s] | [km/s]
He ii 4339Å | 22 | 10.0
He ii 4542Å | 22 | 8.3
He ii 4686Å | 38 | 13.9
N v 4604+4620Å | 15 | 5.6
N v 4944Å(5/7 stars) | 15 | 6.3
N v 4604+4620Å, 4944Å | 14 | 5.4
Narrow He ii and N | 10 | 4.2
Table 2:  Summary of the results of our RV monitoring campaign obtained using N v 4604+4620, 4944 Å lines. We describe in Sect. 6.3 how we obtain the absolute radial velocity $\mathrm{RV}_{\mathrm{abs}}.$ Object |  $\Delta$RV | $\sigma_{\mathrm{RV}}$ |  RVabs
---|---|---|---
| [km/s] | [km/s] | [km/s]
SMC AB1 | 23.0 | 7.0 | $203\pm 8$
SMC AB2 | 13.0 | 4.9 | $149\pm 23$
SMC AB4 | 14.3 | 5.2 | $144\pm 23$
SMC AB9 | 15.4 | 5.2 | $136\pm 3$
SMC AB10 | 5.5 | 2.4 | $152\pm 25$
SMC AB11 | 13.8 | 5.1 | $228\pm 8$
SMC AB12 | 18.4 | 6.9 | $156\pm 22$

###  3.2 Radial velocity variations

In search for binary reflex motion we consider the $\Delta$RV values that we
measured for our individual WR stars. We measure $\Delta$RV values between 6
km/s and 23 km/s (Table 2; individual measurements are tabulated in Appendix
A). These numbers are based on spectra taken during our observing campaign,
but we note that RV measurements based on 15-year-old spectra of Marchenko et
al. (2007) fall within the RV range that we measure using the new spectra. The
$\Delta$RV values that we measure are much lower than for the known SMC WR
binaries, for which F03 measured values in the range
$\Delta\mathrm{RV}=400-600$ km/s.

Figure 4:  Similar to Fig. 3, but showing RV measurements from a previous
monitoring campaign (F03) and the three previous RV measurements of SMC AB12
from Foellmi (2004). These previous measurements have been shifted by the
previously measured average RV for each star. The center and right plots on
the bottom row are cumulative probability distributions.

We compare our $\Delta$RV values to earlier measurements of our targets. We
use Engauge Digitizer111http://markummitchell.github.io/engauge-digitizer/
(Mitchell et al., 2020) to extract RV data from the plots of F03 and three
measurements of SMC AB12 of Foellmi (2004), as these were not tabulated.
Figure 4 shows that we measure much more constant RVs for our target stars
than F03 and Foellmi (2004) did. We further quantify the difference in the
central lower panel of Fig. 4. Our median $\Delta$RV value of 14 km/s is a
factor eight lower than in the earlier RV monitoring campaign, where the
median $\Delta$RV value was 110 km/s ($\Delta\mathrm{RV}$ range: $80-210$
km/s). This proves that at least the majority of the RV scatter seen by F03 is
caused by measurement uncertainties rather than genuine RV motion. Likely,
multiple factors contribute to more precise measurements in our campaign. F03
measurements used the He ii 4686 Å line, which we deemed to be less suitable
than the N v lines for RV measurements in Sect. 3.1. Furthermore, our VLT-UVES
spectra are of higher quality than those available for the previous campaign,
in particular in terms of wavelength calibration (see sect. 2.5 from F03) and
resolving power ($R\sim 40000$ vs. $R\sim 1000$).

AB 9 has been classified as a binary candidate by F03, whose measurements
indicated a value of $\Delta\mathrm{RV}=155$ km/s for this source. These
authors deemed it to be ‘marginally a binary from the RV point of view’. Their
tentative orbital solution has $P_{\mathrm{orb}}=37.6$ d and radial velocity
amplitude of $K=43$ km/s. Here we find that the RV of SMC AB9 is much more
constant ($\Delta\mathrm{RV}=15$km/s). This means that SMC AB9 was most likely
a spurious detection in F03, as these authors already suspected. We note that
when we use the He ii 4686 Å line for our RV measurements of this source, we
find $\Delta\mathrm{RV}=53$km/s, making it likely that line profile
variability (LPV; Fullerton et al., 1996; Dsilva et al., 2022) is at least
partially responsible for the marginal detection in F03.

The median standard deviation of the RV measurements for individual WR stars
is shown in the bottom right panel of Fig. 4. In the campaign of F03, the
median $\sigma_{\mathrm{RV}}=30$ km/s lies somewhat above the 20 km/s that
they quote as typical error for their measurements. Our campaign has a much
lower median $\sigma_{\mathrm{RV}}\approx 5$ km/s, in line with the $\Delta$RV
result. This $\sigma_{\mathrm{RV}}\approx 5$ km/s is, however, larger than the
statistical errors obtained from the fits to the CCFs, which are only of the
order of 1 km/s. The true error is most likely larger than that, for example
because of LPV.

In summary, we measure nearly constant RV values, which strongly reduces the
possible binary parameter space for potential binary companions. Still, we
find some RV variability that could not be explained by the statistical errors
alone. Below we explore two possibilities: that the small RV variability
arises from LPV (Sect. 4), and that our target stars are in binaries with low-
mass and/or far-away companions (Sect. 5).

##  4 Line profile variability

Here we take a more detailed look at spectrum 4 of SMC AB1, taken during
observation night 4, from which we infer an apparent antiphase RV motion
between the N v lines and the He ii absorption lines (Fig. 3). Figure 5 shows
that LPV does take place in SMC AB1: compared to the co-added spectrum,
spectrum 4 exhibits a higher flux at the red side (i.e., at longer
wavelengths) of the N v and He ii features. This excess of red flux causes an
apparent shift to the red for the emission features (left panels of Fig. 5),
which lead us to measure positive values for their relative RVs. Conversely,
the red excess causes an apparent shift to the blue for absorption features
(right panels of Fig. 5), and therefore we measured negative relative RVs for
them.

For the blue side of the N v emission features and the He ii absorption
features in spectrum 4, there seems to be no systematic shift compared to the
co-added spectrum, which would be expected for RV offsets caused by binary
motion. We conclude that LPV rather than binary motion is the most plausible
explanation for the measured RV shifts in spectrum 4 of SMC AB1, and for the
apparent antiphase motion between absorption and emission lines.

Figure 5: Normalized cutouts of the co-added VLT-UVES spectrum of SMC AB1
(black lines).We also show the spectrum taken during observing night 4 in our
observing campaign (orange lines), corrected for barycentric motion.

This absorption/emission antiphase motion is not as clearly seen in all of the
other spectra that we have taken. This could be explained by smaller
variability, as we measured relatively large RV shifts in spectrum 4 of SMC
AB1. Furthermore, in SMC AB4 the He ii 4339 Å and He ii 4542 Å lines are in
emission rather than in absorption, which may explain why the RV shifts
measured with the He ii and N v lines tend to be in the same direction for
individual spectra of this star (Fig. 3).

A value of $\sigma_{\mathrm{RV}}\approx 5$ km/s caused by LPV would be the
same as what was found by Dsilva et al. (2022). These authors studied LPV from
N v lines by taking over 20 spectra in a narrow time interval of the wide
binary star WR 138 in the Milky Way, which is an early-type WN star, like our
target stars. We note that even larger RV variations (measured with a N v line
in spectra of WR 6 in the Milky Way, a WN4 star) have been attributed to LPV
caused by a co-rotating interaction region – rather than binary motion – by
Barclay et al. (2024).

##  5 Possible presence of undetected companions

###  5.1 Sensitivity of our campaign to binary motion

In order explore the allowed binary parameter space for companion stars, we
follow a procedure similar to the one described in Dsilva et al. (2020),
Dsilva et al. (2022), and Shenar et al. (2023a). We summarize the taken
procedure below. We assume that our targeted WR stars have a mass of 20 M⊙
(Hainich et al., 2015; Schootemeijer & Langer, 2018). We explore orbital
periods in the range of $\log(P_{\mathrm{orb}}/\mathrm{d})=[0,0.1,\ldots,3.5]$
and companion masses in the range of $M_{\mathrm{comp}}=[1,2,\ldots,40]$ M⊙
(see Fig. 6). For each of the 1440 combinations of orbital period and
companion mass, we simulate our observing campaign 10000 times, by drawing a
random orientation of the orbital plane (i.e., a random $\cos i$, where $i$ is
the inclination angle to the line of sight) and a random orbital phase for the
first observation. Then we measure the RV of the WR star in this mock binary
at the same dates as in our observing campaign. We set the mock RV measurement
error $\sigma_{\mathrm{RV,\,mock}}$ to zero. This is a conservative
assumption, since we find that setting $\sigma_{\mathrm{RV,\,mock}}>0$
increases, on average, the mock $\Delta$RV values. Then, for each companion
mass and orbital period combination, we counted how often mock binary motion
would result in a $\Delta$RV value larger than what is observed for the WR
star.

Figure 6: Color-coded probability
$p(\Delta\mathrm{RV}_{\mathrm{sim}}>\Delta\mathrm{RV}_{\mathrm{obs}})$ that
binary-induced motion would lead to a RV variation exceeding the observed
$\Delta\mathrm{RV}_{\mathrm{obs}}$ in SMC AB9, for different combinations of
orbital period $P_{\mathrm{orb}}$ and companion mass (see text for details).
The higher this probability is, the less likely is the source an undetected
binary. The dashed blue vertical line indicates $P_{\mathrm{orb}}=1$ yr.
Figure 7: Top: correlation of the time interval $\Delta t_{\mathrm{succ}}$
between two successive RV measurements and their observed $\Delta$RVsucc. The
black crosses represent measurements taken in this work and in Marchenko et
al. (2007). Middle: the colored dots represent measurements of a simulated
population of long-period binaries with orbital periods between 1 and 10 years
(see text for details). For clarity, we show only 2 out of 100000 simulated
observing campaigns. Bottom: histogram of the 100000 fitted slopes. The two
vertical lines indicate the values of the two examples shown in the middle
panel.

The results for SMC AB9 ($\Delta\mathrm{RV}_{\mathrm{obs}}=15$ km/s) are shown
in Fig. 6. This figure shows that we would have most likely measured higher
$\Delta$RV values than observed if SMC AB9 had a binary companion. For
companion masses above 5 M⊙ and orbital periods below one year, this
probability exceeds 95% in 98,6% of the parameter space.

We can obtain an overall probability $\hat{p}$ that binarity would have been
detected by our measurements when we specify underlying orbital period and
mass ratio probability distributions. For flat distributions of
$\log(P_{\mathrm{orb}}/d)$ and mass ratio, this overall probability is just
the average of the probability values displayed in Fig. 6, which becomes
$\hat{p}=0.90$. However, binary evolution models predict that in a scenario
where the WR star formed through binary stripping, configurations with very
low-mass companions, as well as very short or very long orbital periods, are
avoided. When using underlying orbital period and mass ratio distributions
motivated by binary population synthesis models (Renzo et al., 2019; Langer et
al., 2020, see Appendix B.2 for details) we find even higher values of
$\hat{p}>0.97$, while when we truncate these distributions for
$P_{\mathrm{orb}}>30$ d – since those systems have already been detected – we
still find $\hat{p}>0.94$ for SMC AB9.

For our other target stars we obtain similar results (Figs. 15-20; Table 12).
Hence, for each individual WR star we can rule out most of the realistic
parameter space for binaries that may have previously interacted, in
particular where $P_{\mathrm{orb}}<1$ yr. We discuss binaries with
$1\,\mathrm{yr}<P_{\mathrm{orb}}<10\,\mathrm{yr}$ in Sect. 5.2. Performing the
same experiment with the F03 data reveals the possibility that companions of
some tens of Solar masses with $P_{\mathrm{orb}}\approx 1$ yr could have been
missed (Figs. 21-27; Table 14).

We also compute the most likely companion mass of our WR stars under the
assumption that, apart from the small $\sigma_{\mathrm{CCF}}\approx 1$ km/s,
the measured RV variability is caused by binary motion. Figure 14 shows that
for SMC AB9, this is about $1-3$ M⊙ for periods of up to one year, increasing
to beyond 5 M⊙ for periods longer than 3 yr. The most likely companion masses
reside very close to the 0.5 probability contour in Fig 6.

To construct Fig. 6 we assumed circular orbits, since we expect that if binary
interaction has taken place this would have circularized the orbit (Iorio et
al., 2023, eq. 20). As a test, we have repeated the simulations described in
this section with a flat eccentricity distribution in the range $0\leq e\leq
0.9$. This lead to detection probabilities that were smaller but only slightly
so (Table 13 and Fig. 12), and do therefore not change the interpretation of
our results.

For all the considered ranges of the possible binary parameter space described
in Appendix B, we find the lowest values for $\hat{p}$ when excluding orbital
periods below 30 d and including eccentric orbits (Table 13). For the seven
target WR stars, we then find an average of $\hat{p}_{\mathrm{P30+}}$ = 0.801.
This implies a chance that we missed a binary companion for all of our target
WR stars of the order of $(1-0.801)^{7}\approx 10^{-5}$ even for this set of
assumptions.

###  5.2 Long-period binaries

In the last section we have seen that our small observed $\Delta$RV values
make it unlikely that our target stars have binary companions, especially with
orbital periods of a year or less. Because we observed during three successive
semesters, it could be more likely that we missed binary companions with an
orbital period of the order of a few years in our campaign.

Here we perform an additional experiment where we explore the likelihood that
all our target WR stars are in 1-10 yr binaries. If so, we would expect to
measure small RV differences for observations that are separated days or weeks
apart, and larger RV differences for observations taken months or years apart.
In the top panel of Fig. 7 we plot with black crosses the time between
successive observations ($\Delta t_{\mathrm{succ}}$) against the RV difference
between them ($\Delta\mathrm{RV}_{\mathrm{succ}}$). For our seven target
stars, Fig. 7 shows a total of 42 data points for successive VLT-UVES
observations, including both our observations and those from Marchenko et al.
(2007). There is no significant correlation in the observational data, as we
obtain a value of $0.01\pm 0.10$ for the slope of the linear fit ( also shown
in the top panel of Fig. 7). This in line with our measured
$\sigma_{\mathrm{RV}}\approx 5$ km/s being caused by a combination of
measurement errors and LPV on timescales around one day or shorter.

To quantify how likely it is that $\Delta t_{\mathrm{succ}}$ and
$\Delta\mathrm{RV}_{\mathrm{succ}}$ do not show a significant positive
correlation if all our target stars are in $1-10$ yr binaries, we again
simulate our observing campaign as in Sect. 5.1, but now 100000 times, for all
seven target stars. We draw a random logarithm of the orbital period in the
range $0\leq\log(P_{\mathrm{orb}}/\mathrm{yr})\leq 1$ and a random mass ratio
in the range $0.05\leq M_{\mathrm{comp}}/M_{\mathrm{WR}}\leq 2$ (i.e., a
companion mass in the range $1-40$ M⊙), both from flat distributions (Öpik,
1924; Kouwenhoven et al., 2007). We add a Gaussian error to each mock binary
measurement, based on the observed $\sigma_{\mathrm{CCF}}$. The green and
orange dots in the middle panel of Fig. 7 show the results for two individual
simulated observing campaigns; the other 99998 are omitted for visibility
purposes. We find that in all except 13 simulations (99.987%), the linear fit
of $\Delta t_{\mathrm{succ}}$ and $\Delta\mathrm{RV}_{\mathrm{succ}}$ is
larger than the observed value of 0.01 (bottom panel of Fig. 7). Therefore, it
is highly unlikely that all our target stars are long-period binaries with of
$P_{\mathrm{orb}}=1-10$ yr.

###  5.3 Neutron star companions

Neutron stars (NSs) have masses in the range 1-2 M⊙, and hence we could only
detect them as companions to our target WR stars if they had orbital periods
of $\sim$30 days or less (Fig.6). However, NSs are expected to emit X-rays
arising from the accretion of material from the wind of the WR stars. F03
found upper limits of $L_{\mathrm{x}}<1-2\cdot 10^{33}$ erg/s based on ROSAT
data. Using eq. 1 from Shapiro & Lightman (1976) for the NS accretion radii
and the stellar parameters from Hainich et al. (2015), and adopting Bondi-
Hoyle accretion, we find that these X-ray non-detections exclude the presence
of NSs with orbital periods of $P_{\mathrm{orb}}\lesssim 100$ d. However, the
accretion rates, and thus the X-ray flux, could be significantly smaller if
the magnetosphere of the rotating NS star would expel a fraction of the
inflowing wind material.

Independent of this, it appears unlikely that all the apparently single WR
stars have NS companions, for three reasons. First, the observed WR stars in
the SMC are thought to originate from main-sequence stars with $M\gtrsim 40$
M⊙ (Schootemeijer & Langer, 2018; Shenar et al., 2020). Any more evolved
companion, e.g., a compact star, would have an initial mass similar to that or
higher, and therefore would most likely be a BH and not an NS (Schneider et
al., 2023). Second, the majority of NS producing binaries are expected to
break up when the corresponding SN occurs (Eldridge et al., 2011; Renzo et
al., 2019), which makes it unlikely that the number of WR+NS binaries is
comparable to the number of WR+O star progenitors (of which there are five in
the SMC; Shenar et al., 2016). Third, binary interaction in massive non-
disrupted systems with NS companions has been claimed to always lead to a
merger, explaining the absence of detected WR$+$NS binaries (van den Heuvel et
al., 2017; Toalá et al., 2018). Consequently, it appears unlikely for each of
our WR stars to have an NS companion, and we dismiss the possibility that all
our target WR stars have NS companions.

##  6 Other signatures of binarity

###  6.1 Photometric variability

We analyze photometric data that has become available in the last 20 years.
F03 have investigated time-series photometry data of all of the SMC WR stars
but SMC AB12 and the eclipsing binary SMC AB5 (Moffat et al., 1998). They
found no periodic signals except for the apparently single star SMC AB4, for
which they reported a 3.2$\sigma$ detection of a 6.55 d period variability.
This detection was for MACHO-B data (Alcock et al., 1999) but there were non-
detections for MACHO-R and OGLE I data (Udalski et al., 1998).

Using the lightkurve python package (Lightkurve Collaboration et al., 2018),
we analyze OGLE III (Udalski et al., 2008) and OGLE IV (Udalski et al., 2015)
time-series photometry data of all seven of our targets stars (A. Udalski,
private communication). These data cover a timespan of about twenty years.
With the ‘Lomb-Scargle’ method we obtain the power spectrum for each target WR
star, exploring $10^{6}$ equidistant frequencies in the range
$0.001\,\mathrm{d}^{-1}<f<0.67\,\mathrm{d}^{-1}$. Then we used the flatten
function to calculate the SNR spectrum. We provide an overview of the OGLE
data and the highest SNR values calculated from them in Table 11. Nowhere in
the considered frequency range did any of our target stars approach a
threshold S/N of 4.6 (Bowman & Michielsen, 2021) for periodic variability. We
also provide the power spectra and phase-folded light curves in Fig. 11, which
further support the absence of convincing binary-induced signals. We are thus
not able to find a significant periodic signal for any of our target stars,
including SMC AB4.

###  6.2 Imprints of companions in the spectra

SMC AB1, 9, 10, 11, and 12 are hot objects ($T_{*}\geq 80$ kK; Hainich et al.,
2015) and therefore potential cooler companions could be found by their
imprints in the spectra. For example, we look for the He i 4471 Å line, which
appears in stars later than O5 (Walborn et al., 2000). None of the hot WR
stars in our sample show any sign of this He i line (also not after
rebinning). This might be unsurprising since even in the composite spectra of
the five binary SMC WR stars (which have O-star companions of tens of Solar
masses; see Shenar et al., 2016) the He i 4471 Å absorption line does not go
deeper than $0.02-0.1$ times the continuum. For the presence of lower-mass
(and thus dimmer) stars, our RV measurements can be expected to place much
stronger constraints than line strength measurements.

Other lines that we attempt to find in search for companions of hot WR stars
are He i 4387 Å, He i 4713 Å, He i 4922 Å, Si iv 4089 Å, Si iv 4116 Å, and O
ii 4350 Å. We note that these tend to be weaker than the He i 4471 Å line in
the spectra of WR binaries in the Magellanic Clouds (Shenar et al., 2016,
2019). We do not see a signature of any of these lines in the spectra of the
hot WR stars that we targeted. We conclude that we are unable to find imprints
of companion stars in the WR spectra, in line with the results from our RV
measurements.

Finally, we inspect spectra of SMC AB2 and SMC AB4 from the Ultraviolet Legacy
Library of Young Stars as Essential Standards (ULLYSES; Roman-Duval et al.,
2020). We notice that for both stars, the C iv feature around 1550 Å is
saturated, which is indicative of an absence of another hot star (because
another hot star with a weaker wind than the WR star could contribute to the
flux in this line region). The only other SMC WR star in the ULYSSES sample is
SMC AB9, which is much hotter than SMC AB2 and SMC AB4 and lacks a strong C iv
feature.

###  6.3 Absolute radial velocities and proper motions

So far we have found little evidence for our target WR stars currently being
in binary systems. In principle, they may have been in binary systems in the
past, that have broken up or merged (see also Sect. 7.1). If they were in
binary systems that have been disrupted by supernova explosions, this will
result in higher space velocities. To test this, we investigate their absolute
RVs and proper motions.

The absolute RVs are obtained from cross-correlating our co-added spectra
(Sect. 2.2) with the best-fit model atmosphere spectra from Hainich et al.
(2015). Based on where the best fit of this model to the observations is
achieved, we select three line regions for the cross-correlation. Two of these
are the He ii 4339 Å and He ii 4686 Å lines. As the third line region we
select N iv 4058Å for the two coldest objects (SMC AB 2 and SMC AB4), and N v
4944 Å for the other five WR stars. We define our measured absolute RV as the
average measured RV and the error as the standard deviation of the three
measurements (again with 1.5 degree of freedom subtracted in the divisor). We
correct the co-added spectra for barycentric motion by subtracting the average
barycentric velocity, which we obtain with the baryCorr function of the python
package pyasl. (Piskunov & Valenti, 2002) The results are shown in Table 2. We
then compare the absolute RVs of our WR stars to those from stars in their
vicinity. For this comparison, from the GAIA
archive222https://gea.esac.esa.int/archive/ we download all DR3 sources (Gaia
Collaboration et al., 2023) that have measured RVs and reside within 15′ of
our target WR stars. We exclude sources with a parallax larger than five times
the parallax error to remove foreground sources (Aadland et al., 2018;
Schootemeijer et al., 2021). Fig. 8 shows the absolute RVs of our targeted WR
stars and the nearby sources within 15′.

Figure 8:  RVs of our targeted WR stars (vertical orange lines; the shaded
region indicates the error margin). The blue histograms show RVs of GAIA DR3
sources in a 15′radius.

Five out of seven single WR stars have absolute RVs that are consistent with
those of sources in their surroundings. For SMC AB11, its
$\mathrm{RV_{\mathrm{abs}}}$ of $228\pm 8$ km/s is significantly different to
sources in its surroundings, where we find $\mathrm{RV_{\mathrm{abs}}=139\pm
21}$ km/s (the error being the standard deviation of the nearby sources’ RV,
and 139 km/s being their average RV value). This makes SMC AB11 a runaway
star, moving with a relative velocity of $91\pm 23$ km/s in the line-of-sight
direction. Similarly, for SMC AB1 we derive a relative velocity of $68\pm 35$
km/s based on its $\mathrm{RV_{\mathrm{abs}}=203\pm 8}$ km/s, and an
$\mathrm{RV_{\mathrm{abs}}=135\pm 35}$ km/s for sources in its vicinity. This
qualifies SMC AB1 also as a runaway star.

As a check, we also look up the proper motions of SMC WR stars in GAIA DR3 and
compare them to surrounding sources within 5′. We remove foreground sources as
described above, and we discard sources dimmer than $G=17$ or with
$\mathrm{RUWE}>1.4$. We translate the proper motions intro transverse
velocities $v_{\mathrm{trans}}$ adopting an SMC distance of 62.44 kpc (Graczyk
et al., 2020). The results are shown in Fig. 9. The transverse velocities of
most of our WR stars are similar to those of their surrounding sources,
except, again, for SMC AB1 and SMC AB11. We find a relative two-dimensional
$v_{\mathrm{trans}}\approx 107\pm 33$ km/s for SMC AB11 compared to the
average of the surrounding stars (the error follows from the GAIA proper
motion uncertainty of SMC AB11). This confirms its nature as a runaway star.
Similarly, for SMC AB1 we measure a relative $v_{\mathrm{trans}}=78$ km/s, in
line with the result for the absolute RV.

Puzzlingly, we also find a high relative transverse velocity for SMC AB6,
which is unexpected given that this is part of a multiple system (Shenar et
al., 2018). This might indicate that the SMC WR proper motions are more
uncertain than the errors indicate, perhaps because these very massive stars
that tend to be in crowded regions. Additionally, we note that, given the
errors, our method is not sensitive to intermediate-velocity runaway stars.

Figure 9: Velocities transverse to the line of sight. Our targeted WR stars
are shown in orange, and binary SMC WR stars are shown in magenta. The shown
errors are obtained from proper motion errors. GAIA DR3 sources in a 5′ radius
are shown in blue.

We take a closer look at the runaway star SMC AB11, which has a bright red
source only 1″ away that we also put in the slit (Sect. 2.1). From the
composite spectrum, we measure the absolute RV of the red source to see if it
moves in the opposite direction of SMC AB11, which would be expected if they
were in a multiple system that has been broken up. However, using the narrow H
$\alpha$ line we find $\mathrm{RV}_{\mathrm{abs}}\approx 150$ km/s, which is a
very typical value in its environment (Fig. 8). This implies that the red
source is a chance alignment rather than that it originates from a common
multiple system with SMC AB11.

Finally, we investigate the supernova remnant (SNR) population in the SMC
(Maggi et al., 2019; Matsuura et al., 2022). SMC AB11 is closest to SNR
J0052-7236, which resides at a projected distance of 50 pc (3′). It has an age
of about 35 kyr (Leahy & Filipović, 2022). SMC AB11 would need much higher
velocity of $v_{\mathrm{trans}}\approx 1500$ km/s to reach a 3′ separation
within the lifetime of the SNR. In addition, the proper motion vector does not
point back to SNR J0052-7236. Therefore, we cannot associate SMC AB11 to an
SNR. We conclude the same for SMC AB1, which is not close to any known SNR in
the SMC. Our inability to link SMC AB1 and SMC AB11 to a ‘smoking gun’ SNR
does not rule out that they are post-supernova runaways, since the helium-
burning WR lifetime is about ten times longer than the oldest SNR in the SMC
(which is 35 kr; Leahy & Filipović, 2022). We will further discuss the
implications of the runaway nature of SMC AB1 and SMC AB11 in Sect. 7.1.

##  7 Discussion

Schootemeijer & Langer (2018) found a fundamental difference in the internal
constitution between binary and apparently single WR stars in the SMC. The
binary WR stars showed a shallow internal radial H/He-gradient as expected
from a retreating convective core during central hydrogen burning, whereas the
apparently single WR stars were shown to have an up to 10-times steeper H/He
gradient, and a correspondingly less massive H-rich envelope. The shallow
gradient of the WR stars in binaries is well reproduced by the mass
transferring short period binary evolution models of Pauli et al. (2022).
However, the steep H/He-gradients require another WR star formation history.

To form a steep H/He-gradient requires that the region with the naturally
shallow gradient is mixed. As discussed by Schootemeijer & Langer (2018), this
may occur either by allowing the WR progenitor to expand during its post main-
sequence evolution, such that convection induced by either hydrogen shell
burning or by a convective envelope could mix the corresponding layers, or by
increasing the convective core in a hydrogen-burning mass gainer. In both
cases, the WR star might have lost its envelope by binary mass transfer or, as
a single star, by winds or eruptions. Below, we discuss the scenarios proposed
in the literature that allow for the formation of single WR stars.

###  7.1 Formation channel of single WR stars in the SMC

##### Chemically homogeneous evolution.

In CHE models, rotationally-induced mixing brings hydrogen-depleted material
to the surface of exceptionally rapidly spinning stars (e.g., Maeder, 1987;
Brott et al., 2011). This scenario works most efficiently at low metallicity
(Yoon et al., 2006). As such, CHE could in principle explain the presence of
hydrogen-poor WR stars in the SMC (Martins et al., 2009; Hainich et al., 2015;
Ramachandran et al., 2019); and see Koenigsberger et al. (2014) for SMC AB5.
However, five out of seven of the single WR stars (SMC AB1, 9, 10, 11, and 12)
are too hot to be explained by core hydrogen burning models (Schootemeijer &
Langer, 2018, their fig. 2), but have, at the same time, a high surface
hydrogen mass fraction (0.36 on average; Hainich et al., 2015). Their chemical
structure is thus very inhomogeneous. In addition, the apparently single SMC
WR stars show low projected rotational velocities (Martins et al., 2009;
Hainich et al., 2015; Vink & Harries, 2017).

##### Stellar mergers.

We expect that mergers involving H-rich stars do not produce merger products
resembling the observed single SMC WR stars, as these typically contain less
than a Solar mass of hydrogen (Schootemeijer & Langer, 2018). In principle it
is possible to produce a H-poor merger product when both stars develop He
cores and then interact. Such an evolutionary path has been produced for the 2
M⊙ quasi-WR star HD 45166 (Shenar et al., 2023b). However, at high stellar
masses (where stars do not develop degenerate helium cores, meaning that
companions have little time to catch up in their evolution) this might require
fine-tuning of the initial mass ratio and physics assumptions (Pols, 1994;
Wellstein & Langer, 1999). Also, if mergers of two H-poor stars produced all
single SMC WR stars, we would expect that in the WR+MS systems, which are
their progenitors, the MS stars are close to the end of their H-burning
lifetime or close to filling their Roche lobes. However, typically the MS
companions are early O stars that are far from filling their Roche lobes
(Shenar et al., 2016), which are not expected to exhaust H in their cores or
otherwise instigate interaction before the end of the life of their companion.
We are therefore unaware of a merger channel that is likely to produce seven
out of twelve SMC WR stars.

##### Binary stripping.

At first glace it seems that binary companions have not stripped our seven
targeted SMC WR stars, because it is unlikely that we missed them (Sect. 5).
While it is possible that we missed some rather low mass companion in a wide
orbit, it is implausible that such a star could have removed the hydrogen
envelope of a WR star in the past. As the progenitors of our WR stars were
more massive than $\sim$40 M⊙ (Sect. 5.3), their binaries with companion
masses below $\sim$5 M⊙ would have such an extreme initial mass ratio that
stable mass transfer is not expected (Henneco et al., 2023). Unstable mass
transfer in a system with such a mass ratio is predicted to result in a merger
(Kruckow et al., 2016).

It is, in principle, possible that a WR star is stripped by a binary companion
that has since exploded as a supernova, which disrupted the binary system.
However, as discussed for the merger scenario above, this requires one binary
component to end hydrogen burning before the other reaches core collapse. This
is unlikely to happen in most of the initial binary parameter space. Still,
binary stripping might be a minority channel – see also the cases of SMC AB1
and SMC AB11 (Sect. 6.3).

Recently, Pauli et al. (2023) explained an apparent bimodality in the
temperatures of SMC WR stars by proposing that the colder WR stars SMC AB 2
and 4 are accretors in binary systems. This is in contrast to the lack of
detected RV variability in our study.

##### Single star mass loss.

Evolutionary models with SMC or near-SMC metallicity tend to use wind mass
loss recipes in which winds are too weak to strip stars with initial masses
around 40 M⊙ (e.g., Brott et al., 2011; Georgy et al., 2013; Choi et al.,
2016; Limongi & Chieffi, 2018; Schootemeijer et al., 2019). Could the mass
loss rates have been underestimated in these models? For MS stars, the models
typically use the recipe of Vink et al. (2001) for which recently evidence has
built up that it overestimates rather than underestimates MS mass loss (by a
factor $\sim$3; Björklund et al., 2021; Brands et al., 2022). Perhaps more
likely is that stars lose a significant amount of mass as cool stars after
their core-hydrogen burning phase.

One possibility for post-main-sequence self-stripping is eruptive mass loss in
luminous blue variable (LBV) stars. The LBV phenomenon might be related to
stars reaching the Eddington limit (Gräfener et al., 2012; Sanyal et al.,
2017). According to recent predictions, this can happen in evolved SMC stars
with initial masses as low as 30 M⊙ and lead to eruptive mass loss that can
strip their H-rich envelope (Cheng et al., 2024). This seems to be consistent
with the initial masses of $M\gtrsim 30-40$ M⊙ for SMC WR stars (Table 3). The
SMC harbors two stars that show or have shown LBV behavior (see Richardson &
Mehner, 2018, also for two more candidates). One of these (HD 5980) underwent
an LBV outburst in the 1990s and at present manifests itself as a WR star in
the only WR+WR binary in the SMC (Koenigsberger et al., 2010). The other is R
40 (Szeifert et al., 1993; Agliozzo et al., 2021), which is currently in an
LBV eruption episode, and it has an uncertain but seemingly high mass loss
rate of $10^{-4}-10^{-3}$ M⊙/yr (Campagnolo et al., 2018). Both sources imply
that LBV mass loss could play a role in the production of WR stars at low
metallicity. An SMC population of two LBVs and of the order of ten wind-
stripped WR stars would be in line with the LMC and Milky Way, where LBVs are
also more rare than WR stars by a factor of at least a few (Richardson &
Mehner, 2018; van der Hucht, 2001; Shenar et al., 2019). A caveat might be
that LBVs in the LMC seem to be much more isolated than O and WN (although not
WC) stars, which could indicate that (at least in the LMC) LBVs could be
accretors in binary systems (Smith & Tombleson, 2015).

Wind mass loss during a red supergiant (RSG) phase could also be responsible
for stripping the outer layers of our WR progenitors. Mass loss rates of RSGs
are notoriously uncertain and also their metallicity dependence is not
established. Therefore, RSG mass loss rates in the SMC could occupy a wide
range of values – see e.g. fig. 15 of Yang et al. (2023). Some mass loss rate
predictions shown there increase steeply with luminosity and reach values at
$\log(L/\mathrm{L}_{\odot})\approx 5.5$ that are high enough to strip H-rich
envelopes of massive stars on timescales of $\sim$10 kyr (Beasor et al., 2023;
Yang et al., 2023). However, these mass-loss rates at the bright end are the
most uncertain, where RSGs are scarce. Related to this, we note that Kee et
al. (2021) found that high and metallicity-independent mass loss rates can be
achieved if turbulent velocities in RSGs are high.

##### Resumé.

From the discussion above, we conclude that chemically homogeneous evolution,
stellar mergers, and binary stripping are unlikely to constitute the dominant
channel for producing single WR stars in the SMC. It follows that single star
mass loss – most probably in the LBV or RSG phase – is likely responsible for
the loss of the H-rich envelope in the majority of the investigated SMC WR
stars.

Figure 10: Initial mass – final mass relations for SMC stars. Data points of
SMC WR stars are shown with circles (based on Hainich et al., 2015) and
squares (based on Martins et al., 2009; Martins, 2023), the numbers indicating
their identifier (e.g., 1 for SMC AB1). The grey shaded region indicates He
core masses at the terminal-age main sequence ($M_{\mathrm{He\,core,\,TAMS}}$)
from evolutionary models of Schootemeijer et al. (2019), for overshooting
values between 0.22 and 0.44. The colored lines show the following relations
from literature: SEVN (Spera et al., 2015), Schn23 (Schneider et al., 2023),
Bonn (Schootemeijer et al., 2019), Geneva (Georgy et al., 2013), MIST (Dotter,
2016; Choi et al., 2016), Belc10 (Belczynski et al., 2010, data taken from
fig. 2 of Banerjee et al. (2020)), MOBSE (Giacobbo et al., 2018), and Hurl00
(Hurley et al., 2000, data taken from fig. 1 of Belczynski et al. (2010)).

###  7.2 Implications for the outcome of massive binary interaction at low
metallicity.

After showing that binary evolution was likely not important for the majority
of the seven targeted SMC WR stars, let us assume that the detected binaries
constitute the whole SMC WR binary population. All five binaries have short
orbital periods in the range $6\,\mathrm{d}<P_{\mathrm{orb}}<20\,\mathrm{d}$.
We use initial-final orbital separation relations (Soberman et al., 1997;
Tauris & van den Heuvel, 2006) to find possible solutions for the initial
orbital periods for SMC AB6, 7, and 8 (SMC AB3 has a poorly defined mass
ratio, and the initial-final separation formula likely does not apply to the
evolutionary history of SMC AB5; see Koenigsberger et al., 2014; Schootemeijer
& Langer, 2018). Under the assumption that mass stripped away from the donor
either ends up on the accretor or is lost from the accretor, there are no
solutions for $P_{\mathrm{orb,\,ini}}>25$ d, regardless of the adopted mass
transfer efficiency. Even though this is low-number statistics, the apparent
absence of medium- to long-period binaries in the SMC WR population appears
striking. It could imply that massive longer-period binaries at low
metallicity merge upon binary interaction. This might impede the channel for
producing binary BH mergers from isolated binary evolution, which often
involves stars with large initial periods (Belczynski et al., 2016; Kruckow et
al., 2018). Therefore, this needs to be carefully checked in future work.

Interestingly, there are also no known binaries with $P_{\mathrm{orb}}\geq 40$
d among the ‘classical’ WR stars on the nitrogen sequence (subtype earlier
than WN5) in the Large Magellanic Cloud (LMC – Foellmi et al., 2003b; Shenar
et al., 2019). Since there are over ten times more WN stars in the LMC than in
the SMC, this seems to exacerbate the lack of long- and intermediate period WR
binaries at low metallicity. However, the LMC WR stars have not been monitored
with high-quality observations such as those presented in this study. The LMC
would thus be a good place to further investigate this issue.

Additionally, all SMC WR stars in binaries have companions that are of similar
or higher mass than the WR star (with mass ratios between 1 and 3; F03, Shenar
et al., 2016). This may require a non-negligible mass transfer efficiency in
the systems that survived binary interaction.

###  7.3 Implications for BH masses at low metallicity

Our finding that the majority of the apparently single SMC WR stars lost their
envelope without the help of a binary companion has direct consequences for
the BH population produced by the massive stars in the SMC. It makes these
stars uniquely suitable to investigate the initial mass – final mass
($M_{\mathrm{ini}}-M_{\mathrm{final}}$) relation because of two properties: i)
they have H-rich layers near their surfaces, but these contain only $\sim$1 M⊙
of material (Schootemeijer & Langer, 2018), and, ii), they have low mass loss
rates (at which they lose $M\lesssim 1\,\mathrm{M}_{\odot}$ during their WR
lifetime; Hainich et al., 2015). This implies that their current mass, their
He core mass, and their final mass $M_{\mathrm{final}}$ are practically the
same333SMC AB2 and SMC AB4 are relatively cold objects that could also burn H
in their cores (Schootemeijer & Langer, 2018). We therefore exclude them from
this analysis.. We will use this property to obtain the
$M_{\mathrm{ini}}-M_{\mathrm{final}}$ relation of the single SMC WR stars.

First, we obtain the He core masses of the SMC WR stars from the best-fit
models of Schootemeijer & Langer (2018), from their large grid of synthetic
massive star models with pre-defined helium core masses, surface hydrogen
abundances, and H/He gradients. We then use the initial mass – He core mass
relations from stellar evolution models (Schootemeijer et al., 2019, with an
overshooting value of 0.33) to obtain $M_{\mathrm{ini}}-M_{\mathrm{final}}$
relations. To estimate the errors, we assume that the uncertainty in
luminosity (from Hainich et al., 2015; Shenar et al., 2016, 2018) propagates
into the uncertainty in mass. The error on the mass then depends on the slope
of the mass-luminosity relation, which we obtain from Gräfener et al. (2011).
We repeat the analysis based on the observational WR star properties obtained
by Martins et al. (2009) and Martins (2023), who found somewhat lower
luminosities.

The results define the empirical initial-final mass relation shown as the
thick grey line in Fig. 10, whose width reflects the error in this relation
(see also Tab. 3). It is well defined in the initial mass range $40\dots 65$
M⊙ (or $30\dots 50$ M⊙ based on the luminosities of Martins et al.), and
extrapolated to $\sim 100$ M⊙.

In Fig. 10, we compare our result with initial-final mass relation or initial
mass-BH mass relations obtained from detailed stellar evolution calculations,
or used in so-called rapid binary evolution models. To obtain data from works
from the literature that provide no online data apart from their plots, we
used Engauge Digitizer again. Belczynski et al. (2010), Banerjee et al.
(2020), and Giacobbo et al. (2018) provide remnant masses rather than final
stellar masses – for these, to obtain the final masses we divide the remnant
masses by 0.90, as these studies assumed that the gravitational BH mass is 10%
smaller than the baryonic mass. We only show the mass range
$M_{\mathrm{ini}}>32$ M⊙, since for lower masses, where the CO core mass is
lower than 11 M⊙, these codes assume BH fallback fractions lower than unity
(Fryer et al., 2012). The $M_{\mathrm{ini}}-M_{\mathrm{final}}$ relations from
the literature roughly come in two flavors: those that experience strong LBV
mass loss in the mass range $40<M_{\mathrm{ini}}/\mathrm{M}_{\odot}<80$, and
those that do not.

Table 3:  Initial masses and He core masses found for the SMC WR stars. The errors on $M_{\mathrm{core}}$ and $M_{\mathrm{ini}}$ are correlated. Potsdam refers to Hainich et al. (2015) for the single stars and Shenar et al. (2016) for the binaries. Martins refers to Martins et al. (2009) and Martins (2023). Notes have the following meaning. a): object has a surface temperature that matches both H-core and He-core burning models; b): values from Wang et al. (2019). | $M_{\mathrm{core}}/\mathrm{M}_{\odot}$ | $M_{\mathrm{ini}}/\mathrm{M}_{\odot}$ | $M_{\mathrm{core}}/\mathrm{M}_{\odot}$ | $M_{\mathrm{ini}}/\mathrm{M}_{\odot}$
---|---|---|---|---
Based on: | Potsdam | Potsdam | Martins | Martins
Single: |  |  |  |
SMC AB1 |  31.6${}^{+11.4}_{-8.3}$ |  63.8${}^{+18.7}_{-14.1}$ |  21.7${}^{+5.1}_{-4.2}$ |  47.1${}^{+8.7}_{-7.1}$
SMC AB2a) |  16.3${}^{+2.4}_{-2.1}$ |  37.9${}^{+4.1}_{-3.6}$ |  13.3${}^{+1.9}_{-1.7}$ |  32.6${}^{+3.3}_{-2.9}$
SMC AB4a) |  18.4${}^{+2.8}_{-2.4}$ |  41.4${}^{+4.8}_{-4.2}$ |  21.3${}^{+3.4}_{-2.9}$ |  46.5${}^{+5.8}_{-5.0}$
SMC AB9 |  29.6${}^{+10.6}_{-7.8}$ |  60.5${}^{+17.5}_{-13.4}$ |  16.6${}^{+3.9}_{-3.1}$ |  38.4${}^{+6.7}_{-5.5}$
SMC AB10 |  19.5${}^{+6.2}_{-4.7}$ |  43.4${}^{+10.6}_{-8.2}$ |  12.9${}^{+2.9}_{-2.4}$ |  31.9${}^{+5.0}_{-4.1}$
SMC AB11 |  25.5${}^{+8.6}_{-6.4}$ |  53.6${}^{+14.4}_{-11.0}$ |  12.3${}^{+2.7}_{-2.2}$ |  30.9${}^{+4.8}_{-3.9}$
SMC AB12 |  26.5${}^{+9.1}_{-6.8}$ |  55.3${}^{+15.2}_{-11.5}$ |  |
Binary: |  |  |  |
SMC AB3 |  24.7${}^{+1.9}_{-1.8}$ |  52.2${}^{+3.2}_{-3.0}$ |  |
SMC AB5a) |  43.1${}^{+7.8}_{-6.8}$ |  82.8${}^{+12.6}_{-10.8}$ |  |
SMC AB6 |  22.9${}^{+3.6}_{-3.1}$ |  49.2${}^{+6.1}_{-5.3}$ |  |
SMC AB7 |  30.4${}^{+5.1}_{-4.4}$ |  61.8${}^{+8.5}_{-7.3}$ |  |
SMC AB8b) | 35 | 80 |  |

####  7.3.1 Initial – final mass relations with strong LBV mass loss

The mass-loss prescription from Belczynski et al. (2010) includes a strong,
metallicity-independent LBV mass loss rate of $1.5\cdot 10^{-4}$ M⊙/yr, which
in practice removes all the hydrogen-rich layers of stars born more massive
than 40 M⊙. It is widely used to predict BH-BH merger rates in rapid binary
evolution codes (e.g., COMPAS; Riley et al., 2022). It is also used in a non-
default simulation of MOBSE (MOBSE2; Giacobbo et al., 2018) and the triple
evolution code TRES (Toonen et al., 2016; Kummer et al., 2023). Also many
N-body simulations of dynamical BH-BH binary formation include the
prescription of Belczynski et al. (2010), for example simulations of young
star clusters (Chattopadhyay et al., 2022; Fragione et al., 2022a) which
include NBODY7 (Banerjee et al., 2020; Banerjee, 2021) and PeTar (Wang et al.,
2020; Barber et al., 2023), as well as simulations around galactic nuclei
(Antonini & Rasio, 2016; Tagawa et al., 2020; Fragione et al., 2022b), and
simulations of globular clusters (Rodriguez et al., 2016; Antonini & Gieles,
2020). The high amount of late-phase mass loss in these simulations seems to
be in agreement with our findings at SMC metallicity. However, we find (while
assuming negligible WR mass loss) somewhat higher final masses than what is
predicted by Belczynski et al. (2010), most likely because of their high WR
mass loss rates, which are about five times larger than those that are
observed for single SMC WR stars (Hainich et al., 2015).

The mass loss prescription of Hurley et al. (2000) includes, apart from strong
LBV mass loss, metallicity-independent WR winds. These overestimate the mass
loss rates of SMC WR stars by about a factor twenty compared to what is
observed by Hainich et al. (2015), and lead to very low final masses. The
Hurley et al. (2000) prescription is used in BH-BH merger simulations of
globular clusters (Askar et al., 2017; Hong et al., 2018; Samsing et al.,
2018) and star clusters near galactic nuclei (Petrovich & Antonini, 2017).

####  7.3.2 Initial – final mass relations with weak LBV mass loss

Many codes use or obtain $M_{\mathrm{ini}}-M_{\mathrm{final}}$ relations under
the assumption of less drastic late-phase mass loss near SMC metallicity,
which leads to higher final masses than we infer for SMC WR stars. Examples
are SEVN (Spera et al., 2015), which uses PARSEC models (Tang et al., 2014;
Chen et al., 2015), the single-star models of Schneider et al. (2023), MIST
(Dotter, 2016; Choi et al., 2016), Geneva models (Georgy et al., 2013), and
Bonn models (here showing Schootemeijer et al., 2019, which uses the same
mass-loss recipe as Brott et al. (2011) and ComBinE (Kruckow et al., 2018)).
Moreover, in the latest prescription of MOBSE (Giacobbo et al., 2018) called
MOBSE1, the LBV mass loss prescription is $Z$-dependent for stars with initial
masses up to $\sim$80 M⊙ (i.e., up to the point where stars approach the
Eddington limit, following Tang et al., 2014; Chen et al., 2015). This allows
40-80 M⊙ models near SMC metallicity to retain more H-rich layers and produce
much more massive BHs than with the prescription of Belczynski et al. (2010).
The MOBSE1 prescription is used in other rapid binary evolution codes (e.g.,
COSMIC; Breivik et al., 2020) and N-body simulations of young star clusters
(Di Carlo et al., 2019; Santoliquido et al., 2020; Rastello et al., 2021;
Torniamenti et al., 2022), nuclear star clusters (Stephan et al., 2019; Wang
et al., 2021), and globular clusters (Mapelli et al., 2021). Our results imply
that the mass loss rates used in these calculations are mostly too low.

####  7.3.3 Implications for population synthesis of double BH merger
progenitors

Our results indicate that $M_{\mathrm{ini}}-M_{\mathrm{final}}$ relations that
include strong late-phase mass loss (e.g., Belczynski et al., 2010) are
preferable at $Z_{\mathrm{SMC}}$. Relations calculated without strong late-
phase mass loss around SMC metallicity in the initial mass range of 40 M⊙ to
80 M⊙ (e.g., MOBSE1, SEVN) seem to overestimate final masses of single stars.
This result might be of particular importance for simulations of double BH
mergers produced via the dynamical channel, since there stars that lived as
single stars can be involved in double BH mergers.

For double BH merger progenitors produced by isolated binary evolution, the
inferred strong late-phase mass loss might only be of importance if it is LBV
rather than RSG mass loss that sheds the envelope. Unlike strong RSG mass
loss, strong LBV mass loss might prevent very massive stars from expanding to
large radii, and thereby prevent a common-envelope phase that tightens the BH-
BH progenitor system. Our study cannot identify which of the two channels (LBV
or RSG mass loss) is favorable.

##  8 Conclusions

We have used high-quality spectra taken with VLT-UVES during three consecutive
semesters to monitor the RVs of all seven apparently single SMC WR stars, in
search of binary reflex motion. To measure the RVs we used narrow N v lines,
which turned out to be remarkably stable. We found $\sigma_{\mathrm{RV}}$
values of $\sim$5 km/s and all seven WR stars had $\Delta$RV values in the
range of 6 km/s to 23 km/s, including the former binary candidate WR star SMC
AB9. We argue that LPV from atmospheric or wind variability can account for
these small RV variations.

For individual stars, our Monte Carlo simulations exclude the presence of
companions more massive than 5 M⊙ and orbital periods shorter than one year
with $\sim$95% confidence or more. Statistically, this makes it practically
impossible that we missed such companions for all of them. Companions with
smaller masses are thought to rather merge with their primary star than to
strip its envelope. When considering underlying orbital period distributions
up to ten years, we still estimate a probability below $\sim$10-5 that all our
seven target SMC WR stars are binaries. Based on a lack of correlation of
$\Delta$RV measurements between successive exposures and the time elapsed
between them, we can further rule out that all of our target stars are in
long-period ($1-10$ yr) binaries. Moreover, our simulations show that previous
studies were not sensitive to a large part of the binary parameter space. The
fraction of SMC WR stars with detected binary companions remains at 0.42.

SMC AB1 and SMC AB11 appear to be runaway stars with relative line-of-sight
velocities of $\sim$80 km/s, compared to their surroundings. The proper
motions of these sources strengthen this finding. While such high velocities
are not preferred in population synthesis models, Eldridge et al. (2011)
predict runaway WR stars with space velocities up to 120 km/s. Furthermore,
Evans et al. (2011) and Sana et al. (2022) find runaway O stars with up to
$\sim$90 M⊙in the LMC with similarly high runaway velocities.

Our results imply that for SMC stars above $\sim$40 M⊙, single stars lose the
major part of their hydrogen-rich envelope, and that a binary companion is not
required for this. We argue that late-phase mass loss in the LBV or RSG phase
is the most likely mechanism to strip these stars so they become WR stars. Our
results put strong constraints on the maximum black hole mass which can emerge
from stars above $\sim$40 M⊙.

All five binary WR stars in the SMC show orbital periods below 20 d. Although
their number is small, this could imply that massive interacting long-period
binaries at low metallicity merge. While more work is needed to substantiate
this hypothesis, it would suggest that the predictions for the double black
hole merger rate based on isolated evolution of long period binaries are
largely overestimated.

###### Acknowledgements.

We thank the anonymous referee for a constructive report and useful
suggestions. We warmly thank Andrzej Udalski for sending us the OGLE-III and
OGLE-IV data of our target stars. Equally warmly, we thank Rainer Hainich for
providing the best-fit model atmosphere spectra. AS thanks Frank Backs, Sarah
Brands, Michal Pawlak, Philipp Podsiadlowski, and Fabian Schneider for useful
discussions. We thank ESO for delivering the excellent spectra that made this
study possible. The research leading to these results has received funding
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme (grant agreement numbers 772225:
MULTIPLES).

## References

  * Aadland et al. (2018) Aadland, E., Massey, P., Neugent, K. F., & Drout, M. R. 2018, AJ, 156, 294
  * Abbott et al. (2016) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, Physical Review Letters, 116, 061102
  * Adams et al. (2023) Adams, N. J., Conselice, C. J., Ferreira, L., et al. 2023, MNRAS, 518, 4755
  * Agliozzo et al. (2021) Agliozzo, C., Phillips, N., Mehner, A., et al. 2021, A&A, 655, A98
  * Alcock et al. (1999) Alcock, C., Allsman, R. A., Alves, D. R., et al. 1999, PASP, 111, 1539
  * Antonini & Gieles (2020) Antonini, F. & Gieles, M. 2020, Phys. Rev. D, 102, 123016
  * Antonini & Rasio (2016) Antonini, F. & Rasio, F. A. 2016, ApJ, 831, 187
  * Askar et al. (2017) Askar, A., Szkudlarek, M., Gondek-Rosińska, D., Giersz, M., & Bulik, T. 2017, MNRAS, 464, L36
  * Azzopardi & Breysacher (1979) Azzopardi, M. & Breysacher, J. 1979, A&A, 75, 120
  * Banerjee (2021) Banerjee, S. 2021, MNRAS, 500, 3002
  * Banerjee et al. (2020) Banerjee, S., Belczynski, K., Fryer, C. L., et al. 2020, A&A, 639, A41
  * Barber et al. (2023) Barber, J., Chattopadhyay, D., & Antonini, F. 2023, arXiv e-prints, arXiv:2310.02376
  * Barclay et al. (2024) Barclay, K. D. G., Rosu, S., Richardson, N. D., et al. 2024, MNRAS, 527, 2198
  * Bartzakos et al. (2001) Bartzakos, P., Moffat, A. F. J., & Niemela, V. S. 2001, MNRAS, 324, 18
  * Beasor et al. (2023) Beasor, E. R., Davies, B., Smith, N., et al. 2023, MNRAS, 524, 2460
  * Belczynski et al. (2010) Belczynski, K., Dominik, M., Bulik, T., et al. 2010, ApJ, 715, L138
  * Belczynski et al. (2016) Belczynski, K., Holz, D. E., Bulik, T., & O’Shaughnessy, R. 2016, Nature, 534, 512
  * Björklund et al. (2021) Björklund, R., Sundqvist, J. O., Puls, J., & Najarro, F. 2021, A&A, 648, A36
  * Bowman & Michielsen (2021) Bowman, D. M. & Michielsen, M. 2021, A&A, 656, A158
  * Brands et al. (2022) Brands, S. A., de Koter, A., Bestenlehner, J. M., et al. 2022, A&A, 663, A36
  * Breivik et al. (2020) Breivik, K., Coughlin, S., Zevin, M., et al. 2020, ApJ, 898, 71
  * Breysacher & Westerlund (1978) Breysacher, J. & Westerlund, B. E. 1978, A&A, 67, 261
  * Brott et al. (2011) Brott, I., de Mink, S. E., Cantiello, M., et al. 2011, A&A, 530, A115
  * Campagnolo et al. (2018) Campagnolo, J. C. N., Borges Fernandes, M., Drake, N. A., et al. 2018, A&A, 613, A33
  * Castellano et al. (2022) Castellano, M., Fontana, A., Treu, T., et al. 2022, ApJ, 938, L15
  * Chattopadhyay et al. (2022) Chattopadhyay, D., Hurley, J., Stevenson, S., & Raidani, A. 2022, MNRAS, 513, 4527
  * Chen et al. (2015) Chen, Y., Bressan, A., Girardi, L., et al. 2015, MNRAS, 452, 1068
  * Cheng et al. (2024) Cheng, S. J., Goldberg, J. A., Cantiello, M., et al. 2024, arXiv e-prints, arXiv:2405.12274
  * Choi et al. (2016) Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102
  * Dekker et al. (2000) Dekker, H., D’Odorico, S., Kaufer, A., Delabre, B., & Kotzlowski, H. 2000, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4008, Optical and IR Telescope Instrumentation and Detectors, ed. M. Iye & A. F. Moorwood, 534–545
  * Di Carlo et al. (2019) Di Carlo, U. N., Giacobbo, N., Mapelli, M., et al. 2019, MNRAS, 487, 2947
  * Dotter (2016) Dotter, A. 2016, ApJS, 222, 8
  * Dsilva et al. (2020) Dsilva, K., Shenar, T., Sana, H., & Marchant, P. 2020, A&A, 641, A26
  * Dsilva et al. (2022) Dsilva, K., Shenar, T., Sana, H., & Marchant, P. 2022, A&A, 664, A93
  * Dsilva et al. (2023) Dsilva, K., Shenar, T., Sana, H., & Marchant, P. 2023, A&A, 674, A88
  * Eldridge et al. (2011) Eldridge, J. J., Langer, N., & Tout, C. A. 2011, MNRAS, 414, 3501
  * Evans et al. (2011) Evans, C. J., Taylor, W. D., Hénault-Brunet, V., et al. 2011, A&A, 530, A108
  * Foellmi (2004) Foellmi, C. 2004, A&A, 416, 291
  * Foellmi et al. (2003a) Foellmi, C., Moffat, A. F. J., & Guerrero, M. A. 2003a, MNRAS, 338, 360
  * Foellmi et al. (2003b) Foellmi, C., Moffat, A. F. J., & Guerrero, M. A. 2003b, MNRAS, 338, 1025
  * Fragione et al. (2022a) Fragione, G., Kocsis, B., Rasio, F. A., & Silk, J. 2022a, ApJ, 927, 231
  * Fragione et al. (2022b) Fragione, G., Loeb, A., Kocsis, B., & Rasio, F. A. 2022b, ApJ, 933, 170
  * Fryer et al. (2012) Fryer, C. L., Belczynski, K., Wiktorowicz, G., et al. 2012, ApJ, 749, 91
  * Fullerton et al. (1996) Fullerton, A. W., Gies, D. R., & Bolton, C. T. 1996, ApJS, 103, 475
  * Gaia Collaboration et al. (2023) Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, A&A, 674, A1
  * Georgy et al. (2013) Georgy, C., Ekström, S., Eggenberger, P., et al. 2013, A&A, 558, A103
  * Giacobbo et al. (2018) Giacobbo, N., Mapelli, M., & Spera, M. 2018, MNRAS, 474, 2959
  * Graczyk et al. (2020) Graczyk, D., Pietrzyński, G., Thompson, I. B., et al. 2020, ApJ, 904, 13
  * Gräfener et al. (2012) Gräfener, G., Owocki, S. P., & Vink, J. S. 2012, A&A, 538, A40
  * Gräfener et al. (2011) Gräfener, G., Vink, J. S., de Koter, A., & Langer, N. 2011, A&A, 535, A56
  * Hainich et al. (2015) Hainich, R., Pasemann, D., Todt, H., et al. 2015, A&A, 581, A21
  * Henneco et al. (2023) Henneco, J., Schneider, F. R. N., & Laplace, E. 2023, arXiv e-prints, arXiv:2311.12124
  * Hong et al. (2018) Hong, J., Vesperini, E., Askar, A., et al. 2018, MNRAS, 480, 5645
  * Hurley et al. (2000) Hurley, J. R., Pols, O. R., & Tout, C. A. 2000, MNRAS, 315, 543
  * Iorio et al. (2023) Iorio, G., Mapelli, M., Costa, G., et al. 2023, MNRAS, 524, 426
  * Kee et al. (2021) Kee, N. D., Sundqvist, J. O., Decin, L., de Koter, A., & Sana, H. 2021, A&A, 646, A180
  * Koenigsberger et al. (2010) Koenigsberger, G., Georgiev, L., Hillier, D. J., et al. 2010, AJ, 139, 2600
  * Koenigsberger et al. (2014) Koenigsberger, G., Morrell, N., Hillier, D. J., et al. 2014, AJ, 148, 62
  * Kouwenhoven et al. (2007) Kouwenhoven, M. B. N., Brown, A. G. A., Portegies Zwart, S. F., & Kaper, L. 2007, A&A, 474, 77
  * Kruckow et al. (2018) Kruckow, M. U., Tauris, T. M., Langer, N., Kramer, M., & Izzard, R. G. 2018, MNRAS, 481, 1908
  * Kruckow et al. (2016) Kruckow, M. U., Tauris, T. M., Langer, N., et al. 2016, A&A, 596, A58
  * Kummer et al. (2023) Kummer, F., Toonen, S., & de Koter, A. 2023, A&A, 678, A60
  * Langer et al. (2020) Langer, N., Schürmann, C., Stoll, K., et al. 2020, A&A, 638, A39
  * Leahy & Filipović (2022) Leahy, D. A. & Filipović, M. D. 2022, ApJ, 931, 20
  * Lightkurve Collaboration et al. (2018) Lightkurve Collaboration, Cardoso, J. V. d. M., Hedges, C., et al. 2018, Lightkurve: Kepler and TESS time series analysis in Python, Astrophysics Source Code Library, record ascl:1812.013
  * Limongi & Chieffi (2018) Limongi, M. & Chieffi, A. 2018, ApJS, 237, 13
  * Maeder (1987) Maeder, A. 1987, A&A, 178, 159
  * Maggi et al. (2019) Maggi, P., Filipović, M. D., Vukotić, B., et al. 2019, A&A, 631, A127
  * Mapelli et al. (2021) Mapelli, M., Dall’Amico, M., Bouffanais, Y., et al. 2021, MNRAS, 505, 339
  * Marchenko et al. (2007) Marchenko, S. V., Foellmi, C., Moffat, A. F. J., et al. 2007, ApJ, 656, L77
  * Martins (2023) Martins, F. 2023, arXiv e-prints, arXiv:2310.06539
  * Martins et al. (2009) Martins, F., Hillier, D. J., Bouret, J. C., et al. 2009, A&A, 495, 257
  * Massey (2003) Massey, P. 2003, ARA&A, 41, 15
  * Massey & Duffy (2001) Massey, P. & Duffy, A. S. 2001, ApJ, 550, 713
  * Matsuura et al. (2022) Matsuura, M., Ayley, V., Chawner, H., et al. 2022, MNRAS, 513, 1154
  * Mitchell et al. (2020) Mitchell, M., Muftakhidinov, B., Winchen, T., et al. 2020, markummitchell/engauge-digitizer: Nonrelease, Zenodo
  * Moffat (1982) Moffat, A. F. J. 1982, ApJ, 257, 110
  * Moffat (1988) Moffat, A. F. J. 1988, ApJ, 330, 766
  * Moffat et al. (1985) Moffat, A. F. J., Breysacher, J., & Seggewiss, W. 1985, ApJ, 292, 511
  * Moffat et al. (1998) Moffat, A. F. J., Marchenko, S. V., Bartzakos, P., et al. 1998, ApJ, 497, 896
  * Morgan et al. (1991) Morgan, D. H., Vassiliadis, E., & Dopita, M. A. 1991, MNRAS, 251, 51P
  * Neugent et al. (2018) Neugent, K. F., Massey, P., & Morrell, N. 2018, ApJ, 863, 181
  * Öpik (1924) Öpik, E. 1924, Publications of the Tartu Astrofizica Observatory, 25
  * Pauli et al. (2022) Pauli, D., Langer, N., Aguilera-Dena, D. R., Wang, C., & Marchant, P. 2022, A&A, 667, A58
  * Pauli et al. (2023) Pauli, D., Oskinova, L. M., Hamann, W. R., et al. 2023, A&A, 673, A40
  * Petrovich & Antonini (2017) Petrovich, C. & Antonini, F. 2017, ApJ, 846, 146
  * Piskunov & Valenti (2002) Piskunov, N. E. & Valenti, J. A. 2002, A&A, 385, 1095
  * Pols (1994) Pols, O. R. 1994, A&A, 290, 119
  * Ramachandran et al. (2019) Ramachandran, V., Hamann, W. R., Oskinova, L. M., et al. 2019, A&A, 625, A104
  * Rastello et al. (2021) Rastello, S., Mapelli, M., Di Carlo, U. N., et al. 2021, MNRAS, 507, 3612
  * Renzo et al. (2019) Renzo, M., Zapartas, E., de Mink, S. E., et al. 2019, A&A, 624, A66
  * Richardson & Mehner (2018) Richardson, N. D. & Mehner, A. 2018, Research Notes of the American Astronomical Society, 2, 121
  * Riley et al. (2022) Riley, J., Agrawal, P., Barrett, J. W., et al. 2022, ApJS, 258, 34
  * Rodriguez et al. (2016) Rodriguez, C. L., Chatterjee, S., & Rasio, F. A. 2016, Phys. Rev. D, 93, 084029
  * Roman-Duval et al. (2020) Roman-Duval, J., Proffitt, C. R., Taylor, J. M., et al. 2020, Research Notes of the American Astronomical Society, 4, 205
  * Samsing et al. (2018) Samsing, J., Askar, A., & Giersz, M. 2018, ApJ, 855, 124
  * Sana et al. (2012) Sana, H., de Mink, S. E., de Koter, A., et al. 2012, Science, 337, 444
  * Sana et al. (2022) Sana, H., Ramírez-Agudelo, O. H., Hénault-Brunet, V., et al. 2022, A&A, 668, L5
  * Sanduleak (1968) Sanduleak, N. 1968, AJ, 73, 246
  * Sanduleak (1969) Sanduleak, N. 1969, AJ, 74, 877
  * Santoliquido et al. (2020) Santoliquido, F., Mapelli, M., Bouffanais, Y., et al. 2020, ApJ, 898, 152
  * Sanyal et al. (2017) Sanyal, D., Langer, N., Szécsi, D., -C Yoon, S., & Grassitelli, L. 2017, A&A, 597, A71
  * Schneider et al. (2023) Schneider, F. R. N., Podsiadlowski, P., & Laplace, E. 2023, ApJ, 950, L9
  * Schootemeijer & Langer (2018) Schootemeijer, A. & Langer, N. 2018, A&A, 611, A75
  * Schootemeijer et al. (2019) Schootemeijer, A., Langer, N., Grin, N. J., & Wang, C. 2019, A&A, 625, A132
  * Schootemeijer et al. (2021) Schootemeijer, A., Langer, N., Lennon, D., et al. 2021, A&A, 646, A106
  * Shapiro & Lightman (1976) Shapiro, S. L. & Lightman, A. P. 1976, ApJ, 204, 555
  * Shenar et al. (2020) Shenar, T., Gilkis, A., Vink, J. S., Sana, H., & Sander, A. A. C. 2020, A&A, 634, A79
  * Shenar et al. (2018) Shenar, T., Hainich, R., Todt, H., et al. 2018, A&A, 616, A103
  * Shenar et al. (2016) Shenar, T., Hainich, R., Todt, H., et al. 2016, A&A, 591, A22
  * Shenar et al. (2019) Shenar, T., Sablowski, D. P., Hainich, R., et al. 2019, A&A, 627, A151
  * Shenar et al. (2023a) Shenar, T., Sana, H., Crowther, P. A., et al. 2023a, A&A, 679, A36
  * Shenar et al. (2023b) Shenar, T., Wade, G. A., Marchant, P., et al. 2023b, Science, 381, 761
  * Smith (1968) Smith, L. F. 1968, MNRAS, 140, 409
  * Smith & Tombleson (2015) Smith, N. & Tombleson, R. 2015, MNRAS, 447, 598
  * Soberman et al. (1997) Soberman, G. E., Phinney, E. S., & van den Heuvel, E. P. J. 1997, A&A, 327, 620
  * Spera et al. (2015) Spera, M., Mapelli, M., & Bressan, A. 2015, MNRAS, 451, 4086
  * Stephan et al. (2019) Stephan, A. P., Naoz, S., Ghez, A. M., et al. 2019, ApJ, 878, 58
  * Stoehr et al. (2008) Stoehr, F., White, R., Smith, M., et al. 2008, in Astronomical Society of the Pacific Conference Series, Vol. 394, Astronomical Data Analysis Software and Systems XVII, ed. R. W. Argyle, P. S. Bunclark, & J. R. Lewis, 505
  * Szécsi et al. (2015) Szécsi, D., Langer, N., Yoon, S.-C., et al. 2015, A&A, 581, A15
  * Szeifert et al. (1993) Szeifert, T., Stahl, O., Wolf, B., et al. 1993, A&A, 280, 508
  * Tagawa et al. (2020) Tagawa, H., Haiman, Z., & Kocsis, B. 2020, ApJ, 898, 25
  * Tang et al. (2014) Tang, J., Bressan, A., Rosenfield, P., et al. 2014, MNRAS, 445, 4287
  * Tauris & van den Heuvel (2006) Tauris, T. M. & van den Heuvel, E. P. J. 2006, in Compact stellar X-ray sources, Vol. 39, 623–665
  * Toalá et al. (2018) Toalá, J. A., Oskinova, L. M., Hamann, W. R., et al. 2018, ApJ, 869, L11
  * Toonen et al. (2016) Toonen, S., Hamers, A., & Portegies Zwart, S. 2016, Computational Astrophysics and Cosmology, 3, 6
  * Torniamenti et al. (2022) Torniamenti, S., Rastello, S., Mapelli, M., et al. 2022, MNRAS, 517, 2953
  * Udalski et al. (1998) Udalski, A., Szymanski, M., Kubiak, M., et al. 1998, Acta Astron., 48, 147
  * Udalski et al. (2008) Udalski, A., Szymanski, M. K., Soszynski, I., & Poleski, R. 2008, Acta Astron., 58, 69
  * Udalski et al. (2015) Udalski, A., Szymański, M. K., & Szymański, G. 2015, Acta Astron., 65, 1
  * van den Heuvel et al. (2017) van den Heuvel, E. P. J., Portegies Zwart, S. F., & de Mink, S. E. 2017, MNRAS, 471, 4256
  * van der Hucht (2001) van der Hucht, K. A. 2001, New A Rev., 45, 135
  * Venn (1999) Venn, K. A. 1999, ApJ, 518, 405
  * Vink et al. (2001) Vink, J. S., de Koter, A., & Lamers, H. J. G. L. M. 2001, A&A, 369, 574
  * Vink & Harries (2017) Vink, J. S. & Harries, T. J. 2017, A&A, 603, A120
  * Walborn et al. (2000) Walborn, N. R., Lennon, D. J., Heap, S. R., et al. 2000, PASP, 112, 1243
  * Wang et al. (2019) Wang, C., Langer, N., Gräfener, G., & Marchant, P. 2019, IAU Symposium, 346, 78
  * Wang et al. (2021) Wang, H., Stephan, A. P., Naoz, S., Hoang, B.-M., & Breivik, K. 2021, ApJ, 917, 76
  * Wang et al. (2020) Wang, L., Iwasawa, M., Nitadori, K., & Makino, J. 2020, MNRAS, 497, 536
  * Wellstein & Langer (1999) Wellstein, S. & Langer, N. 1999, A&A, 350, 148
  * Westerlund (1964) Westerlund, B. E. 1964, in The Galaxy and the Magellanic Clouds, ed. F. J. Kerr, Vol. 20, 316
  * Yang et al. (2023) Yang, M., Bonanos, A. Z., Jiang, B., et al. 2023, arXiv e-prints, arXiv:2304.01835
  * Yoon et al. (2006) Yoon, S.-C., Langer, N., & Norman, C. 2006, A&A, 460, 199
  * Yusof et al. (2013) Yusof, N., Hirschi, R., Meynet, G., et al. 2013, MNRAS, 433, 1114
  * Zucker (2003) Zucker, S. 2003, MNRAS, 342, 1291

##  Appendix A Observational data

Table 4:  RV measurements of SMC AB 1. We provide the RVs relative to the co-added spectrum obtained using the N v 4604+4620 Å and N v 4944 Å lines. The absolute RVs can be found in Table 2. Measurements before MJD 54000 are based on spectra from Marchenko et al. (2007). The MJD is measured at the start of the observations.  MJD |  RV${}_{\mathrm{N\,V}}$ | $\sigma_{\mathrm{CCF,\,NV}}$
---|---|---
53974.08143819 | -2.8 | 0.8
53974.15281245 | 0.6 | 0.6
53974.22478018 | 0.6 | 0.6
53974.29469711 | 2.4 | 0.8
53974.3652008 | -1.3 | 0.8
53975.06148374 | 3.1 | 0.8
53975.13130698 | 1.4 | 0.7
53975.20126692 | -2.2 | 0.7
53975.27429798 | -0.5 | 0.7
53975.34441279 | 0.8 | 0.7
59522.241237232 | 2.7 | 0.6
59545.140053771 | -13.8 | 0.7
59764.337670124 | -2.1 | 0.8
59790.286486529 | 2.9 | 0.7
59853.155227237 | 9.1 | 0.7
59875.139356245 | -1.4 | 0.7
59881.104598668 | -0.9 | 0.7
Table 5:  Same as Table 4, but for SMC AB2. MJD |  RV${}_{\mathrm{N\,V}}$ | $\sigma_{\mathrm{CCF,\,NV}}$
---|---|---
53974.05813941 | -0.6 | 0.8
53974.12974506 | 3.9 | 0.6
53974.20138833 | 1.7 | 0.8
53974.27175485 | 3.0 | 0.7
53974.341774 | 4.0 | 0.8
53974.41218787 | 2.9 | 1.0
53975.03795807 | -5.6 | 0.8
53975.1080113 | -2.4 | 0.7
53975.17794324 | 2.9 | 0.7
53975.24800434 | 3.1 | 0.7
53975.32088677 | -1.2 | 0.7
53975.390976 | -3.8 | 0.7
59528.098763616 | -4.7 | 0.9
59541.075943645 | -1.5 | 1.4
59725.389460631 | 7.5 | 1.2
59760.400719872 | -5.5 | 0.8
59853.198151712 | 1.0 | 1.0
59884.112963287 | -4.4 | 1.9
Table 6:  Same as Table 4, but for SMC AB4. MJD |  RV${}_{\mathrm{N\,V}}$ | $\sigma_{\mathrm{CCF,\,NV}}$
---|---|---
53974.04129595 | -4.7 | 0.7
53974.11267138 | -1.7 | 0.7
53974.18478773 | -1.2 | 0.6
53974.25548055 | -2.4 | 0.7
53974.32562773 | -2.9 | 0.7
53974.39570418 | -2.7 | 0.8
53975.02142447 | -6.2 | 0.9
53975.09172646 | -3.2 | 0.8
53975.16159462 | -2.1 | 0.6
53975.23155571 | 1.3 | 0.7
53975.30485427 | 1.2 | 0.7
53975.3747245 | 0.8 | 0.6
59521.267478467 | -4.2 | 1.0
59544.223161156 | 4.6 | 1.1
59739.40864374 | 1.7 | 1.1
59763.390003412 | 3.8 | 1.0
59855.126683198 | 10.1 | 1.2
59898.151501907 | 8.8 | 1.0
Table 7:  Same as Table 4, but for SMC AB9.  MJD |  RV${}_{\mathrm{N\,V}}$ | $\sigma_{\mathrm{CCF,\,NV}}$
---|---|---
59527.235134485 | -8.6 | 0.6
59545.182260154 | 2.7 | 0.6
59767.263809198 | 6.8 | 1.0
59790.329875426 | 0.1 | 0.7
59855.137487622 | 1.1 | 0.8
59903.169638222 | -2.5 | 0.7
Table 8:  Same as Table 4, but for SMC AB10.  MJD |  RV${}_{\mathrm{N\,V}}$ | $\sigma_{\mathrm{CCF,\,NV}}$
---|---|---
59524.15053773 | 1.3 | 0.8
59552.13830278 | -2.9 | 0.8
59767.310841186 | 1.9 | 0.9
59788.20105391 | 1.4 | 1.0
59855.18150306 | -3.7 | 0.8
59885.044768909 | -0.4 | 0.9
Table 9:  Same as Table 4, but for SMC AB11. MJD |  RV${}_{\mathrm{N\,V}}$ | $\sigma_{\mathrm{CCF,\,NV}}$
---|---|---
59524.210034441 | 4.0 | 1.2
59544.171211109 | -9.8 | 0.8
59790.372793215 | -0.6 | 0.7
59811.253413358 | 2.9 | 0.9
59903.113894512 | 2.8 | 0.9
59933.070686762 | 0.1 | 1.1
Table 10:  Same as Table 4, but for SMC AB12. MJD |  RV${}_{\mathrm{N\,V}}$ | $\sigma_{\mathrm{CCF,\,NV}}$
---|---|---
59527.176912331 | -0.6 | 0.8
59561.038740174 | 3.1 | 0.9
59791.223275872 | -3.1 | 1.0
59811.306473575 | 6.6 | 0.9
59896.068647057 | -11.8 | 1.0
59906.06373527 | 6.0 | 0.9
Figure 11:  Left panels: Lomb-Scargle periodograms of our target stars, based on OGLE data. The orange line shows the frequency at which the highest power is achieved, and the corresponding period is written in each panel. Right panels: phase-folded light curves of I-band magnitudes, adopting the periods mentioned in the left panels. Table 11:  Overview of the OGLE-III and OGLE-IV time-series photometry data that was analyzed in this work, showing the dates of the first and last observation, the number of observations ($N_{\mathrm{obs}}$), and the maximum SNR (SNRmax) found for each source. No OGLE-IV data is available for SMC AB4. Object | First observation | Last observation | $N_{\mathrm{obs}}$ |  SNRmax
---|---|---|---|---
SMC AB1 | 07-2001 | 11-2022 | 1503 | 2.41
SMC AB2 | 06-2001 | 11-2022 | 1796 | 2.28
SMC AB4 | 06-2001 | 01-2009 | 718 | 2.47
SMC AB9 | 06-2001 | 11-2022 | 1441 | 2.40
SMC AB10 | 06-2001 | 11-2022 | 1785 | 2.48
SMC AB11 | 06-2001 | 11-2022 | 1889 | 2.89
SMC AB12 | 07-2001 | 11-2022 | 1764 | 2.26

##  Appendix B Detection probabilities

Figure 12:  As Fig. 6, but assuming a flat eccentricity distribution instead
of circular orbits.  Figure 13:  Histograms of simulated $\Delta$RV values for
SMC AB9, where we show the scattered observations and a mock binary with
$P_{\mathrm{orb}}=501.2$ d and a 5 M⊙ companion.  Figure 14:  Same as Fig. 6,
but instead of calculating the probability that $\Delta$RVsim exceeds the
threshold value $\Delta$RVobs, we calculate the probability $p_{\mathrm{m}}$
that $\Delta$RVsim matches $\Delta$RVobs (see text for details).

###  B.1 More detection probability figures

Figure 12 is the same figure as Fig. 6, but taking into account eccentricity.
Specifically, eccentricities were drawn from a flat distribution between $e=0$
and $e=0.9$. The argument of periastron is given a random orientation for each
mock binary.

We do another experiment where we try to match the observed $\Delta$RV value
of SMC AB 9 with binary motion, rather than using a $\Delta$RV cutoff. This is
illustrated in Fig. 13. We first estimate the error on the observed $\Delta$RV
by simulating our observing campaign of SMC AB9 105times. We give each RV
measurement a random Gaussian offset based on its $\sigma_{\mathrm{CCF}}$,
which causes a modest spread around the originally measured $\Delta$RV value
(black line in Fig. 13). The observed $\Delta$RV error is then inferred from
fitting a Gaussian to the distribution of $\Delta$RV values. We then compare
this with $\Delta$RV values from a mock binary with a certain orbital period
and companion mass (e.g., $P_{\mathrm{orb}}=251.2$ d and a 5 M⊙ in Fig. 13),
for which we assume that the only uncertainty affecting $\Delta$RV is given by
$\sigma_{\mathrm{CCF}}$. Finally, we define $p_{\mathrm{m}}$ as the fractional
overlap of the black and orange $\Delta$RV histograms (shaded orange region in
Fig. 13). Figure 14 shows the $p_{\mathrm{m}}$ values that we obtain over the
whole binary parameter space in the range $0\leq\log(P_{\mathrm{orb}}/d)\leq
3.5$ and $1\leq M_{2}/M_{\odot}\leq 40$.

###  B.2 Integrated detection probabilities

We can calculate the weighted average detection probability $\hat{p}$ by
summing over all the probabilities $p_{\mathrm{ij}}$ we calculated (shown in
Fig. 6 and similar figures in Appendix B.1), weighted by the individual
weights $w_{\mathrm{P,i}}$ (based on the orbital period) and
$w_{\mathrm{M,j}}$ (based the companion mass) and normalized by the sum of the
individual weights:

$\hat{p}=\frac{\Sigma_{i,j=1,1}^{36,40}\,p_{\mathrm{i,j}}w_{\mathrm{P,i}}w_{\mathrm{M,j}}}{\Sigma_{i,j=1,1}^{36,40}\,w_{\mathrm{P,i}}w_{\mathrm{M,j}}}.$
(1)

We sum from $i=1$ to $i=36$ because we explore 36 values for
$\log(P_{\mathrm{orb}}/d)$ in Sect. 5, and we sum over 40 values of $j$
because we explore 40 values for the companion mass. For flat probability
distributions, individual weights $w_{\mathrm{P,i}}$ and $w_{\mathrm{M,j}}$
have a value of unity if the orbital period and companion mass are within the
range we consider (else they are zero). In two cases ($\hat{p}_{\mathrm{L20}}$
and $\hat{p}_{\mathrm{R19}}$), we use non-flat probability distributions of
$\log(P_{\mathrm{orb}}/d)$ and $M_{\mathrm{2}}$, which are based on previous
binary populations synthesis works (Langer et al. 2020; Renzo et al. 2019).
Below, we describe how we obtain these probability distributions.

To calculate $\hat{p}_{\mathrm{L20}}$, we used the orbital period and mass
ratio distributions obtained from the population synthesis calculations of
Langer et al. (2020) for O star-black hole binaries in the LMC (cf., their
figs. 4 and 6). The mass transfer in most of the underlying detailed binary
models was highly inefficient. Since black hole formation kicks were neglected
in this work, the predicted parameters of the O star-black hole binaries are
expected to be similar the those of their O star-WR binaries progenitors.

We also investigate the theoretical orbital period and mass ratio distribution
based on models with conservative mass transfer (Renzo et al. 2019), which
produce wider binaries (more difficult to detect) with more massive companions
(easier to detect). To calculate $\hat{p}_{\mathrm{R19}}$, we used Engauge
Digitizer to tabulate the semi-major axis distribution of $M>15$ M⊙ accretor
stars in fig. D.2 of Renzo et al. (2019). We converted this to an orbital
period distribution assuming typical masses of $M_{\mathrm{WR}}=20$ M⊙ and
$M_{\mathrm{comp}}=35$ M⊙. This lead to a period distribution with peaks
around $\log(P_{\mathrm{orb}}/d)=1$ and $\log(P_{\mathrm{orb}}/d)=2.8$, going
towards zero for $\log(P_{\mathrm{orb}}/d)<0.2$ and
$\log(P_{\mathrm{orb}}/d)>3.5$. The companion mass distribution was estimated
using the following assumptions: i) upon interaction, a 40 M⊙ star gets
stripped to a 20 M⊙ WR star; ii) the minimum value of $q_{\mathrm{crit}}=0.4$
for stable mass transfer adopted by Renzo et al. (2019); fully conservative
mass transfer, i.e., the 20 M⊙ lost by the WR star end up on the accretor
star; iv) a flat initial mass ratio distribution. This leads to a flat
companion mass distribution between 36 M⊙ and 60 M⊙. To take a slightly
conservative approach to estimate $\hat{p}_{\mathrm{R19}}$ and to bring this
in line with the parameter space explored before, we instead adopt a flat
companion mass distribution between 30 M⊙ and 40 M⊙.

We find that $\hat{p}_{\mathrm{L20}}$ and $\hat{p}_{\mathrm{R19}}$ are even
higher ($\sim$98%) than $\hat{p}$ ($\sim$90%). The reason is that the binary
models do not predict very low mass companions, and the binary models of
Langer et al. (2020) avoid orbital periods above $\log(P_{\mathrm{orb}}/d)=3$.

For both $\hat{p}_{\mathrm{L20}}$ and $\hat{p}_{\mathrm{R19}}$, we repeat the
exercise but we take the systems with $P_{\mathrm{orb}}<30$ d out of the
distribution, with the motivation that all of these might have been detected
already. This shifts the probability distributions of the remaining binaries
towards higher periods and hence the values for $\hat{p}_{\mathrm{L20}}$ and
$\hat{p}_{\mathrm{R19}}$ become smaller. However, we still find average
detection probabilities in excess of 95%.

Table 12:  Probabilities that binaries would have caused a larger $\Delta$RV than observed. We integrated over different parts of the binary parameter space; if no extra restriction is mentioned, we consider the range $1\leq M_{\mathrm{2}}/\mathrm{M}_{\odot}\leq 40$ and $0\leq\log(P_{\mathrm{orb}}/\mathrm{d})\leq 3.5$. To calculate $\hat{p}_{\mathrm{L20}}$ and $\hat{p}_{\mathrm{R19}}$, we used orbital period and and mass ratio distributions based on Langer et al. (2020) and Renzo et al. (2019), respectively (see text for details).  | $\hat{p}$ | $\hat{p}_{\mathrm{M5+}}$ | $\hat{p}_{\mathrm{P1yr-}}$ | $\hat{p}_{\mathrm{M5+P1yr-}}$ | $\hat{p}_{\mathrm{P30d+}}$ | $\hat{p}_{\mathrm{L20}}$ | $\hat{p}_{\mathrm{L20,P30d+}}$ | $\hat{p}_{\mathrm{R19}}$ | $\hat{p}_{\mathrm{R19,P30d+}}$
---|---|---|---|---|---|---|---|---|---
$M_{\mathrm{2}}$ restr. |  | $M_{\mathrm{2}}>5$ |  | $M_{\mathrm{2}}>5$ |  |  |  |  |
1$P_{\mathrm{orb}}$ restr. |  |  |  $P_{\mathrm{orb}}<1$ yr |  $P_{\mathrm{orb}}<1$ yr |  $P_{\mathrm{orb}}>30$ d |  |  $P_{\mathrm{orb}}>30$ d |  |  $P_{\mathrm{orb}}>30$ d
SMC AB1 | 0.866 | 0.916 | 0.952 | 0.989 | 0.785 | 0.969 | 0.963 | 0.970 | 0.949
SMC AB2 | 0.927 | 0.963 | 0.978 | 0.996 | 0.880 | 0.990 | 0.988 | 0.988 | 0.980
SMC AB4 | 0.908 | 0.950 | 0.968 | 0.994 | 0.850 | 0.985 | 0.982 | 0.983 | 0.972
SMC AB9 | 0.887 | 0.916 | 0.967 | 0.993 | 0.798 | 0.985 | 0.982 | 0.965 | 0.942
SMC AB10 | 0.956 | 0.974 | 0.995 | 0.999 | 0.924 | 0.998 | 0.997 | 0.992 | 0.986
SMC AB11 | 0.892 | 0.929 | 0.971 | 0.994 | 0.822 | 0.983 | 0.980 | 0.973 | 0.955
SMC AB12 | 0.852 | 0.896 | 0.956 | 0.988 | 0.760 | 0.972 | 0.966 | 0.954 | 0.924
Average | 0.901 | 0.935 | 0.970 | 0.993 | 0.831 | 0.983 | 0.980 | 0.975 | 0.958
Table 13:  Same as Table 12, but considering eccentric orbits instead of only circular orbits. The eccentricity is drawn from a flat distribution between 0 and 0.9.  | $\hat{p}$ | $\hat{p}_{\mathrm{M5+}}$ | $\hat{p}_{\mathrm{P1yr-}}$ | $\hat{p}_{\mathrm{M5+P1yr-}}$ | $\hat{p}_{\mathrm{P30d+}}$ | $\hat{p}_{\mathrm{L20}}$ | $\hat{p}_{\mathrm{L20,P30d+}}$ | $\hat{p}_{\mathrm{R19}}$ | $\hat{p}_{\mathrm{R19,P30d+}}$
---|---|---|---|---|---|---|---|---|---
$M_{\mathrm{2}}$ restr. |  | $M_{\mathrm{2}}>5$ |  | $M_{\mathrm{2}}>5$ |  |  |  |  |
1$P_{\mathrm{orb}}$ restr. |  |  |  $P_{\mathrm{orb}}<1$ yr |  $P_{\mathrm{orb}}<1$ yr |  $P_{\mathrm{orb}}>30$ d |  |  $P_{\mathrm{orb}}>30$ d |  |  $P_{\mathrm{orb}}>30$ d
SMC AB1 | 0.847 | 0.895 | 0.943 | 0.980 | 0.756 | 0.948 | 0.937 | 0.950 | 0.918
SMC AB2 | 0.911 | 0.946 | 0.970 | 0.991 | 0.853 | 0.978 | 0.973 | 0.977 | 0.963
SMC AB4 | 0.889 | 0.931 | 0.959 | 0.987 | 0.822 | 0.969 | 0.963 | 0.970 | 0.951
SMC AB9 | 0.856 | 0.895 | 0.955 | 0.984 | 0.769 | 0.966 | 0.960 | 0.946 | 0.913
SMC AB10 | 0.941 | 0.960 | 0.991 | 0.998 | 0.901 | 0.994 | 0.993 | 0.984 | 0.973
SMC AB11 | 0.870 | 0.908 | 0.959 | 0.985 | 0.790 | 0.964 | 0.956 | 0.955 | 0.926
SMC AB12 | 0.831 | 0.874 | 0.941 | 0.976 | 0.729 | 0.945 | 0.933 | 0.933 | 0.890
Average | 0.878 | 0.916 | 0.960 | 0.986 | 0.803 | 0.966 | 0.968 | 0.959 | 0.933
Table 14:  Same as Table 12, but for the observing campaign of Foellmi et al. (2003a) and, for SMC AB12, Foellmi (2004).  | $\hat{p}$ | $\hat{p}_{\mathrm{M5+}}$ | $\hat{p}_{\mathrm{P1yr-}}$ | $\hat{p}_{\mathrm{M5+P1yr-}}$ | $\hat{p}_{\mathrm{P30d+}}$ | $\hat{p}_{\mathrm{L20}}$ | $\hat{p}_{\mathrm{L20,P30d+}}$ | $\hat{p}_{\mathrm{R19}}$ | $\hat{p}_{\mathrm{R19,P30d+}}$
---|---|---|---|---|---|---|---|---|---
$M_{\mathrm{2}}$ restr. |  | $M_{\mathrm{2}}>5$ |  | $M_{\mathrm{2}}>5$ |  |  |  |  |
1$P_{\mathrm{orb}}$ restr. |  |  |  $P_{\mathrm{orb}}<1$ yr |  $P_{\mathrm{orb}}<1$ yr |  $P_{\mathrm{orb}}>30$ d |  |  $P_{\mathrm{orb}}>30$ d |  |  $P_{\mathrm{orb}}>30$ d
SMC AB1 | 0.479 | 0.537 | 0.660 | 0.740 | 0.229 | 0.401 | 0.282 | 0.634 | 0.413
SMC AB2 | 0.491 | 0.548 | 0.675 | 0.753 | 0.234 | 0.423 | 0.302 | 0.635 | 0.410
SMC AB4 | 0.526 | 0.584 | 0.716 | 0.794 | 0.278 | 0.491 | 0.380 | 0.679 | 0.481
SMC AB9 | 0.396 | 0.448 | 0.548 | 0.620 | 0.134 | 0.236 | 0.111 | 0.545 | 0.283
SMC AB10 | 0.543 | 0.599 | 0.734 | 0.810 | 0.296 | 0.540 | 0.438 | 0.684 | 0.485
SMC AB11 | 0.295 | 0.345 | 0.408 | 0.478 | 0.052 | 0.103 | 0.019 | 0.432 | 0.138
SMC AB12 | 0.384 | 0.423 | 0.532 | 0.588 | 0.100 | 0.254 | 0.125 | 0.452 | 0.159
Average | 0.445 | 0.498 | 0.610 | 0.683 | 0.189 | 0.350 | 0.237 | 0.580 | 0.338
Figure 15:  Same as Fig. 6, but for SMC AB1.  Figure 16:  Same as Fig. 6, but
for SMC AB2.  Figure 17:  Same as Fig. 6, but for SMC AB4.  Figure 18:  Same
as Fig. 6, but for SMC AB10.  Figure 19:  Same as Fig. 6, but for SMC AB11.
Figure 20:  Same as Fig. 6, but for SMC AB12.  Figure 21:  Probability that
binary-induced motion would exceed the observed
$\Delta\mathrm{RV}_{\mathrm{obs}}$ for measurements performed by F03 on SMC
AB1. Figure 22:  Same as Fig. 21, but for SMC AB2.  Figure 23:  Same as Fig.
21, but for SMC AB4.  Figure 24:  Same as Fig. 21, but for SMC AB9.  Figure
25:  Same as Fig. 21, but for SMC AB10.  Figure 26:  Same as Fig. 21, but for
SMC AB11.  Figure 27:  Same as Fig. 21, but for SMC AB12. In this case,
$\Delta\mathrm{RV}_{\mathrm{obs}}$ is based on measurements of Foellmi (2004).