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A&A, 692, A192 (2024)
https://doi.org/10.1051/0004-6361/202450176
c
The Authors 2024
Astronomy
&
Astrophysics
The rotation rate of single- and double-lined southern O stars
Determining what increases the rotation rate in binaries
Susanne Blex1,?, Martin Haas1,?, and Rolf Chini1,2,3, ?
1Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), 44780 Bochum, Germany
2Polish Academy of Sciences, Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warszawa, Poland
3Universidad Católica del Norte, Instituto de Astronomía, Avenida Angamos 0610, Antofagasta, Chile
Received 29 March 2024 /Accepted 14 October 2024
ABSTRACT
We determined the projected rotational velocity (vsin i) of 238 southern O stars selected from the Galactic O-star Survey. The sample
contains 130 spectroscopic single stars (C), 36 single-lined binaries (SB1), and 72 SB2 systems (including eight triples). We applied
the Fourier method to high-resolution spectra taken at Cerro Murphy, Chile, and supplemented by archival spectra. The overall vsin i
statistics peaks at slow rotators (40–100km/s) with a tail towards medium (100–200km/s) and fast rotators (200–400 km/s). Binaries,
on average, show increased rotation, which differs for close (Porb <10 d) and wide binaries (10 d<Porb <3700d), and for primaries
and secondaries. The spin-up of close binaries is well explained by the superposition of spin-orbit synchronisation and mass transfer
via Roche-lobe overflow. The increased rotation of wide binaries, however, needs another explanation. Therefore, we discuss various
spin-up mechanisms. Timescale arguments lead us to favour a scenario where wide O binaries are spun-up by a combination of cloud
or disk fragmentation, which lays the basis of triple and multiple stars, and the subsequent merging or swallowing of low-mass by
higher-mass stars or proto-stars.
Key words. binaries: spectroscopic – stars: evolution – stars: massive – stars: rotation
1. Introduction
O stars are rare, but are massive and luminous. They strongly
influence the evolution of a galaxy. Located near their birth
clouds, they trigger the formation of lower mass stars and of
the next generation of stars. O stars are the main factories of
higher chemical elements blown into the interstellar medium by
fast winds and at the end of the stars’ life as a supernova. The
initial angular momentum of a cloud core is at least three orders
of magnitude larger than the rotational angular momentum of
the resulting star and must be redistributed or removed dur-
ing collapse (Zinnecker & Yorke 2007). The majority of O stars
are found in binaries or multiple systems (e.g. Sana et al. 2012;
Chini et al. 2012). Then the angular momentum is distributed
on the rotational and orbital parts. In a binary the presence
of a nearby companion induces tidal forces (Zahn 1975,1977;
Tassoul & Tassoul 1997) as well as mass-exchange and trans-
fer of angular momentum between the primary and the sec-
ondary star (Kriz & Harmanec 1975;de Mink et al. 2013, and
references therein).
For single O stars in the Milky Way, the projected rota-
tional velocity, veq sin(i) (hereafter vsin i), was derived from
the broadening of spectroscopic line profiles (Slettebak 1956;
Conti & Ebbets 1977;Penny 1996;Howarth et al. 1997). The
line broadening is caused by rotation and turbulence, being
disentangled by various techniques (e.g. Simón-Díaz & Herrero
2007,2014, and references therein). Additional contribution to
line broadening comes from stellar pulsations (Aerts et al. 2014)
and winds (Howarth & Prinja 1989;Kudritzki & Puls 2000;
?Corresponding authors; sublex@astro.rub.de;
haas@astro.rub.de;chini@astro.rub.de
Martins et al. 2005a). Overall, the vsin idistribution is skewed
with a peak at ∼75 km/s and a tail extending to ∼500 km/s. Here
we divide the stars along vsin iinto three groups: slow, medium,
and fast rotators, separated by 100 km/s and 200 km/s.
We define ‘increased rotators’ as medium or fast rotators
(vsin i>100 km/s).
The Galactic O-star Survey (GOS, Maíz-Apellániz et al.
2004;Maíz Apellániz et al. 2016) established a basis to expand
vsin istudies to binaries. Sota et al. (2011,2014) classified the
spectral types and binary nature of GOS stars, based on low-
resolution (R ∼2500) spectra, supplemented by high-resolution
information when available. Observationally, large databases of
high-resolution spectra are being gathered in the course of sev-
eral projects. Most prominent are the spectroscopic survey of
Galactic OB stars at the IAC (IACOB, PI Sergio Simón Díaz,
IAC) and the spectroscopic survey of Galactic O and WN stars
(OWN, initiated by Rodolfo Barbá, La Serena). In addition, a
dedicated survey on the southern O stars has been performed
by our group, aimed at finding eclipsing or spectroscopic bina-
ries, involving photometric and high-resolution spectral monitor-
ing since 2009 from Cerro Murphy1(PI Rolf Chini, Bochum).
Finally, as a low-metallicity starburst complement, the VLT
FLAMES Tarantula Survey (VFTS) observed 800 massive stars
in the H II region 30 Doradus in the Large Magellanic Cloud
(Evans et al. 2011).
1Cerro Murphy is a subsidiary summit of Cerro Armazones in the
Chilean Atacama desert. Since 2020 the observatory is run by the Niko-
laus Kopernikus Center, Warsaw, Poland. Hosted within ESO’s Paranal
Observatory, its name is Rolf Chini Cerro Murphy Observatory.
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This article is published in open access under the Subscribe to Open model.Subscribe to A&A to support open access publication.
A192, page 1 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
Based on the GOS, the IACOB+OWN survey documented
the progress in understanding the O stars by dozens of papers.
Recently, Holgado et al. (2022) presented a statistical study on
the projected rotational velocity (vsin i) of 285 Galactic single-
line O stars in the full northern and southern hemisphere. This
sample contains 230 single stars ‘C’ with non-detected radial
velocity (RV) variations, adopted to have constant RV, and 55
single-lined binaries (SB1) with detected secure RV variations.
In the binaries the detected component likely refers to the pri-
mary, whereas the secondary is not seen. Compared to the sin-
gle stars, the primaries show a more pronounced tail of medium
rotators, but a deficit of fast rotators. Holgado et al. provide
empirical evidence supporting that the tail of fast rotators is
mainly produced by binary interactions. Stars with extreme rota-
tion (>300 km/s) appear as single stars that are located in the
lower zone of the spectroscopic Hertzsprung–Russell diagram
(sHRD). The rotation rates of the youngest observed stars favour
an empirical initial velocity distribution with .20% of the criti-
cal velocity.
Britavskiy et al. (2023) searched for empirical signatures of
binarity in fast-rotating O-type stars (vsin i>200 km/s). They
expanded the IACOB+OWN sample of single-lined O stars
(Holgado et al. 2022) by including eight double-lined binaries
(SB2) which contain at least one fast rotator (vsin i>200 km/s).
In addition, they imployed Gaia and TESS data for astromet-
ric and photometric (e.g. eclipse) information. Their empirical
results seem to be in good agreement with the assumption that
the tail of fast-rotating O-type stars is mostly populated by post-
interaction binary products.
The VFTS presented vsin iof O stars in the LMC for 216 sin-
gle stars (Ramírez-Agudelo et al. 2013), 85 SB1s, and 31 SB2s
(Ramírez-Agudelo et al. 2015). The overall vsin idistribution of
114 (high-quality) primary stars resembles that of single stars,
but it differs in two ways: in binaries the distribution is broader
and slightly shifted to higher values. This shift is mostly due to
short-period binaries (Porb .10 d). Second, the vsin idistribu-
tion of primaries lacks a significant population of stars spinning
faster than 300 km/s, while such a population is clearly present in
the single-star sample. The orbital periods were not directly mea-
sured, but have been inferred from Monte Carlo simulations of
the amplitude of the radial velocity variations, max_dRV, of five
to eight epochs (Sana et al. 2013). The higher average spin rate
of stars in short-period binaries may either be explained by spin-
up through tides in such close binary systems, or by spin-down of
a fraction of the presumed-single stars and long-period binaries
through magnetic braking (Ramírez-Agudelo et al. 2015). The
fraction of SB2s in the VFTS is surprisingly low (31/332 =9%),
only one-third of that found in the Milky Way (Holgado et al.
2022, this work), while that of SB1s is very high (26%), sug-
gesting that the majority of SB2s in the VFTS have escaped
detection. Follow-up spectroscopy of 51 SB1s by the Tarantula
Massive Binary Monitoring (TMBM) revealed that the SB2 frac-
tion is indeed at least 50% higher (Shenar et al. 2022;Sana et al.
2022).
Comprehensive vsin istatistics of a complete sample of
Galactic O stars that includes all SB2s is still desired. In the
course of case studies of targeted Galactic clusters and SB2s,
the rotation rate of disentangled primaries and secondaries
have been derived by various authors using different methods
(Table D.2, available at the CDS, lists a compilation of 49 south-
ern SB2s). For this paper we used high-resolution spectra of a
GOS-based sample of ∼250 southern O stars monitored with
our telescopes from Cerro Murphy and supplemented by archival
spectra from FEROS at ESO. We decided to analyse not only the
SB2s, but the entire sample in a homogeneous way, to ensure
that any methodical biases were minimised. Section 2lists the
sample and the spectra. Section 3describes the SB classifica-
tion and the determination of vsin iwith the Fourier technique.
We used several spectral lines to determine vsin iand checked
for consistency. Section 4presents the vsin iresults obtained for
different subsamples. In Sect. 5we discuss the vsin idifference
and spin-up mechanisms for close and wide binaries, primaries,
and secondaries. Section 6presents a summary and our conclu-
sions.
2. Sample and spectra
2.1. Sample
We performed a comprehensive spectroscopic survey on a large
representative sample of 249 O-type stars south of Declination
+20◦, visible from Cerro Murphy. Chini et al. (2012) presented
early results on the multiplicity.
The sample is drawn from the Galactic O Star Catalogue
(GOSC Version 2.0; Sota et al. 2008), which is assumed to be
complete to V≤8 mag. Sota et al. (2011,2014) provided a com-
prehensive spectral classification. Two stars with four or more
components are rejected from the sample, because the lines are
faint and it is difficult to obtain reliable vsin ifor them; these
stars are: HD 101205 (SB7, Zasche et al. 2022), and the multi-
ple system Herschel 36 with no less than 10 stellar components
in a radius of 400 Maíz Apellániz et al. (2015), Campillay et al.
(2019). In addition, the complex system HD 57060 (=UW CMa)
is excluded: our six spectra show at least three components,
two narrow and one broad, with partly eclipses, hindering
us to accurately disentangle the components. The O2 super-
giant HD 093129 Aab shows at least two components (see also
Maíz Apellániz et al. 2017) but unique disentangling of our data
was not possible. We therefore rejected this source from the sam-
ple. Seven stars with spectral line profiles dominated by extreme
winds preventing any reliable vsin idetermination are also
rejected from the sample: HD 39680, HD 45314, HD 169515,
HD 313846, LSS 2063, LSS 4067, and Pismis 24-17. The SB2
system HD 104649 was rejected, because it turned out to be an
early B-type pair. HD93161 AB is now splitted into HD93161 A
and HD 93161 B.
The resulting sample of 238 stars is listed in Tables D.1
and D.2 (available at the CDS), splitted into spectroscopic binary
types (C, SB1, SB2 and SB3), newly classified as described in
Sect. 3.1. The sample is not strictly complete but it constitutes
a representative sample suited for statistical studies, for exam-
ple on vsin idifferences between single- and double-line stars,
early- and late-type stars, giants (luminosity class LC I-III) and
dwarfs (LC IV+V).
2.2. Spectra
The vsin istudy here is based on 3424 high-resolution multi-
epoch optical spectra. About half (1762 spectra) were taken
with the Bochum Echelle Spectrograph BESO (Fuhrmann et al.
2011) at the Universitätssternwarte Bochum on Cerro Murphy.
BESO was mounted at the 1.5 m Hexapod-Telescope in the years
from 2009 until 2012 and thereafter until 2020 at the 0.8 m IRIS
telescope. BESO is a twin of the ESO FEROS spectrograph
(Kaufer et al. 1997,1999). The spectra comprise a wavelength
range from 3620 to 8530 Å with a mean spectral resolution of
R=50 000. The entrance aperture of the star fibre is 3.4 arcsec.
A192, page 2 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
The integration time per spectrum was adapted to the published
visual brightness of each star. It was our primary goal to monitor
a large number of stars rather than to obtain a very high S/N for
individual stars. All data were reduced with a pipeline based on
the MIDAS package developed for FEROS (Stahl et al. 1999).
We complemented the data set with archival high-resolution
spectra taken with FEROS at ESO (N=1628), with UVES at
ESO (N =7) (Dekker et al. 2000), and with ELODIE (N =28)
(Baranne et al. 1996) at the 1.93 m telescope of the Observatoire
de Haute-Provence2.
Figure 1(top) depicts the number of spectra per star; the
median is 10 and the minimum is 4 obtained for 3 stars:
HD 95589 is a C, BD +223782 marginally failed our SB1 crite-
ria (Sect. 3.1) and we assigned C, while HD 93161 B is a robust
SB1. Certainly, our SB classification suffers from incomplete-
ness caused in case of small radial velocity differences (dRV),
in other words a low mass ratio M2/M1 and/or a large separa-
tion of the binary components. This leads to a well-known bias
favouring class C as discussed by Sana et al. (2012), among oth-
ers. Likewise, a faint SB2 companion may escape detection, in
particular if it has flat broad lines (e.g. Mahy et al. 2022). The
detection of an SB system depends also on the quality of the
spectra and their number. The fraction of detected SB systems,
fSB, in the sample is about 40–45%. Figure 1(bottom) compares
the number of spectra for Cs and SBs. Cs have on the median
about 9/12 fewer spectra than SBs. We note that we have fin-
ished the monitoring of a star once the obtained spectra estab-
lish its SB nature, even if there are fewer than ten spectra. This
may explain the high fraction fSB &35% for the subsample with
only four to ten spectra, in particular if they have been caught by
chance at large dRV.
3. Analysis
Whenever it was possible, we used the OIII 5592 line and in
addition suited lines among the master set of He I (4026, 4387,
4471, 4713, 4922, 5876, 6678, 7065) and He II lines (4200,
4541, 4686, 5411). Whether a line is suited depends on several
factors and the spectral type of a star: for instance, He I 4387
is too faint in early O-type stars, while O III 5592 and He II
lines are absent in B-type companions. He I5876 is the brightest
line but often suffers from wind features. He I 4026, 4387, 6678,
7065 are often noisy in BESO spectra. We took care to reject
possible blends which may mimic a binary, e.g. OII 4924.5 near
He I 4922, and He II 6683 near He I 6678. In addition, for some
objects and spectra the He I 4471 line shows suspicious wings
not seen in other helium lines, and in these cases we removed
He I 4471 from further analysis. Our aim is to obtain robust vsin i
by using as many lines as necessary, rather than using as many
lines as possible. After visual inspection, we rejected asymmet-
ric and suspicious cases. Table D.4 (available at the CDS) lists
for each line how many stars use the line.
3.1. SB classification
The spectroscopic binary (SB) classification by Chini et al.
(2012) was based on a limited number of spectra per star. Now
about twice the number of spectra are available, allowing us to
improve the former SB classification. Here, we re-inspected all
spectra. A number of suspected binaries turn out to be likely
single-lined stars with line profile variations (LPV). We have
detected two new SB1s (CPD−58 2620, HD 093160) but no new
2http://atlas.obs-hp.fr/elodie/
0 50 100 150 200 250
star index
0
20
40
60
80
100
120
Number of spectra
median = 10
224
Number of all stars = 238
All Spectra 3424
BESO 1762
0 5 10 15 20 25 30
number of spectra
0.0
0.2
0.4
0.6
0.8
1.0
cumulative fraction
C
SB 1+2+3
Fig. 1. Statistics of the spectra. Top: Number of spectra per star. The
star index is sorted in ascending order of the total number of spectra per
star (Ntot, black), and for equal Nt ot in ascending order of the number
of BESO spectra (red). Bottom: Cumulative fraction of the number of
spectra for Cs and SBs, zoomed in to the range 0–30. The dotted lines
mark the median; Cs have statistically (about 9/12) fewer spectra than
SBs.
SB2s; in other words: all (except of the two) SB1s and all SB2s
found here have meanwhile been reported in publications apart
from (Chini et al. 2012). For some SB2s we give the spectral
types, if they are not yet published or if they differ from previous
works. We here find 135 Cs, 41 SB1s, 54 SB2s and 8 SB3s. The
inclusion of literature information yields a sample of 130 Cs, 36
SB1s, 64 SB2s and 8 SB3s for the scientific analysis.
We begin the description with stars with double (triple)
lines; they are classified SB2 (SB3) accordingly. We required
that the radial velocity (RV) of the binary components is con-
sistent for at least four lines. A varying fast wind may mimic
two stellar components with a “false faint secondary” lying in
all spectra on the blue side of the line peak. Therefore, as a
conservative approach, for an SB2 we required that the sec-
ondary should switch the position for different spectral epochs,
for example from the blue to the red side of the primary. In
addition, the spectral types of the disentangled components,
derived from the He I 4471/He II 4541 equivalent width (EW)
ratio (Conti & Alschuler 1971;Martins 2018), have to be con-
sistent with literature results. To derive/check the luminos-
ity classes we used the EW ratios He II 4686/He I 4713 and
Si IV 4089/He I 4026 (Martins 2018). More details on the dis-
entangling are given in Appendix A.
A192, page 3 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
The stars not classified as SB2 or SB3 are adopted as sin-
gle line stars. To identify a spectroscopic binary, SB1, it should
exhibit reliable dRV. The detectability of RV variations depends
on the line width (here denoted as FWHM of the inverted line
profile), the line depth (or equivalently EW/FWHM), the signal-
to-noise ratio (S/N) of the spectrum and on the number of spec-
tra, whereby some good luck is needed to catch the target at two
RVs with a large separation. Based on the experience with our
data, in case of good S/N and EW, the uncertainty of RV is about
3% of the FWHM. We determined the FWHM of the line pro-
file and dRVmax, the maximum of dRV for the set of spectra,
using the four lines above. In general, the OIII 5592 line is the
sharpest but has a small EW, while He lines suffer more from
LPVs. On the other hand, He I 5876 next to the interstellar lines
Na5890 provides low wavelength calibration errors (which are
typically about 3 km/s). This way we obtain an SB1 candidate, if
dRV >3+0.03 ·FWHM km/s.
The S/N of the spectra is not homogeneous; this holds for
both FEROS and BESO spectra as well. In particular, numerous
BESO spectra taken during the early HPT operational phase suf-
fer from a poor focus. For our data, the S/N inhomogeneity limits
the use of σRV, the RV dispersion. Therefore, all SB1 candidates
have been visually inspected by at least two of the three authors
independently, and they had to agree on the SB1 classification.
RV variations are easier detected for a star with narrow
lines (i.e. small vsin i). This inevitably poses a bias against the
detection of SB1s among fast rotators (vsin i>200 km/s). The
remaining stars (not classified as SB1, SB2, SB3) are adopted
as C. Nearly all of them show line profile variations (also in
O III 5592) which could be real or caused by the relatively large
noise. Therefore, we here do not distinguish between C and LPV
(as Holgado et al. 2022 did).
3.2. Determination of vsin i
3.2.1. Overview and caveats
The broadening of a spectral line profile is essentially caused by
the atmospheric turbulence and the stellar rotation (e.g. Slettebak
1956;Conti & Ebbets 1977). It is widely assumed that the line
profile Pline can be written as convolution of the stellar rotation
profile Prot with the atmospheric turbulence profile Pturb ,
Pline =Prot ×Pturb ,(1)
where Prot has a round elliptical shape, and Pturb looks more
triangular and can be approximated by a Lorentzian profile
(or a Gaussian or a combination of both). The different shape
“round” and “triangle” has been used to disentangle turbulence
and rotation by obtaining a best fit of Eq. (1) to the spectral line
(Goodness-Of-Fit method, GOF). Modern techniques use syn-
thetic model spectra for Pturb derived from e.g. the FASTWIND
tool (Puls et al. 2005;Rivero González et al. 2012).
The Fourier transform (FT) spectrum of a Lorentzian or
Gaussian is featureless, but the FT spectrum of Prot exhibits
characteristic minima related to vsin i(Carroll 1933;Gray
1973,2005), making the FT method ideally suited for deter-
mining vsin i.Royer (2005) has successfully applied the
FT method to F- and A-type stars. Ryans et al. (2002) and
Simón-Díaz & Herrero (2007,2014) introduced the FT method
to single O- and B-type stars and performed a thorough com-
parison of vsin iderived by FT and GOF. Both methods yield
consistent vsin ivalues. Despite the great success achieved so
far, the most important caveat is that other effects than stellar
rotation may produce FT minima as well, potentially leading to
biased vsin ivalues.
Conti & Ebbets (1977) already realised a lack of narrow-
lined, slow rotating O stars with vsin i.50 km/s which would
be expected for small inclination angles of the rotation axis with
respect to the line-of-sight (i<10◦). Indeed, there exist slowly
rotating magnetic O stars with well-determined rotation peri-
ods implying vsin i≤1 km/s, but both the FT and GOF meth-
ods yield vsin i≈45 km/s (Sundqvist et al. 2013). As suggested
by these authors and others, the severe vsin ioverestimates for
slow rotators are most likely caused by an insufficient treat-
ment of the competing broadening mechanisms often referred
to as micro- and macro-turbulence. Aerts et al. (2014) also cau-
tion the blind application of the FT method to stars with con-
siderable pulsational line broadening. Finally, O-type stars are
well known to produce a strong wind (e.g. Howarth & Prinja
1989;Kudritzki & Puls 2000). We suggest that a slow wind
(Martins et al. 2005a) or an expanding halo may cause spurious
effects on vsin ifor slow rotators; we describe some ideas on that
in Appendix B. Winds might have only little influence on vsin i
for medium and fast rotators which are at the focus of this paper.
3.2.2. Fourier transform method applied here
Compared to Helium and Balmer lines, the metal lines like
O III 5592 are well known to be less affected by wind and
turbulence, but often weak. Therefore, for each star, we have
determined vsin ifor O III 5592, several Helium lines. We
applied the FT method following the recipes described in
Simón-Díaz & Herrero (2007,2014). We checked that the first
FT minimum occurs at a frequency range where the median
FT amplitude lies well above the noise level (see Fig. 2 in
Simón-Díaz & Herrero 2007); data not fulfilling that criterium
are rejected.
Almost all lines show an asymmetric profile. We rejected
asymmetric lines where the profile is certainly dominated by
a strong wind (P Cyg). Profiles with a mild asymmetry were
accepted. Simón-Díaz et al. (2017) have extensively investigated
asymmetric profiles and their skewness; profiles with blue or red
asymmetries (negative and positive skewness, respectrively) may
indicate the presence of stellar oscillations as well.
In order to increase the S/N, we averaged the line profiles
from the 2–5 best spectra, shifted for each line to radial velocity
RV =0. We applied the FT method both to single and averaged
spectra. The averaged spectra not only show a better S/N but
the main advantage is: For the disentangled SB2+SB3 compo-
nents, averaging the spectra reduces the residual wiggles which
arise from a non-perfect disentangling (illustrated in Sect. 3.2.3).
Likewise, for pulsating stars the line profile variations may aver-
age out.
We found in general that the first FT minimum is shallow,
sometimes barely recognisable; this holds in particular for sin-
gle spectra, where sometimes the first FT minimum escaped
detection. Fourier theory predicts that (even small) profile asym-
metry may smear out the first FT minimum. To understand the
effect of profile asymmetry on the vsin idetermination, we artifi-
cially symmetrised the profiles (i.e. consider mirrored profiles);
we here expand the symmetrising approach of Sundqvist et al.
(2013). The mirroring axis has been determined from a parabola
fit of the profile trough. This allows us to examine four profile
types and derive their vsin i, as illustrated with the example in
Fig. 2(top):
(1) original profile (black dashed)
(2) blue half mirrored (blue)
A192, page 4 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
(3) red half mirrored (red)
(4) averaged red and blue mirrored (green).
The original profile shows a small asymmetry with a blue
absorption which likely arises from a slow wind (v < 150 km/s).
The three mirrored profiles are symmetric.
Figure 2(middle panel) depicts the FT amplitudes: The first
FT minimum of the original asymmetric profile (black dashed)
is shallow, while the symmetric mirrored profiles yield a sharp
minimum (red, blue, green). The green and black FT minima
yield essentially the same vsin ivalues. This gives us confidence
that the shallow black minimum is indeed related to the sought-
for signature of rotation. However, the red and blue profiles yield
smaller and larger vsin i, respectively. This is expected, if the
blue profile suffers from the additional line broadening (i.e. the
asymmetric wing of the original profile is caused by a wind).
The bottom panel of Fig. 2plots the FT of the original pro-
file in the complex Fourier plane. Mathematically, the FT of a
symmetric profile is real (i.e. the imaginary part is zero). For an
asymmetric profile, however, the imaginary part is non-zero; as
a consequence the FT data points move in the complex Fourier
plane around the (0,0) origin, leading to shallow rather than
sharp amplitude minima.
The example of Fig. 2suggests that the effect of a wind-
caused asymmetry may easily be reduced by taking vsin iof the
red mirrored profile (see also Sundqvist et al. 2013). However,
this strategy seems to be not applicable in general for our data,
because negative and positive skewness occur. One reason may
be that the skewness is sensitive to the uncertainty of the mirror-
ing axis, in particular for faint noisy lines. Thus, for about half
of the lines the red mirrored profile yields a larger vsin ithan the
blue mirrored profile, contrary to what one would expect, if the
blue wing is due to wind.
We take the median of these four vsin ivalues and the stan-
dard deviation of the median as an error estimate (for an indi-
vidual line, e.g. O III 5592). This way, outliers among the four
values have little influence on the adopted vsin i, but a large
error warns us that the vsin icalculation may suffer from asym-
metric profiles. Indeed, the mirroring method enabled us to
reject numerous uncertain cases. Further error considerations are
addressed in Sect. 3.2.4.
We were able to determine vsin ifor all Cs and SB1s in our
sample, and in the 64 SB2 and 8 SB3 systems for all stars except
10 secondaries and 2 tertiaries.
The detection limit for slow vsin ihas to be calculated in
Fourier space, since we seek for the first minimum of the FT
amplitude. The crucial point is not only the spectral resolution
(in our case R ∼50 000) and the noise level, but also the addi-
tional line broadening by turbulence vturb which is strong in
O stars (Sect. 2.1. in Simón-Díaz & Herrero 2007). This steep-
ens the decline of the FT amplitude, so that the noise level is
reached at larger vsin i, compared to a negligible vturb. We esti-
mate a lower limit of vsin iabout 20–30 km/s (consistent with
Fig. 4 in Simón-Díaz & Herrero 2007).
3.2.3. Residuals of the disentangling
For SB2s (and SB3s), the Gaussian decomposition may lead
to residual wiggles in the resulting line profiles (Appendix A).
Mostly the residuals are small, within the noise of the spectra.
We have examined possible effects on vsin i, also using simu-
lated line profiles, finding indeed that any effects on the deter-
mination of vsin iare negligible. However, a few SB2s show
strong residuals, clearly exceeding the noise of the spectra and
Fig. 3shows one of the worst cases, HD152219 (DT 5). In
CPD−58 2611 HeI5876
−200 −100 0 100 200
v [ km/s ]
0.7
0.8
0.9
1.0
spectrum
averaged red and blue
blue mirrored
red mirrored
original
asymmetric wing
40 60 80 100
vsini [ km/s ]
−5
−4
−3
−2
−1
0
log FT amplitude
CPD−58 2611 HeI5876
1. min
Complex Fourier Plane CPD−58 2611 HeI5876
0 200 400 600 800 1000
real part x 1000
−20
−15
−10
−5
0
5
imaginary part x 1000
1. min
Fig. 2. Mirrored asymmetric line profiles and their Fourier transforms.
Example from CPD−58◦2611 in the He I 5876 line. Top: Original line
profile with an asymmetric blue wing (black dashed), as well as the
three types of symmetric line profiles: mirroring the red (v > 0 km/s)
and blue (v < 0 km/s) halves of the line profile, and in green the aver-
aged mirrored line profile. Middle: FT amplitude of the original line
profile (black dashed), and the mirrored profiles (red, blue, green). Bot-
tom: FT of the original asymmetric profile resembles a smooth chain of
data points in the complex plane with non-zero imaginary part. The data
point of the first minimum of the FT amplitude is marked in red.
brief, the take-away messages from the top panels of the figure
are:
(1) Residuals can be characterised by Ares /Aline &0.3, where
Ares is the peak-to-peak amplitude of the residual wiggles
and Aline is the Gaussian height of the line. Any residuals
produced by the faint component (with small Aline) are neg-
ligible for the bright component (with large Aline).
(2) Relevant residuals are produced by the bright component
(here the primary P) and play a role for the faint com-
ponent (here the secondary S). Significant effects occur, if
Aline(P)/Aline (S) &3.
(3) Averaging of N spectra with different dRV is able to quickly
smooth the residuals. For instance, Ares/Aline of S declines by
about N1/2.
(4) If dRV between P and S is sufficiently large, then most of the
wiggles lie outside of the line-core of S. Then cutting-out a
A192, page 5 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
HD152219 , HeI4922 , 2004-May-09
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
intensity
Primary
FWHM = 294
A line = 0.110
A res = 0.011
Secondary
FWHM = 162
A line = 0.022
A res = 0.020
residuals
HD152219 , HeI4922 , 2006-May-06
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
intensity
Primary
FWHM = 284
A line = 0.112
A res = 0.011
Secondary
FWHM = 212
A line = 0.020
A res = 0.017
residuals
HD152219 , HeI4922 , avg_2
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
intensity
Primary
FWHM = 282
A line = 0.110
A res = 0.009
Secondary
FWHM = 157
A line = 0.023
A res = 0.010
HD152219 , HeI4922 , avg_9
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
intensity
Primary
FWHM = 284
A line = 0.108
A res = 0.003
Secondary
FWHM = 169
A line = 0.019
A res = 0.007
HD152219 , HeI4922 , avg_9
50 100 150 200 250 300
vsini [ km/s ]
-5
-4
-3
-2
-1
0
1
log FT amplitude
Primary:
residuals included
vsini = 178+/- 4
HD152219 , HeI4922 , avg_9
50 100 150 200 250 300
vsini [ km/s ]
-5
-4
-3
-2
-1
0
1
log FT amplitude
FWHM
Secondary:
residuals included
vsini = 139+/- 4
HD152219 , HeI4922 , avg_9
50 100 150 200 250 300
vsini [ km/s ]
-5
-4
-3
-2
-1
0
1
log FT amplitude
FWHM
Secondary:
residuals excluded
vsini = 119+/- 2
Fig. 3. One of the worst cases (DT5) of the effects of disentangling residuals of HD 152219 in He I4922. Top, two left panels: Single spectra as
observed on 2004 May 09 and on 2006 May 06. The two right panels show the average of the two spectra and of all nine spectra used, whereby
each component has been shifted to RV=0 before averaging. The black lines mark Gaussian fits to the profiles, with FWHM and height Aline
labelled. Likewise, the peak-to-peak amplitude of the residual wiggles, Ares, is given, determined outside of the velocity range ±3σof each profile
marked by the coloured vertical bars. Averaging reduces the residual wiggles. Bottom: Fourier transform amplitude of the average of all nine
spectra, for the primary (left panel), and for the secondary including and excluding residuals (middle and right panel, respectively). The details are
explained in Sect. 3.2.3.
small window (typically ±3σ) around the line profile of S
excludes most of the wiggles.
The bottom row of Fig. 3depicts the Fourier amplitudes of the
average of all 9 spectra used, for the primary (left panel), and for
the secondary including and excluding the residuals (middle and
right panel), respectively. The four colours refer to the four cases
of the mirroring technique (black =original line, red mirrored,
blue mirrored and averaged red and blue mirrored). The take-
away messages for vsin iare:
(5) The primary is well measured, even with included residuals;
excluding the residuals does not change vsin ias expected
because they are small and not visible within the noise (FT
plot not shown).
(6) vsin iof the secondary changes from 139 ±4 km/s to
119 ±2 km/s when the residuals are excluded. The small
errors are due to the large symmetry of the profiles. The
comparison with FWHM of the line profile (169 km/s, ver-
tical dashed line) strongly supports that the exclusion of the
residuals yields a trustable vsin i3.
(7) The quite well-isolated secondary of HD 152219 allows us
to estimate a potential vsin ibias of about 20% (factor of
139/119, i.e. ∼1.2), when the residuals are not excluded. In
addition, we have explored artificial SB2s using modelled
line profiles of rotating stars with turbulence, finding a vsin i
bias of typically less than 5% when the residuals are small
but reaching up to 20% in worst cases with strong residuals.
Such an estimate is useful for binaries where the Gaussian
decomposition yields strong residuals for a faint broad sec-
ondary. If the secondary is sufficiently broad, then the resid-
uals produced by the primary lie within the line core and
cannot be excluded via cutting-out a window. CPD −59 2600
is the worst example of such binaries (see DT 6 in Fig. A.1).
Notably, whenever an SB2 in our sample exhibits a bright
3Typical ratios vsin i/FWHM lie in a valid range between
0.3 and 0.7, as we derived from modelled line profiles of
rotating stars with turbulence. For the secondary we obtain:
vsin i/FWHM =139/169 =0.82 (wiggles included) which lies outside
the valid range and vsin i/FWHM =119/169 =0.7 (wiggles excluded)
inside the valid range.
narrow and a flat broad component, the residuals and the
potential vsin ibias only affect the flat component. We
checked that the bias does not change the main result, namely
that the flat component is significantly broader than the nar-
row component.
To summarise, we have checked that for our SB2 sample the
Gaussian decomposition mostly yields small residuals and neg-
ligible effects on vsin i. However, a few cases may suffer from
increased uncertainties and a potential bias up to 20%. They do
not change the main result qualitatively.
3.2.4. Error estimates
For a given star and a given line (e.g. He I 5876), the mirror-
ing technique yields a formal error (i.e. standard deviation) on
vsin iwhich refers to the symmetry of the line. Often these errors
appear unrealistically small, <5%. For instance, Holgado et al.
(2022) found vsin idifferences down to ∼10% between the
FT method and the Goodness-of-fit method (which we did not
apply here). An alternative realistic error may be obtained from
the average of vsin iindependently measured in several HeI
lines. Therefore we compared the errors from the two methods:
a) from averaging over several lines and b) from the mirroring
technique4.
Figure 4shows the distribution of vsin ierrors in percentage,
separated for two samples: 148 single-lined stars and 140 stars in
SB2+SB3 systems. All stars in SB2+SB3 systems have a vsin i
measurement in at least two He I lines, which allows error calcu-
lation via method (a). Eighteen of the 166 single-lined stars were
omitted in this comparison because they were measured in less
than two He I lines. The reason is that these 18 stars show strong
winds and P Cyg profiles in most He I lines which were rejected
from the vsin idetermination. Notably, such sources are not used
for the binary disentangling as well, and this likely explains why
binaries lack P Cyg profiles in most He I lines, which enables
a successful vsin imeasurement in these lines. For method (b),
4The error comparison of the He II lines is similar to that of the He I
lines. For O III we have only one line at 5592 Å.
A192, page 6 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
C+SB1 , Helium I lines
0 10 20 30 40 50
relative vsini error [ % ]
0
20
40
60
80
Number
356 line measurements
std dev from avg
148 stars
SB2+SB3 , Helium I lines
0 10 20 30 40 50
relative vsini error [ % ]
0
20
40
60
80
Number
382 line measurements
std dev from avg
140 stars
Fig. 4. Distribution of vsin ierrors in the heliumI lines for single-lined
stars (left) and for stars in SB2+SB3 systems (right). The width of the
histogram bins is 3%. The details are explained in Sect. 3.2.4.
the black histograms show the error distribution of the mirroring
technique, whereby the number of line measurements (356 and
385, resp.) gives the number of used HeI lines integrated over all
stars. Basically, for both methods and both samples the error his-
tograms look similar with a peak around 6% and a tail towards
30%. For method (b) the fraction of very small (<6%) errors is
larger. For method (a), choosing a minimum error of 6% may
still be optimistic, because vsin iwas calculated only for good
He I lines (poor lines were excluded from the calculation).
We conclude that realistic vsin ierrors are at least 10%. For
the disentangled binaries we have assigned quality flags with
a likely vsin ierror between 10% and 30% (Appendix A). For
both samples, the proposed errors agree also with the differences
between literature vsin ivalues and ours (Sect. 4.2). Further-
more, for slow rotators with vsin i<100 km/s a minimum error
of 10 km/s might be realistic to better account for potential sys-
tematic effects.
In this work, we do not use the vsin ierrors further but we
note that the statistical differences of vsin i(and of line FWHM)
between single stars (C) and binaries and between primaries and
secondaries are so large that the main results of this paper remain
unchanged, even if vsin ierrors are taken into account.
4. Results
The large sample allows us to build well-defined subsamples, in
order to explore how vsin idepends on the spectral line used, the
spectral type, luminosity class, and the SB type.
4.1. v sin i from different spectral lines
For each star we have determined vsin iusing several lines with
good S/N. In order to compare the statistics of the rotation rate
for subsamples (C, SB1, SB2, etc.), we need a “representative”
vsin ivalue for each star. To this end, we used the value from
O III (118 C+SB1, 54 stars in SB2+SB3); if O III is not avail-
able, then we take the average of HeI lines (48 C+SB1, 61 O
stars in SB2+SB3, 25 B-type companions). This way we obtain
for each star a vsin ivalue (called best vsin i) used for the sta-
tistical comparison. Table D.3 (available at the CDS) list vsin i
for O III and the average of He I lines and of He II lines, and the
values for each Helium line5.
The best vsin iis based on either O III or He I, raising the
question on a potential bias. To that end, we sort the stars along
rising best vsin iand then plot for each star the average vsin i
values of the lines used (Fig. 5). The horizontal lines mark vsin i
5We find that vsin iobtained for HeI4471 is consistent with that for
other He I lines. It turns out that wings in HeI4471 have little influence
on the determination of vsin i, when using the mirroring technique.
ranges, and the corresponding vertical lines visualise the fraction
of stars in these ranges. It documents a systematic trend that – for
rotators below about 100km/s – vsin itypically increases from
O III over He I to He II lines. This apparent “stratification” dis-
appears for medium and fast rotators, where rotation dominates
other line broadening effects. This gives us confidence that for
rotators with true vsin i>150 km/s the best vsin iis unbiased.
On the other hand, for slow rotators with true vsin i<100 km/s
the best vsin imay depend on whether it is obtained by O III or
He I.
How strong is the effect of a mix of O III and He I based
best vsin ion the comparison between subsamples, e.g. Cs
and SB2s? Of particular interest here is the “transition region”
around vsin i=100 km/s between slow and increased (i.e.
medium+fast) rotators. Figure 6compares the vsin idistribution
derived from OIII and He I, respectively, for stars with measure-
ments in both lines. The results are the following:
(1) The distributions peak at slow vsin i. For He I they shift to
slightly larger vsin ithan for O III (by less than one his-
togram bin). The modes of the distributions differ by 6 km/s
for single-lined stars and by 10 km/s double-lined stars. The
shifts are small, within typical vsin ierrors.
(2) When using He I instead of O III, the number (fraction) of
stars shifting from slow to increased rotators is 4 out of 107
and 5 out of 54 (i.e. about 4% and 10%) for single- and
double-lined stars, respectively. Hence the fractional differ-
ence is 6%.
(3) The fraction of stars with He I based best vsin iis larger for
SB2+SB3 than for single-lined stars (61/115 =53% versus
48/166 ∼29%, from Fig. 5). One reason is that O III 5592 is
too faint in many double-lined stars and about a third of them
have B-type companions.
(4) By combining the differences in items 2 and 3, we obtain a
statistical bias of about 6% ×53/29 =11%.
To conclude, in the “transition region” around vsin i=100 km/s
the mixed use of O III and He I leads to a small bias. For com-
paring the number of slow and increased rotators between single
and double-lined stars, the bias is about 11%. For each com-
parison in the sections below, we have carefully checked, if the
differences between subsamples may be due to the (partial) use
of non-metal lines and how strong this effect is. We found that
the exclusive use of HeI instead of O III does not alter the results
and conclusions drawn below.
4.2. v sin i comparison with literature
A comparison with the literature allowed us to check the quality
and reliability of our results.
We compared the vsin iresults of our 166 C/SB1 stars with
those of the IACOB & OWN surveys (285 C/SB1). We found
a match of 135 stars with the latest catalog by Holgado et al.
(2022), who have carefully derived vsin ifrom a combination of
the FT method with GOF. Figure 7illustrates the overall good
agreement. Two notable differences are:
(1) Among slow rotators (<100 km/s) there is a group of six
stars where we find larger vsin ithan Holgado et al. Our vsin i
value of these stars is based on He I instead of O III. A detailed
check shows that these are mostly single stars (C) and equally
balanced among dwarfs and giants. We do not expect that the use
of He I instead of O III alters the results and conclusions below.
(2) We find 7 more slow SB1 and Holgado et al. find 5 more
medium and fast SB1. This difference could be due to spectra
observed at different orbital phases with better RV separation.
An additional explanation could be, that our SB1 criterion uses
A192, page 7 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
C + SB1, N = 166
0 50 100 150
object number sorted by best vsini
0
100
200
300
400
vsini [ km/s ]
HeII
HeI , N = 48
OIII , N = 118
O-stars in SB2 + SB3 , N = 115
0 20 40 60 80 100 120
object number sorted by best vsini
0
100
200
300
400
vsini [ km/s ]
HeII
HeI , N = 61
OIII , N = 54
Fig. 5. vsin idistribution of different lines. Left for single-lined stars. Right for stars in SB2+SB3 systems, whereby the 25 B-type companions
are excluded because their vsin ihas been derived solely from HeI lines. The legends also list the number of stars with the best vsin ifrom O III
(green) and He I (red).
C + SB1, comparison OIII and HeI
0 100 200 300 400
vsini [ km/s ]
0
10
20
30
40
Number of stars
107 OIII
107 HeI
84
80
19
23
4
4
mode = 64 +- 21
mode = 58 +- 22
SB2 + SB3, comparison OIII and HeI
0 100 200 300 400
vsini [ km/s ]
0
5
10
15
20
Number of stars
54 OIII
54 HeI
44
39
10
15
0
0
mode = 80 +- 22
mode = 70 +- 22
Fig. 6. vsin idistribution derived from OIII (green) and HeI (red) for
single-lined stars (left) and for stars in SB2+SB3 systems (right). Only
stars with vsin imeasurements in both O III and He I are used. The
histogram bin size is 20 km/s. The vertical dotted lines separate slow,
medium, and fast rotators. The number of stars in the slow, medium,
and fast bins and in total are given, as well as the modes of the distribu-
tions.
0 100 200 300 400
vsini Holgado+2022
0
100
200
300
400
vsini this work
20/103 matched SB1/C
7 SB1 this work, C H22
5 SB1 H22, C this work
38 HeI, this work
11 HeI, H22
Fig. 7. Comparison of vsin iof C and SB1 stars from this work with
those measured by Holgado et al. (2022). The values scatter around
unity (solid diagonal line). Regarding the C/SB1 classification, the
Holgado et al. sources with line profile variation (LPV) are matched
with our Cs. Stars with different C/SB1 classifications are marked: we
find more slow SB1s (red) and Holgado et al. find more fast SB1s (blue).
0 100 200 300 400
vsini literature [ km/s ]
0
100
200
300
400
vsini this work [ km/s ]
30%
SB2 + SB3, N = 85
Primary (49)
Secondary (36)
Fig. 8. Comparison of vsin iof disentangled SB2 and SB3 components
from this work with collected values in the literature (Table D.2). The
values scatter around unity (solid diagonal line), within ∼30% (dashed
lines). The black arrows are upper limits.
the FWHM of the line profile (i.e. a flexible threshold), while
Holgado et al. switched from a flexible to a fixed threshold, if
vsin i<180 km/s. Without going into details we emphasise that
the results and conclusions drawn from our and Holgado et al.’s
sample largely agree.
The stars classified as C by our spectra but as SB1 by Hol-
gado et al. are listed as ‘C (SB1)’ in Table D.1. To use the most
actual classification in the scientific analysis, for the rest of the
paper we treat these stars as SB1s.
In the literature we found a match for 85 stars with listed
vsin ivalues. The literature compilation is listed in Table D.2.
The vsin icomparison is shown in Fig. 8, illustrating overall
good agreement within 30%. A bias towards larger literature
vsin ivalues may be explained as follows: Many literature vsin i
values were estimated from the line FWHM (e.g. Penny 1996;
Howarth et al. 1997), while we used the FT method; likewise
A192, page 8 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
C , dwarfs LC = IV+V, N = 57
0 100 200 300 400
vsini [ km/s ]
0
5
10
15
20
Number of stars
SpT > O7.5 ( 26 )
SpT <= O7.5 ( 31 )
slow medium fast
39 7 11
23 2 6
16 5 5
v peak = 57.9+- 13.0
v peak = 38.6+- 25.4
C , giants LC = I+II+III, N = 73
0 100 200 300 400
vsini [ km/s ]
0
5
10
15
20
25
Number of stars
SpT > O7.5 ( 42 )
SpT <= O7.5 ( 31 )
slow medium fast
51 13 9
20 7 4
31 6 5
v peak = 58.7+- 18.9
v peak = 58.6+- 17.7
Fig. 9. Dependence of vsin iof single stars (C) on spectral type and
luminosity class. Among giants and dwarfs, respectively, the early-type
stars (blue) show a ∼10 km/s faster v_peak than the late-type stars (red).
Likewise, the v_peak of giants is faster than that of dwarfs.
our vsin iis largely based on O III 5592, while in the literature
He I lines were used.
4.3. v sin i dependence on spectral type and luminosity class
The aim is to see whether spectral type or luminosity class have
a significant effect on vsin i, and whether this may influence the
vsin idependence on SB type. Compared to single stars vsin i
of binaries is on average increased (Sect. 4.4), so that trends
with spectral type or luminosity class may be blurred. There-
fore, we restrict the vsin idependence on spectral type and lumi-
nosity class to the 130 Cs (presumably mostly single stars, but
see Fig. 3 of de Mink et al. 2014). We consider v_peak and its
standard deviation, calculated via the mode of the vsin idistri-
butions.
We separate giants and dwarfs. Within each group, stars
of spectral type earlier or equal O7.5 show a ∼10 km/s faster
v_peak than later spectral types (vsin i-mass trend, Fig. 9). For
instance, the highest blue histogram bar is shifted by one his-
togram bin (20 km/s) against the highest red histogram bar.
So far, however, only tentative observational evidence for that
trend between early- and late-type stars has been presented,
C, N = 130
0 10 20 30 40
rotation period / sin(i) [ d ]
0
5
10
15
Number of stars
I+II ( 54 )
III ( 19 )
IV+V ( 57 )
short
31
19
8
4
median (P)
14.3 d
12.8 d
10.0 d
Fig. 10. Rotational period of single stars. Prot =2πR∗/v. The vertical
dotted line at Prot =5 d corresponds to v≈100 km/s for an O7.5V star.
Dwarfs exhibit a clear short-period peak. The median period is given
for the bulk of stars with long periods >5 d.
partly because the vsin idistributions are broad and the sam-
ples were small. Examples are given by Conti & Ebbets (1977,
their Fig. 6), and by Simón-Díaz & Herrero (2014, their Figs. 14
and 15). We see the vsin i-mass trend clearly in our data, but in
view of the broad standard deviations the statistical significance
is small. We mention three possible explanations:
(1) An earlier (and more massive) dwarf has a larger radius R∗,
so that for a given angular velocity, ω, the earlier dwarf
exhibits a larger vsin i; for giants, however, this explanation
does not work because R∗decreases from late to early giants.
(2) If early-type stars have a stronger macroturbulence or wind
component than late-type stars, then a bias, as explained in
Appendix B, may lead to a larger vsin iin the earlier types.
(3) Compared to late-type stars, early-type stars are, on average,
a factor of ∼2 more massive and a factor of ∼5 more lumi-
nous. This makes it harder to detect a companion, either by
dRV or by discerning the spectral lines of the companion.
Therefore, the fraction of Cs with detection-escaped com-
panions might be higher among early-type stars. Such Cs are
then actually binaries and may have a larger vsin i(Sect. 4.4).
The vsin i-mass trend may provide interesting clues to processes
in O stars, but a detailed exploration is beyond the scope of this
paper.
Similar to the vsin i-mass trend, Fig. 9shows also a vsin i-LC
trend: for a given spectral type range, the dwarfs show a slightly
slower vsin ipeak than giants and supergiants. This trend was
previously seen with a stronger difference between dwarfs and
giants (Conti & Ebbets 1977 and subsequent works). We note
here that giants have a larger radius than dwarfs (a factor of 2-
4 depending on the spectral type, Martins et al. 2005a). For the
angular momentum (a fundamental quantity with conservation
law) the angular velocity, ω, is relevant. Here we use alterna-
tively the rotational period Prot =2πR∗/v, with stellar radius R∗
from Martins et al. 2005a. Figure 10 shows the distribution of
Prot . In terms of period, on average, the giants and SGs rotate
significantly (30–40%) slower than dwarfs. This result supports
the consensus view that the rotation is braked during the stellar
evolution.
Another topic is the distribution of dwarfs in Fig. 10: above
and below the 5d line it appears split into two different compo-
nents. We suggest that this bimodal appearance is real, but diffi-
cult to interpret. It is certainly of formal (mathematical) nature:
A192, page 9 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
C, N = 130
0 100 200 300 400
vsini [ km/s ]
0
10
20
30
40
Number of stars
I+II ( 54 )
III ( 19 )
IV+V ( 57 )
slow medium fast
90 20 20
39 7 11
11 2 6
40 11 3
v peak = 54.8+- 22.8
v peak = 56.9+- 20.3
SB1, N = 36
0 100 200 300 400
vsini [ km/s ]
0
2
4
6
8
Number of stars
I+II ( 12 )
III ( 7 )
IV+V ( 17 )
slow medium fast
21 13 2
11 5 1
3 3 1
7 5 0
v peak = 47.4+- 27.5
v peak = 62.7+- 13.9
SB2+SB3 , N = 140
0 100 200 300 400
vsini [ km/s ]
0
5
10
15
20
25
30
Number of stars
I+II ( 11 )
III ( 25 )
IV+V ( 104 )
slow medium fast
77 49 14
51 41 12
17 7 1
9 1 1
v peak = 91.5+- 52.9
v peak = 72.2+- 14.6
Fig. 11. vsin idistribution of the spectroscopic binary types. The 140
stars in SB2+SB3 systems are 72 primaries, 62 secondaries, and 6 ter-
tiaries. v_peak gives the mode and standard deviation of the vsin idis-
tributions for LC I+II (red) and IV+V (blue); LC III has too few data
points.
when plotted versus Prot , the view onto slow rotators is zoomed,
but all increased rotators (vsin i>100 km/s) are “compressed”
into a small range leading to the strong histogram peak below
the 5 d line. Further details are beyond the scope of this paper.
Both vsin i-mass and vsin i-LC trends are also present when
vsin ihas been measured only with the O III 5592 line and not
with He I lines; this shows that the usage of He I lines does not
significantly biases the vsin istatistics of our samples. In addi-
tion, both trends are weak, suggesting that they have only little
effect compared to other vsin itrends and statistical vsin idiffer-
ences below.
4.4. v sin i dependence on SB type
So far the sample, for which we have determined vsin i, consists
of 130 Cs, 36 SB1s, and 140 stars in 64 SB2 and 8 SB3 systems
(72 primaries, 62 secondaries and 6 tertiaries). Figure 11 shows
the vsin idependence on SB-type:
(1) For each SB-type, most stars are slow rotators with vsin i
around 40–100 km/s.
(2) Spectroscopic binaries (SB1/SB2/SB3) show a factor of 2
larger fraction at medium vsin ibetween 100 and 200 km/s,
compared to single stars (C).
(3) The singles stars show a larger fraction (15%) of fast rota-
tors (vsin i>200 km/s) than the binaries (SB1 5%, 10%
SB2+SB3).
(4) The separation of luminosity classes into dwarfs (IV+V), III
and I+II shows that dwarfs dominate the binary population
(dwarf fraction 121/176 =69%), compared to the Cs (dwarf
fraction 57/130 =44%, giant/SG fraction 56%). The expla-
nation for the larger giant/SG fraction in Cs is not straight-
forward: Fast evolution of one of the I/II components in a
binary could make the system to appear as C or SB1 later.
Also an observational bias against giants/SGs in SB2s could
occur, since a giant/SG is more luminous making it harder to
detect a less luminous companion.
(5) The results and trends are already seen in the pure O III sam-
ple (Fig. 6) but they are more pronounced in the full sample.
This is due to the larger sample size and the fact that typically
medium and fast rotators are harder to measure in O III.
(6) The vsin idistribution of the 27 B-type secondaries appears
similar to that of the O stars in binaries; differences might
be due to the low number statistics. The inclusion of B-type
secondaries does not significantly alter the results.
For the further analysis and discussion we use the full sample
including the B-type companions.
5. Increased rotation rate in binaries
The rotation rate of binaries shows a clear excess at medium
vsin i(100 km/s< v sin i<200 km/s) compared to stars classi-
fied as C (Fig. 11). On the other hand, fast rotators (vsin i>
200 km/s) are more frequent in Cs than in binaries at all. Cs may
be true single stars or intrinsic binaries which escaped detection
due to small dRV arising from poorly inclined orbital axis, very
wide orbits and very low mass ratio M2/M1. The aim is to under-
stand the observed rotation difference between binaries and Cs,
with the key question of what leads to the increased rotation in
binaries.
Holgado et al. (2022) have reported a medium vsin iexcess
in SB1s and ascribed it to binary interaction during massive star
evolution. Britavskiy et al. (2023) have investigated the post-
interaction nature of fast rotators (vsin i>200 km/s) and their
rareness among binaries compared to Cs. Sana et al. (2012) have
analysed the O star population (31 C, 7 SB1 and 33 SB2) of six
nearby Galactic open stellar clusters. To conclude on the role
of interactions on the binary evolution, they performed Monte
Carlo simulations of interacting massive stars and compared
A192, page 10 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
them with the orbital period, eccentricity and mass ratio mea-
sured for the SBs. They briefly addressed spin-up of the sec-
ondary by donor-gainer mass transfer. However, they did not
include measurements of vsin i.
Ramírez-Agudelo et al. (2015) have found in the VLT-
Flames study of the 30 Doradus region of the LMC, that SB1
and SB2 rotate faster, if they have a large radial velocity dif-
ference (dRV >200 km/s) which serves as a proxy for a short
orbital period (Porb <10 d) and a small separation of the compo-
nents. Our O-star sample provides a large statistical vsin istudy
of SB2s in the Milky Way and we use measured orbital periods.
Of course, the detection of a spectroscopic binary is affected
by the inclination of the orbital axis; likewise, vsin iof individ-
ual stars is affected by the inclination of the rotational axis. Thus,
it is tempting to “unify” binaries and Cs via orientation, where
at least part of the Cs are misaligned binaries. Statistically, how-
ever, inclination plays a minor role on vsin i(in the range of
10%) and fails to explain the rotation differences between the
samples; more details are given in Appendix C.
Therefore, interaction between binary components has to
be considered, in other words spin-up both by tidal synchro-
nisation when the components approach (Zahn 1975,1977;
Tassoul & Tassoul 1997) and by mass exchange and merging
(de Mink et al. 2013). The components in our sample are sep-
arated by less than 30 AU down to 0.1AU (Fig. 12). For com-
parison, the distance between Neptun and the Sun is 30 AU.
If the common assumption holds that O-type binaries are born
with a component separation of at least several hundred AU (e.g.
the massive proto-binary found by Zhang et al. 2019), then it is
tempting that our data provide insight to the spin evolution dur-
ing the approach of the components.
We here focus on how the rotation of primary P and sec-
ondary S changes from wide to close binaries6. The separation of
the components is inferred from the orbital period, Porb. Because
for most SB systems our spectra are sparsely monitored, the
period information is collected from the literature (Tables D.1
and D.2). This yields subsamples with vsin iand Porb containing
24 out of 36 SB1s (67%) and 70 out of 72 SB2s (97%, including
the 8 SB3s). The respective vsin ihistograms (for SB1, P and S)
of the subsamples look – apart from absolute numbers – similar
to those of the original SB samples, suggesting that the subsam-
ples provide robust statistical implications and conclusions.
5.1. Rotation difference between close and wide binaries
We divide the sample into close and wide binaries applying
a threshold of Porb =10 d which is justified further below;
this threshold has also been applied by Ramírez-Agudelo et al.
(2015) for O stars in the LMC. We converted Porb into semi-
major axis, a, using the (re-written Keplerian) equation
a=4.208 ·(P2
orb ·(M1+M2))(1/3),(2)
with Porb in days, mass M1+M2of the two components in
Mand ain R. For M1and M2we used the “calibration
mass”, which is the stellar mass from the (interpolated) tables
in Martins et al. (2005b), corresponding to the spectral type and
luminosity class of the SB components (listed in Tables D.1
and D.2). For SB1s we adopted M2=0.1·M1. For one
SB2, V 961 Cen, we used the mass from Doppler tomography
6Seven of the 8 SB3 systems host a close P-S pair. The tertiaries
(associated with a close P-S pair) have a long period, if measured, or
a small RV amplitude indicating a large separation. The tertiaries are
not included in the analysis.
(Penny et al. 2002) which is about a factor of 2 smaller than the
calibration mass and leads here to more consistent results.
Figure 12 (top) shows stellar mass versus Porb. Across the
entire Porb range, SB1 and SB2 have similar median primary
mass M1of about 25 M. For SB2s the median secondary mass
M2is about 15 M(i.e. 2/3 of M1). The number of long period
SB1s, and SB2s with M2/M1<0.5 declines at Porb &100 d.
This is likely an observational bias against detection of wide
binaries, but it does not affect the findings and conclusions on
the wide binaries below (see Sect. 5.3).
Figure 12 (bottom) shows aversus Porb. The least-squares fit
to the (logarithmic) data agrees well with a∝P2/3
orb . The mod-
erate scatter around the fit is due to deviations of the calibration
mass from the real one. For our sample, Porb =10 d corresponds
to a≈60 R, and a=1 AU to Porb ≈70 d.
Figure 13 shows vsin iversus Porb of the SB systems. It
reveals:
(1) For SB1s, all medium rotators lie at Porb <30 d, while slow
rotators distribute evenly across the entire logarithmic period
scale (top left panel).
(2) For SB2s, the components distribute across the entire param-
eter space (top right panel). There are clear trends – like for
SB1s – when P and S are plotted separately (two bottom pan-
els): the medium rotators concentrate at short Porb <10 d,
and most of the slow rotators are at long Porb >10 d.
(3) Among SB2s, there are few fast rotators. Notably, fast pri-
maries are preferrably located among close SB2s, and fast
secondaries among wide SB2s. (Among SB1s, the period of
the only one fast rotator, HD041997, is not known.)
(4) Giants distribute across the entire period range. The giant
incidence is larger in SB1s than SB2s and among SB2s it is
larger among primaries and among close SB2s. Among the
wide SB2 primaries, all but one are slow rotators.
The trends are corroborated by the vsin ihistograms in Fig. 14.
Top row: Both close primaries and secondaries reveal a strong
excess of medium rotators. Bottom row: The vsin idistribution
of wide primaries is dominated by slow rotators and appears
similar to that of Cs (cf. Fig. 11). However, wide secondaries
are different exhibiting a strong fast tail. Removing B-type sec-
ondaries yields similar histograms. In addition, separating close
and wide binaries by Porb =20 d (instead of 10d) yields similar
histograms. To summarise, close binaries exhibit a pronounced
spin-up further analysed in Sect. 5.2. On the other hand, wide
primaries, on average, appear similar to Cs dominated by slow
rotators, and wide secondaries appear nearly bi-modal with a
dominant slow rotator peak and a strong spin-up peak; this is
discussed in Sect. 5.3.
5.2. Synchronisation and mass transfer in close binaries
The two mechanisms discussed are:
(1) Spin-orbit synchronisation aligns the rotational and
orbital axes and equalises the periods (Porb ≈Pprim
rot ≈Psec
rot ). It
may increase but also brake the stellar rotation (e.g. Zahn 1977).
It predicts an increased equatorial stellar rotation (v > 100 km/s),
if Porb .10 d. On the other hand, at Porb =10 d spin-orbit syn-
chronisation will brake a fast rotating O9 I star with v=300 km/s
(Prot =4 d).
(2) Mass transfer (MT) from an expanded mass donor (typi-
cally the primary) to a mass accretor (the gainer, typically the
secondary) increases the spin of the gainer, while that of the
donor may be reduced. Since the mass flow is in the orbital plane,
MT may align the rotational and orbital axes (and aligned axes
have been generally assumed in simulations, e.g. Wellstein 2001;
de Mink et al. 2013).
A192, page 11 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
1 10 100 1000
Orbital period [ d ]
0
10
20
30
40
50
60
Mass [ M sun ]
24 SB1
median mass = 24.5
1 10 100 1000
Orbital period [ d ]
0
10
20
30
40
50
60
Mass M1, M2 [ M sun ]
70 SB2
Primary
Secondary
median = 24.4
median = 15.4
1 10 100 1000
Orbital period [ d ]
10
100
1000
Orbit semi-major axis [ R sun ]
SB1, N = 24
y = 1.111 + 0.665 x
1 AU
1 10 100 1000
Orbital period [ d ]
10
100
1000
Orbit semi-major axis [ R sun ]
M2 / M1 > 0.5
M2 / M1 < 0.5
26 16 10 45 8 1 0
70 SB2
y = 1.150 + 0.675 x
10 AU
1 AU
Fig. 12. Calibration mass and semi-major axis vs. orbital period. Left for SB1 and right for SB2. Top: Mass, the horizontal line marks the median
mass. Bottom: Semi-major axis, the solid line marks a least-squares fit to the data (logarithmic, with the fit equation labelled). The vertical and
horizontal lines are for guidance (dashed and dotted). Bottom right: Blue and red distinguish the mass ratio, the coloured numbers give the number
of SB2s in the Porb bins separated by Porb =10, 100, 1000.
If both mechanisms synchronisation and mass transfer are at
work, we expect (1) the primary period to equal that of the orbit
(neglecting spin down of the mass donor) and (2) the secondary
period to be equal or shorter than that of the orbit.
We will not examine each SB system individually but per-
form a statistical analysis. Our chain of reasoning begins with
assuming – as a guide line – that the binaries are synchronised,
testing how far the data are consistent with synchronisation using
the simple criterion that the periods should be equal. Then, we
look how far the deviations between data and the ideal synchro-
nisation picture are consistent with the mass transfer scenario.
We will use suited examples for illustration.
To check for synchronisation, we will compare Porb and Prot.
Porb is precisely available from radial velocity curves (with an
error smaller than 1%). However, direct measurements of Prot
are not available. We derived Prot from the measured vsin iand
calculated the auxiliary quantity
Psini
rot =Prot/sin(i)=2πR∗/v sin i,(3)
which still depends on the inclination. The inclination will be
rectified below (Sect. 5.2.2). The error of vsin ilies between
10% and 30%. The uncertainty of Prot is likely dominated by
the choice of R∗.
For R∗we take the “calibration radius” (i.e. the stellar radius)
from the logarithmically interpolated tables for spectral type
and luminosity class in Martins et al. (2005b) (see Tables D.1
and D.2). For 19 SB2s in our sample, the actual stellar radius
was measured by various authors, for example with eclipses
or inferred from luminosity considerations, if the distance was
known. We denote it the literature radius Rlit. It is listed together
with references in Table D.2. Figure 15 shows that Rlit is on
average ∼10% smaller than R∗; this has already been noted for
some O binaries by Rauw et al. (2001a). The standard deviation
of Rlit/R∗indicates that the uncertainty of R∗is at least 20%.
Nevertheless, we here use the calibration radius and discuss the
outcome. An exception is V 961 Cen where we used the literature
radius (Penny et al. 2002), which is a factor 2–3 smaller than the
calibration radius and here yields more consistent results.
For comparison, we have calculated the effective Roche
radius Rroche of all stars using the formula by Eggleton (1983). If
Roche lobe overflow (RLOF) is frequent in the primaries, then
one would expect for them that Rlit ∼Rroche which may reach
up to 2 R∗. Notably, this is not seen in Fig. 15. Furthermore, if
RLOF is frequent in close but not in wide binaries, then one
may expect a trend of increasing Rlit/R∗with decreasing orbital
period. Therefore, in Fig. 15 we have plotted Rlit/R∗versus
Porb. However, such a trend is only marginally visible
A192, page 12 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
1 10 100 1000
Orbital period [ d ]
0
50
100
150
200
250
300
350
vsini [ km/s ]
24 SB1
I+II+III x
slow medium fast
1 10 100 1000
Orbital period [ d ]
0
50
100
150
200
250
300
350
vsini [ km/s ]
70 SB2
Primary
Secondary
I+II+III x
slow medium fast
1 10 100 1000
Orbital period [ d ]
0
50
100
150
200
250
300
350
vsini [ km/s ]
70 SB2
Primary
I+II+III x
slow medium fast
1 10 100 1000
Orbital period [ d ]
0
50
100
150
200
250
300
350
vsini [ km/s ]
B-type o
70 SB2
Secondary
I+II+III x
slow medium fast
Fig. 13. vsin ivs. orbital period. Top: For SB1 (left) and SB2 (right), each primary–secondary pair is connected with a vertical dotted line. Bottom:
SB2 primaries and secondaries plotted separately; for the secondaries the black encircled symbols mark B-type companions.
close SB2, only P, N = 31
0 100 200 300 400
vsini [ km/s ]
0
2
4
6
8
Number of stars
I+II ( 2 )
III ( 7 )
IV+V ( 22 )
slow medium fast
12 15 4
7 12 3
4 3 0
1 0 1
v peak = 123.6+- 42.5
close SB2, only S, N = 31
0 100 200 300 400
vsini [ km/s ]
0
2
4
6
Number of stars
I+II ( 2 )
III ( 4 )
IV+V ( 25 )
slow medium fast
14 16 1
10 14 1
3 1 0
1 1 0
v peak = 104.1+- 38.0
close SB2 P+S, N = 62
0 100 200 300 400
vsini [ km/s ]
0
2
4
6
8
10
12
14
Number of stars
I+II ( 4 )
III ( 11 )
IV+V ( 47 )
slow medium fast
26 31 5
17 26 4
7 4 0
2 1 1
v peak = 110.8+- 39.0
wide SB2, only P, N = 39
0 100 200 300 400
vsini [ km/s ]
0
5
10
15
Number of stars
I+II ( 6 )
III ( 9 )
IV+V ( 24 )
slow medium fast
29 8 2
15 7 2
8 1 0
6 0 0
v peak = 59.4+- 14.6
wide SB2, only S, N = 39
0 100 200 300 400
vsini [ km/s ]
0
2
4
6
Number of stars
I+II ( 1 )
III ( 5 )
IV+V ( 33 )
slow medium fast
22 10 7
19 8 6
2 2 1
1 0 0
v peak = 63.9+- 38.4
wide SB2 P+S, N = 78
0 100 200 300 400
vsini [ km/s ]
0
5
10
15
20
Number of stars
I+II ( 7 )
III ( 14 )
IV+V ( 57 )
slow medium fast
51 18 9
34 15 8
10 3 1
7 0 0
v peak = 52.1+- 33.1
Fig. 14. Histograms of vsin ifor close and wide SB2s separated by Porb =10 d, from left to right for primaries, secondaries, and the entire samples.
Top row: Close SB2s. Bottom row: Wide SB2s.
A192, page 13 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
1 10 100 1000
Orbital period [ d ]
1
Literature / calibration radius
avg = 0.92+/-0.17
0.5
2
Primary
Secondary
Fig. 15. Ratio of literature to calibration radius Rlit/R∗of the stars vs.
orbital period for the 19 SB2s where a literature radius is available
(Table D.2). For each SB2 the vertical dotted line connects the pri-
mary and secondary. Unity is marked by the horizontal solid line, and
the average and its standard deviation by the horizontal long-dashed
and dash-dotted lines, respectively. The blue three-dot-dashed line is an
ordinary least-squares fit (log Y vs. log X) of the primary data.
(blue 3-dot-dashed line) and appears not yet of statistical signifi-
cance in this small sample of 19 SB2. A precise measurement of
the actual stellar radius is challenging. In view of the uncertain-
ties, we think that the results on Rlit/R∗obtained so far do not
argue striktly against RLOF in close binaries, rather they appear
consistent with partial Roche lobe filling of the primaries.
5.2.1. Close SB1s
We begin with how far synchronisation may lead to a spin-up of
vsin ifrom slow to medium rotators (i.e. crossing the threshold
of 100 km/s). We consider the close SB1s in Fig. 13 (top left)
and Fig. 16 (top left):
(1) four giants with medium vsin i(blue dots with black cross)
(2) two dwarfs with medium vsin i(blue dots without black
cross)
(3) three dwarfs with slow vsin i(red dots without black cross).
Despite having different vsin i, all of these stars (except one dis-
cussed below) lie in a narrow range between the sync line defined
by Prot/sin(i)=Porb (solid diagonal line) and the dotted line
labelled i=30 (Fig. 16). Inclination, if present, shifts the stars
down by a factor f=1/sin(i), e.g. f =2 for i=30◦. The incli-
nation of the SB1s is not known and cannot be corrected here.
Nevertheless, it appears consistent that SB1s which are close to
or below the sync line are synchronised. Then it is tempting to
speculate that the SB1s with medium vsin i(blue dots) originally
(i.e. in the past) were slow rotators and received a spin-up during
the approach and synchronisation of the components. We will
discuss this possibility further for those SB2s where inclination
is corrected.
The examples illustrate also that the analysis of vsin ialone
is not an ideal spin-orbit indicator, because the stellar radius
plays a crucial role on Prot. The radius increases from late to
early dwarfs by about a factor of 2. Giants have a factor of 2-
4 larger radius than dwarfs. Even slow rotating SB1s may be
synchronised, if their radius is small enough. Synchronisation
may begin already at Porb =30 d but there the stars would be
slow rotators. The threshold of vsin i=100 km/s corresponds to
Prot/sin(i) between 5 and 10 d, depending on the stellar radius;
this can be seen from the vertical distribution of the blue and red
symbols in Fig. 16 top left.
One close SB1 lies a factor of 2 above the sync line
(HD 053975 vsin i≈180 km/s, Fig. 16 top left). Obviously it
rotates too fast for the pure synchronisation scenario. We address
two possibilities which could bring HD 053975 into a consistent
picture:
(1) Fast rotation deformes a spherical star to a lenticular star with
increased equatorial radius Req ∼1.5 ·Rpolar. However, this
requires between 40–50% and up to 90% of the critical rota-
tion (i.e. veq between 250 and 500 km/s) (Abdul-Masih 2023;
Maeder & Meynet 2000;Maeder 2009). For HD 053975 the
inclination of the rotation axis is not known; if irot =30◦
then veq ≈360 km/s, in the range required for deformation
but then HD 053975 will lie a factor of 4 above the sync
line, exceeding a possible down-shift by the factor 1.5 due
to deformation. Thus, an additional mechanism is required.
(2) In case of (partial) Roche-lobe filling the actual radius is
larger than the calibration radius used to convert vsin i
to Psini
rot . This may shift HD 053975 towards the sync
line.
(3) On the other hand, if the R∗(used for the plot) is indeed
the actual stellar radius, then HD 053975 is not (yet) spin-
orbit synchronised. Then it has already a high spin, some-
how obtained in the past. One may speculate that the ongoing
tidal forces may spin-down HD 053975. This and/or a poten-
tial further approach of the components may finally move
HD 053975 to the synchronisation line.
To conclude, the close SB1s populate a range of Prot/sin(i) vs.
Porb which is consistent with spin-orbit synchronisation. In some
cases an expanded stellar radius, e.g. by (partial) Roche-lobe fill-
ing, of the SB1 primary may be implied.
5.2.2. Close SB2s
Figure 16 (top right) displays Prot/sin(i) vs. Porb. To understand
what happens in the close SB2s, we first compare with the wide
SB2s. Wide binaries with Porb >10 d exhibit a large difference
(on the logarithmic scale) between Pprim
rot /sin(i) and Psec
rot /sin(i),
marked by the vertical dotted lines connecting the SB2 compo-
nents. This difference provides evidence that wide SB2s are not
synchronised, because the components have different periods or
inclinations. However, for close binaries the rotational periods of
the components converge.
The convergence of the rotational periods is visualised by
the ratio PA/B
rot of the rotational period, either Pprim
rot /Psec
rot or
Psec
rot /Pprim
rot , whereof the value <1 is used (Fig. 16, middle left).
The blue dots mark SB2s, where (in terms of angular velocity)
the primary spins faster than the secondary, and vice versa for
the red crosses. The advantage of the ratio is that the effect of
inclination cancels, if the rotation axes are parallel as assumed
to be the case for synchronisation; therefore we have omitted
the term sin(i). The median PA/B
rot is about 0.5 for wide binaries
(in two bins separated by Porb =100 d) and increases steeply to
0.8 for close binaries. Given the uncertainty of about 20% in the
stellar radius (Fig. 15) and allowing for a small inclination differ-
ence, it is tempting to accept PA/B
rot ≈0.8 as consistent with unity.
This argues in favour of synchronised rotation in close SB2s.
The sudden rise for PA/B
rot at Porb .10 d also supports the choice
Porb =10 d to separate between close and wide binaries. We
note that the ratio is suited for a consistency check (i.e. whether
synchronisation could be present), but it requires that the
A192, page 14 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
1 10 100 1000
Orbital period [ d ]
10
1
P rot / sin (i) [ d ]
close wide
sync
i=30
24 SB1 vsini < 100 km/s
vsini > 100 km/s
I+II+III x
HD053975
HD101191
1 10 100 1000
Orbital period [ d ]
10
1
P rot / sin (i) [ d ]
close wide
70 SB2
sync
i=30
Primary
Secondary
HD048099
HD167771
1 10 100 1000
Orbital period [ d ]
0.0
0.2
0.4
0.6
0.8
1.0
Ratio of rotational period
70 SB2
per(P) > per(S)
per(S) > per(P)
53 SB2 with known orbit inclination
1 10 100 1000
Orbital period [ d ]
10
1
Corrected P rot [ d ]
close wide
known MT
sync +/- 30%
Primary
Secondary
HD093161A HD163892
HD165246
HD149404
HD152218
HD093343
HD054662
HD152246
HD047129
HD115455
HD093403
HD096264
1 10 100 1000
Orbital period [ d ]
0
100
200
300
400
500
600
vsini [ km/s ]
53 SB2
Primary
Secondary
I+II+III x
slow medium fast
1 10 100 1000
Orbital period [ d ]
0
100
200
300
400
500
600
v equatorial [ km/s ]
inclination
corrected
53 SB2
Primary
Secondary
I+II+III x
slow medium fast
Fig. 16. Dependence of rotational properties on the orbital period. Top: Rotational period vs. orbital period for SB1s (left) and SB2s (right). The
y-axis of Fig. 16 is inverted in order to preserve that faster (slower) rotating stars are plotted up (down). The vertical long-dashed line separates
close and wide binaries. The solid diagonal line labelled ‘sync’ marks equal periods, as required for synchonised rotation, whereby an inclination
i=30◦of the rotational axis shifts the data points from this line by a factor of 1/sin(30◦)=2 to the dotted inclination line labelled ‘i =30’. The two
SB2s below the dotted inclination line indeed have known iorb <20◦(HD048099 and HD167771). For SB2s, the components are connected with
a vertical dotted line; for most wide binaries at Porb >10 d, Prot differs strongly between the components but the difference reduces and disappears
for close binaries. Middle left: Ratio of Prot of the binary components vs. Porb ; the horizontal lines mark the median in each Porb range. Middle
right: Prot vs. Porb for those 53 SB2s with known inclination of the orbital axis iorb;Prot is corrected for inclination assuming parallel orbital and
rotational axes. The solid and dotted diagonal lines mark the synchronisation range with a width of 30% and a factor of 2, respectively. The green
triangle marks the range above the sync line discussed in the text. Bottom left: vsin ivs. Porb for the 53 SB2s with known iorb. Bottom right: veq vs.
Porb after correction for the inclination.
A192, page 15 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
stellar radius (used to convert vsin ito rotational period) is pre-
cise enough.
Therefore, we further examine the effect of inclination,
sin(i), visible in Fig. 16 (top right). Like for the SB1s, inclination
shifts the close SB2s below the “sync” line. Synchronised rota-
tional and orbital axes are assumed to be parallel (iorb ≈irot),
so that knowledge of iorb allows us to correct for irot. For 53
SB2s in our sample the inclination of the orbital axis, iorb, is
known (Table D.2). iorb has been inferred from Msin(i)3mea-
surements and the calibration mass (Martins et al. 2005b). For
these SB2s, we corrected Prot via multiplication with sin(i). We
note that wide binaries may not be synchronised, and the incli-
nation correction for them should be considered with caution.
The result is shown in Fig. 16, middle right. After correction,
about 40 of the 62 close binaries lie inside the sync range ±30%
around the “sync” line. Given the uncertainty of vsin i, stellar
radius and inclination, it appears reasonable to adopt an uncer-
tainty of 30% for Prot. Strikingly, there are no stars in the region
below the sync range. Such stars would rotate too slowly for
being synchronised. For the 53 SB2s with known iorb, Fig. 16
bottom right and left show vsin ias observed and the equato-
rial velocity veq after correcting for the inclination. Indeed, the
close SB2s lack slow rotators; almost all are increased rotators
with veq >100 km/s (see also the histograms in Fig. C.1). On the
other hand, there exist many slow rotators among wide SB2s. It
is plausible that the wide SB2s evolve to become close SB2s.
This strongly suggests that any originally slow rotating compo-
nents increased their rotation rate during the approach and syn-
chronisation.
About 20 of the 62 close SB2 components (∼30%) lie above
the sync range in a region marked by the green triangle (Fig. 16
middle right); 7 components lie even above the upper dotted sync
line. These stars appear to rotate too fast to match synchroni-
sation. This suggests that additional mechanisms play a crucial
role. To shift the stars in the green triangle down to the sync
range, we have to increase Prot, hence enlarge the used calibra-
tion radius to the synchronised radius Rsync. To obtain a consis-
tent picture, this implies:
(1) For fast rotators: the equatorial radius is increased, but this
is limited to a factor of ∼1.5 (Abdul-Masih 2023). However,
most stars in the green triangle are not fast rotators.
(2) For the primaries: the radius is increased by (partial) Roche-
lobe filling or Roche-lobe overflow (RLOF). This strongly
supports the presence of mass transfer in these SB2s.
(3) For the secondaries: spin-up by mass transfer likely acts
against spin-orbit synchronisation. Therefore, it is consistent
that secondaries are seen also above the sync range. There
is no need to shift the secondaries towards the sync line. In
addition, the duration of the spin-up phase is short (cf. Fig. 2
in de Mink et al. 2013). Therefore, not all secondaries must
be seen currently in a spin-up phase. As a consequence, sec-
ondaries may reside both close to and above the sync line.
We illustrate the increase of the radius with few examples, where
the primary lies above the sync line. They are labelled in Fig. 16,
middle right:
(1) HD 152218 at Porb =5.6 d , Pprim
rot =3.3 d, Psec
rot =2.3 d, with
i∼71◦. This is a binary with clear evidence for wind-wind
interaction, hence mass located between the components
supporting that MT is indeed present (Sana et al. 2008a). To
shift the primary to the sync line requires an increase of the
radius from R∗=10 Rby a factor of 5.6/3.3 =1.7, yielding
Rsync =17 R, lying well inside the Roche-Radius Rroche =
22.8 R. Both components are medium rotators with inclina-
tion corrected equatorial rotational velocities v∼150 km/s.
(2) HD 149404 consists of two super-giants with Roche lobe
overflow episodes during the past (O7.5 I, vsin i=88 km/s,
ON9.7 I, vsin i=71 km/s), Porb =9.8 d, Pprim
rot =4.8 d,
Psec
rot =6.3 d, with i=24◦(Rauw et al. 2001b;Raucq et al.
2016). Both components are medium-fast rotators with incli-
nation corrected equatorial rotational velocities v=216 km/s
and 175 km/s. To shift the primary to the sync line requires
an increase of the radius by a factor of 9.8/4.8 =2.04, yield-
ing Rsync =42 R, still consistent with Rroche =40 R.
(3) HD 165246 has a fast primary (O8 V, vsin i=221 km/s)
and a slow-medium secondary (B0: V, vsin i=100 km/s),
with i=83◦,Porb =4.6 d, Pprim
rot =1.9 d, Psec
rot =1.7 d
(Mahy et al. 2022). To shift the primary to the sync line
requires an increase of the radius by a factor of 4.6/1.9 =2.4,
yielding Rsync =20.3 R, slightly below the Rroche =22.7 R.
(4) HD 163892 has a medium-fast primary (O9.5 IV, vsin i=
183 km/s) and a very slow secondary (B0: V, vsin i=
39 km/s), with i=70◦,Porb =7.8 d, Pprim
rot =2.5 d, Psec
rot =
5.0 d (Mahy et al. 2022). To shift the primary to the sync line
requires an increase of the radius by a factor of 7.8/2.5 =3.1,
yielding Rsync =30 R, similar to Rroche =29.5 R.
In the two last examples, the secondary has a moderate incli-
nation corrected equatorial rotational velocity v(HD 165246:
v∼102 km/s, HD 163892: v∼41 km/s). In terms of angular
velocity, however, the spin of the secondary is a factor (2.7, 1.5)
faster than that of the orbit. If the radius of the secondary is
indeed correct (and thus Psec
rot ), this implies a spin-up by mass
transfer even in slow rotating secondaries.
For the stars in the green triangle we have seen that the actual
radius (in particular of the primary) may be larger than the cali-
bration radius used. (If a star in the green triangle were correctly
positioned, then it had already spun-up by any mechanism and
we discuss this possibility in Sect. 5.3.) In principle, also the
remaining stars close to the sync range may have an increased
actual radius. If true, this may shift them below the sync range.
Then they rotate too slow for being synchronised. The timescale
to reach synchronisation is between 0.5 and 5 Myr depending
(besides other parameters) on the separation a/R∗(Fig. 3 of Zahn
1977). Thus the stars could still be on the way to become syn-
chronised.
The distinction between sync range and green triangle arises
during our chain of reasoning. Therefore, we consider it unlikely
that the close SB2s split into two physically distinct populations,
one with calibration radius perfectly matching the sync line and
one needing an expanded radius. This leads to the conclusion,
that the spin is not only controlled by synchronisation. Rather,
binary interaction involving Roche-lobe filling and mass transfer
(MT) plays a strong and presumably dominant role for spinning-
up.
For HD 093161 A at Porb =8.6d the O7.5 V primary appears
on the sync line, but the O9 V secondary lies below the sync
range suggesting that it is not yet synchronised. A plausible
explanation is that the rotational axis of the secondary is not
aligned with the orbit axis. The orbit inclination is 84◦. The sys-
tem is a well isolated SB2 (DT 1). Both components are slow
rotators with vsin i=48 km/s and 31 km/s, robustly measured in
O III 5592.
Finally, we pay attention to HD047129 (Plaskett’s star), a
well studied binary considered to be a prototype of spin-up by
mass transfer. With Porb =14.4 d (a=96 R), the system
lies near the border between close and wide binaries. It con-
sists of a slow rotating primary (O8 I, vsin i=80 km/s) and
a fast secondary (O8.5 fp, vsin i=236 km/s), with i=71◦,
A192, page 16 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
Pprim
rot =12.5 d, Psec
rot =2.7 d (Bagnuolo et al. 1992;Linder et al.
2008). The primary is a super-giant with a calibration radius of
about 40–50% of the Roche radius. The primary lies perfectly in
the sync range, while the secondary has spun-up by a factor of
∼5 compared to Porb.
For slow rotating stars synchronisation is possible at 10 <
Porb/d<30, but the range Porb <10 d is relevant for a
synchronisation-induced spin-up (vsin i&100 km/s). Therefore
we have chosen Porb =10 d as threshold between close and wide
binaries Furthermore, if a star lies above the sync range, then in
principle, synchronisation may brake its rotation.
To conclude, close SB2s contain slow, medium and a very
few fast rotators. Conversion of vsin ito Prot and correcting for
inclination, yields the formal result that 70% of the stars lie in
an ideal range for spin-orbit synchronisation. About 30% of the
stars lies above that sync range, implying that the primaries have
an expanded radius (e.g. due to RLOF), consistent with the pres-
ence of mass transfer which in turn spins-up the secondaries.
The separation of the stars in two distinct groups is likely of for-
mal nature. From our statistical point of view, the spin-up of the
close SB2s can fully be explained by a combination of ongoing
synchronisation and mass transfer. The scarcity of fast rotators
remains puzzling.
5.3. Increased rotators in wide binaries
The sample contains 15/24 (62%) wide SB1s and 39/70 (56%)
wide SB2s. Compared to the SB1s, the larger number of SB2s
allows for better statistics and the comparison of primary and
secondary provides further insight. Therefore, we here restrict
the discussion on the SB2s and, if suited, mention SB1s in com-
parison with SB2 primaries.
The rotation rate of wide SB2s (primaries and secondaries
together) shows a pronounced peak at vsin i∼50−80 km/s
(Fig. 14, bottom right). At first glance, it is reminiscent to that
of Cs (Fig. 9, top). About 35% (27/78) of the wide SB2 compo-
nents are increased rotators (Fig. 14). What increased the rota-
tion in the medium and fast wide binaries? Or were they born
with increased rotation? The period difference between primary
and secondary rejects an increase by spin-orbit synchronisation
(Fig. 13). We first discuss how far the wide binaries could be
spun-up by mass transfer (Sect. 5.3.1). This turns out to be
not satisfying. Therefore, we discuss alternative spin-up mecha-
nisms in Sect. 5.3.3.
5.3.1. Spin-up by mass transfer
A testable prediction of MT-based spin-up in wide binaries is
that the gainer rotates faster than the donor. We first assume
that the more massive component (the primary) is the donor.
We count the number of increased rotators among the secon-
daries: 17/39 (∼44%) from Fig. 13 and Fig. 14 (bottom mid-
dle panel). All of them rotate faster than the primary; we call
these binaries “seco-fast” (they are marked by large red sym-
bols in Fig. 17). The mass ratios M2/M1 of the seco-fast bina-
ries distribute between 0.27 and 1. At first glance, this altogether
appears consistent with expectations for MT-based spin-up.
On the other hand, 10/39 (∼26%) of the wide primaries are
increased rotators (Fig. 14). While 4 of these 10 have a secondary
which rotates faster (and are plotted with a large red symbol in
Fig. 17), 6 of these 10 wide primaries rotate even faster than the
secondary (large blue symbols in Fig. 17); we call them “prim-
fast”. If they are donors in the MT-scenario, one may wonder
how they increased vsin ito >100 km/s. At first glance, they
1 10 100 1000
Orbital period [ d ]
0.0
0.2
0.4
0.6
0.8
1.0
Mass ratio M2 / M1
70 SB2
31 close 39 wide vsini(S) > vsini(P)
vsini(P) > vsini(S)
Fig. 17. Mass ratio vs. orbital period. Blue and red indicate whether
the primary or secondary is the faster component (measured by vsin i).
Among wide binaries, the large symbols mark those pairs where at least
one component is a medium or fast rotator (6 blue and 17 red pairs; see
Sect. 5.3.1).
appear to contradict the expectation of MT-based spin-up, where
the gainer should rotate faster than the donor. Seeking to solve
this puzzle, we find: The mass ratios M2/M1 of these 6 prim-fast
binaries are closer to 1 compared to that of the seco-fast above.
The similar mass ratio makes the distinction between donor and
gainer uncertain. In the remaining 16 (=39–17–6) wide bina-
ries both primary and secondary are slow rotators. So far, the
data statistics appears consistent with expectations for MT-based
spin-up in wide binaries.
However, MT becomes inefficient for wide binaries with sep-
aration a&100 R, corresponding to Porb ∼20 d and RRoche ∼
50 R. In addition, a significant spin-up of the secondary by cap-
ture of wind mass lossed by the primary can be excluded:
(1) MT in binary stars occurs when (a) the star expands to fill
its Roche-lobe or (b) the orbit, and thus Roche-lobe, shrinks
until R∗<RRL. RLOF is grouped into three cases: (A)
MT while donor is on the MS, (B) donor is in the Red
Giant phase, (C) Super-Giant phase. (For details we refer
to de Mink et al. 2013.) The calibration radius R∗of O- and
early B-stars lies between 7 and 15 R(II-V), and reaches
25 Rfor luminosity class I (Martins et al. 2005b). While
RLOF provides an efficient MT-based spin-up for close bina-
ries, MT by RLOF becomes inefficient beyond a∼100 R.
(2) Next we consider a possible mass and momentum transferred
by a stellar wind. To increase the stellar spin by a factor of
2–5 (e.g. from vsin i=60 km/s to 120 or 300 km/s, respec-
tively), in an ideal acretion disk scenario a total accretion
Macc of a few percent of the star’s mass Mis needed (Packet
1981). We adopt the optimistic case Macc /M=1%, hence for
a 20 Mstar a required Macc ∼0.2 M. The typical mass loss
carried by a wind lies at 10−6–10−8M/yr (Howarth & Prinja
1989;Martins et al. 2005a;Vink 2022). For an isotropic
wind, only a fraction of the wind mass ejected by the pri-
mary is captured by the secondary. For simplicity, this frac-
tion is estimated by the solid angle Ω = (2πR2
cap)/(4πa2).
For a=100 R, adopting Rcap ≈2R∗and R∗=10 R(an
O5 V or O8 IV secondary), we obtain Ω≈0.02. Adopting
the optimistic case of a strong primary wind of 10−6M/yr,
the required Macc is reached after 107yr, exceeding the star’s
life time.
A192, page 17 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
Roche-lobe filling can be measured, for example by modelling
eclipsing light curves for close binaries but hardly for wide bina-
ries. To observationally establish the presence of a mass flow
located between the binary components, one possibility is via the
Struve-Sahade (S-S) effect: the apparent strengthening of a star’s
spectrum as it approaches and weakening as it recedes (Struve
1937;Sahade 1959). Such investigations are challenging (e.g.
Linder et al. 2007) and beyond the scope of this work.
Depending on the stellar mass and radius (luminosity class)
MT-based spin-up is possible up to P.30 d, and some examples
are:
–HD 047129, an O8 I primary and an O8.5 fp secondary,
Porb =14.4 d, with a pronounced S-S effect (Linder et al.
2008).
–HD 093403, O5.5 I +O7 V, Porb =15.1 d (Rauw et al. 2002)
We note that both examples are formally wide binaries but near
to the border to close binaries (Fig. 16, middle right). On the
other hand, there exist examples against MT-based spin-up for
wide binaries:
–HD 093343 is comprised of two O7.5 Vz stars in a wide and
eccentric orbit (Porb =50 d, e =0.4, Putkuri et al. 2018).
–HD 096264, an O9.2 IV primary and a B0 V(n) secondary
(Porb =124 d, e =0.2 Putkuri et al. 2021).
–HD 054662, an O6.5 V(n)z and O7.5 Vz, notably the primary
is a fast rotator while the secondary has slow vsin i∼40 km/s
(Porb =2113 d, Barbá et al. 2020).
To explain the increased rotation in the three counter-examples,
the authors argued:
(1) The wide separation in combination with the young age
(∼2.5 to 4.5 Myr, implying small radii) excludes spin-up by
RLOF. This conclusion is supported by our estimates of the
low MT efficiency above. Spin-up by tidal interaction (spin-orbit
synchronisation mechanisms) can also be excluded.
(2) The non-synchronous rotation of the components and
the increased rotation of one component is likely a – not fur-
ther specified – consequence of the stellar formation process.
We note that this may lead to a puzzling picture as it implies
that the fast spinning component has not suffered from rota-
tional braking since birth, while the slow rotation of the coeval
component requires a braking (or an unfavourable inclination of
the rotational axis). Therefore, we consider possible scenarios in
Sects. 5.3.2 and 5.3.3.
5.3.2. Initial velocity and rotational braking
After the birth of a single star, its initial rotational velocity vini
decreases, and the braking rate depends on the combination of
angular momentum losses due to stellar winds and the internal
angular momentum transport from the core to the envelope. One
may expect that this common view for single stars applies also
to wide binaries, at least to those which are not affected by tides
and mass transfer events.
Braking depends not only on the wind but also on inter-
nal momentum redistribution. The complex details require
sophisticated modelling (Maeder & Meynet 2000;Langer 2012;
Ekström et al. 2020). The rotational behaviour during stellar
evolution is studied by theoretical models which are calibrated
against observations.
Holgado et al. (2022) performed a thorough investigation
using 255 single O stars with well-measured rotational velocity
and well-determined position in the spectroscopic Hertzsprung–
Rusell diagram (sHRD). They compared the empirical results to
the predictions – regarding current and initial rotational veloci-
ties – of two sets of stellar evolutionary models (Ekström et al.
-1.0 -0.5 0.0 0.5 1.0
log ( rotational period P / S )
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Mass ratio M2 / M1
Primary faster Secondary faster
39 wide SB2
10<orb<100
orb>100
-1.0 -0.5 0.0 0.5 1.0
log ( rotational period P / S )
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Mass ratio M2 / M1
Primary faster Secondary faster
slow+slow excl.
23 wide SB2
10<orb<100
orb>100
-1.0 -0.5 0.0 0.5 1.0
log ( rotational period P / S )
0
2
4
6
8
10
12
14
N
Primary faster Secondary faster
10
5
5
29
19
10
39 wide
10<orb<100
orb>100
-1.0 -0.5 0.0 0.5 1.0
log ( rotational period P / S )
0
2
4
6
8
10
12
14
N
Primary faster Secondary faster
slow+slow excl.
5
2
3
18
9
9
23 wide
10<orb<100
orb>100
Fig. 18. Rotational asymmetry between primaries and secondaries, for
all wide binaries (left column), and for a subset where at least one com-
ponent is an increased rotator (right column). Medium wide and very
wide binaries are coloured in blue and red, respectively. Top: Ratio of
the rotation periods P/S vs. mass ratio; the size of the circles indicates
the orbital period, i.e. component separation, on a logarithmic scale.
Bottom: Histograms.
2012;Brott et al. 2011), that have been commonly used by the
massive star community in the last decade. To summarise their
results, both model sets do not provide a satisfying match to
the data. In particular for numerous young slowly rotating stars
(veq <75 km/s) the models overpredict veq.Holgado et al. (2022)
concluded that the models need a slower vini <0.2vcrit, compared
to 0.4vcrit typically adopted, or a stronger braking in the first 2–
4 Myr after birth7. Therefore, the current models do not yet allow
us to exclude (or imply) a substantial braking in the first 2–4 Myr
after birth.
What can we learn from our wide SB2s about vini and rota-
tional braking? A striking feature of our wide SB2s is the high
fraction (∼70%) of non-synchronous rotation and, in addition, a
strong “rotational asymmetry”: For most wide binaries the sec-
ondary rotates faster than the primary (Fig. 18). This feature
holds for the set of 39 wide binaries and for the subset of 23
binaries where at least one of the components is an increased
rotator (i.e. slow+slow pairs are excluded). The exclusion of
the slow+slow pairs reduces potential noise in the distribution
of the period ratios. Likewise, the trend holds for splitting the
sample into 24 medium wide (10 d <Porb <100 d, blue) and 15
very wide binaries (Porb >100 d, red). Despite the small num-
ber statistics, incompleteness and potential biases of the sample,
there is little doubt on the rotational asymmetry.
A statistical analysis of non-synchronous wide SB2s may
distinguish between two potential scenarios:
(1) At birth or shortly after birth (∼2 Myr), both components are
fast rotators and the currently observed slow component is
braked or misaligned.
7The models predict a surprisingly smooth braking curve over time.
However, a massive star exhibits line profile variations indicating irreg-
ular episodes of strong winds and/or re-arrangement of internal angular
momentum. As a promising future step for the models, we suggest to
include stochastic mass loss episodes.
A192, page 18 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
Table 1. Joint probability, Pjoint, of finding N or more misaligned sys-
tems among wide SB2s in a sample size Nsample (see Sect. 5.3.2 for
details).
Nsample NPjoint NPjoint NPjoint
39 5 0.61 10 0.029 15 0.00008
24 5 0.21 10 0.0006 15 <0.000001
15 5 0.04 10 0.0000026 15 <0.000001
(2) At birth or shortly after birth, both components are already
slow rotators and the currently observed fast component had
received a spin-up by any other mechanism.
We outline a tentative test to distinguish between the scenarios.
The strategy is to consider scenario 1 for a subsample of non-
synchronous SB2s with suited plausible assumptions so that only
orientation differences between slow and fast component are left
over to explain any difference of vsin i. Then we estimate the
joint probability, Pjoint, of finding that many non-synchronous
SB2s with a misaligned component as observed. We call these
systems “misaligned”. At the end, a low Pjoint may argue against
or in favour of scenario1.
In detail, we assume (1) for each SB2 that the components A
and B are coeval without tidal and mass transfer events, and (2)
that A and B have not yet suffered from significant rotational
braking, so that any increased rotation must be from birth. (A
can be either the primary or the secondary, respectively.) Fur-
thermore, if A and B are coeval and of sufficiently similar mass,
any initial braking in the first ∼2 Myr should affect both com-
ponents equally strong, thus we assume that both components
have the same veq. (Indeed, the last assumption is supportd by
the fact that the samples contain also pairs where both compo-
nents have similar vsin i; mostly these are slow rotators.) With
this set of plausible assumptions, we will consider the spin dif-
ference between A and B. Since other mechanism are excluded
by construction of the test, the spin difference can only be due to
the inclination difference of the rotational axes iA
rot and iB
rot.
In ∼70% of the wide SB2s, B spins a factor of >2 slower
than A. The factor 2 implies, for instance, if iA
rot ∼90◦, then
iB
rot <30◦, or if iA
rot ∼60◦, then iB
rot <25◦. To keep the experiment
manageable, we define a common threshold for the inclination
difference and iB
rot. Conservatively adopting iA
rot ∼90◦, yields
iB
rot <30◦. Then the probability of finding a single misaligned
axis is i=30◦and P<0.134 (adopting randomly oriented rota-
tion axes; see Appendix C).
We determined the joint probability Pjoint of finding N or
more misaligned systems among Nsample systems by calculat-
ing ten million experiments using a random number generator.
Table 1lists the Pjoint results for three samples (of Nsample =39,
24, 15 SB2s) and N =5, 10, 15. Two direct conclusions can be
drawn:
(1) To find just few (e.g. N =5) misaligned systems, is rel-
atively easy with Pjoint up to 0.61. However, the whole
population of misaligned systems contains up to ∼70% of
the samples. Conservatively adopting 50%, we have to use
N∼0.5 Nsample. Then Pjoint is very small (below 0.0001)
rejecting scenario 1.
(2) This leads us to favour scenario 2. It suggests a further gen-
eralisation to single O stars: without other interaction events,
already shortly after birth (∼2 Myr) any O stars have become
slow rotators, quasi as a “ground state of stellar rotation”.
Then any increased rotation is likely not due to birth but
requires other spin-up mechanisms.
The outcome of this test appears to consistently expand the con-
clusions of Holgado et al. (2022) above. A thorough implemen-
tation of this test needs a completeness and bias analysis of the
wide SB2 sample, including a careful selection of those non-
synchronous SB2 which certainly fulfil the assumptions, and a
refinement of the assumptions. This is beyond the scope of our
paper.
5.3.3. Spin-up mechanisms for wide binaries
Here, we consider possible spin-up mechanisms for wide SB2s.
Each scenario should fulfil the following criteria:
(1) It should be able to reach spin-up by factors of 2–5 (e.g.
increasing vsin ifrom 60 to 120 or 300 km/s).
(2) It should allow for “no spin-up”; in 16 (∼40%) of the 39 wide
SB2 systems both components are slow rotators, whereby
inclination is unlikely the main cause.
(3) It should be consistent with the rotational asymmetry
between primaries and secondaries (Fig. 18).
(4) The timescale should be short (.5 Myr) because of the short
life time of massive stars.
Here, we discuss four possible scenarios to explain the increased
rotators in wide SB2 systems:
(1) We begin with revisiting the case that the binary stars are
born with different spins (Fig. 19). Cause for the unequal
rotation could be different cloud (or disk) fragmentation sit-
uations. Thus, two massive stars are possible that each bring
their spin from birth.
For simulations of the formation of binaries from accretion
disks, we refer to Krumholz et al. (2009) and Oliva & Kuiper
(2020), and for observations of large accretion disks and
their fragmentation to Chini et al. (2004) and Ahmadi et al.
(2023).
Figure 19 intentionally shows the increased rotator being
formed not from one single cloud fragment but from the
merging of several (three) cloud fragments. This would eas-
ily explain the rotational asymmetry. However, it appears
more likely that each cloud fragment forms its own star, and
we come back to this further below.
The O stars in our sample have already reached or evolved
beyond the zero age main sequence (ZAMS). Then rotational
braking could have taken effect, as discussed in Sect. 5.3.2. If
the enlarged spin is from birth, the two questions that remain
are identifying the mechanism that maintained the spin dif-
ference of the binary components so large and the reason
why mostly the secondary spins faster than the primary.
Next we assume that the stars have evolved beyond the ZAMS,
have become slow rotators, which then spin-up by one of the
following mechanisms:
(2) Hidden SB1 mechanism: The wide SB2 system is actu-
ally a hierarchical (i.e. stable) triple system (Fig. 20). The
increased rotator (primary or secondary) of the SB2 system
is itself a close SB1 with a hidden (lower mass) component
(tertiary C). Our sample contains two potential examples: (a)
HD 152246 is a hierarchical triple system, consisting of a
wide (PA+B
orb =470 d) SB2 with a hidden tertiary which is
bound to the secondary B in a close circular orbit with a
short period of PB+C
orb =6 d (Nasseri et al. 2014). This makes
the secondary actually an SB1 with 18.4 and 4.4 M(the
components are named Ba and Bb in Nasseri et al.). The
short period suggests that the SB1 subsystem is synchro-
nised, like the O-dwarfs plotted in red in the top left panel
of Fig. 13.PB+C
orb and radius of component B (spectral type
O9 V) yield vB
sync =63.5 km/s, consistent with the observed
vsin i(B) =52.7 km/s. This argues for synchronisation of this
SB1 subsystem, if iB+C
orb &70◦(we recall that iA+B
orb =68◦).
A192, page 19 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
Fig. 19. Stars are born with different spins. Before the main sequence
phase, fragmented cloud cores merged and the resulting stars feature
slow (red) and increased (blue) spin.
Fig. 20. Hierarchical triple system. The two massive stars form a wide
SB2, consisting of one slow and one increased rotator (red and blue,
respectively). Left: Increased rotator itself is a close SB1 with a hidden
lower mass companion, the tertiary star (green). The tertiary may pro-
duce strong radial velocity variations. Right: Increased rotator has two
hidden lower mass components (green) orbiting such that they balance
out strong radial velocity variations of the high-mass star.
Slow
Rotator
Increased
Rotator
Ejection and
tidal interaction
Fig. 21. Fly-by in an unstable triple system. The binary system of two
massive slow rotators (red) captured a lower mass star (green) that inter-
acted in a close fly-by with one of the massive stars before being ejected.
The spin of the closely flown-by star may be increased a bit (blue).
Fig. 22. Swallowing of a low-mass star in a triple system: A low-mass
star (green) initially moves in the gravitational field of a wide high-mass
binary with two slow rotators (red). When the orbit of the low-mass
star is disturbed, it spirals on to one of the high-mass components. The
resulting post-merger star has an increased mass and spin (blue).
Then B is still a slow rotator without need for spin-up. If
B and C approach further, then B will become an increased
rotator. This example demonstrates that a stable synchro-
nised SB1 subsystem with increased rotator may well exist
in a triple system.
The detection of an SB1 subsystem of a wide SB2 requires
precise RV curves which are well sampled in time, in order to
discern the contribution of the long and short orbital periods.
The RV amplitude of HD 152246 C is about 75 km/s, hence
well detected. The RV curves of HD 152246 (analysed by
Nasseri et al. 2014) contain 49 data points. The RV curves of
most SBs in our sample have typically about 20 data points
or less (cf. Mayer et al. 2014a,2017;Mahy et al. 2022).
Often the RV curves show a large scatter (up to 20km/s) and
the periodograms display several peaks. Therefore, it is well
possible that these are SB1 subsystems which escaped detec-
tion.
In addition, one could hypothetically think of two low-mass
companions, orbiting such that they balance out strong radial
velocity variations of the high-mass star. This special – albeit
somewhat artificial – configuration is scetched in Fig. 20,
left.
Regarding spin-up mechanisms, the remaining puzzle for
HD 152246 is that the primary A is the increased medium-
fast rotator; its spin-up needs another explanation.
(b) The second example, HD 123056, is a wide binary
(PA+B
orb =1314 d) with moderate (vsin i∼100 km/s)
rotators A and B, hence litte spin-up. It is possibly a hierar-
chical SB3, and the tertiary could be a fast rotator (225 km/s,
Mayer et al. 2017). The short period (PB+C
orb .2 d) sug-
gests that the B+C system is in a phase of spin-orbit-
synchonisation or MT. Further observations are needed.
Alltogether, it appears that the hidden SB1 mechanism is rare
and plays a minor role for the spin-up of wide binaries. A char-
acteristic of the hidden SB1 mechanism is that the observations
catch the binary in a phase where the spin-up is in the making.
In contrast, the following spin-up mechanisms must have taken
place somewhere in the past before we observed the stars:
(3) Fly-by event in an unstable triple system (Fig. 21): The
SB2 system of two massive stars had captured a third
(lower mass) star. If the system is unstable, the tertiary may
approach one of the massive stars before being ejected in a
close fly-by. An open issue is whether the fly-by event trans-
fers enough angular momentum to spin-up the massive star
(or whether it may lead to a spin-down). We suggest that this
mechanism plays a minor role for spinning-up wide binary
components.
(4) Post-merger scenario: In this scenario, the increased rotator
had undergone a merging event in the past. Hence the sys-
tem can be considered as a former triple, where two of the
stars have merged, resulting in a post-merger product with
increased rotation.
Merging of two stars is expected to lead to an excessive spin-
up of the post-merger product; accretion of only a few pecent
of the primary’s mass is needed (see Packet 1981). How-
ever, a scenario in which two massive stars merge and after-
wards finds a new companion to form a new wide SB2 prob-
ably requires too large a time frame (>5 Myr; cf. Fig. 2 in
de Mink et al. 2013), exceeding the young age of our O stars.
The timescale problem challenges the post-merger scenario
to spin-up the increased rotator of a wide binary by the merg-
ing of two massive stars.
To alleviate the timescale problem, an alternative could be
the merging of a massive star with a low-mass star (.1 M)
which we call “swallowing” (Fig. 22). We suggest that a low-
mass star could be swallowed in much shorter time (.1 Myr).
For instance, consider an initially distant (>300 AU) mas-
sive binary with components A and B where a low-mass
star C orbits around component B (the host star). Assume
that either A or B is the primary8. When A and B approach
each other, the gravitational field of A disturbs the orbit of C,
possibly leading to an infall onto the host star which in turn
gains angular momentum. The swallowing of a low-mass star
might be very efficient. In addition, the high-mass compo-
nents of a wide SB2 could have several low-mass compan-
8The statistical balance between capture probability Pcap and spin gain
gspin is controlled by (1) the spatial distribution of tertiaries Cs relative
to A and B, (2) a higher likelyhood that the more massive primary wins
the capture competition (Pcap ∝M), and (3) a higher spin gain for the
less massive secondary (gspin ∝1/(MR2), with mass Mand radius Rof
A and B). Details are beyond the scope of this paper.
A192, page 20 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
Fig. 23. Fragmentation of a cloud or disk into three cores of different
size and mass. They form three stars (or proto-stars) of high, medium,
and low mass, respectively. During subsequent evolution, the stars inter-
act and merge.
ions. This allows for multiple swallowing. Then, both com-
ponents A and B may become increased rotators.
Each of the four scenarios has its advantages and drawbacks.
We suggest that there is no simple best scenario applying for all
wide SB2s, rather each SB2 may be best explained by one of
the scenarios, and all four scenarios are needed to explain the
spin-ups in the entire sample.
We focus the further discussion on the swallowing scenario
(item 4 above). It appears plausible, but it depends on the exis-
tence of low-mass companions that are at least 8 mag fainter
than O stars and can hardly be detected by observations. The
short main sequence (MS) time of O stars is similar to the time
which low-mass (solar mass) stars need to reach the zero-age
main sequence. Thus, the question follows of whether O stars
can be surrounded by solar mass stars. We find three arguments
supporting that this is indeed probable:
(1) The fragmentation of a large disk (or cloud) around an O-
star leads to substructures (Fig. 23). These start to form lower
mass stars, say an A0-star with final MS mass of 2 M. When
the O star finished accretion and its radiation/wind blows off
the parental cloud and disk, then the mass accretion of the A-
star has not yet completed. However, the parental cloud and
disk is blown away too, leading to cease further mass accre-
tion. This results in a “truncated” A-star with a mass which is
presumably a factor of 2–3 lower than the final MS mass. So
far, simulations (cf. scenario 1 above) and observations reach
bright companions, but with better sensitivity and resolution
faint companions should become detectable.
(2) About 40% of B-type stars are SB1 systems (e.g. Chini et al.
2012; a re-investigation of the same sample using additional
spectra yields 52% SB1s, Symietz 2024). They have a mass
between 3 and 18 M. For SB1 companions, one expects
M2≈0.1·M1 of the host star, so that M2 lies in the range of
the sought-for low-mass stars. This shows that solar mass
stars likely exist around B stars. Since O and B stars are
seen mostly in common clusters or star forming regions, it
is likely that also O stars are surrounded by solar mass stars.
(3) If indeed several merging events with low-mass stars are at
work for O-type stars, then it is possible that the MS time of
an O star is also prolonged as it rejuvenates with every merg-
ing event (e.g. de Mink et al. 2013). In this case, “coeval”
O stars and solar mass stars can exist.
To conclude, the likelihood for (many) low-mass stars around an
O star is high, also indicated by Initial Mass Functions.
Bally & Zinnecker (2005) suggested that two effects can
increase the probability of a merging event. Gravitational focus-
ing describes the effect that the true cross section for interac-
tions is much larger than only the projected area of the star.
Disk-assisted protostellar capture can further increase the cross
section of interaction. The merging event itself is expected to be
a very fast process.
The timescale problem for merging/swallowing may be fur-
ther alleviated, if the configuration of the triple system is already
founded before the stars reached the main sequence (Fig. 23).
If the triple is already present in an earlier cloud/disk era of the
stars, this shortens the overall time needed to achieve the merg-
ing.Bonnell et al. (2001) proposed the “competitive accretion”
model. It explains the stellar mass spectrum of a cluster, as well
as the fact that high-mass stars gain mass as they gain com-
panions, implying a direct causal relationship between the clus-
ter formation process and the formation of higher-mass stars
therein. We note that the accretion processes take place before
the stars reach the main sequence. The low-mass merging sce-
nario does not question the competitive accretion model. Rather
it provides a valuable additive mechanism during a later evolu-
tionary phase.
To summarise, the post-merger scenario involving the merg-
ing of low-mass stars (“swallowing”) has remarkable advan-
tages. In principle, the spin-up is not limited. Swallowing can
explain the wide range from slow to fast rotators in wide SB2
systems. For a given input angular momentum by the captured
low-mass star the spin-up of the host star increases with declin-
ing mass of the host star. Thus, swallowing can naturally explain
why mostly the less massive secondary rotates faster then the
primary. Swallowing likely reduces the timescale problem of
merging of high-mass stars. The interplay/combination with
cloud/disk fragmentation is likely and efficient. The existence
of low-mass stars “co-eval” with the O star is probable.
As an outlook we note: If the swallowing scenario substan-
tiates, it can indeed help to answer a fundamental question of
massive star formation (Zinnecker & Yorke 2007), as it supports
the hypothesis that massive stars are grown by the merger of low-
mass with medium-mass stars (Bonnell et al. 1998). The progen-
itor medium-mass stars may be formed by accretion or by merg-
ing of several low-mass components.
5.4. Comparison of wide binaries with single stars
The fraction of giants is larger in Cs (57%) than in wide SB2s
(32%). Part of the difference could be an observational bias.
Giants are more luminous than dwarfs impeding the detection
of a companion. We do not follow up on at that here.
Wide binaries (P+S combined9) and C-stars show a similar
fraction of slow, medium and fast rotators. This suggests that
most C-stars are intrinsically similar to components of (very)
wide binaries or of binaries with a (very) small mass ratio
M2/M1, both beyond detectability.
In detail, there are some differences, for instance: the frac-
tion of fast rotators among wide binaries (11%) is lower than
that of Cs (15%). However, about 50% of the single fast rota-
tors are runaways assumed to be post-interaction binary products
(Britavskiy et al. 2023, their Fig. 9). These are rather evolved
“old” stars (>5 Myr) but the wide binaries are almost all “young”
(<5 Myr). We want to compare only matched samples (i.e. young
stars) here. By removing the “old” runaway stars (i.e. the 50%
above), the fraction of young single fast rotators reduces from
15% to about 7.5%, slightly below that of the wide SB2s.
Does a comparison with the Cs favour or discard one of the
4 spin-up mechanisms proposed for wide binaries? We consider
how far these (and other small) statistical differences are consis-
tent with the spin-up scenarios:
9If P and S were not gravitationally bound, then each would be a single
star, and therefore we use the statistics for P+S combined.
A192, page 21 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
(1) The binary stars are born with different spins: If this scenario
is true, then we assume that also singles stars are born with a
broad range of spins. As long as the stars are young and did
not suffer from magnetic braking or momentum loss through
winds during the star’s ageing, the vsin idistribution and the
fraction of fast rotators should be equal (similar) for wide
binaries and single stars, consistent with the data.
(2) Hidden SB1 in a hierarchical triple system: for a single star,
this spin-up mechanism reduces to spin-orbit synchronisa-
tion in a close SB1 which is undetected (likely due to a very
low mass ratio) and appears therefore as C. At the moment,
the comparison of wide SB2s with Cs does not reject this
mechanism.
(3) Fly-by event: for a single star this spin-up mechanism
reduced to a very unlikely rare fly-by of another star. As for
wide binaries, this mechanism might play a minor role.
(4) The low-mass merger scenario requires that the orbit of the
low-mass star is strongly disturbed. In a binary the massive
companion is an ideal disturber. However, a single star lacks
that massive disturber, suggesting that Cs have fewer spun-
up rotators than wide SB2s.
In principle, the presence of another low-mass star may be
sufficient to disturb the orbit, but the effect is weaker. Then,
part of the Cs can be spun-up in a similar manner as the wide
SB2s.
Some of the fast rotators Cs are indeed the merge of two
massive stars, and they are older but appear rejuvenated after
the merger (Schneider et al. 2014;Wang et al. 2020).
To conclude, the vsin idistributions of wide binaries and Cs
appear consistent with all 4 spin-up scenarios, not allowing us
to reject a scenario.
6. Summary and conclusions
We determined the projected rotational velocity (vsin i) of 238
southern O stars selected from the Galactic O-star Survey. The
sample contains 130 spectroscopic single stars (C), 36 single-
lined binaries (SB1s), 64 SB2s, and 8 SB3 systems. We care-
fully applied the Fourier method to high-resolution spectra taken
with BESO at Cerro Murphy and supplemented by archival spec-
tra. New results were obtained, in particular for the double-lined
binaries (SB2s, including SB3s):
(1) The overall vsin istatistics peaks at slow rotators (40–
100 km/s) with a tail towards medium (100–200km/s) and
fast (200–400 km/s) rotators. The medium rotators are four
times more frequent in binaries than in single stars. Fast rota-
tors are more frequent in single stars. These results are likely
intrinsic and not mimicked by inclination effects. We take
slow rotation as a standard against which we compare any
cases of increased rotation (>100 km/s).
(2) For 70 SB2s the orbital periods Porb are known, allowing us
to explore the relation between rotation and binary separa-
tion. The vsin idistributions differ for close (Porb <10 d)
and wide SB2s (10 d <Porb <3700d) and for primaries and
secondaries:
(a) For the 29 close SB2s, both primaries and secondaries
show a spin-up with respect to slow rotators, and for
most binaries the secondary spins faster than the pri-
mary (in terms of angular velocity). The spin-up is well
explained by a combination of both spin-orbit synchroni-
sation and mass transfer (Roche-lobe overflow, RLOF).
This is particularly demonstrated for a subsample of 53
SB2s with known orbit inclination where almost all com-
ponents rotate with equatorial velocity veq >100 km/s.
(b) For the 39 wide SB2s, the overall vsin idistribution
appears similar to that of single stars (Cs), but primaries
differ from secondaries. Mostly, the primary is a slow
rotator (75%), but 25% are medium and fast rotators.
Mostly, the secondary (∼75%) rotates faster than the
primary. A high fraction (43%) of the secondaries are
medium and fast rotators. To explain the increased rota-
tion in both primaries and secondaries, mechanisms like
spin-orbit synchronisation, RLOF mass transfer, and cap-
tured wind material cannot be applied and do not work
(at least for Porb &30 d).
(3) We discussed possible spin-up scenarios for the wide SB2s:
(A) The stars are simply born with different spins. Fragmen-
tation of the birth cloud or the disk may have formed
(proto)-stars with different spin. However, it is puzzling
why mostly the secondary rotates faster than the primary,
and why single stars lack the increased rotation seen in
the secondaries.
(B) One (or both) of the massive binary components is a hid-
den close SB1 in a hierarchical triple (or multiple) sys-
tem. The short period suggests ongoing spin-orbit syn-
chronisation and mass transfer. Our sample contains two
examples, yet they do not exhibit a remarkable spin-up.
This brings into question whether this scenario plays a
major role.
(C) One of the two massive components was spun-up by a
fly-by of an ejected third star in an unstable triple sys-
tem. It is unclear whether a fly-by transfers enough angu-
lar momentum to the massive star. Many such events
might be required. Therefore, statistically this mecha-
nism might play a minor role.
(D) Post-merger: In this picture, the increased rotator was
spun-up by a merging event in the past. However, a sce-
nario in which two high-mass stars merge and afterwards
find a new massive companion to form a new wide SB2
probably requires time frame that is too large and that
exceeds the young age of our O stars. Two ways (or a
combination of them) alleviate the timescale problem:
(a) The capture and merging (swallowing) of low-mass
stars (M.1 M) is very efficient and explains why
the secondary mostly rotates faster than the primary.
(b) The basics of the triple system are already founded
before the stars reached the main sequence, in an
earlier cloud or disk era of the stars. This shortens
the overall time needed to achieve the merging.
This leads us to conclude that the spin-up of wide O binaries
is best explained by a combination of disk fragmentation,
which lays the basis of the triple, and the subsequent swal-
lowing of low-mass by higher-mass (proto)-stars.
(4) If the swallowing of low-mass stars substantiates, then (in
adition to accretion) the formation of high-mass stars bene-
fits from merging.
The rotation rate of the binary components indeed provides valu-
able insights to early star formation processes.
Data availability
Tables D.1–D.4 are available at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://
cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/692/A192
Acknowledgements. The start of the young massive stars project was funded
by the Akademie der Wissenschaften und der Künste Nord-Rhein-Westfalen,
Germany. This work was supported by the Deutsche Forschungsgemeinschaft,
A192, page 22 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
DFG project number CH71/33-1. This research uses data obtained from the ESO
Science Archive Facility and the ELODIE archive at Observatoire de Haute-
Provence (OHP). This research has also made use of the SIMBAD database,
operated at the CDS, Strasbourg, France. Most of the early BESO spectra were
taken and optimised with endurance by Vera Hoffmeister. Anita Nasseri and
Noemi Roggero helped with the FEROS and BESO pipelines. We thank Petr
Hamanec, Hans Zinnecker and Gonzalo Holgado for fruitful discussions and the
anonymous referee for plenty of constructive suggestions.
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Blex, S., et al.: A&A, 692, A192 (2024)
Appendix A: Disentangling SB2 and SB3 line
profiles
We disentangled the components of SB2/SB3 systems using a
decomposition of the observed profiles into 2-3 Gaussian pro-
files. The line profile of one component is then obtained after
subtracting the Gaussian fit of the complementary component(s)
from the entire observed line profile.
In more detail, the line profiles of single O stars are to first
order well approximated by a Gaussian function. Small devi-
ations are due to the fact that the line shape is rounder for
a rotation-dominated star and more triangular with Lorentzian
wings for a turbulence-dominated star (e.g. Simón-Díaz et al.
2017). For the chi square minimising Gaussian fit, we used the
SIMPLEX algorithm with a suite of starting parameters in order
to avoid false local chi square minima. For a given line, we
first determined the parameters of the Gaussian profiles for those
spectra, where individual line profiles were best separated in RV.
Whenever it was possible, we then kept these parameters fixed
and changed only the RV of individual profiles to get the best fit.
However, when keeping width and height fixed, this often leads
to uncomfortable residuals in the resulting decomposed line pro-
files. The residuals become worse for small dRV between the
components. The residuals may affect the vsin idetermination.
The technique with fixed Gaussian widths and heights for all
spectra is widely used to determine RV curves (e.g. Mayer et al.
2017; problems near conjunction are discussed in Rosu et al.
2022). However, our aim here is slightly different from the deter-
mination of RV curves, since we are seeking for a best removal
of the complementary component(s). Therefore, we did not keep
width and height fixed for all spectra. Exceptions are a few very
faint companions, where a fixed width and/or height stabilised
the fit. Rather in most cases, we fit each spectrum individually
with free parameters, and then rejected outliers (typically those
spectra with small dRV). In the net effect, we used only those
(mostly 2-7) spectra with best separated components.
Binaries with narrow components are easier resolved than
with broad components. To account for this, we quantify the sep-
aration of the components A and B relative to their mean width,
avgFWHM =(FWHM(A) +FWHM(B))/2, via the dimension-
less number:
Sep =dRV /avgFWHM (A.1)
In addition to Sep, we take into account the depth (i.e. Gaussian
height) of the components. In a given spectrum with constant
noise level, a deep component allows for a better fit than a flat
component. Overall, the quality of the disentangling may affect
the quality of the vsin idetermination. Therefore, we assigned
a quality flag QF to each SB2/SB3 component, based on visual
inspection of the spectra, Sep and the component depths: high
(QF=A), medium (B) and low (C). For each QF we estimate a
likely uncertainty of vsin iof about 10% (A), 20% (B) and 30%
(C). Further details on the errors are given in Sect. 3.2.4.
We group the typical observed line shapes into 8 disentan-
gling types (DT). They cover the range of separation, line width
(narrow and broad), and height (faint and flat, or bright and
sharp) of our sample. Examples for the 8 DTs are depicted in
Fig. A.1. They include the best and worst cases for SB2s (DT
1-6) and SB3s (DT 7-8). The next subsections describe the 8
DTs, followed by a summary of the disentangling (Sect. A.9).
Primaries, Secondaries and Tertiaries are abbreviated by P,
S and T. All plots mentioned refer to the panel blocks in
Fig. A.1.
A.1. DT 1
HD 075759, two isolated components with wide separation, one
of the best examples. From the 30 available spectra (i.e. epochs)
we show only the best one.
Left panel: Both components are seen in O III and He II,
hence are O-Stars (we determined spectral types O9 IV and O9.5
V; previous studies have reported O9 V and B0 V, e.g. Sota et al.
2014;Holgado et al. 2020).
Right panel: The profiles are well fit by Gaussian functions.
Subtracting Gaussian fits results in negligible residual wiggles
with amplitude <3% of the profile depths. The residuals are
about/below the noise level. Therefore, their effect on the vsin i
determination is negligible.
We assign QF =A to both components P and S.
A.2. DT 2
HD 117856, two components of similar width with wide separa-
tion (Sep =1.4) but slightly overlapping in RV. This is still a very
good example. The important feature of the observed profiles is
the clear dip seen between the two components (in the Helium
lines). The dip constrains the Gaussian decomposition, yielding
two components of similar width. From the 11 available spectra
we here show only the best one.
Left panel: The secondary is barely seen in O III but both
components are well seen in He II 4686 (as well as He II 4541
and 5411 with broader profiles not shown here). This argues for
O-type stars (not B-stars) and we determined spectral types O9.5
III and O9.7 V for P and S, respectively.
Right panel: The profiles are well fit by Gaussian functions.
Gaussian decomposition results in negligible residual wiggles,
suggesting no influence on the vsin idetermination.
We assign QF =A to both components.
A.3. DT 3
HD 168075, still a good example, two resolved components of
similar width with small separation (Sep ≈1.1). The line profiles
strongly overlap in velocity. The secondary is relatively faint and
produces only a wing in the He lines without dip. The wing
is exclusively one-sided (sometimes left, sometimes right). We
show 2 of the 10 available spectra.
2006-Aug-21: In O III, the companion is "isolated" by a dip
in the observed line profile. He I 5876 shows a clear wing of
a companion right-hand of P. The HeII 4686 line is generally
broader leading to a small asymmetry to the right.
2004-May-07: near conjuction (dRV close to 0 and the joint
profile depth lies near the maximum). O III appears quite sym-
metric, He I and He II show a very small asymmetry to the left.
Gaussian decomposition was performed using free param-
eters (height, width, position) for P, but for S we kept height
and/or width fixed with some tolerance (after examining the
range for suited height and width). The width of P and S, respec-
tively, differs between the two spectra by about 10%. We have
allowed for this difference, to keep the residuals small. Indeed,
for both spectra, the residuals lie within the noise suggesting no
influence on the vsin idetermination.
We assign QF =A and B to P and S, respectively.
Note: already these two spectra reject the possibility that the
right tail seen on 2006-Aug-21 is due to a broad S with a narrow
P on top. Furthermore, from Gaussian decomposition of the 10
available spectra we were able to reproduce the orbital elements,
period (∼43.6 d Barbá et al. 2010;Sota et al. 2014) as well as
A192, page 24 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
❉❚ ✶✮
-400 -200 0 200 400
v [ km/s ]
0.7
0.8
0.9
1.0
1.1
normed spectrum
HD 075759 2015-Dec-04
OIII 5592
HeI 5876
HeII 4686
-400 -200 0 200 400
v [ km/s ]
0.7
0.8
0.9
1.0
1.1
normed spectrum
HD 075759 , HeI 5876 2015-Dec-04
dRV=251
FWHM=95,83
Sep = 2.83
Prim SecSec
❉❚ ✷✮
-400 -200 0 200 400
v [ km/s ]
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
normed spectrum
HD 117856 2011-May-18
OIII 5592
HeI 5876
HeII 4686
V
dip
-400 -200 0 200 400
v [ km/s ]
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
normed spectrum
HD 117856 , HeI 5876 2011-May-18
dRV=239
FWHM=182,159
Sep = 1.40
PrimSec Sum
❉❚ ✸
✮
-400 -200 0 200 400
v [ km/s ]
0.8
0.9
1.0
1.1
normed spectrum
HD 168075 2006-Aug-21
OIII 5592
HeI 5876
HeII 4686
V
companion
-400 -200 0 200 400
v [ km/s ]
0.8
0.9
1.0
1.1
normed spectrum
HD 168075 , HeI 5876 2006-Aug-21
dRV=132
FWHM=132,106
Sep = 1.11
Prim SecSum
-400 -200 0 200 400
v [ km/s ]
0.8
0.9
1.0
1.1
normed spectrum
HD 168075 2004-May-07
OIII 5592
HeI 5876
HeII 4686
-400 -200 0 200 400
v [ km/s ]
0.8
0.9
1.0
1.1
normed spectrum
HD 168075 , HeI 5876 2004-May-07
dRV=75
FWHM=113,104
Sep = 0.69
PrimSecSum
❉❚ ✹
✮
-400 -200 0 200 400
v [ km/s ]
0.8
0.9
1.0
1.1
normed spectrum
HD 150135 2015-Jun-25
OIII 5592
HeI 5876
HeII 4686
V
companion
-400 -200 0 200 400
v [ km/s ]
0.8
0.9
1.0
1.1
normed spectrum
HD 150135 , HeI 5876 2007-Apr-21
dRV=64
FWHM=112,259
Sep = 0.35
PrimSecSum
-400 -200 0 200 400
v [ km/s ]
0.8
0.9
1.0
1.1
normed spectrum
HD 150135 , HeI 5876 2009-Feb-20
dRV=100
FWHM=100,259
Sep = 0.56
Prim SecSum
-400 -200 0 200 400
v [ km/s ]
0.8
0.9
1.0
1.1
normed spectrum
HD 150135 , HeI 5876 2013-Jun-25
dRV=32
FWHM=116,257
Sep = 0.17
Prim SecSum
❉❚ ✺
✮
-600 -400 -200 0 200 400 600
v [ km/s ]
0.80
0.85
0.90
0.95
1.00
1.05
1.10
normed spectrum
HD 152219 2004-May-05
HeI 4713
HeI 4922
HeI 5876
V
comp.
-600 -400 -200 0 200 400 600
v [ km/s ]
0.80
0.85
0.90
0.95
1.00
1.05
1.10
normed spectrum
HD 152219 2006-May-06
HeI 4713
HeI 4922
HeI 5876
V
companion
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
normed spectrum
HD 152219 , HeI 4922 2004-May-05
dRV=373
FWHM=294,146
Sep = 1.69 Prim
Sec
Sum
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
normed spectrum
HD 152219 , HeI 4922 2006-May-06
dRV=399
FWHM=283,212
Sep = 1.61
Prim
Sec
Sum
❉❚ ✻
✮
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
normed spectrum
CPD -59 2600 2004-Feb-05
OIII 5592
HeI 4471
HeII 4541
V
tail
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
normed spectrum
CPD -59 2600 2008-Jun-10
OIII 5592
HeI 4471
HeII 4541
V
tail
-600 -400 -200 0 200 400
v [ km/s ]
0.90
0.95
1.00
normed spectrum
CPD -59 2600, HeI 5876 2004-Feb-05
dRV=121
FWHM=198,398
Sep = 0.41
PrimSec Sum
-600 -400 -200 0 200 400
v [ km/s ]
0.90
0.95
1.00
normed spectrum
CPD -59 2600, HeI 5876 2008-Jun-10
dRV=156
FWHM=205,398
Sep = 0.52
Prim SecSum
❉❚ ✼
✮❛✮❜
✮
-400 -200 0 200 400
v [ km/s ]
0.90
0.95
1.00
normed spectrum
CPD -59 2636ABC 2008-Jun-09
OIII 5592
HeI 5876
HeII 4686
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
normed spectrum
CPD -59 2603 2015-Jun-26
OIII 5592
HeI 5876
HeII 4541
-400 -200 0 200 400
v [ km/s ]
0.90
0.95
1.00
normed spectrum
CPD592636ABC , HeI 5876 2008-Jun-09
Prim SecTert
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
normed spectrum
CPD -59 2603 , HeI 5876 2015-Jun-26
Prim SecTert
❉❚ ✽
✮
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
normed spectrum
δ Cir 2009-May-02
OIII 5592
HeI 4922
HeII 4541
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
normed spectrum
δ Cir 2009-Aug-28
OIII 5592
HeI 4922
HeII 4541
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
normed spectrum
δ Cir , HeI 4922 2009-May-02
PrimSec TertSum
-600 -400 -200 0 200 400 600
v [ km/s ]
0.90
0.95
1.00
normed spectrum
δ Cir , HeI 4922 2009-Aug-28
Prim SecTertSum
Fig. A.1. Examples of disentangling types (DT) for SB2s (DT 1-6) and SB3s (DT 7-8). Details are described in Sect. A. Radial velocities “v” are
as observed and not corrected for systemic velocities.
K1 ≈35 km/s and K2 ≈70 km/s. This supports the correctness
of the Gaussian decomposition and that both components have
similar (narrow) width.
We identified 6 SB2s of DT 3. They have between 9 and
50 spectra available. Near conjunction, all these SB2s lack the
two-sided wing indicating a broad S (an exception may be
HD 153426; see below). This gives us confidence in the decom-
position with regard to vsin i.
A.4. DT 4
HD 150135, a narrow primary component overplotted on a broad
component is nevertheless a good example. The components
largely overlap in velocity and have a very small separation (Sep
<0.8). In the 17 available spectra, the relatively faint S produces
only a wing without dip. The wing switches to the opposite RV
side of the narrow P (sometimes left, sometimes right) but near
conjunction the wing appears simultaneously to both sides. This
strongly suggests a broad S.
We show 4 spectra: 2015-Jun-25: clear asymmetry in O III,
He I 5876 and He II 4686, suggesting an O-type companion.
None of the spectra shows a dip hence the companion is not con-
strained to be narrow, rather it might be broad.
The next 3 panels show the Gaussian decomposition in
He I5876 using free parameters. It provides evidence for a broad
companion:
A192, page 25 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
2007-Apr-21: S left-hand of P, Sep =0.35.
2009-Feb-20: S right-hand of P, Sep =0.56.
2013-Jun-25 (S close to P, Sep =0.19): In this epoch, RV
of P lies between the two extremes above, indicating near-
conjunction. There, the observed profile exhibits wings on both
sides left and right of the narrow P. This (and the absence of a
dip at larger dRV) strongly supports a broad S: If S were narrow
(similar to P), then only a one-sided wing would be expected.
For most spectra, the residuals from the Gaussian decompo-
sition with free parameters are small. We assign QF(P) =A and
QF(S) =B, to account for a few spectra with noticable wiggles
for S.
With regard to vsin i, the narrow P and broad S are trustable.
Additional evidence for a broad S comes from the spectral types
and the orbits:
a) Altogether, from the decomposition of several lines and
their EW ratios we estimated spectral types about O6 V and O7
III for P and S, respectively, consistent with spectral type O6.5
V ((f))z reported for the combined system (Sota et al. 2014).
We were able to reproduce the known orbital period of 181 d
reported by Sota et al. (2014). From the 17 spectra we derived
RV curves for both components, yielding a nearly circular orbit
with K1 ≈60 ±5 km/s and K2 ≈75 ±10 km/s, hence a mass
ratio M2/M1 ≈0.8, which is within the uncertainties consistent
with that of an O6 +O7 binary (Martins 2018).
b) If we fit S with a narrow (instead of a broad) compo-
nent, this would result in a noisy RV curve with K2 >120 km/s,
yielding M2/M1 ≈0.5. If P has M1 ≈28 M(adopting O6.5 V,
Sota et al. 2014), this would imply M2 ≈14 M, hence a B-type
secondary which would not show OIII and He II lines, contrary
to the observed spectra.
The examples in DT 3 and DT 4 appear similar by showing
wings without a dip, but they differ by the width of the com-
panion (narrow or broad). We have identified 15 DT4 systems.
Based on 9 - 45 available spectra, each DT4 system shows the
switch between one-sided broad wings near maximal dRV and
two-sided wings near conjunction.
A.5. DT 5
HD 152219, a prominent broad P and a faint narrower S. It is an
example for a poor secondary in our sample, and the only one
of this DT. While the components are well discerned by their
separation (Sep ≈1.6), the brightness ratio is large (P/S&5).
For Gaussian profiles, a large P/S is not harmful as we found
in many other cases. In HD 152219, however, the line profile of
P is rotation-dominated (i.e. round), and deviates from a Gaus-
sian. The combination of the large brightness ratio and the round
shape of P lead to exceptionally large residuals for S.
S is a B-type star (Sana et al. 2008b;Rosu et al. 2022) and
neither seen in O III nor He II. Here we show 2 out of 24 spec-
tra in He I 4713, 4922, 5876 (top panel row of the block) on
2004-May-05 and 2006-May-06 (S on the left and right of P,
respectively).
Gaussian decomposition using free parameters is shown for
He I 4922 (bottom panels of the block). The large brightness ratio
leads to strong residual wiggles for S, when subtracting a Gaus-
sian fit of P. The amplitude of the wiggles reaches the depth of
S. As mentioned above, the line profile deviates from a
Gaussian, and this difference leads to the residual wiggles. The
relative strength of the wiggles depends on the depth ratio (Gaus-
sian height). Therefore P does not receive noticeable wiggles
produced by the faint S, but S receives strong wiggles from the
bright P.
Notably Sep is sufficiently large, so that the wiggles caused
by P lie essentially outside of the profile of S. They do not
directly change the (core of the) profile of S. This enables us to
essentially reduce the effect of the residuals on the vsin ideter-
mination by using a small cut-out window around the profile of
S (±3×standard deviation ≈1.25 ×FWHM of S). More details
are explained in Sect. 3.2.3.
We assign QF =A and C to P and S, respectively.
A.6. DT 6
CPD −59 2600 has a bright medium-narrow P and a faint broad
S, with very small Sep .0.5. This is the worst and most uncer-
tain SB2 example in our sample. Sota et al. (2014) noted that this
system is an O6 V ((f)) SB1 with a 626 d period, and possibly an
SB2.
Here we show 2 out of 15 spectra in O III 5592, He I 4471
and He II 4541 (top panels of the block) on 2004-Feb-05 (tail
left) and 2008-Jun-10 (tail right). The observed line profiles
show a tail to the left and right of P, respectively, but no dip.
The tail is also seen in other sufficiently strong lines (He I 4026,
5876 and He II 4686). Whenever visible, the tails are consistent
in all lines.
The tail lies at the opposite RV side of P (e.g. if P is on
the left, then the tail to the right, vice versa). As for HD 150135
(DT 4), the observed profile appears – near conjunction – more
symmetric with two-sided tails left and right of P. This tail
behaviour questions a wind as explanation. Rather it suggests
the presence of an O-type companion with broad and flat line
profile.
The Gaussian decomposition yields consistent results, shown
for He I 5876 in the bottom panels of the block in Fig. A.1.
After several attempts with free Gaussian parameters, we kept
the depth and width of the broad S fixed.
Applying Gaussian decomposition, the known period helped
us to derive (noisy) RV curves for P and S. They indicate an
eccentric orbit with K1 ≈50 km/s, K2 ≈75 km/s, and a mass
ratio M2/M1 ≈0.67. The EW ratios He I4471/He II4541 for P
and S are uncertain but consistent with spectral types O6 V and
O8 V, respectively, yielding M2/M1 ≈0.67. The overall agree-
ments provide further support for the broad S.
vsin iof P is well determined, but S is affected by strong
residual noise with an amplitude up to 30% of the profile depth.
We assign QF =B and C to P and S, respectively.
A.7. DT 7
DT 7 and DT 8 systems refer to SB3s. DT7 show two dips allow-
ing us to clearly decompose the three components, but DT 8 have
only one dip. While a DT 8 is a clear SB3, it is harder compared
to a DT 7 to distinguish between narrow or broad components
and more spectra are needed to decompose the three compo-
nents. In this section we illustrate the DT 7 with two examples (a
and b):
a) CPD −59 2636ABC, this is a very good example for a triple
system. Albacete Colombo et al. (2002) found spectral types O7
V, O8 V and a period of 3.6 d for P and S, and a third star T of
spectral type O9 V with a 5.03 d period.
Here we show one spectrum out of five. Top panel: in all lines
the three components are well separated with a clear dip, remi-
A192, page 26 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
niscent of DT 2 above. P, S and T are identified via EW ratios of
diagnostic lines.
Bottom panel: the Gaussian decomposition yields moderate
FWHM for all components, and any residuals lie inside the noise
level.
Regarding vsin i, we assign QF =A to each component.
b) CPD −59 2603, a known hierarchical triple system consist-
ing of a short-period (2.15 d) eclipsing O7 V +O9.5 V binary
bound to a B0.2 IV star (Rauw et al. 2001a). Among 10 spectra,
it is still a good example with three well isolated components in
He I (top panel). However, as for DT5 above, the bright rotation-
dominated round profile of P leads to strong residuals for S and
T (bottom panel). Therefore, we assign QF =A, B, and C to P, S
and T, respectively.
A.8. DT 8
δCir is a known triple consisting of an eclipsing binary O7 III-V
+O9.5 V bound to a wide B0.5 V (Penny et al. 2001 based on
IUE spectra; Mayer et al. 2014b essentially based on 95 HARPS
spectra10). Based on the BESO and FEROS spectra, however,
it is a relatively poor SB3 example, because a unique decom-
position requires an additional second dip between components
which is not seen.
We show two spectra with large dRV out of 16, one with
FEROS on 2009-May-02 and a noisier one with BESO on 2009-
Aug-28. We consider the He I 4922 spectra: The FEROS spec-
trum exhibits one clear dip, which reveals S well isolated from
P. However, Gaussian decomposition implies a third component
T which is poorly constrained between S and P. The BESO spec-
trum shows two marginal dips, which could be the signatures
between P, S and T. However, in other HeI lines we see only
one dip and He I 4922 could be contaminated by O II 4924.5 as
noted by Mayer et al. 2014b. While any residuals appear small,
the Gaussian parameters width and height of the resulting com-
ponents differ between the two different spectra. This holds also
for other He I lines. Therefore, we assign QF =B, C, and C to P,
S and T, respectively.
A.9. Disentangling summary
Using Gaussian decomposition, we were able to disentangle 62
SB2+SB3 systems (54 SB2 and 8 SB3). The number of systems
in the disentangling types are: 18 (DT 1), 13 (DT2), 6 (DT 3), 15
(DT 4), 1 (DT 5), 1 (DT 6), 4 (DT 7), 4 (DT 8). The DTs cover
the range from well resolved pairs to pairs with small separation;
they also cover the full range of component width and their peak
height. The SB2 decomposition is possible down to very small
separation (Sep ∼0.5) which is is reached, if a sharply-peaked
profile sits on top of a broad flat profile. A larger separation is
required (Sep &1), if the components have similar width and
height.
Among our sample, two SB2 systems are of poor disentan-
gling quality (DT 5, DT 6). In addition, two SB2s appear uncer-
tain: HD 125206 (DT 3) could be an SB3, and in HD 153426
(DT 3) the secondary could be even broader (DT4). In two SB3s,
HD 092206 C and δCir, we determined vsin ionly for P and S,
because the tertiary is too noisy/uncertain. We are confident that
10 HARPS has a spectral resolution R ∼100000, twice that of BESO,
FEROS, UVES and Elodie. To keep our sample homogeneous, we did
not use HARPS spectra here.
the disentangling of the vast majority of systems (58/62) allows
us to obtain robust vsin iresults for P and S.
In addition, we identified 10 SB1s which are reported in the
literature as SB2 systems with reliable vsin ifor both P and S
(Tab. D.2). For the scientific analysis in Sect. 5we included
these 10 systems as SB2s; S is faint and shows mostly a flat
broad profile below our detectability. We determined vsin iof P
from the total line profile and took vsin iof S from the literature;
we assigned DT =0 and QF (P,S) =(D,N), with a likely uncer-
tainty of vsin i(P) of about 15% (QF=D) and QF=N means “not
known”.
Tab. D.2 lists the entire sample of 72 SB2+SB3s together
with spectral types and luminosity classes; if they have not been
published or differ from previous works, we list those derived
by our data and add the reference “this work”. There we also
give the disentangling types DT and the quality flags QF for the
components.
Appendix B: The effect of a slow wind or an
expanding halo on the usin idetermination
In a nutshell, an isotropic stellar wind (or an expanding halo) is
a priori independent of rotation and turbulence. The wind has to
be treated as an additional component Pwind to the line profile
such that eq. 1is expanded as
Pline =Prot ×Pturb +Pwind (B.1)
We note that Pwind is not part of the mathematical convolution
term (eq. 1). A convolution term will not lead to a bias in the
vsin idetermination; the only exception is that a large vturb leads
to a steeply declining FT amplitude and may shift the FT min-
imum into the noise level biasing vsin ito larger values (see
Fig. 2. of Simón-Díaz & Herrero 2007). The problematic issue
is that the wind is an additive and asymmetric contribution to
the line profile. Any asymmetric profile leads to a complicated
behaviour in the Fourier domain and may produce FT minima
which are hard to predict.
In more detail, Pwind is composed of an absorption compo-
nent Pw_abs seen by the observer in front of the stellar disk and
an emission component Pw_em surrounding the star
Pwind =Pw_abs +Pw_em (B.2)
Pw_abs and Pw_em may be somehow connected, but details are still
complicated and subject to intense research. On the modelling
side, we note that TLUSTY models (Lanz & Hubeny 2003) are
static plane-parallel models with full non-LTE metal line blan-
keting. They are suitable for O stars that have weak stellar winds.
On the other hand, FASTWIND model atmospheres (Puls et al.
2005;Rivero González et al. 2012) are not truly hydrodynamic
as the programme does not solve for the wind structure but
adopts a wind law.
To illustrate the effect of a wind on the vsin idetermination,
we assume 1) that both the wind intensity and velocity are low
(“weak wind” Martins et al. 2004,2005a), 2) that both compo-
nents Pw_abs and Pw_em are optically thin, 3) that the contribu-
tion of Pw_em behind the stellar disk is negligible. Notably an
isotropic wind emission component Pw_em with constant velocity
has a round line profile shape, similar to the shape of a rotating
star, and it will likewise produce charactieristic FT minima as
well.
To keep the illustration simple, we further assume that the
blue absorption wing Pw_abs is absent, so that the line profiles
are symmetric and the FT is real (i.e. imaginary part =0). Such
A192, page 27 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
-300 -200 -100 0 100 200 300
v [ km/s ]
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
line profile
star vsini = 0
star + wind
wind v = 50
FWHM=100
-300 -200 -100 0 100 200 300
v [ km/s ]
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
line profile
star vsini = 100
star + wind
wind v = 50
FWHM=174
20 40 60 80 100
vsini [ km/s ]
-10
-8
-6
-4
-2
0
log amplitude
star vsini = 0
star + wind
vsini~30
wind
V
40 60 80 100 120 140
vsini [ km/s ]
-6
-5
-4
-3
-2
-1
0
log amplitude
vsini = 109
star vsini = 100
star + wind
wind
Fig. B.1. Effect of the emission of an isotropic slow wind on the line
profile (top) and its Fourier transform (FT, bottom), for two rotation
speeds vsin i=0 km/s (“null rotator” left) and vsin i=100 km/s
(“mid rotator” right). Top: The blue solid line shows (1) the rotating
star, a rotationally broadened line profile convolved with Lorentzian,
vrot =50 km/s, FWHM vturb =50 km/s. The red solid line shows (2) an
isotropic slow wind halo with vwind =50 km/s and dispersion ≈20 km/s
(total FWHM ≈65 km/s). The wind strength is 5% of the stellar equiv-
alent width. The dash-dotted black line shows the combined profile of
star and wind (sum of (1) and (2)). We note that the combined profile
has a rounder (i.e. more boxy) shape than the pure stellar profile. Bot-
tom: FT amplitude of the rotating star (blue), of the wind halo (red), and
of the combined profile (star and wind, black).
a configuration resembles that of an extended slowly expanding
halo with Rhalo >100 Rstar, where the solid angle for produc-
ing the blue absorption wing Pw_abs becomes negligibly small.
Figure B.1 shows two model examples, both having vturb =
50 km/s (Lorentz FWHM), vwind =50 km/s (convolved with a
Gaussian, but details on that do not affect the illustration), and
Pw_abs contributes 5% to the original line profile:
1) a “null” rotator with vsin i=0 km/s,
2) a “mid” rotator with vsin i=100 km/s.
We note that FWHM of the wind profile is smaller than FWHM
of the stellar profile. The two striking features of the examples
are
1) The wind rounds-offthe tip of the line profile (top panels).
2) The wind produces FT minima (bottom panels, red), similar
to what a rotating star does.
3) For the “null” rotator: at high frequencies (low vsin ion the
x-axis) the FT amplitude of the wind exceeds that of the star,
because the FWHM of line profile of the wind is smaller than
that of the star. As a consequence of this “cross-over” of the
FT amplitudes, the combined profile shows high frequency
FT minima which are only due to the wind and have little to
do with the stellar rotation.
4) For the “mid” rotator, the wind effects are present but much
weaker, so that the FT minima of the combined star+wind
profile are moderately shifted to lower frequencies (right
hand direction in the plot), resulting in a bias of vsin ito
about 10% larger values.
The model examples are simple but give us a warning on poten-
tial biases when determining vsin iin the presence of a slow
wind. For a fast rotating star (vrot ∼200 km/s) the line profile
is broad and shallow. Then a slow wind emission (of 5% inten-
sity) creates a bump, so that the line profiles mimick a double
star and can be rejected for rotational analysis.
We note that both the vsin iartefact and the vsin ibias are not
a failure of the FT method. Rather the FT method is particularly
sensitive to features affecting the line profile, probably more sen-
sitive than the GOF method. The basic effect of the wind is to
round-offthe tip of the line profile. If the rounded-offline profile
is analysed with the GOF method, a similar vsin ibias will be
obtained.
A fast wind (emission component) has a broad line profile
and a steeply declining FT amplitude. If the wind’s FWHM is
larger than the stellar FWHM, no “cross-over” of the FT ampli-
tudes occurs. Likewise, we note that typically the line profile of a
wind absorption component Pw_abs is narrower than the emission
component, but blue-shifted. This asymmetry leads to a compli-
cated behaviour of the FT amplitude, spiraling around the origin
of the FT plane, but in general no “cross-over” of the FT ampli-
tudes is expected.
For real data, even with simplified assumptions, a proper
solution of Eqs. B.1 and B.2 has many free parameters and
requires extremely precise spectral line profiles, exceeding the
quality of our observational data. For instance, the continuum
level next to the line profile exhibits low-frequency noise lead-
ing to uncertainties and preventing the required precision.
A detailed treatment and modelling of the wind effects with
regard to the determination of vsin iis beyond the frame of this
paper; we hope that the examples encourage specialists in the
field to find practicable solutions. For our purpose here we keep
in mind that a weak wind may bias vsin iof slow+medium rotat-
ing stars to larger values. For fast rotators a slow wind produces
an easily detectable dip in the line profile, unless the wind inten-
sity is extremely small; for such a weak and slow wind any bias
on vsin imight be small (<10%). The example of the “null” rota-
tor with a slow wind may provide a key to solve the puzzling
vsin iresults reported by Sundqvist et al. (2013). In addition, if
a slow wind is more pronounced in Helium lines compared to
metal lines, it may explain (at least partially) why the determined
vsin iof slow rotators is often larger for Helium lines than metal
lines.
Appendix C: The effect of inclination on vsin i
We distinguish between the inclination iorb of the orbital axis
of a binary and the rotation axis irot of the individual stars; the
inclination angle iis measured against the line-of-sight.
C.1. Random inclination
Figure C.1 illustrates the effect of inclination on the observed
velocity distribution. It holds for both iorb and irot. The intrinsic
orbital and rotational velocities (vorb and vrot ) are known to be
in the order of 100 km/s and we adopt this value to illustrate the
argumentation.
For randomly oriented rotation axes, the probability of find-
ing an axis inclination φwithin a cone angle iwith the line-of-
sight is P(φ < i)=1−cos (i). This is illustrated in Fig. C.1,
panel A. Marked is the case of i=30◦and P=0.134. This
yields the cumulative fraction for vsin i, in other words the frac-
tion of measured vsin(φ) cumulated for all inclinations φ < i
(panel B).
From this we can derive the fraction of vsin ifor equidistant
narrow bins of 1km/s (panel C). For example, 13.4% of a sam-
ple with intrinsic v=100 km/s and randomly oriented axes will
show vsin i<50 km/s. Remarkable is also the inclination range
i>60◦, where 50% of the sample show vsin i>87 km/s (i.e.
very little affected by inclination).
A192, page 28 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
0 20 40 60 80 100
inclination i [ degree ]
0.0
0.2
0.4
0.6
0.8
1.0
P ( ϕ < i )
A) 1 − cos (i)
0.134
0.5
0 20 40 60 80 100
v sin (i) [ km/s ]
0
20
40
60
80
100
cumulative fraction [ % ]
13.4
50
B) v = 100
0 20 40 60 80 100
v sin (i) [ km/s ]
0
5
10
15
fraction [ % ]
C) d vsini = 1
13.4
50
0 50 100 150 200 250
[ km/s ]
0
5
10
15
20
25
Number
D)
v_eq
vsini
0 50 100 150 200 250
[ km/s ]
0
5
10
15
20
25
Number
E) v_eq
i > 30
i < 30
0 100 200 300 400
[ km/s ]
0
5
10
15
Number
F)v_eq
i > 30
i < 30
0 100 200 300 400
[ km/s ]
0
5
10
15
Number
G)v_eq
i > 30
i < 30
tol = 30
close SB2, i known, P+S, N = 56
0 100 200 300 400
vsini [ km/s ]
0
2
4
6
8
10
12
Number of stars
I+II ( 4 )
III ( 7 )
IV+V ( 45 )
slow medium fast
21 30 5
16 25 4
3 4 0
2 1 1
v peak = 117.1+- 45.9
close SB2, i corr, P+S, N = 56
0 100 200 300 400
v equatorial [ km/s ]
0
2
4
6
8
10
12
14
Number of stars
I+II ( 4 )
III ( 7 )
IV+V ( 45 )
slow medium fast
4 39 13
3 33 9
1 4 2
0 2 2
v peak = 150.2+- 49.4
Fig. C.1. Effect of inclination. A: Fraction of axes at inclination φ < ivs i, assuming random inclinations. The red and blue dots mark two
inclinations used in the discussion. B: Same as for A) but vs vsin i, adopting equatorial v=100 km/s. C: Fraction of vsin ivalues for equidistant bins
dvsin i=1 km/s. The red and blue numbers give the cumulative contributions of the red and blue shaded area. D: Adopted constant distribution of
stars with equatorial veq between 100 and 200 km/s (red shaded rectangle) and vsin idistribution resulting for random inclinations (blue histogram).
The two black curves are the distribution in panel C scaled in vfor 100 and 200 km/s. The blue histogram is constructed from the integration of all
such black curves, it is scaled to the same area as the red histogram. About 30% of the stars from the rectangular red histogram will be observed
with vsin i<100 km/s. E: Same as for D) but separated with an inclination cut i>30◦(blue) and i<30◦(black). F: Same as for E) but adopting
a triangular distribution of equatorial vbetween 50 and 350 km/s (red shaded). G: Same as for F) but allowing for a deviation between iorb and irot
with a tolerance of 30◦for co-axial rotation (Sect. C.3). The last two panels show the rotational velocity distribution of stars in close SB2 systems
with known orbital inclination, without and with inclination correction (middle panel: vsin i, right: v equatorial).
So far, we have adopted a sample with a fixed intrinsic veloc-
ity (veq =100 km/s). There the histogram peaks at 100 km/s but
random inclination produces a declining tail towards smaller
vsin i. How will the vsin idistribution appear for a range of veq?
Panel D considers a sample with a constant distribution of
the intrinsic velocity veq between 100 and 200 km/s (red shaded);
this velocity range is typically observed for vorb. The blue his-
togram shows the predicted vsin idistribution which will be
observed for randomly oriented rotational axes.11 The striking
histogram feature is the steep rise from 200 km/s to 100 km/s
followed by a steep decline towards smaller vsin i. Compared
to panel C with a declining tail towards smaller vsin i, the his-
togram of panel D appears more symmetric showing two tails.
For about 30% of this sample the predicted vsin ilies below
100 km/s.
11 For each bin of width 1 km/s at velocity v_bin the distribution of
panel C is scaled from 100 km/s to v_bin (this is illustrated by the thin
black curves at 100 km/s and 200 km/s). Then these distributions are
coadded, yielding the observed histogram for vsin i(blue).
Panel E assumes a sample with the same veq input distribu-
tion as Panel D, but the randomly oriented rotational axes are
separated for favourable (i>30◦, blue) and misaligned inclina-
tions (i<30◦, black).
Cases D and E consider an isolated block of input veq, which
does not take into account the influence of neighbouring faster
or slower veq populations onto the predicted vsin idistribution,
for example.
Panel F assumes a triangular input distribution of the intrin-
sic velocity veq between 50 and 350 km/s (red shaded) seen with
random inclinations 0◦<i<90◦. Such an input distribution
is motivated by the observed triangle-shaped vsin idistribution
of SBs. Strikingly, the predicted vsin idistribution (blue, for
i>30◦) largely resembles the input distribution, with only a
small tail of vsin ibelow 50 km/s which comprises a fraction of
about 10%. The black histogram refers to i<30◦.
Panel G shows the same as Panel F but assuming a mild co-
axial rotation, explained in Sect. C.3.
For comparison, the observed vsin idistributions for SB and
C dwarfs are depicted in last two panels.
A192, page 29 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
C.2. Inclination of the orbit axis
There are four possibilities for a system to be classified as C:
1) it is intrinsically a single star,
2) it is intrinsically an SB but the distance between primary and
companion is too large,
3) it is intrinsically an SB but the mass ratio companion/primary
is too small,
4) it is intrinsically an SB but has too a small inclination of the
orbit axis.
In cases 2-4, the observed RV variation may be too small so that
the system escapes SB detection.
We first discuss the influence of iorb on the SB classification.
To detect an SB, iorb must be sufficiently large because otherwise
any RV variations (dRV) would be too small to be detectable. If
iorb is too small, then the system would be classified as C.
To estimate the inclination threshold at which the transition
from an SB to a C likely occurs, we adopt an intrinsic vorb =
100 km/s. For iorb =15◦one obtains sin (iorb)=26 km/s and a
cumulative probability P=3.5% (Fig. C.1, panel B). To detect
an SB, a dRV of ∼25 km/s may be sufficient but requires good
luck with the observations. Therefore, we choose a conserva-
tive “worst case” threshold iorb =30◦, yielding vorb sin(iorb)=
50 km/s and P≈13% (Fig. C.1, panel B).
With this threshold, and assuming randomly oriented orbit
axes, the detected SBs comprise (at least) 87% of the entire SB
population, and (at most) 13% of the binaries are misaligned and
observed (i.e. hidden) as C. This yields a “hiding fraction” h f =
13/87 =0.15. For a range of 100 km/s< vorb <200 km/s about
8% of the binaries will appear as a C (black histogram of case E
in Fig. C.1).
To compare with observed numbers, our sample contains 130
Cs and 108 detected SB systems (36 SB1, 64 SB2, 8 SB3).
Applying the hiding fraction above yields the number of pre-
dicted misaligned SBs as 108 ·0.15 ∼16. This comprises only
12% of the Cs (16/130). It rejects the possibility that a signifi-
cant fraction of the Cs are misaligned SBs (more precisely: are
the misaligned counterparts of our detected SB systems).
We conclude that the vast majority of the Cs (likely more
than 85%) are either single stars or binaries with small vorb (i.e.
with wide orbits or small companion/primary mass ratios). The
majority of Cs in our sample must be intrinsically different from
the SBs systems. This conclusion is further supported by the dif-
ferent fraction of dwarfs (44% vs 69% in Cs and SBs, respec-
tively, Fig. 11).
C.3. Inclination of the rotation axis
The aim is to quantify the effect of inclination also with respect
to the three categories slow/medium/fast rotators.
The samples comprise in total 130 Cs and 176 stars in SB
systems. The number of fast rotators in both SBs and Cs (16
and 20 stars, respectively) is small making it unlikely that their
inclined counterparts have a significant effect on the statistics
of the medium rotators. The number of medium rotators in SBs
is 62 stars (13 in SB1, 49 in SB2+SB3) compared to 20 in Cs;
when restricting to dwarfs, the numbers are 46 in SBs and 7 in
Cs (Fig. 11). Using all stars, the observed fraction of medium
rotators is 62/176 =35% for SBs and 20/130 =15% for Cs. This
yields an excess of medium rotators in SB stars by a factor of
35/15 =2.3 compared to Cs.
Firstly, we discuss the number of medium rotators and their
observed excess for SB stars compared to Cs, assuming that irot
is randomly oriented in all stars. For both samples we consider
the fraction of intrinsic medium rotators which is hidden (i.e.
shifted to slow rotators due to inclination). We adopt the same
threshold i=30◦used in the discussion of iorb (Sect. C.2).
Applying the hiding fraction h f =0.15 as above, we obtain, for
instance, 9 hidden stars in SBs (62 x 0.15) and 3 hidden stars in
Cs (20 x 0.15). A correction for the hidden medium rotators lies
in the range below 10%. It would mildly increase the medium
rotator excess.
However, it has been widely assumed or speculated that
the spin axes of stellar rotation and orbital rotation are roughly
aligned (i.e. co-axial). Because the SB systems are preferentially
oriented at large iorb >30◦, co-axial rotation implies that irot and
vsin iis larger for the detected SB stars than for those Cs, which
are misaligned SBs. Then the stars of misaligned SBs are shifted
to slow rotating Cs, because sin(i)<0.5 (e.g. 200 km/s goes to
100 km/s or less). For a rough estimate, we adopt that 16 SBs
are misaligned and classified as C (Sect. C.2). We assume as an
upper limit that each of them contains two intrinsic medium rota-
tors which appear now as slow rotating Cs. Then the number of
true Cs may reduce from 130 to 98 (i.e. by ∼25%). A correction
will mildly lower the medium rotator excess.
The above estimate assumed that rotation axis and orbit axis
are parallel. It did not take into account the possible effect caused
by a deviation of rotation axis from the orbit axis. Therefore we
refined the estimate, whereby we need some empirical model
assumptions. We consider two cases of co-axial rotation:
1) Strict co-axial rotation assumes that rotation axis and orbit
axis are parallel, implying irot =iorb .
2) Mild co-axial rotation allows for a deviation δiof rotation
axis from the orbit axis by a tolerance up to 30◦. (This thresh-
old appears reasonable: the inclination of the earth’s rota-
tion axis against the ecliptic axis is 23◦.) To parameterise the
deviation distribution, we have chosen a cosine function fd ev
=cos (3 ·δi) for −30 < δi<30.
In both cases we assume a triangular input distribution of veq
and calculate the predicted vsin idistributions. Panels F and G
of Fig. C.1 illustrate the results for strict and mild co-axial rota-
tion, respectively. The blue histograms show the predicted vsin i
distribution for the SBs (iorb >30), the black histograms for
the Cs (iorb <30). The difference between panels F and G is
marginal; G predicts slightly more slow rotators. Notably, the
blue histograms of F and G appear similar to the observed SB
distribution. This suggests that panels F and G describe a realis-
tic case for the SBs. Then the black histograms would quantify
how the misaligned SBs may contribute to Cs.
We also compared with literature results on inclination
effects. Assuming that the rotation axes are randomly distributed,
Ramírez-Agudelo et al. (2013,2015) deconvolved the observed
vsin idistributions with a Bayesian approach and presented an
intrinsic rotation distribution of the Cs and the SB2 Primaries
in the VFTS, respectively. Instead of the simple triangular input
distribution of veq used by us in panels F and G, their adopted
input distribution is the sum of a Gaussian and a Gamma distri-
bution. This refined input model has in total (at least) six free
parameters (3 for the Gamma distribution and 3 for the Gaus-
sian). If the rotation axis angles are not randomly distributed and
our vsin ihistograms with about 18 bins (i.e. data points) are
noisy, any refinements of the intrinsic rotation distribution com-
pared to our simple triangle estimate should be considered with
care. Therefore, the simple triangle estimate is sufficient for our
purpose here to estimate the inclination effects (i.e. the shift of
medium rotators to slow rotators).
To summarise the theoretical considerations, we have per-
formed a simple estimate with minimal assumptions and by
A192, page 30 of 31
Blex, S., et al.: A&A, 692, A192 (2024)
calculating the predicted vsin idistribution from two empirical
models, a rectangular and a triangular input distribution of veq.
In both cases, we conclude that the medium rotator excess in
SBs compared to Cs is real and not caused by inclination effects.
Likewise, the predicted fraction of increased rotators which will
be shifted by inclination to slow rotators is moderate, less than
30%. We note that fractions of this range are by far too small
to significantly reduce the observed excess of medium rotators
in SB stars compared to Cs by the factor 2.3, mentioned at the
begin of this section.
To compare with real data, our sample contains 53 SB2s
with known orbital inclination. Assuming that rotation axis and
orbit axis are parallel, we derived the inclination corrected “true”
rotational velocities. These and the observed rotational veloci-
ties are shown versus the orbital period in the bottom panels of
Fig. 16. The sample splits into 28 close SB2s with presumably
well-determined veq and 25 wide SB2s with uncertain veq due
to uncertain parallelity of spin and orbit axes. The results for the
close SB2s are shown in Fig. C.1: The observed and true number
counts for slow/medium/fast rotators are 21/30/5 and 4/39/13,
respectively. This yields 52 (=39 +13) true increased rota-
tors, and 17 (=21 - 4) observed slow rotators which are shifted
by inclination into the slow bin. The shift fraction of 17/52 =
33% appears well consistent with the fraction of 30% predicted
above from the rectangular model. The corresponding number
counts for the wide SB2s are 32/11/7 and 20/17/13, respectively.
This yields 30 true increased rotators and 12 shifted slow rota-
tors, hence a shift fraction of 40%, exceeding the model pre-
dictions. The inclination correction of the 25 wide SB2s may
be erroneous, and we consider their results with some caution.
Notably, the histograms of the true rotational velocities of the
close and wide SB2s differ in shape, being rather rectangular
and triangular respectively. So far, we can only speculate that
the rectangular-triangular difference is not simply a matter of the
small number statistics, but has a physical origin. More detailed
inclination studies require larger data samples, which is beyond
the scope of this paper.
A192, page 31 of 31
