© ESO, 2015

1. Introduction

The spin rate that a massive star acquires at birth carries information about the poorly constrained processes involved in its formation (e.g., Zinnecker & Yorke 2007; Rosen et al. 2012; Krumholz 2015). Once assembled, the spin of a massive star influences its nucleosynthesis and lifetime and more generally its evolutionary scenario (e.g., Maeder & Meynet 2000; Brott et al. 2011; Ekström et al. 2012; Georgy et al. 2012) and supernova characteristics (e.g., Yoon & Langer 2005; Woosley & Heger 2006; Yusof et al. 2013; Groh et al. 2014). For a review of these topics we refer to Langer (2012).

To date, studies of the spin properties of massive stars have focused on the presumed-single-star population with only some reporting measurements of spectroscopic binaries (e.g., Penny 1996; Penny & Gies 2009). Here we present an analysis of the projected equatorial rotational velocity (ν_esini) properties of the O-type spectroscopic binary population observed in the context of the VLT-FLAMES Tarantula Survey (VFTS; Evans et al. 2011b), a multi-epoch optical spectroscopic campaign targeting 360 O-type objects and hundreds of cooler stars in the 30 Doradus region in the Large Magellanic Cloud. Overviews of the VFTS project and the results obtained so far are provided in e.g., Evans et al. (2011a), de Koter et al. (2011, 2013) and Sana (2013).

For the single O-type stars in the VFTS survey, Ramírez-Agudelo et al. (2013) found a ν_esini distribution rotational peaking at about 80 kms^-1. After deconvolution assuming arbitrary orientations of stellar spin vectors this peak shifts to about 100 kms^-1, corresponding to ~20% of the break-up speed. Because spin-down through stellar winds is limited for the vast majority of the sample, Ramírez-Agudelo et al. (2013) suggested that this peak was representative of the initial distribution of spin rates of single stars, i.e., of the outcome of the formation process.

Dufton et al. (2013) investigated the VFTS spin distribution of non-supergiant early-B stars and found a strong bi-modality. The distribution presents a low-velocity peak at ν_e< 100kms^-1 and a broad and even stronger high-velocity component extending from ν_e ≈ 170 to 350 kms^-1. The low and high rotation velocity regimes in the B-star sample are separated by a clear ν_esini gap extending from 100 to about 170 kms^-1. The strong bi-modality of the spin-rate distribution seen in the VFTS B-star sample is generally not observed in the VFTS single O-type stars (Ramírez-Agudelo et al. 2013). A hint of bi-modality may, however, be present in the O9 distribution and may suggest a smooth transition from the O-star to the B-star regimes. The weakly pronounced second O-star peak is, however, located at a slightly lower projected spin velocity and is not as broad as the one seen in the B-star distribution. Further data would also be needed to validate the statistical significance of such a peak in the O9 star population.

According to the authors, the low spin velocity peak in the B-star distribution may be the result of magnetic spin-down, with long-lived magnetic fields possibly generated through stellar mergers in close binary systems. Since close binaries are unlikely to contain stellar merger products, such a scenario would predict that the fraction of binary components with the lowest spin rates is lower in the spin distribution of binary components.

The observed ν_esini distribution of O stars further presents a declining high-velocity tail at ν_esini ≳ 300 kms^-1. By comparison with the results of recent population synthesis (de Mink et al. 2013), Ramírez-Agudelo et al. (2013) suggested that the high rotational velocity tail may be dominated by spun-up post-interaction objects. Such binary evolution products are indeed expected to be seen as single stars in radial velocity (RV) surveys due to the absence of large RV variations, either because of coalescence of the binary components or because the mass donor has lost so much mass that the induced RV variations of the mass gainer – that dominates the system optical brightness after mass transfer – are below any practical detection limit for spectroscopic binaries (de Mink et al. 2014). The mass gainer may also be truly single in cases where the binary is disrupted by the supernova explosion of the primary component.

The O-type binaries in the VFTS are identified by their RV variations (Sana et al. 2013) and hence will be dominated by pre-interaction systems (de Mink et al. 2014). To first order, this sample provides thus a view on the rotation properties of massive stars which is unaffected by strong binary interactions, either through mass transfer or merging. In this respects, results from the binary sample may thus be confronted to that from presumed-single stars, where binary interaction effects are expected to lead to strong signals (de Mink et al. 2013, 2014).

Furthermore, we may look at the role of tidal interactions in systems with periods in the range 1–10 days. This tends to synchronize the rotation of the components with the orbital motion, leading to a spin-up of stars that at first have modest rotation rates, and to the synchronization of the angular velocities of primaries and secondaries in the same binary (de Mink et al. 2013).

Finally, we can take out the strong binary effects from the single-star spin distribution, that of tides from the binary component spin distribution, and then relate the differences, if any, in the two resulting distributions to differences in the star formation processes of single and binary stars.

This paper is organized as follows. Section 2 introduces our sample. Section 3 describes the adopted methodology to estimate the projected spin rates of the components of the VFTS O-type spectroscopic binaries. Section 4 presents and analyzes the obtained ν_esini distributions. Our results are discussed in Sect. 5 and our conclusions are summarized in Sect. 6.

2. VFTS data and O-type binary sample

The VFTS campaign has been described in Evans et al. (2011b). All spectra discussed here were obtained with the Medusa fibre-feed to the Giraffe spectrograph. The VFTS Medusa sample contains 332 O-type objects. Spectral classification and RV measurements were presented in Walborn et al. (2014) and Sana et al. (2013), respectively.

The method adopted for RV measurements was based on Gaussian fitting and, for each O star in the sample, provided Doppler shift, amplitude and full width at half maximum (FWHM) values for a subset of He i and He ii lines. 116 objects were classified as spectroscopic binaries (SB) on the basis of their statistically significant and large-amplitude RV variations (ΔRV > 20 kms^-1).

The identified O-type binary sample contains 85 single-lined (SB1) and 31 double-lined (SB2) spectroscopic binary systems. Figure 1 shows the spatial distribution of the binary stars. The systems are concentrated in the two clusters, NGC 2070 and NGC 2060 (6.7′ southwest of NGC 2070), although a sizeable fraction is distributed throughout the field of view.

3. Projected rotational velocities

The projected rotational velocity, ν_esini, of stars can be measured from the broadening of their spectral lines (Carroll 1933; Gray 1976). For OB-type stars commonly used methods include the direct measurement of the full width at half maximum (e.g., Slettebak et al. 1975; Herrero et al. 1992; Abt et al. 2002), cross-correlation of the observed spectrum against a template spectrum (e.g., Penny 1996; Howarth et al. 1997), and comparison with synthetic lines calculated from model atmospheres (e.g., Mokiem et al. 2006). Simón-Díaz & Herrero (2007, 2014) apply a Fourier transform (FT) and a line-profile fitting method.

The phase coverage of the VFTS binaries does not allow us to disentangle the spectra of the binary components, hence one cannot easily apply FT or line profile fitting methods. To obtain the projected rotational velocities of the binary sample in a homogeneous way, we thus focus on the FWHM of specific spectral lines. The FWHMs are indeed easily quantifiable and less affected by line blending and/or the presence of faint or undetected companions than more sophisticated methods. The drawbacks are relatively large error bars on individual measurements (see below) and a limited discriminative power at ν_esini_~< 100kms^-1 due to the fact that the method does not disentangle rotational broadening from other types of broadening such as macro-turbulence and Stark broadening.

3.1. FWHM −ν_e sin i calibration

For the single O-type stars in the VFTS sample, we used the projected rotational velocities previously obtained using both a FT and a line profile fitting method (Ramírez-Agudelo et al. 2013). The FT method is able to disentangle rotational broadening from other broadening agents (such as broadening due to macro-turbulent velocity fields) in cases where rotational effects dominate. For stars where rotation does not dominate the broadening, the line fitting method also takes into consideration the presence of additional broadening (for more details see Ramírez-Agudelo et al. 2013).

The He iλ4387, 4922, 4713, and He iiλ4541 lines studied in Ramírez-Agudelo et al. (2013) have also been used for RV measurements (Sana et al. 2013) and thus have both ν_esini and FWHM measurements available. This allows a calibration of the FWHM values in terms of ν_esini.

Figure 2 compares ν_esini vs. FWHM for the helium lines listed above. The number of stars for which this comparison can be made differs per spectral line and depends on spectral type and data quality. The main reason for this is that the lines need to pass a set of quality criteria to ensure reliable measurements, i.e., they should be of sufficient strength relative to the signal-to-noise ratio of the spectrum and to have a ν_esini measurement that is above the resolution limit (Ramírez-Agudelo et al. 2013).

All four lines show a clear trend between ν_esini and FWHM. The scatter in the relation for He ii is considerably larger at relatively low ν_esini than for the three He i lines because the Stark broadening is more pronounced in He ii lines, and because a treatment of this broadening is not properly included in Ramírez-Agudelo et al. (2013).

To calibrate FWHM as a function of ν_esini we obtain a phenomenological description of the trends seen in Fig. 2. We successively adjusted polynomials of order 1, 2 and 3 as well as a power-law function with and without an offset. We perform F-tests to decide whether additional fit parameters result in significantly better representations of the data. All fit functions, except the first order polynomial, give similar quality results. As the power law without an offset has the fewest free parameters, we opted for this representation¹, i.e., ${\nu }_{\mathrm{e}}\sin \mathit{i}\mathrm{=}\mathit{a}\mathrm{\times }{\mathit{FWHM}}^{\mathit{b}}\mathit{,}$(1)where a and b are the fitting parameters.

Table 1 summarizes the best fit values and their uncertainties for each of the lines. The root mean square (rms) residuals are about 15 to 25 kms^-1 for the He i lines and reach 35 kms^-1 for the He iiλ4541 line. In the next section, we lay out our strategy in using these four relations and investigate the uncertainty in the ν_esini determinations that follow from this approach.

3.2. FWHM−based ν_e sin i values for the single-star sample

We first apply the FWHM−ν_esini calibration to the sample of 216 presumably single O-type stars, as this provides an internal consistency test. For four stars (VFTS 400, 444, 565, and 587), no FWHM measurements of any of the four diagnostic lines are available. These have been excluded from our comparison, leaving 212 stars.

In Ramírez-Agudelo et al. (2013) the diagnostic lines have been divided into two groups, one of relatively good quality lines (Group A) and one of lesser quality lines (Group B). The Group A lines include the He i lines at 4387, 4713, and 4922 Å. If these lines are very weak or not present, ν_esini relies on He iiλ4541. These latter measurements are referred to as Group B. We also adopt this division of measurements in two groups in order to decide which lines provide the most reliable value of ν_esini. If more than one He i line is available in a Group A source, we use the unweighted average to obtain the mean projected rotational velocity. As this mean is based on at most three values, the standard deviation tends to underestimate the true dispersion. We thus adopt the rms residuals of Table 1 as error bars for our measurements, unless the standard deviation of a given star is larger. In that case we adopt the computed value.

Figure 3 compares the ν_esini values based on the FWHM−ν_esini calibration with those obtained in Ramírez-Agudelo et al. (2013). The systematic difference is less than 5 kms^-1 and the standard deviation is about 40 kms^-1. For slowly spinning stars, the calibration presented here tends to somewhat overestimate ν_esini. Indeed in this low-velocity regime the FWHM−ν_esini calibration fails to disentangle the contribution of rotational broadening from other broadening mechanisms (see Sect. 3.1). At large velocity (ν_esini  > 400 kms^-1), our FWHM-based measurements tend to underestimate ν_esini compared to values obtained with the FT or profile fitting method. It may suggest the presence of a higher-degree component in the calibrations but our previous tests show that, if present, the latter could not be firmly identified considering our data quality.

Figure 4 compares the ν_esini probability density distribution (top panel) and ν_esini frequency distribution (bottom panel) of Ramírez-Agudelo et al. (2013) with that derived from the FWHM−ν_esini calibration, adopting a bin size of 40  kms^-1 as in Ramírez-Agudelo et al. (2013). The first bin of the new calibration is depopulated relative to the Ramírez-Agudelo et al. (2013) analysis because of the neglect of additional broadening processes. Such extra broadening may affect ν_esini for stars rotating below 150 kms^-1, and also explains the shift of the peak of the distribution to somewhat higher ν_esini values. At velocities above 200 kms^-1, the distribution is not affected in a statistically significant way, and the small (one bin) changes that occur can be explained by the random uncertainties in the two methods.

Importantly, the general shape of the distribution – a peak at about 100 kms^-1 and a high-velocity tail – is preserved using the FWHM−ν_esini method. Despite relatively large individual measurement errors, this method still allows us to statistically investigate the rotational properties of massive stars, including those of binaries for which more sophisticated methods are more cumbersome to apply.

4. The ν_e sin i distribution of spectroscopic binaries

In this section, we construct and investigate the ν_esini distribution of stars in O-type binary systems. Out of 116 primaries ν_esini values could not be established in two cases, VFTS 531 and VFTS 810, as the spectra do not pass the quality criteria described in Sect. 3.1 and thus these two stars were excluded from our analysis.

From here on, we only use ν_esini values obtained using the FWHM−ν_esini calibration, i.e., also for the set of single stars. This allows for a direct and consistent comparison of the single and binary samples.

We also perform statistical comparisons of the ν_esini distribution from different subsamples in order to identify statistically significant signals in our data. We use Kuiper’s test of hypothesis (KP, Kuiper 1960) to search for significant differences between the parent populations of the considered samples. Results are reported in Table 2, where we list the sum D of the absolute values of the most positive and most negative differences between the two cumulative distribution functions that are being compared and the Kuiper probability p_K. Specifically, KP tests allow us to test the null hypothesis that two observed distributions are randomly drawn from the same parent population. This test is similar to the more well-known Kolmogorov-Smirnov test but presents the advantage that it is as sensitive to variations in the tail as it is to variations in the middle of the distributions.

In Table 3 we also present the frequency of stars in a given subpopulation that are spinning faster than the specified limit. We will use these statistics to assess the significance of the presence of a high-velocity tail in these subpopulations.

4.1. Primaries vs. single stars

4.1.1. Observed distributions

Figure 5 compares the ν_esini distributions of presumed-single stars and primaries. There is a surprising overall qualitative similarity between the two distributions. Both are dominated by a low-velocity peak (<200 kms^-1) which contains ~80% of the samples, and show a shoulder at intermediate velocities (200 kms^-1 ≤ ν_esini ≤ 300kms^-1). The distribution of single stars also features a clearly identifiable high-velocity (>300 kms^-1) tail, which is hardly seen in the distribution of primaries.

Though qualitatively similar the main low-velocity peaks are not identical. The main peak in the distribution of the primaries is wider than that of the presumed-single sample and overpopulates the distribution in the region between 100 and 300 kms^-1 by 18%. At large projected spin rates (ν_esini> 300kms^-1), there is a deficiency of very fast rotating primaries with respect to the presumed-single stars. The presumed-single sample harbor 22 stars out of 212 that exhibit projected rotational velocities larger than 300 kms^-1 (10.4 ± 1.9% of that sample) while the sample of primaries only has 3 stars out of 114 (2.6 ± 1.8%, see Table 3). A KP test confirms that parent ν_esini distributions of primaries and of single stars are statistically different (p_K = 3%).

4.1.2. Impact of inclination distributions

Binary systems that are seen closer to edge-on, i.e., that have a large inclination angle, are more easily identified as binaries because of larger projected radial velocity variations. This may introduce an observational bias in the spin rate distribution of binaries relative to that of single stars if the spin axis of the components is preferentially oriented along the orbital rotation axis. For identical intrinsic spin distributions, the observed distribution of single stars might thus be shifted to lower ν_esini than that of binaries, potentially explaining part of the shift that we see in Fig. 5.

An estimate of the most extreme effect of this bias is provided by comparing the deconvolved ν_e distribution of single stars (Fig. 16 of Ramírez-Agudelo et al. 2013) to the observed ν_esini  distribution of binaries in Fig. 5. The latter indeed corresponds to the intrinsic distribution of binaries under maximum detection likelihood corresponding to i = 90° for all systems. Even in these conditions, the main peak of the binary distribution is still broader and shifted towards higher ν_e than the main peak of the single-star distribution. Furthermore, we show in Appendix A that the actual distribution of inclination angle of the binaries in our sample is not sufficiently different to that expected from random orientation in 3D space to significantly contribute to differences seen in Fig. 5. We conclude that the possible bias induced by a preferred binary detection for higher inclination system cannot be invoked to explain the difference of the ν_e distributions in the primaries and presumed-single star samples.

4.1.3. Intrinsic ν_e distributions

In Appendix A, we use a simulated distribution of orbital inclination of the binaries in our sample and a Bayesian method to compute the intrinsic distribution of rotational velocities of the primaries in our sample. We adopt a general analytical form of the rotation velocity probability density function f(ν_e), similar to the one found for single O stars within VFTS (Ramírez-Agudelo et al. 2013), i.e., a combination of a gamma distribution and of a normal distribution, $\mathit{f}\mathrm{\left(}{\nu }_{\mathrm{e}}\mathrm{\right)}\mathrm{=}{\mathit{I}}_{\mathit{\gamma }}\mathit{g}\mathrm{\left(}{\nu }_{\mathrm{e}}\mathrm{;}\mathit{\alpha ,\beta }\mathrm{\right)}\mathrm{+}\mathrm{(}\mathrm{1}\mathrm{-}{\mathit{I}}_{\mathit{\gamma }}\mathrm{)}\mathit{N}\mathrm{\left(}{\nu }_{\mathrm{e}}\mathrm{;}\mathit{\mu ,}{\mathit{\sigma }}^{\mathrm{2}}\mathrm{\right)}\mathit{,}$(2)where

$\begin{array}{ccc}& & \\ & & \end{array}$In Eq. (4), erf is the error function. It is used to renormalize the Gaussian component to unity between x = 0 and ∞. Similarly, I_γ and (1−I_γ) are the relative contributions of both distributions to f(v_e), such that f(v_e) is also normalized to unity when integrated over positive ν_e. Such renormalizations were not implemented in the same way in Ramírez-Agudelo et al. (2013). For consistency in the comparison, we thus also applied the Bayesian analysis method to the single-star ν_esini measurements obtained through our FWHM calibration curves.

Table 4 provides the 50th percentiles as well as the 68% confidence interval of the marginalized posterior distributions of each of the parameters of Eq. (2). It also provides the mean (μ_γ = α/β) and standard deviation (${\mathit{\sigma }}_{\mathit{\gamma }}\mathrm{=}\sqrt{\mathit{\alpha }\mathit{/}{\mathit{\beta }}^{\mathrm{2}}}$) of the gamma-component in the rotation distributions. Projections of the corresponding marginalized posterior probability distributions are shown in Figs. A.3 and A.4. Finally, Fig. 6 compares a histogram of the observed ν_esini data with the obtained best-fit rotational velocity distributions (see Appendix A for details).

For the single-star distribution, the best-fit parameters are different from those obtained by Ramírez-Agudelo et al. (2013). This likely originates from the different analytical forms and fitting methodology used. It may also have origins in the derivation of the ν_esini measurements (now based on FWHM for consistency). Still, the overall shape of the recovered intrinsic rotational velocity distribution is in good agreement (see Fig. A.2), which allows us to validate the method.

The Bayesian analysis allows us to confirm and quantify our previous conclusions:

• (i)

  The peak of the intrinsic rotation distribution(ν_e) is shifted by almost 50 kms^-1 towards larger values in the sample of primaries.

• (ii)

  The main peak is 50% broader in the distribution of primaries (σ_γ = 59 kms^-1 compared to 41 kms^-1 in the single-star distribution).

• (iii)

  The high-ν_esini tail is very limited or altogether absent in the sample of primaries².

Our analysis is the most complete, bias-corrected and consistent comparison of the rotation properties of single and binary massive stars. It reveals small but fundamental difference between both samples, which we will discuss in Sect. 5. We also derive a suitable analytical representation of the intrinsic distribution of rotation rates that can be used, e.g., in population synthesis. In doing so, two aspects have to be kept in mind: (i) the derived distributions are present-day distributions, not initial ones; (ii) the lack of extremely slow rotators (ν_esini_~< 40kms^-1) in the distributions results from the limitation of our ν_esini measurement method. Higher resolution data and, for the binaries, spectral disentangling, would be needed to further our analysis at low spin rates.

4.2. Secondaries vs. primaries

In Fig. 7 we show the distributions for primaries (114 stars) and secondaries (31 stars). Qualitatively, there is no clear difference between both distributions, i.e., the peak is at a similar rotational velocity and both distributions are equally broad. The primary distribution presents two stars with ν_esini  ≥ 400 kms^-1. These two outliers represent less than 2 ± 2% of the total primary sample and do not affect a comparison of the two distributions. A KP test shows that both distributions are compatible with one another (p_K = 95%).

Similarly, Fig. 8 displays the probability density distributions of the primaries and of the secondaries in our sample of 31 SB2 binaries. These systems mostly have large radial velocity variations (ΔRV > 200kms^-1) indicating that the population is dominated by close binaries. A Kuiper test yields D = 0.16 and p_K = 99%. This is thus consistent with the hypothesis of tidal synchronization, assuming the sizes of the accompanying primaries and secondaries are comparable.

4.3. Small vs. large RV variation systems

The size of the sample of primaries allows us to explore ν_esini distributions for subpopulations. By selecting a limit for the amplitude of the radial velocity variations ΔRV_limit of 200 kms^-1 we divide the primary sample into two subsamples, i.e., stars that display ΔRV ≤ ΔRV_limit (henceforth low-ΔRV) and stars with ΔRV > ΔRV_limit (high-ΔRV). The latter are systems for which tidal synchronization is expected to be important during the main-sequence phase. Indeed, given our sample properties and using Monte Carlo simulations described in Appendix A.2, we can show that 88% of systems with ΔRV_limit> 200 kms^-1 have orbital periods less than 10 days (see Sect. 5.1). Most of the systems in the low-ΔRV subsample will not suffer from tides – 75% of them are expected to have P> 10 d and/or q< 0.5. The exception is of course for short-period similar-mass systems that are seen at low inclination angles or for which the RV-curve has been poorly sampled.

Figure 9 plots the ν_esini distributions of the low-ΔRV (85 stars) and high-ΔRV (29 stars) subsamples. At low rotation velocities the distribution of high-ΔRV, compared to the low-ΔRV, appears shifted by one bin (≈40kms^-1) to higher rotational velocities. The high-velocity tail (i.e., ν_esini  > 300 kms^-1) of the low-ΔRV distribution has three primaries (3.5 ± 2.4% of that sample), while there is no star in the high-ΔRV distribution. Despite these apparent differences, a KP test does not allow us to reject the hypothesis that both samples are drawn from the same parent distribution (p_K = 57%), possibly as a result of the limited size of the high-ΔRV sample. The weighted mean of the samples (130 ± 2kms^-1 and 153 ± 5 kms^-1 for the low- and high-ΔRV samples, respectively) are however significantly different from one another, confirming a higher average spin rate of the high-ΔRV (short-period) systems.

4.4. Single stars vs. low-Δ RV binaries

We compare the subsample of low-ΔRV sources with that of the presumed-single stars in Fig. 10. As pointed out in Sect. 4.3 the primary stars in the sample of long-period systems are not likely to interact with their companion during their main-sequence lives (where we have used a low-ΔRV as a proxy for long periods, barring low inclination systems and orbit sampling effects; see Sect. 4.3). As the bulk of the binary systems and the presumed-single systems are on the main sequence, a comparison of the two may thus help to address the question whether or not the primaries in wide (enough) binaries and presumed-single stars leave the formation process with similar spin-rate distributions. In Ramírez-Agudelo et al. (2013) we claimed the low-velocity peak to be the representative of the outcome of the formation process where in (projected) rotational velocity it extended up to ~160 kms^-1. Indeed, for ν_esini ≤ 170 kms^-1 both distributions are very similar and a KP shows that both distributions are compatible with each other (p_K = 76%, see Table 2).

For the binaries with projected rotational velocities larger than 170 kms^-1 both distributions behave differently. The sample of presumed-single stars presents a significant contribution of stars rotating more rapidly than 300 kms^-1, while the wide-binary sample does not.

5. Discussion

The dominant feature in the ν_esini distributions of the O-type primaries and secondaries in 30 Dor is their low-velocity peak. Compared to the rotational distribution of single stars, there is a low occurrence of stars spinning faster than 300 kms^-1 in the distribution of primaries. The well developed high-velocity tail seen in the presumed-single stars is thus nearly absent. In this section we consider possible physical mechanisms that may be affecting the global shape of the initial ν_esini distribution of binary components, notably tidal interaction and mass transfer.

5.1. Signatures of tidal interaction

As discussed earlier, the bulk of our binary sample is expected to consist of pre-interaction systems. de Mink et al. (2013) study the effects of binary evolution on the rotation rates of primaries and secondaries as a function of initial period and initial mass ratio. Prior to the primary star filling its Roche lobe, changes in stellar structure, mass loss and tides may affect the spin rates.

Following Weidner & Vink (2010), we estimate that about 20% of the stars in our total sample have masses above 40 M_⊙, with the most massive stars (accounting for ~4% of the total) being roughly 60–80  M_⊙. It is these stars that develop winds that are strong enough to facilitate significant angular momentum loss during their main-sequence evolution (see Vink et al. 2001). As we focus on the ensemble properties, this implies that the bulk of the stars will not have stellar winds strong enough to spin them down significantly on the main-sequence timescale. Exceptions may be stars with strong large-scale magnetic fields, which we discuss separately below.

Stellar expansion due to changes in internal structure will only have a minor effect on the spin rate when the star approaches the end of its main-sequence lifetime, and can be further neglected (Brott et al. 2011; de Mink et al. 2013). The most important effect on the spin rate before Roche lobe overflow takes over is therefore that of tides. In systems where the radii of the components is comparable to the separation between the stars (de Mink et al. 2013), tidal interaction strives to synchronize the rotation of the binary components with the orbit. For massive binaries this corresponds to systems with orbital periods up to about a week. Once synchronized the rotational velocity of the co-rotating primary increases mainly because of secular stellar expansion.

Like the single O-type stars in 30 Dor, the ν_esini distributions of primaries and secondaries also present a low-velocity peak at around 100 kms^-1, though it is broader and skewed to somewhat higher spin rates (see Figs. 5−8). Under the assumption that the initial spin distributions of binary components and single stars are the same, this difference is consistent with tidal interaction playing a role in our binary sample. Effects of tides and, in a minority of cases, possible slow Case A mass transfer (see Figs. 2 and 3; de Mink et al. 2013), appear most visible in the contrast between the projected spin-rate distributions of low-ΔRV vs. high-ΔRV systems (see Fig. 9), the latter corresponding to closer binaries and displaying larger spin rates (see Sect. 4.3).

To investigate the effect of tidal locking we make use of a diagram showing the relation between projected rotational velocity and amplitude of the RV variations (see Figs. 11 and 12). The timescale of tidal synchronization is a function of the primary mass M₁, the mass ratio of the primary and secondary q = M₂/M₁, and the orbital period P_orb. For our sample we lack information about the latter and therefore we use the amplitude of the radial velocity variations ΔRV as a proxy of the period.

For a (hypothetical) fully synchronized binary system ν_esini and ΔRV are fully determined by P_orb, M₁, the radius R of the primary and q through Kepler’s law. The thick green line in Fig. 11 indicates this relation for a typical system assuming a synchronized orbital and spin period between 10 days (lower left) down to the minimum period at which the primary star would be filling its Roche lobe (upper right). Here we adopted a primary mass of 20 M_⊙, with R = 8 R_⊙, and a mass ratio q = 0.5. The shaded region shows the area spanned by similar relations varying the mass ratio between q = 0.25 (left side boundary of the green area) and q = 1 (right side boundary). The thick gray line and gray shaded area show the same adopting a primary mass of 60 M_⊙  with R = 10  R_⊙. It stretches out to both higher rotation rates because ν_esini scales with the stellar radius which is larger for more massive stars and to larger ΔRV which is proportional to ${\mathit{M}}_{\mathrm{1}}^{\mathrm{1}\mathit{/}\mathrm{3}}$.

Tides are most effective for short-period systems (Zahn 1975, 1977) with a weak preference for equal mass systems. Stars in the upper part of the shaded areas are expected to be fully synchronized as the timescale for synchronization becomes (much) less than 1 percent of the main-sequence lifetime (de Mink et al. 2009). For binary systems with periods larger than about 10 days and/or extreme mass ratios we do not expect tides to have a significant effect during the main sequence evolution.

Forty VFTS primaries fall within the shaded zones. According to the above argument, they are expected to be synchronized or to tend towards synchronization. Given their ν_esini properties, these systems are sufficiently numerous to explain the overpopulation of primaries in the 100−300 kms^-1 range compared to the single-star sample (Sect. 4.1).

For the SB2 systems one can in principle check whether the two stars have synchronized. In Fig. 12 we show the 31 stars for which we have obtained ΔRV and ν_esini for both the primary and the secondary components. Most of the systems fall in the green zone. This is a selection effect as SB2s are most easily recognized if the radial velocity shifts are large (Sana et al. 2011). All primary and secondary pairs have been connected with a straight line. If a pair is synchronized the angular spin velocities are the same. In that case one expects the lines to be horizontal in case of equal sized stars, and about horizontal when allowing for size differences. Indeed, by far the most systems show an about horizontal line connecting the two components and we thus interpret this property as the signature of tidal synchronization. Two distinct outliers are both located in the lower left corner of the diagram, where the timescale for tidal synchronization is the longest.

Using population synthesis de Mink et al. (2013) show that tides are expected to only lead to relatively few systems with velocities larger than 200 kms^-1, and that ~300 kms^-1 is the limit that tides can achieve. Close binary systems that potentially could have acquired spin rates much in excess of 300 kms^-1 at birth would be spun down to ~200 kms^-1. Indeed, this expectation too seems consistent with our findings: none of the components of the close systems (using ΔRV > 200kms^-1 as a proxy for identifying such systems) spin faster than this limit (see Sect. 5.3). When mass transfer occurs later-on in the evolution, the secondary may reach spin rates of up to about 600 kms^-1.

5.2. Magnetic braking

If a massive star possesses a sufficiently strong magnetic field, the interplay of the stellar winds with the stellar magnetosphere may lead to a rapid spin-down of the star. The efficiency of the spin-down depends both on the strength of the magnetic field and on that of the winds (Ud-Doula et al. 2009). For known magnetic galactic O stars, the timescale of magnetic spin-down is on the order of several 0.1 Myr to 3 Myr (Petit et al. 2013). Given the lower metallicity, hence lower wind strengths of stars in the 30 Doradus region, their spin-down timescales will be longer. Using Ud-Doula et al. formalism, we estimated them to be on the order of 6 to 25 Myr for a 1 kG field, and a factor of ten shorter for a 10 kG field.

Possibly up to 10% of the Galactic O stars host a long-lived large scale magnetic field (Morel et al. 2015; Wade et al. 2014). However, only one has a known magnetic field stronger than 10 kG (NGC 1624-2, Wade et al. 2012; Petit et al. 2013), a condition that seems required but rarely met for magnetic braking to efficiently affect LMC O stars. While the detected magnetic fields in Galactic OB stars are believed to be in a stable configuration, Fossati et al. (in prep.) argue that the decay timescales in massive main-sequence stars may be shorter than their lifetimes. If the field decay timescale is longer than the magnetic breaking one (for stars to have time to spin-down), but fast enough for most strong magnetic fields to elude detection, a substantially larger fraction of stars may have been affected by magnetic breaking than what the current detection rate of strong magnetic fields suggests. As a consequence one needs to consider the possibility that magnetic braking may have some impact on our sample.

The origin of the magnetic fields in massive stars is still under debate. One possibility is that of a fossil field that is frozen into a stable large-scale configuration during the star-formation process (Duez & Mathis 2010). Another is that the fields are formed during a strong binary interaction, e.g., a stellar merger (e.g., Wickramasinghe et al. 2014). Merger products are not expected to be found in close binaries, as then the pre-merger triple star would have been highly unstable. This agrees with the fact that the vast majority of magnetic massive stars seems to lack a close binary component, in contrast with their non-magnetic counterparts (Alecian et al. 2015).

If magnetic spin-down is affecting our sample and if magnetic fields are generated by binary interaction, one thus expects a lower number of slowly spinning components in close binaries than in single stars or wide binaries. However, tidal interaction in close binaries leads to synchronization, such that this effect is not detectable. The resulting slowly spinning post-interaction products would mostly show up in the sample of presumed-single O stars. This would move the main peak in their initial spin distribution towards lower spin velocity. A similar effect may affect the wide binary distribution. Indeed, systems born as hierarchical triples may now appear as a wide binary, after the close pair has interacted. The post-interaction products may spin slowly because of magnetic spin-down.

To conclude, this discussion illustrates the possibility that the observed shift of the spin distribution main peaks in the primaries and in the single-star samples may partly be due to magnetic braking in post-binary interaction products. Similarly magnetic braking may also have contributed to the larger average spin rate observed in the short- compared to the long-period systems. To test this scenario, further higher resolution spectra would be necessary as they would enable the search for magnetic spin-down signatures in the form of substructures in the low-velocity part of the spin distributions.

5.3. Nature of the high-velocity tail and the single-star channel towards long-duration GRBs

Ramírez-Agudelo et al. (2013) suggested that the high-ν_esini tail identified in the projected spin-rate distribution of the presumed-single O-type stars in the VFTS is populated by binary interaction products that have been spun up through mass transfer or coalescence. A test of this scenario is to determine the projected spin-rate distribution of primaries and to check whether a high-velocity tail is present or not. If the tail in the sample of presumed-single stars is dominated by post-interaction products one may anticipate that it is less pronounced or even absent in the binary sample. Indeed tides in pre-interaction binaries cannot spin the components up beyond 300 kms^-1 (see Sect. 5.1). We find that 2.6 ± 1.8% of the primaries spin faster than 300 kms^-1, compared to 10.4 ± 1.9% in the presumed-single sample. This supports the hypothesis that the high-velocity tail is actually dominated by post-interaction products and that genuine single stars spinning faster than 300 kms^-1 are rare and maybe absent altogether.

Rapidly spinning single stars have been considered as a potential channel towards the production of long-duration gamma-ray bursts (GRBs; Yoon & Langer2005; Woosley & Heger2006). If stars are born with rotational speeds of at least 300−400 kms^-1, depending on metallicity, they will evolve quasi-chemically homogeneously (e.g., Maeder 1987; Brott et al. 2011), i.e., they remain compact throughout their hydrogen burning phase and become Wolf-Rayet stars directly afterwards. In this way their stellar cores may retain enough angular momentum to fulfill the requirements of the collapsar model towards GRBs (Woosley 1993), especially in low-metallicity environments where stellar winds of Wolf-Rayet stars are weaker, hence remove less angular momentum (Vink & de Koter 2005).

If rapidly spinning single stars are rarely formed or not formed at all this particular channel towards GRB progenitors may not occur in nature. It is however too early to tell whether or not this implies that single stars may not produce GRBs altogether, as stars with low magnetic torques (Woosley 2011) or core collapse early-on in the Wolf-Rayet phase (Vink et al. 2011; Gräfener et al. 2012) might provide alternative routes. Spun-up mass gainers in close binaries might evolve chemically homogeneously and provide a GRB at the end of their life (Cantiello et al. 2007), such that the total number of GRBs predicted from chemically homogeneous evolution does not necessarily change.

5.4. The nature of the low-velocity peak and the formation of massive stars

The low-ΔRV sources are mostly SB1 systems with relatively long orbital periods (with probable periods ~10−1000 d) and/or low mass ratio systems, neglecting inclination and sampling effects. They are likely dominated by pre-interaction systems that do not undergo significant tidal synchronization (Fig. 11). The few systems that are located in the region of the parameters where tidal interaction is expected to take place (shaded areas in Fig. 11) occupy the lower left corner, where the tidal synchronization timescale is the longest. Moreover, the stellar winds of the bulk of this sample are too weak to induce significant angular-momentum loss, and spin-down due to envelope expansion as a result of secular evolution may be neglected. This points to the idea that the spin rates of the low-ΔRV, wide and low-q spectroscopic binaries have seldom been modified since the stars were formed and thus reflect the outcome of the formation process.

While the formation process of massive stars is heavily debated (e.g., Zinnecker & Yorke 2007; Tan et al. 2014; Krumholz 2015), most theories agree on the need for disk-mediated accretion. In its simplest form, the collapse of the initial natal cloud and the gas-accretion phase concentrate by far more angular momentum in the central region than can be stored in a single star. Therefore, the spin rate of genuinely single massive stars may naively be expected to be close to the star break up velocity (the so-called angular momentum problem). Gravitational torques between star and disk appear capable of limiting the stellar spin rates to about half the break-up speed ν_crit (Lin et al. 2011). This value is however still substantially higher than the 0.1−0.3 ν_crit that is typical for the observed low-velocity peak (e.g., Conti & Ebbets 1977; Huang & Gies 2006; Penny & Gies 2009; Ramírez-Agudelo et al. 2013). Magnetic breaking might aid in a further spin-down of the stars, either by self-induced fields or by fossil fields. Alternatively, the formation of massive stars as part of a multiple system may provide a reservoir to store angular momentum from the natal cloud into the orbit of the components. This too may alleviate the need to have a large initial stellar spin.

The low-velocity peak of the presumed-single stars studied by Ramírez-Agudelo et al. (2013) may also reflect the initial conditions. Here we compare the low-ν_esini regime of the two samples to assess whether the spin distribution of presumed-single stars is similar to that of low-ΔRV primaries in binary systems. A Kuiper’s test selecting the stars having ν_esini ≤ 170 kms^-1 in both samples fails to identify significant differences and we henceforth accept that these samples may be drawn from the same parent distribution.

The resemblance of the low-velocity peak in the ν_esini distributions of the long-period and/or low-q spectroscopic binaries and the presumed-single stars thus suggests similarities in their formation process. This could be the case if primordial magnetic fields control the birth spin properties of massive stars, be they either single or part of a binary system. If the initial spin properties are controlled by multiplicity, i.e., by a formation process that proceeds through the creation of pairs or higher order multiples, the similarity in the spin distributions of presumed-single stars and primaries might suggest that true single stars do not actually exist. Interestingly, a high angular resolution survey of galactic massive stars combined with spectroscopic results from the literature has revealed an extreme multiplicity fraction (>90%), suggesting that all massive stars may form as part of a binary or higher order system and that higher hierarchical systems are not uncommon (Sana et al. 2014). Therefore, genuine single massive stars may not exist.

6. Conclusions

Based on a sample of presumed-single stars in the 30 Doradus region in the LMC presented by Ramírez-Agudelo et al. (2013) we have constructed a calibration that allows to estimate the projected spin rates of O-type stars from FWHM measurements. We have applied this ν_esini–FWHM calibration to determine the projected spin rates of the components of 83 SB1 and 31 SB2 O-type binary systems in 30 Dor.

Given the uncertainties in individual measurements (20−40 kms^-1) we have focused on the spin-rate distribution, and have compared it to that of presumed-single stars. The sample is large enough to define subsamples that could be compared to each other. This has lead to the following conclusions:

• 1.

  The overall spin distribution of the O-type binary componentsqualitatively resembles that of the single O-type stars in the30 Dor region (see Ramírez-Agudelo et al. 2013). Both aredominated by a low-velocity peak (<200 kms^-1) which contains ~80% of the samples, and both show a shoulder at intermediate velocities (200 kms^-1 ≤ ν_esini ≤ 300kms^-1). However, the spin distribution of presumed-single stars features a clearly identifiable high-velocity tail (>300 kms^-1), which is hardly populated in the binary population.

• 2.

  The main peak of the binary distribution, centered at ~100 kms^-1, is broader and skewed to somewhat higher spin rates than in the single-star distribution. Similarly, the spin distribution of primaries in systems with ΔRV > 200kms^-1, which we associate with close binaries with P_orb~< 10 days, is skewed towards higher spin rates than that of primaries in the presumably wider systems with lower ΔRVs.

• 3.

  About one third of our binaries are so close that the spins of their components are expected to be tidally synchronized with the orbital motion. The current spin rates in such close binaries are thus expected to be independent of their birth spins. Tidal synchronization may be responsible for the larger average spin rates that we observe between components of close binaries and that of single stars and of stars in longer period systems. A part of this shift might also result from magnetic braking affecting a fraction of the single-star and long-period binary population. The latter scenario would be consistent with magnetic fields being generated by strong binary interaction in an earlier epoch of evolution of these systems.

• 4.

  The SB2 systems in our sample mostly show radial velocity variation amplitudes above 200 kms^-1 and have tidal synchronization timescales that are shorter than their main-sequence timescale. On average, we find that the ν_esini values of SB2 primaries and secondaries are similar, as expected for tidally synchronized binaries.

• 5.

  Among the primaries of our binary systems, stars spinning faster than ν_esini = 300 kms^-1are about four times as rare as in the single-star population, in relative terms. This is consistent with the hypothesis, proposed by Ramírez-Agudelo et al. (2013) and de Mink et al. (2013), that this tail in the (presumed) single-star distribution is dominated by post-interaction binary products. In particular, the population of wide binaries shows hardly any components spinning faster than 300 kms^-1  suggesting that massive star formation rarely produces such systems, if at all.

• 6.

  The low-velocity peak in the sample of primaries in wide spectroscopic binaries that orbit their companion at distances in the range 1–10 AU, is very similar to that of the presumed-single stars. This, together with the overall qualitative similarity in the spin distribution of single stars and binary components, and plausible scenarios for even explaining the quantitative differences through binary evolution, suggests that the spins of massive stars are set independently of whether they form in single stars or binary systems and are controlled by similar physics.

The results presented in this work focus on the overall spin distribution of our binary O-star sample. A detailed analysis of the properties of multiple systems – yielding periods rather than ΔRV as proxies of these periods – and their relation to stellar parameters is a logical next step in assessing the role of tides and mass-transfer in binary systems and the differences and similarities in the intrinsic rotational properties of single and binary systems. Higher spectral resolution data and an emphasis on metallic lines would also offer the possibility of obtaining more accurate measurements in the low spin regime. These are needed to investigate possible sub-structures in the low-velocity peak and their relation to, e.g., magnetic spin-down.

Online material

Appendix A: A Bayesian derivation of the intrinsic rotation distribution

Appendix A.1: Analytical form of the ν_e distribution

As described in the main text, we adopt a two-component general analytical form for the probability density function (PDF) of the intrinsic rotational velocity distribution. Following Ramírez-Agudelo et al. (2013), the adopted PDF is composed of a gamma distribution and a normal distribution (see Eq. (2)). The five free parameters in Eq. (2) (α,β,μ,σ,I_γ) also allow us to represent PDFs with a variety of shapes, e.g., a velocity peak with varying skewness or peak location, and a high-velocity tail that can vary from negligible to strong. For example, in the deconvolved rotational velocity distribution of the sample of single O-type stars from the VFTS (Ramírez-Agudelo et al. 2013), the gamma distribution allowed us to represent a low-velocity peak and the normal distribution was used to model an additional high-velocity contribution.

Appendix A.2: sin i distribution

We use the intrinsic orbital parameters distributions of the VFTS binary sample and the VFTS binary detection probability as a function of orbital period, mass-ratio, eccentricity, sampling and accuracy of the RV measurements obtained in Sana et al. (2013) to compute the intrinsic distribution of the orbital inclinations of the detected binaries in the VFTS sample. The obtained inclination distribution shows a quasi-absence (<5%) of systems with i< 20° and an over abundance of systems with i> 50° compared to a distribution computed with random orientation of the binary plane in the 3D space (Fig. A.1). While the difference between the random-orientation distribution and the detected-binary distribution are real, the overall effect is small enough that we can show that – within the quality of our data – our results do not depend of which distribution is adopted. For consistency, we nevertheless proceed by adopting the detected-binary distribution for the binary sample and the random-orientation one for the single stars.

Appendix A.3: ν_e sin i distribution

Given a PDF for the intrinsic rotational velocity distribution, the associated projected rotational velocity distribution will depend on the distribution of orientations of the rotation axis with respect to the line of sight. In what follows, we consider two different PDFs for the sini distribution. The PDF for a case where the rotation axes are randomly oriented can be written as: $\mathit{h}\mathrm{\left(}\sin \mathit{i}\mathrm{\right)}\mathrm{=}\frac{\mathrm{1}}{\mathrm{(}\mathrm{1}\mathit{/}{\sin }^{\mathrm{2}}\mathit{i}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathrm{1}\mathit{/}\mathrm{2}}}\mathrm{·}$(A.1)For the sample of primaries of O-type binaries from VFTS, we also consider the PDF of sini computed for the detected spectroscopic binary systems in the VFTS (see Sect. A.2). Both sini distributions considered are actually very similar, as shown in Fig. A.1.

The PDF of the projected rotational velocity, q(ν_esini), is obtained by convolving the intrinsic rotational velocity PDF with the sini distribution in the following way: $\mathit{q}\mathrm{\left(}{\nu }_{\mathrm{e}}\sin \mathit{i}\mathrm{\right)}\mathrm{=}{\mathrm{\int }}_{{\nu }_{\mathrm{e}}\sin \mathit{i}}^{\mathrm{\infty }}\mathit{f}\mathrm{\left(}{\mathit{v}}^{\mathrm{\prime }}\mathrm{\right)}\mathit{h}\left(\frac{{\nu }_{\mathrm{e}}\sin \mathit{i}}{{\mathit{v}}^{\mathrm{\prime }}}\right)\frac{\mathrm{1}}{{\mathit{v}}^{\mathrm{\prime }}}\hspace{0.17em}\mathrm{d}\mathit{v}\prime$(A.2)where f is as defined in Eq. (2), and h can be either of the two sini PDFs discussed above. The likelihood of an individual ν_esini measurement given the above model and a measurement uncertainty σ_v is then obtained by the convolution of the above PDF and a Gaussian distribution (assuming normally distributed errors): ${\mathit{l}}_{\mathrm{obs}}\mathrm{\left(}{\nu }_{\mathrm{e}}\sin \mathit{i}\mathrm{\right)}\mathrm{=}{\mathrm{\int }}_{\mathrm{-}\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{1}}{\sqrt{\mathrm{2}\mathit{\pi }}{\mathit{\sigma }}_{\mathit{v}}}\exp \left(\frac{\mathrm{-}\mathrm{(}{\nu }_{\mathrm{e}}\sin \mathit{i}{\mathrm{)}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\sigma }}_{\mathit{v}}^{\mathrm{2}}}\right)\mathit{q}\mathrm{(}{\nu }_{\mathrm{e}}\sin \mathit{i}\mathrm{-}{\mathit{v}}^{\mathrm{\prime }}\mathrm{)}\mathrm{d}{\mathit{v}}^{\mathrm{\prime }}\mathit{,}$(A.3)where q is as defined above.

To illustrate the effect of inclination and measurement uncertainties on the rotational velocity distribution, we show in Fig. A.2 the intrinsic, projected, and observable velocity distribution for a chosen set of parameters that were identified by Ramírez-Agudelo et al. (2013) as providing the best representation of the rotational velocity distribution of the single O-type stars in VFTS (α = 4.82,β = 1 / 25,μ = 205 kms^-1,σ = 190 kms^-1,I_γ = 0.43). For this illustration, we assume a measurement uncertainty of σ_v = 20 kms^-1. We show an example adopting each of the sini distributions discussed above. As expected, in both cases, the projected rotational velocity distribution is shifted to lower velocities due to the effect of inclination, and observational errors slightly broaden the distribution.

Appendix A.4: Bayesian analysis and posterior sampling

We are interested in the posterior probability density function of the model parameters (Θ = ^{α,β,μ,σ,I_γ^}) given the data (D: a set of ν_esini measurements with associated uncertainties σ_v). In a Bayesian framework, the posterior probability density can be written as: $\mathit{p}\mathrm{\left(}\mathrm{\Theta }\mathrm{\right|}\mathit{D}\mathrm{)}\mathrm{=}\frac{\mathrm{1}}{\mathit{Z}}\mathit{p}\mathrm{(}\mathit{D}\mathrm{\left|}\mathrm{\Theta }\mathrm{\right)}\mathit{p}\mathrm{\left(}\mathrm{\Theta }\mathrm{\right)}\mathit{,}$(A.4)where p(D | Θ) is the likelihood function and p(Θ) is the prior distribution. The normalization Z = p(D) is independent of Θ for a given choice of the form of the generative model, so for the problem that we are interested in here, we can simply sample from p(Θ | D) without computing Z. Our likelihood function is simply the product of individual likelihoods as given by Eq. (A.3), i.e., $\mathit{p}\mathrm{\left(}\mathit{D}\mathrm{\right|}\mathrm{\Theta }\mathrm{)}\mathrm{=}\underset{\mathit{i}}{\mathrm{􏽙}}{\mathit{l}}_{\mathrm{obs}\mathit{,i}}\mathrm{(}{\nu }_{\mathrm{e}}\sin {\mathit{i}}_{\mathit{i}}\mathit{,}{\mathit{\sigma }}_{\mathit{v,i}}\mathrm{;}\mathit{\alpha ,}\mathit{\beta ,}\mathit{\mu ,}\mathit{\sigma ,}{\mathit{I}}_{\mathit{\gamma }}\mathrm{)}\mathit{,}$(A.5)

where ν_esini_i corresponds to each individual measurement with uncertainty σ_v,i. We assume uniform priors for all five model parameters over the following ranges: 0.1 <α< 10, 0 <β< 0.4, 0 <μ< 500, 30 <σ< 450, 0 <I_γ< 1.

Finally, to sample efficiently from the posterior distribution and obtain a sampling approximation to the posterior PDF, we use the Python implementation of the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) available through the emcee code (Foreman-Mackey et al. 2013). We use 100 walkers and run the chains for 3500 steps with a burn-in phase of 500 steps. Our final results for the parameters of the rotational probability density functions for the primaries and for the single stars are given in Table 4 and illustrated in Fig. 6.

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Acknowledgments

The authors thank Sergio Simón-Díaz for constructive discussions. S.d.M. acknowledges support by NASA through an Einstein Fellowship grant. V.H.B. is grateful to the “Fonds the recherche du Québec − Nature et technologies” (FRQNT) for financial support through a postdoctoral fellowship grant. The authors are grateful to the anonymous referee for comments and suggestions that helped improve the quality of the paper.

References

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