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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 twocomponent general analytical form for the probability density function (PDF) of the intrinsic rotational velocity distribution. Following RamírezAgudelo 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 highvelocity tail that can vary from negligible to strong. For example, in the deconvolved rotational velocity distribution of the sample of single Otype stars from the VFTS (RamírezAgudelo et al. 2013), the gamma distribution allowed us to represent a lowvelocity peak and the normal distribution was used to model an additional highvelocity 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, massratio, 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 quasiabsence (<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 randomorientation distribution and the detectedbinary 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 detectedbinary distribution for the binary sample and the randomorientation 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: (A.1)For the sample of primaries of Otype 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.
Fig. A.1
sini distribution for the two different cases considered: a random orientation for the rotation axis, or a rotation axis aligned with the binary axis which is computed for the detected spectroscopic binaries in the VFTS sample. 

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The PDF of the projected rotational velocity, q(ν_{e}sini), is obtained by convolving the intrinsic rotational velocity PDF with the sini distribution in the following way: (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_{e}sini 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): (A.3)where q is as defined above.
Fig. A.2
PDF of the intrinsic (red), projected (green), and observable (blue) rotational velocity distribution for a model with α = 4.82,β = 1 / 25,μ = 205 kms^{1},σ^{2} = ^{(}190 kms^{1}^{)}^{2},I_{γ} = 0.43, and assuming σ_{v} = 20 kms^{1}. Upper panel: sini distribution assuming random orientation of the rotation axes. Lower panel: sini distribution computed for the detected spectroscopic binaries in VFTS is assumed. 

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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 RamírezAgudelo et al. (2013) as providing the best representation of the rotational velocity distribution of the single Otype 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_{e}sini measurements with associated uncertainties σ_{v}). In a Bayesian framework, the posterior probability density can be written as: (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., (A.5)
where ν_{e}sini_{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 affineinvariant ensemble sampler for Markov chain Monte Carlo (MCMC) available through the emcee code (ForemanMackey et al. 2013). We use 100 walkers and run the chains for 3500 steps with a burnin 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.
Fig. A.3
One and twodimensional projections of the posterior probability distributions of the rotational velocity distribution parameters for the sample of primaries of Otype binaries, assuming the sini distribution computed for the detected binaries in VFTS. 

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Fig. A.4
One and twodimensional projections of the posterior probability distributions of the rotational velocity distribution parameters for the sample of apparently single Otype stars, assuming a random orientation of the rotation axis. 

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© ESO, 2015