EDP Sciences
Free Access
Issue
A&A
Volume 580, August 2015
Article Number A92
Number of page(s) 17
Section Stellar structure and evolution
DOI https://doi.org/10.1051/0004-6361/201425424
Published online 07 August 2015

Online material

Appendix A: A Bayesian derivation of the intrinsic rotation distribution

Appendix A.1: Analytical form of the νe 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,Iγ) 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: νe 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 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.

thumbnail 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, qesini), 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 νesini measurement given the above model and a measurement uncertainty σv 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.

thumbnail 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,σ2 = (190 kms-1)2,Iγ = 0.43, and assuming σv = 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 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,Iγ = 0.43). For this illustration, we assume a measurement uncertainty of σv = 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,Iγ}) given the data (D: a set of νesini measurements with associated uncertainties σv). 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 νesinii corresponds to each individual measurement with uncertainty σv,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 <Iγ< 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.

thumbnail Fig. A.3

One- and two-dimensional projections of the posterior probability distributions of the rotational velocity distribution parameters for the sample of primaries of O-type binaries, assuming the sini distribution computed for the detected binaries in VFTS.

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thumbnail Fig. A.4

One- and two-dimensional projections of the posterior probability distributions of the rotational velocity distribution parameters for the sample of apparently single O-type stars, assuming a random orientation of the rotation axis.

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