Volume 582, October 2015
|Number of page(s)
|Stellar structure and evolution
|30 September 2015
Frequencies for Star A.
Frequencies for Star B.
Mode linewidths, mode heights and mode amplitude for Star A.
Mode linewidths, mode heights and mode amplitude for Star B.
Speckle interferometric data of the orbital position of the binary.
The coordinates of the orbit on the plane of the sky (x,y) are derived as follows: where A,B,F,G are the Thiele-Innes elements, and (X,Y) are the position of the object in the plane of the orbit. The Thieles-Innes elements are related to the semi-major axis a, to the inclination of the orbit i on the plane of sky and to the argument of the periastron (ω) and the longitude of the ascending nodes (Ω) as: The position of the orbit in the plane of the orbit is given by: where E is the eccentric anomaly and e is the orbit eccentricity. The eccentric anomaly can be found by solving the following equation: (B.9)where T0 is the orbit periodicity and T1 is the time of passage at periastron.
For the derivation of the orbit, we computed the log likelihood of the data D given the orbital parameters (=(Ω,ω,i,a,e,T1,T0)), as: (C.1)where D are the data, and are the output of the astrometric model of Appendix A at time ti, and and are the observations at time ti, and the σi are the estimated errors of the observations. There are two approaches used for deriving the orbital parameters:
Frequentist: minimisation of the log likelihood
Bayesian: Monte Carlo Markov Chain using the Metropolis Hasting algorithm.
The first approach is equivalent to the use of a non-linear least square fit. The error bars are derived using the inverse of the Hessian matrix. This is the classical way of estimating parameters from a model and observation. The error bars were also verified using a Monte-Carlo simulation of the orbit. When the estimated parameters are close to a minimum, the error bars from the Hessian and the Monte-Carlo simulations are similar (See Appourchaux 2014).
Unfortunately, the error bars returned by the two methods gave inconsistent results which is why we implemented the second approach which is based upon a Bayesian framework. In that case, we derived the posterior probability () of the parameters using Bayes’ theorem as: (C.2)where is the a priori probability of the orbital parameters, and P(D) is the global normalisation likelihood. The derivation of the posterior probabilities can be done using the Metropolis Hasting algorithm (See as a starting point, Appourchaux 2014). We use a Markov Chain for exploring the space to go from a set to another set , assuming that either set have the same probabilities, i.e. . The Metropolis-Hasting algorithm then requires that we accept the new set using the following ratio: (C.3)This is simply the ratio of the likelihood given in Eq. (B.1). The new value is accepted if r<α with probability α (drawn from a uniform distribution) otherwise it is rejected.
We set 10 chains of 10 million points each, with the following starting points taken randomly: (C.4)The new set of parameters is computed from a random walk from the previous value as: (C.5)where is given by a multinomial normal distribution with independent parameters: (C.6)where αrate is an adjustable parameter that is reduced by a factor 2 until the rate of acceptance of the new value t′ is above 25%. The proper convergence of the chains was verified using the Gelman-Rubin test (Gelman & Rubin 1992) as implemented by Ford (2006). After rejecting the initial burn-in phase (10% of the chain), all values of the test of the 7 sets of parameters were below 1.1. Then the chains provide the posterior probability for each parameter. For all chains of each parameter, we computed the median and the credible intervals at 16% and 84%, corresponding to a 1-σ interval for a normal distribution. The advantage of this
percentile definition over the mode (maximum of the posterior distribution) or the mean (average of the distribution) is that it is conservative with respect to any change of variable over these parameters.
Knowing the distance (via the parallax π), the semi-major axis a and the period of the system T0, we can deduce from the Kepler’s third law the total mass of the binary system in units of the solar mass as (D.1)where π is in mas. The error bars can be computed assuming that the semi-major axis, the orbital period, and parallax are independent of each other as: (D.2)In order to derive the corresponding credible intervals for the mass of the system, we use a Monte-Carlo simulation using the chains for the semi-major axis and the period from our Bayesian analysis, and use a randomised parallax as inferred from van Leeuwen (2007) for HIP 93511. The use of the MCMC chains explicitly includes the correlation between the orbital period and the semi-major axis for the final error propagation. The three values were then injected in Eq. (D.1) for getting the median and the credible intervals.
© ESO, 2015
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