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Volume 557, September 2013
Article Number A79
Number of page(s) 14
Section Stellar structure and evolution
Published online 04 September 2013

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Appendix A: Determination of the flux ratios

Appendix A.1: Basics

Let Fk,  k = 1,2 be the line fluxes of the two components of the binary and the corresponding continuum fluxes. The observed composite spectrum normalised to the common continuum of all components is (A.1)KOREL delivers the decomposed spectra normalised to the common continuum of the composite input spectrum, (A.2)The wavelength-dependent ratios of the single continuum fluxes to the total flux are (A.3)We want to calculate the decomposed spectra normalised to the continua of the single components, . Introducing the line depths, and rk = 1 −Rk, we simply get (A.4)For the comparison with the synthetic spectra, Sk, we replace the rk in Eq. (A.4) by sk = 1 −Sk. As a measure of the goodness of fit, we use (A.5)where the stand for the accuracy of the line depths in the decomposed spectra. Since we compare the spectra on the scale of the non-normalised decomposed ones, it will be σ1 = σ2. Setting f1 = 1 −f2 and derivating Eq. (A.5) with respect to f2, we finally get (A.6)from which we can determine the optimum value of f2 (usually by developing f2 into a polynomial of low degree in λ and solving the resulting system of linear equations).

Appendix A.2: The optimum, continuum corrected solution

During the spectrum reduction, the observed composite spectrum is not exactly normalised to the real local continuum but to some pseudo-continuum. We introduce correction factors αk(λ) in the sense (A.7)where are the wrongly normalised decomposed spectra. The αk can be determined from spline fits to the uppermost points in the decomposed spectra and to those in the synthetic spectra Sk. To compute the pseudo-continua on the scale of the non-normalised decomposed spectra, the latter spline fit has to be rescaled according to Eq. (A.4) and we get (A.8)

The occurrence of the so far unknown fk in Eq. (A.8) does not complicate the calculations because the continuum correction can be included into Eq. (A.5) to solve for the optimum values: inserting the continuum-corrected into Eq. (A.5) and derivating with respect to f2 gives (A.9)After solving Eq. (A.9) for f2, χ2 is calculated from Eq. (A.5). This can be done on a grid of synthetic spectra sk to find the minimum in χ2 and the corresponding optimum atmospheric parameters. The renormalised decomposed spectra follow from Eqs. (A.4) and (A.7) to (A.10)

Appendix B: The Roche model

The Roche model is valid for centrally condensed stars where the gravitational potential can be considered as the potential of point masses and, in the standard case, for stars that rotate synchronously with the orbit and with rotation axes perpendicularly aligned to the orbital plane. For reference, we give here the expression for the Roche potential in the reference frame co-rotating with star 1 for the more general case that star 1 rotates non-synchronously (see e.g. Bisikalo et al. 1999): (B.1)All coordinates have their origin in the centre of star 1 and are in units of the separation a between the two stars. x points to star 2, z along the rotation axis of star 1, and y lies in the orbital plane to span a right-handed coordinate system. q = M2/M1 is the mass ratio, the distance to the centre of star 1, the distance to star 2, and xc = q/(1 +q). The synchronisation factor is s = Ωrotorb.

The Roche potential given by Eq. (B.1), multiplied by −1, can be written as (B.2)The shapes of the stars follow from the equipotential surfaces passing through the substellar surface points P: (B.3)The position of the inner Lagrangian point L1 is calculated from setting the derivative dΨ(x,0,0)/dx to zero and solving (B.4)numerically. The radius rs that describes a sphere of the same volume as the critical Roche lobe (the surface of the Roche potential through L1) can be approximated with an accuracy of 1 % (Eggleton 1983) by (B.5)

For a synchronously rotating star, this radius can be used to check if the star fills its critical Roche lobe or not. Instead, we used the filling factor f = P2/L1 as the more accurate criterion, where P2 is the substellar surface point of the secondary component (see Fig. 15). It is f = 1 when the star fills its critical Roche lobe, i.e. when P2 reaches the Lagrangian point L1.

© ESO, 2013

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