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Appendix A: Determination of the flux ratios

Appendix A.1: Basics

Let F_k,  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 r_k = 1 −R_k, we simply get (A.4)For the comparison with the synthetic spectra, S_k, we replace the r_k in Eq. (A.4) by s_k = 1 −S_k. 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 α_k(λ) in the sense (A.7)where are the wrongly normalised decomposed spectra. The α_k can be determined from spline fits to the uppermost points in the decomposed spectra and to those in the synthetic spectra S_k. 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 f_k 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 s_k 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 x_c = q/(1 +q). The synchronisation factor is s = Ω_rot/Ω_orb.

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 r_s 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