Free Access
Volume 550, February 2013
Article Number A93
Number of page(s) 13
Section Stellar structure and evolution
Published online 31 January 2013

Online material

Appendix A: Simple model of V383 Sco

To make the model of the 2007/8 eclipse of V383 Sco, which we describe briefly below, many simplifying assumptions were made. Stars were considered as spheres with radii Rhot, and effective temperatures of photosferes Thot, Tcool for the hot F-type and cool M-type components, respectively. Limb darkening was neglected and stellar fluxes were approximated as black bodies. We describe the main eclipse (of the hot component by the cool supergiant) by taking into account an impact parameter D which measures the projected distance between the centres of stellar disks at the mid-eclipse point, as well as changes in the radius of the cool star in the range  ± ΔR from the mean value of the radius as a result of the pulsations (Fig. A.1). Pulsational changes in the radius and brightness of the cool component were described simply, similar to the approach used for modelling pulsations in Mira χ Cyg by Reid & Goldston (2002). In our model we assume that the changes in magnitudes and radius of the cool star can be expressed with cosine functions where ΔM is the semi-amplitude of brightness changes caused by pulsations (from the mean value), φpul is a pulsation phase, and Δφpul is a phase shift which expresses by how much the moment of maximum radius of the M star precedes the moment of maximum of its brightness. The period and zero moment of pulsation maxima were adopted from the ephemeris (Eq. (4)). Several parameters were treated as fixed, not subject to change in the process of solution. The effective temperature of the hot component Thot was set to 7500 K and its radius Rhot = 58   R was estimated with the Stefan-Boltzmann law by adopting the luminosity Lhot extracted from the SED (see Table 6). The broad, atmospheric parts of the eclipse were included by introducing a light-absorbing envelope that changes the density distribution as a function of distance from its centre r as r-2. Its radius Renv was roughly estimated from the total duration of the atmospheric eclipse and fixed at 440 R. We assumed that the intensity of radiation I, measured after passing through the envelope, changes from an initial value I0 according to (A.3)where τ(r) is an optical depth calculated (A.4)The absorption coefficients κV and κI were chosen via visual comparison of synthetic curves with photometric observational data and were fixed at the values 0.03 and 0.015 for V and I bands, respectively, assuming that the integration constant C is equal to 1. We carried out the solutions for several temperatures of the cool component Tcool in the range from 2600 K to 3500 K with a step of 100 K. For each Tcool value we calculated the star’s radius from the Stefan-Boltzmann law using the luminosity extracted from the SED from the spectrum of RX Boo fitted in the place of the cool component (see Table 6). The adjustable, free parameters were: the moment of mid-eclipse defined as the time at which the centres of stellar disks are at the minimum

separation JD0, the reciprocal tangential velocity of the stars Vt, the semi-amplitude of brightness changes caused by pulsations ΔMV, ΔMI for V and I bands, respectively, and three parameters described earlier: D, ΔR, and Δφpul. To reduce the number of free parameters we combined the semi-amplitudes of ΔMV, ΔMI using an empirical relationship ΔMI ≈ 0.478   ΔMV that we found when analysing the AAVSO photometric data for Miras- and SR-type stars (see Fig. A.2). The best solution, i.e. corresponding to the minimum value of the sum of square residuals, was obtained for Tcool = 2800 K and . The complete set of resulting parameters of the model is shown in Table A.1.

thumbnail Fig. A.1

Schematic explanation for the geometric parameters of the model.

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thumbnail Fig. A.2

Dependence of the pulsation amplitude of Mira/Semi-Regular pulsating stars in the V photometric band on the amplitude in the I band obtained from AAVSO light curves.

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Table A.1

Input (left) and output (right) parameters of the model of the 2007/8 eclipse of V383 Sco.

© ESO, 2013

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