F

Here At is the bolometric albedo of the target star specifying the local ratio of the reradiated to the incident energy over all wavelengths. The local reflection factor, R, is the ratio of the total radiated flux (including the fraction due to reflection) and the internal flux according to the gravity brightening law, so that R 1. For atmospheres in radiative equilibrium, and therefore for local energy conservation, A = 1. For stars in convective equilibrium, the albedo may be lower (0 < A < 1) which follows from the thermodynamical requirement that the entropy in deeper convection regions is the same on both irradiated and nonirradiated hemispheres. It is reasonable to follow Rucinski (1969) and set A = 0.5 for convective atmospheres. This value has been derived from computational experiments. We can interpret A = 0.5 as follows: A star with a convective envelope locally reradiates about half of the external heating energy, while the rest emerges from the entire surface.

In order to compute Rt we need to compute the ratio of Fs / Ft of bolometric fluxes. Let T{ be the increased temperature at rl due to the absorbed and reprocessed flux from the other component. Further, let p be the distance between the point, rl, and the center of the irradiating source star; Ft(Tl), the local bolometric flux at rl; Fs, the bolometric irradiance flux from star s received at rl; and At, the bolometric albedo (the fraction of Fs which is "reflected"). Once Tl' is known the monochromatic flux follows from Planck's law or from a model atmosphere.

The computation of T{ is based on the following assumption: The temperature, Tl', and the sum of the effective irradiance and internal flux, AtFs + Ft, are coupled by Stefan-Boltzmann's law according to

Ft V Ft

Ft is computed similarly to (3.2.13). Flux conservation guarantees that

The accuracy of reflection modeling depends on how the incident flux, Fs , is computed. First consider a simple inverse square law treatment, and some corrections for penumbral and ellipsoidal effects. If it is assumed that bolometric flux decreases with the square of the distance p, the incident flux Fs is given by31

where £ is the angle between the direction toward rl and the normal vector n(rl). The ratio Fs/Ft is

In Wilson et al. (1972) a physically more realistic model is presented in which ellipsoidal geometry is assumed for the irradiating star. It is explicitly considered that the irradiating star might only be partially above the local horizon. Relation (3.2.44) for Fs /Ft is therefore modified by two factors E (for ellipsoidal correction) and P (for penumbra correction) which describe a more detailed geometry. In Appendix E.29 a detailed derivation of the relation for E and P is given, which leads to

Ft "bol4np2 |V Q |g where £ is also defined in Appendix E.29. The computation of the ratio of bolometric luminosities is further discussed in Appendix E.5. So, finally, we can compute the reflection factor

Eventually, in Kitamura & Yamasaki (1984) and Wilson (1990), we find an accurate computation of the incident flux integrated over the visible surface of the irradiating

31 This is the case, for instance, if the irradiating star is spherical.

star. In that case (compare Appendix E.25 where e is replaced by ya ) we have an expression of the form f cosy cose

Fs = Fs (r) = / I (cos y ; g, T, A) / 2 da, (3.2.47)

where S'" indicates that we integrate over that part of the irradiating star's surface that is visible from r. Note that all quantities in the integrand of (3.2.47) depend on source star properties.

Figure 3.19 illustrates the reflection effect and how light curves change when the albedo, A1, of the hotter star is varied from 0 to 1.

The impinging radiation is not merely reflected from the receiving star but heats up the impacted surface, which then reradiates32 toward the irradiating star (source star). Multiple reflections are thus needed to treat the effect properly, and the flux at each point of both components must be integrated to consider each subsequent reflection. The process is, of course, iterative because each reflection produces higher temperatures on both initial target and source stars. Iterations are stopped when the multiple reflection computations come to a constant distribution of surface effective temperature, or we might simply ask for a certain number of iterations.

Fig. 3.19 Light curves with albedo varying from 0 to 1. Temperatures were T1 = 20,000 and T2 = 3,600 K. Light curves were produced with Binary Maker 2.0 (Bradstreet 1993) with the albedo A2 varying between 0 and 1

32 The multiple reflection effect is part of the Wilson-Devinney model of the early 1990s (Wilson 1990). Multiple reflection seems to be significant only in a small fraction of binaries.

The multiple reflection effect involves many iterative computations and thus much computing time, it is advised to pay close attention to the structure and logic of the computations (see Appendix E.25).

3.2.6 Integrated Monochromatic Flux

Eventually, the monochromatic flux or light from component j is lj (0) = j x (rs) I (cos y; g, T, A) cos y da, (3.2.48)

where xs is the characteristic function defined in (3.3.2) ensuring that we only consider those points rs, or equivalently, we integrate only over those parts of the stellar surface which are visible to the observer on Earth. In spherical polar coordinates (3.2.48) takes the form n 2n cos y 2

lj (0 ) = j jx (rs ) I (cos y ; g, r,^)^^ r2 sin 0d<pd0. (3.2.49)

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