The light l3 of a third source, usually a third star far away from the binary system, is ordinarily assumed to be independent of time or phase 0. Astrophysical interpretations of third light are discussed in Sect. 3.4.1.

3.3 Modeling Aspect and Eclipses

Exitus acta probat (The end justifies the means)

The computation of I j (0) in (3.2.49) requires not only the stellar surface S' defined in (3.2.1), but in addition, for considering eclipses, the visible stellar surface S":

S" = S"(q, ├čo, F, e; 0) := (rs | rs e S' and rs visible}. (3.3.1)

The attribute "visible" means rs e S' so that the observer receives flux emitted from a point rs. In particular, this requires that rs be on the side of the star facing the observer and that the other component does not eclipse the point rs. The characteristic function, x (rs), summarizes this condition:

This function is the product of two characteristic functions, x A(rs) and x B (rs). The first one is the horizon and self-eclipse function

A )i 1, rs is not beyond the component's own horizon, (3 3 3)

The second one is the companion eclipse function

B(r )._ { 1, rs is not eclipsed by the companion, (3 3

According to the orientation of the normal vector, n, and the line-of-sight vector, s, introduced in Sect. 3.1.1, and as illustrated in Fig. 3.20, for convex surfaces, i.e., positive curvature on the whole surface, x A(rs) can be expressed by the aspect angle y and the relation x v.)=10,er y< (3.3.5)


The situation is more complicated for surfaces that have local negative curvatures such as over-contact equipotentials. In that case, self-eclipses are possible. We have outlined in a formal way how to restrict the range of integration over the visible part, but we also need to explain how x (rs) is computed in real light curve programs. Thus we need a procedure that indicates whether or not a point on a star is visible from Earth. A necessary condition for a surface point rs to be visible is that its associated angle y = y ($) between s and n(rs) computed according to (3.1.16) fulfills the condition cos y < 0. (3.3.6)

Thus, in principle all those points fulfilling (3.3.6) on both stars can be identified and establish our function xA(rs). Depending on phase an eclipse might occur or not. In the trigonometric relations involving the phase, $, we rather need the true phase angle, 0, defined in Sect. 3.1.2. Thus, whenever a term such as sin $ or cos $ occurs, it means rather sin 0 or cos 0, where 0 is computed according to (3.1.19) in the circular case and according to (3.1.37) in the eccentric case. Figure 3.21 shows the eclipse geometry and units. Although treating the eclipse geometry requires many subtle details the basic ideas are simple. The first step is a sort of global checking whether at phase $ an eclipse is possible at all or not. For spherical stars with relative radii r1 and r2, an eclipse occurs only if 5, the projected (plane-of-sky) distance between the centers of the components, fulfills the relation n + r2 R j

5 < , rj , j = 1, 2, (3.3.7) da where 5 is computed according to (3.1.9) in Sect. 3.1.1, i.e.,

52 = (ys)2 + (zs)2 = d2($) (sin2 $ + sin2 i cos2 $). (3.3.8)

plane of sky d sin sin $

plane of sky d sin sin $

Fig. 3.21 Projected plane-of-sky distance. The projected plane-of-sky distance can be used to test whether an eclipse can occur

Within the Roche model the largest value for rj(0, p) might be used as the radius r} and then (3.3.7) might be applied.

If an eclipse is not excluded, the next step is to identify which star is in front (this star cannot be eclipsed at this phase). In Appendix E.21 we describe how this test is performed in the WD program. The next step is to compute and represent the horizon of the front star in the plane-of-sky coordinates; this step is full of tricky details related to numerical accuracy and varies among different light curve programs. Once this representation is available, the grid points representing the surface of the distant star can be tested w.r.t. eclipse by comparing its plane-of-sky coordinates with those of the horizon. This procedure establishes the function x B (rs).

3.4 Sources and Treatment of Perturbations

In omnia paratus (Ready for all things)

3.4.1 Third Light

The light l3 of a third source, usually a third star far away from the binary system,33 is assumed to be independent of time or phase 0. Third light has become more interesting now that companions can be discovered with the new generation of interferometers. The presence of the light of a third star decreases the depths of both eclipses because addition of a constant to a positive function diminishes its "fractional" or "percent" variation. As third light decreases the depth of eclipses it roughly simulates a system with lower inclination. A hot companion may make a greater relative contribution to system flux in the ultraviolet, whereas a cooler companion will be a stronger contributor in the infrared. In the intermediate or far infrared, emission from circumstellar dust can contribute to the background. If the third body has a spectral type different from the close binary components, the added flux will be different in different passbands, and the third light may be modeled to determine the nature and apparent brightness of the third star. In 44i Bootis, third light is contributed by the primary component of a visual binary star system in which the secondary is the EB. The angular semi-major axis is only ~3.8 arc-sec (Linshan et al. 1985), and the separation over the next half-century will not exceed ~2.5 arc-sec.

The third component is brighter than the hotter, more luminous component of the over-contact binary (Hill et al. 1989) by at least a magnitude in V and so contributes a significant amount of "third light," unless the component somehow can be excluded from the measurement. It is also difficult to exclude its scattered light from spectra.

33 The binary system and the third star may establish a gravitationally bound triple system. By "far away," however, we mean compared to the separation of the eclipsing binary components. Typically, the third component may be seen as a very close visual binary with the combined light of the eclipsing system, or may not even be resolved optically.

Sometimes, as is the case for VV Orionis, there is spectroscopic evidence for a third body in the form of disturbances of the radial velocity curve [cf. Scarfe et al. (1994)]. In that particular case, however, Van Hamme & Wilson (2007) have shown that the light and radial velocity curves of VV Orionis can be fitted without third light. It may be possible to demonstrate the existence of a third body by analysis of times of minima. A linear relation between Observed-Computed times of minimum plotted versus time implies a simple correction to the period; a parabola implies a constant rate of period change; and a sinusoid implies apsidal motion34 or variation in arrival time (light-time effect) due to orbital motion of the close binary around the system center of mass (binary plus third body). Apsidal motion can be due to gravitational perturbations (e.g., by a third body), finite nonspherical mass distribution of the stars (not point masses), and general relativistic effects [cf. Quataert et al. (1996)].

In light curve modeling, third light l3 is usually added to the computed light as a constant

The partial derivative 9l/9l3 is therefore constant, namely

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