5 NB: in the ebop notation, I and L mean something different than in the general EB literature.

f (0) = 0.2 + 0.4cos 0 + 0.2cos2 0, sx = cos 0 = sin i cos 0, (6.2.19)

where cos 0 is the direction cosine of the line-of-sight w.r.t. the line joining both stars, and 0 is the true phase angle calculated according to (3.1.19) or (3.1.37). The light of the primary component varies according to

L 'v = Lp + ALp, ALp = Spf (0), Sp = 0.4 ApLsfp,, (6.2.20)

where Sp is the contribution of the illuminated hemisphere of the primary and Ap is the bolometric albedo introduced in Sect. 3.2.5. A similar expression holds for the secondary component. Due to the phase shift of n, a negative cosine term occurs.

The total brightness variation caused by reflection is obtained by adding the contributions of the stars. Because the contribution of the star in front is very small, and only increases when the eclipsed area, a, becomes small, the observer sees the brightness variation

= (1 - a) [ 1 (Ss + Sp) - (Ss - Sp) cos 0 + 1 (Ss + Sp) cos2 0], where A Le is the light reflected from the eclipsed star. Obviously, the reflection effect produces a brightness variation outside eclipse. The Fourier series representation of light variation outside eclipse generally contains the terms cos 0 and cos2 0. According to (6.2.21), the term cos 0 vanishes if the surface brightnesses Ss and Sp are equal. In the case Ss > Sp, the effect of the cos 0 term is that the brightness of the binary system increases when approaching the secondary minimum.

In order to reproduce the brightness at quadrature (0 = 90°), a free normalization parameter mq is added to the least-squares problem. In a system with no significant reflection effect, mq is identical with the luminosity corresponding to the brightness at quadrature.

In practice, EBOP seems to be sufficiently accurate for relatively uncomplicated detached systems with an average oblateness, e < 0.04. This program was attractive in the 1970s and 1980s because of its high integration accuracy, and a computational time which saves a factor of 15-40 when compared with the more sophisticated Wood (1972) WINK program. Therefore it is possible to use it for sensitivity analysis (see Appendix B.2) of small parameter changes.

EBOP is still popular and frequently used [cf. Devor (2005), Devor & Charbon-neau (2006a), Tamuz et al. (2006) for analyzing large number of light curves, or Southworth (2008) studying transiting extrasolar planets]. It has been enhanced by individual users [cf. Southworth et al. (2004a, b, c), and Southworth et al. (2007a, b) who have incorporated the Levenberg-Marquardt algorithm (MRQMIN; Press et al. 1992, p. 678), an improved treatment of limb darkening, and extensive error analysis techniques].

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