Fig. 6.4 Basic eclipse geometry. The figure shows the distances £ and p of a point P to components 1 and 2, the distance A between the centers of the stars, and S(A)
where rc denotes the radius of the luminous disk. The luminosity of the system is normalized to unity. See Fig. 6.4 for the geometry. During the eclipse of star p, the flux Ic(p) originating from surface area element da, and beamed into solid angle element d^, is absorbed by the extended atmosphere of star £ to the extent
where t(£) = a(r)dx, where a(r) is the absorption coefficient per unit volume in star £, and r is the distance from the center of this star. The absorbed flux must be reradiated in the extended atmosphere of star £, and this is asserted to be the equivalent of the reflection effect [Goncharsky et al. (1978)]. The light loss due to eclipse is determined by the integration over the overlapping surface SA
L£ + Lp - li(a) = 1 - li(4) = / Ic(p) [1 - e-T(£)] da, (6.4.3)
jsa where ^(4) is the light seen when the star centers are separated by plane-of-sky distance 42 = cos2 i + sin2 i sin2 0, and S denotes the overlap region. During the secondary minimum, the light absorbed by the atmosphere of star p is written, analogously, as
L£ + Lp - ¿2(4) = 1 - ¿2(4) = f Ic(£)[1 - e-T(p)]da. (6.4.4)
Cherepashchuk refers to the quantities [1-e-T(£)] and [1-e-T(p)] as Ia(£) and Ia(p), respectively. The relations among Ic and Ia require two a priori relations in addition to (6.4.3) and (6.4.4). These relations must involve the choice of a model of particular structure for each component. Models of stars of unknown structure will not suf fice. Cherepashchuk suggests two types of models: Classical and semi-classical. The following brief exposition of the treatment of these two cases is intended to convey the flavor of the Russian school's concern about the determinability of light curve solutions. The classical model assumes spherical stars with opaque disks and thin atmospheres. The functions in this model are
__J 1, if 0 < f < rjaJ 1, if 0 < p < rPa,
It is the "standard model" with arbitrary limb-darkening laws described by the functions Ic(f), and Ic(p), found from solving the integral equations (6.4.3) and (6.4.4).
The semi-classical model comprises a classical model star and a "peculiar star" with extended atmosphere. In Cherepashchuk's formulation, the functions Ic(p) and Ia(p) describing the "normal" star are known. The radiative and absorption properties of the peculiar component, viz., Ic (f) and Ia (f), are not assumed a priori but are determined by solving the equations (6.4.3) and (6.4.4). The latter equations determine only two functions which depend on the parameters of the light curve: The stellar radii and the orbital inclination. To permit the determination of other elements, the luminosity normalization equation (6.4.1) must be solved as well. The number of additional equations required to provide system parameters depends on the maximum value of the overlap region, i.e., on whether the eclipses are partial or total. Cherepashchuk considers two cases:
(a) cos i > rp: where rp is the radius of the normal component. In this case, at the moment of conjunction (# = 0), the limb of star p does not reach the center of the disk of component f and the light curve does not contain information about the functions Ic(f) and Ia (f) for the central regions of the f -component, described by the expression: f < cos i — rp. In such a case, even though it is possible in principle to derive the parameters of the normal star, a unique solution of (6.4.1), (6.4.3), and (6.4.4) is not possible for the peculiar binary. In the second case, on the other hand, where
(b) cos i < rp: all parts of component f are eclipsed, and functions Ic(f) and Ia(f) are determined by the troika of equations for all values of f : 0 < f < r^ac. The accuracy of the light curve and the specific values for rp and i determine the efficiency with which (6.4.3) and (6.4.4) can determine the parameters.
On the basis of a "determinability" analysis for the classical and semi-classical models, Cherepashchuk (1971) concluded that unique solutions are determinable only in the following cases:
1. Classical models:
• partial eclipses when for each minimum, cos i < rocc, where rocc is the radius of the eclipsing star.
2. Semi-classical models:
• total eclipse of the peculiar star by the normal component;
• partial eclipses with the condition: cos i < rp, where rp is the radius of the normal component.
3. Semi-classical models including opaque cores:
• total eclipses of the peculiar star by the normal component;
• total eclipses of the normal star by the core of the peculiar star;
Cherepashchuk recommends that external sources of information, such as spectrophotometry light ratios, be used to determine if the cos i < rp condition holds. In order to keep the problem well-posed and therefore to be able to solve the light curve of extended atmosphere systems, it is assumed that any unknown functions, such as the center-to-limb variation, are monotonic and nonnegative.
Perhaps the line of attack is best illustrated by an analysis of the light curves of the WN5+O6 system V444 Cygni, studied in detail by Cherepashchuk and his colleagues. It is described as a typical semi-classical system with an extended atmosphere (disk) around the Wolf-Rayet WN5 component, which is in front at primary minimum. The absorption of the O6 star's light by the disk of the WN5 star is expressed as ip(f) = I0p [1 - e-T«)], (6.4.6)
where ip is the brightness of the O6 component at the disk center, and t(f) is the optical depth of the disk of the Wolf-Rayet component. The data are assumed to be in the form of normal points in a rectified light curve. The adopted formalism requires solutions of a series of equations:
1 - ) = j a, rp)IP(f )df, if cos i < a < % + rp, (6.4.7)
1 - ¿2(0) =J *2(f, a, rp)Ic(f)df, if cos i < a < Rfc + rp, (6.4.9)
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