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1 + q 1 + q where q = M2/M1 is the mass ratio of the binary, and the radius vector, d, within the elliptic orbit, is given by

d = d (0) = .-t-t-- = —-= 1 — e cos E. (3.1.36)

Finally, we can relate u to the (true) longitude in orbit and also to the (true) phase angle v = u — o, 0 = 0 (0) = u + o — 90°. (3.1.37)

So, finally, we coupled the orbital phase 0 and the geometrical phase, 0, through the mean anomaly, M, and Kepler's equation. In the circular case we just had the simple relation (3.1.19).

3.1.3 Spherical Models

De mortuis nihil nisi bonum (Of the dead, say nothing but good)

Diogenes Laertios, I.3n.2, 70

Binaries with two slowly rotating stars sufficiently detached from their limiting lobes are accurately represented by spheres. Stars with radii of the order of 1015% of their separation as an upper limit fall into this category, and main sequence examples are reasonably common.

The model described here is closely related to the Russell-Merrill model (see Sect. 6.2.1 and Appendix D.1) and its modern counterpart, the Nelson-Davis-Etzel (NDE) model by Nelson & Davis (1972). It involves two spherical stars that move on ellipses around the center of mass. In a binary system with spherical components moving on circular orbits we may encounter a situation as illustrated in Fig. 3.10 and in Fig. 4.8 on page 200. The normal vector, n, and p are simply given by n = er, p = 0.

Fig. 3.10 Schematic light curve in the spherical model. The figure shows the light curve of a binary system with spheroidal components moving on circular orbits. The relative orbit of star 2 around star 1 is shown. The inclination is 90°. Note that the light curve has no curvature; however, it would if the figure would be more than schematic. Outside eclipses and during totality the light is constant. Reproduced from Fig. 2 in McVean (1994, p. 7)

Fig. 3.10 Schematic light curve in the spherical model. The figure shows the light curve of a binary system with spheroidal components moving on circular orbits. The relative orbit of star 2 around star 1 is shown. The inclination is 90°. Note that the light curve has no curvature; however, it would if the figure would be more than schematic. Outside eclipses and during totality the light is constant. Reproduced from Fig. 2 in McVean (1994, p. 7)

In the framework of spherical models, the component eclipsed at the deeper minimum is traditionally called the primary component13 and is labeled with subscript p. In most cases the primary is the one with higher surface brightness14 (note that for e = 0 this is not necessarily true; however, exceptions are rare). The secondary star is labeled with subscript s.

In the spherical model, the light curve of an EB depends on

1. the relative radius rp of the primary measured in units of the semi-major axis a of the orbit;

13 Note that in most parts of the book we adopt the Wilson-Devinney convention that star 1 is the one eclipsed near phase zero.

14 Surface brightness has the physical dimension of energy/time/solid angle/wavelength unit/unit area. Surface brightness is intensity as "seen" by the observer as he/she looks at the surface of the object.

3. the fractional luminosity, Lp/(Lp + Ls), of the primary;

4. the inclination i;

5. the center-to-limb variation15 of surface brightness (limb-darkening coefficients xp and xs as in the Russell-Merrill model);

6. the eccentricity e of the orbit and the argument of periastron, m; and

7. third light, l3 (extra light of an optical or physical component).

The distinction between third light, l3, and third luminosity, L3, is commonly ignored in the spherical and ellipsoidal models, leading to some inconsistencies. Whereas the luminosities Lp and Ls of the components are independent of phase, we really want to compare the total phase-dependent flux l with observed light curves. Thus, although l3 is independent of phase, it has to be defined consistently with the phase-dependent fluxes lp and ls and it has to be added to these quantities as is done in (3.2.50).

Thus, the usual convention in spherical and ellipsoidal models, which normalizes luminosity by

has to be carefully checked to keep track of the proper physics. If (3.1.39) is used to normalize luminosity, then Ls need not be specified. Alternatively to Lp, we could also use the mean surface brightness, Js, of the secondary while fixing Jp = 1. Besides numerical reasons related to the modeling of limb-darkening effects, this approach has the following advantage: The ratio of mean surface brightnesses is approximately the ratio of the eclipse depths for stars moving on circular orbits. Using (3.2.31 ) we get the following expression for the unnormalized luminosity:

Lp = 4n (1 - ^ Jpr2p, Ls = 4n (1 - ^ Jsrs2, (3.1.40)

which shows that the luminosity ratio and the surface brightness relation are connected by

In spherical models, the computation of light works as follows: For a given phase, the distance, d, between the centers of the stars is computed according to (3.1.36). Next, the projected distance, 5, follows from (3.1.10). If the eclipse condition (3.3.7) is violated, total light is equal to third light plus the flux received from both components. If it is fulfilled, we have to subtract the amount of light lost due

15 Limb darkening is a physical phenomenon in which the intensity is progressively dimmer toward the limb (edge of the visible disk) of a star. The discussion of limb darkening is postponed to Sect. 3.2.4 but, here, we already use some formulas describing this phenomenon.

to the eclipse. If we neglect limb darkening for a moment and consider only stellar disks of uniform surface brightness, the light loss during eclipse is the product of the surface brightness of the eclipsed star and its eclipsed surface area. The orbital computation allows us to decide which component is eclipsed and which is in front. Once we answer this question the problem is reduced to calculating the area of a segment of a circle, i.e., the area between an arc of a circle and its subtending chord. Analytical formulas for this task are found in Nelson & Davis (1972, pp. 618-619).

If we want to treat limb-darkened stars the light loss during eclipse is the flux integral of the surface brightness of the eclipsed star over its eclipsed surface area. We can follow Nelson and Davis's approach evaluating the stellar luminosities by integrating over concentric limb-darkened rings projected onto the stellar disk. Further details about the NDE model and its associated program EBOP are given in Sect. 6.2.2.

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