M

with [compare term on page 213]

Note that I, the normal emergent intensity at the pole, does not depend on ^ and 0, and thus is extracted from the sum in (6.3.10). Due to (6.3.7) we note, in addition, the relation

As already mentioned in Sect. 4.1.1.4, the WD model provides several modes to specify the geometry of the binary system or to add constraints or relations between parameters. The WD operations are summarized in the following table (the vector pG contains the geometrical parameters):

-1 satisfy X-ray eclipse duration &e = &e(i, q, e, F)

0 approximate the R-M model L1, L2, T1, and T2 uncoupled

1 over-contact binary with T2 = f (T1, pG) = > Qc, T2 = f (T1)

2 L2 is a function of (L1, T1, T2) L2 = L2(L 1, T1, T2)

3 over-contact binary (circular case only) Q2 = Q1, x2 = x1, g2 = g1

4 primary star fills its limiting lobe Q1 = Qc

5 secondary star fills its limiting lobe Q2 = Qc

6 double-contact binaries Q1 = QC1, Q2 = QC2

The function f (applied in mode 1 but not in mode 3)

r /SSpV

Vg1p/

relates the mean polar effective temperatures T1 and T2 such that the local effective temperatures of the stars are equal at their interface on the neck region (Wilson & Devinney, 1973, p. 542). g1p and g2p are the surface gravity accelerations at the poles, and P is the exponent defined in (3.2.16); for over-contact binaries we have p = p1 = p2-

The model described so far is called mode 0, in which the luminosities L1 and L2, as well as the temperatures T1 and T2, are mutually independent. This is similar to the assumptions by Russell & Merrill (1952) in the Russell-Merrill model. In all other modes, except mode -1, L2 is a function of L1 and the temperatures T1 and T2:

From (6.3.16), it follows that the luminosity ratio L2/L1 is a constant depending explicitly on T1, T2, L1, x1, and x2, and implicitly on geometric parameters. Due to (6.3.16) the total flux scales with L1 and thus we can compute the partial derivative as

d L1 L1

According to (6.3.16), the luminosity L2 is no longer a free parameter. The condition expressed by (6.3.17) allows the interpretation that the free parameter L1 acts as a scaling factor to couple the calculated light curve with the observed data. Although many modelers publish only the monochromatic luminosity ratio,8 L2/L1, the individual luminosities are coupled self-consistently to the output fluxes. This point is made obvious by the units of L 1>2 and l($>). For example, the user may understand an input L1 to be in the unit 1033 erg/s/^m. Thus an entered L1 of 6.0000 would mean 6 ■ 1033 erg/s/^m. Corresponding units for l($>) would then be 1033 erg/s/^m/ d2/cm2, where d is the distance of the binary in centimeters. In the program, d is assumed to be equal to the semi-major axis a which in turn is measured in solar radii. Naturally, when dealing with arbitrarily scaled data, we need not worry about these absolute meanings.

The extension of mode 0 by the relation (6.3.16) is realized in all modes above 0. Additional constraints are added in mode 1 and mode 3,9 appropriate for over-contact systems, where «1>2 > «(rc; q, F),

and rc is the coordinate vector of the equilibrium point Lp. In Cartesian coordinates, r

= (xLp, 0, 0)T, i.e., xLp follows from the condition

8 Note that L2/L1 is the monochromatic luminosity ratio and not the bolometric luminosity ratio Lbol/Lbo needed in the computation of the reflection effect. Appendix E.5 shows how to compute L2bol/Lb1ol as a function of L2/L1.

9 Note that from the 2003 version on the wd program, the equalities g2 = g1, A2 = A1, and x2 = x1 are not enforced any longer in mode 3.

Appendix E.12 shows in detail how to compute xLp. In the coordinate system of the secondary star, the equilibrium point Lp has the coordinate xLp = 1 - xLp. (6.3.20)

Li 1

In contrast to mode 3, mode 1 requires, in addition to (6.3.18), the condition that T2 = f (Tp, pG) which allows the modeling of an over-contact system in thermal contact. pG denotes the geometrical parameters. Note that there are different temperatures on the two stars, but the local temperatures are equal where their surfaces meet on the neck of the binary (for idealized thermal contact).

Explicit modeling of semi-detached systems is forced by mode 4 and mode 5 which apply the constraints = Qc or Q2 = Qc, respectively. In mode 4 the primary and in mode 5 the secondary fill its limiting lobe - that is, the Roche lobe - assuming circular orbits and synchronous rotation.

Let rjole, rjoint, rside, and rjack denote, respectively, the local radii of component j, at its pole, in the direction of the other component: In the direction perpendicular to that direction and in the plane of the orbit; and in the direction opposite to the other component. A useful mean radius rj can be defined by means of the volume:

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