J

In (6.3.21) the symmetry of the Roche geometry is reflected by the factor 4. For detached and semi-detached systems, the function r(0, y) is well defined for all 0 and y. The volumes computed according to (6.3.21) are dimensionless. Multiplication by R3 yields the physical volume. For particular cases other means, such as harmonic, may be used also. As described in Appendix E.30, relation (6.3.21) also can be used to compute the associated Roche potential Qj for an estimated radius rJJ by solving the equation rJ (ß J) = r;. (6.3.22)

The WD program consists of two main programs named LC and DC, and approximately two dozen subroutines further discussed in Appendix D.1.

6.3.6.2 New Features in the 1999-2007 Models

The 2003 version of the WD program followed the first edition of this book. It provided stellar atmosphere approximation functions by Van Hamme & Wilson (2003). The functions are based on model stellar atmospheres by Kurucz (1993). The radiative atmosphere implementation is in terms of photometric bands, with 25 bands now accommodated.

Input for the radiative treatment includes log g (to allow for the handling giants, sub-giants, etc., in addition to main sequence stars) and 19 chemical compositions in addition to effective temperature, Teff. Influences of Teff and log g on radiative output are applied locally on all surface elements (not merely with one correction for the entire star as in some programs). See Van Hamme & Wilson (2003) for information about smooth transitions from atmosphere to blackbody treatment at the limits of the atmosphere tables and for other specifics of the radiative atmosphere application.

In versions of 1998 and earlier using the old stellar atmosphere routine, one could enter any period, P, or orbit size, a, without affecting light curves, as the scaling of observable light (output) from luminosities (input) involved only temperature but not log g or chemical composition. In the newer versions, this is different as log g is derived from GM/R2 (strictly speaking, from local conditions, including effects of rotation and the other star's gravity) with M and R dependent on period and absolute size. Thus, realistic guesses for P, a, and [M/H]s should be used with the new radiative treatment (pure blackbody computations remain unaffected by absolute masses and dimensions). In a non-simultaneous light-velocity solution, the final semi-major axis a from the velocities should be the same as used for the light curves. That condition will be satisfied automatically in a simultaneous solution. The programs LC and DC are made to be mutually consistent but will have different L2s and light if absolute dimensions and masses differ between the two programs. Naturally the foregoing warnings do not apply for blackbody computations, where the programs' light curves are unaffected by absolute masses and dimensions.

Another significant change from earlier versions concerns MODE=3 operations. The parameters A2, g2, x2, and y2 are free parameters, not set equal to Ai, gi, xi, and y1. The reason for this relaxation is that T2 may differ considerably from T1; TU Muscae is a good example of a hot over-contact binary with different temperatures [cf. Andersen & Gr0nbech (1975)]. As A2, g2, x2, and y2 depend on temperature, it would be hard to argue why these parameters should be set equal to A1, g1, x1, and y1.

The WD model in its most recent version (2007, at this writing) includes two major new features (Van Hamme & Wilson 2007):

• an alternative method to derive the ephemeris by considering whole light and radial velocity curves yields the time of conjunction, period, rate of period change, and orbital rotation (apsidal motion)

• light-time and velocity shifts due to a third body (or several of them) lead to six additional adjustable third-body parameters (heliocentric reference time or epoch T0,3b (time of conjunction) and period P0,3b, eccentricity e3b, argument of periastron Q3b, orbital semi-major axis a3b, and inclination i3 of the third-body orbit relative to plane of sky)

• LC and DC now can interpolate (locally) in [Teff, log g] for x and y limb-darkening coefficients from the Van Hamme (1993) tables for any of 19 compositions ([M/H]). Negative values of the control integers LD1 and LD2 activate interpolation while positive values enforce the use of fixed limb-darkening coefficients.

Each of the features is a major step in completing the picture of EB analysis. The essential basis of the third-body procedure [ cf. Wilson (2007, Sect. 3.1) and Van Hamme & Wilson (2007,)] is to fit multiple curves in time rather than phase. The most problematic part of the analysis is the determination of the orbit period of the third body.

Direct distance estimation as described in Sect. 5.1.4 and Wilson (2008) will be available in the 2009 version of the WD program. This is based on absolute, calibrated light curves which allow us to determine the temperatures of both components and to compare absolute, computed distance-dependent flux with the observed flux.

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