This subroutine now works also for very extreme mass ratios, e.g., q = 10
FOUR computes a Fourier series for the representation of the boundaries of the eclipsed regions. In the 1998 and later versions this subroutine is replaced by FOURLS.
FOURLS computes the Fourier coefficients by solving a least-squares problem. It replaces subroutine FOUR present in versions older than 1998. The new subroutine fits the horizon points by least squares and avoids the Fourier approach. It is more accurate and the sorting routine is no longer needed.
E.15 GABS - Polar Gravity Acceleration
GABS computes the polar acceleration due to effective gravity in cm2/s.
Subroutine JDPH allows to convert phase into Julian date but also Julian Date into phase. It computes a phase (phout) based on an input JD (xjdin), reference epoch (t0), period (p0), and dP/dt (dpdt). It also computes a JD (xjdout) from an input phase (phin) and the same ephemeris.
This subroutine solves Kepler's equation (3.1.28) with the Newton-Raphson scheme (see Appendix E.12) with initial value E(0) = M. Iterations are stopped when Ej < 10-10. Eventually, the true anomaly is computed according to (3.1.27). If u < 0, then 2n is added to ensure that 0 < u < 2n.
These are the main programs of the Wilson-Devinney program. LC solves the direct problem: From a given set of parameters, phased light and radial velocity curves, spectral line profiles, star dimensions, or sky coordinates for producing images are computed. DC solves the inverse problem: The parameters are derived from observations. The structure of the main programs is illustrated in Figs. E.2 and E.3.
E.19 LCR - Aspect Independent Surface Computations
This subroutine (see Fig. E.4) oversees all aspect-independent computations of the stellar surfaces by calling other subroutines in proper sequence. This includes the shapes of the components and the potential gradient on both stars. It updates those quantities if they change in an eccentric orbit. In that case, LCR computes the volume from the potential Qp at periastron and finds the phase-specific potential for that volume. Then, the polar temperature is computed by (3.2.19) from the average surface effective temperature. Eventually, LCR calls subroutine LUM, and either OLUMP or LUMP.
Based on the recursive relationship this subroutine evaluates the Legendre polynomials used in the atmosphere calculation. LEGENDRE reconstructs values for intensity from the precomputed coefficients stored in atmcof.dat and atmcofplanck.dat.
E.21 LIGHT - Projections, Horizon, and Eclipse Effects
Subroutine LIGHT performs the aspect computations involving the projections, horizon, and eclipse effects and summation over the visible surface S" of each star. The WD program uses a normal vector, n, pointing inward and a line-of-sight vector, s, pointing from the binary to the observer. In the first part of subroutine light there is a test which decides which star is in front. If the phase 0 is close to 0.5,
then the primary is in front, otherwise the secondary is in front. For the star in front, subroutine LIGHT checks all grid points for cos y < 0.
For the orientation of n and s reviewed above, the grid point is visible if the condition is fulfilled. The whole horizon is then represented by an array (0H, pH), the nearest points to the horizon. These points are identified when integrating the star in front and detecting a change in the sign of cos y. LIGHT fits a Fourier representation p(0) of the horizon by least-squares in subroutine FOURLS.
The star in the background, possibly eclipsed by the star in front, is treated similarly. Visible grid points are identified (E.21.2). A quick test is then applied to ascertain if a visible point is eclipsed. This test is followed by a fine-tuned test using p (0) which is improved by a fractional area correction if the boundary lies between adjacent grid points.
The related program WD93K93, developed in Calgary, integrates the flux by Simpson integration for points visible on the "distant" star.
E.22 LimbDark - Limb Darkening
LimbDark supports the computation of local limb darkening by interpolating in Van Hamme's (1993) band-specific limb-darkening tables.
Added in the 1998 version of the WD program, subroutine LinPro computes spectral line profiles of absorption and emission lines are generated for MPAGE=3 in LC (not DC). The profiles are for rotation only, although other broadening mechanisms may be added later. Blending is incorporated, including blending of mixed absorption and emission lines. Lines can originate either from an entire star or from designated sub-areas of the surface, as explained below. Spectra are formed by binning, with the spectra of the two stars formed separately. The user can add them (weighted by observable flux) if spectra of the binary are needed.
The computation of the flux from each surface element is based on the local intensity. In order to integrate star brightness (in subroutine LIGHT) we need to know the normal intensity at a reference point on the surface, ordinarily the pole. However, the input parameter to the WD program is the luminosity in units determined by the user. Subroutine LUM accomplishes the necessary inversion such that luminosity becomes input and reference intensity is output. It uses (6.3.7) to compute the polar normal intensity Ij required to yield the relative monochromatic luminosity when the local fluxes are suitably integrated over the surface. For both components LUM also computes and stores the local bolometric and monochromatic ratios Gj (rs) of normal intensities at all local points rs to that at the pole according to (6.3.8). LUM also implements model atmosphere corrections.
As the multiple reflection effect involves many iterative computations it is strongly recommended to pay some attention to the structure and logic of computations. As in Wilson (1990) we now put the formulas in a more symmetrical form and label the stars A and B. The effective irradiance fluxes are denoted by primes (F'), while the
"intrinsic" fluxes (those which would exist in the absence of the reflection effect) are unprimed.
Let us start by computing the irradiance flux FB' from component B received at a surface point on component A. Combining ( 6.3.10) and (3.2.47) we get x—v x—v i cos yA cos YB 2 1
pgi P cos fo J
where y denotes, at a given surface point, the angle between the local surface normal and the line-of-sight to a given surface element on the other star, and p is the distance between that point and the surface element. If, in the common coordinate system, the surface point of component A and the surface element of the other star have the coordinates rA and rB, cos ya follows simply as cos YA = nA ■ rB-—, p = |rB - rA|. (E.25.2)
The effective irradiance flux considers the local bolometric albedo, Aa, and givesus
The intrinsic flux, FA, is given by
Fa = da Ib Ga, where D is the effective bolometric limb-darkening factor introduced on page 122. Wilson (1990) expresses the bolometric flux ratio FB/FA in the kth iteration as
FA = FA- (Ra ) = GB £ ? GaKaRAk) GabD(ya), (E.25.4)
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