Linnells Model

Linnell's model as described in Linnell (1984, 1993) is also based on Roche geometry. The initial 1984 version assumed circular orbits, but in the current version, this has been generalized to include eccentric orbits. The 1984 version required the input data specifying the photospheric potentials to be in physical units. The purpose was to permit some model other than the Roche model (such as a polytropic model) to be used; this was effected by an input switch to program PGA that computes the shapes and surfaces of the stars. A built-in alternative assumes a spherical model and permits testing of integration accuracy.

To begin the computations, the program CALPT accepts dimensionless Roche potentials and calculates the corresponding physical potentials. A component's rotation may be nonsynchronous. The software package consists of a series of programs. PGA defines the geometric properties of the photospheres, including direction cosines and values of photospheric grid radii, gravity values, and related quantities. PGB accepts an orbital inclination and either a single orbital longitude or an array of them. It projects the components onto the plane of the sky and fits an array of overlapping parabolas to the horizon points of each component. These quantities are used in turn to calculate shadow boundaries on a given component during an eclipse. PGB determines direction cosines for the line-of-sight to the observer and calculates the zenith distance and its cosine, for that line-of-sight, at all photospheric grid points. PGC determines the radiation characteristics of each component. The program interfaces with an external data file for evaluation of limb-darkening coefficients. PGC calculates linear or quadratic limb-darkening coefficients appropriate to each grid point by fast interpolation in an external file tabulated in (X, g, T). The data file reference is passed to PGC by name, permitting easy change to an updated table of limb-darkening coefficients. The external file of limb-darkening coefficients is based on Kurucz model atmospheres.

The effective temperature Teff is distinguished from the boundary temperature T0. According to Chandrasekhar (1950, Chap. XI, Eq. (31)), Teff and T0 are related by

The temperatures stored for the grid points follow the specification of (3.2.16), but the equation for grid point intensities uses T0. The reflection effect is modeled with high geometric accuracy which increases not only the reliability of the results but also the computing time.

PGC provides several alternatives for calculating continuum intensities. The least accurate, but fastest, alternative uses the Planck law. A model atmospheres option interfaces to external files of continuum fluxes tabulated by (X, g, Teff), and interpolates, as with limb darkening. In the midst 1990s, Linnell & Hubeny (1994) developed a much more accurate self-consistent procedure that interfaces directly to a spectrum synthesis program. In this procedure, an accurate spectrum synthesis is carried out. It encompasses distorted, irradiated binary star components and permits the computation of intensities at all local grid points at the effective wavelengths of the observational data. This information, in turn, permits determination of limb-darkening coefficients for the same data set that represents the component spectrum.

In common with other light synthesis procedures, this approach calculates monochromatic light curves. As noted earlier, photometric observations involve an integration over the passband of the product of the composite stellar spectrum, the transmission function of the Earth's atmosphere, and the response function of the optics and detector. Monochromatic light curves represent only an approximation to the photometric quantities actually obtained observationally. Buser (1978) has studied this problem from the standpoint of single stars and has introduced the term synthetic photometry to describe his representation of single star colors. Linnell et al. (1998) have extended the use of synthetic spectra for binary stars to include synthetic photometry.

PGD integrates over the components to calculate surface areas, total emergent flux, and flux toward the observer. The program also integrates between shadow boundaries to determine light lost by eclipse. The integration uses Simpson's rule rather than summations of light contributions from discrete surface elements.

SPT, originally called PGE, calculates theoretical light values at the times of observation. It does this in the following way. Starting with PGB the program uses an array of fiducial orbital longitudes. The output of PGD is an array of theoretical light values at the same orbital longitudes. SPT uses an accurate nonlinear interpolation algorithm, with the output of PGD, to calculate theoretical light values at the times of observation. The array of fiducial longitudes is chosen so that no interpolation is required across phases where the light derivative is discontinuous. The reason for this arrangement is that it is possible to calculate theoretical light values for comparison with an indefinitely large data set while preserving the basic accuracy of the light synthesis calculation. This in turn means that it is unnecessary to combine observations into normal points (NB: This is true for all light curve programs now that computers have gotten so fast). The reason for the change in name, from PGE to SPT, is that SPT permits placement of dark (or bright) spots on the component photospheres (Linnell, 1991). The accurate light curve computations consider a variety of horizon and eclipse problems, including over-contact "self-eclipses."

Linnell's differential correction program DIFCORR (Linnell 1989) produces corrections for the parameters i, Q1; Q2, q, M, A2, g1; g2, T1; T2 (polar temperatures), S1, S2 (limb-darkening scaling coefficients), and U , a light level normalizing factor to fit the observed light curve to the calculated light curve. The program can handle multicolor light curves and so, in principle, can determine both T1 and T2. For the sake of self-consistency, the values of L1 and L2 are defined as derived quantities, calculated in PGD. The program makes no provision to decouple L1 and L2 from T1 and T2, as the Wilson-Devinney model does. A separate differential correction program determines star spot parameters to fit residual light curve effects. It is also possible to adjust the entire primary minimum by a shift-parameter tp. This parameter should not be confused with a shift due to possible orbital eccentricity.

A major difference exists between the calculation of light derivatives in Linnell's program and other light synthesis programs. In Linnell's program, there is a central reference set of parameters which determine a central reference light curve at each observed wavelength. For each parameter, a displacement for that parameter from the central reference value is chosen and two outlying parameter sets are established, symmetric with the central reference set, and displaced by the chosen value. Care must be exercised to avoid physically impossible parameters. The three light curves for each parameter determine two first differences and one second difference, and these remain fixed for successive iterations. Only the coefficient of the second difference changes in the calculation of new first derivatives. See Linnell (1989, Eq. (30)).

At the cost of substantial initial calculations, the computing time for successive iterations is greatly reduced. In addition, the accuracy of the first derivatives is quite high.

A justification for the separate, multiple program approach (rather than subroutines in a single program) is flexibility. A diagnostic run with a modified T2, say, is possible without recomputing the geometry. The running of PGB is the most time consuming part of the calculation. A call to a batch Fortran program, that oversees the whole process, runs all the separate programs in sequence. Implementation of the synthetic photometry program (Linnell et al. 1998) would have been impractical without the division of the entire project into separate programs.

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