1. Original Aims of Our Light Curve Modeling Program
  2. The idea is to better understand physical processes in stars and their environments through the development of improved methods to analyse and acquire the light and spectra of eclipsing variable stars. To that end, a highly capable team has been assembled to help. See Groups.

    This project aims:
    1) to get more efficient and accurate convergence and better search algorithms to explore many-dimensional parameter space for the deepest minimum;
    2) to model eclipses of translucent clouds of material produced by collision of stellar winds in the atmospheres of high-temperature stars where plasma dominates over normal matter,
    3) to incorporate hydrodynamic flow code to model systems with gas streams between stars;
    4) to include techniques to explore the effects of star spots and other phenomena associated with active regions;
    5) to survey star clusters and associations for new variables; and
    6) to follow up these discoveries with further study with RAO instruments.

    Aim 5) requires an upgrade of a high-quality optical telescope originally designed for satellite tracking and the development of cameras and software, and 6) requires an infrared camera. Observations of starlight are to be combined with high quality spectroscopic data from the outside groups such as Latham's group based at CFA- Harvard. We plan to extend the modeling tools and techniques to young, massive objects which are still close to the regions of their births in the expanding stellar associations of the Milky Way; the location of such systems in ensembles provides a bootstrap for exploring both the systems and their wider environment and so the circumstances of their origins. The data capable of being modeled with the new tools will be from a wide variety of wavelengths from x-rays to microwaves, involving data from space and satellites as well as ground-based observatories.

    Note that an important part of the program over the years has been to convert the RAO's Baker-Nunn satellite-tracking camera into a wide-field imaging camera to permit a broader search for variables stars among the young open clusters and associations of the Milky Way, and the construction of an IR camera.

    Nearly all the long-range goals have been achieved, but the last (an IR camera developed with a donated engineering array has, unfortunately, not proven to be of sufficiently high sensitivity to be useful for stellar work).

    Finally, an improved set of IR passbands (Young, Milone, & Stagg 1994; Milone & Young 2005, 2007) is now available to get the best possible precision from IR photometry, especially when carried out from sites other than the highest and driest in the world.

    (See: Astronomical Instrumentation & Techniques,

    Optical & Infrared Photometry, and

    Patrol Camera image)

  3. Improved Tools
    • WD93K93
    • The modeling of high-precision data requires the most accurate physics. We begin with the Roche model of stellar shapes in close binary systems, first suggested by Zdenek Kopal in the 1950's. We have used the Wilson-Devinney program, which is based on this model, into a precise tool for the exploration of the basic properties of stars. Tidal, reradiation, and limb-darkening effects have been dealt with ever more successfully, and the latest version of our code, WD93K93 incorporates not only these improvements, but also allows for metallicity variation -- an increasing source of uncertainty in star cluster and variable star studies.

      WD93K93 now employs Kurucz atmosphere models for a range of chemical composition, each integrated over selected photometric passbands, and for narrow ranges of ultraviolet flux from satellite observations. The improvements permit full analysis of high-precision data from the far UV to the intermediate infrared for a range of chemical compositions for the first time.

      It is easy to provide an example of the superiority of these modern tools: in an ongoing study of the 24.6d evolved but otherwise uncomplicated eclipsing system AI Phoenicis, the masses and radii of the component stars have been determined now to better than 1%. These are among the best stellar determinations ever obtained, especially for evolved stars, but we intend to go further. Further improvements can be anticipated as the calibration of the temperature scale improves. Even now, the data place sharp constraints on model mixing length, age, and chemical composition.

      For close binary stars, the case is usually complicated by interaction effects, so special tools are needed to produce precise results. In the case of contact systems, stars in the process of merging, in which an outer envelope is shared between the component stars, evolution calculations must be invoked to estimate the loss in mass from the system and the degree to which mass has transferred from one component to the other. Here, too, however, work by Rucinski (1994, 1995) suggests that such objects may be usable as standard candles.

      The historical development of modern light curve analysis codes and our contributions to it are discussed in several papers in Milone (1993), which also describe the current state of the art of light curve modeling. As excellent as our existing modeling tools and observing capabilities are, however, there are important areas that need further improvement before their full potentials can be reached.

    • Simplex Modeling
    • For solution uniqueness investigations, Josef Kallrath has upgraded his simplex code with this enhanced version during a recent visit to Calgary. Simplex has been a formal part of our light curve solving strategy for several years now, but it is not a fool-proof method of finding the 'best' solution because deep minima may hide in narrow valleys amid mountainous regions of multi-dimensional parameter space. Thus, while we have successfully used this program for many studies (see Kallrath 1993 for a summary and further citations of the uses and justification for this procedure) we propose to experiment further to find even more effective and efficient techniques for parameter space exploration. Jason McVean plans to develop neural net techniques to explore initial parameter constraints to narrow search regions as part of his PhD work at the University of Calgary. In recent years, we have also explored less efficent but more robust techniques. One of these is Simulated Annealing, We describe the progrss of this work in Milone & Kallrath (2007).

    • WD95
      • Kernel and Modules.
      • First, the I/O format of the current program is more suitable for batch running than for interactive, default-driven input. The philosophy behind the new coding which we propose to carry out is described in a book by Kallrath and Milone (1999). The package gives a choice of light curve tools available from a common front-end. The front end was coded by Dirk C. Terrell, now at SWRI, Boulder. This user-friendly front-end is explicitly for a variety of PC but may be used as a front end for UNIX machines as well.

      • Improved Convergence Algorithms.
      • The original Wilson-Devinney program was written around 1971, and much of the basic structure of that era is retained. This makes it relatively unrewarding to vectorise the code, for example, although we reported some progress in this area (Milone et al. 1989). Another is the lack of automatic iteration in the code. Although this omission was done on purpose (to avoid blind searches into physically unrealistic realms of parameter space), the lack of such iteration greatly slows down the solution-determination process for those who understand the limitations of the modeling process.

        An efficient procedure for locating a solution once the general properties, and thus the parameter constraints, are known is an important goal. Terrell developed PC front-end software for use with WD93K93 which is capable of selecting solution subsets with the smallest error predictions and resubmitting an altered input file. However, as university servers become swamped because memory upgrades are outpaced by increases in numbers of users, even slight inefficiencies will begin to tell. For example, a typical simplex set of 150 iterations could take as much as a day or two to run during Fall or Winter term sessions, despite the high benchmarks of the IBM RS6000 system in use at the University of Calgary to late ~2004. Thus, it is imperative that the code be highly efficient for any platform on which it may be run.

        Damped least squares (DLS) was invented a considerable while ago (Levenberg 1944) and has been independently reinvented many times since then (e.g., by Girard 1958; Wynne 1959; Marquardt 1963; Hoerl and Kennard 1970). The method has been variously described as an interpolation between gradient methods and simple least-squares differential corrections, as a maximum- neighborhood confidence region technique, or as a step-limiting method to prevent solution blow-up in ill-conditioned problems. But it is best viewed as a means of selecting only the well-determined eigenvectors of the matrix of the normal equations (cf. Matsui and Tanaka 1992, 1994, 1995 as examples). Numerous comparisons of methods for function optimisation (e.g., Meiron 1965; Pitha and Jones 1966; Kidger and Wynne 1967; Bard 1970; Gans 1976; Hiebert 1981) have shown that DLS with multiplicative damping is among the very best methods for nonlinear problems, and it is now generally recognized as the first choice for nonlinear least-squares problems.

        Young has used DLS for nearly 20 years to solve nonlinear least-squares problems in planetary physics, spectroscopy, and astronomical photometry (Young 1982, Young & Young 1977a,b, 1978, 1979). It is now routinely used in both Wilson's own updated version of his software, and in our versions of it.

      • Hydrodynamic Codes and Stream Modeling.
      • This work, undertaken by Dirk Terrell (1993, 1994) as part of his PhD thesis, can be incorporated into the code to model semi-detached systems in which a stream of gas leaves the inner Lagrangian point of one component to orbit the other component. Variations of stream behaviour, and the depth of the potential well into which it falls, determine a variety of phenomena in late stages of stellar evolution. The code computes a predicted spectrum which can be compared to observations. Wilson and Terrell have also written a hydrodynamic-flow and polarisation program which computes the Stokes Q and U quantities for comparison with observations.

      • Treatment of Atmospheric Eclipses.
      • Work by Cherepashchuk (1993) in Russia and by Moffet and colleagues at Montreal on the analysis of eclipsing Wolf-Rayet systems has demonstrated the importance of atmospheric eclipse phenomena to the understanding of very young stars. Advances by Kallrath and Milone and most recently by Wilson, have used the Roche model to better model the geometry of these systems and allow for eclipses of atmospheric clouds as additional light-curve determinants. Detailed treatment of the transfer of energy through such clouds and the extent of their optical thinness requires spectroscopic information. Levels of modeling detail therefore must be made dependent on the information available. The orginator and maintainer of the Wilson-Devinney code, and a leading authority on light curve analysis, Robert E. Wilson, and Josef Kallrath have worked on aspects of this code, and limited modeling of this sort can now be done.

      • Added Benefits.
      • Improved convergence, global solution searches and uniqueness tests benefit all sciences and have considerable application in industry. The modular approach here may also prove equally useful.

    • WD98.
    • Josef Kallrath completed the incorporation of the innovations produced for WD95 in Wilson's 1998 program. The package was called WD98. It had all the upgrades of WD95 and permitted atmospheric eclipses, three types of limb-darkening treatment and advanced treatment of the reflection effect. As with Wilson's own version of the Wilson-Devinney program, the new package permitted time as well as phase based data analysis, and included the period and epoch as adjustable parameters.

    • WD2007
    • Currently, we are using WD2007 (and non-self-iterating Unix version, wd98k93h). This version has the improvements of all previous versions, but incorporates additional improvements provided by Kallrath, C.R. Stagg, and M.D. Williams. These include:
      • stellar atmosphere corrections for individual grid elements;
      • dimensioning increase to accommodate transit eclipses by objects as small as Saturn-sized planets;
      • looping in particular elements (such as q, i, or P) which may be difficult to adjust to convergence in runs with multi-parameter adjustments.

  4. Other Packages.
  5. We have been very impressed over the years with the use of stellar tomography, the use of high resolution spectra to explore the velocities of components of Algol and other binary star systems that have circumstellar material. One of the most successful of the codes of which we are aware is SHELLSPEC developed by Jan Budaj and Mercedes Richards starting about 2004. The simultaneous use of WD-type codes with this type of software tool may prove invaluable for future investigations. In fact Miller, et al. (2007) seems to do just this. Details about SHELLSPEC may be found at


    which contains the latest fully documented and public version, SHELLSPEC07.

    Josef and I have summarized the current state of light curve modeling as we know it in our paper "The Tools of the Trade and the Products they Produce: Modeling of Eclipsing Binary Observable," in Short-Period Binary Stars: Observations, Analyses, and Results," (2007), eds. E.F. Milone, D.A. Leahy, & D.W. Hobill, (Dordrecht: Springer), pp. 195-219.


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Davidge, T., and Milone, E.F. (1984) ApJS, 55, 571.

Gans, P. (1976). Coord. Chem. Rev., 19, 99.

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