with phase of first contact, 01. A theoretical expression for A2m is

where a is the fractional loss of light, « indicates the separation of the components so that «0 = cos i, namely, the value of « at closest approach (at 0 = 0), and 51j2 mark separations at first and second contact, respectively. The analysis also requires the determination of the coefficients cj of the light curve representation

Although his approach and the notation used above is based mostly on uniform, circular disks, some corrections are applied for limb- and gravity-darkened, tidally distorted nonspherical stars with light curve perturbations (Kopal, 1989, 1990). The analysis of the moments yields the elements of the system. For perturbed light curves, the prescriptive procedure is as follows:

1. determine the necessary number of moments of the light curves (at least 4);

2. evaluate the necessary number of constants cj=1n (at least 4) from the light curve maxima and take their weighted sum;

3. evaluate the (normalized) moments;

4. evaluate the r1j2, i, and L 1j2 from the moments;

5. compute the light curves from the elements and evaluate the perturbations, P2m;

6. use the perturbations to obtain improved moments; and

7. iterate Steps 3-6 to the final set of elements.

Kopal (1979, p. 193) asserts that no "time-domain" analyses , by which he characterizes all the other methods described here, can make use of empirical data to convert observed light curves into a form from which the elements can be derived directly; keeping in mind that the whole idea of least-squares procedures is to extract parameters indirectly, it is not clear why the direct determination of parameters should be an advantage. Interpreting the assertion to mean that a closed analytic solution cannot be found for such cases, the question remains whether this method can do so either. Its claimed superiority has yet to be demonstrated.

The Fourier analysis frequency domain method is a rather formal mathematical approach suffering from the following disadvantages:

1. the components are assumed to be spheres, or at best, corrected spheres (Kopal 1989, 1990);

2. it is doubtful that a few Fourier coefficients can reveal fully the information in a light curve, generally. The Fourier analysis method uses an integral moment11 of the observations, thus masking the contribution of each point to the value of A2m, particularly for small sin2m 0 as m increases. That means that perturbation terms for proximity effects are evaluated essentially from the uneclipsed portion of the light curve;

3. it is necessary to specify the luminosity scaling;

4. the value of the angle 01 of external tangency must be specified; and

5. it is not easy to introduce additional physical effects into the model in this analysis.

11 This process smoothes the observations. Strictly speaking, information is thrown away in the process.

Budding (1993, p. 211) notes similarly that although the method is straightforward, in practical situations complications arise, especially because the proximity effect representation must be very precise, but the cj quantities "are not well suited to accurate numerical derivation." He also notes that the procedure loses the advantage of simplicity for partial eclipses.

Light curve analysis based on the solution of an inverse problem with an underlying physical model as its core is not limited by these disadvantages. Physical models are open to the implementation of improved physics.

6.5.3 Mochnacki's General Synthesis Code, GENSYN

The GENSYN code was developed by Mochnacki and Doughty in the early 1970s, of Mochnacki & Doughty (1972a, b), and further improved around 1983. Mochnacki's work began by using Lucy's (1968) code, which he found to be reliable but slow due to its use of a "ray-tracing" algorithm. The GENSYN code was intended to do both light curve and line profile synthesis. It used a cylindrical coordinate scheme making the radius vector a single-valued function for all configurations: contact and noncontact. This might be an advantage, although not a crucial one, because there is no problem finding the correct surface in spherical coordinates either. The program was designed to be compact, fast, and numerically stable but, unlike the Wilson-Devinney LC program, did not incorporate an accurate correction scheme for horizon visibility. GENSYN was applied originally to totally eclipsing A-type contact systems (such as AW Ursae Majoris and V566 Ophiuchi). According to Mochnacki, it was the first Roche geometry-based light curve code to incorporate full mutual irradiation by mapping each surface element to all those illuminating it.

6.5.4 Collier-Mochnacki-Hendry GDDSYN Spotted General Synthesis Code

Andrew Collier and Stefan Mochnacki combined their programs SPOTTY and GENSYN in 1987 to analyze spotted eclipsing systems. An improved geodesic grid system with triangular elements was introduced by Paul Hendry as part of his MSc thesis at the University of Toronto under Mochnacki's direction. Mochnacki states that the resulting code, GDDSYN, is both faster and more accurate than WD93. All grid elements have comparable surface areas and there is little aliasing due to projected symmetries (Hendry & Mochnacki, 1992). In addition, Hendry wrote a program to determine the most likely spot distribution which makes use of a maximum entropy algorithm, and combined it with GDDSYN and SPOTTY to determine spot distributions in contact systems (Hendry et al. 1992). Finally, Hendry more recently wrote a program to fit both orbital elements and spot distributions to photometric and spectroscopic data. Mochnacki notes that this code was used to analyze several years of DDO observations as part of Hendry's PhD thesis.

6.5.5 Other Spot Analysis Methods

A long-standing tradition at the Osservatorio Astrofisico di Catania, on the island of Sicily, has been the study of the effects of star spots on light curves. Their work has focused on the RS CVn (e.g., Rodono et al. 1995) and to the similar systems AR Lac, II Peg, and SZ Psc. This work was then extended to the somewhat more complicated system RTLac, discussed earlier. The spot modeling technique is described in several papers; see Lanza et al. (1998, 2002) and references contained therein. Their goal has been to observe the spot distribution as a function of longitude on the star's surface and to study spot migrations over longer intervals of time and modulation over shorter intervals. Information of this sort can lead to an understanding of magnetic structures and their relationship to the rotational and orbital dynamics of the system. Eclipsing systems provide the necessary spatial resolution to explore the brightness distribution on the surface of the eclipsed component, which is divided into several hundred pixels, each of which is allowed to vary within constrained limits. Fittings were obtained with the use of a priori assumptions supplied by a maximum entropy criterion, and, in an independent check on the results, a Tikhonov regularization criterion. According to the authors, the use of such criteria leads to stable and unique solutions of the distribution over the surfaces of the stars. A lengthy series of observations extending over decades provided the database, and the maximum brightness of the system observed during that time served as a kind of unspotted calibration for the pixels. In the case of AR Lac, data were available for 20 annual light curves in the interval 1968-1992. The system is subject to variations exceeding 0.05 magnitudes in a single day, possibly due to flares on timescales of tens of minutes. The reference level was established for the system as observed in 1987 when V = 6m 030 ± 0 m005 at orbital phase 07395. The EB model for the system, at least for the earlier studies, involved tri-axial ellipsoids for the stellar shapes but also Kopal's (1959) geometric treatment of ellipsoids, reflection, and gravity darkening (Lanza et al. 1998). The resulting longitude distributions are obtained for each component and show general agreement for particular years for each of the two restrictive criteria.

6.6 Selected Bibliography

This section is intended to guide the reader to recommended books or articles on light curve models and programs.

• The review article by Wilson (1994) gives an excellent overview on light curve models. It provides a historical view and discusses the underlying astrophysics.

• An early overview on modern Light Curve Modeling of Eclipsing Binaries is provided by Milone (1993). In this collection, several authors briefly describe or comment on the latest versions of some of the better-known light curve models and programs.

• The Determination of the Elements of Eclipsing Binaries by Russell & Merrill (1952) is probably the best and most complete description of material related to the Russell-Merrill model.

• In his review Close Binary Star Observables: Modeling Innovations 2003-06, Wilson (2007) summarizes the developments of the WD program during 2003 and 2006.

• The journal publication Eclipsing Binary Solutions in Physical Units and Direct Distance Estimation by Wilson (2008) is a pleasant-to-read article on the assumption and implications of direct distance estimation.


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