The Matching Approach

In their work to establish the best distance indicators among detached and semidetached binaries in the Small Magellanic Cloud, Wyithe & Wilson (2002a,b) and Wilson & Wyithe (2003) obtained starting parameters for the rigorous WD model by comparing each light curve with a set of archived model light curves, and then sending the best match to an automated version of the WD differential corrector program DC.

In ongoing work, Kallrath & Wilson (2057) are extending this approach by an inner linear regression loop, incorporating a priori information, adding interpolation techniques, and increasing storage and numerical efficiency. This approach now supports all WD parameters.

For a given binary system, let l0ic be any observed value for observable c, c = 1... C, at phase 0. Correspondingly, l°ick denotes the computed value at the same phase 0i for the archive1 curve k, k = 1... K. Note that K might easily be a large number such as 1010. The matching approach returns the number of the best fitting archive light curve, a scaling parameter, a , and a shift parameter, b, by solving the

7 Synonomously, we use the terms stored, library, or template light curves.

following nested minimization problem:

min k

Note that the inner minimization problem only requires solution of a linear regression problem. Thus, for each k, there exists an analytic solution for the unknown parameters a and b. Note that the icicn values are obtained by interpolation. The archive light curves are generated in such a way that they are well covered in the eclipses, while a few points will do in those phases that show only small variation. Thus, there is a non-equidistant distribution of grid points that is well interpolated by cubic polynomials.

5.3.2.1 Solving Linear Regression Problems

Although solving linear regression problems is not difficult as such, one should exploit a priori knowledge of light curve parameters when looping over k. If a priori knowledge is available, for instance, on the mass ratio, q, or the temperatures T1 and T2, then certain k values can be excluded. The analysis of C observables (radial velocity curves and light curves) requires to solve C linear regression problems. If the observable is a radial velocity curve, the additive constant b gives the systemic velocity y . For light curves, a returns the WD scaling quantity L 1 and b is third light, I3

5.3.2.2 Generation and Storage of the Archive Curves

Archive generation requires appropriate looping and proper interfacing to subroutine LC of the WD program. Special attention should be paid to the way the stored sets can be accessed. If a priori knowledge is available in connection with the Roche potentials Ü1 and for instance, on the mass ratio, q, we should exclude unphys-ical configurations and ensure that certain values of k can be excluded.

An additional aspect is the storage of the computed archive light curves. For each light curve and observable (wavelength), we need rk = 4 x C x Ik bytes, where we consider 4 bytes, Ik phases and C bands. Note that we may have different numbers of phases depending on the shape and amplitude of the light curve (used in our interpolation scheme). The total memory requirement is then R := J2K= 1 rk. Note that R may easily reach the order of 4C ■ 108 light curves if all reasonable combinations of the photometric parameters e,d, i, q,Q1,Q1, T1, T2, and log g are considered.

The choice of unadjusted parameters A1, A2, g1, and g2 depends on T1 and T2. L1 can be set arbitrarily to L1 = 1 because the matching problem involves the scaling parameter anyway. L2 follows as a function of L1. l3 is covered by the linear regression in the matching problem. Limb-darkening parameters also can be chosen, from, for instance, Van Hamme's (1993) limb-darkening coefficients. As the computation of limb-darkening coefficients depends on log g, we have added this as a parameter. Great care is necessary when involving the eccentric orbit parameters. Both eccentricity, e, and length of the perihel, m, need a very fine grid.

In addition to the sets generated automatically, we add all the light curve parameters sets for those EBs for which a light curve solution is available. This way, when we find a match to an observed light curve, we are able to provide not only some reasonable light curve parameters but, in addition, also a candidate similar to the current EB.

One might think to store the library light curves in a type of database. However, database techniques become very poor when talking about 1010 light curves. Therefore, a flat storage scheme is used. In the simplest case, for each k we store the physical and geometric parameters, then those parameters describing observable c, and then the values of the observable.

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