Advanced Topics and Techniques

This chapter addresses some of the many improvements and extensions of ideas and techniques in EB research over the last decade: The direct distance estimation through the analyses of EBs and the derivation of ephemerides and third-body orbital parameters from light and radial velocity curves. The Kepler mission [cf. Koch et al. (2006)] launched on 6th of March 2009, the GAIA mission with a target launch data in 2012, or the ground based survey LSST in discussion for after 2015 will add a new challenge to the field: The analysis of a large number of EB light curves from surveys. Detection of extrasolar planets by transit methods is a field where EB methods have been used successfully.

5.1 Extended Sets of Observables and Parameters

O 6eóg ecpKiaae rov koa^o ki elne: 'Onoxei ^vakó ag nopeuerai.'

God made the world and said: "He who has a brain will go on."

(Cephalonian proverb)

In EB analysis, the observables are usually the light curves and radial velocity curves, whereas the adjustable parameters are typically dimensionless quantities such as the mass ratio, Roche potential, and inclination, and also a few parameters in physical units such as the semi-major axis, and mean temperatures. In this section, we discuss absolute masses, temperatures, and distance as adjustable parameters and treat, for instance, the mass ratio as an additional observable. An important consequence and advantage is that the two-step procedure is replaced by one consistent least-squares analysis. We also learn that adjustable parameters can have observables as direct counterparts.

After outlining a general concept of extended sets of observables and parameters, especially useful in the context of direct absolute parameters estimation (Sect. 5.1.1), in Sect. 5.1.2.1 we focus on the following problem: Most light curve models require the temperature of at least one star as input. One of the main difficulties of modeling EBs is the accurate determination of the individual temperatures. In Sect. 5.1.2.3 we describe how to evaluate light curves in at least two passbands to determine

J. Kallrath, E.F. Milone, Eclipsing Binary Stars: Modeling and Analysis, Astronomy 221

and Astrophysics Library, DOI 10.1007/978-1-4419-0699-1.5, (9 Springer Science+Business Media, LLC 2009

both temperatures and the distance (Sect. 5.1.4). This overcomes some limits of the traditional two-step approach to estimate distances of EBs.

The section is ended by a discussion of using main sequence constraints to reduce correlations among adjustable parameters and the incorporation of intrinsic variables into light curve models.

5.1.1 Inclusion of Absolute Parameters in Light Curve Analysis

In Sect. 4.4.4.1 the absolute parameters and, especially, the distance of a star were computed a posteriori. Here we outline some ideas of how absolute parameters can be made part of the least-squares analysis providing their consistent standard errors, and thus improving on the standard two-step scheme in which the absolute parameters are computed only after the light curve solution has been performed. We introduce the absolute masses, M1 and M2, as examples. Later we shall present recent ideas that derive, for instance, both temperatures and the distance as part of the overall least-squares algorithm.

To illustrate the addition of adjustable parameters (not only absolute parameters, but any new ones) and relations to an existing model that already includes adjustable parameters pk, we show how to add M1 and M2 as free parameters to an existing model, in which the semi-major axis a, and the mass ratio q are already adjustable (there might be others as well). Based on the relationships (4.4.14,4.4.15 and 4.4.16), we obtain two equations

1 4n2 a3 q 4n2 a3

for the two new parameters M1 and M2. In the least-squares analysis, we need to consider the extended equation of condition

9 M1 9 M2 ^ 9 pk and the new partial derivatives 9o°v/9M1 and 9o°v/dM2, where o^ and o°v denote observed and calculated values of some observable at time tv. As (5.1.1) and (4.4.15) are explicit equations, this is rather simple and is given by

The q-derivatives are given by dq 1 q 9 q 1

Computing the a-derivatives leads to a3

and thus da G P2

9Mj 4n2 Ba2

This procedure has the advantage that we do not need to compute any additional numerical derivatives with respect to M1 or M2, as the derivatives 9o^/9a and 9o£/9q are known already.

The situation becomes more complicated if the period P is also an adjustable parameter, as in that case we have no explicit relationship P(Mj). Therefore, we need to resort to numerical derivatives for 9o£/dMj, although the ones for q and a are available. Thus, to conclude this illustration, we summarize that for adding additional adjustable parameters one has to check whether full explicit relations are available (that is, no further numerical derivatives are needed), or whether the additional relationships do not allow derivation of analytic derivatives. There are three major advantages of a consistent one-step approach:

1. It is impersonal, driven only by the input data and their standard deviations.

2. The least-squares analysis automatically provides standard errors of the estimated new adjustable parameters.

3. It allows to include estimations and standard deviations of these parameters from other sources, e.g., from published data.

We conclude this section by stressing the advantage of including parameter estimations from other data sources when they are specified with reliable standard errors. Distance or mass ratio serves as examples. In such cases, an adjustable parameter p can be compared to a direct counterpart observable pm with standard error em. Index m, m = 1,..., M, refers to several published values, where M is usually a small number, say, M < 3. To consider the observable pm in the least-squares analysis, we rename p into p c and obtain M equations of condition pm - pc = Spm, m = 1,..., M, (5.1.7)

or the corresponding least-squares term

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