Determining Individual Temperatures

One of the main difficulties of modeling EBs is the accurate determination of the individual temperatures. Frequent practice in the literature is to assume the temperature of one star, or better to obtain it from spectra or color indices in an a priori step as described on page 199, after which the other star's temperature follows by fitting the light curve model to the data. In this two-step approach, the accuracy of the estimated temperature depends either on the how well the individual spectra can be extracted from the composite binary spectrum or the reliability of deriving that temperature from the composite time- or phase-dependent color index. As it is difficult to estimate accurately the contribution of only one star in advance, in Sect. 5.1.2.1 we exploit extra information to derive estimates of the temperatures, Sect. 5.1.2.2 uses color indices to determine the temperature, while in Sect. 5.1.2.3 we describe how to evaluate light curves in at least two passbands to determine both temperatures and the distance (Sect. 5.1.4) which overcomes some limits of the traditional two-step approach to estimate distances of EBs (Sect. 5.1.4).

5.1.2.1 Temperature Estimations

One way to estimate individual temperatures is to exploit extra information about the system. If the stars under consideration are known members of a stellar ensemble, for example, and the ensemble has been well studied, the condition is met. By well studied, we mean that the color-magnitude (CMD) is well established, the distance has been found, and there is a well-fitted isochrone available. With these criteria, a method has been devised by Milone et al. (2004b) which utilizes the properties of the isochrone to determine the properties of both components of an EB, in the case that no spectroscopy and only two-passband photometry is available. In the case of the 47 Tuc survey for extrasolar planets, cf. Gilliland et al. (2000); Milone et al. (2004b), this proved to be the case. The idea is basically this: From the Russell-Merrill treatment in Sect. 6.2.1, we obtain a relation for eclipse depths. The depth of mid-eclipse for primary (p) and secondary (s) minimum can be written as dp = 1 - Ip = (L1 + L2) - [L1 + (1 - a0)L 1] = «0L1 ds (5.1.9) = 1 - Is = (L1 + L2) - [L1 + (1 - Oi)L2k2] = a0k2L2, (5.1.10)

where I refersluminosity of component j, a0 is the fraction of light lost at mid-eclipse, and k = R2/R1. The passband luminosity is the product of the surface brightness and the area of the disk, so the ratio of depths in any passband is dp _ L1 _ ai ds [ L 2k2 ct2

where Oj is the surface brightness of component j. Comparing now the ratio of depth ratios for two different passbands

Rearranging this

But this ratio is directly related to the color index difference between components:

What is observed, however, is the net color index of the system, C12, and, if the cluster has been well-enough studied, the well-known cluster ambiguities of interstellar and cluster reddening and metallicity have been resolved, so that the intrinsic color indices and the absolute visual magnitude can be assumed known for the system. This is where the isochrone comes in: In the case of the detached systems, from which the independent properties of the system may, indeed, represent those of independently evolved stars, we assume that one of the two components lies on the isochrone, if the system is a cluster member. If the eclipse is total, the occultation eclipse provides an individual CI; if partial, one may assume initially that either component is on the isochrone. Proximity of the system to either the isochrone or to the "binary main sequence" indicates that the components are either dominated by one component or are equally luminous. Thus the difference between the system's MV and that of the isochrone provides vital clues for the modeling of the light curves. The full procedure for the determination of the full set of parameters and the preliminary results for the 47 Tuc cluster are further discussed in Milone et al. (2004b). The bottom line of this method is that the accuracy and precision of the derived parameters, including the temperatures of the components, are strongly dependent on the accuracy and precision of the model for the cluster.

5.1.2.2 Color Indices as Individual Temperature Indicators

Prsa & Zwitter (2005b) explicitly discuss an approach that is often taken by light curve modeling astronomers, to determine the individual effective temperatures, T1 and T2, from standard photometry observations without perhaps fewer a priori assumptions than other approaches. This approach involves the system's binary effective temperature, TB. Observationally, an unresolved binary may be regarded as a point-source with a time- or phase-dependent effective temperature TB = TB(t) that could be compared to an observed color index curve. Both components contribute to TB(t) according to their sizes and individual temperatures, T1 and T2, and inclination. If a model is to accurately reproduce observations, the composed contributions of both components must match this behavior.

An initial value of TB(t) may be obtained from a color-temperature calibration. For the Johnson B — V color, Prsa & Zwitter (2005b) in their program PHOEBE compute the empirical color-based effective temperature, Teff(B — V), by a polynomial of degree 7

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