If the distance is measured in parsecs and the parallax is measured in arc-seconds, the constant is kn = 1. To couple the parallax to the binary model it is more convenient to measure the distance in units of the semi-major axis a.

First, we include the parallax both as an observable and also as an adjustable parameter. Second, instead of the normalized light or flux l (0) usually used in light curve analysis, the flux lD (0 ) in absolute physical dimensions [energy/time/ wavelength/unit receiver area] must be used in the least-squares analysis. Note that this requires absolute calibration of the photometric systems as discussed in Sect.

The addition of parallaxes to light curve analysis slightly extends the least-squares function (see Chap. 4). The contribution of parallax as an observable is

¿w (on'.' - of)2 = ¿ w (nf - n -l)2, n " = n, (3.9.2)

where the weights are derived from the standard deviations of the parallaxes according to (A.3.4). With a derivative-based least-squares algorithm the analytic derivatives are

9 onal 9 orcval dlD n n = 1, ) = 0, (0) =-2—Id (0), (3.9.3)

lr-n n _ 1 rv dn ' 9n dn k which follows directly from (3.2.51) and (3.2.1). As the computation of radial velocities does not explicitly depend on the parallax, the radial velocities' partial derivatives are zero. An obvious point is that, in the absence of radial velocity curves (semi-major axis a is not known in that case) n cannot be determined because a/D = an/kn = const and a is unconstrained.

3.10 Chromospheric and Coronal Modeling

The extension of observables to include spectrometric data and very narrow line profile information, the availability of X-ray and ultraviolet data from space platforms, infrared and even radio data, make it possible to model the details of stellar chromospheres and coronae with improved accuracy.

It is well known that strong spectral lines, such as Ha, or Ca II H&K, originate much higher in the photosphere than does the continuum. From rocket and space platforms, such as NRLs stigmatic solar spectrograph data on Skylab, the far ultraviolet emission regions have been mapped in great detail. Of particular interest are the He II spectroheliographic images at 30.4 nm which map coronal holes. Figure 3.31 above shows the Sun in the far-ultraviolet region.

Ability to model these features requires capability in a light curve code that does not exist. The disk of the Sun, for example, is silhouetted by the active regions and chromospheric network behind the limb; it is dark in the far-ultraviolet and X-ray regions. The emission in many passbands comes exclusively from active regions, and the optical depth may be so low outside these regions that only patches of the Sun may be visible in an otherwise dark field. The emission may arise

Fig. 3.31 He II spectroheliogravarPhic image of the Sun. This Skylab photograph (Experiment S082A) shows the Sun at 30.4 nm. Courtesy R. Tousey, US Naval Research Laboratory (NRL)

solely from an annulus around the dark disk. There is every reason to suspect that a similar situation is to be found in other stars with convective envelopes, at least of solar type. As described in Strassmeier (1997, Chap. 9), many observational data, e.g., the Ca II H&K lines, are also available for active stars and active chromospheres.

3.11 Spectral Energy Distribution

Bona diagnosis, bona curatio (Good diagnosis, good cure)

Plotting flux (essentially brightness) versus frequency or wavelength of light of an astronomical object gives its spectral energy distribution (SED). Studies of individual EBs (cf. Siviero et al. (2004) and Marrese et al. (2005) for example) have shown that including flattened SEDs may be used as an external check of the model solution, where individual spectral lines of echelle spectra are compared with Kurucz (1993) model atmospheres. Flattened spectra are used instead of flux-calibrated spectra because observed echelle spectra are virtually impossible to flux-calibrate. There are good ways to calibrate B&C spectra but not echelle spectra, and the latter are mainly used for RV studies. Hence flattened spectra. Despite the fact that some information is lost, fortunately there are many spectral lines and their profiles and equivalent widths are strongly dependent on reff, log(g/g0), vrot, and [M/H].

SED data are useful in solving the inverse EB problem as discussed by Prsa & Zwitter (2005b). Their program PHOEBE described in Sect. 8.2 already takes a step in that direction by using a synthetic spectra database to test whether flattened, wavelength-calibrated spectra match synthetic spectra within a given level of significance. As the spectra depend on |Teff, log(g/g0), urot]1j2 and metallicity, they can in favorite cases provide valuable insight to break the often experienced problem of degeneracy among light curve parameters (often, Roche potentials , and inclination i), or to support (for well determined radii) the determination of the yielded synchronicity parameters F1 and F2, because the only way to compensate for the change in rotational velocities for any predetermined radii is to break the corotation presumption. This may be especially important in analysis of well-detached systems, as demonstrated by Siviero et al. (2004).

3.12 Interstellar Extinction

To ay^a 0i ano piKpo ayKuXty vet (A thorn stings even if it's small)

In the field of EB analysis interstellar extinction and reddening usually have not been treated as part of EB models. A binary star appears fainter if its light passes through regions of the interstellar medium filled with dust and gas particles causing absorption and scattering. If scattering by dust or grain solids is the main cause the process is roughly described by Mie scattering. As in Mie scattering the amount of scattered light in optical wavelengths decreases with wavelength, more blue light is removed, i.e., the apparent B brightness decreases (percentwise) more than the apparent V brightness, and objects appear reddened. Because the difference B - V increases with extinction, the color excess is a useful measure of interstellar extinction. The quantity (B - V)0 is the intrinsic color index of the object. The (U - B) color excess is defined similarly. The effect of interstellar extinction on the V band is described by the attenuation, AV, expressed in magnitudes. As the ratio R = AV/E(B - V) « 3.1 is a good approximation to most directions across the sky, an estimated value of AV is derived directly from the observed color excess by

Regardless of how AV (or other passband attenuations) has been estimated, in EB analysis it is traditionally subtracted uniformly for all phases from photometric magnitudes, i.e.,

where MV is the absolute magnitude in the V band and D is distance. Prsa & Zwitter (2005a, b) questioned whether this type of correction is adequate, especially if interstellar extinction and the color difference between the binary components are large.

Instead, Prsa & Zwitter (2005a) treated reddening systematically in the context of data fitting. They determined E(B - V) from multi-color EB light curves by comparing several color indices in-and-out of eclipse and demonstrated that estimation of E(B - V) from least-squares analysis requires light curves in three or more bands. As interstellar extinction and reddening depend on wavelength, one has to integrate over the wavelength of a passband instead of using a simple effective wavelength, A.eff, in the calculations.

Wilson (2008) remarks in his development of the direct distance estimation scheme that, although interstellar extinction increases distance estimates, its associated reddening decreases temperature estimates. Reduced theoretical temperatures reduce predicted absolute fluxes and so decrease distance estimates. Thus, in regard to distance determined from light curve analyses, extinction and reddening partly offset one another and, accordingly, the overall effect of extinction on distance determination is less than one might suppose. Wilson (2008, Sect. 7) also investigated the possibility of determining the attenuation A through the least-squares analysis. Note that attenuations in different passbands are connected through the Jason Cardelli & Mathis approximation functions and thus only one attenuation A is a free parameter. Although this is indeed possible given accurate absolute light curves in three passbands, it is not very practical as the sensitivity with respect to the calibration of the light curves is too strong. Small deviations in the calibration lead to significantly wrong values of A. The situation might be improved if the three bands are widely separated in wavelength.

3.13 Selected Bibliography

This section is intended to guide the reader to recommended books or articles on the physics involved in modeling EBs, or binaries in general.

• The review article by Wilson (1994) gives an excellent overview of Light Curve Models. It provides a historical view and discusses the embedded astrophysics.

• The Proceedings of IAU Symposium 51 provides many useful contributions on gas streams (Batten, 1973b).

• Readers interested in the Structure and Evolution of Close Binary Systems are pointed to the Proceedings of IAU Symposium 73 (Eggleton et al. 1976).

• On the topic of stellar atmospheres, Alter's (1963) Astrophysics: The Atmospheres of the Sun and Stars has excellent physical insights. Theoretical Astrophysics by Ambartsumyan (1958) is a classic work; an excellent book. Stellar Atmospheres by Mihalas (1978) is the most detailed reference on this topic.

• A compact source on the Theory of Rotating Stars is provided by Tassoul's (1978) book.

• Warner (1995) provides in his book Cataclysmic Variable Stars useful material on mass transfer and accretion disks. This book also contains background on other astrophysical topics relevant to EB modeling.

• The Tidal Evolution in Close Binary Systems is discussed and quantitatively investigated in the excellent paper by Hut (1981).

• The book Magnetohydrodynamics in Binary Stars by Campbell (1997) gives an outline of early work in binary stars and introduces the fundamentals of magnetohydrodynamics and binary star theory. It also covers X-ray binary pul-sars,accretion disk magnetism, and stellar and disk winds.

• A brief review onX-ray binaries, their classification, and observational facts is given by Krautter (1997).

• A useful introduction into the field of active stars, stellar spots, active chromospheres, and stellar magnetic fields is provided in the book Aktive Sterne by Strassmeier (1997).

• New Techniques and Limitations of Light Curve Analysis by Hadrava (2005) provides an overview on light curve modeling and analysis with a historical introduction very recommended to the reader.


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