5 For a comprehensive review of adaptive optics, see Beckers (1993).

even from sites which are nonoptimal both astrometrically and photometrically [see Baldwin et al. (1996) for a description of work by the Cambridge Optical Aperture Synthesis Telescope (COAST) group].

• Intensity interferometry: Brown et al. (1974a, b) measured the diameters of blue stars with an intensity interferometer at a multiple telescope observatory at Narrabri, Australia, beginning in the 1950s. The technique involves determining the correlation between the light received by several collectors (in the Narrabri case, 6.5 m incoherent dishes). See Brown (1968) for a basic review of these techniques as well as the use of lunar occultations.

• Speckle interferometry:6 Speckle observations involve the determination of pattern parameters in which the atmosphere acts as a diffusing screen [see Schlosser et al. (1991, pp. 119-123), for a simple but clear exposition]. The method has proven very fruitful for visual binary work. By the mid-1990s, the CHARA group at Georgia State University, led by Harold A. McAlister, made more than 40,000 speckle observations of more than 7,300 stars or systems, and many more observations were being carried out (Mason et al. 1996).

Long-baseline interferometry permits the resolution of many spectroscopic binaries. As is the case for all well-determined visual binaries, coupled with high-precision radial velocity data, the parameters can yield all the geometric elements of the orbits. To a certain degree, the relative brightness of the components can also be obtained and in combination with photometry can provide a distance. An excellent example of such collaboration can be found in the work of Scarfe et al. (1994) and Van Hamme et al. (1994). Interferometric observations from space offer many advantages, among them a spectral range from the far-ultraviolet to the far-infrared. This means the possibility of observations of objects such as protostar binaries which radiate in the far-infrared. The techniques of long-baseline optical and infrared interferometry were reviewed by Shao & Colavita (1992).

A calibration of stellar surface brightness making use of the measured sizes of stars was carried out first by Wesselink (1969). This information source can be useful in several ways, e.g., initial values for radii (given a spectroscopic estimate of luminosity and photometric color index) for light curve modeling might be obtained from his Fig. 2. This plot shows the radii of stars superimposed on a color-magnitude array. A short catalogue of derived stellar sizes was compiled by Wesselink et al. (1972). Spectroscopic Binaries

The detection and analysis of spectroscopic binaries is not subject to geometrical resolution limits as are angular measurements. With sufficient light gathering power, it is possible to investigate spectroscopic binaries even in nearby galaxies and to derive the luminosity ratio and mass ratio.

6 Although speckle interferometry is called interferometry, the reader should be aware that the concept is completely different from the others kinds of interferometry mentioned above. Whereas the latter involves a certain base line, speckle interferometry is rather a correlation method.

The luminosity ratio, i.e., the relative luminosities of the component stars, can be derived from spectra alone, by a method developed by Petrie (1939) at the Dominion Astrophysical Observatory. The determination of the mass ratio is more difficult. It can be derived only spectroscopically under favorable conditions, namely where the components have similar luminosities (say within a factor of 5). In that case, a radial velocity curve1 may be observed for each component; both radial velocity curves enable us to compute the spectroscopic mass ratio. Note that spectro-scopists usually define the mass ratio as the more massive over the less massive star.8 Such a system is called a double-lined spectroscopic binary (SB2). If only one component can be observed spectroscopically, the system is called a single-lined spectroscopic binary (SB1). In this case, a useful quantity defined in (4.4.34) and known as the mass function can still be obtained which, according to (4.4.38), provides a lower bound on the sum of masses and gives a lower bound on the unobserved mass in any case (because the observed star cannot have mass less than zero).

The calculation of masses and radii requires the inclination, i, which cannot be found from spectroscopic data alone. In the SB1 case, the mass ratio also is not known. If i is sufficiently large9 and the separation of the components is sufficiently small, the binary appears as an EB. Eclipsing Binaries

A variable star observer measures a time-dependent flux, the display of which versus time or phase (the repeated foldings of the time into the period of variation) is known as the light curve. The acquisition and reduction of photometric observations will be discussed in Sect. 2.1. EBs establish a special class of variables stars. For the nomenclature and classification of variable stars we refer the reader to the book Light Curves of Variable Stars by Sterken (1997) and Wilson (2001). Whereas eruptive, pulsating, rotating, and cataclysmic variables are said to be intrinsic variables caused by different physical mechanisms, EBs are extrinsic variables requiring models including both astrophysics and geometry.

As we have indicated, an eclipsing variable is a binary system whose orbital motion is in a plane sufficiently edge-on to the observer for eclipses to occur. The smaller the orbit relative to the sizes of the stars, the greater the likelihood of eclipses. For a special subgroup of EBs (so-called over-contact binaries, with a common envelope) eclipses may occur, although perhaps not perceptibly, even if the

7 A radial velocity curve is a plot of the star's velocity component toward (or away from) the observer versus time or orbital phase (essentially the fraction of an orbital cycle). See Fig. 3.27 for an example and Sect. 2.2 for details.

8 See Sect. 2.8 for terminology concerning stars 1 and 2, as well as the uses of the term "primary" in referring to components and eclipses.

9 The inclination i is defined such that an edge-on orbit has i = 90° .

inclination is as small as 35°. Illustrations demonstrating the visibility of eclipses at low inclinations for an over-contact system are in Sect. 8.1 (Figs. 8.1, 8.2, and 8.3). These binaries usually have orbital periods of less than 10 days and in most cases less than 1 day. Among the exceptions are some rare cases of hot and/or developed systems. The longest period EB known at present is e Aurigae [see, for instance, Caroll et al. (1991)] with an orbital period of 27.1 years. According to Kepler's third law this binary has an orbit relatively large10 compared to the sizes of the components. Historically, considerations concerning the likelihood of eclipses lead to a connection between EBs and "close binaries." In the early days, a "close binary" was defined as a binary with component radii not small compared to the stars' separation. This definition was later replaced by a more physical definition related to the evolution of the components by Plavec (1968), which we discuss at the end of Sect. 1.2.3.

EB studies often involve the combination of photometric (light curve) and spec-troscopic (mainly, radial velocity curve) data. Analysis of the light curve yields, in principle, the orbital inclination and eccentricity, relative stellar sizes and shapes, the mass ratio in a few cases, the ratio of surface brightnesses, and brightness distributions of the components among other quantities. If radial velocities are available, the masses and semi-major axis may also be determinable. Many other parameters describing the system and component stars may be determined, in principle, if the light curve data have high enough precision and the stars do not differ greatly from the assumed model. The prediction of the information content of particular light curves has been a major topic of concern in binary star studies; the exposition of this topic is an important component of the present work also.

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