Note that Tp and Teff are equal for spherical stars. Now that we have the local temperature, Tl , and local surface gravityacceleration, gl , we are able to compute the monochromatic or bolometric fluxes and intensities from stellar atmosphere models, briefly as follows. From the gravity brightening law we compute the local bolometric flux. The local bolometric flux plus a stellar atmosphere model enable us to compute the bolometric and monochromatic intensities.

3.2.2 Stellar Atmosphere Models

For a given chemical composition, stellar atmosphere models allow us to compute the monochromatic intensity 4(7^, log g, y) as a function of the effective temperature, Teff, the logarithm, log g, of the surface gravity, and the aspect angle, y■ They allow integration over photometric passbands for the computation of bolo-metric intensities. Frequently used stellar atmosphere models are those by Mihalas (1965), the Uppsala model atmospheres by Gustafsson et al. (1975), and the Kurucz (1979, 1993) stellar atmosphere models. Some light curve programs (e.g., the Wilson-Devinney program) have included stellar atmosphere corrections. Others have empirical or semi-empirical corrections [Hill & Rucinski (1993); Linnell (1991)]. Milone et al. (1992) and Van Hamme & Wilson (2003) apply Kurucz's stellar atmospheres to the Wilson-Devinney light curve model. The requirement to use an accurate model for the radiation physics becomes crucial when the binary components have very different temperatures. The use of stellar atmospheres is most valuable to analyze light curves simultaneously at two or more wavelengths. In order to have a consistent model it is important to incorporate log g correctly. Otherwise there would be only one radiative parameter, namely Teff. The consequence would be that a computed eclipse may be too deep in one passband and not deep enough in another.

3.2.3 Analytic Approximations for Computing Intensities

Light curves directly show relative radiative power or observed radiative flux because the telescope collects integrated light from the stellar system and does not resolve the details of the surface. However, it is useful to think of the process of emission at the stars' surfaces.

The computation of intensities by means of stellar atmosphere models is very time consuming, so usually a simple analytical approximation is used to calculate the specific intensity [cf. Mihalas (1978, p. 2)] at the surface of the stars, namely, the local monochromatic intensity Il (X) which has units of energy/unit surface area/time/solid angle/wavelength. In the simplest case the computation starts with blackbody radiation,29 i.e., Il (Tl, X) = BX(T) with BX(T) being the Planck function

2hc2 1

29 A blackbody is a (hypothetical) perfect radiator of light that absorbs and reemits all radiation incident upon it.

where Planck's constant, h = 6.62608 ■ 10_34 Js, Boltzmann's constant k = 1.3807■ 10_23 J/K, and the speed of light, c = 2.9979 ■ 108 m/s. Alternatively, we also write the Planck function in the form

2nv3 1

The blackbody assumption holds strictly only where there is no net flux of radiation (and thus in no real star, and, after all, we see the radiation that emerges from its surface). It is a useful starting point, and sometimes not a bad approximation to real surfaces, but it is only a very rough approximation for most real stars. Although an ideal radiator has no limb darkening, we can regard (3.2.20) as representing the emergent intensity normal to the surface and introduce a limb-darkening factor, as in Sect. 3.2.4.

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