Limb Darkening U1 And U2 Coefficients

with the effective limb-darkening factor, D,

D =-2 = 1(1 - x)siny cos y dyd< + x sin y cos2 yl dy d<,

we eventually get (assuming a unit disk, i.e., R = 1)

The linear limb-darkening law is a one-parameter law. It is only a very rough representation of the actual emergent intensity. Accuracy is increased if we consider two-parameter, nonlinear limb-darkening laws. These laws and their coefficients are derived from stellar atmosphere models [see Van Hamme (1993) and references therein], e.g., by least-squares fitting of the chosen expression to the normalized intensities of the atmosphere model, tabulated as a function of m. Some approaches impose the condition of conservation of total emergent flux and some do not. The most simple class of nonlinear relations involves polynomials, such as

For (3.2.32) the associated flux F(P) over the entire disk is

where I0 is again the normal emergent intensity.

In particular, Linnell (1984) has used the quadratic limb-darkening law (p = 2)

D(m) = 1 - x(1 - m) - y(1 - b)2 =: 1 - u1 - u2 + u1 cos y + u2 cos2 y (3.2.34)

with limb-darkening coefficients u 1 = x + 2y and u2 = -y. In that case the monochromatic flux F(2) received from the stellar disk displayed in Fig. 3.17 is given by

F(2) = n (1 - 1 x - 1 y) I0 = n (1 - 1 u 1 - 1 u2) I0. (3.2.35)

The Wilson-Devinney program includes the logarithmic law

DO) = 1 - x(1 - m) - ym ln m, Flog = n (1 - ±x + §y) I0, (3.2.36)

proposed by Klinglesmith & Sobieski (1970). The square-root law

DO) = 1 - x(1 - m) - y (1 - VM) , FSRL = n (1 - 3x - 1 y) Io, (3.2.37)

has been investigated by Diaz-Cordoves & Gimenez (1992) and is now also included as an option in the WD-program.

Note that the nonlinear limb-darkening laws reduce to the linear law (3.2.23) if y = 0. Unlike x in the linear law, the coefficients y are not restricted to nonnegative values. In least-squares analyses we should check how strongly x and y are correlated. Usually, we should not adjust both.

Whatever limb-darkening law is used, the local intensity I follows:

I = I (cos y ; g, T ,X) = Dx (m) In(cos y = 1; g, T ,X), (3.2.38)

where IN(cos y = 1; g, T, X) is the local normal monochromatic intensity, and y, g, and T are also local quantities. The most simple case is to assume IN to be equal to the blackbody radiation defined in (3.2.20). More accurate modeling requires that I be computed from a model atmosphere, with such local effects as spots, prominences, faculae, and gas streams .

More complicated limb-darkening laws have been proposed for the Sun, and the form of the limb darkening varies with wavelength, especially when the radiation comes predominantly from regions other than the visible photosphere. Thus the center-to-limb variation for the Sun from 200 to 300 nm may be fitted with logarithmic among other limb-darkening laws; cf. Kjeldseth-Moe & Milone (1978). In the far-ultraviolet, below ~ 160 nm, limb brightening occurs because ultraviolet arises primarily from the chromosphere where temperature increases with height.

Bolometric limb-darkening coefficients can be obtained by numerically integrating the model monochromatic intensities over all wavelengths. Bolometric coefficients can then be derived similar to the monochromatic coefficients. Van Hamme (1993a) lists bolometric limb-darkening coefficients as well as monochromatic coefficients derived from Kurucz's model atmospheres.

3.2.5 Reflection Effect

In a binary system the presence of a companion star leads to an increased radiative brightness on the side that faces toward the companion. The cause is heating by the radiant energy of the companion. That in turn leads to an increase of the temperature calculated according to (3.2.15). Because heating caused by mutual irradiation is the physical cause, it is somewhat misleading to use the expression reflection effect. However, in very hot binaries a considerable fraction of the incident radiation is simply scattered by free electrons, so in that case the term reflection effect is reasonably appropriate.

As illustrated in Fig. 3.19, one effect of reflection on binary star light curves is to raise the light around the secondary eclipse relative to that near the primary eclipse. Another is to produce a concave-upward curvature between eclipses (curvature opposite to that from ellipsoidal variation). The reflection effect is usually modeled by mean global parameters such as the bolometric albedo, without consideration of the microphysics. For binaries whose components have similar temperatures and are close to but not actually over-contact, it may be necessary to consider multiple reflection [see, for instance, Kitamura and Yamasaki (1984) or Wilson (1990)]. BFAurigae [cf. Kallrath & Kamper (1992), Van Hamme (1993b), Kallrath & Strassmeier (2000)] is an example for such a binary. The first star heats the second star, and the (now warmer) second star then heats the first star more than otherwise expected because of its own raised temperature. This process is iterative, leading to higher temperatures on the facing hemispheres. Tassoul & Tassoul (1983) investigated gradient-induced diffusion as another potentially important effect, at least close to the terminator regions of the reflection-illuminated hemispheres.

Heating caused by reflection can be a strong effect. The current champion is HZ Herculis with a 1m5 reflection amplitude (cf. Lyutyi et al. 1973). Figure 3.18 shows the light variation of the close binary V664 Cassiopeiae, the nucleus of the planetary nebulae HFG 1. According to Grauer et al. (1987) , Bond et al. (1989), and Bond & Livio (1990), an extremely hot primary heats one hemisphere of a larger and cooler main sequence companion in this noneclipsing binary classified as a reflection variable according to the General Catalog of Variable Stars [cf. Sterken & Jaschek (1997)]. The separation is sufficiently small so that reflection produces significant variability of the total light.

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