1 Introduction

Limb-darkening coefficients can only be directly determined for very few stars apart from the Sun. Due to that, when computing light curves (LC) synthesis programs, those coefficients are usually interpolated from tables of theoretical values calculated from atmosphere models.

Over the past decades many analytical approximations have been proposed to describe the variation of the intensity over a stellar surface. Initially, the most adopted was the linear limb-darkening law (Milne 1921):

$$R_{\lambda}(\mu)=\frac{I_{\lambda}(\mu)}{I_{\lambda}(1)}=1-x_{\lambda}(1-\mu)$$ (1)

where $I_{\lambda}$ is the beam intensity at the wavelength $\lambda$,$\mu$ is the co-sinus of $\gamma$, the angle between the atmosphere normal and the beam direction (the line of sight angle, see Fig.1) and $x_{\lambda}$ is the so-called limb-darkening coefficient. This law gives a good description of the limb-darkening in the solar atmosphere that, due to its temperature and physical conditions, is reasonably well represented by a grey atmosphere (for which the limb-darkening is well adjusted by the linear law). Lately, mainly due to theoretical studies on stellar atmospheres, other non-linear laws were proposed, as the polynomial approximations, that better described the effect away from the solar temperature range:

$$R_{\lambda}(\mu)=\frac{I_{\lambda}(\mu)}{I_{\lambda}(1)}=1-x_{\lambda}(1-\mu)-y_{\lambda}(1-\mu)^n$$ (2)

where $y_{\lambda}$ are the non-linear limb-darkening coefficients, n=2 (Wade & Rucinski 1985, using the version 1979 of ATLAS by Kurucz 1979; Manduca et al. 1977 and Claret & Giménez 1990, using versions of UMA, see Gustafsson et al. 1975; Bell et al. 1976) for the quadratic case and n=3 (Van't Veer 1960, quoted by Díaz-Cordovés & Giménez 1992) for the cubic law.

The logarithmic approximation, proposed by Klingesmith & Sobieski (1970) gave very good results in representing their theoretical models, valid for the interval $10\,000\,{\rm K}<T_{\rm eff}<40\,000\,{\rm K}$:

$$R_{\lambda}(\mu)=\frac{I_{\lambda}(\mu)}{I_{\lambda}(1)}=1-x_{\lambda}(1-\mu)-y_{\lambda}\mu\ln{\mu},$$ (3)

confirmed by Van Hamme (VH, 1993), with ATLAS (version 1991). Díaz-Cordovés & Giménez (1992, using ATLAS version 1979) proposed a new non-linear approximation, a square root limb-darkening law:

  

Figure 1: Geometry of illumination in a double star system

$$R_{\lambda}(\mu)=\frac{I_{\lambda}(\mu)}{I_{\lambda}(1)}=1-x_{\lambda}(1-\mu)-y_{\lambda}(1-\sqrt{\mu}).$$ (4)

In subsequent papers (Claret et al. 1995 and Díaz-Cordovés et al. 1995, both using ATLAS version 1991), studies are made only with the square root law and the quadratic and linear ones. In that case, the square root approximation seems almost always to be the best one.

According to VH, the logarithmic law gives the best approximation in the UV, while the square root law is the best in the IR and longer wavelengths. In the optical region, cooler stars are better represented by the logarithmic law and high temperature stars by a square root law.

In close binary systems, the mutual irradiation affects the light curves and the spectra of both stars. The reflection effect present in close binary systems has a strong influence on limb-darkening coefficients, as already noticed by Vaz & Nordlund (1985), Nordlund & Vaz (1990) and Claret & Giménez (CG, 1990). One effect concerns the values of the limb-darkening coefficients, which are changed by the infalling flux. In fact, even the limb-darkening law which best represents the variation of the flux with the line of sight angle may change due to the external illumination. Another important difference, as compared to the normal stellar atmospheres (for which the limb-darkening laws and coefficients are valid over the whole stellar surface), is that the illuminated atmospheres show different limb-darkening coefficients (and laws) for different points on the stellar surface, due to the dependence of these on the incidence angle of the infalling flux. Therefore, the best representation for the center-to-limb variation of the surface brightness of eclipsing binary components in the synthetic LC generation studies is to use the coefficients calculated for the local configuration (i.e. considering the apparent radius, the direction of illumination and the temperature of the companion, see below).

Our goal is to understand how the external illumination affects the limb-darkening laws and coefficients and to present the results in a way to easily account for the effect in LC synthesis programs. We made calculations of bolometric, passband-specific and monochromatic limb-darkening coefficients for all the laws presented above.

In real systems both stars are affected by the mutual illumination. However, in the present study we do not consider the effect of the irradiation from the illuminated (or reflecting) star on the illuminating one (also referred to as heating or source star), i.e. the small changing in the effective temperature of the illuminating star due to the "reflection effect'' from its companion. As we are considering mostly systems for which the illuminated star is cooler than the illuminating one, this is probably a minor contribution. Consequently, we did not take second order effects into account (back reflection), either.

In Sect.2 we describe the method and study the effect of illumination on the limb-darkening laws and coefficients (bolometric, monochromatic and for the passband specific filters of the Strömgren and of the Johnson-Morgan photometric systems), presenting the results as polynomial expressions and tables. We discuss the results in Sect.3, and present our conclusions in Sect.4.