Astron. Astrophys. 363, 1081-1190 (2000)

1. Introduction

The importance of the limb-darkening coefficients (LDC) in light curves analysis of eclipsing binary systems is well known but there are also other fields of the astrophysics where they are essential. Among these additional applications we can quote the studies on the stellar diameters and on the line profiles in rotating stars. A more recent field of research where the LDC is important was explored, for example, by Alcock et al. (1997) who used them in the investigation of the gravitational micro-lensing. These coefficients are also needed to detect extra solar planets (see, for example, Mazeh et al. 2000).

The list of papers dedicated to the calculation of such coefficients is large and the covered range of effective temperature embraces hot and moderately cold stellar atmosphere models (see for example, Wade & Rucinski 1985; Claret & Giménez 1990; Van Hamme 1993; Díaz-Cordovés et al. 1995; Claret et al. 1995). In a recent paper, Claret 1998 investigated, for the first time, the behavior of intensity distribution for very cold models specially indicated for brown dwarfs (Allard & Hauschildt 1995; Allard et al. 1997; Hauschildt et al. 1997a; Hauschildt et al. 1997b).

In the first attempts to describe the intensity variation over the disk a linear law was used (Milne 1921) and even today, unfortunately, some authors do prefer such an approximation. However, with the advent of more modern stellar atmosphere models it was shown that this simple law was not adequate (Kinglesmith & Sobieski 1970; Manduca et al. 1977; Claret & Giménez 1990; Díaz-Cordobés & Giménez 1992; Van Hamme 1993). Particularly, for very cold models the 's corresponding to the linear approximation may be as high as 0.15 (Claret 1998).

Once accepted that the limb-darkening is not a linear phenomenon, alternative laws were proposed: quadratic (Manduca et al. 1977; Wade & Rucinski 1985; Claret & Giménez 1990); square root (Díaz-Cordovés & Giménez 1992); logarithmic (Kinglesmith & Sobieski 1970). These laws are represented by the following equations:

Linear

Quadratic

Square root

Logarithmic

where I(1) is the specific intensity at the center of the disk, are the corresponding LDC and µ = , being the angle between the line of sight and the emergent intensity.

On the other hand, the numerical method used to compute the LDC was always matter of discussion. Grygar (1965), Grygar et al. (1972), Manduca et al. (1977), Claret & Giménez (1990), Díaz-Cordovés et al. (1995), Claret et al. (1995), Claret (1998) adopted the Least-Square Method. Another method, based on the flux conservation, was used for example by Kinglesmith & Sobieski (1970), Wade & Rucisnki (1985) and Van Hamme (1993).

In this paper we present a new non-linear approximation for the specific intensity distribution which describes it very accurately and as a consequence, the flux is conserved with high accuracy. In fact, even for a few worse fitting, the flux is conserved with a mean accuracy better than 0.05 per cent. The least-squares method (LSM) was used to fit the model specific intensity distribution. In Sect. 2 we describe the details of the computational method to derive the LDC and Sect. 3 is devoted to analyze the results and compare them with previous calculations, mainly those based on the flux conservation method (FCM). Finally, we give in Sect. 4 a summary of the available data and how they can be retrieved.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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