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

2. Numerical method and the models of stellar atmosphere

Before applying the LSM, we integrate the specific intensities for each µ for the most commonly used photometric passbands following the equation:

where I_a(µ) is the specific intensity in the band a, I(,µ) is the monochromatic specific intensity and S() is the response function which considers the terrestrial atmospheric transmission, filter transmission curves, detector sensitivity and reflection from the aluminum coated mirror. Atmospheric transmission and reflection from aluminum mirror were taken from Allen (1976). For R and I, the transmission curves were taken from Bessel (1990) and the sensitivity of the CCD Tektronics was taken from Peletier (1994). For J, H, and K, the transmission curves were taken from Alonso et al. (1994) (the sensitivity of the In-Sb detector has been included). A small dependence with the spectral type of the effective wavelength was detected for the filters . The sensitivity of the detectors must be compatible with the transmission curves of the filters since, in general, older detectors are not sensitive in the ultraviolet and/or in the infrared. The monochromatic calculations were performed following their classical definition.

2.1. A new non-linear law

It is known that the behavior of the distribution of the specific intensities depends on the effective temperature and local gravity. For example, Díaz-Cordovés (1990) concludes that the square root law is more adequate to hot stars while for colder ones the quadratic approximation represents the distribution better. Only for effective temperatures of the order of 5500 K the linear approximation can be considered as marginally good. Similar conclusions were obtained by different authors too. It is then clear that it would be very useful a single and accurate law for the whole spectrum of effective temperature and gravities. In order to improve the situation we introduce in this paper a new non-linear law which, as we shall see later, is capable of reproducing the specific intensities over the whole disk and that simultaneously preserves the flux with high accuracy.

The law that we propose can be written as

or, in a more compact form

A criticism one can make on such an approximation is that now we have four coefficients instead of one or two. However, this is the price one has to pay to describe correctly a non-linear phenomenon. As shown in previous works, simple laws with two coefficients are not good enough to reproduce the intensities though they do it better than the linear one. Even in the hypothetical case that such laws could do it, there is not a single law, i.e., one has to separate the HR diagram in "laws" in order to use the LDC adequately. Moreover, Eq. (6) produces 's of the order of 2 magnitudes smaller than any of the laws quoted previously. The implementation of this equation in the codes of light curve synthesis is easy as well as in the other fields of the astrophysics where the LDC are need. The apparent complexity of Eq. (6) is highly compensated by accurate calculations and by the use of a single law for the whole HR diagram. In fact, the four coefficients are not an extra weight to light curves investigators. The modern light curves codes tend to process all the information from the atmosphere models and, in the near future, the LDC may be even computed internally by those codes.

At first sight, it would seem that just by increasing the number of parameters or the degree of the polynomial, we would get better matches to the intensity distribution. We have performed some numerical experiments to investigate that. For example, if we take the expression 1 - the resulting 's are higher than those with Eq. (6). Others combinations yield similar results.

2.2. Least-squares and flux conservation methods

The controversy on the best method to derive the LDC is an old matter of discussion. Summarizing the question, the LSM gives a better fit to the intensity distribution but the resulting flux, as computed using the fit, does not always equal the actual flux obtained directly from the model. On the other hand, the flux conservation method (FCM) uses this constraint to derive the LDC. However, although by definition the flux is conserved, the distribution of the intensities is not well described as required and the respective 's are usually high, mainly near the limb.

The disagreements I(model) - I(fitted) derived from the LSM and FCM formulations are often only qualitatively discussed. In the present work we shall present an exhaustive analysis of this crucial point. As we will see later, the 's associated to the FCM are as large as 1000 times those provided by the LSM and Eq. (6). This is a serious restriction to the FCM since a good match is necessary to compute the loss (gain) of light during the eclipses, for example. Another weak point of the FCM is that if a non-linear law is used, an extra condition must be introduced. Often, this extra condition is arbitrary: some authors imposed that the limb-darkening law in question should produce the same mean intensity as the atmosphere model. Others make use of a different procedure. This implies that the derived LCD will depend on the selected extra conditions.

The FCM does not make use of any direct information on how the specific intensity is distributed over the disk; its contribution is realized only through integrations. An additional criticism to the FCM is that it does not distinguish, a priori, one law from another since for any adopted law (linear or not) the flux is conserved. The users of such a numerical procedure have to use the corresponding 's to select the best law. It is interesting to note that the test is then decisive whereas it is not when comparing the capability of reproducing the specific intensity distribution of the two competing numerical methods.

It is important to keep in mind that not only the flux conservation is essential to any limb-darkening law. That law must match two conditions: it must be capable of describing accurately the intensity distribution when the stellar disk is scanned and this must lead to the flux conservation. The FCM, by definition, is only capable of fulfilling the second condition. Once obtained, the LCD cannot be adjusted to improve the fitting to the model intensities. On the other hand, the LSM, using an adequate law, can fulfill both conditions. Therefore, we search for a compromise between the best fit to the points and the flux conservation within a small tolerance limit. One should criticize this numerical limitation but it should be remembered that this concept, flux conservation, must be considered with care. In order to check the accuracy of the actual flux calculation, we have performed some numerical experiments by changing the number and position of the µ points. The results, based on the ATLAS grid with log [M/H] = 0, reveal that the errors introduced in the actual fluxes due to these changes are of the same order - sometimes much larger - as the differences between the actual fluxes and those computed using the new limb-darkening law. Other numerical limitations are recognized even by the FCM users (see Wade & Rucinski 1985) who found differences sometimes larger than 2 per cent between the fluxes they computed and those provided directly from the Kurucz's grids (1979). Such numerical limitations shall restrict the flux conservation within certain tolerance, which obviously must be as small as possible. Remember that the numerical limitations due to the flux calculations also affect the LDC computed following the FCM.

With these remarks on mind, we decide to adopt the LSM in the present work using the new law expressed by Eq. (6). One is interested in a approximation that:

1. uses a single law which would be valid for the whole HR diagram

2. would be capable to reproduce very well the intensity distribution

3. the flux would be conserved within a very small tolerance

4. would be applicable to different filters as well as to monochromatic values

5. would be applicable to different chemical compositions, effective temperatures, local gravities and microturbulent velocities.

2.3. Description of the stellar atmosphere models

We shall use an extensive amount of data which we separate following their origin. In all cases the adopted geometry is plane-parallel. Hydrostatic equilibrium and LTE are assumed. Most of these comprehensive data were derived from the ATLAS code and were kindly prepared for the present work by Kurucz (2000). It should be emphasized that such ATLAS models incorporate several improvements with respect to the 1993 set. The effective temperatures are between 3500 and 50000 K with log g varying from 0.0 up to 5.0. The adopted mixing-length parameter was 1.25 and 1221 wavelengths were used at 17 values of µ. The present calculations are not limited to the solar composition but they were also performed for 19 metallicities. The logarithms of the metal/hydrogen ratio are: -5.0, -4.5, -4.0, -3.5, -3.0, -2.5, -2.0, -1.5, -1.0, -0.5, -0.3, -0.2, -0.1, 0.0, +0.1, +0.2, +0.3, +0.5, +1.0, for a microturbulent velocity of 2 km/s. With these characteristics it will be possible, for the first time, to study systematically the influence of the metal content on the LDC. Of course, the importance of this fact is not only in the theoretical intercomparison but also under the observational point of view given that observations of eclipsing binaries and gravitational micro-lensing investigations are now being carried out in chemical environments different from the solar one. For the solar abundance, calculations were performed for four additional values of the microturbulent velocity: 0, 1, 4 and 8 km/s. With this comprehensive set of data it is possible to analyze the influence of this parameter on the LCD. There are about 400 models for each metallicity, completing about 9500 models.

The second set of models is derived from the PHOENIX code and they were the subject of a previous study (Claret 1998) but they were included here in order to test the law proposed in Eq. (6) at temperatures lower than in the ATLAS grid. Moreover, we will also compute the monochromatic and bolometric LCD for such models since in the mentioned work only the results for 12 passbands were presented. Let us remember some of its basic input physics. One hundred molecules and about 2 million atomic lines were included. Dust formation and dust opacities are not taken into account. This affects mainly models with effective temperature smaller than 3000 K. The range of effective temperature is 2000 K log T_eff 9800 K and log g is between 3.5 and 5.0 and only models with the solar metallicity were considered. The selected mixing-length parameter was 1.0 and the microturbulent velocity is 2 km/s. The number of wavelengths is 5399 ranging from 300 nm up to 2999.5 nm. The number of emergence angles is similar to that of ATLAS (16).

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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