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

3. Discussion of the results

In Sect. 3.1 we analyze the results for the ATLAS models concerning the 12 passbands and the monochromatic calculations. In addition, we discuss the monochromatic comparison of the LSM and FCM results for the Sun while the models calculated with the PHOENIX code will be the matter of the Sect. 3.2.

3.1. The effects of metallicity and microturbulent velocity on the LDC: ATLAS models

Let us first check if the new approximation we have proposed fulfills the requirements described above. In order to do that, we shall use two criteria: the merit function and the flux conservation. We emphasize that the flux conservation in our formulation is a consequence of the goodness-of-the-fit and it is not imposed a priori. The merit function is written as

where is the model intensity at the point i, the fitted function at the same point, N is the number of points and M is the number of coefficients to be adjusted.

In order to compare the LSM and FCM results, we define the function which gives the ratio of the 's obtained using the FCM and LSM. The upper panels in Fig. 1 show the resulting log for the 12 passbands, including all available values of log g, log T_eff and for 9 values of log [M/H] (around 400 models per metallicity). It is clear that the quality of the LSM fitting is much better than that obtained with FCM for any filter, metallicity, effective temperature or log g. In some cases, the ratio can achieve 1000. We have adopted the logarithmic approximation for the FCM case but similar results are derived when considering the root square or quadratic approximation. The accuracy of the fitting can be also translated to the parameter F'/F, the ratio of the actual flux at the passband a and that using the new limb-darkening law. Those ratios, again for the 12 filters, log g, log T_eff and 9 values of log [M/H], are drawn in the lower panels in Fig. 1. The flux conservation is guaranteed within a high precision. The worst points correspond generally to convective models with log T_eff 3.9. The maximum difference is smaller than 0.0005, a result certainly better than those obtained with any bi-parametric approximations and perfectly within the current numerical limitations. It should be also mentioned that a slightly larger scattering is detected for less metallic models.

Fig. 1. Upper panels: The log of the ratio of the error in the FCM fit to the error in the LCM fit for the 12 passbands and all available log g. Lower panels: The F'/F test. Only the results for 9 metallicities are shown. ATLAS models, microturbulent velocity=2 km/s. Note the logarithmic scale for .

In Fig. 2 we can inspect directly how the angular intensity distribution depends on the metallicity for a few models. Note that there are changes in the behavior of the model with T_eff = 5000 K when going from smaller to higher metallicities. The angular distribution is more pronounced for more metallic models, that is, the linear component predominates. For high effective temperatures, the dependence with the metal content is not so large. Fig. 3 displays how , , and depend on the metal content and effective temperature for a fixed value of log g. For the half-integral values of the exponent of µ in Eq. (6), the corresponding coefficients increase with the metallicity while for the integer ones the contrary occurs. As pointed out above, the LDC for hot models do not depend strongly on the metallicity. The behavior of the LDC is more complicated when one goes from the hot models to cold ones. The discontinuity, around log T_eff = 3.9 is detected as already noted by Díaz-Corbovés et al. (1995) (their Fig. 11) and Claret et al. (1995) (see their Fig. 1). However, such a gap is a characteristic of the atmosphere models themselves: it is due to the onset of convection. Note that the discontinuity in the is dependent on the metal content, being relatively smaller for more metallic models. This discontinuity depends also on the passband, being smaller for the larger effective wavelengths. In Fig. 4 we can see in more detail the influence of the metallicity on the LCD for some selected models. For the models with T_eff = 10000 K and 20000 K, the coefficients are practically independent of the metallicity if, log [M/H] -2.

Fig. 2. The angular distribution of the specific intensity as a function of the metallicity for models with Teff = 5000 K (left) and 30000 K (right) for log g =4.5. Continuous line denotes log [M/H] = -3.0, dashed log [M/H] = -1.0, dotted-dashed log [M/H] = 0.0 and dotted log [M/H] = +1.0. ATLAS models, microturbulent velocity = 2 km/s. Filter y.

Fig. 3. The four LDC as a function of log Teff and metallicity. The symbols have the following meaning: (continuous) log [M/H] = -5.0; (- - -) log [M/H] = -1.0; (-. -.) log [M/H] = 0.0; (....) log [M/H] = +1.0. Results for the y filter and log g = 4.5. ATLAS models, microturbulent velocity=2 km/s.

Fig. 4. The behavior of limb-darkening coefficients as a function of metallicity. The represented models have Teff = 5000 K, 10000 K and 20000 K for log g = 4.5. Filter y.

The influence of the microturbulent velocity on the LDC can be examined by inspecting Fig. 5. As expected, the contribution of the microturbulent velocity is small for the hotter models. For this reason, we only draw the results for the colder ones. The a₁ and a₃ coefficients increase as the microturbulent velocity increase. A similar effect occurs if the metallic content is increased for a fixed microturbulent velocity.

Fig. 5. The influence of the microturbulent velocity on the coefficients of Eq. (6). The symbols are: (continuous, 0 km/s), (- - -, 1 km/s), (-.-, 2 km/s), (..., 4 km/s) and (-...-, 8 km/s). The represented models have log g = 4.5. Filter y.

With reference to the Sun, we have performed monochromatic and band calculations in order to compare with the results obtained with the FCM. The first two frames in Fig. 6 illustrate the angular distribution of the intensities for two wavelengths: 220.5 nm and 2545 nm. The asterisks represent the fit using Eq. (6) while crosses denote the logarithmic law as computed following the FCM. The superiority of our proposed law can be clearly seen. The corresponding flux ratios, F'/F, are 1.0005 and 0.99994 respectively, perfectly within the flux conservation criterion. The F'/F ratio is very good also for other spectral regions as we can see in the third frame of Fig. 6. The high level of disagreement between the intensities model and those computed following the FCM prescription can be checked in the fourth frame of the Fig. 6. We have plotted the difference between the fitted intensity near the limb (for µ=0.01) and that extracted directly from the atmosphere model for the Sun as a function of the wavelength. The quality of our fitting (dashed line) if compared with the FCM one (continuous line) is obvious. The FCM produces discrepancies as large as 0.3 at the ultraviolet. Such differences are smaller for larger wavelengths but they are always larger than those derived using the LSM and Eq. (6). On the contrary, the LSM gives a very good fit for any spectral region. Similar behavior was found for other ints of the disk.

Fig. 6. Comparison between the predictions provided by the LSM (asterisks) and FCM (crosses) for the Sun. The two first frames refer to monochromatic calculations for 220.5 and 2545 nm. Continuous line represents the model intensities. The third picture shows the F'/F test for the Sun while the fourth frame illustrates the deviation of the intensity (model-fitted) at µ = 0.01 computed using LSM-Eq. (7) (dashed) and FCM (continuous, Eq. 4). Sun model atmosphere from ATLAS code. Fifth frame: the intensity distribution for a model with Teff = 10500 K, log g = 4.5, log [M/H] = -5.0 at = 106.5 nm showing limb-brightening.

Note that the logarithmic LDC as computed using FCM (first frame of Fig. 6) predicts a limb-brightening at = 220.5 nm for the solar model, which is in contradiction with the actual intensities. However, limb-brightening can be effectively found for some models. For example, for log [M/H] = -5.0 we have identified this effect for several wavelengths between 23 and 509 nm for effective temperatures between 4250 K and 50000 K. The fifth frame of Fig. 6 shows the intensity distribution for a model with T_eff = 10500 K, log g = 4.5 and = 106.5 nm. Although the effect of limb-brightening is not too large, it can be easily observed in that figure. The limb-brightening we have detected in this model may not be real (Kurucz 2000). Real limb-brightening, may be present in irradiated atmospheres. In 1990 and 1992, Claret & Giménez studied the effects of irradiation on the limb-darkening, effective temperatures, albedo, etc. Under those conditions of irradiation, they have not found limb-brightening in their irradiated models. They only quoted a tendency of the intensity distribution of an irradiated atmosphere to be more uniform. In fact, none of their figures indicated limb-brightening since the intensities in the outer parts of the disk are always smaller or, at least, equal to the previous points. Buerger (1972) found a similar effect.

3.2. PHOENIX models: monochromatic and 12 passbands calculations

The same numerical procedure was used to investigate the angular distribution of the intensities for the PHOENIX models. In the upper panels in Fig. 7 we represent log against log T_eff. All available values of log g for the 12 filters are plotted. Again, the superiority of Eq. (6) and of LSM fitting is notorious given that log , as in the case of the ATLAS models, can achieve 3 orders of magnitude in some cases. The lower panels in Fig. 7 allow us to inspect the quality of the fitting on the ratio of the fluxes. In fact, the worse value of this ratio is around 0.9998. Within this numerical discrepancy, we can consider that the requirement of the flux conservation is fulfilled by Eq. (6) also for the PHOENIX models for the 12 passbands.

Fig. 7. Upper panels: The log of the ratio of the error in the FCM fit to the error in the LCM fit for all models and passbands. Lower panels: The F'/F test. The gap around log Teff = 3.8 is due to that there are not available models in this range. PHOENIX models. Note the logarithmic scale for .

The monochromatic calculations required more than 750000 individual fits ranging from 300 nm up to 2999.5 nm. The goodness-of-the fit is somewhat inferior to the band matches but they are perfectly acceptable. In the upper panels in Fig. 8 we show how the flux ratios depend on the wavelength for four selected models with effective temperatures of 4600 K, 5400 K and 8000 K. The fittings are almost perfect for the hotter models for all wavelengths considered. For colder models, there is a dependence of the fitting quality with . For shorter wavelengths, F'/F depends on the spectral region in a more complicated way. These remarks indicate that there is not only a dependence of the fluxes ratio with but also with the effective temperatures. On the other hand, by comparing the first and the second frames of the Fig. 8 we note a slight dependence of F'/F on log g since for the visible this ratio approximates a little more to the unity for the model with smaller radius.

Fig. 8. Upper panels: The F'/F test for some selected models as a function of the wavelength and log g. Lower panels: the deviation of the intensity (model-fitted) at µ = 0.005 computed using LSM (dashed) and FCM (continuous, Eq. 4). PHOENIX models.

The lower panels in Fig. 8 show the fitting quality using LSM and FCM for the same models at , that is, near the border of the disk. As previously commented, this is a crucial test. It clearly shows, once again, the superiority of the LSM against FCM predictions. Therefore, also for the PHOENIX models, the use of LDC based on the FCM may introduce severe errors. The high quality of the fitting provided by LSM guarantees that the flux is conserved within a desired accuracy and simultaneously that the procedure is able to describe very well the intensity distribution at the limb as well as in any part of the disk for any passband or wavelength.

As a final remark, it should be mentioned that limb-brightening was detected for some colder models ( 2600 K). However, as commented in Sect. 2.3, the models in this range of effective temperature are not enough realistic. The CH₄ may cause temperature inversions which would be responsible for the limb-brightening (Hauschildt 2000).

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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