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Astron. Astrophys. 327, 207-214 (1997)

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4. Discussion

As mentioned in Sect. 1, an accurate estimation of the mass, radius and surface gravity for single stars can be performed through stellar atmosphere and evolutionary models by means of Strömgren-Crawford photometric indices. Of course, an accurate determination can be obtained only in double-lined eclipsing binaries. Nevertheless, Table  1 shows that the calibrations presented in this paper, Eqs. (2), (3) and (4), reach a similar level of precision to that attained by direct interpolation in the models, with a much simpler procedure. This is in principle an unexpected result since, apparently, the enormous amount of information contained in the atmosphere models is not used at all. In fact, this information is present in the reference relation, which gives the physical parameters of a star for a well defined evolutionary status. The good accuracy of the calibrations is strongly related to the observed linearity of the relations, and it is useful that the physical parameters of different stars at the same effective temperature change in such a simple way as a function of the evolutionary status, at least within the main sequence.

[FORMULA] Only stars with [FORMULA] > 2.72

Table 1. Mean residuals in [FORMULA], [FORMULA] and [FORMULA] of different calibrations with respect to the empirical determinations. The calibrations used are, on one hand, those including the interpolation in MD grids (applying the corrections given in Paper I) and in the evolutionary models by SSMM, and on the other hand, the calibrations presented in this work (Eqs. (2), (3) and (4))

However, we still cannot match the internal dispersion of the entry data although a carefully selected sample of detached eclipsing binaries was used. Some uncorrected abundance effects may have a negative influence on the accuracy. In the late region, three stars show a particularly large residual, in [FORMULA], [FORMULA] and [FORMULA], two of which (AI Hya A, KW Hya A) have metallic spectra and the third (GZ CMa B) belongs to a system with a metallic-spectrum primary component. If these three stars are not taken into account, the mean residuals obtained from Eq. (4) are [FORMULA]  0.017 dex, [FORMULA]  0.021 dex and [FORMULA]  0.03 dex, a very significant improvement, which leads to precisions of 4-5% in both masses and radii. However, YZ Cas A, also quoted as an Am star, does not show a large residual. Abundance effects were also observed in the intermediate region (VV Pyx) but could not be corrected for due to the small number of stars in the sample. Another source of uncertainty is the precision of the photometric data. Photometry of eclipsing binaries is, in general, of lower quality than that of single stars. The reason for this is inherent to the calculation of the individual indices through the luminosity ratios which carries larger errors, especially for the least luminous star in the system. The overall result is thus that the suggested simple relations are remarkably linear, but may not be able to describe the physical parameters of a star with the accuracy of the entry data.

The values of the [FORMULA] coefficients computed using both real stars and synthetic stars are not exactly the same as those quoted by Andersen (1991). To explain this discrepancy we reformulated Eq. (1) using [FORMULA] as the evolutionary term instead of [FORMULA]. Similar calculations to those presented in Sect. 3 for both real and synthetic stars were performed. The absolute magnitudes for the real stars were taken from Andersen (1991), Nordström & Johansen (1994a, 1994b) and Clausen (1996), and for the synthetic stars they were computed from the luminosity quoted in the evolutionary models and the bolometric correction of Schmidt-Kaler (1982). The bolometric correction has negligible effect over the final result, as we computed differences of absolute magnitude nearly at the same effective temperature, and so, [FORMULA]. The evolutionary coefficients ([FORMULA]) obtained are presented in Table 2 for the synthetic and real stars. No computation was performed in the intermediate region for the real stars because there were few of them. The agreement between the simulations with evolutionary models and the real stars is again very good. The evolutionary coefficients obtained are compatible with those quoted by Andersen (1991) if we assume a -10 factor when relating [FORMULA] and [FORMULA]. In Table 2 it is also shown that this factor is not constant for all regions. In the late region the coefficient is in better agreement with the -9 value found by Crawford (1979). For the intermediate region, the slopes are slightly larger in absolute value than -17, which was quoted by Strömgren (1966). The discrepancy in the coefficient for the early region is explained, as previously stated, by the qualitatively different construction of Crawford's (1978) absolute magnitude calibration. He used [FORMULA] and our definition is [FORMULA].

a: Simulations with evolutionary tracks; b: Stars in the sample

Table 2. Evolutionary coefficients for the synthetic and real stars using the same expressions as those quoted in Eq. (1) but using [FORMULA] as evolutionary term. [FORMULA] represents the correlation coefficient of the linear least squares fit. [FORMULA], using [FORMULA] for the late region. The values quoted by Andersen (1991) are [FORMULA], [FORMULA] and [FORMULA] using [FORMULA] in all photometric regions


Table 3. ZAMS relation for the early region. [FORMULA], M, R and [FORMULA] from SSMM and [FORMULA] and [FORMULA] from MD grids and NSW interpolation using the correction in [FORMULA] proposed in Paper I


Table 4. ZAMS relation for the intermediate region. [FORMULA], M, R and [FORMULA] from SSMM and [FORMULA] and r from MD grids and NSW interpolation

The full analysis and calculation of the coeffcicients in Eq. (1) was also performed using the ZAMS and the synthetic stars from the evolutionary models computed by Claret & Giménez (1992) and Bressan et al. (1993) at [FORMULA]. The linear trend of [FORMULA] with respect to [FORMULA] was again clearly present. The [FORMULA] coefficients slightly change since the reference line is different, but the final accuracies of the calibrations, estimated through the residuals of real stars were almost identical.


Table 5. ZAMS relation for the late region ([FORMULA]) ([FORMULA] 7000 K). [FORMULA], M, R and [FORMULA] from SSMM, [FORMULA] and [FORMULA] from MD grids and NSW interpolation and [FORMULA]  ([FORMULA]) from Crawford (1979)

Balona (1994) suggested mass and surface gravity calibrations computed through polynomial fits to the evolutionary models of Claret & Giménez (1992) and SSMM. The polynomial expression for the mass is a function of the effective temperature and the luminosity of the star, whereas the surface gravity is obtained though the [FORMULA] and [FORMULA] indices. By using these calibrations, we computed masses and surface gravities for the stars in our sample. To obtain the luminosites we used Strömgren-Crawford photometric indices, the absolute magnitude calibrations of Balona & Shobbrook (1984), Strömgren (1966) and Crawford (1979) for the early, intermediate and late regions, respectively, and the bolometric correction calibration of Balona (1994). The mean residual in [FORMULA] was equivalent to that derived from the biparametric calibrations proposed in the present work, except for the late region where the residual is almost a factor of two larger, probably due to the abundance effects that were not taken into account by Balona (1994). The mean residuals in [FORMULA] and, accordingly, in [FORMULA] are much larger, but, as pointed out by the author, the surface gravity calibrations suggested in that work may not be reliable enough.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998