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

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3. Calibrations

Taking into account the qualitative similarities of the evolutionary changes in radius and luminosity for a given mass, in terms of the effective temperature, when compared to absolute magnitude, Andersen (1991) suggested the construction of mass, radius and surface gravity calibrations in an equivalent way to the absolute magnitude calibrations of Crawford (1975, 1978, 1979), i.e. using a relation for the ZAMS and taking into account the evolutionary stage of each star.

Thus, the general form of the proposed equations becomes:

[EQUATION]

where q denotes in each case M, R and g in units of [FORMULA], [FORMULA] and cgs, respectively, [FORMULA] and [FORMULA] are intrinsic photometric indices related to effective temperature and surface gravity, [FORMULA] stands for the relation of the ZAMS stars as a function of [FORMULA], and [FORMULA] is the term that accounts for the evolution of the star. Depending on the photometric region, [FORMULA] and [FORMULA] have different correspondences with Strömgren-Crawford indices. Thus, [FORMULA] and [FORMULA], [FORMULA] and r, [FORMULA] and [FORMULA] are the [FORMULA] and [FORMULA] indices for the early, intermediate and late regions, respectively.

Andersen (1991), considering that the coefficient of the evolutionary term in Crawford's (1975, 1978, 1979) absolute magnitude calibrations was always close to -10, assumed that [FORMULA] and [FORMULA], defined as [FORMULA] and [FORMULA], both have the same evolutionary scale factor. Thus, [FORMULA], [FORMULA] and [FORMULA] ought to be independent of the photometric region. Due to the lack of complete [FORMULA] individual photometry for many of the binaries in his data sample and the small number of systems near to the ZAMS, he used stars, belonging to different binaries, with almost identical effective temperatures but different degrees of evolution to obtain differences in both the physical quantities and the absolute magnitude. In this way, assuming -10 for all regions when relating [FORMULA] and [FORMULA], he estimated the evolutionary factors to be [FORMULA], [FORMULA] and [FORMULA], constant to about 10%. However, it is advisable to approach the construction of the calibrations separately for each region, since, as we will see in Sect. 4, the assumed -10 factor is not constant for all photometric regions.

The equations for [FORMULA], [FORMULA] and [FORMULA] are obviously not independent, since there exists a linear relationship between them: [FORMULA]. Thus, from Eq. (1) and the previous expression we obtain that [FORMULA]. Nevertheless, the three equations for [FORMULA], [FORMULA] and [FORMULA] will be explicitly shown in the following sections basically to facilitate their application.

As commented in Paper I, the number of evolved stars within the sample is low, especially for the early region. Therefore, to provide an appropriate coverage of the different evolutionary phases from the ZAMS to the termination age (TAMS) of the main sequence, we have also used points in the form of synthetic stars from the evolutionary tracks by SSMM (adopting Z=0.02 for the metal content). The Strömgren-Crawford indices for these points were computed by using the theoretical predictions for [FORMULA] and [FORMULA] from the models and the interpolation of Napiwotzky et al. (1993, hereafter NSW) applied to the photometric grids of Moon & Dworetsky (1985, hereafter MD). Moreover, we considered the correction to the photometric surface gravity of the early region given by Eq. (2) of Paper I. As we will see in the next subsections, we thus obtained a uniform distribution of "stars" spread over a wide range of evolutionary stages with such a procedure. These synthetic stars were used first to check the expected agreement with the observational data from our sample as well as the potential relevance of chemical composition differences. After this, they were used to compute the [FORMULA] evolutionary coefficients.

3.1. ZAMS relations

To establish the unknown coefficients of Eq. (1), a relation between masses, radii, surface gravities and photometric indices ([FORMULA] and [FORMULA]) is needed. Even though we have adopted a ZAMS as reference relation, this is not strictly necessary. The validity of the formal expressions should remain when changing the reference line, i.e. we only need the [FORMULA] coefficients to be consistent with the relation adopted.

The best way to define a ZAMS relation in general is by using a sample of non-evolved eclipsing binary stars to trace a lower envelope, defining all needed parameters in a purely observational way and thus, completely model-independent. However, the number of eclipsing binary systems in the sample near the ZAMS is too low to attempt such a definition. In this situation, there are two different, although not completely independent, ways to define the ZAMS. On one hand, stellar evolutionary models provide a "theoretical" ZAMS giving M, R and [FORMULA] for different [FORMULA], which can be related to the photometric indices by means of the photometric grids. On the other hand, the "observational" ZAMS defined from colour-colour diagrams can be related to M, R and [FORMULA] through the photometric grids and the evolutionary models. These latter calculations obviously cannot be performed when the observational ZAMS is below the theoretical one (i.e. if for a given effective temperature, the surface gravity of the observational ZAMS is larger than that of the theoretical value). To avoid this, we have adopted the first theoretical ZAMS definition using ZAMS of SSMM evolutionary models for solar metallicity (i.e. [FORMULA]). The ZAMS relations for the early, intermediate and late regions are listed in Tables 3, 4 and 5, respectively.

3.2. Early region

As stated in Paper I, stellar atmosphere models (Kurucz 1979, 1991) used to build photometric grids may not provide reliable surface gravity determinations for stars with [FORMULA] 20000 K. For stars with 11000 K [FORMULA] 20000 K, the correction proposed in Paper I yields an accuracy in the determination of surface gravity of 0.07 dex. Due to the expected uncertainties for stars hotter than 20000 K, we restricted our study to [FORMULA] lower than this value.

Fig. 1 shows the relation between [FORMULA] and [FORMULA] for both synthetic stars and 17 real stars. The agreement between the synthetic and the real stars is remarkable, even for the more evolved cases in the sample. The linearity of the relation is also clearly shown, and the use of the correction given in Paper I substantially improves both the linearity of the synthetic stars and their compatibility with the real ones.

[FIGURE] Fig. 1. Mass, radius and surface gravity calibrations as a function of an evolutionary term for the early region. Filled circles are real stars whereas open circles are synthetic stars

The calibration obtained for each physical parameter has the following expression:

[EQUATION]

These expressions are valid for main sequence stars with 11000 K [FORMULA] 20000 K. The mean residuals of the real stars in the sample are [FORMULA]  0.027 dex, [FORMULA]  0.046 dex and [FORMULA]  0.08 dex. Metallicity effects in this early region can be ignored.

3.3. Intermediate region

As shown in Fig. 2 , the number of stars in the observational sample that belong to the intermediate region is very low. Nevertheless, we again find a good agreement between synthetic and real stars. Therefore, we were able to compute the calibrations for each of the physical parameters and obtain the following values:

[FIGURE] Fig. 2. Same as Fig. 1 for the intermediate region

[EQUATION]

where [FORMULA]. These calibrations are valid for main sequence stars with 8500 K [FORMULA] 11000 K. The mean residuals of the real stars in the sample are: [FORMULA]  0.041 dex, [FORMULA]  0.057 dex and [FORMULA]  0.10 dex.

The small number of stars with good photometric data and absolute dimensions in this region did not allow us to analyze corrections for the abundance effects, though their influence may not be negligible. For example, Andersen (1991) pointed out important metallicity effects for VV Pyx AB, and this system shows the largest residual in the [FORMULA] in Fig. 2. However, the discrepancy is not observed in the [FORMULA] or [FORMULA] plots.

3.4. Late region

The late region can be divided into two subregions (splitting at [FORMULA], i.e. [FORMULA] 7000 K), roughly A and F stars, with different trends, as shown in Paper I. First, we will focus on the hotter subregion including stars with [FORMULA] (7000 K [FORMULA] 8500 K, A stars) and later in this section we will discuss the behaviour of stars with [FORMULA] (6000 K [FORMULA] 7000 K, F stars).

As seen in Fig. 3, synthetic stars in the hot subregion show a well defined linear behaviour, but the real stars have a much larger dispersion, even though the general trend is similar. The reason for this different behaviour is changes in chemical composition. While all the synthetic stars have the same chemical composition (Z = 0.02), the stars in our observational sample present important variations.

[FIGURE] Fig. 3. Same as Fig. 1 for the late region ([FORMULA])

A linear least squares fit of the synthetic stars for each of the physical parameters provides values of 0.64, 1.80 and 2.97 for the [FORMULA] coefficients. The derived mean residuals for the real stars are then: [FORMULA]  0.039 dex, [FORMULA]  0.116 dex and [FORMULA]  0.19 dex, which are roughly a factor of two larger than those of the early region. Since the cause of the dispersion is likely to be the presence of stars with different metallicities, we further analyzed the residuals as a function of a photometric index related to the metallicity. Fig. 4 shows a of the aforementioned residuals with respect to [FORMULA], and the correlation is obvious. Thus, an additional linear term with [FORMULA] was considered in the calibrations to obtain:

[FIGURE] Fig. 4. Differences between the empirical [FORMULA] and the determination using [FORMULA], [FORMULA] and [FORMULA] vs. [FORMULA] for the late region ([FORMULA]). The linear least squares fit to the stars is shown as a straight line

[EQUATION]

which are applicable to stars in the main sequence belonging to the late region and with [FORMULA] 7000 K. The derived mean residuals are [FORMULA]  0.021 dex, [FORMULA]  0.050 dex and [FORMULA]  0.08 dex for the real stars in the data sample. These values are now very similar to those obtained for the early and intermediate regions. The zero point in [FORMULA] takes into account that SSMM evolutionary models were built for solar metallicity ([FORMULA]) yielding [FORMULA] for both the synthetic stars and the ZAMS (using Smalley's (1993)) expression relating [FORMULA] to [FORMULA]).

Guthrie (1987) studied the influence of the metallicity effects on the evolutionary index [FORMULA], and defined a new [FORMULA], which only accounts for the evolution of the star. By using this definition, the previous equations can be reformulated to include the contribution of [FORMULA] both in the ZAMS relation and in the [FORMULA] index:

[EQUATION]

and considering new reference lines [FORMULA] that are dependent on [FORMULA]:

[EQUATION]

This new formulation in terms of [FORMULA] and [FORMULA] allows us to plot Fig. 5, which is similar to Fig. 3, but takes metallicity effects into account. The initial dispersion has been greatly reduced by including a new term depending on [FORMULA], and the agreement between real and synthetic stars is quite good.

[FIGURE] Fig. 5. Same as Fig. 1 for the late region ([FORMULA]). Metallicity effects are included in the definition of [FORMULA], [FORMULA], [FORMULA] and [FORMULA] (see text)

[FORMULA] are actually different ZAMS relations for different metallicities, showing how the physical parameters of the ZAMS change as a function of [FORMULA] for a given [FORMULA]. These corrections to the ZAMS depending on the chemical composition were tested by using the evolutionary models computed by the Geneva group (SSMM, Schaerer et al. 1993a, 1993b, and Charbonnel et al. 1993) for several metallicities ranging from [FORMULA] to [FORMULA]. We used Smalley's (1993) calibration relating the [FORMULA] parameter (in our case computed through Z) to the [FORMULA] index. For 7000 K [FORMULA] 8500 K the stellar surface gravity of a ZAMS relation is nearly constant in [FORMULA] but dependent on the metallicity (of the model). We performed a linear least squares fit to the mean value of the surface gravity of each ZAMS as a function of [FORMULA], deriving a slope of [FORMULA], with a correlation coefficient of [FORMULA]. This shows how the surface gravity of the stars on the ZAMS changes as a function of [FORMULA] for a given [FORMULA]. It is important to notice that in the definition of [FORMULA] the free parameter is [FORMULA] instead of [FORMULA]. For a given metallicity, [FORMULA] means [FORMULA], but in this case we consider different metallicities and this is no longer satisfied. In any case, since [FORMULA] is nearly constant for a range of [FORMULA], the value of the previous slope can be directly compared with the value of 2.21 derived for the real stars. The verification of the [FORMULA] and [FORMULA] expressions is not possible, unless we know how the [FORMULA] index varies as a function of the metallicity, since the mass and the radius for the ZAMS are not just a function of the metallicity but also of the [FORMULA].

Finally, we will briefly deal with the coolest stars in this region, having [FORMULA], i.e. 6000 K [FORMULA] 7000 K. Fig. 6 shows no clear trend for the real stars and a relatively large dispersion of the synthetic stars with non-linear behaviour. When plotting the residuals of the real stars from a least squares fit to the synthetic ones, no clear correlation with the metallicity index [FORMULA] was observed. The large uncertainties in the photometric surface gravity determination through model atmospheres, which are discussed in Paper I, may have a negative influence on the precision of the ZAMS relation, the synthetic stars, and, accordingly, on the attempted calibrations. The lack of linearity could also be due to the physics involved, which may not allow us to establish simple linear relations. Thus, in the present situation it is not advisable to suggest any calibration for this subregion.

[FIGURE] Fig. 6. Same as Fig. 1 for the late region ([FORMULA])
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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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