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2. The B stars

2.1. Methods and results

Following Cramer & Maeder (1979), we use here the reddening-free parameters X and Y which have, as these authors showed, the optimum efficiency for determining the effective temperature and the surface gravity respectively. Although the definition of the X and Y parameters is given in several papers (Cramer & Maeder 1979; NN90), we recall it here for convenience:
eqnarray216

eqnarray218
where U, B1, B2, V1 and G stand for the Geneva colour indices [U-B], [B1-B], [B2-B] etc. Let us recall that the Z parameter allows the separation of the Bp stars (mostly of the Si and SiCr types) from the normal B stars (Cramer & Maeder 1980); it will not be used here, however.

The synthetic colours U, B1 etc. have been computed by one of us (BN) using recent Kurucz models with scaled solar metallicities and a constant microturbulent velocity tex2html_wrap_inline3048 (Kurucz 1993). The passbands used were those determined by Rufener & Nicolet (1988). The X and Y parameters computed in this way are very similar to those obtained for older models, because the additional line opacity of the new models affects essentially the ultraviolet rather than the visible part of the energy distribution. As before, these synthetic parameters do not reproduce exactly the observations and should be corrected. However, we adopted for this particular point another philosophy than that generally adopted to date (Lester et al. 1986; Moon & Dworetsky 1985 etc.). Instead of comparing the observed colour indices with those interpolated in the ``direct'' grids of synthetic colours from the known fundamental parameters, we preferred to compare the fundamental physical parameters with those interpolated in the inverted grid from the observed colours. Briefly, the inversion of a grid implies an iterative, two-dimensional spline interpolation in the ``direct'' grid (where colours are given for regularly spaced values of physical parameters like tex2html_wrap_inline3054 and log g) and results in an ``inverted'' grid giving the physical parameters for regularly spaced values of the photometric parameters. In other words, we first invert the grid of the synthetic X and Y parameters once and for all, following the method described by NN90; then, we obtain for the standard stars interpolated physical parameters, which can be compared with the fundamental ones. For effective temperature, we use the quantity tex2html_wrap_inline3062 rather than tex2html_wrap_inline3064 itself because tex2html_wrap_inline3066 varies linearly with the X parameter and the range of tex2html_wrap_inline3070 is large. In this way, the rms scatter around the mean trend is roughly constant, while it would vary strongly if we used tex2html_wrap_inline3072 directly; this is much safer from the point of view of the least-squares fit, and is equivalent to give a lower weight to the high effective temperatures. We obtain
equation235
and we plot tex2html_wrap_inline3074 vs. tex2html_wrap_inline3076 in Fig. 1 (click here). The trend can be fitted by a straight horizontal line in the present case, because the slope indicated by the least-squares method is smaller than its uncertainty. The interpolated reciprocal effective temperature will then be corrected using the formula:
equation245
where tex2html_wrap_inline3078 stands for the interpolated value of tex2html_wrap_inline3080. The advantage of this method over the previous one is that the grids need to be inverted only once, while different corrections can be tried thereafter, for example as new fundamental data are published. The fundamental stars are those used by NN90, supplemented by new data from Adelman (1988) and Adelman et al. (1993). The Adelman effective temperatures cannot be considered as purely fundamental because they are partly based on a comparison between the observed energy distribution and a theoretical one. However, the Balmer lines were also used to estimate these temperatures, which appear a posteriori quite consistent with the purely fundamental ones of Code et al. (1976). In any case, these temperatures are evidently independent from any possible systematic error in the passbands of the Geneva system.

  figure253
Figure 1: Difference between interpolated and fundamental tex2html_wrap_inline3082 values vs. fundamental tex2html_wrap_inline3084 for the hot stars. The fitted horizontal line is shown; see Table 1 for the key to the symbols

The fundamental tex2html_wrap_inline3086 values are listed in Table 5 (click here), together with the interpolated and corrected values. The uncertainties of the fundamental values are quoted from their authors, while those of the interpolated values are estimated from the photometric errors (for a photometric weight P = 1), as described in NN90. The tex2html_wrap_inline3090 and tex2html_wrap_inline3092 values obtained from the observed colours by interpolation in the corrected grids are compared with their respective fundamental values in Figs. 2 (click here)a and 2 (click here)b.

  figure267
Figure 2: a) Difference between photometric and fundamental tex2html_wrap_inline3094 values vs. the X parameter. The continuous line is the mean while the broken lines define the average rms scatter; see Table 1 for the key to the symbols. b) Same as a), but for tex2html_wrap_inline3098. The horizontal line is arbitrarily set to zero. Notice the large increase of the scatter towards small values of X, i.e. towards the hotter stars

  figure277
Figure 3: Difference between interpolated and fundamental log g values vs. the photometric tex2html_wrap_inline3104 for the hot stars. The regression line is the adopted correction

  figure284
Figure 4: Difference between photometric and fundamental log g values vs. the X parameter. The continuous line is the mean while the broken lines define the average rms scatter

One clearly sees in Fig. 2 (click here)b that for tex2html_wrap_inline3110, the scatter increases strongly towards small values of the X parameter, i.e. towards high temperatures, where the sensitivity of the photometry to temperature is known to strongly decrease. On average, the rms scatter of the difference amounts to 751 K. For X > 0.4 (tex2html_wrap_inline3116), the scatter reduces to 386 K, while it increases to 1388 K for X < 0.4. This scatter is mostly attributable to errors in the fundamental data. Their contributions amount to about 96% of the total scatter. Photometric errors induce only a small dispersion. There is a small systematic zero-point shift of tex2html_wrap_inline3120, essentially due to the hot stars, which were weighted differently by using tex2html_wrap_inline3122 instead of tex2html_wrap_inline3124 to define the correction. A shift of about tex2html_wrap_inline3126 was present with the previous calibration of NN90.

The difference between the interpolated and fundamental log g values follows the trend shown in Fig. 3 (click here), and the interpolated values have to be corrected according to the equation
equation298
where tex2html_wrap_inline3130 is the interpolated surface gravity while tex2html_wrap_inline3132 is the interpolated and corrected tex2html_wrap_inline3134. The fundamental values are listed in Tables 2 (click here) and 3 (click here), as well as the interpolated and corrected ones with their standard deviations. Table 2 (click here) lists the eclipsing binaries, for which the most fundamental values of log g can be determined, and the non-eclipsing but well-known stars Sirius and Vega. In Table 3 (click here) we list the members of the Orion OB1 association, whose surface gravity is inferred from the models of internal structure of Schaller et al. (1992) for isochrones with log t = 6.8 (subgroup c) and 5.7 (subgroup d). Compared with the values given by NN90, the fundamental log g values given here are about 0.06 dex smaller. This is due to the new opacities used by Schaller et al. (1992). Figure 4 (click here) compares the photometric and fundamental values of log g; we see that a very good accuracy can be achieved, of the order of 0.10 dex, provided the star is not too hot. On the whole range of B stars, the rms scatter of the residuals is only tex2html_wrap_inline3146 dex and it is chiefly due to errors in photometric data.

Finally, the inverted and corrected grid for solar metallicity is shown in Fig. 5 (click here), in the form of a diagram log g vs. tex2html_wrap_inline3150 containing the lines of constant X and constant Y parameters. Although such a diagram is unusual, it allows graphical interpolation with the same efficiency as the usual photometric diagrams where lines of constant physical parameters are shown.

2.2. Accuracy of the numerical interpolation

The reliability of the inverted grids and of the bicubic spline interpolation used to determine the physical parameters has been tested in the following way. Knowing the synthetic colours of each atmosphere model, we determined the corresponding physical parameters tex2html_wrap_inline3156 and log g by interpolation in the inverted, but uncorrected grid. Then, we could verify that the interpolated tex2html_wrap_inline3160 and log g values correspond to those defining the model to within 1 K for tex2html_wrap_inline3164 and 0.01 dex for log g. This holds true not only for this particular grid, but also for all other grids presented below; the metallicity [M/H], for cool stars, is also interpolated with an accuracy better than 0.01 dex.


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