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4. Determination of effective temperatures

It is firmly established that the Balmer lines are optimum indicators for tex2html_wrap_inline2653 because of their virtually null gravity dependence (Smalley & Dworetsky 1993; Furhmann et al. 1994). In this work, effective temperatures for La Palma observations have been obtained from the comparisons, using a least-squares fitting technique, between the observed Htex2html_wrap_inline2655 profile and a grid of synthetic profiles generated using ATLAS8 (Kurucz 1979a). The gravity and metallicity were fixed to tex2html_wrap_inline2657 and [M/H]=0.0 respectively. Previous to any comparison it was necessary to convolve the synthetic spectra with the instrumental and rotational profiles. The instrumental profile was calculated from the Thorium-Argon calibration spectra taken every night whereas the rotational profile was derived for each spectra by using Eq. (1), where tex2html_wrap_inline2661 was calculated following the method described in the previous section. A step of 100 K was assumed between models.

Likewise, we tried to apply this method to McDonald observations. However, the spectral range covered by the order where tex2html_wrap_inline2663 appears was not wide enough to embrace the wings of the line. This produced a continuum indetermination giving rise large errors in tex2html_wrap_inline2665. An alternative method based on the variations in intensity of the wings of tex2html_wrap_inline2667 at two wavelengths (tex2html_wrap_inline2669, tex2html_wrap_inline2671) was proposed. The selection of the wavelengths was done in such a way that they lie in the region of the tex2html_wrap_inline2673 profile with the greatest dependency with temperature. Using the set of non-variable stars and variable stars with tex2html_wrap_inline2675 (which would correspond to a variation in temperature of tex2html_wrap_inline2677, that is, the assumed step in tex2html_wrap_inline2679) observed in McDonald and whose temperatures were determined using tex2html_wrap_inline2681 from La Palma spectra, we obtained a linear relationship between the intensity ratio and temperatures (Fig. 3.1 (click here)). An error of tex2html_wrap_inline2683 was assumed for La Palma spectra and tex2html_wrap_inline2685 for McDonald observations.

Table 5: Influence of the limb-darkening coefficient for different rotation velocities

Effective temperatures of the observed stars are displayed in Tables 6, 7. The effective temperature of UZ Lyn was not calculated using tex2html_wrap_inline2699 since the photometric calibrations indicate a temperature tex2html_wrap_inline2701: at these temperatures, Balmer lines also depend on surface gravity and they cannot be used. Moreover, the effective temperature of CY Aqr has not been calculated using tex2html_wrap_inline2703 due to the low signal-to-noise ratio which prevented us from deriving its projected rotation velocity. Also, some McDonald spectra have no tex2html_wrap_inline2705 using tex2html_wrap_inline2707 due to problems in the flatfields of one of the nights.

4.1. Comparison with integrated flux methods

The most natural way of checking the method of calculating tex2html_wrap_inline2713 based on Balmer lines would be to use it for standard stars with accurate values of tex2html_wrap_inline2715. Code et al. (1976) and Hayes (1978) gave a list of stars with fundamentally determined values of tex2html_wrap_inline2717. Three stars from Code et al. (1976) were also observed by us (Table 8): no systematic differences were found between the tex2html_wrap_inline2719 derived from tex2html_wrap_inline2721 and those given in Code et al. (1976).

Table 8: Comparison between the effective temperatures given in Code et al. (1976) and those derived using tex2html_wrap_inline2723. An error of tex2html_wrap_inline2725 was assumed for the tex2html_wrap_inline2727 measurements

4.2. Comparison with tex2html_wrap_inline2739 derived via photometric calibrations

The temperatures obtained using tex2html_wrap_inline2741 have been compared with those obtained from the following tex2html_wrap_inline2743 photometric calibrations: Petersen & Jørgensen (1972) (PJ72), Philip & Relyea (1979) (PR79), Moon & Dworetsky (1985) (MD85), Lester et al. (1986) (LGK86a,b) and Balona (1994) (B94). Except for PJ72 which relies on the scale of temperatures tex2html_wrap_inline2745 (Popper et al. 1970) together with the relation tex2html_wrap_inline2747 (Crawford & Perry 1966), the remaining calibrations use the ATLAS8 code (Kurucz 1979a) to generate the synthetic indices and colours except for LGK86b who also used a different version of ATLAS8 (Kurucz 1979b) with a modified treatment of convection (Lester et al. 1982). Temperatures provided by the Petersen & Jørgensen (1972) and Balona (1994) calibrations have been derived using an interpolation formulae whereas temperatures calculated with the rest of photometric calibrations have been derived from grids assuming a step of 50 K. All calibrations use tex2html_wrap_inline2749 as the temperature indicator except PR79 who use (b-y). The tex2html_wrap_inline2753 and (b-y) values of the tex2html_wrap_inline2757 Sct stars have been taken from Rodríguez et al. (1994) except for UZ Lyn which does not appeared in this catalogue and its values were taken from García et al. (1993). For the sample of non-variable stars the tex2html_wrap_inline2759 and (b-y) were taken from the catalogue of Hauck & Mermilliod (1990). Both for variable and for non-variable stars, the dereddened indices were obtained using the dereddening law given by Philip & Relyea (1979). A comparison between the values of tex2html_wrap_inline2763 derived using tex2html_wrap_inline2765 and the photometric calibrations given above appears in Fig. 4.2 (click here). Those stars with variations in amplitude tex2html_wrap_inline2767 which would correspond to variations tex2html_wrap_inline2769 along the pulsation cycle were not considered.


PJ72 calculated effective temperatures which are, on average, 196 K higher than the tex2html_wrap_inline2779 values. These differences can be interpreted in terms of the old relationships and the few stars which this calibration is based on. In fact, these systematic differences disappear when a more modern calibration with more complete bolometric corrections and more numerous and precise angular diameters is applied (Bohm-Vitense 1981). The effective temperatures derived from PR79 are, on average, 155 K hotter than the Htex2html_wrap_inline2781 temperatures, this difference being larger when tex2html_wrap_inline2783 is hotter which can be attributed to the fact that only one star, Vega, with tex2html_wrap_inline2785 well outside of our range of temperatures (9 400 K), has been used in the transformation from synthetic to observed colors. MD85 provides the closest values to tex2html_wrap_inline2787 temperatures with a mean difference of 22 K and a standard deviation of tex2html_wrap_inline2789, consistent with the error of tex2html_wrap_inline2791 adopted in the tex2html_wrap_inline2793 values whereas the mean difference between LGK86a and tex2html_wrap_inline2795 is tex2html_wrap_inline2797 with a standard deviation of tex2html_wrap_inline2799. Although this difference is not significant with respect to the assumed error in the tex2html_wrap_inline2801 temperatures (tex2html_wrap_inline2803), Fig. 4.2 (click here) illustrates that the difference tends to be greater in the region of lower temperature. This trend can be again caused by the lack of calibration stars in our range of temperatures (only one, Procyon, and just in the edge of the interval (tex2html_wrap_inline2805 tex2html_wrap_inline2807 K)). The differences are even more relevant when the LGK86b is used (tex2html_wrap_inline2809, tex2html_wrap_inline2811 tex2html_wrap_inline2813). Finally, the mean difference between B94 and tex2html_wrap_inline2815 is 96 K with a standard deviation of 164 K. This difference shows a similar trend to LGK86a which is reasonable since B94 is based on the synthetic colours derived by Lester et al. (1986a). Moreover, the greatest differences, which appear in the interval tex2html_wrap_inline2817, can be explained in terms of how the calibration was defined: B94 used three different calibrations for three different intervals of tex2html_wrap_inline2819: tex2html_wrap_inline2821 tex2html_wrap_inline2823 tex2html_wrap_inline2825; tex2html_wrap_inline2827; tex2html_wrap_inline2829. In this last group the calibrating stars have tex2html_wrap_inline2831 tex2html_wrap_inline2833, which means that effective temperatures in the range tex2html_wrap_inline2835 were calculated by extrapolation and thus the error being greater. Moreover, we see that, for tex2html_wrap_inline2837 tex2html_wrap_inline2839, when a new calibration is used, the differences are much less significant.

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