3 Stellar atmospheric parameters

3.1 Effective temperature and metallicity

The effective temperature was determined from the Strömgren indices (b-y and c₁) and [Fe/H] using the calibration of Alonso et al. ([1996]). If the color excess E(b-y), as calculated from the H$_{\beta}$ index calibration by Olsen ([1988]), is larger than 0.01, then a reddening correction was applied.

The metallicity, required in the input for temperature and abundance calculation, was first derived from the Strömgren m₁ index using the calibrations of Schuster & Nissen ([1989]). But the spectroscopic metallicity obtained later was used to iterate the whole procedure.

The errors of the photometric data are $\sigma(b-y) = 0.004$ and $\sigma(c_1) = 0.008$ according to Olsen ([1993]). Adopting $\sigma(\mbox{\rm [Fe/H]})$ = 0.1 from the spectroscopic analysis, the statistical error of $T_{\rm eff}$ is estimated to about $\pm 50$ K. Considering a possible error of $\pm$50 K in the calibration, the error in temperature could reach $\pm 70$ K. We do not adopt the excitation temperature, determined from a consistent abundance derived from Fe I lines with different excitation potentials, because errors induced by incorrect damping parameters (Ryan [1998]) or non-LTE effects can be strongly dependent on excitation potential, leading to an error in effective temperature as high as 100 K.

   
3.2 Gravity

In most works, gravities are determined from the abundance analysis by requiring that Fe I and Fe II lines give the same iron abundance. But it is well known that the derivation of iron abundance from Fe I and Fe II lines may be affected by many factors such as unreliable oscillator strengths, possible non-LTE effects and uncertainties in the temperature structure of the model atmospheres. From the Hipparcos parallaxes, we can determine more reliable gravities using the relations:

 

$$\log \frac{g}{g_{\odot}} \log \frac{\mathcal M}{{\mathcal M}_{\odot}} +4 \log \frac{\mbox{... ...{\rm eff}$ }_{\odot}}+ 0.4 \left(M_{\mathrm{bol}}-M_{\mathrm{bol},\odot}\right)$$ = (1)

and

 

$$V+BC+5 \log \pi + 5,$$ M_bol = (2)

where, ${\mathcal M}$ is the stellar mass, M_bol the absolute bolometric magnitude, V the visual magnitude, BCthe bolometric correction, and $\pi$ the parallax.

The parallax is taken from the Hipparcos Satellite observations (ESA [1997]). For most program stars, the relative error in the parallax is of the order of 5%. Only two stars in our sample have errors larger than 10%. From these accurate parallaxes, stellar distances and absolute magnitudes were obtained. Note, however, that our sample includes some binaries, for which the absolute magnitude from the Hipparcos parallax could be significantly in error. An offset of -0.75 mag will be introduced for a binary with equal components through the visual magnitude in Eq. (2). Thus, we also calculated absolute magnitudes from the photometric indices $\beta$ and c₁ using the relations found by EAGLNT. Although the absolute magnitude of a binary derived by the photometric method is also not very accurate due to different spectral types and thus different flux distributions of the components, it may be better than the value from the parallax method. Hence, for a few stars with large differences in absolute magnitudes between the photometry and parallax determination, we adopt the photometric values.

The bolometric correction was interpolated from the new BC grids of Alonso et al. ([1995]) determined from line-blanketed flux distributions of ATLAS9 models. It is noted that the zero-point of the bolometric correction adopted by Alonso et al., $BC_{\mathrm{\odot}}=-0.12$, is not consistent with the bolometric magnitude of the Sun, $M_{\mathrm{bol},\odot}$=4.75, recently recommended by the IAU ([1999]). But the gravity determination from the Eq. (1) only depends on the M_bol difference between the stars and the Sun and thus the zero-point is irrelevant.

The derivation of mass is described in Sect. 6. The estimated error of 0.06 $M_{\odot}$ in mass corresponds to an error of 0.03 dex in gravity, while errors of 0.05 mag in BC and 70 K in temperature each leads to an uncertainty of 0.02 dex in logg. The largest uncertainty of the gravity comes from the parallax. A typical relative error of 5% corresponds to an error of 0.04 dex in logg. In total, the error of logg is less than 0.10 dex.

The surface gravity was also estimated from the Balmer discontinuity index c₁ as described in EAGLNT. We find a small systematical shift (about 0.1 dex) between the two sets of logg, with lower gravities from the parallaxes. There is no corresponding shift between M_V(par) and M_V(phot). The mean deviation is 0.03 mag only, which indicates that the systematic deviation in logg comes from the gravity calibration in EAGLNT.

   
3.3 Microturbulence

The microturbulence, $\xi_{{\rm t}}$, was determined from the abundance analysis by requiring a zero slope of [Fe/H] vs. EW. The large number of Fe I lines in this study enables us to choose a set of lines with accurate oscillator strengths, similar excitation potentials ( $\chi_{{\rm low}} \geq 4.0$ eV) and a large range of equivalent widths (10 - 100 mÅ) for the determination. With this selection, we hope to reduce the errors from oscillator strengths and potential non-LTE effects for Fe I lines with low excitation potentials. The error of the microturbulence is about 0.3 kms^-1.

The relation of $\xi_{{\rm t}}$ as a function of $T_{\rm eff}$ and logg derived by EAGLNT corresponds to about 0.3 kms^-1 lower values than those derived from our spectroscopic analysis. No obvious dependence of the difference on temperature, gravity and metallicity can be found. In particular, the value for the Sun in our work is 1.44 kms^-1, also 0.3 kms^-1 higher than the value of 1.15 found from the EAGLNT relation. The difference in $\xi_{{\rm t}}$ between EAGLNT and the present work is probably related to the difference in equivalent widths of intermediate-strong lines discussed in Sect. 2.3. EAGLNT measured these lines by fitting a Gaussian function and hence underestimated their equivalent widths, leading to a lower microturbulence.

Finally, given that the atmospheric parameters were not determined independently, the whole procedure of deriving $T_{\rm eff}$, logg, [Fe/H] and $\xi_{{\rm t}}$ was iterated to consistency. The atmospheric parameters of 90 stars are presented in Table 3. The uncertainties of the parameters are: $\sigma(\mbox{$T_{\rm eff}$ }) = 70$ K, $\sigma(\mbox{{\rm log}$g$ }) = 0.1$, $\sigma(\mbox{\rm [Fe/H]}) = 0.1$, and $\sigma(\xi_{t})=0.3$ kms^-1.