3. Analysis and results
Atmospheric parameters and element abundances have been derived using the plane parallel, flux constant, LTE model atmospheres described and discussed in detail by Edvardsson et al. (1993). In order to perform a differential analysis relative to the Sun these models have been used for both the analysed stars and the Sun. The adopted solar atmospheric parameters were = 5777 K (Neckel, 1986), log g = 4.44, [Fe/H] = 0.00, = 1.00 km s-1 and n(He)/n(H) = 0.10.
Gf-values were derived from solar equivalent widths, measured on solar spectra taken in the same conditions as the stellar ones, by forcing the calculated solar element abundances to agree with those determined by Anders & Grevesse (1989), and Grevesse & Noels (1993). This ensures the differential character of our analysis.
3.1. Atmospheric parameters
3.1.1. Effective temperatures
The effective temperatures have been derived by four independent criteria: excitation equilibrium, photometric indices calibration and fitting of and profiles. The excitation equilibrium has been determined through the analysis of the FeI lines. Using the model atmospheres, "solar" gf values and atomic data we calculate the iron abundances line by line, using a program kindly supplied by M. Spite (Observatoire de Paris-Meudon, France). The theoretical abundance for a given line is changed iteratively until the obtained equivalent width equals the measured one. If the effective temperature is correct the abundance of Fe should be independent of the excitation potential of the lines. So if we plot an iron abundance vs. excitation potential graph for all FeI lines and fit a straight line, this line should present a negligible (within the expected error) angular coeficient. If a significant positive or negative slope is measured, the model effective temperature is increased or decreased, respectively, and the iron abundances recalculated. The effective temperatures determined by this method are hereafter designated excitation temperatures .
We estimated a 55 K probable error for the excitation temperatures by changing the model atmosphere temperature until the angular coeficient of the linear fit equalled the error of its determination.
We derived effective temperature estimations from the (B-V), (b-y), (V-K), (R-I) and photometric indices using the calibrations by Porto de Mello (1996) for solar-type stars, which take into account the stars' metallicities. According to the author of the calibrations the use of 5 different indices simultaneously warrants a probable error as low as 70 K. Table 1 summarizes photometric data for and Ret.
Table 1. Photometric data for and Reticuli (in the Johnson and Strömgren systems).
The and profiles have been shown, in the case of cool stars, to be rather insensitive to surface gravity, microturbulence velocity and metallicity variations, at least for quasi-solar abundance stars (da Silva, 1975; Gehren, 1981; Fuhrmann et al., 1993). For the early G stars these profiles are, notwithstanding, very sensible to the effective temperature of the atmosphere. By comparing the theoretical profiles with the observational ones we can estimate the temperatures (Fig. 2). A program kindly made available by Praderie (1967) was used to compute the Balmer lines theoretical profiles. It takes into account the convolution of Stark, Doppler and self-resonance broadenings. Stark broadening has been computed using the method of Vidal et al. (1971), and self-resonance broadening has been included according to the prescription of Cayrel & Traving (1960). The probable internal error of these determinations, reflecting the uncertainties of the continuum determination and personal judgment of the fitting of the observed profiles, is estimated by us as being 45 K.
The adopted effective temperatures of and Ret were calculated by averaging the four previously cited estimations (Table 2). The probable error of the adopted temperatures is 27 K, calculated using the known expression for the probable error of an average of independent estimations. The standard deviations of the four criteria are 5 K for Ret and 13 for Ret, which demonstrates the excellent internal agreement.
Table 2. Effective temperatures according to four criteria.
3.1.2. Surface gravities
The surface gravities have been derived by three criteria: ionization equilibrium and using the astrometric parallaxes with masses obtained from a mass-luminosity relation and from evolutionary tracks. We obtained spectroscopic surface gravities by requiring that the ionization equilibrium be satisfied. When this happens, the abundances derived from the lines of ionized and neutral species of any element should be equal. We changed the model atmosphere surface gravity until the iron abundances derived from FeI and FeII lines matched. We also required the TiI and TiII abundances to match. The logaritmic gravities have an estimated probable error of 0.07 dex for Ret and 0.03 dex for Ret. These errors have been estimated by changing the model atmosphere surface gravity until the ionization equilibrium was no longer verified at the 1 level.
We also obtained surface gravity estimations using the stellar effective temperatures determined by us, luminosities and masses. To obtain the absolute bolometric magnitudes we used apparent visual magnitudes from Hoffleit & Jaschek (1982), with parallaxes from the Hipparcos catalogue (ESA, 1997) and the Habets & Heintze (1981) bolometric corrections. The latter were adjusted so that the solar absolute bolometric magnitude was . Masses were derived using either the Böhm (1989) mass-luminosity relation for main sequence stars or the Schaller et al. (1992), Schaerer et al. (1993) and Charbonnel et al. (1993) (hereafter Gen92/93) evolutionary trajectories. The gravities were calculated according to the known equation
Table 3 contains the obtained values along with probable errors. The standard deviations of the three criteria are 0.05 dex for Ret and 0.04 dex for Ret.
Table 3. Surface gravities according to three criteria.
3.1.3. Microturbulence velocity
The procedure used to derive the microturbulence velocity is very similar to the one we used to derive the excitation temperature. We plot an iron abundance vs. equivalent width graph for all FeI lines and fit a straight line. If the angular coeficient of this line is negligible (within the expected error), then we have chosen the correct microturbulence velocity. Again if the slope is significantly positive or negative we increase or decrease the microturbulence, respectively, and recalculate the iron abundancies until convergence is attained.
A probable error of 0.19 km s-1 for Ret and 0.14 km s-1 for Ret was estimated for the microturbulence velocity by changing this parameter until the angular coeficient of the linear fit equaled the error of its determination.
The adopted stellar parameters for and Ret are listed in Table 4.
Table 4. Adopted stellar parameters for and Reticuli.
3.2. Element abundances
The abundances of iron, calculated from the FeI and FeII lines, were obtained during the atmospheric parameters determination procedure. For Ret we measured 25 FeI lines and 3 FeII lines. For Ret we measured 29 FeI lines and 4 FeII lines. Here we have to call attention to the fact that our method of determining the temperatures by excitation equilibrium and and profile fitting, the surface gravities by ionization equilibrium and the Fe abundances is completely iterative in the sense that a change in one of these parameters makes it necessary to recalculate the other ones, until all values obtained are entirely consistent with one another. Chemical abundances were calculated for AlI , CeII , CrI , NiI , SiI , TiI , TiII and VI the same way that iron abundances were, i.e., line by line (using the adopted atmospheric parameters previously described). Since the analysis was a differential one, the abundances were derived relatively to the Sun and given in the usual notation [element/H].
Both stars were found to be a little metal deficient ([Fe/H] = -0.22). To estimate the internal probable errors of iron-relative abundances we analysed the influence of the uncertainties in the adopted atmospheric parameters and also the equivalent width measuring error according to the following procedure:
i) We recalculated [element/Fe] for all studied elements with the atmospheric parameters added to their probable errors. The effective temperature, surface gravity, microturbulence velocity and metallicity were changed one by one, independently, while the other parameters remained constant.
ii) To determine the influence of dispersion on the equivalent widths we once again compared our solar equivalent widths with those of the Kurucz solar atlas (which we supposed to be completely free of dispersion).
We plotted a ( - )/ vs. graph, whose ordinate gives the percentage error of the measurements made on the sky spectra. The weaker lines (W 30 mÅ) present a 7.0% dispersion,while lines stronger than 30 mÅ present a 3.7% dispersion. We increased the equivalent widths of a percentage value equal to the obtained dispersions and recalculated the abundances.
Thus after the determination of the individual probable errors (due to the uncertainties of the individual parameters) we determined the total probable errors:
Table 5 contains the abundances of all analysed elements, relatively to iron. These abundances are shown on Fig. 3 (the abundance of titanium is the average of the abundances obtained through the analysis of the TiI and TiII lines). The error bars are the estimated probable errors. Both stars have the abundances of all elements compatible with each other and with the Sun. The abundance of Ce for Ret can be regarded as marginally (within 2 ) compatible with the abundance of Ret. The good agreement of the abundances of and Ret strengthens the idea of a common origin for these two stars.
Table 5. Chemical abundances of and Reticuli, relatively to iron. The number of lines used is given for each star.
© European Southern Observatory (ESO) 2000
Online publication: June 26, 2000