5 Abundances and their uncertainties

5.1 Model atmospheres and abundance calculations

The abundance analysis is based on a grid of flux constant, homogeneous, LTE model atmospheres, kindly supplied by Bengt Edvardsson (Uppsala). The models were computed with the MARCS code using the updated continuous opacities by Asplund et al. ([1997]) including UV line blanketing by millions of absorption lines and many molecular lines.

The abundance was calculated with the program EQWIDTH (also made available from the stellar atmospheric group in Uppsala) by requiring that the calculated equivalent width from the model should match the observed value. The calculation includes natural broadening, thermal broadening, van der Waals damping, and the microturbulent Doppler broadening. The mean abundance was derived from all available lines by giving equal weight to each line. Finally, solar abundances, calculated from the Moon spectrum, were used to derive stellar abundances relative to solar values (Table 5). Such differential abundances are generally more reliable than absolute abundances because many systematic errors nearly cancel out.

   
5.2 Uncertainties of abundances

There are two kinds of uncertainties in the abundance determination: one acts on individual lines, and includes random errors of equivalent widths, oscillator strengths, and damping constants; another acts on the whole set of lines with the main uncertainties coming from the atmospheric parameters.

5.2.1 Errors from equivalent widths and atomic data

The comparison of equivalent widths in Sect. 2.3 indicates that the typical uncertainty of the equivalent width is about 3 mÅ, which leads to an error of about 0.06 dex in the elemental ratio X/H derived from a single line with an equivalent width around 50 mÅ. For an element represented by N lines, the error is decreased by a factor $\sqrt{N}$. In this way, the errors from equivalent widths were estimated for elements with only one or a few lines. Alternatively, the scatter of the deduced abundances from a large number of lines with reliable oscillator strengths gives another estimate of the uncertainty from equivalent widths. With over 100 Fe I lines for most stars, the scatter varies somewhat from star to star with a mean value of 0.07 dex, corresponding to an error of 0.007 dex in [Fe/H]. Other elements with significant numbers of lines, such as Ca, Ni and Si, have even smaller mean line-to-line scatters.

The uncertainties in atomic data are more difficult to evaluate. But any error in the differential abundance caused by errors in the gf values is nearly excluded due to the correction of some experimental or theoretical gf values and the adoption of mean gf values from 10 "standard'' stars. Concerning the uncertainties in the damping constants, we have estimated their effects by increasing the adopted enhancement factors by 50%. The microturbulence was accordingly adjusted because of the coupling between the two parameters. The net effect on the differential abundances with respect to the Sun is rather small as seen from Table 1.

5.2.2 Consistency check of atmospheric parameters

As a check of the photometric temperature, the derived iron abundance from individual Fe I lines was studied as a function of the excitation potential. To reduce the influence of microturbulence, only lines with equivalent widths less than 70 mÅ were included. A linear least squares fit to the abundance derived from each line vs. low excitation potential determines the slope in the relation $\mbox{\rm [Fe/H]}= a + b \cdot \chi_{{\rm low}}$. The mean slope coefficient for all stars is $b=0.004 \pm 0.013$. There is only a very small (if any) dependence of b on effective temperature, surface gravity or metallicity. A suspected binary, HD15814, has a very deviating slope coefficient (b=-0.056) and is excluded from further analysis.

The agreement of iron abundances derived from Fe I and Fe II lines is satisfactory when gravities based on Hipparcos parallaxes are used (see Fig. 3). The deviation is less than 0.1 dex for most stars with a mean value of $-0.009 \pm 0.07$ dex. From $\mbox{\rm [Fe/H]}=0.0$ to $\mbox{\rm [Fe/H]}=-0.5$, the mean deviation ( $\mbox{\rm [Fe/H]}_{\mathrm{II}}-\mbox{\rm [Fe/H]}_\mathrm{I}$) seems, however, to increase by about 0.1 dex in rough agreement with predictions from non-LTE computations (see Sect. 5.3).

Figure 3: Difference in iron abundances derived from Fe I and Fe II lines vs. [Fe/H] with suspected binaries marked by a square around the filled circles

The deviation in iron abundances based on Fe I and Fe II abundance provides a way to identify binaries and to estimate the influence of the component on the primary. The suspected binaries are marked with an additional square around the filled circles in Fig. 3. It shows that there is no significant influence from the component for these binaries except in the case of HD15814, which was already excluded on the basis of it's b-coefficient in the excitation equilibrium of Fe I lines. Thus, the other possible binaries were included in our analysis. It is, however, surprising that HD186257 show a higher iron abundance based on Fe II lines than that from Fe I lines with a deviation as large as 0.28 dex. We discard this star in the final analysis and thus have 90 stars left in our sample.

   
5.2.3 Errors in resulting abundances

Table 1 shows the effects on the derived abundances of a change by 70 K in effective temperature, 0.1 dex in gravity, 0.1 dex in metallicity, and 0.3 $\mbox{\rm\,km\,s$^{-1}$ }$ in microturbulence, along with errors from equivalent widths and enhancement factors, for two representative stars.

   

Table 1: Abundance errors. The last column gives the total error assuming that the individual errors are uncorrelated

HD 142373  $\mbox{$T_{\rm eff}$ }=5920~\mbox{{\rm log}$g$ }=4.27~\mbox{\rm [Fe/H]}= -0.39~\xi_{\rm t}=1.48$
	$\frac{\sigma_{{\rm EW}}}{\sqrt{{\rm N}}}$	$\Delta \mbox{$T_{\rm eff}$ }$	$\Delta \mbox{{\rm log}$g$ }$	$\Delta \mbox{${\rm [\frac{Fe}{H}]}$ }$	$\Delta \xi_{\rm t}$	$\Delta E_{\gamma}$	$\sigma_{{\rm tot}}$
		+70K	+0.1	+0.1	+0.3	50%
$\Delta \mbox{${\rm [\frac{Fe}{H}]}$ }$	.009	.048	-.005	.001	-.040	.015	.065
$\Delta \mbox{${\rm [\frac{Fe}{H}]}$ }_{{\rm II}}$	.030	-.012	.031	.018	-.046	.007	.067
$\Delta \mbox{${\rm [\frac{O}{Fe}]}$ }$	.035	-.104	.024	.004	.020	.013	.115
$\Delta \mbox{${\rm [\frac{Na}{Fe}]}$ }$	.042	-.017	.004	-.001	.034	.011	.058
$\Delta \mbox{${\rm [\frac{Mg}{Fe}]}$ }$	.042	-.018	-.005	.005	.019	.015	.052
$\Delta \mbox{${\rm [\frac{Al}{Fe}]}$ }$	.035	-.023	.003	-.001	.033	.015	.055
$\Delta \mbox{${\rm [\frac{Si}{Fe}]}$ }$	.015	-.026	.005	.003	.025	.015	.042
$\Delta \mbox{${\rm [\frac{Ca}{Fe}]}$ }$	.023	-.002	-.008	.002	-.005	.011	.028
$\Delta \mbox{${\rm [\frac{Ti}{Fe}]}$ }$	.024	.017	.004	.000	.023	.009	.039
$\Delta \mbox{${\rm [\frac{V}{Fe}]}$ }$	.035	.019	.004	.000	.031	.011	.051
$\Delta \mbox{${\rm [\frac{Cr}{Fe}]}$ }$	.035	-.007	.001	.000	.025	.015	.053
$\Delta \mbox{${\rm [\frac{Ni}{Fe}]}$ }$	.011	.002	.004	.003	.013	.023	.029
$\Delta \mbox{${\rm [\frac{Ba}{Fe}]}$ }$	.060	.039	-.008	.005	-.041	.012	.084
$\Delta \mbox{${\rm [\frac{K}{Fe}]}$ }$	.060	.013	-.024	.013	-.029	.012	.074

HD 106516  $\mbox{$T_{\rm eff}$ }=6135~\mbox{{\rm log}$g$ }=4.34~\mbox{\rm [Fe/H]}= -0.71~\xi_{\rm t}=1.48$
	$\frac{\sigma_{{\rm EW}}}{\sqrt{{\rm N}}}$	$\Delta \mbox{$T_{\rm eff}$ }$	$\Delta \mbox{{\rm log}$g$ }$	$\Delta \mbox{${\rm [\frac{Fe}{H}]}$ }$	$\Delta \xi_{\rm t}$	$\Delta E_{\gamma}$	$\sigma_{{\rm tot}}$
		+70 K	+0.1	+0.1	+0.3	50%
$\Delta \mbox{${\rm [\frac{Fe}{H}]}$ }_{{\rm I}}$	.012	.042	-.003	.006	-.029	.022	.057
$\Delta \mbox{${\rm [\frac{Fe}{H}]}$ }_{{\rm II}}$	.023	.000	.032	.008	-.025	.016	.050
$\Delta \mbox{${\rm [\frac{O}{Fe}]}$ }$	.042	-.079	.017	-.003	.007	-.003	.091
$\Delta \mbox{${\rm [\frac{Na}{Fe}]}$ }$	.042	-.018	.002	-.003	.026	.013	.054
$\Delta \mbox{${\rm [\frac{Mg}{Fe}]}$ }$	.042	-.014	-.009	.001	.012	.022	.052
$\Delta \mbox{${\rm [\frac{Si}{Fe}]}$ }$	.019	-.021	.004	-.001	.019	.023	.041
$\Delta \mbox{${\rm [\frac{Ca}{Fe}]}$ }$	.015	-.001	-.009	.002	-.009	.014	.024
$\Delta \mbox{${\rm [\frac{Ti}{Fe}]}$ }$	.030	.010	.003	.000	.021	.010	.039
$\Delta \mbox{${\rm [\frac{V}{Fe}]}$ }$	.042	.014	.003	.000	.025	.006	.051
$\Delta \mbox{${\rm [\frac{Cr}{Fe}]}$ }$	.035	-.006	.001	-.001	.018	.034	.052
$\Delta \mbox{${\rm [\frac{Ni}{Fe}]}$ }$	.012	-.001	.002	-.001	.013	.029	.034
$\Delta \mbox{${\rm [\frac{Ba}{Fe}]}$ }$	.042	.041	-.012	.008	-.047	.008	.077
$\Delta \mbox{${\rm [\frac{K}{Fe}]}$ }$	.060	.016	-.026	.010	-.039	.024	.082

It is seen that the relative abundances with respect to iron are quite insensitive to variations of the atmospheric parameters. One exception is [O/Fe] due to the well known fact that the oxygen abundance derived from the infrared triplet has an opposite dependence on temperature to that of the iron abundance. After rescaling of our oxygen abundances to results from the forbidden line at $\lambda 6300$(see next section) the error is somewhat reduced. Therefore, the error for [O/Fe] in Table 1 might be overestimated.

In all, the uncertainties of the atmospheric parameters give errors of less than 0.06 dex in the resulting [Fe/H] values and less than 0.04 dex in the relative abundance ratios. For an elemental abundance derived from many lines, this is the dominant error, while for an abundance derived from a few lines, the uncertainty in the equivalent widths may be more significant. Note that the uncertainties of equivalent widths for V and Cr (possibly also Ti) might be underestimated given that their lines are generally weak in this work. In addition, with only one strong line for the K abundance determination, the errors from equivalent widths, microturbulence and atomic line data are relatively large.

Lastly, we have explored the HFS effect on one Al I line at $\lambda$6698, one Mg I line at $\lambda$5711, and two Ba II lines at $\lambda$6141 and $\lambda$6496. The HFS data are taken from three sources: Biehl ([1976]) for Al, Steffen ([1985]) for Mg and François ([1996]) for Ba. The results indicate that the HFS effects are very small for all these lines with a value less than 0.01 dex.

   
5.3 Non-LTE effects and inhomogeneous models

The assumption of LTE and the use of homogeneous model atmospheres may introduce systematic errors, especially on the slope of various abundance ratios [X/Fe] vs. [Fe/H]. These problems were discussed at quite some length by EAGLNT. Here we add some remarks based on recent non-LTE studies and computations of 3D hydrodynamical model atmospheres.

Based on a number of studies, EAGLNT concluded that the maximum non-LTE correction of [Fe/H], as derived from Fe I lines, is 0.05 to 0.1 dex for metal-poor F and G disk dwarfs. Recently, Thévenin & Idiart ([1999]) computed non-LTE corrections on the order of 0.1 to 0.2 dex at $\mbox{\rm [Fe/H]}= -1.0$. Figure 3 suggests that the maximium correction to $\mbox{\rm [Fe/H]}$derived from Fe I lines is around 0.1 dex, but we emphasize that this empirical check may depend on the adopted $\mbox{$T_{\rm eff}$ }$ calibration as a function of $\mbox{\rm [Fe/H]}$.

The oxygen infrared triplet lines are suspected to be affected by non-LTE formation, because they give systematically higher abundances than forbidden lines. Recent work by Reetz ([1999]) indicates that non-LTE effects are insignificant (< 0.05 dex) for metal-poor and cool stars, but become important for warm and metal-rich stars. For stars with $\mbox{\rm [Fe/H]}> -0.5$ and $\mbox{$T_{\rm eff}$ }> 6000$ K in our sample, non-LTE effects could reduce the oxygen abundances by 0.1-0.2dex. For this reason, we use Eq. (11) of EAGLNT to scale the oxygen abundances derived from the infrared triplet to those derived by Nissen & Edvardsson ([1992]) from the forbidden [O I] $\lambda$6300. The two weak Na I lines ( $\lambda 6154$ and $\lambda 6160$) used for our Na abundance determinations, are only marginally affected by deviations from LTE formation (Baumüller et al. [1998]). The situation for Al may, however, be different. The non-LTE analysis by Baumüller & Gehren ([1997]) of one of the Al I lines used in the present work ( $\lambda 6698$) leads to about 0.15 dex higher Al abundances for the metal-poor disk dwarfs than those calculated from LTE. No non-LTE study for the other two lines used in the present work is available. We find, however, that the derived Al abundances depend on $\mbox{$T_{\rm eff}$ }$ with lower [Al/Fe] for higher temperature stars. This may be due to the neglect of non-LTE effects in our work. Hence, we suspect that the trend of [Al/Fe] vs. [Fe/H] could be seriously affected by non-LTE effects.

The recent non-LTE analysis of neutral magnesium in the solar atmosphere by Zhao et al. ([1998]) and in metal-poor stars by Zhao & Gehren ([1999]) leads to non-LTE corrections of 0.05 dex for the Sun and 0.10 dex for a $\mbox{\rm [Fe/H]}= -1.0$ dwarf, when the abundance of Mg is derived from the $\lambda 5711$ Mg I line. Similar corrections are obtained for some of the other lines used in the present work. Hence, we conclude that the derived trend of [Mg/Fe] vs. [Fe/H] is not significantly affected by non-LTE.

The line-profile analysis of the K I resonance line at $\lambda$7699 by Takeda et al. ([1996]) shows that the non-LTE correction is -0.4 dex for the Sun and -0.7 dex for Procyon. There are no computations for metal-poor stars, but given the very large corrections for the Sun and Procyon one may expect that the slope of [K/Fe] vs. [Fe/H] could be seriously affected by differential non-LTE effects between the Sun and metal-poor stars.

The non-LTE study of Ba lines by Mashonkina et al. ([1999]), which includes two of our three Ba II lines ($\lambda$5853 and $\lambda$6496), give rather small corrections (<0.10 dex) to the LTE abundances, and the corrections are very similar for solar metallicity and $\mbox{\rm [Fe/H]}\simeq -1.0$dwarfs. Hence, [Ba/Fe] is not affected significantly.

In addition to possible non-LTE effects, the derived abundances may also be affected by the representation of the stellar atmospheres by plane-parallel, homogeneous models. The recent 3D hydrodynamical model atmospheres of metal-poor stars by Asplund et al. ([1999]) have substantial lower temperatures in the upper photosphere than 1D models due to the dominance of adiabatic cooling over radiative heating. Consequently, the iron abundance derived from Fe I lines in a star like HD84937 ( $\mbox{$T_{\rm eff}$ }\simeq 6300$ K, $\mbox{{\rm log}$g$ }\simeq 4.0$ and $\mbox{\rm [Fe/H]}\simeq -2.3$) is 0.4 dex lower than the value based on a 1D model. Although the effect will be smaller in a $\mbox{\rm [Fe/H]}\simeq -1.0$ star, and the derived abundance ratios are not so sensitive to the temperature structure of the model, we clearly have to worry about this problem.

5.4 Abundance comparison of this work with EAGLNT

A comparison in abundances between this work and EAGLNT for the 25 stars in common provides an independent estimate of the errors of the derived abundances. The results are summarized in Table 2.

 

 

Table 2: Mean abundance differences (this work-EAGLNT) and standard deviations. N is the number of stars, for which a comparison was possible

	$<\Delta>$	$\sigma$	N
$\mbox{\rm [Fe/H]}_{\rm I}$	-0.020	0.068	25
$\mbox{\rm [Fe/H]}_{{\rm II}}$	-0.004	0.090	25
$\mbox{\rm [O/Fe]}$	0.147	0.064	5
$\mbox{\rm [Na/Fe]}$	-0.079	0.057	21
$\mbox{\rm [Mg/Fe]}$	-0.020	0.080	21
$\mbox{\rm [Al/Fe]}$	-0.033	0.080	16
$\mbox{\rm [Si/Fe]}$	-0.012	0.054	23
$\mbox{\rm [Ca/Fe]}$	0.055	0.039	25
$\mbox{\rm [Ti/Fe]}$	-0.093	0.099	24
$\mbox{\rm [Ni/Fe]}$	-0.008	0.045	25
$\mbox{\rm [Ba/Fe]}$	0.068	0.081	25

The agreement in iron abundance derived from Fe I lines is satisfactory with deviations within $\pm$0.1 dex for the 25 common stars. These small deviations are mainly explained by different temperatures given the fact that the abundance differences increase with temperature deviations between the two works. The rms deviation in iron abundance derived from Fe II lines are slightly larger than that from Fe I lines. The usage of different gravities partly explain this. But the small line-to-line scatter from 8 Fe II lines in our work indicates a more reliable abundance than that of EAGLNT who used 2 Fe II lines only.

Our oxygen abundances are systematically higher by 0.15 dex than those of EAGLNT for 5 common stars. Clearly, the temperature deviation is the main reason. The systematically lower value of 70 K in our work increases [O/Fe] by 0.10 dex (see Table 1).

The mean abundance differences for Mg, Al, Si, Ca and Ni between the two works are hardly significant. The systematical differences (this work - EAGLNT) of -0.08 dex for [Na/Fe] and [Ti/Fe] and +0.07 dex for [Ba/Fe] are difficult to explain, but we note that when the abundances are based on a few lines only, a systematic offset of the stars relative to the Sun may occur simply because of errors in the solar equivalent widths.