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Astron. Astrophys. 364, 102-136 (2000)

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9. Uncertainties

In this section we consider the effect on our abundances of uncertainties in [FORMULA], [FORMULA], and the microturbulent velocity [FORMULA]. We also consider errors in log gf and NLTE effects.

9.1. Uncertainty in the the stellar parameters [FORMULA], [FORMULA] 

The quantitative dependence of the derived abundances on differences [FORMULA][FORMULA] = [FORMULA]100 K and [FORMULA][FORMULA] = [FORMULA]0.1 dex in the stellar parameters is given for the stars HD 161817, HD 139961, HD 167105, and HD 87112 in Table 11. These stars are representative of stars having [FORMULA] around 7500 K, 8500 K, 9000 K, and 9750 K respectively. It is seen that the uncertainty in [FORMULA] affects the abundances more than the uncertainty in [FORMULA]. Furthermore, the species most affected by uncertainties in the parameters are Cr I , Fe I  and Ba II . Their abundance changes by about 0.1 dex for [FORMULA][FORMULA] = [FORMULA]100 K. The effect on Mg II [FORMULA] 4481 is small and decreases with increasing [FORMULA].


Table 11. Abundance changes produced by uncertainties of [FORMULA]100 K in [FORMULA] and [FORMULA]0.1 in [FORMULA]

9.2. Uncertainty in [FORMULA]

The value of [FORMULA] was assumed for the spectra of the two stars BD 00+00 145 and HD 14829 and for all the ESO-CAT spectra because there were too few lines in these spectra to determine this quantity. Table 12 gives the abundances of the different species in these stars for microturbulent velocities [FORMULA] = 2 km s[FORMULA], 3 km s[FORMULA] and 4 km s[FORMULA]. For a change [FORMULA][FORMULA] = 1 km s[FORMULA], the abundance derived from the Mg II  [FORMULA] 4481 line changes by about 0.2 dex for the stars observed at ESO and about 0.05 dex for the two weaker-lined stars observed at KPNO (BD +00 145 and HD 14829). The abundance derived from the Ti II  lines is also affected by the value of [FORMULA]; the change varies from 0.2 dex for HD 4850 and HD 13780 to 0.05 dex for HD 106304. The effect of [FORMULA] on the Fe I  and Fe II  abundances is very small in all these stars.


Table 12. The effect of the microturbulent velocity [FORMULA] on the abundances

9.3. Errors in [FORMULA]

The errors in [FORMULA] can be a significant source of uncertainty if only a few lines of a species are available for measurement. This can happen if the star is very metal-poor (e.g. HD 008376) so that only the strongest lines are measurable or, as with the ESO-CAT spectra, the observed waveband is not large. We inferred the presence of these errors in [FORMULA] as follows.

Our first estimate of the abundance was made by fitting the measured equivalent widths ([FORMULA]) of the apparently unblended lines to those computed by Kurucz's WIDTH program. We therefore have an abundance for each line and the difference between this abundance and the mean for that species in a given star is called [FORMULA][m/H]. This quantity, when averaged over all our program stars, ([FORMULA]) is shown in Fig. 8 for both the Fe II  and Ti II  lines. It shows little correlation with equivalent width (the [FORMULA] on the left of Fig. 8 are those for HD 93329 which has an intermediate [FORMULA]).

[FIGURE] Fig. 8. [FORMULA][m/H] is the difference between the abundance derived from a given line and the mean abundance of that species for that star. [FORMULA] is the mean value of this quantity for each [Fe II ] line (above) and [Ti II ] line (below). This mean is plotted against the equivalent width (on the left) and against the same quantity from the BHB spectra of Lambert et al. (1992) (on the right).

[FORMULA] was also computed (for the same lines) from the BHB star data of Lambert et al. (1992) and is called [FORMULA]. It is seen that there is a correlation between the values of [FORMULA] determined from our data and those of Lambert et al.; moreover the range in this quantity is markedly greater for the Ti II  lines than for the Fe II  ones. This scatter in [FORMULA] is greater than can be accounted for by measuring errors (the vertical error bars) and must be caused by a factor that is intrinsic to each species and which is common to both our calculations and those of Lambert et al. It seems most likely that it is caused by errors in the assumed [FORMULA].

9.4. Non-LTE effects

The models used to derive our abundances assume LTE conditions. In hot stars, however, UV radiation can cause the Fe I  states to be underpopulated while the Fe II  lines are relatively unaffected; the effect is expected to increase with decreasing metallicity. Lambert et al. (1992) tried to allow for this effect by adjusting their stellar parameters so as to make [Fe I ] - [Fe II ] = -0.2. Cohen & McCarthy (1997), however, made no non-LTE corrections in deriving the abundances of BHB stars in M 92. The [FORMULA] of their stars were in the range 7 500 K to 9 375 K and were derived from their [FORMULA] and [FORMULA] colours. They found a mean value for [FORMULA] of only -0.08; this suggests that non-LTE effects are not significant. The abundances, moreover, which they found for their BHB stars were in excellent agreement with those previously found for red giants in the same cluster. We find [FORMULA] = 0.01[FORMULA]0.01 for the 27 spectra where we measured both Fe I  and Fe II  lines. We therefore feel that it is unlikely that our iron abundances are significantly compromised by non-LTE effects. Our barium abundances (Table 9) were derived from the Ba II [FORMULA]4554.03 line alone and gave a mean LTE abundance of [Ba/Fe] from nine stars of -0.08[FORMULA]0.05; hyperfine broadening was not taken into account and significant non-LTE effects may be expected for this line (Mashonkina & Bikmaev 1996; Belyakova et al. 1998).

9.5. Convection

For the coolest stars of our sample ([FORMULA][FORMULA]8 000 K) there may be a problem with the treatment of the convection in the model atmospheres. Uncertainties of the order of 200 K in [FORMULA] can be expected in the sense that [FORMULA] is higher for the mixing-length parameter l/H = 1.25 that we adopted than for the lower value l/H =0.5 suggested by Fuhrmann et al. (1993, 1994). Also, a different convection theory, like that of Canuto & Mazzitelli (1992) leads to a very low convection (or no convection) in stars hotter than 7 000 K, so that [FORMULA] derived by adopting this theory may be lower than that derived by us. We feel, however, that more accurate observations that allow a more precise location of the continuum and more discussions on the theories adopted to compute the Balmer profiles are needed in order to confirm the superiority of other convections over that adopted by us. The effect of convection on the colour indices and Balmer profiles, and therefore on the [FORMULA] derived from them, has been discussed by Smalley & Kupka (1997), van't Veer-Menneret & Méssier (1996), Castelli et al. (1997), and Gardiner et al. (1999).

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Online publication: December 15, 2000