5 The fitting procedure

The fitting procedure is based on that described by Lane & Lester (1984) in which the observed energy distribution is fitted to the model which yields the minimum rms difference. The search for the minimum rms difference is made by interpolating in the grid of computed fluxes. The computed fluxes are sampled in steps of 50 K in $T_{\rm eff}$ and in steps of 0.1 dex in $\log\,g$, so the finer sampling was obtained by linear interpolation. The error $\sigma(\lambda)$ of the flux associated to the INES spectra was used at each wavelength to weight the square differences between the observed flux and the computed flux. In this way, the parameters derived by fitting a selected image to the models are almost independent from the limits of the wavelength interval adopted for the fit. Bad pixels were excluded from the fit.

 

 

Table 2: The parameters derived from the fit of the IUE fluxes to the new-ODF models.

Star	[M/H]	$\xi$	E(B-V)	$T_{\rm eff}$	$\log\,g$	$T_{\rm eff}$	$\log\,g$	$T_{\rm eff}$	$\log\,g$
				1200-1978 Å		1979-3300 Å		1200-3300 Å
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
HD 2857	[-1.75a]	4.0 (3.0)	0.022	7650	2.5	7550	3.0	7600	2.8
HD 4850	[-1.25a]	2.0	0.009	8450	2.8	8500	2.2	8450	2.7
HD 8376	[-2.501]	1.0	0.0202	8050	2.6
HD 13780	[-1.50]	2.0	0.0002	7900	2.7	7650	2.9	7900	2.7
HD 14829	[-2.50a]	2.0	0.018	8900	3.1	8900	3.1	8900	3.1
HD 31943	[-1.00a]	4.0	0.006	7850	3.1	7900	3.0	7850	3.1
HD 60778	[-1.50a]	4.0 (3.0)	0.028	8250	2.9	8400	2.6	8250	2.9
HD 74721	[-1.50a]	2.0 (4.0)	0.012	8800	3.2	8550	3.8	8800	3.2
HD 78913	[-1.50a]	2.0	0.034	8750	2.9	8600	2.9	8700	2.8
HD 86986	[-1.75a]	2.0 (2.5)	0.022	8100	2.7	7650	3.3	8100	2.8
HD 87047	[-2.50a]	2.0	0.006	7900	2.7	7800	2.9	7900	2.8
HD 87112	[-1.50a]	2.0	0.003	9700	3.6
HD 93329	[-1.50a]	2.0	0.014	8250	2.8	8250	2.9	8250	2.9
HD 106304	[-1.25a]	2.0	0.038	9600	3.5
HD 109995	[-1.75a]	2.0(3.0)	0.0102	8500	3.1	8250	3.3	8500	3.0
HD 117880	[-1.50a]	2.0	0.077	9350	3.5	9000	4.0	9350	3.3
HD 128801	[-1.50a]	2.0	0.010	10200	3.5	10000	3.7	10200	3.5
HD 130095	[-1.75a]	2.0	0.072	9100	3.1	9200	2.9	9100	3.2
HD 130201	[-1.00a]	2.0	0.035	8900	2.5	8750	2.8	8900	2.5
HD 139961	[-1.75a]	4.0 (3.0)	0.051	8600	3.0	8550	2.6	8600	2.8
HD 161817	[-1.50a]	4.0 (3.0)	0.000	7600	2.6	7250	3.1	7600	2.7
HD 167105	[-1.50a]	2.0 (3.0)	0.024	8900	3.2	8950	2.8	9000	3.1
HD 180903	[-1.50a]	4.0 (3.0)	0.098	7850	2.8	7550	3.0	7800	2.9
HD 202759	[-2.00a]	2.0	0.072	7500	2.9	7400	2.8	7500	2.8
HD 213468	[-1.75a]	2.0	0.008	9100	3.4	8900	3.6	9100	3.3
HD 252940	[-1.75a]	4.0 (3.5)	0.048	7650	2.6	7500	2.8	7650	2.7
BD+00 145a	[-2.50a]	2.0	0.018	9900	3.2	9750	3.0	9900	3.1
BD+32 2188	[-1.00]	1.0	0.007	10300	2.1	10250	2.1	10300	2.2
BD+42 2309	[-1.75a]	2.0	0.013	8750	3.0

¹ [Fe/H] from KCC is [-2.95] for HD 8376.
² E(B-V) from KCC is 0.041 for HD 8376, 0.014 for HD 13780, and 0.022 for HD 109995.

For the fitting procedure, the IUE spectra were dereddened for the E(B-V) listed in Col. 4 of Table 2. The reddening E(B-V) was taken from KCC for all the stars, except HD 8376, HD 13780, HD 109995. The new values were fixed on the basis of the better agreement between the observed and computed ultraviolet fluxes yielded by them. The interstellar extinction, as a function of wavelength, was taken from Mathis (1990). We adopted R_V=3.1. The dereddened IUE fluxes and the computed fluxes were normalized at 5556 Å. The observed flux at 5556 Å was obtained by means of the relation $\log F_{\lambda}=-0.400V-8.456$ from Gray (1976, p. 202) and it was then dereddened according to the procedure used for the ultraviolet fluxes.

For each star, a grid of new-ODF models computed for a given metallicity [M/H] and a given microturbulent velocity $\xi$ was selected for the fit. The metallicity [M/H] is that listed in Col. 2 of Table 2. For all the stars, except HD 8376, HD 93329, and HD 117880, it approximates within 0.125 dex the iron abundance [Fe/H] obtained by KCC and given in Col. 7 of Table 1. Because the abundance analysis performed by KCC indicated that all the stars, except BD+32 2188, have the Mg and Ti abundances enhanced, on average, by 0.4 dex over the iron, we started by assuming that all the alpha elements are equally enhanced. Therefore we adopted $\alpha$-enhanced ATLAS9 models and fluxes for all the stars, except BD+32 2188.

The microturbulent velocity $\xi$ given in Col. 3 of Table 2 is based on that derived by KCC. Because ODFs are computed only for $\xi=0.0$, 1.0, 2.0, 4.0 and 8.0 km s^-1, we approximated the microturbulent velocities $\xi=2.5$ km s^-1, 3.0 km s^-1, and 3.5 km s^-1 obtained by KCC for some stars with $\xi=2.0$ km s^-1 or $\xi=4.0$ km s^-1, rather than interpolating the ODFs for the microturbulent velocity. In fact, the uncertainty in $\xi$ is not less than 1 km s^-1. As explained in Sect. 8 in Kinman et al. (2000), the microturbulent velocity in KCC was derived from Fe I, Fe II, and Ti II lines by assuming that, for a given element, the abundance is independent of the equivalent widths. The uncertainty, however, both in the equivalent widths of the weak lines and in the $\log~gf$ values (especially for the lines of Ti II, which are the most numerous) severely limits this method of obtaining $\xi$. The adopted lines, their measured equivalent widths, the adopted $\log\,gf$ and their sources are given for each star in Tables 4 and 5 in Kinman et al. (2000). The value of $\xi$ derived from the equivalent widths was then refined by KCC by comparing the observed spectra against a series of synthetic spectra in which $\xi$ was sampled in steps of 1 km s^-1; only in a few cases an intermediate step of 0.5 km s^-1 was used.

Furthermore, for BD+00 145, HD 14829 and all the stars observed only at ESO (HD 4850, HD 13780, HD 78913, HD 106304, HD 130201, HD 213468) $\xi$ was assumed "a priori'' to be equal to 2 km s^-1, because there were too few lines in the spectra of these stars even for an estimate of $\xi$.

The comparison of the observed and computed ultraviolet energy distributions has led us to modify the starting parameters for some stars. In particular the changes are:
HD 8376: $\rm [M/H]=-2.50$, instead of $\rm [M/H]=-2.95$a;
HD 13780: $\rm [M/H]=-1.50$, instead of $\rm [M/H]=-1.50$a;
HD 93329: $\rm [M/H]=-1.50$a, instead of $\rm [M/H]=-1.25$a;
HD 117880: $\rm [M/H]=-1.50$a, $\xi =2$ km s^-1, instead of $\rm [M/H]=-1.75$a, and $\xi =4$ km s^-1.

The metallicity was modified mostly on the basis of the comparison between the observed and computed energy distributions shortward of 1500 Å, where the intensity of the emitted flux is related with the size of the Si I discontinuity at 1525 Å. Also for HD 87047, a silicon abundance higher than the adopted one of -6.59 dex, could have improved the agreement between observations and computations for $\lambda< 1500$ Å (Fig. A.12, Appendix A). However, for all the stars, a detailed abundance analysis is needed in order to fix the silicon abundance best reproducing the observations.

Figure 4: Upper panel: $T_{\rm eff}$ differences between KCC and new-ODF models as functions of $T_{\rm eff}$, as derived from the new-ODF models. Lower panel: same as upper panel, for $\log\,g$.

In Table 2 we list the parameters derived by fitting the short-wavelength part (1200-1978 Å) of the UV energy distribution (Cols. 5 and 6), the long-wavelength part (1979-3300 Å) (Cols. 7 and 8), and the whole IUE spectrum (1200-3300 Å) (Cols. 9 and 10) to the new-ODF models. The parameters derived from the entire IUE energy distribution are very similar to those derived from the short-wavelength range, confirming the results of Fig. 3 that shows the stronger dependence of the parameters on the short-wavelength spectrum.

The differences in $T_{\rm eff}$ and $\log\,g$ obtained from the short- and long-wavelength regions are, on average, on the order of 150 K in $T_{\rm eff}$ and 0.3 dex in $\log\,g$. $T_{\rm eff}$ from the 1200-1978 Å spectrum is generally higher than that from the 1979-3300 Å spectrum. The differences in the parameters are due to the difficulty in obtaining both $T_{\rm eff}$ and $\log\,g$ from the IUE long-wavelength region and also probably to possible inconsistencies between the short- and long IUE spectra (Sect. 2) or to some inaccuracy in the models. For instance, the models were computed with "a priori'' abundances for all the elements, except magnesium, titanium, and iron for which the actual abundances derived by KCC were used; in addition, the models are affected by a lower line blanketing than the real energy distributions, owing to several missing lines, in particular in the UV.

For each star, a grid of old-ODF fluxes computed for the same metallicity and microturbulent velocity adopted for the new-ODF fluxes (Table 2, Cols. 2 and 3) was used in order to derive parameters from the old-ODF models by means of the fitting procedure. Table 3 compares the parameters from the new-ODF models and the whole IUE wavelength region, the parameters from the old-ODF models and the whole IUE wavelength region, and the parameters from KCC. The stars are ordered by decreasing $T_{\rm eff}$, as they were derived from the new-ODF models, and as they are ordered in Appendix A, where the observed energy distributions are compared with energy distributions computed from the old-ODF models, from the new-ODF models, and from KCC. This order allows a better estimate of the dependence of the ultraviolet energy distribution on $T_{\rm eff}$, in particular of the H-H⁺ and H-H quasi-molecular absorptions near 1400 Å and 1600 Å.

Table 3 shows that, for most of the stars, the parameters from old-ODF models and new-ODF models are the same within the uncertainty of the fit which is on the order of 50 K in $T_{\rm eff}$ and 0.1 dex in $\log\,g$. Instead, Figs. A.1-A.15 in Appendix A show that the short-wavelength energy distributions are reproduced by the new-ODF models much better, especially for stars cooler than 8000 K. Therefore, in spite of the parameters being almost the same, the rms from the fit is lower when the new-ODF models are used. In general, the gravities from the old-ODF models are higher than those from the new-ODF models. The largest difference, which amounts to 0.3 dex, occurs for BD+00 145, HD 117880, and HD 213468. Because no model is able to reproduce the core of Lyman-$\alpha$ of BD+00 185, the gravity is different depending on the wavelength interval (1200-3300 Å or 1300-3300 Å) used for the fit. In the second case, the model reproduces the observed absorption at 1400 Å well, which is instead predicted too low in the first case. Therefore we adopted for BD+00 185 the parameters derived by fitting the 1300-3300 Å region.

 

 

Table 3: The comparison of parameters from new-ODF, old-ODF models and KCC.

Star	E(B-V)	[M/H]	$\xi$	$T_{\rm eff}$	$\log\,g$	$T_{\rm eff}$	$\log\,g$	$T_{\rm eff}$	$\log\,g$
				new-ODFs		old-ODFs		KCC
BD+32 2188	0.007	[-1.00]	1.0	10300	2.2	10300	2.2	10450	2.10
HD 128801	0.010	[-1.50a]	2.0	10200	3.5	10200	3.5	10300	3.55
BD+00 145	0.018	[-2.50a]	2.0	9900	3.2	9900	2.9	9700	4.00
"1				9850	3.8	9800	4.1
HD 87112	0.003	[-1.50a]	2.0	9700	3.6	9650	3.8	9750	3.50
HD 106304	0.038	[-1.25a]	2.0	9600	3.5	9600	3.5	9750	3.50
HD 117880	0.077	[-1.50a]	2.0	9350	3.3	9300	3.6	9300	3.30
HD 213468	0.008	[-1.75a]	2.0	9100	3.3	9050	3.6	9150	3.30
HD 130095	0.072	[-1.75a]	2.0	9100	3.2	9100	3.3	9000	3.30
HD 167105	0.024	[-1.50a]	2.0	9000	3.1	9000	3.2	9050	3.30
HD 14829	0.018	[-2.50a]	2.0	8900	3.1	8900	3.1	8900	3.20
HD 130201	0.035	[-1.00a]	2.0	8900	2.5	8900	2.6	8650	3.50
HD 74721	0.012	[-1.50a]	2.0	8800	3.2	8800	3.3	8900	3.30
BD+42 2309	0.013	[-1.75a]	2.0	8750	3.0	8700	3.2	8800	3.20
HD 78913	0.034	[-1.50a]	2.0	8700	2.8	8700	2.9	8500	3.25
HD 139961	0.051	[-1.75a]	4.0	8600	2.8	8600	2.8	8500	3.20
HD 109995	0.010	[-1.75a]	2.0	8500	3.0	8450	3.1	8500	3.10
HD 4850	0.009	[-1.25a]	2.0	8450	2.7	8400	2.9	8450	3.20
HD 93329	0.014	[-1.50a]	2.0	8250	2.9	8250	2.9	8250	3.10
HD 60778	0.028	[-1.50a]	4.0	8250	2.9	8200	3.1	8050	3.10
HD 86986	0.022	[-1.75a]	2.0	8100	2.8	8050	2.9	7950	3.20
HD 8376	0.020	[-2.50]	1.0	8050	2.6	8100	2.6	8150	3.30
HD 13780	0.000	[-1.50]	2.0	7900	2.7	7900	2.8	7950	3.10
HD 87047	0.006	[-2.50a]	2.0	7900	2.8	7850	2.9	7850	3.10
HD 31943	0.006	[-1.00a]	4.0	7850	3.1	7850	3.1	7900	3.20
HD 180903	0.098	[-1.50a]	4.0	7800	2.9	7800	2.9	7700	3.10
HD 252940	0.048	[-1.75a]	4.0	7650	2.7	7650	2.7	7550	2.95
HD 2857	0.022	[-1.75a]	4.0	7600	2.8	7600	2.8	7550	3.00
HD 161817	0.000	[-1.50a]	4.0	7600	2.7	7600	2.7	7550	3.00
HD 202759	0.072	[-2.00a]	2.0	7500	2.8	7550	2.8	7500	3.05

¹ Parameters for a fitting range starting at 1300 Å instead of at 1200 Å.

The last two columns of Table 3 list the parameters found by KCC. Figure 4 illustrates, in the upper panel, the differences between $T_{\rm eff}$ from KCC and $T_{\rm eff}$ from the IUE fluxes and new-ODF models and, in the lower panel, the differences between $\log\,g$ from KCC and $\log\,g$ from the IUE fluxes and new-ODF models. Both differences are plotted as a function of $T_{\rm eff}$, as derived from the new-ODF models. The temperatures agree within 150 K for all the stars, except for HD 130201 ($\Delta$ $T_{\rm eff}$=-250 K), HD 78913 ($\Delta$ $T_{\rm eff}$=-200 K), and HD 60778 ($\Delta$ $T_{\rm eff}$=-200 K). No trend of $\Delta$ $T_{\rm eff}$ with $T_{\rm eff}$ is manifest. The gravities agree within 0.2 dex for the stars hotter than 8700 K (except HD 130201), but for the other stars the gravities from the whole IUE flux and new-ODF models are systamatically lower than those from KCC, with an average difference of about 0.3 dex.