3 Method of analysis

The spectra were analysed using a differential model atmosphere technique. The method of analysis and atomic line parameters are the same as used recently by Tautvaisiene et al. (2000, Paper I), where the chemical composition of evolved stars in the open cluster M 67 was investigated. The Eqwidth and Spectrum programme packages, developed at Uppsala Astronomical Observatory, were used to carry out the calculations of abundances from measured equivalent widths and synthetic spectra, respectively. A set of plane parallel, line-blanketed, flux constant LTE model atmospheres with solar abundance ratios was computed by M. Asplund (Uppsala Astronomical Observatory) with the updated version of the MARCS code (Gustafsson et al. 1975) using continuous opacities from Asplund et al. (1997) and including UV line blanketing as described by Edvardsson et al. (1993). The solar model atmosphere for the differential analysis was also calculated in Uppsala (Edvardsson et al. 1993).

Figure 1: Synthetic (dashed and dotted curves) and observed (solid curve with dots) spectra for the 1-0 C2 region near $\lambda 5635$ Å of BD+34$^\circ$2371. The syntheses were generated with [C/H] =-0.3, -0.4, and -0.5 (dashed-dotted, dotted and dashed curves, respectively).

The effective temperatures for the programme stars were initially taken from Tautvaisiene (1996b), where they were derived using the intrinsic colour index (Y-V)₀ of the Vilnius photometric system. For the star BD+28$^\circ$2079 the (Y-V)₀ was taken from Bartkevicius & Lazauskaite (1997) and the same procedure applied. In the work by Tautvaisiene (1996b) as well as in the work by Norris (1987) the interstellar reddening for these stars was accepted to be zero. Although these stars are located in the direction of the North Galactic Pole, where the reddening should be small, Bartkevicius & Lazauskaite (1997) have found that some of the stars are affected. We decided to introduce a spectroscopic method to solve the problem. We corrected, when needed, the effective temperatures by achieving the LTE exitation balance in the iron abundance results. For nine stars the effective temperatures were adjusted by -70-+110K.

The surface gravities were found by forcing Fe I and Fe II to yield the same iron abundances, 47 Fe I and 5 Fe II lines were used. The microturbulent velocities were determined by forcing Fe I line abundances to be independent of the equivalent width. The derived atmospheric parameters are listed in Table 2.

Abundances of carbon and nitrogen were determined using the spectrum synthesis technique. The interval of 5632-5636 Å was synthesized and compared with observations in the vicinity of the ${\rm C}_2$ Swan 0-1 band head at 5635.5 Å. The 5635.5 Å ${\rm C}_2$ band head is strong enough in our spectra and is quite sensitive to changes of the carbon abundance (see Fig. 1 for illustration). The same atomic data of ${\rm C}_2$ as used by Gonzalez et al. (1998) and in Paper I were adopted for the analysis.

 

 

Table 2: Atmospheric parameters derived for the field RHB stars. The last two columns give numbers of spectra observed with the resolving power $R{\rm 1}\approx 30\, 000$ and $R{\rm 2}\approx 60\, 000$.

BD/HD	$T_{\rm e},{\rm K}$	$\log g$	[Fe/H]	${ v_{\rm t},{\rm km\,s^{-1}}}$	$R{\rm 1}$	$R{\rm 2}$
+25$^\circ$2436	4990	2.4	-0.48	1.7	2	2
+25$^\circ$2459	4980	2.5	-0.35	1.5	1	2
+25$^\circ$2502	5090	2.2	-0.74	1.3		2
+27$^\circ$2057	4840	2.1	-0.60	1.7	1	2
+28$^\circ$2079	4950	2.5	-0.44	2.0	2
+29$^\circ$2231	5060	2.5	-0.39	1.9	2	2
+29$^\circ$2294	5020	2.1	-0.54	1.7	1
+29$^\circ$2321	4980	2.3	-0.50	1.9	2
+33$^\circ$2280	5000	2.4	-0.48	1.6		1
+34$^\circ$2371	4980	2.5	-0.18	1.6	1
+36$^\circ$2303	4700	1.8	-0.76	2.0	2
104783	5140	2.4	-0.55	1.5	1	3
105944	5090	2.1	-0.37	1.4	2

The intervals of 7980-8130 Å with $R\approx 30\,000$ and 8380-8430 Å with $R\approx 60\,000$, containing strong CN features, were analysed in order to determine the nitrogen abundance. The ¹²C/¹³C determination was based on the 8004.728 Å ¹³CN feature. 11 other weaker ¹³CN features ($\lambda$ 7989.45, 8010.4, 8011.2, 8016.35, 8022.65, 8036.15, 8043.2, 8048.3, 8051.8, 8056.4, 8058.2 and 8065.0 Å) were used for error estimation. The molecular data for ¹²C¹⁴N and ¹³C¹⁴N were taken from ab initio calculations of CN isotopic line strengths, energy levels and wavelengths by Plez (1999), with all gf values increased by +0.03 dex in order to fit our model spectrum to the solar atlas of Kurucz et al. (1984). The ¹³CN line wavelengths were, however, adopted from laboratory measurements by Wyller (1966). Parameters of atomic lines in the spectral synthesis intervals were adopted from the VALD database (Piskunov et al. 1995). In order to check the correctness of the input data, synthetic spectra of the Sun were compared to the solar atlas of Kurucz et al. (1984) and necessary adjustments were made to the line data.

Figure 2 illustrates the enhancement of the ¹³CN line at 8004.7 Å in a spectrum of the star BD+27$^\circ$2057.

Figure 2: A small portion of the 8000 Å wavelength interval showing the 8004.7 Å $^{13}{\rm CN}$ feature in the star BD +27$^\circ$2057. The dotted line shows a synthetic spectrum with [N/H] =-0.28 and $^{12}{\rm C}/^{13}{\rm C}=7$, the dashed line shows a synthetic spectrum with [N/H]  =-0.22 and $^{12}{\rm C}/^{13}{\rm C}=3$. The dots indicate features dominated by $^{13}{\rm CN}$, and the crosses mark features dominated by $^{12}{\rm CN}$. Unfitted features belong to the Earth atmosphere.

Abundances of oxygen were determined using equivalent widths of the [O I] forbidden line at 6300 Å, widely used in analyses of other late-type stars. This line was recently reexamined in the solar spectrum with a three-dimensional time-dependent hydrodynamical model solar atmosphere and implications of the Ni I blend on oxygen abundances discussed (Prieto et al. 2001). Our test calculations showed that in our sample of stars the influence of the Ni line is very small (oxygen abundance changes do not exceed 0.01-0.03 dex).

The interval of 6643-6648 Å, containing the Eu II line at 6645 Å, was computed in order to determine the europium abundance (see Fig. 3 for illustration). The oscillator strength of the Eu II line, $\log gf=0.17$, was adopted from Gurtovenko & Kostik (1989). The solar abundance of europium, later used for the differential analysis, $\log A({\rm Eu})\odot=0.49$, was determined by fitting of the Kurucz et al. (1984) solar flux spectrum. Parameters of other lines in the interval were compiled from the VALD database. CN lines were also included, but none of them seems to affect the europium line significantly.

Figure 3: Synthetic and observed (thick solid curve and dots) spectra for the region around the Eu II line at $\lambda$ 6645 Å in BD+27$^\circ$2057. The syntheses are generated with [Eu/Fe]$\, =0.4$, 0.5 and 0.6 (dashed, dotted and long-dashed curves, respectively).

 

 

Table 3: Effects on derived abundances resulting from model changes for the star BD+25$^\circ$2436. The table entries show the effects on the logarithmic abundances relative to hydrogen, $\Delta [A/\rm H]$. Note that the effects on "relative'' abundances, for example [A/Fe], are often considerably smaller than abundances relative to hydrogen, [A/H].

Ion	${ \Delta T_{\rm eff} }\atop{ -100 {\rm ~K} }$	${ \Delta \log g }\atop{ -0.3 }$	${ \Delta v_{\rm t} }\atop{ -0.3 {\rm km~s}^{-1}}$
C (C2)	0.02	-0.03	0.00
N (CN)	-0.10	-0.03	0.00
O I	-0.01	-0.13	0.00
Na I	-0.07	0.01	-0.05
Mg I	-0.04	-0.01	-0.03
Al I	-0.05	0.01	-0.02
Si I	0.01	-0.04	0.03
Ca I	-0.10	0.01	-0.11
Sc I	-0.12	0.00	0.02
Sc II	0.02	-0.13	0.10
Ti I	-0.14	0.01	0.09
Ti II	0.01	-0.12	0.08
V I	-0.16	0.00	0.03
Cr I	-0.11	0.01	-0.09
Mn I	-0.08	-0.01	0.04
Fe I	-0.08	-0.02	0.06
Fe II	0.09	-0.14	0.10
Co I	-0.08	-0.02	-0.02
Ni I	-0.05	-0.03	0.08
Y I	-0.17	-0.01	0.02
Y II	0.00	-0.14	0.13
Zr I	-0.17	0.00	-0.01
Ba II	-0.02	-0.11	0.27
La II	-0.01	-0.13	0.01
Sm II	-0.02	-0.14	0.03
Eu II	0.00	-0.10	-0.01

Typical internal error estimates for the atmospheric parameters are: $\pm$100 K for $T_{\rm eff}$, $\pm$0.3 dex for $\log g$ and ${\pm} 0.3~{\rm km~s}^{-1}$ for $v_{\rm t}$. The sensitivity of the abundance estimates to changes in the atmospheric parameters by the assumed errors is illustrated for the star BD+25$^\circ$2436 (Table 3). It is seen that our estimated parameter uncertainties do not affect the abundances seriously; the element-to-iron ratios, which we use in our discussion, are even less sensitive. The small differences between the chemical composition of the models and the final abundance results have a neglible effect on the results. The $^{12}{\rm C}/^{13}{\rm C}$ ratios are not sensitive to the model parameters or errors in the $\log gf$ values since they are determined after fitting the $^{12}{\rm CN}$ features.

The scatter of the deduced line abundances $\sigma$, presented in Table 4, gives an estimate of the uncertainty coming from the random errors in the line parameters (e.g. random errors in equivalent widths, oscillator strengths and possible undetected line blends). The approximate value of these uncertainties amounts in the mean to $\sigma=0.10$ dex. Other sources of observational errors, such as continuum placement or background subtraction problems are partly included in the equivalent width uncertainties. The nitrogen abundance is less dependent on line measurement uncertainties because, depending on the number of spectra observed, the number of CN lines used for the analysis was ranging from 34 to 162.

Figure 4: [C/Fe] as a function of [Fe/H]. Results of this paper are indicated by filled circles, results obtained for dwarf stars of the galactic disk (Gustafsson et al. 1999) are indicated by " plus'' signs and the solid line. The relative underabundance in the He-core burning stars is clearly seen.

Since the abundances of C, N and O are tied together by the molecular equilibria in the stellar atmospheres and the abundances were determined in the sequence O (from [O I]) $\Rightarrow$ C (from C₂) $\Rightarrow$ N (from CN), we have investigated how an error in one of them would typically affect our abundance determinations of the others. Calculations for BD+25$^\circ$2436: a change of the oxygen abundance of $\Delta{\rm [O/H]}=-0.10$ would result in $\Delta{\rm [C/H]}=-0.04$, $\Delta{\rm [N/H]}=-0.01$ and thus $\Delta{\rm [C/N]}=-0.03$; a change in $\Delta{\rm [C/H]}=-0.10$ would cause $\Delta{\rm [N/H]}=+0.10$, $\Delta{\rm [C/N]}=-0.20$ and $\Delta{\rm [O/H]}=-0.03$; $\Delta{\rm [N/H]}=-0.10$ has no appreciable effect on either the oxygen or the carbon molecular equilibria (except for CN). Note in particular that the C/N ratios are sensitive to the carbon abundance uncertainties squared.