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Astron. Astrophys. 348, 211-221 (1999)

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4. The isotope ratio 6Li/7Li

4.1. The center-of-gravity wavelength of the LiI line

As discussed by e.g. Smith et al. (1998) the 6Li/7Li ratio can be determined by two methods: From the center-of-gravity (cog) of the LiI 6708 Å line or from a detailed model atmosphere synthesis of the profile. The isotopic shift of the 6Li doublet is +0.158 Å relative to the 7Li doublet. Addition of 6Li therefore shifts the LiI line to longer wavelengths and increases the FWHM. The cog-method relies in principle on a very simple and straightforward measurement, but its accuracy is limited by possible errors in the laboratory wavelengths of the lithium line and the reference lines needed to correct for the radial velocity shift of the star. The errors in the laboratory wavelengths are typically [FORMULA] mÅ. Differences in convective blueshifts may well be of the same order of size (Dravins 1987). To this should be added the uncertainty in measuring the cog-[FORMULA] for an asymmetric lithium line, which is slightly blended by a weak Fe I line in the blue wing. According to our experience the cog-[FORMULA] will be uncertain by about [FORMULA] mÅ. This translates to a one-sigma error of [FORMULA] in 6Li/7Li, which is inferior to what may be obtained with the profile method. Hence, only the profile method will be used, and the wavelength of the LiI line will be considered as a free parameter in the comparison between synthetic and observed profiles.

4.2. Synthetic spectra

The synthetic spectra have been obtained with the Uppsala Synthetic Spectrum Package, which computes the flux in a given wavelength region for an OSMARC model atmosphere of the star (Edvardsson et al. 1993). LTE is assumed and thermal and microturbulent line broadening as well as pressure broadening is included. The resulting spectrum is folded with a line broadening function that can be either Gaussian, radial-tangential (Gray 1978), rotational (Gray 1992) or any combination of these functions. As a check, one star (HD 68284) has also been analyzed with a Kurucz ATLAS9 model atmosphere and the SYNTHE code (Kurucz 1993).

Wavelengths and gf-values for the 6Li and 7Li components of the LiI doublet are taken from Table 3 of Smith et al. (1998). As discussed by Smith et al. these values are not a source of significant errors in connection with determinations of the isotopic abundances of Li.

In the case of halo turnoff stars with [FORMULA] there is no significant blending of the LiI line by other lines, but already at [FORMULA] one has to worry about contributions from other absorption lines. To get more detailed information about line blending we therefore started by synthesizing the solar spectrum, which is well suited for studying this problem because of the weakness of the LiI line. Fig. 2 shows the solar flux spectrum from the atlas of Kurucz et al. (1984) compared to a synthetic spectrum based on the OSMARC solar model and a list of CN and metal lines from Müller et al. (1975), who obtained log [FORMULA](Li) = 1.0 and 6Li/7Li [FORMULA] 0.0 from a synthesis analysis of the solar LiI feature. We adopted these values and adjusted the gf-values of the other lines to get the best possible fit of the synthetic spectrum (broadened by a radial-tangential function with a FWHM = 2.5 km s-1) to the Solar Flux Atlas. As seen from Fig. 2 the fit is quite satisfactory. The main problem is the two unidentified weak lines at 6708.02 and 6708.28 Å, which have equivalent widths of 0.6 and 1.1 mÅ, respectively. The first line has nearly the same wavelength as the weak component of the 6Li doublet and therefore makes the determination of 6Li/7Li very tricky for solar-type metallicities, whereas the unidentified line at 6708.28 Å is further away and only affects the determination of the lithium isotope ratio marginally.

[FIGURE] Fig. 2. The solar flux spectrum around the LiI line. The full drawn line is the observed spectrum from the Solar Flux Atlas of Kurucz et al. (1984). The dashed line is the synthetic spectrum computed for the model atmosphere of the Sun. The dotted line is the synthetic spectrum without the LiI line

Assuming that the two unidentified lines are neutral metal lines they will have equivalent widths less than 0.15 and 0.3 mÅ, respectively, in solar-type stars with [FORMULA]. This is too small to have any significant effect on the determination of the lithium isotope ratio. If the lines are ionized metal lines they will be somewhat stronger and affect the determination of 6Li/7Li marginally at the 1% level. The probability for lines in solar-type spectra being from ionized metals instead of neutral is, however, small, and we have therefore chosen to exclude the two unidentified lines from our model atmosphere synthesis of the metal-poor disk stars.

The CN lines seen in Fig. 2 play no rôle in the spectra of the program stars mainly because both C and N scales almost linearly with Fe and partly because the stars have somewhat higher effective temperatures than the Sun. Hence, there is no reason to include these lines. The only line that has a significant effect on the synthesis of the lithium line is the Fe I line at 6707.43 Å. This line was included with the gf-value derived from the fit to the solar spectrum.

As discussed in Sect. 4.3 the Fe I lines at 6703.6 and 6705.1 Å have been used to determine the instrumental and stellar atmospheric line broadening. The two lines are blended by several faint CN lines in the solar spectrum as seen from Fig. 2 of Brault & Müller (1975), but the synthesis of the solar and stellar spectra shows that these lines disappear beyond detection in the program stars. The same technique has been used to define a number of spectral windows practically free of lines in the program stars. These regions are marked by `C' in Fig. 1, and are used to set the continuum and to estimate the S/N in connection with the chi-square analysis of the lines.

4.3. [FORMULA] analysis

The synthetic spectra were computed for model atmospheres with the parameters given in Table 1. Line broadening due to macroturbulence and rotation was determined from the two Fe I lines at 6703.6 and 6705.1 Å with [FORMULA] and 4.61 eV, respectively. The synthetic spectrum was first folded with a Gaussian function representing the instrumental profile and then with either a Gaussian, a radial-tangential or a rotational profile. Various combinations of these profiles was also tried, but in no cases the fit to the iron lines was better than that obtained with a Gaussian function. The final analysis was therefore carried out with a single Gaussian representing the combined effect of instrumental and stellar atmospheric broadening.

The FWHM of the Gaussian broadening profile, [FORMULA], has been determined with the following procedure. First the continuum is set from the two windows on each side of a Fe I line and at the same time the standard deviation, [FORMULA] of the spectrum is estimated. Then the chi-square function is computed:

[EQUATION]

where Oi is the observed spectral point and Si is the synthesis. The summation is performed over the spectral interval marked by `L' in Fig. 1 corresponding to N = 25 datapoints. In addition to [FORMULA], the equivalent width and the exact wavelength of the Fe I line are considered as free parameters. [FORMULA] is varied in steps of 0.1 or 0.2 km s-1 and the other two parameters are optimized for each value of [FORMULA] to find the lowest [FORMULA]. This results in a parabolic variation of [FORMULA] as shown in Fig. 3. The most probable value of [FORMULA] corresponds to the minimum of [FORMULA], and [FORMULA] = 1, 4 and 9 correspond to the 1-, 2-, and 3-[FORMULA] confidence limits of determining [FORMULA] alone (Bevington & Robinson 1992).

[FIGURE] Fig. 3. Variation of the [FORMULA] of the fit to the Fe I 6705.1 Å line as a function of the FWHM of the Gaussian broadening function

Table 3 summarizes the results of the [FORMULA] fitting of the two Fe I lines. Note, that the reduced chi-square ([FORMULA], where [FORMULA] is the number of degrees of freedom in the fit) is satisfactorily close to 1. Furthermore, the values of [FORMULA] from the two Fe I lines agree rather well, although there is a tendency that the 6705.1 Å line gives slightly higher values of [FORMULA] than the 6703.6 Å line.


[TABLE]

Table 3. Results from the [FORMULA] analysis of the Fe I lines at 6703.6 and 6705.1 Å. [FORMULA] and [FORMULA] are the FWHM of the Gaussian broadening function (instrumental + rotation + macroturbulence) applied to the synthetic lines. The errors given are the formal one-sigma errors resulting from the [FORMULA] analysis


Adopting a weighted average of [FORMULA] from Table 3 the lithium isotope ratio is determined from a [FORMULA] analysis of the fit between the computed and observed profile of the LiI line. The free parameters in the fit are the total lithium abundance of the star, log[FORMULA](Li), and the 6Li fraction, f(6Li) = N(6Li)/N(Li). Furthermore, a wavelength shift, [FORMULA], of the Li line relative to the wavelengths given in Table 3 of Smith et al. (1998) is allowed. The continuum and the S/N are determined from the two adjacent windows shown in Fig. 1 (resulting in nearly the same S/N values as given in Table 3) and the [FORMULA] analysis is carried out over the region marked by `L', i.e. over N=33 datapoints. Note, that the weak Fe I line at 6707.43 Å has only a small effect on the flux in this line region.

The relative 6Li abundance is varied in steps of 0.01 and the other two parameters, log[FORMULA](Li) and [FORMULA], are optimized for each value of f(6Li) to find the lowest [FORMULA]. In order to study the behaviour of the [FORMULA] function around f(6Li) = 0.00 `negative' values of f(6Li) have been simulated by subtracting the line absorption coefficient due to 6Li from the continuous absorption coefficient instead of adding it.

The results of the [FORMULA] analysis are summarized in Table 4 and the variation of [FORMULA] with f(6Li) is shown in Fig. 4. As seen, the three `HR' stars have f(6Li) close to zero, whereas 6Li has been detected in the two `HD' stars at a high confidence level. As a further illustration of this result Figs. 5 and 6 show the fits to the Fe I and the LiI for two stars - one with f(6Li) [FORMULA] 0.0 and the other one with f(6Li) [FORMULA] 0.06 - and Fig. 7 shows a plot of the residuals in the observations after subtracting the 7Li and Fe I part of the synthesis. Although there is a disturbing periodic noise in the residuals with an amplitude of 0.1 to 0.2% in addition to the shot noise (the reason for which remains unexplained) a clear residual absorption at the wavelength of the 6Li doublet is seen in the spectra of HD 68284 and HD 130551. Note, that although the S/N of the spectrum of HD 68284 is inferior to that of HD 130551, the error of f(6Li) is nearly the same for the two stars, because the Li line is about a factor of two stronger in the spectrum of HD 68284 than in the case of HD 130551.


[TABLE]

Table 4. Results from the [FORMULA] analysis of the LiI line. [FORMULA] is the weighted average of the FWHM of the Gaussian broadening function determined from the Fe I lines at 6703.6 and 6705.1 Å (Table 3). log[FORMULA](Li) is the total Li abundance and f(6Li) is the relative abundance of 6Li. The errors given are the formal one-sigma errors resulting from the [FORMULA] analysis. The last column gives the reduced chi-square of the fit


[FIGURE] Fig. 4. Variation of the [FORMULA] of the fit to the LiI 6707.8 Å line as a function of the relative abundance of 6Li

[FIGURE] Fig. 5. The model atmosphere synthesis of the Fe I 6705.1 Å and the LiI 6707.8 Å line in the spectrum of HR 8181. The datapoints are shown with open circles. In the upper figure the full drawn line corresponds to a Gaussian broadening parameter of [FORMULA] km s-1, whereas the dotted and dashed lines correspond to [FORMULA] and 6.2 km s-1, respectively. In the lower figure [FORMULA] km s-1 has been applied. The full drawn line corresponds to f(6Li) = 0.0 and the dashed line to f(6Li) = 0.05. Note, that when [FORMULA] and f(6Li) are varied the other free parameters in the fits, the wavelengths and the equivalent widths of the lines, have been optimized to get the best possible fits

[FIGURE] Fig. 6. Same as Fig. 4 for HD 130551. In the upper figure the full drawn line corresponds to a Gaussian broadening parameter of [FORMULA] km s-1, and the dotted and dashed lines to [FORMULA] and 7.2 km s-1, respectively. In the lower figure [FORMULA] km s-1 has been applied. Here the full drawn line corresponds to f(6Li) = 0.06 and the dotted and dashed lines to f(6Li) = 0.00 and 0.10, respectively

[FIGURE] Fig. 7. The residuals of the observations after subtraction of the 7Li and Fe I 6707.43 Å part of the synthesis of the LiI line. For comparison the synthesis of the 6Li doublet is shown with a full drawn line

In order to check that there are no significant systematic differences between the results obtained for the three observing periods the individual spectra of HR 3578 were analyzed separately including the determination of [FORMULA]. The [FORMULA] analysis gave the following values: f(6Li) = -0.018 [FORMULA] 0.012, +0.008 [FORMULA] 0.010, and -0.006 [FORMULA] 0.014 for the Oct. 92, June 93 and Feb. 95 spectra. Within the errors these values agree satisfactorily with the value (f(6Li) = 0.000 [FORMULA] 0.006) derived for the averaged spectrum.

4.4. Systematic errors of 6Li/7Li

The errors of f(6Li) given in Table 4 are purely statistical. In addition we must consider possible systematic errors resulting from approximations in the model atmospheres and the spectrum synthesis of the Fe I and LiI lines.

First we note that the exact values of the atmospheric parameters of the models are not critical for the lithium isotope ratio derived. An increase of [FORMULA] with +100 K changes log[FORMULA](Li) with +0.07 dex but f(6Li) is practically unchanged. Changes in the gravity within reasonable limits have completely negligible effects. A decrease of the microturbulence by say 0.4 km s-1 is compensated by a slight increase of [FORMULA] and the change of f(6Li) is less than 0.003. A more significant, but still rather small change, results from the use of Kurucz's model atmospheres (Kurucz 1993) instead of the OSMARC models. As an example a model atmosphere with the parameters of HD 68284 was interpolated between the closest models in the Kurucz ATLAS9 grid (re-computed with the "approximate" overshooting option switched off) and the SYNTHE code was used to compute profiles of the Fe I and LiI lines. This resulted in an increase of [FORMULA] from 5.6 to 5.9 km s-1 and a change of f(6Li) from 0.041 to 0.033. Hence, the class of plane-parallel model atmospheres adopted has some effect on the value of the lithium isotope ratio derived, but not large enough to question the detection of 6Li in HD 68284 and HD 130551.

A more critical problem is whether classical plane-parallel model atmospheres are good enough for a determination of f(6Li) with an accuracy of say [FORMULA]. It is well known that convective motions in real atmospheres produce slightly asymmetric line profiles, i.e. curved bisectors with a shape that depends on the depth of line formation (e.g. Dravins 1987). The question is therefore if the atmospheric velocity broadening determined from the two Fe I lines is also valid for the LiI line. Due to the low excitation potential of the lithium resonance line it is probably formed somewhat higher in the atmosphere than the iron lines. In the case of HD 130551 a drastic increase of [FORMULA] to 7.5 km s-1 instead of the value derived from the Fe I lines, (6.5 km s-1) decreases f(6Li) to about zero. One could also imagine that the LiI line in HD 68284 and HD 130551 has a red asymmetry that mimics the 6Li doublet although no such asymmetry is seen in the profiles of the Fe I lines. However, such possible effects have to occur for HD 68284 and HD 130551 only, because we can not allow a reduction of f(6Li) for the other three stars which have f(6Li) [FORMULA] 0.00, when the atmospheric velocity broadening of the Fe I lines is adopted.

As seen from Table 1 there are indeed significant differences in the atmospheric parameters of the two `HD' stars and the three `HR' stars that could induce some differential effects in the broadening of the Fe I and LiI lines. HD 68284 has a lower gravity and HD 130551 has a higher [FORMULA] than the other three stars. A hint that there may be systematic differences in the convective pattern between the two groups of stars comes from the small changes of the laboratory wavelengths needed to optimize the [FORMULA] fit of the Fe I and LiI lines (see Table 5). The apparent heliocentric radial velocities (including gravitational redshift and convective blueshift) is determined from the two Fe I lines. Hence, the sum of the wavelength shifts for these two lines is zero by definition. As seen, the wavelength shift of the LiI line is always positive. The average value is 4.4 mÅ (corresponding to a redshift of 0.20 km s-1 of the LiI line relative to the Fe I lines) and the rms scatter is 3.4 mÅ. This is more than the expected error of the laboratory wavelengths. In particular, we note that the redshift for HD 68284 and HD 130551 are lower than the redshift for the other three stars. Although this could be accidental, it is a warning that there may be differential effects in the convective line broadening. Clearly, this problem should be further studied by applying recently constructed inhomogeneous 3D hydrodynamical model atmospheres (Asplund et al. 1999) in the analysis of the LiI line.


[TABLE]

Table 5. Radial velocities of the stars as derived from the accurate wavelengths of the Fe I lines (6703.567 and 6705.102 Å) measured by Nave et al. (1995). The wavelengths shifts given are obtained from the [FORMULA] fits of the synthetic spectra to the individual lines


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Online publication: July 16, 1999
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