Despite the rather large uncertainties on the derived Li abundances, we have a number of interesting results.

Starting from the bottom of the AGB, we can conclude that the early AGB stars in NGC 1866 do show lithium, although the derived abundance is very low ( $\log N({\rm Li})\simeq 0$ ). This is consistent with the fact that for a large progenitor mass, we do not expect lithium to be destroyed in the evolutionary phases preceding AGB but only to be diluted, when the convective envelope sinks into the star. Figure 1 shows, however, that the residual abundance of our models (for a 4 $M_\odot$ ) is $\log N({\rm Li})\sim 2.0$ , i.e. much larger than the observed one. It has to be noticed that these models have a solar-type initial abundance ( $\log N({\rm Li})\simeq 3.3$ ), which by comparison with the early AGB value, implies a dilution factor $\sim$ 20. If the initial abundance were somewhat lower, say $\log N({\rm Li})\simeq 2.7$ , the diluted abundance would be accordingly scaled. That, however, cannot fully explain the discrepancy: it is highly plausible that an additional mixing mechanism providing further dilution is at work. The occurrence of such a mechanism is well established, at least for lower mass red giants, by the low ¹²C/ ¹³C and low lithium found on the red giant branch (e.g. Charbonnel 1995). We could also suggest that the lower abundances are due to a mass loss larger than that included in the Ventura et al. (2000) models. Anyway, the models can still be used for comparison, as the memory of the initial value is lost after the first lithium dilution phase.

In spite of this quantitative discrepancy, the presence of lithium in the early AGB stars proves that in this phase the star preserves some lithium, as expected from theory. The models do not predict any lithium production here, so we regard this result as a firm point: prior to the AGB, lithium is present but heavily diluted, and its abundance in the envelope sets an upper limit to its content during the following evolutionary phases, in absence of further production.

The spectrum of the cooler star N1866#3 shows the TiO bands and a relatively strong lithium line. However, this star is not yet as luminous as we would expect for lithium production by HBB. It has to be noticed that because of the somewhat lower quality of the spectra, the uncertainties on the abundance are rather large (see Table 2), and we cannot safely assume for this star a lithium content larger than that of the other, hotter, early AGB ones.

Looking at the most luminous stars N1866#1 and #2, we find a totally different situation: star #2 has an abundance definitely larger than that of the early AGB ones ( $\log N({\rm Li}) \lower.5ex\hbox{$\; \buildrel > \over \sim \;$ }1.5 \pm 0.5$ ), so that we conclude that we are witnessing lithium production. No lithium line is detected in the spectra of the other star. This yields an upper limit at $\log N({\rm Li})< -0.5$ ; on this basis we can certainly suggest that lithium in this star has been destroyed.

The amount of lithium manufactured in NGC 1866#2 is not particularly large, but because of the error size, we must wait for high dispersion spectra to set more stringent limits. The abundance is, however, in the range expected from the models (Fig. 1), also taking into account that it varies with the thermal pulse phase.

The absolute bolometric magnitude of this star is actually at the lower boundary of what we expect for the occurrence of HBB: ( $\mbox{$M_{\rm bol}$ }=-6$ ), and that might be an additional reason for the relatively low log N(Li) value. Figure 6 shows the Li vs. $M_{\rm bol}$ along the same 4 $M_\odot$ evolutionary track of Fig. 1. The points corresponding to the observed stars are also plotted. We see that indeed the star #2 in NGC 1866 is at the phase in which the residual lithium is completely destroyed, while the star #1 is in a phase in which it is manufactured by HBB. The agreement with the theoretical models is quite satisfactory. Things are however less clear for the cluster NGC 2031.

$\begin{figure} \par {\resizebox{8.8cm}{!}{\rotatebox{0}{\includegraphics{h3885f6.eps}}} }\end{figure}$

Figure 6: Lithium abundance versus $M_{\rm bol}$ for the target stars in NGC 1866 (squares) and NGC 2031 (triangles). The arrows represent upper limits. The evolutionary track of Fig. 1 is also shown.

6.2 NGC 2031

The stars in this cluster show a Li abundance behaviour similar to that in NGC 1866, but both the quality of some spectra and the fact that this cluster is much less populated than NGC 1866 make the interpretation less stringent.

First of all we have , for the stars in early AGB phase, only one reliable Li determination. This is for star (#5), that clearly shows a lithium line of similar strength to those in the corresponding stars of NGC 1866. The abundance analysis gives $\log~N{\rm (Li)}=0.2 \pm 0.2$ . Star (#6) turned out to have a spectral type later than that expected on the basis of its photometry, a fact that casts doubts on its cluster membership. For star #3, whose spectrum has no lithium line, we could only get an upper limit.

Concerning the later AGB objects, star #1 seems to be very similar to star #1 in NGC 1866: it shows no lithium (abundance <1) and it is at about the right luminosity to be in the first phase of lithium destruction due to HBB. The analysis of star #2, whose spectra have however a relatively lower S/Nratio, provides a high lithium abundance ( $\sim$ $2.5\pm0 .5$ ) but, according to the models, this star is not luminous enough to be in the HBB phase! Mould et al. (1993) suggested, on the base their photometry and the very red B-R (>3), that this star could be a very bright carbon star: in this case its large lithium abundance would imply a classification as J-subtype carbon star (Bouigue 1954). This possibility is ruled out, however, by the fact that its spectrum looks oxygen-rich.

6 Comparison with theoretical models

6.1 NGC 1866

6.2 NGC 2031

6.3 NGC 2214 and NGC 2107