The sodium overabundances [Na/Fe] derived from our spectra are summarized in Table 4 for our seven stars. They are small in NGC 2360, the oldest of the two clusters, with [Na/Fe]= 0.07 - 0.13 dex, and amount to 0.15 - 0.22 dex in NGC 2447.
The error in the Na overabundance [Na/Fe] due to the uncertainties on , and are not large because Na i and Fe i have similar behaviors with respect to those parameters (e.g. Table 9 of Luck 1994) such that their effect on the [Na/Fe] ratio cancels out. The errors on the [Na/Fe] ratio due to the continuum placement, on the one hand, and to the photon noise, on the other hand, are estimated in the following way. For each star, the equivalent widths of the two Na i lines are measured five times, each time after having renormalized the spectrum of the relevant region. The standard deviations of the equivalent widths are then computed from these data and propagated into standard deviations of abundances using the MOOG code. The resulting dispersion on the Na abundance amounts to 0.02-0.05 dex, depending on the S/N ratio. A synthetic spectrum of a typical red giant (with K, , [M/H]=0.0 but [Na/Fe]=+0.15) is then produced, and a Gaussian noise added to it for four representative S/N values, five spectra being produced independently with the same S/N ratio. The equivalent widths of both Na lines are measured (by a Gaussian fit) for these 20 spectra and their standard deviation computed for each S/N ratio. The deviations are found to lie between 2 and 3.3 percent for S/N ratios between 230 and 90. They are translated into abundance errors (after dividing them by since there are two lines) and added quadratically to the errors due to the continuum position. Finally, we assume that the error on the continuum position for Fe i is similar to that for Na i and, admitting that they are independent of each other, add them quadratically to the total error on the Na abundance 2 (in fact there is probably a correlation between the continuum placement for the Fe lines and that for the Na lines, but neglecting it only results in an overestimate of the error, so that we stay on the safe side). The resulting estimated errors amount to about 0.03-0.07 dex, the lowest ones pertaining to NGC 2447.
Among the systematic errors which may affect the Na and Fe abundances, some are negligible for the [Na/Fe] abundance ratio because of their mutual cancelation, as mentioned above. This is not the case, however, for the oscillator strengths adopted in the synthetic spectra. If the values of the two Na i lines are slightly in error while those of Fe i are statistically correct, for instance, then the absolute values of [Na/Fe] would be wrong, though their relative values (i.e. their differences) would remain valid.
Finally, let us consider the systematic errors due to the assumption of local thermodynamic equilibrium (LTE) in the MOOG program. Non-LTE (NLTE) calculations performed by Gratton et al. (1999) in atmospheric conditions relevant to red giants show that the NLTE effect on the Na i line (one of the two lines used in this paper) strongly depends on surface gravity and slightly on effective temperature. This is illustrated in Fig. 4, which reveals that the NLTE correction should be small in the range characterizing our stars. Indeed, the average NLTE corrections are found to amount to 0.006-0.035 dex for the stars in NGC 2360 and of 0.032-0.048 dex for NGC 2447 using Table 11 of Gratton et al. (quadratically interpolated in and linearly in , and assuming similar corrections for both and lines of Na i). A similar calculation for iron 3 leads to NLTE corrections for Fe of 0.015-0.021 dex for NGC 2360 and 0.027-0.028 dex for NGC 2447. The resulting NLTE effects on [Na/Fe] thus ranges between -0.013 and 0.014 dex for NGC 2360 and between 0.005 and 0.020 for NGC 2447. Of course, these values are only approximate since they are derived from only one iron line assumed to be representative of the 55 lines observed for that element, but they do suggest that the errors brought by our LTE approximation are much smaller than the above mentioned random error bars.
As a conclusion, the main abundance errors, besides the possible systematic errors due to the oscillator strengths 4, are due to the equivalent width and continuum measurements, which amount up to 0.07 dex. The error bars on the masses of each star, on the other hand, are estimated by considering a 0.05 dex error on the age of the clusters. The surface sodium overabundances as a function of stellar mass are shown in Fig. 5 by rectangles taking into account the above mentioned uncertainties.
Sodium production during H-burning results from the transformation of into by proton capture. This reaction occurs very efficiently at the temperatures characterizing the core of MS stars (see Appendix A of Mowlavi 1999). First dredge-up then mixes some of the synthesized Na from the deep layers to the surface. This scenario is confirmed by the observation of sodium overabundances at the surface of many giants and supergiants (e.g. Luck 1994; Boyarchuk et al. 1996; Takeda & Takada-Hidai 1994).
The surface Na overabundance predicted by stellar model calculations (without core overshooting) as a function of stellar mass is shown in Fig. 5 by filled circles connected with solid line. The models have a metallicity 0.05 dex above solar (which is the metallicity of NGC 2447) and are followed from the pre-MS up to the completion of the 1DUP. The stellar evolution code is the same as in Mowlavi (1999), except that the NACRE reaction rates (Arnould et al. 1999) are used to follow the nucleosynthesis and that the formalism of Canuto et al. (1996) is used to describe the energy transport in convective zones. The surface sodium abundance is seen to be very sensitive to stellar mass in the range. It increases from no Na enhancement below to an overabundance of 0.15 dex at , and keeps this value for stellar masses up to before increasing again with stellar mass (see, e.g., Mowlavi 1998).
The sensitivity of those predictions to core overshooting, convection prescription and stellar metallicity is explored by computing extra models from the pre-MS to the 1DUP. None of those parameters, however, turns out to have a significant impact on the surface Na abundance. Models with core overshooting (with an extra-mixing extent of 0.20 times the pressure scale at the core boundary) predict a 0.02 dex enhancement (open circles connected with solid line in Fig. 5) compared to predictions without core overshooting. Increasing the metallicity by 0.17 dex does not change the surface [Na/Fe] prediction after 1DUP by more than 0.01 dex. And using the mixing length theory (with a mixing length of 1.5 times the pressure scale height) instead of the Canuto, Goldmann & Mazzitelli formalism does not change the surface abundance predictions within 0.001 dex.
Let us now explore the uncertainties linked to nuclear reaction rates. Both the and p-capture reactions on are still subject to large uncertainties (Arnould et al. 1999). In order to assess their impact on our surface Na abundance predictions, several models are recomputed from the pre-MS up to the 1DUP with the upper/lower limits for the rates provided by the NACRE compilation (cf. Arnould et al.), as appropriate to minimize/maximize production. The results in the `minimal' and `maximal' production cases are shown in filled circles connected with dotted lines in Fig. 5. They reveal a variation in the surface Na abundance predictions of up to 0.08 dex relative to the `nominal' case where the recommended NACRE rates are used. Nuclear reaction rate uncertainties thus dominate the uncertainties associated with stellar metallicity and convection prescriptions for sodium predictions in red giants.
Fig. 5 shows a very good agreement between our Na abundance predictions in model stars and those observed in NGC 2360. The nominal predictions of the model star, on the other hand, seem a little too low compared to the abundances observed in NGC 2447. The predictions in the maximal case of production would fit the highest Na abundance measured among the three stars observed in NGC 2447. Those predictions, however, would not be compatible with the Na abundances measured in the red giants of NGC 2360. Fig. 5 thus suggests that the nominal Ne-Na reaction rates should not be too much altered, if at all. The solution to the discrepancy between our observed Na abundance observations in NGC2447 and predictions should be found in other(s) mechanism(s) such as, possibly, meridional mixing induced by stellar rotation. Further theoretical and observational investigations should be performed before being able to draw a firm conclusion.
Finally, let us mention that the positive sodium abundance - stellar mass correlation translates, at a given effective temperature into a sodium abundance - surface gravity anti-correlation (or sodium abundance - luminosity correlation). This is well known in the literature, and shown in Fig. 6 where our data are displayed together with those of Luck (1994) and Boyarchuck et al. (1996). The dependence on effective temperature is small (Luck 1994). The computation of our 2.2 to models with core overshooting is carried on up to the clump in the core helium burning phase. The clump is defined as the point where the stellar luminosity (gravity) reaches its minimum (maximum) value after core helium ignition. The sodium abundance (which is not altered between the 1DUP and the clump) predictions for those clump models are shown by a solid line in Fig. 6.
© European Southern Observatory (ESO) 2000
Online publication: August 17, 2000