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Astron. Astrophys. 357, 61-65 (2000)

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3. Modelling the evolution of [CO] in a SSP

The integrated CO index of a SSP with a given metallicity and age can be most conveniently determined from the integrated synthetic spectrum using the same methods adopted for measuring the index from the observational data. The integrated synthetic spectrum can be expressed as


where T=[FORMULA], g=[FORMULA] and [FORMULA] are the stellar temperature, gravity and monochromatic luminosity  1 along the isochrone. These parameters are explicitly tabulated by the Padua's models (Bertelli et al. 1994) while can be derived from the Geneva's evolutionary tracks as outlined in Leitherer et al. (1999).

The shape of the initial mass function, [FORMULA], has only a minor effect on the derived indices and, for all practical purposes, can be approximated by a standard Salpeter's IMF ([FORMULA]). The function [FORMULA] is the spectrum (normalized to unity at 2.2 µm) appropriate for a star of the given metallicity, temperature and gravity. This function is often evaluated using observations of field stars whose spectral types are converted into surface temperatures and luminosities using empirical calibrations. However, this method is limited to stars with quasi-solar metallicity and is particularly uncertain for supergiants and AGB stars.

A more objective estimate of [FORMULA] can be obtained from model stellar atmospheres which yield a synthetic spectrum with the desired resolution for a given set of metallicity (Z), effective temperature (T), surface gravity (g), microturbulent velocity ([FORMULA]) and carbon relative abundance ([C/Fe]). The last two quantities should be treated as free parameters because they cannot be unequivocally related to other physical parameters of the star. A detailed description of the procedure used to construct the synthetic spectra can be found in Origlia et al. (1993).

The time evolution of the CO index simply follows from Eq. (4) using theoretical isochrones at different ages. The results are plotted in Fig. 1 where we also evaluate the effect of using different stellar evolutionary models. These agree in predicting strong [CO] at early times, when the near IR emission is dominated by red supergiants (i.e. [FORMULA] Myr). Indeed, the comparison with observations is not yet satisfactory because models predict too warm red supergiants, especially at sub-solar metallicities (e.g. Oliva & Origlia 1998, Origlia et al. 1999).

[FIGURE] Fig. 1. Predicted evolution of the CO index for different metallicities and using different stellar evolutionary models. The left panels are based on the Geneva's tracks (Schaller et al. 1992, Charbonnel et al. 1993, Schaerer et al. 1993) while those in the right hand panels are based on the Padua's tracks (Bertelli et al. 1994). The solid lines represent our `reference models' which are computed adopting an empirical relationship between microturbulent velocity and bolometric luminosity, i.e. using Eq. (3). For comparison, the dotted curves show the results obtained by assuming a constant value of microturbulent velocity [FORMULA]=4.0 (see Sects. 2.2,3 for details). Note the very different behaviours beyond [FORMULA]100 Myr, i.e. when the near IR emission is dominated by stars evolving on the AGB. This simply reflects the different extent of the AGB in the two sets of models (see Fig. 2).

The most striking result in Fig. 1 is the very different behaviour in the AGB phase (i.e. [FORMULA] Myr) where the predicted CO index varies by large ([FORMULA]3) factors depending on the adopted stellar evolutionary tracks. In particular, the steady decay of [CO] in the curves based on the Geneva's models contrasts with the increase at the onset of the AGB predicted by the Padua's tracks. Interestingly, the latter predict strong CO at low metallicities where the other models give [CO][FORMULA]0.

To investigate the reason(s) for the very different behaviours in the AGB phase, it is instructive to compare the model isochrones which are displayed in Fig. 2 for two representative ages at solar metallicity. The Geneva's curves stop at relatively low luminosities and high temperatures ([FORMULA]3600 K), this implies that the stars dominating the IR emission are warm enough to dissociate CO, have quite large surface gravities and relatively low luminosity. Thus the CO bands are weak because the CO/C relative abundance, the column density of the photosphere and the microturbulent velocity are all relatively small (see Sect. 2.2). The AGB in the Padua's models, on the contrary, extends to very high luminosities and low temperatures. This implies low surface gravities and large microturbulent velocities, i.e. deep CO features.

[FIGURE] Fig. 2. HR diagram with theoretical isochrones from different stellar evolutionary models, all curves are for solar metallicities. The solid lines are based on the Geneva's tracks (Schaller et al. 1992, Charbonnel et al. 1993, Schaerer et al. 1993) which stop the computation at the onset of thermal pulses and, therefore, cuts the AGB at unrealistically high temperatures and low luminosities. The dotted curves are from Bertelli et al. (1994) which follow the evolution through the thermal pulse phase and hence produce more luminous and cooler AGB stars. However, the extent of the AGB is probably overestimated by these models which assume quite low mass loss rates along the AGB (see Sect. 3). The shaded region shows the loci of AGB stars in intermediate age Magellanic Cloud clusters (Aaronson & Mould 1982, Frogel et al. 1990).

The different extent of the AGB in Fig. 2 primarily follows from assumptions made by the models. The Geneva's tracks arbitrarily stop at the onset of thermal pulses, while the Padua's computations follow this phase up to the very end of the double shell burning using semi-analytical approximations. However, both approaches are unrealistic because the evolution along the AGB, which for sure extends well into the thermal pulses phase, is regulated and shortened by the strong mass-loss experienced by AGB stars, a parameter not included in the Bertelli et al. (1994) tracks. Moreover, one should keep in mind that the predicted stellar temperatures are also quite uncertain and, probably, too warm (Chieffi et al. 1995, Oliva & Origlia 1998). This effect is visible in Fig. 2 where one can notice that, within the part of the AGB covered by both models, the Padua's tracks are systematically warmer than those of Geneva. Indeed, recent developments of the theory of convective energy transfer indicate that all the temperatures of red stars in the Padua's tracks should be decreased by 200-300 K (Bressan, private communication).

Therefore, the "true" AGB is likely to be less extended but redder than predicted by the Padua's tracks and the two effects on the [CO] should compensate each others. In practice, the "true" CO index during the AGB phase is probably similar to that plotted in the right hand panels of Fig. 1. For the moment being, assuming [CO][FORMULA]constant with time is probably a fair approximation which, at least, does not lead to far reaching conclusions on the age of stellar systems.

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© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000