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Astron. Astrophys. 354, 150-156 (2000)

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5. Model differences

It appears of obvious relevance to address the problem of the discrepancies between the models by G98 and C99, trying to understand them in terms of different descriptions of the input physics. We leave out differences in the actual treatment of the input physics (e.g. interpolation in and between opacity tables), which we think are of smaller influence and might contribute to a minor part of the variations in the clump luminosities as shown in Fig. 2.

Since the most important quantity determining the core helium burning luminosity is the core mass at the helium flash, we will concentrate on this parameter. Fig. 5 shows that the 1 [FORMULA] G98 models have core masses lower by [FORMULA] compared to C99 at the time of helium ignition at the RGB tip. Fig. 5 is made for solar composition but we checked that even for other metallicities and helium abundances the differences in the He core mass are very similar.

[FIGURE] Fig. 5. Comparison between the helium core mass at helium ignition from G98 and C99 models. Chemical composition as labeled.

We have identified the following differences in the input physics used in the two evolutionary programs ("Padua" for G98 and "FRANEC" for C99 models) under consideration (other aspects such as reaction rates, electron screening, mixing-length formalism, etc., are to great extent identical):

  1. Plasma neutrino emission : G98 use Munakata et al. (1985) for [FORMULA], while C99 use Haft et al. (1994);

  2. Electron conduction : G98 employ Hubbard & Lampe (1969), C99 Itoh et al. (1983);

  3. Radiative opacities : the '92 tables by OPAL (Rogers & Iglesias 1992) are used in G98, while the '96 ones (Iglesias & Rogers 1996) are those in C99;

  4. Equation of state : C99 use the new OPAL-EOS tables (Rogers et al. 1996) extended by that of Straniero (1988) in those regions where the OPAL-EOS does not exist (see Cassisi et al. 1998 for more details). G98 use their own analytic EOS, which takes into account partial ionization of hydrogen and helium, degeneracy and Coulomb-effects, and which has been described in Girardi et al. (1996).

The influence of some of these differences could be investigated quite easily because the two codes to some extent allow the selection of several sources of input physics. The tests were done for different cases of initial chemical composition and mass.

Neutrino emission. For the composition [FORMULA], [FORMULA] and an initial mass of [FORMULA] we find that [FORMULA] decreases from 0.482 to 0.476 (-0.006) [FORMULA], if we switch from the Haft et al. (1994) back to the older Munakata et al. (1985) neutrino emission rates (FRANEC code). This is the same result as Cassisi et al. (1998; models 8 and 7 in their Table 1) obtained for an [FORMULA] model with [FORMULA], [FORMULA] and also identical to what we find in the case of the Padua-code for the same model. In the latter case, we also verified that using the Munakata et al. (1985) emission down to [FORMULA] increases the He core mass by [FORMULA]. Therefore, the total budget of core mass reduction due to the G98 treatment of neutrino emission physics amounts to [FORMULA]. Recall that for the [FORMULA] model, the G98 value for [FORMULA] is [FORMULA] smaller than that of C99.

Opacities. We checked the effect of electron conduction opacity by switching in the FRANEC code from Hubbard & Lampe (1969) to Itoh et al. (1983) electron conduction opacities. The radiative opacities in these test cases are of a generation older than OPAL, but the differential effect of changing condution opacities can safely be assumed to be largely independent of the radiative opacities. The test model was a [FORMULA] star of [FORMULA], [FORMULA]. The use of the older conductive opacities (i.e. those used by G98) leads to a reduction of [FORMULA] in [FORMULA].

The influence of switching from the '92 to the '96 OPAL opacities was not tested, but according to our experience it should be minor compared to that of the electron conduction.

Equation of state. The EOS being a complicated part of both programs, we could not easily exchange one for the other. However, we could perform the following test: we took the pressure and temperature stratification of a C99-model on the RGB and applied the G99-EOS to it in order to obtain the density. We compared with the original C99 density stratification. The result is shown in Fig. 6 for a RGB model with Y=0.238, Z=0.004: while the core would have densities higher by about 1.5% in the Padua code, the envelope would be less dense by 1-2%. Both effects reflect the evolutionary changes along the RGB, such that a G98-model would appear to be more evolved than a C99 one. Therefore, also the difference in the EOS is expected to lead to a lower core mass at helium ignition for the G98 calculations.

[FIGURE] Fig. 6. Relative difference in density stratification between C99 and G98 EOS for an RGB model (see text).

From these three investigations we conclude that with the differences in neutrino emission rates and conductive opacities we can explain almost 60% of the discrepancy in core mass. This moves the G98 core masses already within the general spread of results. At least part of the remaining difference can be ascribed to the EOS.

Finally, we have compared results with three further codes, which use almost identical input physics as C99, in particular with regard to radiative opacities, neutrino emission and electron conduction. With regard to the Garching stellar evolution code (see, e.g., Schlattl & Weiss 1999) we find that [FORMULA] is [FORMULA] larger in the C99 models for a [FORMULA] star of [FORMULA] and two metallicities, [FORMULA]. This small difference can be understood as being a consequence of the fact that more (0.004) helium is dredged up in the C99 models.

Finally, evolutionary calculations by Pols et al. (1998) and Dominguez et al. (1999) with their respective codes (Pols et al. 1998; Straniero et al. 1997) reveal for various chemical compositions a very high degree of agreement with those of C99 as shown in Fig 7, including [FORMULA], which differs by less than 0.01 [FORMULA] between Dominguez and C99.

[FIGURE] Fig. 7. Upper panel: comparison between the 1 [FORMULA] model with solar compositions from Pols et al. (1998) and from C99; lower panel: comparison between the 1.5 [FORMULA] model with solar compositions from Dominguez et al. (1999) and from C99.

We therefore can conclude that more than half of the difference in the core mass at helium ignition between G98 and C99 can be removed by adopting the same neutrino emission rates and electron conduction opacities; using identical EOS would lead to further convergence, although we cannot quantify this point. At the same time, codes with identical physics do indeed result in very similar core masses which differ at a level of a few [FORMULA] only. Therefore, the differences between the G98 and C99 results concerning this quantity are well understood as a consequence of different physical inputs. Note that taking from the literature (see e.g. Sweigart & Gross 1978) [FORMULA]logL/[FORMULA] [FORMULA] for stars with degenerate progenitors, the difference in [FORMULA] explains the difference in luminosity among the various models in Fig. 2, with the exception of predictions by Bressan et al. (1993) that reveal the contribution of some other difference than the core mass only. As a conclusion, the different predicted luminosities we are dealing with appear as the natural results of evolutionary codes with different - but in both cases reasonable - input physics and thus as an example of the intrinsic unavoidable uncertainties in any current evolutionary scenario.

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© European Southern Observatory (ESO) 2000

Online publication: January 31, 2000