Core Mass at the 1 st TP

shooting behave differently in this respect: when convective overshooting is not included, a He-flash tends to occur for stars of higher mass.

Figure 6.7 presents the highest helium-burning luminosity reached on the RGB during central helium ignition for all our models (here only the ones below 2.6M are shown)

with solar-scaled metallicities. The maximum extent of the shell of highest energy release is also plotted in the bottom panel. We see that the trend is similar independent of metallic- ity and that no models above 2Mgo through the He-flash. While the severity of the flash

increases - higher helium-burning luminosities and core temperatures because of stronger degeneracy - for lower Z values at masses from 1Mto 1.6M, higher metallicity models

seem to continue having moderately high helium luminosities for higher initial mass. In- deed for 2Mstars, the strongest energy release happens at the center for all values of Z,

but the two higher metallicity models reach 105and 107L, which is much higher than the

two lower Z ones. Actually for all stars with initial masses higher than 1.8M, helium is

ignited at the center. In our models, which include overshooting, the boundary between low- and intermediate-mass stars is thus between 1.8Mand 2M. Only the two higher

metallicity 2Mmodels experience “mild” (105 and 107L) flashes for a short time. In

comparison, models of K03 give a maximum initial mass for a He-flash to occur at about 2.25M. This value is higher than ours but she does not include overshooting. Models with

core overshooting obtain lower values: Bertelli et al. (1986) get about 1.6Mand Herwig

(2005a) sets according to his models the low-mass star boundary at 1.8M. This agrees

well with our value (for further discussion on the metallicity dependent low/intermediate- mass star boundary in section 7.3).

6.4. Core Mass at the 1

TP

After helium core exhaustion, the star is said to be on the E-AGB and starts to climb once more in the giant region of the HR diagram. As it reaches the end of the E-AGB, before the onset of the instability flashes, one property characterizing the structure of the star, that will play an important role in the subsequent evolution, is the mass of the core. The interest on the size of the core comes from the fact that at the end of the TP-AGB the remaining bare core will become the future WD.

During the TP-AGB, on the one hand the piled-up nuclear burning ashes increase the size of the central inert core, but on the other the mass lost by the envelope reduces the available fuel and ultimately stops the growth of the central region. It is thus the compe- tition between these two phenomena that mainly regulate the final outcome and the end of the TP-AGB. As nuclear burning is more or less constant, it is the rate at which the envelope material is lost that becomes important. A weak mass-loss rate will give the core time to grow and a strong one should result in a post-AGB object of smaller size. Param- eters indirectly influencing the mass-loss rate are, temperature, luminosity, composition, mass of the star, TDU, HBB, opacities etc. But some of these quantities can also directly influence the size of the core. For example, as part of the material is dredged-up at each TDU episode and mixed into the envelope, if the TDU is efficient enough - compared to

the growth of the central part after the last deep penetration of the envelope - the size of the core can be reduced.

When studying the TP-AGB evolution the question is therefore to see if, with respect to its mass at the beginning of the TP-cycle, the size of the core will actually grow, remain the same or decrease. For this reason it is necessary to know the size of the core we start offwith at the end of the E-AGB phase. Additionally, the time spent on the TP-AGB also depends on the initial remaining envelope mass that needs to be ejected between the 1stTP and the beginning of the post-AGB evolution when the star moves to higher temperatures. The total mass of the star and the core mass at the 1st TP are given in the last two columns of Tables 6.1 - 6.3. Low-mass stars have already lost a considerable amount of their envelope (up to 32% of the total mass for the 1M, Z=0.04/ α-enhanced model)

during the previous evolution - mainly on the RGB. But the more massive intermediate- mass models remain almost as massive as on the MS - they lose on average only a few percent of their total mass - because they spend less time as giants.

Figure 6.8.:Mass (in M) of the core at the 1stTP with respect to initial ZAMS mass (in

M).

This is the reason why, as presented in Chapter 3, we have chosen a different Reimers mass-loss parameterηRfor the two cases. All our models up to 1.6MhaveηR=0.4 and the ones starting at 1.8Mare calculated withηR=1.0. This is similar to the prescription

adopted by Blöcker (1995b) for calculations leading to TP-AGB models. He uses 0.5 for the 1 Mcase and 1.0 for 3 Mon the RGB. K03 uses only a 0.4 Reimers parameter for

6.4 Core Mass at the 1stTP

all stars. On the other hand VW93 does not include mass-loss before the AGB forM_ZAMS >1M. Finally, Herwig et al. (2000) also use an almost identical prescription to ours with

a 0.5 value forM₅1.7Mand 1.0 for all higher mass. Therefore, our choice corresponds

well to what has been previously used, especially in the most recent calculations, and for the pre-AGB evolution was mainly made for compatibility with the early evolution of existing TP-AGB grids.

In Figure 6.8 we present the core mass at the 1stTP with respect to the initial ZAMS mass of the models. The colored curves represent the five different metallicities. Solid lines are for the solar-scaled compositions and the dashed ones for theα-enhanced mixtures.

For masses between 1M and 2M, core sizes are very similar for all compositions.

After a drop in core mass for stars withM_ZAMS=2M, a metallicity dependence is clearly

visible: lower metallicity models render higher mass cores. But the increasing behavior is the same for all Z. Small differences exist between the solar-scaled and theα-enhanced models, the later ones having cores slightly larger for the most part. The largest spread observed between models of different Z values is for the 3Mand the 6Mcases. In order

of increasing metallicity for the five solar type mixtures, the sizes (in M) are 0.780, 0.744,

0.672, 0.596 and 0.596 in the 3Mcase, and 1.128, 1.060, 0.991, 0.928 and 0.893 in the

6M. These correspond to spreads respectively of 24 and 21%.

We have also compared our core masses with existing sets of models in the left side of Figure 6.4. From top to bottom, the red dotted curves represents results given by K03 for the 0.02, 0.008 and 0.004 metallicities. For all three cases our values match the trend and the values between 2.6Mand 5Mvery well. For the low-mass stars, agreement is

increasingly better as we decrease the initial mass. The biggest discrepancies for all three metallicities arise for the 2Mmodel. As an example, for the solar mixture case our result

is 0.478 compared to 0.553 from K03, thus a value lower by 14%. The 6Mtracks for

LMC and SMC metallicities also do not agree very well with the K03 values.

For the solar composition case, the models of Miller Bertolami (2006) (blue diamonds in the top right panel of Figure 6.4), who use a similar code, include the same overshooting prescriptions and RGB mass-loss as we do, agree even better then the K03 ones. In ad- dition, the Miller Bertolami (2006) results also match very well the core of our low-mass stars (1Mand 2.2M).

We also plot the core masses for some models calculated with no overshooting and no mass-loss until the 1st TP. These are the black triangles in Figure 6.4. The change is significant as the mass of the core drops in all three cases by 12, 6.5 and 5% for the 3, 5 and 6Mcases. Our masses when overshooting is included, agree therefore better with the

K03 ones even if this treatment for the mixing at the convective boundaries is ignored in her models.

When our tracks reach the 1stTP on the AGB, their core masses and lifetimes (due to the dominance of the MS) agree well with previous works that use different evolution codes. This is true even for calculations that do not include overshooting from the ZAMS up to

the TP-AGB. Our surface carbon isotopic12C/13C ratios, modified due to the FDU of the convective envelope, are also coherent with models that do not include overshooting.

No convergence problems were encountered during the entire evolution until the 1stTP was reached. The average timestep used during these pre TP-AGB phases is around 10 000 yrs and the number of models needed to reach the TP-AGB range anywhere from 3000 to 25 000 (depending on the initial stellar masses). The number of iterations for a model to converge is typically 3. The timestep and the number of iterations can vary during the different evolutionary phases. The best example is the He-flash, where it may take up to a few hundred iterations for models to reach convergence and the timesteps can be as low as 10−4years; this however occurs without any numerical problems.