## 7. Dredge-up and the formation of white dwarfsThe dredge-up laws established in Sects. 5 and 6 allow an approximate study of the structural evolution of AGB stars without actually performing full AGB model calculations. This is the so called `synthetic' AGB calculation technique. Such a study is outside the scope of the present paper. However, a preliminary analysis of the synthetic evolution of a star with dredge-up is presented in Sects. 7.2 and 7.3 using simplified assumptions. But before doing that, the implications of the linear dependence of the dredge-up rate on the core mass is first discussed in Sect. 7.1. ## 7.1. Core mass evolution and the formation of white dwarfsThe analysis presented in Sect. 6 supports a linear relation between and . I assume that such a linear relation remains valid between and even when the feedback from the dredge-ups on the structural AGB evolution is taken into account. Such a linear relation between and
has an important consequence on the
core mass reached at the end of the AGB evolution. Eq. 10 predicts
that reaches unity at
. It is easy to see that the
feedback between and
leads to the evolution of
towards an asymptotic value
. Indeed, if
, then
and the core mass increases from
one pulse to the next. If , then
and the core mass ## 7.2. Structural evolutionAccording to Eq. 9, dredge-up begins to operate in our star around the 10th pulse, at . This is where we begin our synthetic evolution. We assume that the star has reached its asymptotic regime, which is a good enough assumption (see Fig. 6) for our purposes. The evolution of is governed, on the one hand, by its increase due to hydrogen burning during the interpulse. For a solar metallicity star, is given by (all quantities in this section are expressed in solar units and in years) where the luminosity of the H-burning shell is approximated by the surface luminosity (at the time of the maximum extension of the next pulse). The factor is introduced in order to account for the fact that the mean surface luminosity during the interpulse is actually lower than that at the the next pulse. From the standard model calculations, we find . After each pulse, on the other hand, decreases as a result of the 3DUP. The amplitude of this decrease is given by , where is estimated from Eq. 10. The resulting net increase in from one pulse to the next is then In model calculations without dredge-up, the change in luminosity from one pulse to the next in the asymptotic regime is given by Eq. 4, while the change in the core radius is given by Eq. 8. In the presence of dredge-up, those relations must be modified. For , a dependence on is suggested from Eq. 14, and we adopt For , we make the assumption that
the Finally, the interpulse duration is assumed not to be altered by the dredge-up episodes, and is given by Eq. 7. The predictions of our synthetic calculations are displayed in
Figs. 11 (time evolution) and 12
(-
The assumptions underlying those synthetic calculations certainly are too simplistic. For example, the feedbacks of dredge-up on , or have been neglected, and relations 14 and 15 need to be confirmed by model calculations following the dredge-ups all along the AGB evolution. Yet, the main conclusions, such as the existence of a limiting towards which the core mass evolves asymptotically, should qualitatively be correct.
## 7.3. Carbon star formationLet us now follow through our synthetic calculations the carbon abundance predicted at the surface of a star. The amount of carbon dredged-up to the surface, , is given by where is the mass fraction left over by the pulse in the intershell layers. From the values reported in Table 1, a mean value of can be assumed. The initial and mass fractions in the envelope are taken from our standard star at the beginning of the TP-AGB phase, and are equal to and , respectively. The resulting evolution of the surface C/O number ratio is shown in Fig. 13. It is seen that a star satisfying Eq. 10 becomes a C star after about twenty dredge-up episodes, at . Furthermore, it is known that the luminosity decreases by a factor of almost two during about 20% of the interpulse phase. This means that the star displayed in Fig. 13 could be observed as a C-star at luminosities as low as . I should stress that the effects of mass loss have been neglected in those synthetic calculations, being outside the scope of this article. Mass loss would decrease the dilution factor of into the envelope, but would probably also decrease the dredge-up efficiency. Those effects should be considered in a future, more detailed, study of C star formation. © European Southern Observatory (ESO) 1999 Online publication: March 18, 1999 |