7. Dredge-up and the formation of white dwarfs
The 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 dwarfs
The 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 decreases from one pulse to the next. At , the amount of material dredged-up from the core during the 3DUP equals that added to the core by H-burning during the interpulse phase, and remains constant from one pulse to the next. We thus expect to level off at unity, and to reach . This puts an upper limit to the mass of white dwarf remnants (which is for our star), and helps to constrain the predicted initial-final mass relation for white dwarfs.
7.2. Structural evolution
According 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 time evolution of remains unaffected by the dredge-ups. Eqs. 6 and 8 then determine as a function of time t.
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 (-L evolution). As expected from the discussion in Sect. 7.1, increases towards unity and the core mass towards the asymptotic value of . The luminosity, however, increases continuously. This results from the fact that L depends not only on but also on according to Eq. 14. Had we used a linear relation, then L would have evolved towards an asymptotic value (of about for our star) in a similar way as does . The non-linearity of L with is made clear in Fig. 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.
Effects of mass loss. A last word about the effects of mass-loss. The amplitude of mass loss depends mainly on the surface radius R and effective temperature (and thus on L). Those are expected to be modified by the dredge-ups (Fig. 7). However, the time evolution of L (Fig. 11c) reveals that they are not as much affected as is . As a result, mass loss is expected to be higher in models with dredge-up than in those without, at a given . In other words, mass loss will eject the AGB's envelope at a much lower value of in the presence of dredge-up. For example, let us suppose that a superwind suddenly ejects all the envelope of the star at . Fig. 13 then predicts that the mass of the white dwarf 's remnant would be in the presence of dredge-ups. This is much lower than the predicted white dwarf 's mass of predicted in the absence of dredge-up. Mass loss combined with dredge-ups thus further helps to constrain the mass of white dwarf remnants.
7.3. Carbon star formation
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