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Astron. Astrophys. 344, 617-631 (1999)

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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 [FORMULA] 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 [FORMULA] and [FORMULA]. I assume that such a linear relation remains valid between [FORMULA] and [FORMULA] even when the feedback from the dredge-ups on the structural AGB evolution is taken into account.

Such a linear relation between [FORMULA] and [FORMULA] has an important consequence on the core mass reached at the end of the AGB evolution. Eq. 10 predicts that [FORMULA] reaches unity at [FORMULA]. It is easy to see that the feedback between [FORMULA] and [FORMULA] leads to the evolution of [FORMULA] towards an asymptotic value [FORMULA]. Indeed, if [FORMULA], then [FORMULA] and the core mass increases from one pulse to the next. If [FORMULA], then [FORMULA] and the core mass decreases from one pulse to the next. At [FORMULA], 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 [FORMULA] remains constant from one pulse to the next. We thus expect [FORMULA] to level off at unity, and [FORMULA] to reach [FORMULA]. This puts an upper limit to the mass of white dwarf remnants (which is [FORMULA] for our [FORMULA] 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 [FORMULA] star around the 10th pulse, at [FORMULA]. 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 [FORMULA] is governed, on the one hand, by its increase [FORMULA] due to hydrogen burning during the interpulse. For a solar metallicity star, [FORMULA] is given by (all quantities in this section are expressed in solar units and in years)

[EQUATION]

where the luminosity [FORMULA] of the H-burning shell is approximated by the surface luminosity (at the time of the maximum extension of the next pulse). The factor [FORMULA] 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 [FORMULA]. After each pulse, on the other hand, [FORMULA] decreases as a result of the 3DUP. The amplitude of this decrease is given by [FORMULA], where [FORMULA] is estimated from Eq. 10. The resulting net increase in [FORMULA] from one pulse to the next is then

[EQUATION]

In model calculations without dredge-up, the change in luminosity [FORMULA] from one pulse to the next in the asymptotic regime is given by Eq. 4, while the change in the core radius [FORMULA] is given by Eq. 8. In the presence of dredge-up, those relations must be modified. For [FORMULA], a dependence on [FORMULA] is suggested from Eq. 14, and we adopt

[EQUATION]

For [FORMULA], we make the assumption that the time evolution of [FORMULA] remains unaffected by the dredge-ups. Eqs. 6 and 8 then determine [FORMULA] 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 ([FORMULA]-L evolution). As expected from the discussion in Sect. 7.1, [FORMULA] increases towards unity and the core mass towards the asymptotic value of [FORMULA]. The luminosity, however, increases continuously. This results from the fact that L depends not only on [FORMULA] but also on [FORMULA] according to Eq. 14. Had we used a linear [FORMULA] relation, then L would have evolved towards an asymptotic value (of about [FORMULA] for our [FORMULA] star) in a similar way as does [FORMULA]. The non-linearity of L with [FORMULA] is made clear in Fig. 12.

[FIGURE] Fig. 11a-c. Predictions from synthetic calculations for a [FORMULA] AGB star with (filled circles) and without (filled triangles) dredge-up. a  Dredge-up efficiency, b  core mass, and c surface luminosity as a function of time. The first displayed pulse (at time [FORMULA]) corresponds to the 10th pulse of the standard [FORMULA] models. The horizontal dashed line in b shows the asymptotic core mass predicted for models experiencing dredge-up.

[FIGURE] Fig. 12. Luminosity as a function of core mass predicted by the synthetic calculations for a [FORMULA] star when dredge-up is (filled circles) or is not (filled triangles) taken into account.

The assumptions underlying those synthetic calculations certainly are too simplistic. For example, the feedbacks of dredge-up on [FORMULA], [FORMULA] or [FORMULA] 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 [FORMULA] 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 [FORMULA] (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 [FORMULA]. As a result, mass loss is expected to be higher in models with dredge-up than in those without, at a given [FORMULA]. In other words, mass loss will eject the AGB's envelope at a much lower value of [FORMULA] in the presence of dredge-up. For example, let us suppose that a superwind suddenly ejects all the envelope of the [FORMULA] star at [FORMULA]. Fig. 13 then predicts that the mass of the white dwarf 's remnant would be [FORMULA] in the presence of dredge-ups. This is much lower than the predicted white dwarf 's mass of [FORMULA] predicted in the absence of dredge-up. Mass loss combined with dredge-ups thus further helps to constrain the mass of white dwarf remnants.

[FIGURE] Fig. 13. Same as Fig. 12, but for the surface C/O ratio as a function of surface luminosity at maximum pulse extension (solid line) or at the luminosity dip during the interpulse phase (taken here to be half of the luminosity at maximum pulse extension, dashed line). The dotted line divides the regions where C stars (C/O[FORMULA]1) and M stars (C/O[FORMULA]1) are. The hatched area indicates the range of luminosity predicted for given C/O ratios.

7.3. Carbon star formation

Let us now follow through our synthetic calculations the carbon abundance predicted at the surface of a [FORMULA] star. The amount of carbon dredged-up to the surface, [FORMULA], is given by

[EQUATION]

where [FORMULA] is the [FORMULA] mass fraction left over by the pulse in the intershell layers. From the values reported in Table 1, a mean value of [FORMULA] can be assumed. The initial [FORMULA] and [FORMULA] mass fractions in the envelope are taken from our standard [FORMULA] star at the beginning of the TP-AGB phase, and are equal to [FORMULA] and [FORMULA], respectively. The resulting evolution of the surface C/O number ratio is shown in Fig. 13. It is seen that a [FORMULA] star satisfying Eq. 10 becomes a C star after about twenty dredge-up episodes, at [FORMULA]. 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 [FORMULA].

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 [FORMULA] 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.

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999
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