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

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4. Models without extra-mixing

4.1. Failure to obtain dredge-up

A set of standard 3 [FORMULA] models of solar metallicity is computed with [FORMULA] from the pre-main sequence phase up to the 32nd pulse on the TP-AGB phase. The maximum depth reached by the convective envelope after each pulse is shown in Fig. 2. The depth is expressed in terms of the mass separating the lower convective envelope boundary to either the H-He discontinuity (lower solid line) or the He-C one (upper solid lines). It is seen that the envelope does not penetrate into the He-rich layers (and a fortiori into the C-rich layers) during the first thirty two pulses.

[FIGURE] Fig. 2. Depth reached, as a function of the core mass, by the envelope of [FORMULA], Z=0.02 models (where Z is the metallicity) during the first 20 to 30 thermal pulses (identified by the filled circles and squares on each curve). The two lower curves, labeled [FORMULA], indicate the mass, in solar mass units, separating the lower boundary of the envelope to the location of the H-He discontinuity. The two upper curves give a similar information, but to the location of the He-C discontinuity. Solid and dashed lines refer to models calculated with [FORMULA] and 2.2, respectively.

The non-occurrence of the third dredge-up in those models is easily understood as resulting from the discontinuity in the H abundance when the convective envelope reaches the H-He discontinuity (case b described in Sect. 2.2). In those regions, the opacity is mainly determined by the Thompson opacity [FORMULA], where X is the hydrogen mass fraction. It presents a jump across the composition discontinuity, with the H-rich layers being more opaque to radiation than the He-rich layers (Fig. 3b). This discontinuity in [FORMULA] translates into a similar discontinuity in [FORMULA] (Fig. 3a), which prevents further penetration of the envelope into the H-depleted regions when no extra-mixing is allowed in the models .

[FIGURE] Fig. 3. a  Radiative gradient profile at several times, as labeled (in yr) on the curves, after the 19th pulse of the standard [FORMULA] case. The time origin is arbitrary but identical to that of Fig. 7. The dashed dotted line is the adiabatic gradient taken at the time corresponding to the solid line; b  Same as a , but for the opacity profile. The extents of the H-, He- and C-rich regions are shown at the bottom of b , separated by vertical lines

It is instructive to recall that the structural readjustments characterizing the afterpulse phases result from the evacuation of a gravothermal energy (defined by [FORMULA], where P, [FORMULA] and e are the pressure, density and internal energy, respectively) which develops at the bottom of the former pulse (Paczynski 1977). This translates into a positive gravothermal energy wave propagating outwards and reaching the H-rich layers in about 250 yr during the 19th afterpulse in our [FORMULA] models (see Fig. 4). The luminosity at the bottom of the envelope, [FORMULA], increases concomitantly with time until the gravothermal energy has been evacuated (Fig. 4b). As a result, [FORMULA] increases too, and the convective envelope penetrates inwards until it reaches the H-He discontinuity. A further deepening of the envelope could be possible if the increase in [FORMULA] overcomes the discontinuity in [FORMULA]. In the standard [FORMULA] models, however, this does not happen during the first thirty two pulses calculated so far.

[FIGURE] Fig. 4. a  Gravothermal energy profiles at 5 different times, as labeled (in yr) on the curves, after the 19th pulse of the standard [FORMULA], solar metallicity star. The time origin is taken at maximum pulse extension. The He-rich region is identified by the thick vertical line separating the H- and C-rich regions; b  Same as a , but for the luminosity profiles. The thick dots on the curves indicate the location of the lower convective envelope boundary

Effect of mixing length parameter. The mixing length parameter is known to be very influential on the depth of convective envelopes (Wood 1981). In order to analyze its role in obtaining 3DUP in AGB models without extra-mixing, the standard [FORMULA] star is recalculated from the first to the 23rd pulse with [FORMULA] (i.e. similar to the value used by Straniero et al. 1997). The results are shown in dotted lines in Fig. 2. Indeed the convective envelope reaches deeper layers with [FORMULA] than with 1.5. But they are not more successful in penetrating the He-rich layers. This contrasts with the results of Straniero et al. who claim to obtain dredge-up in their models without any extra-mixing.

Effect of stellar mass. Finally, a last set of standard models is calculated for a [FORMULA] star of solar metallicity up to its 22nd pulse. It is indeed known that 3DUP should also be favored in more massive AGB stars (Wood 1981). The results of our model calculations are shown in Fig. 5. Again, no dredge-up is found in those models calculated without any extra-mixing, for reasons similar to those put forward for the [FORMULA] standard models.

[FIGURE] Fig. 5. Same as Fig. 2, but for [FORMULA], Z=0.02 models with [FORMULA]

Effect of numerical accuracy. The analysis presented in this section shows that models using the local Schwarzschild criterion without extra-mixing do not lead to the occurrence of third dredge-up because of the [FORMULA] discontinuity at the core's edge. That conclusion should thus not be sensitive to the numerical accuracy of the models. Indeed, test calculations performed on the 19th afterpulse without extra-mixing, but with increased or decreased accuracies on both the time-step and mesh resolution lead to results identical to those presented in this section.

4.2. Useful relations

Some relations characterizing the evolution of the standard [FORMULA] models are presented in this section. These will be useful in Sect. 6.

The first relation describes the evolution of the surface luminosity L with core mass growth. The values of those quantities before the onset of each pulse of our standard [FORMULA] models are displayed as filled circles in Fig. 6a. It is well known that AGB models not experiencing dredge-up reach an asymptotic regime characterized by a linear relation between L and [FORMULA]. This is the famous [FORMULA] relation, which writes from our [FORMULA] models (dotted line in Fig. 6a)


L and [FORMULA] being given in solar units. A correction has to be applied to this relation in order to account for the lower luminosities of the first pulses. This correction is found to be well reproduced by an exponential function, and Eq. 4 becomes


This relation is shown in solid line in Fig. 6a, and is seen to fit very well model predictions.

[FIGURE] Fig. 6. a  Stellar luminosities at maximum pulse extensions, b  interpulse pulse duration between two maximum pulse extensions and c  location of the lower envelope boundary at its first minimum after each pulse in the standard [FORMULA] model calculations (filled circles in each panel). The dashed lines show the linear relations fitting the data in each panel (Eqs. 4, 6 and 8). The solid lines in panels a and b show the [FORMULA] (Eq. 5) and [FORMULA] (Eq. 7) relations, respectively, when the deviation from linearity of the first pulses is taken into account.

The second useful relation expresses the interpulse duration [FORMULA] as a function of core mass growth. From our standard [FORMULA] model calculations, shown in filled circles in Fig. 6b, we find an asymptotic [FORMULA] relation given by (dotted line in Fig. 6b)


[FORMULA] being expressed in yr. Again, a correction has to be applied during the first pulses. The resulting relation writes (solid line in Fig. 6b)


Finally, the location [FORMULA] of the lower envelope boundary at its first minimum after the pulse is displayed in filled circles in Fig. 6c as a function of [FORMULA]. A linear relation fits the data in the asymptotic regime (dotted line in Fig. 6c), which writes in solar units


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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999