4. Models without extra-mixing
4.1. Failure to obtain dredge-up
A set of standard 3 models of solar metallicity is computed with 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.
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 , 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 translates into a similar discontinuity in (Fig. 3a), which prevents further penetration of the envelope into the H-depleted regions when no extra-mixing is allowed in the models .
It is instructive to recall that the structural readjustments characterizing the afterpulse phases result from the evacuation of a gravothermal energy (defined by , where P, 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 models (see Fig. 4). The luminosity at the bottom of the envelope, , increases concomitantly with time until the gravothermal energy has been evacuated (Fig. 4b). As a result, 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 overcomes the discontinuity in . In the standard models, however, this does not happen during the first thirty two pulses calculated so far.
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 star is recalculated from the first to the 23rd pulse with (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 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 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 standard models.
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 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 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 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 . This is the famous relation, which writes from our models (dotted line in Fig. 6a)
L and 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.
The second useful relation expresses the interpulse duration as a function of core mass growth. From our standard model calculations, shown in filled circles in Fig. 6b, we find an asymptotic relation given by (dotted line in Fig. 6b)
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 of the lower envelope boundary at its first minimum after the pulse is displayed in filled circles in Fig. 6c as a function of . A linear relation fits the data in the asymptotic regime (dotted line in Fig. 6c), which writes in solar units
© European Southern Observatory (ESO) 1999
Online publication: March 18, 1999