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

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5. Dredge-up in models with extra-mixing

5.1. Model results

The structural evolution of the 19th afterpulse of the standard [FORMULA] star is recalculated in a set of models using the overshooting prescription described in Sect. 3.2, with [FORMULA]. The structural evolution of the intershell layers of these models is shown in long dashed line in Fig. 7a.

[FIGURE] Fig. 7a-e. Structural evolution of several quantities during and after the 19th pulse of the [FORMULA] star in the standard case (solid line) and in models with envelope overshooting (dotted line: [FORMULA]; long dashed line: [FORMULA]; short dashed line: [FORMULA]). a  Mass location of the convective boundaries and of the layer of maximum nuclear energy production (dashed-dotted lines) in the H and He burning shells, as labeled in the graph. Hatched regions correspond to convective zones. The extent of the overshooting regions is less than [FORMULA] and cannot be distinguished in the figure; b  surface [FORMULA] mass fraction; c  stellar luminosity; d  stellar radius; e  location of the lower boundary of the envelope. The time origin is taken at maximum pulse extension.

Clearly, the use of extra-mixing leads to the penetration of the convective envelope into the H-depleted regions, as expected from the discussion in Sect. 2.2. The role of overshooting in the 3DUP was already addressed by Iben (1976) and Paczynski (1977). The latter author, in particular, showed the essential role of extra-mixing in obtaining 3DUP. The envelope is seen in our models to deepen at a maximum dredge-up rate of [FORMULA]/yr. The dredge-up lasts for about 120 yr, which corresponds to the time necessary for the gravothermal energy to evacuate from the bottom of the formal pulse to the stellar surface (see Fig. 4). [FORMULA] of C-rich material is mixed into the envelope, which leads to a 16% increase in the surface carbon mass fraction.

It is instructive to stress that the extent of the overshooting region below the envelope comprises less than [FORMULA]. This is 40 times less than the total amount of material dredged-up. Overshooting alone can thus not explain the amount of material dredged-up. Indeed, the occurrence of dredge-up essentially results from the unstable character of the convection border in the standard models (see Sect. 2.2). Extra-mixing has the role of triggering the penetration of the envelope into the H-rich layers. Had we, at any time during the dredge-up, suppressed the extra-mixing procedure in the model calculations, the envelope would have stopped from further penetration for reasons described in Sect. 4.

The use of a diffusive extra-mixing procedure results in models displaying smooth [FORMULA] profiles as shown in Fig. 8a. This enables to locate properly the Schwarzschild layer. It should be noted, however, that the Schwarzschild layer is still unstable in the sense that more mixing of hydrogen from the envelope into the deeper layers renders them unstable against convection, leading thereby to further penetration of the envelope.

[FIGURE] Fig. 8. a  Same as Fig. 3a, but for models with overshooting ([FORMULA], [FORMULA]). b , c and d same as a , but for the opacity, the gravothermal energy and the radius, respectively.

The question then arises of the sensitivity of the dredge-up characteristics to the extra-mixing parameters. In order to answer that question, two other sets of models are calculated, one with increased extra-mixing extent and efficiency of [FORMULA], and one with lower extra-mixing extent and efficiency of [FORMULA]. The predictions from those sets are displayed in Fig. 7 in dotted and short-dashed lines, respectively. It is seen that the dredge-up characteristics are rather insensitive to the extra-mixing parameters . In particular, the dredge-up rate is independent of the extra-mixing parameters, and equals [FORMULA]/yr in all three sets of calculations. The only small difference between the different sets is the earlier time, in models with higher [FORMULA], at which the envelope reaches the H-depleted regions.

The important conclusion that the dredge-up characteristics are rather insensitive to the extra-mixing parameters can be understood from basic considerations on the dredge-up process, developed in Sect. 6. It allows, in particular, to analyze the 3DUP phenomenon without worrying much about the exact values of the overshooting parameters. In order to perform such an analysis as a function of stellar parameters, additional calculations with extra-mixing are performed on the 12th, 15th, 25th and 32nd afterpulses of the standard [FORMULA] star. The resulting structural evolution of the intershell layers are shown in Fig. 9, and some of their characteristics summarized in Table 1.

[FIGURE] Fig. 9. Same as Fig. 7a, but for, respectively from top to bottom, the 15th, 19th, 25th and 32nd pulses of the [FORMULA] models with [FORMULA] and [FORMULA]. The size of the mass ordinate is similar in all four figures.


Table 1. Several quantities characterizing the afterpulse phases of the 12th, 15th, 19th, 25th and 32nd pulses of the [FORMULA] star computed with the overshooting parameters [FORMULA] and [FORMULA]: the core mass [FORMULA] and the luminosity L at maximum pulse extension, the luminosity dip [FORMULA] during the afterpulse phase (see Fig. 7c), the maximum dredge-up rate of the envelope [FORMULA], the core mass increase [FORMULA] due to nuclear burning during the preceding interpulse phase, the mass of C-rich material dredged-up [FORMULA], the efficiency parameter [FORMULA], the carbon mass fraction [FORMULA] in the intershell layers left over by the pulse, and the increase in the envelope [FORMULA] mass fraction [FORMULA] resulting from each dredge-up.

Dredge-up rate. The dredge-up rate [FORMULA] is seen from Table 1 to be an increasing function of the pulse number. A linear relation between [FORMULA] and [FORMULA] is found from the model calculations, as shown in Fig. 10a. It translates into the following equation for our [FORMULA] star (dashed line in Fig. 10)

[FIGURE] Fig. 10. a  Dredge-up rate and b  dredge-up efficiency as a function of core mass in the models with [FORMULA] and [FORMULA] (filled dots). The dotted lines represent a linear fit to the data (Eqs. 9 and 10 for [FORMULA] and [FORMULA], respectively). The crosses and plus signs in panel b are the data predicted by Straniero et al. (1997) and Herwig et al. (1997), respectively.


[FORMULA] and [FORMULA] being expressed in [FORMULA] and [FORMULA] yr-1, respectively.

Dredge-up efficiency. The depth reached in mass coordinates by the envelope after each pulse depends on both the time over which it penetrates and the dredge-up rate. The former is mainly determined by the thermal time-scale for the evacuation of the gravothermal energy accumulated at the bottom of the pulse (Fig. 4), while the latter is determined by the time-scale of the thermal readjustment of the envelope. The resulting mass of C-rich material dredged-up in the envelope, [FORMULA], is reported in Table 1.

The dredge-up efficiency is traditionally expressed by a number [FORMULA], which is the ratio of [FORMULA] to the core mass increase [FORMULA] due to hydrogen burning during two successive pulses. A value of [FORMULA], for example, implies a constant core mass from one pulse to the next. From the model calculations, [FORMULA] is found to also increase linearly with [FORMULA] (filled circles in Fig. 10b). The linear relation writes (dotted line in Fig. 10b)


[FORMULA] being expressed in solar mass.

Evolution of surface characteristics during the dredge-up. The evolution of the surface luminosity and radius during the 3DUP is shown in Fig. 7c,d (solid lines), and compared to that predicted by models without dredge-up (dotted lines). It is seen that dredge-up leads to surface luminosities lower than those obtained in otherwise similar models but without dredge-up. That the luminosity should be lower in models with dredge-up is easily understood from the above discussion, since the penetration of the envelope absorbs energy in order to lift the dredged-up material into the envelope (see also Paczynski 1977). Dredge-up also leads to lower surface radii.

5.2. Influence of convection prescription

While the dredge-up characteristics are found to be rather insensitive to the extra-mixing parameters, a dependency of those characteristics on the convection prescription in the envelope should be discussed. Higher values of [FORMULA] are known to favor deeper convective envelopes (Wood 1981). The envelope would thus reach the H-depleted layers sooner and lead, for an otherwise similar dredge-up rate, to more C-rich material being dredged-up. Moreover, the dredge-up rate itself should also be affected by the convection prescription. The analysis presented in Sect. 6.1 stresses on the role of the thermal response of the envelope in the dredge-up process. The thermal time-scale of the envelope depends on its thermal structure, which is imposed by the surface conditions and the [FORMULA] parameter. We thus expect the dredge-up rate to be also dependent on [FORMULA]. A test calculation on the 19th afterpulse of the [FORMULA] computed with [FORMULA] leads to [FORMULA] /yr with [FORMULA]=[FORMULA]. This has to be compared with a predicted dredge-up rate of [FORMULA] /yr at that core mass in models with [FORMULA] (from Eq. 9). Models with [FORMULA] thus predict higher dredge-up rates by about 16% than those in models with [FORMULA].

5.3. Comparison with other published works

The calculations presented in Sects. 4 and 5 lead to a simple conclusion about AGB models using the Schwarzschild criterion: models computed without any extra-mixing do not lead to dredge-up while those using an extra-mixing procedure lead to efficient dredge-up.

How does this compare to model calculations published in the literature? The answer to that question requires the knowledge of the precise numerical procedures used by each AGB modeler in their code. Such information is often not available in the literature. For this reason - and because it is anyway outside the scope of this paper to do so - it is impossible to compare all published models. Some of them are however discussed below.

Many AGB calculations of the last five years do not report the occurrence of the 3DUP (Vassiliadis & Wood 1993, Wagenhuber & Weiss 1994, Blöcker 1995, Forestini & Charbonnel 1997). This is consistent with their working hypothesis of the Schwarzschild criterion without extra-mixing.

Several codes use an extra-mixing procedure based on a purely numerical technique. Boothroyd and Sackmann (1988a), for example, mix the radiative layer adjacent to the convective zone before checking for its stability. If the [FORMULA] ratio of that (formally stable) layer is found to be greater than one after mixing, then they declare it convective. As this mixing procedure is performed once per model calculation, it allows a numerical propagation of the convective envelope during the 3DUP. This might be the reason for their success in obtaining low-mass carbon stars (Boothroyd & Sackmann 1988b). A similar technique was already used by Paczynski (1977). This last author obtained dredge-up in [FORMULA] AGB models with such a numerical `overshoot' procedure, while no dredge-up resulted in models not including that overshoot. The use of extra-mixing procedures based on a purely numerical technique suffers from at least two shortcomings. First, they assume instantaneous mixing of the layer(s) added to the envelope. Besides being unphysical, it may lead to convergence difficulties. Second, they would probably depend on the numerical accuracies of the models such as the time-step and mesh re-zoning. As a result, such models may or may not lead to efficient dredge-up.

Herwig et al. (1997) use an overshooting algorithm which assumes an exponentially decreasing bubble velocity field in the extra-mixing region. Their algorithm is very similar to mine, and enables a direct comparison of their predictions with mine. These are reported in Fig. 10b (+ signs). Interestingly, they find higher dredge-up efficiencies than I do at a given stellar luminosity. We note that they use [FORMULA] in their calculations. This could explain part of the differences. We further note that their core masses are lower by about [FORMULA] than mine at a given pulse number, which could imply a dependence of the dredge-up efficiencies on the initial conditions at the onset of the AGB.

Finally, the recent work of Straniero et al. (1997) should be mentioned. These authors obtain dredge-up in models of 1.5 and [FORMULA] stars without using any extra-mixing procedure in their calculation. This seems in contradiction with my conclusions. Private discussions with some of the authors in Straniero et al. (1997) seemed to confirm that they did not use any extra-mixing in their model calculations. They explain their results by the higher temporal and spatial resolution of their models (see their paper for more details). This conclusion, too, is not supported by my calculations reported in Sect. 4. While I do not understand their results, I cannot push the analysis much further. In any case, the dredge-up efficiencies found in their models are lower than those predicted from my calculations including extra-mixing. Their predictions are reported in Fig. 10b ([FORMULA] signs).

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999