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

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6. Third dredge-up laws

6.1. The dredge-up process

The important conclusion that the dredge-up rate is insensitive to the extra-mixing parameters can be understood from the following considerations. The AGB star is basically formed by an [FORMULA]-degenerate core of 0.10-0.15 [FORMULA] surrounded by an extended H-rich envelope. As dredge-up proceeds, the outermost H-depleted layers of the core become part of the envelope and expand (which translates into a negative gravothermal energy production in those layers, see Fig. 8c). This results in an increase of the potential energy of those layers engulfed in the convective envelope, which must be supplied by the luminosity provided by the inner layers. The maximum dredge-up rate is then determined by the thermal relaxation time-scale of those layers expanding into the envelope during the dredge-up.

Let us estimate that dredge-up rate from simplified arguments. The potential energy [FORMULA] of a mass [FORMULA] located at the H-He discontinuity is given by

[EQUATION]

where [FORMULA] is the mass of the H-depleted core and [FORMULA] its radius (i.e. [FORMULA] during dredge-up). The energy necessary to lift the dredged-up material into the envelope is provided by the luminosity at the core edge, [FORMULA]. The thermal time-scale [FORMULA] for the dredge-up is then

[EQUATION]

All quantities in this section are expressed in solar units and years. The dredge-up rate can then be estimated by

[EQUATION]

Typical values for the [FORMULA] star are [FORMULA] (Fig. 7a), [FORMULA] (Fig. 8d) and [FORMULA] (Fig. 4b). Eq. 13 then gives [FORMULA]/yr. This estimate of the dredge-up rate, while giving only an order of magnitude, convincingly supports the rate of [FORMULA]/yr obtained in the full evolutionary models (Sect. 5). It is also consistent with the results obtained by Iben (1976), who estimated the dredge-up rate to be about [FORMULA]/yr in a [FORMULA] star with [FORMULA], and with those of Paczynski (1977), who found [FORMULA]/yr in a [FORMULA] star with [FORMULA].

The question of why the dredge-up rate is independent of the extra-mixing parameters can further be understood by comparing the time-scale of the thermal readjustment of the envelope with the typical time-scale of convective bubbles to cross the H-He transition zone. The velocity of the convective bubbles as they approach the H-He discontinuity is of the order of [FORMULA]/yr, which translates in terms of mass to about [FORMULA]/yr (from Fig. 8d). This rate is much higher than the dredge-up rate of [FORMULA]/yr established above. The deepening of the envelope into the H-depleted layers is thus primarily fixed by the thermal relaxation time-scale of the envelope, rather than by the speed of the convective bubbles penetrating into the H-depleted layers.

6.2. Dredge-up characteristics

Eq. 13 reveals a formal dependence of the dredge-up rate on [FORMULA], [FORMULA] and [FORMULA].

Those three stellar parameters are not independent of each other. Refsdal & Weigert (1970) and Kippenhahn (1981), for example, show from homology considerations that most red giant star properties are functions of the H-depleted core mass and radius. As the burning shell advances, the structure of the intershell layers evolves, to first approximation, like homologous transformations. Let us consider two times t and [FORMULA] characterized by core masses, radii and luminosities of [FORMULA], [FORMULA] and L, respectively, at time t, and [FORMULA], [FORMULA] and [FORMULA], respectively, at time [FORMULA]. Then the homology transformations applied to a [FORMULA] AGB star lead to (Kippenhahn 1981, Herwig et al. 1998)

[EQUATION]

The dredge-up rate [FORMULA] at time [FORMULA] can then easily be evaluated from that at time t, [FORMULA]. Eqs. 13 and 14 lead to

[EQUATION]

The homology transformations thus suggest a linear relation between the dredge-up rate and the core mass .

In standard AGB calculations (i.e. without dredge-up), [FORMULA] is a linear function of [FORMULA] (Eq. 8 for our [FORMULA] star). Eq. 14 then recovers the classical linear [FORMULA]-L relation. In that case, [FORMULA] is a linear function of either [FORMULA], [FORMULA] or L. The dredge-up predictions presented in Sect. 5 essentially obey these rules (Table 1), since the dredge-up calculations have been performed on selected afterpulses of the standard [FORMULA] models and do not include the feedback of the dredge-ups on the AGB evolution.

When the feedback of dredge-up on the structural evolution is taken into account, the core radius is no more a linear relation of the core mass, and the linear [FORMULA]-L does not hold any more (Herwig et al. 1998). It must be replaced by a [FORMULA]-[FORMULA]-L relation as suggested by Eq. 14. Eq. 15, however, reveals that the dredge-up rate keeps a linear dependency on [FORMULA] even in models experiencing dredge-up.

The implications of the linear [FORMULA]-[FORMULA] relation on the core mass evolution are analyzed in Sect. 7.1.

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999
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