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

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3. The stellar evolution code

The stellar evolution code used to compute the models is described in detail in Mowlavi (1995). Among the modifications brought to it since its conception in 1995, let me mention the use of interior radiative opacities taken from Iglesias & Rogers (1996) and the use of low-temperature ones from Alexander & Ferguson (1994). The convection energy transport is described as usually by the MLT, with the mixing length parameter [FORMULA] taken equal to 1.5 in units of pressure scale height. The time-step and mesh allocation procedures relevant to our study are presented in more details in Sect. 3.1, while the mixing schemes of chemical elements are described in Sect. 3.2.

3.1. Time-step and mesh distribution

The time step between each model is determined, in most cases, by the relative variation of the dependent variables (radius, temperature, pressure and luminosity) from one model to the next such that it does not exceed 10% at any layer in the star. On the AGB phase, the time step is further limited to a maximum of [FORMULA], where [FORMULA] is the time between two successive pulses. The time-step during the 19th interpulse phase, for example, is of the order of 25 y, and can decrease down to few hours during a pulse.

The mesh allocation is based on the constitutive differential equations, and ensures a relative accuracy in the finite difference method 2 of better than [FORMULA] (see Mowlavi 1995). Special care is taken around boundaries of convective regions and in regions with steep chemical abundance profiles in order to avoid spurious numerical chemical diffusion. Our models are characterized by about 3000 meshes during the interpulse phases, and by up to 8000 meshes during pulses.

3.2. Mixing

Instantaneous mixing is assumed in all convective zones in models with no extra-mixing. When a discontinuity in the [FORMULA] profile is detected at the convection border, care is taken not to mix the material of the mesh comprising that discontinuity with the convective zone. This prevents a spurious numerical propagation of the convective zone into the radiative layers in unstable cases such as case b described in Sect. 2.2, and guarantees a correct application of the Schwarzschild criterion described in Sect. 2.

In models with extra-mixing, a specific overshooting prescription is used beyond the Schwarzschild layer. The bubble velocity field [FORMULA] in the radiative field is assumed to decay exponentially with the distance to the Schwarzschild layer, in the form

[EQUATION]

with

[EQUATION]

where subscripts sch and ov refer to quantities evaluated at the Schwarzschild boundary [FORMULA] and at the edge of the overshooting region [FORMULA], respectively. The function [FORMULA] ensures that [FORMULA] vanishes at the outer edge of the overshooting region. It is close to unity in all the overshooting region except close to [FORMULA] where it vanishes. Mixing in the convective + overshooting regions is performed through an algorithm coupling diffusive mixing and nucleosynthesis (see Mowlavi 1995) with a diffusion coefficient equal to [FORMULA], where [FORMULA] ([FORMULA] being the pressure scale height). The bubble speed at [FORMULA], [FORMULA], is determined from the bubble velocity profile in the convective zone as given by the MLT, while that at [FORMULA], [FORMULA], is determined by an `efficiency' parameter [FORMULA] according to Eq. 3. The extent of the overshooting region [FORMULA] is determined by a second parameter, [FORMULA], defined such that [FORMULA] locates the layer where [FORMULA], [FORMULA] and [FORMULA] being the pressure at [FORMULA] and [FORMULA], respectively.

In those models with extra-mixing, the mesh distribution is increased in the overshooting regions by about 100 meshes in order to ensure a good description of the abundance profiles. The time-step is further reduced to about [FORMULA] y during a 3DUP. A dredge-up episode is then covered by about 2000-3000 models.

Let us remark that a velocity field of the type of Eq. 2 is supported by recent hydrodynamical simulations of stellar convection in solar-type stars and white dwarfs (Freytag et al. 1996). However, we do not claim that such an overshooting pattern also applies to the envelopes of AGB stars, in which the physical conditions are quite different from those characterizing solar-type stars or white dwarfs. That remains to be confirmed by future hydrodynamical studies. We rather use such a prescription merely as a way to simulate an extra-mixing below the envelope and to analyze its effects on the 3DUP predictions. The extra-mixing could very well be associated with other `non-standard' mixing processes such as, for example, diffusion induced by rotation or shear mixing. Actually, the results presented in Sect. 5 show that the dredge-up predictions are quite insensitive to the extra-mixing parameters.

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999
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