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Astron. Astrophys. 337, 149-177 (1998)

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3. Stellar evolution with mass loss

For a complete definition of the problem, we finally have to specify the basic properties of the central star as a function of time. These are [FORMULA], [FORMULA], which together define the stellar radius [FORMULA], and [FORMULA]. The mass of the central star, [FORMULA], then changes in accordance with the mass loss rate.

For the solar-composition stellar model adopted as a typical example in this work, the changes of the stellar parameters during the different stages of evolution, from the main sequence through the red giant branch, the asymptotic giant branch, the planetary nebula phase and finally towards the White Dwarf regime, are illustrated in the corresponding Hertzsprung-Russell-Diagram in the upper panel of Fig. 1.

[FIGURE] Fig. 1. Top : Evolutionary track of a stellar model with an initial mass of [FORMULA] in the Hertzsprung-Russell-Diagram according to stellar evolution calculations with mass loss by Blöcker (1995). This diagram covers the basic stages of evolution, from the main sequence (MS) through the red giant branch (RGB), the early AGB (EAGB), the the so-called "thermally pulsing AGB" (TP-AGB, boxed region at upper right) and finally the proto-planetary nebula (PPN) and planetary nebula (PN) phases leading towards the White Dwarf (WD) regime. Open diamonds are plotted in time intervals of 1000 yrs (beginning at [FORMULA] yrs) for the first 10 000 yrs of post-AGB evolution, and in time intervals of 10 000 yrs thereafter. Bottom : Close-up view of the "thermally pulsing AGB". During this phase of evolution, the model suffers a total of 17 helium shell flashes (thermal pulses) over a period of about [FORMULA] yrs and looses about 80% of its initial mass. The final four thermal pulses are labeled "14" - "17".

According to stellar evolution calculations with mass loss by Blöcker (1995), this model suffers a total of 17 helium shell flashes (thermal pulses) on the so-called "thermally pulsing AGB" (TP-AGB) and looses about 80% of its initial mass during this stage. The end of AGB evolution and transition to the phase of a proto-planetary nebula (PPN) is signaled by a rapid increase of [FORMULA] at constant [FORMULA]. After a few thousand years the star will be found at the center of a planetary nebula (PN). When the nuclear energy generation ceases at [FORMULA] K, the luminosity begins to drop and the star eventually cools into the White Dwarf (WD) regime.

In the context of the hydrodynamical simulations of the circumstellar gas/dust shell described in the remainder of this work, we will only consider the final phases of evolution on the TP-AGB, covering the last four thermal pulses (number 14 to 17, cf. lower panel of Fig. 1). For the sake of simplicity we will henceforth refer to pulses number 14, 15, 16 and 17 as the `first', `second', `third' and `fourth' (or final) thermal pulse, respectively.

The temporal evolution of luminosity , effective temperature , mass loss rate and stellar mass during the last 350 000 years on the AGB and the following few thousand years of the post-AGB evolution is plotted in Fig. 2. The final thermal pulses on the AGB recur on a typical interpulse time scale of roughly 90 000 yrs.

[FIGURE] Fig. 2. Stellar luminosity, effective temperature, mass loss rate and mass (from top to bottom) vs. time for a [FORMULA] model sequence after Blöcker (1995), including the final four thermal-pulse cycles on the AGB (negative times). For mass loss rates between the dotted horizontal lines (middle frame), the dust condensation fraction is assumed to drop smoothly by a factor of 100 (see Eq. 20). At time zero the stellar effective temperature rises sharply (see second panel), marking the beginning of the model's post-AGB evolution. Labels A to J are referred to in the text. For details of [FORMULA] see also Figs. 10 and 19

In the corresponding stellar evolution calculations the effect of mass loss on the stellar evolution is taken into account through a parameterized mass loss description:

[EQUATION]

where the Reimers (1975) mass loss rate is

[EQUATION]

This mass loss formula (Eq. 23) is based on hydrodynamical calculations of shock-driven winds in the atmospheres of long-period Mira-like stars by Bowen (1988), taking into account radiation pressure on dust in a schematic way. The resulting AGB mass loss rates dramatically increase with the stellar luminosity. As a consequence, most of the mass is lost during the last few thermal pulses before the end of the AGB evolution (cf. lower panel of Fig. 2). In our example, the initial mass of [FORMULA] is reduced to a final mass of [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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