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Astron. Astrophys. 346, L37-L40 (1999)

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3. Results

3.1. Evolution towards the TP-AGB

For our pilot study, we chose to compute the evolution of a 3[FORMULA] star of roughly solar composition with an initial equatorial rotation velocity of 250[FORMULA], which is typical for late B main sequence stars (Fukuda 1982). This choice renders effects of magnetic braking and the core helium flash unimportant. The evolution of our model in the HR diagram, together with that of a non-rotating reference model, is shown in Fig. 1.

[FIGURE] Fig. 1. Evolutionary track of our rotating 3[FORMULA] model (solid line) and of a non-rotating reference model (dotted line) in the HR diagram. The tracks start at the zero age main sequence and end on the TP-AGB. The beginning of the thermal pulses is marked.

As in massive stars (Heger et al. 1999), the dominant rotational mixing process on the main sequence is the Eddington-Sweet circulation. It leads to a 12C/13C-ratio after the first dredge-up of 9.4, compared to a value of 19.4 in our non-rotating model (cf., Boothroyd & Sackmann 1999).

Fig. 2 sketches the evolution of the angular momentum distribution in the innermost 0.8[FORMULA] of our rotating 3[FORMULA] model. It shows that the core specific angular momentum decreases continuously during the evolution. We can give a first quantitative prediction of a white dwarf rotation rate: [FORMULA] and [FORMULA] yields [FORMULA]. This is of the same order as current observational upper limits (Heber et al. 1997, Koester et al. 1998).

[FIGURE] Fig. 2. Upper panel: logarithm of the local specific angular momentum (in cm2s-1) as function of the mass coordinate for three models of our rotating 3[FORMULA] sequence; at core hydrogen exhaustion ([FORMULA]yr, solid line), during core helium burning ([FORMULA]yr, dotted line), and between the 14th and the 15th thermal pulse ([FORMULA]yr, dashed line). Lower panel: logarithm of the local angular velocity (in rad/s) as function of the mass coordinate for the same models which are displayed in the upper panel.

3.2. Mixing and nucleosynthesis on the TP-AGB

Fig. 3 shows the evolution of the internal structure during and after the 25th thermal pulse of our rotating model. It shows that the tip of the pulse-driven convection zone leaves after its decay a region of strong rotational mixing. This mixing becomes even stronger when the convective envelope extends downward during the third dredge-up event (cf. also Fig. 4). The reason is that convection enforces close-to-rigid rotation (cf. Heger et al. 1999), with an envelope rotation rate which is many orders of magnitude smaller than that of the core. The resulting strongly differential rotation (cf. also Fig. 2) allows the Goldreich-Schubert-Fricke instability, and to a lesser extent the shear instability and Eddington-Sweet circulations, to produce a considerable amount of mixing between the carbon-rich layer and the hydrogen envelope.

[FIGURE] Fig. 3. Section of the internal structure during and after the 25th thermal pulse of our rotating 3[FORMULA] sequence. Diagonal hatching denotes convection. The convective envelope extends down to [FORMULA]. The pulse driven convective shell is located at [FORMULA] and [FORMULA]yr. Vertical hatching denotes regions of significant nuclear energy generation, i.e., the hydrogen burning shell (at [FORMULA] and [FORMULA]yr) and the helium burning shell ([FORMULA] and [FORMULA]yr). Gray shading marks regions of significant rotationally induced mixing (see scale on the right side of the figure). Vertical marks at the bottom of the figure denote the time resolution of the calculation, where every fifth time step is indicated. Cf. also Fig. 4. During this thermal pulse, the maximum energy generation rate of the helium burning shell was [FORMULA].

[FIGURE] Fig. 4. Same as Fig. 3, for the same time interval, but magnifying the dredge-up of the convective envelope. Note the hydrogen burning shell source at [FORMULA] and [FORMULA]yr, and the extension of the pulse driven convection zone up to [FORMULA] at [FORMULA]yr.

Fig. 5 depicts the resulting hydrogen and 12C abundance profiles after the convective envelope has receded. It shows a layer of several [FORMULA] containing a large mass fraction of protons and 12C at the same time. Several 1000 yr after the pulse, this layer heats up and 13C is formed through proton capture on 12C. Fig. 5 shows the resulting 13C profiles for four different times. A maximum 13C mass fraction of almost 4% is achieved.

[FIGURE] Fig. 5. Chemical profiles at the location of the maximum depth of the convective envelope during the 25th thermal pulse (cf. Fig. 4) of our rotating 3[FORMULA] sequence. The dotted and dashed lines mark the hydrogen and the 12C mass fractions at [FORMULA]yr, with [FORMULA] defined as in Figs. 3 and 4. The fat solid line denotes the 13C mass fraction at the same time. The three thin solid lines represent the 13C mass fractions at [FORMULA]yr, [FORMULA]yr, and [FORMULA]yr, with a later time corresponding to a smaller peak abundance. The maximum 13C mass fractions of 3.6% occurs at [FORMULA]yr. The 13C peak moved inwards in the time interval from [FORMULA]yr to [FORMULA]yr due to continued proton captures on both, 12C and 13C.

Starting some [FORMULA]yr after the pulse, the 13C-rich layer becomes hot enough for [FORMULA]-captures on 13C to occur (Straniero et al. 1995). Fig. 6 shows the resulting neutron densities [FORMULA] for three different times. Note that we did not include the reaction 14N(n,p)14C in our network. Although most of the resulting protons may form new 13C, it may be an effective neutron sink (Jorissen & Arnould 1989), in particular as also a large abundance of (primary) 14N is produced in the 13C-rich layer. Thus, our neutron densities can only be considered as an order of magnitude estimate. With [FORMULA]cm-3 for [FORMULA]yr, we obtain a neutron irradiation of [FORMULA]neutrons/cm2 which results roughly in a number of neutron captures per iron seed of [FORMULA], i.e. a main component s-process (cf. Figs. 7.22 and 7.23 of Clayton, 1968).

[FIGURE] Fig. 6. Neutron density as function of the mass coordinate for three models of our rotating [FORMULA] sequence after the 25th pulse. The fat solid line corresponds to [FORMULA]yr, the other two (thin solid lines) to [FORMULA]yr (peak at largest mass coordinate) and [FORMULA]yr, with [FORMULA] defined as in Figs. 34 and 5. The time span between the 25th and the 26th thermal pulse is [FORMULA]yr.

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© European Southern Observatory (ESO) 1999

Online publication: June 17, 1999