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Astron. Astrophys. 320, 460-468 (1997)

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3. Yields of [FORMULA] from WR stars

The production of [FORMULA] takes place during the core H-burning phase of the WR stars. It results from the MgAl chain operating at central temperatures between 35 and [FORMULA], while the central densities are of the order of a few [FORMULA]. In such conditions, [FORMULA] is synthesized by proton captures on [FORMULA], and mainly destroyed by [FORMULA] -decay, the [FORMULA] channel being much less probable. As a result, and as made clear in Figs. 1 and 2, the core [FORMULA] mass fraction [FORMULA] goes through a maximum [FORMULA] at [FORMULA], where [FORMULA] is the [FORMULA] [FORMULA] -decay lifetime. The destruction of [FORMULA] is accelerated at the beginning of the He-burning phase as a result of [FORMULA] neutron captures in the convective He-burning core. As a consequence, the late WC-WO evolutionary phases cannot contribute to the [FORMULA] production (Prantzos & Cassé 1986).

3.1. Dependence of the yields on the initial mass and metallicity

In the nucleosynthesis pattern sketched above, a first Z -dependent effect concerns the importance of the [FORMULA] production channel, which increases with the initial amount of [FORMULA], and thus with metallicity. In contrast, its [FORMULA] -decay destruction is Z -independent. The net increase of the core [FORMULA] content with Z is illustrated in Fig. 1, where it is seen that [FORMULA] increases by a factor of about 1.5 when Z is doubled.

A second consequence of a metallicity variation concerns the instant [FORMULA] at which [FORMULA] is reached. Fig. 1 shows that this time increases with Z. For example, it goes from about [FORMULA] to approximately [FORMULA] when Z increases from 0.008 to 0.04. This effect relates directly to the fact that, for a given initial mass, a Z increase imposes a lower central temperature (due to greater CNO content), and consequently a slower building up of [FORMULA] at the beginning of the H-burning phase.

Finally, a Z variation has an impact on the evolution of the post-maximum [FORMULA] values. More specifically, Fig. 1 indicates that [FORMULA] decreases more slowly for higher metallicities. This effect relates to the smaller convective cores found at higher Z as a consequence of both lower central temperatures and stronger [FORMULA]. The post-maximum [FORMULA] values are influenced by the extent of the convective core in the following way: convection mixes the innermost [FORMULA] -rich layers with zones that are poorer and poorer in [FORMULA] as they are more external. Indeed, these cooler outer zones produce less [FORMULA] through proton captures on [FORMULA], while the [FORMULA] [FORMULA] -decay flow is temperature-independent. In other words, smaller convective cores have a tendancy to admix less [FORMULA] -poor layers, and thus to keep [FORMULA] at a higher level, than more extended convective cores.

Fig. 2 complements Fig. 1 by illustrating the influence of [FORMULA] on the evolution of [FORMULA] for our [FORMULA] model stars. We note in particular that [FORMULA] is obtained earlier and reaches higher values with increasing [FORMULA]. This results directly from the fact that larger mass stars develop hotter cores, which in turn allows [FORMULA] to be more quickly and efficiently transformed into [FORMULA]. More specifically, [FORMULA] corresponds to 16% and 30% of the initial [FORMULA] abundance in the 25 and [FORMULA] model stars, respectively. On the other hand, the post-maximum [FORMULA] decrease is seen to decrease less steeply with decreasing [FORMULA]. Lower mass stars indeed develop less extended convective cores, which favours a slower time decrease of [FORMULA], as already explained above.

As displayed in Figs. 1 and 2, the [FORMULA] produced in the H-burning core appears at the stellar surface at times [FORMULA], its mass fraction [FORMULA] building up during the WNL and WNE phases of the evolution of the WR stars. It is eventually ejected by the wind into the interstellar medium, where its [FORMULA] -decay contributes to the observed 1.8 MeV emission line.

As demonstrated by Figs. 1 and 2, larger [FORMULA] can be obtained with increasing Z and [FORMULA]. This trend is well in line with the [FORMULA] and Z dependence of [FORMULA] discussed above, and is strengthened further by the enhancement of the mass loss rate with [FORMULA] and Z. This indeed leads to an earlier emergence of [FORMULA] at the stellar surface, and correspondingly to a reduction of its loss by [FORMULA] -decay before its ejection into the interstellar medium.

At this point, let us note that a metallicity increase and the concomitant mass loss enhancement have the additional effect of decreasing the critical initial mass [FORMULA] below which the WR stage cannot be entered. This has some galactic implications to be discussed in Sect. 4.1.

All in all, one may thus conclude that an increase of Z and [FORMULA] leads to an increase of [FORMULA] in a larger and larger variety of stars.

Table 2 and Fig. 3 provide our calculated values of the mass

[EQUATION]

of [FORMULA] ejected by a star of initial mass [FORMULA] and metallicity Z during the time span [FORMULA] between its Zero Age Main Sequence (ZAMS) and the end of its WC-WO phase (which is in practice its presupernova stage).


[TABLE]

Table 2. Wind ejected mass [FORMULA] ([FORMULA],Z) of 26 Al in units of 10-4 [FORMULA].


[FIGURE] Fig. 3. Masses [FORMULA] of [FORMULA] ejected by a star of initial mass [FORMULA] and metallicity [FORMULA], 0.02 or 0.04 between its ZAMS and the end of its WC-WO phase. The predictions of [FORMULA] WR models computed by Langer et al. (1995) (see main text) are also displayed for comparison.

The [FORMULA] yields are clearly seen to increase with [FORMULA] and Z. More specifically, a Z increase from 0.008 to 0.02 leads to a [FORMULA] enhancement by a factor between 5 and 8, depending on [FORMULA]. This increase ranges from 2 to 4 when Z goes from 0.02 to 0.04. This translates approximately into the power law expression

[EQUATION]

This result agrees with the [FORMULA] dependence proposed by Walter & Maeder (1989) on grounds of qualitative arguments, but predicts a weaker Z dependence than the [FORMULA] one derived by Meynet & Arnould (1993c) from detailed computations of two [FORMULA] stellar models with [FORMULA] and [FORMULA] based on lower mass loss rates than the ones adopted here, as well as on different opacities and initial abundances.

For [FORMULA], our present [FORMULA] yields are a factor of about 4 to 6 higher than those derived by Walter & Maeder (1989), and a factor of 3 larger than the ones reported by Prantzos (1991) for [FORMULA] models. These increased yields mainly relate to our doubling of the mass loss rates, as discussed below.

3.2. Dependence of the yields on the mass loss rates and on the mixing prescription

A faster wind removal of the original envelope has the effect of shortening the time delay between the production of [FORMULA] in the stellar core and its ejection into the interstellar medium. This leads to an increase of the amount of ejected [FORMULA], all other things being equal. The influence of the mass loss rates on the yields can be estimated more quantitatively by comparing the present results for the [FORMULA] model star at [FORMULA] with the yields calculated by Meynet & Arnould (1993c) for the same star, but with the adoption of a two times lower mass loss rate during the pre-WR and WNL phases 2. This difference leads to an increase of our present yields over the previous ones by a factor of about two.

An increase of the convective core size, and thus a reduction of the envelope mass, is also expected to produce an enhanced [FORMULA] yield, as the [FORMULA] -loaded material is more readily ejected by the wind. Langer et al. (1995) have studied more quantitatively the impact of various mixing schemes on the [FORMULA] yields. In particular, they have assumed either that the boundary of the convective core is defined by the Ledoux criterion and that diffusive semiconvective mixing develops in the semiconvective zone, or that the convective core is defined by the Schwarzschild criterion with/without overshooting. They have also explored the possible role of rotation-induced mixing.

Langer et al. (1995) find that a change in the mixing scheme may affect the [FORMULA] yields for a given star by a factor of at most 3, more extended convective cores leading in general to larger yields, as expected. However, in some of the explored cases, the reverse effect is obtained. We interpret this result as the consequence of two opposite effects. On the one hand, a larger convective core favors the early appearence of [FORMULA] at the surface. On the other hand, it may increase the time lag [FORMULA] because [FORMULA] increases as a result of longer main sequence lifetimes.

For the [FORMULA] model stars, Fig. 3 shows that our yields are similar to those predicted by the models of Langer et al. (1995) computed with the standard mass loss rates of de Jager et al. (1988), the Ledoux criterion and diffusive semiconvection. At first sight, this similarity of the yields predicted from quite different models looks surprising. A plausible explanation is that a subtle balance is at work in the considered model stars between the effect of a change of [FORMULA] (whose enhancement increases the [FORMULA] yields) and the impact of a different mixing scheme. This comparison illustrates the intricate dependence of the [FORMULA] yields on the various physical ingredients of the models.

3.3. Effects of changes in some key nuclear reaction rates

The dependence of the WR [FORMULA] yields on the [FORMULA] rate has already been discussed by Prantzos (1991) and by Meynet & Arnould (1993c). Let us simply recall here that the rate proposed by Iliadis et al. (1990) is about 5 times lower than the one recommended by Caughlan & Fowler (1988) for temperatures below [FORMULA]. In contrast to what one might conclude naively, this rate decrease leads to an enhancement of the [FORMULA] yield. A lower rate is indeed responsible for the production of [FORMULA] later in the H-burning phase, so that the time span between [FORMULA] production and wind ejection, and thus the quantity of [FORMULA] destroyed by [FORMULA] -decay, is reduced. According to Meynet & Arnould (1993c), the use of the rate proposed by Iliadis et al. (1990) leads to [FORMULA] yields that are about twice higher than the ones derived for the same star with the adoption of the rate of Caughlan & Fowler (1988).

We have performed additional test computations in order to study the impact of the rate proposed by Champagne et al. (1993) for the reaction [FORMULA] which competes with the [FORMULA] -decay destruction channel. This rate is about a factor of 6 higher than the rate of Vogelaar (1989) for a typical central temperature [FORMULA] K during the H-burning phase of the considered model stars. In spite of the differences between the two predicted rates, the [FORMULA] yields change by less than 4% for stars with [FORMULA] between 60 and [FORMULA] and [FORMULA] and 0.04. This convergence in the yield predictions results from the fact that the [FORMULA] -decay rate of [FORMULA] is larger than the (p, [FORMULA]) rate of Vogelaar (1989) or of Champagne et al. (1993) in the derived H-burning conditions. Recently, a new estimate for the [FORMULA] rate has been proposed (see Arnould et al. 1995). The impact of this revision and of the estimated remaining nuclear uncertainties on the [FORMULA] yields has been evaluated by Arnould et al. (1995) in the framework of parametrized astrophysical models. From this analysis, one can confidently conclude that the WR [FORMULA] yields reported here cannot be drastically affected by the use of this revisited [FORMULA] rate.

In conclusion, the lack of a precise knowledge of the mass loss rates and the limited reliability of the mixing prescriptions are the main factors limiting the accuracy of the WR [FORMULA] yield predictions. Each of these sources of uncertainties may lead to variations in the calculated production of that radionuclide that can amount to a factor of the order of 2 to 3. In contrast, more limited uncertainties arise from a purely nuclear physics origin. Of course, important changes to the situation depicted here and associated additional uncertainties may result from mass transfer by Roche lobe overflow in close binaries, or from rotational mixing in rapidly rotating massive stars (e.g. Walter & Maeder 1989, Braun & Langer 1995, Langer et al. 1995).

In spite of the sources of uncertainties discussed above and of the possible additional intricacies eluded in this paper, the predictions of the [FORMULA] yields from other astrophysical sources are even much less secure than the WR estimates. This results from the fact that the modelling of WR stars in a stage of hydrostatic central H or He burning is much simpler than the one of asymptotic giant branch stars, novae or supernovae. Uncertainties of purely nuclear origin might also be much larger in these various types of objects than in the WR situation (see e.g. Arnould et al. 1995 for an illustration of this statement in the case of asymptotic giant branch stars).

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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