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Astron. Astrophys. 346, 798-804 (1999)

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3. Uncertainties in the predicted Th abundance

The r-process production of Th is obviously model dependent. However, for cosmochronological purposes, it is of fundamental importance to know to what extent the remaining uncertainties in the r-process modelling can affect the Th synthesis. From Eq. (1), it can be seen that the Th abundance has to be determined within less than 16% if we hope to predict the age of star within less than 3 Gyr. A high accuracy has already been achieved observationally with errors reduced to [FORMULA] affecting the stellar age by about 3.7 Gyr. Unfortunately, other uncertainties still need to be solved. These mainly concern the r-process modelling, but before focussing on this subject, it is of interest to stress that the normalization to the Eu abundance is not free from uncertainties. As shown by Goriely (1999), even if the r-process models would be able to reproduce exactly the Eu solar r-abundance, this value is still uncertain by about 20% (the observed abundance in the ultra-metal-poor stars is known within 35%) leading to an error in [FORMULA] of about 6.6 Gyr. Note that with respect to normalization procedure, it might be safer to use the Ho abundance, since Ho is made of one stable isotope only and the s-contribution to its solar abundance is even smaller than for Eu, i.e the error bars on its solar abundance (about 13%) are smaller than in the Eu case. The additional uncertainties related to the predicted Eu r-abundance are neglected in the present study by normalizing the calculated Th abundance to the solar r-abundance of Eu. The complete absence of correlation in the production of Th and Eu in the canonical approach of the r-process justifies this choice.

The sensitivity of the calculated r-process abundances, and in particular of the Th abundance, to the different crucial inputs used in the multi-event model is now examined and the impact of such uncertainties on the age of the CS 22892-052 star ([FORMULA]) is discussed.

3.1. Sensitivity to astrophysics conditions

Among the different thermodynamic parameters entering the canonical model, the most critical one affecting the Th synthesis is obviously the maximum number of neutrons captured by the initial seed nuclei, [FORMULA], which defines the strong component of the r-process. In analogy with the s-process nucleosynthesis, we can define a main r-process component responsible for the production of all the elements up to the [FORMULA] peak and part of the Pb peak. This main component requires a value of [FORMULA] up to about 140. Till now, there is no constraint from realistic models on the largest value that [FORMULA] can take, i.e in analogy with the s-process, on a strong r-component responsible for the bulk production of Pb and Bi. Considering canonical events with values of [FORMULA] would lead to the production of the Pb-peak elements, as well as Th, without affecting the synthesis of the lower-mass elements. To illustrate such a sensitivity, multi-event calculations are performed considering canonical events with a maximum number of neutrons captured of [FORMULA]140, 145, 150 and 200 (Fig. 1). An excellent fit is obtained for all isotopes with [FORMULA] and seen not to be affected at all by the change in the maximum value of [FORMULA] considered. On the contrary, increasing the [FORMULA] above 140 leads to an increase in the production of the Pb-peak, Th and U elements. The large uncertainties in the Pb-peak r-abundances cannot favour one or another fit. The complete absence of a strong r-component, i.e a maximum value of [FORMULA], leads to a negative age (when derived from Eq. 1) of the CS 22892-052 star, and can obviously be rejected. Including a strong r-component with [FORMULA], 150 and 200 leads to a Th abundance that implies a star age [FORMULA], 22.9 and 28.9 Gyr, respectively. Since the fit is constrained by the upper value of the Pb and Bi abundances, increasing [FORMULA] above 200 does not affect the upper value of the Th abundance. More precisely, no event with a value of [FORMULA] contributes to the fit to the solar system abundances with our adopted nuclear inputs. In summary, any age below about 29 Gyr can thus be obtained just by adjusting the strength of such a strong r-process component, unless the s- or r-origin of the Pb and Bi can be determined with a greater accuracy.

[FIGURE] Fig. 1. Comparison between the solar system r-abundances (Goriely 1999) and the distribution predicted by our standard multi-event superposition of events characterized by maximum values of [FORMULA]140 (dashed), 145 (dot-dash), 150 (dotted) and 200 (full curve). The square corresponds to the Th abundance observed in CS 22892-052 (Sneden et al. 1996). The Bi and Th abundance are connected by a straight line to visualize the extrapolation predicted by the respective models. The vertical lines correspond to error bars in the solar system abundances taken from Goriely (1999).

3.2. Sensitivity to nuclear physics input

The most fundamental nuclear input to r-process models is well known to be the nuclear masses. Various mass models are available, but for practical reasons we only consider here in addition to the ETFSI model, the ETFSI-Q model (Pearson et al. 1996) which takes into account the strong shell-quenching found in some microscopic calculations on highly neutron-rich nuclei, the popular FRDM model of Möller et al. (1995) and the recently-developed model of Duflo & Zuker (1995), hereafter DZ, based on a very different approach than the previously cited models and which has proven its remarkable ability to predict experimentally known masses. Many studies have compared the quality of these models and their differences in the prediction of masses far away from the valley of [FORMULA]-stability. Their impact on the r-process nucleosynthesis has also been analyzed in various papers (e.g Goriely & Arnould 1992), so that we will restrict ourselves to analyze their respective predictions of the Th abundance.

Multi-event calculations are now performed making use of the 4 above-cited mass models. The resulting fits are shown in Fig. 2. The fits to the stable nuclei are of the same quality, and in particular in the Pb region, no major differences in the predicted r-abundances can be observed. In particular, it should be emphasized that no major deficiency in the fit is obtained in the pre-peak regions at [FORMULA] and [FORMULA] whatever mass model is used. Given the absence of realistic r-process models, there is obviously no reason to favour one or another mass formula on grounds of parametric fits to the solar r-abundance distribution, especially when dealing with the Th abundance predictions which exclusively depend on the r-process paths in the [FORMULA] region. The extrapolation to the Th abundance appears to be highly affected by the mass model used. The estimate of the star age amounts to [FORMULA], 23.8, 42.2 and 8.7 Gyr for the ETFSI, ETFSI-Q, FRDM and DZ models, respectively. Such differences are not surprising, since it is well known that the r-process paths for these mass models are significantly different, in particular in the [FORMULA] region where the strength of the shell correction energy around the [FORMULA] shell closure can be very different. This is not the case for the ETFSI and ETFSI-Q models, because the shell quenching introduced in the ETFSI-Q model in the vicinity of the [FORMULA] shell closure is small. Therefore, the abundance predictions in the Pb and actinide regions, and consequently the stellar age predictions, are globally similar when making use of ETFSI or ETFSI-Q. Compared with the other models, the DZ formula is characterized by a steep slope of the mass parabola and a weak shell effect around [FORMULA], so that the progenitors responsible for the final Pb abundance are found in a lower mass region by-passing partially Th and U. The Th abundance obtained with FRDM model is higher than with the ETFSI mass models, because of a more widely spread shell effect in the vicinity of the [FORMULA] shell closure affecting the r-process path down to [FORMULA]. The abundance peak around [FORMULA] before freeze-out is consequently flattened to lower masses than in the ETFSI case and is less affected by fission processes after freeze-out. A higher Th abundance predicted with the FRDM model leads to a higher age estimate. The high sensitivity of the predicted Th abundance to the mass model will not be resolved before improving our mass predictions in the heavy ([FORMULA]) neutron-rich region.

[FIGURE] Fig. 2. Same as Fig. 1 where the predictions are obtained with the ETFSI (full), ETFSI-Q (dashed), FRDM (dot-dash) and Duflo & Zuker (dotted) mass models.

Another important ingredient in the Th nucleosynthesis concerns the fission processes, i.e the spontaneous, neutron-induced and [FORMULA]-delayed fission. Most of the r-process calculations do not include the fission processes at all or only partially. However, when dealing with Th cosmochronometry, fission processes must be included in the most careful way in order to describe the competing processes responsible for the final Th abundance (namely [FORMULA]-decays, [FORMULA]-decays and fissions) correctly. The recent large scale calculation of ETFSI fission barriers (Mamdouh et al. 1998) up to [FORMULA] is used to test the significance of fission on the Th cosmochronometry. The spontaneous, neutron-induced and [FORMULA]-delayed fission probabilities are determined in the same way as in Kodoma & Takahashi (1975). It should, however, be stressed that when not available experimentally, the spontaneous fission rates are derived from a new regression fit to experimental data based on our fission barrier predictions. Fig. 3 shows the nuclear regions where the different fission modes influence the r-process flows. Because of the strong ETFSI shell effect on the fission barriers around [FORMULA], no fission recycling is found during the neutron irradiation, at least before crossing the [FORMULA] closure. Only r-process paths characterized by an astrophysical parameter [FORMULA] (for more details about the astrophysical parameter, see Goriely & Arnould 1992) are stopped by neutron-induced fission. Spontaneous fission can also affect such r-process paths before neutron freeze-out. [FORMULA]-delayed fission is found to be of small importance compared with the other decaying modes, even after freeze-out. On the contrary, the spontaneous fission is found to be faster than the [FORMULA]-decay for almost all isobaric chains above [FORMULA] and crucial in estimating the final r-abundances in the Pb and actinide region. In the specific fits studied in the present paper, the fission fragments do not affect the low-mass abundance distribution. Obviously, all the above conclusions should be taken with care, because of the uncertainties remaining in the determination of the fission barriers and fission probabilities, which are to be studied in a forthcoming paper.

[FIGURE] Fig. 3. Representation in the nuclear chart of the dominating fission modes affecting the r-process flow, as given by the legend in the figure. Three r-process paths at [FORMULA], 2 and 3 MeV (full line) are drawn for illustrative purposes, as well as the neutron drip line (dashed line).

In order to quantify the impact of the fission processes on the Th cosmochronometry, a multi-event calculation is reiterated switching off all the fission processes. This numerical test just aims at illustrating the largest error possibly made when neglecting fission, but should not be regarded as a sensible test case for the Th prediction. It is found that the neglect of fission gives rise to an increase of the age of CS 22892-052 by 12.3 Gyr on grounds of the abundance distribution shown in Fig. 4. It can also be seen that when including fission processes, the fission fragments do not modify the global abundance distribution. Obviously, a complete and consistent treatment of the fission processes (especially spontaneous fission) is required to build a reliable cosmochronometry on the actinides.

[FIGURE] Fig. 4. Same as Fig. 1 where the predictions are obtained using the GT2 [FORMULA]-decay and [FORMULA]-delayed neutron emission rates with (full) or without (dash-dot) fission processes. The dotted curve corresponds to the use of the QRPA [FORMULA]-decay and [FORMULA]-delayed neutron emission rates including fission.

Fig. 4 also presents uncertainties associated with [FORMULA]-decays and [FORMULA]-delayed neutron emission by comparing the solar fits obtained with the GT2 model and the QRPA model of Möller et al. (1997). Both models have been extensively used in previous works dedicated to the r-process nucleosynthesis, so that it is of interest to study their influence on the Th cosmochronometry. If use is made of the QRPA model instead of the GT2 model, a reduction from 28.9 Gyr down to 15.1 Gyr is obtained for the age of CS 22892-052.Once again, it should be added that although the fit to the solar distribution obtained with the QRPA model is slightly worse than the one obtained with the GT2 approach, it cannot be rejected a priori , since other nuclear or astrophysics shortcomings of the model can be responsible for the observed discrepancies (for example in the [FORMULA] region). As stated previously, given our poor understanding of the r-process nucleosynthesis (especially of the astrophysical site) the quality of nuclear models should not be tested on astrophysics arguments like fits to the solar abundance distribution.

3.3. Uncertainties in the solar r-abundance

The uncertainties still affecting the s-process model are responsible for non-negligible imprecisions in the determination of the residual r-abundances of the solar system content (Goriely 1999), in particular for the so-called s-dominant isotopes and Pb-Bi isotopes (see the large error bars in Fig. 1). The principal source of uncertainty in the solar r-abundance of Pb and Bi lies in our ignorance of the relative s- and r-contribution to their production. Both processes can produce Pb-Bi almost entirely, so that the r-contribution cannot be estimated in a reliable way on grounds of s-process calculations. This problematic aspect of the solar abundance splitting in the Pb region can only be resolved through realistic modelling of the s-process (or accurate abundance determination of Pb at the surface of ultra-metal-poor stars provided the assumption of the r-process universality be confirmed). It should be kept in mind that the prediction of the Th and Pb-Bi abundances are strongly correlated, so that any uncertainty in the solar r-abundances of the Pb and Bi elements is translated into the exponentially dependent uncertainty in the star age. Among the [FORMULA] r-isotopes, [FORMULA], [FORMULA] and [FORMULA] play an important role since their r-abundances are better determined than for their neighbours. The solar r-abundance of [FORMULA] is indeed well determined, so that it seems logical to constrain the fit in such a way as to reproduce the [FORMULA] solar r-abundance. As regards [FORMULA] and [FORMULA], they are characterized by a relatively well-determined abundance to which the Th production is directly correlated. However, reproducing the recommended solar abundance of [FORMULA], [FORMULA] and [FORMULA] simultaneously appears to be impossible without strongly deteriorating the fit to the [FORMULA] peak. For this reason, we reiterate multi-event calculations in which the fitting procedure is constrained (with an extra statistical weight) to the solar abundance of [FORMULA] and 206 in one case and [FORMULA] and 209 in the other case, using two different mass models, namely the ETFSI and DZ models (Fig. 5). Although a negative age is obtained with the DZ masses when constraining the fit to [FORMULA], the other predicted ages are 30.4 Gyr for the ETFSI calculation constrained to [FORMULA] and 10.5 Gyr in the two remaining cases. So, in addition to the large sensitivity of the stellar age to the mass models as studied in the previous section, the uncertainties in the solar r-abundances appear to affect the age determination by about 20 Gyr. As long as the s- or r-origin of the Pb and Bi solar abundance is not determined with high accuracy, the Th cosmochronometry will not provide any reliable age estimate.

[FIGURE] Fig. 5. Same as Fig. 1 when the fit is constrained on [FORMULA] and [FORMULA] solar abundances with the ETFSI (full) or DZ (dashed) mass models. The dash-dot and dotted lines correspond to the constrain on [FORMULA] and [FORMULA] abundances with ETFSI and DZ mass models, respectively.

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© European Southern Observatory (ESO) 1999

Online publication: June 17, 1999
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