Astron. Astrophys. 363, 555-567 (2000)

3. The model and the main parameters

3.1. The nucleosynthesis and mixing in AGB stars

AGB star consists of three layers: a C-O core of degenerate electron, a He and H double burning shell, and a convective envelope. Heavy-elements synthesized in the He-intershell are dredged up to the convective envelope by the convection that appears in the cooling contraction after a thermal pulse, and show up as the observed overabundances of the heavy-elements and ¹²C. With the evolution of the star, its core mass steadily increases with increasing thermal pulses, while the mass of the convective envelope decreases correspondingly via wind mass loss, until it is completely exhausted, while marks the life span of the thermal pulse AGB star.

During every thermal pulse, we calculate the s-process nucleosynthesis. ⁵⁶Fe is taken to be the seed nucleus for the s-process. The reaction ²⁰⁹Bi(n,)²¹⁰Bi²⁰⁶Pb(²¹⁰Bi has an -decay half-life of 5.01 days) terminates the reaction chain of the slow process of neutron capture. The branch s-process path is adopted in the calculation. The neutron capture cross sections of most of nuclei, are taken from Beer et al. (1992). The -decay rates and electron capture rates of nuclei are taken from Takahashi & Yokoi (1987). More recent values of some nuclei are taken from the same literatures as those cited by Gallino et al. (1998). For the initial abundances, we use standard red giant abundances, which differ from the solar abundances only by a constant factor: . At present, the solar system abundance distribution is the most detailed abundance distribution obtained by us. According to the results of Straniero et al. (1997) and Gallino et al. (1998), the main parameters adopted in calculation are as follows:

(1) Core mass: At the onset of the thermal pulse, the mass of the C-O core is 0.572. With successive pulses, both the H burning and He burning shells move outwards (in mass coordinates) and so does the C-O core. The core mass is up to 0.611 at the 8th pulse, during which the third dredge-up begins. The variation of the core mass influence directly the wind mass loss. The core masses of stars in different TP periods are taken from Table 4 of Straniero et al. (1997).

(2) Mass loss through wind: Wind mass loss is one of the most important ingredients in the computation of both the nucleosynthesis occurring in TP-AGB stars and the wind accretion on the secondary component of a binary system. Here, we adopt the mass loss rates given by Straniero et al. (1997) using Reimers formula (Reimers 1975) based on the TP-AGB stellar models (their Table 4).

The mass lost during the time , the interpulse period, is

which is taken from Table 4 of Straniero et al. (1997).

(3) Dilution Factor: The physical process that influences significantly the surface abundance is the third dredge-up. The dilution factor f is defined as the ratio of the mass dredged in every pulse to the mass of the envelope after the third dredge-up begins, namely,

where is the mass dredged up from He-intershell at every pulse since TDU begins, and is the mass of the envelope. The cumulative effect of the various TDU, , have been given in Fig. 9a of Gallino et al. (1998).

(4) Overlap factor: The overlap factor rr, , is the ratio of the number of a kind of nuclide that is exposed to neutron in two successive pulses to the number exposed in the previous pulse, that is, the overlap fraction of the convective shells between the two successive pulses. Where is the increment of the core mass during the previous interpulse period, and is the mass of the convective inter-shell. The variation of rr with core mass was given by Iben (1977) and Renzini & Voli (1981). According to suggestion of Straniero et al. (1995), rr steadily decreases with increasing core mass. In this paper, we adopt the values given by Fig. 9c of Gallino et al. (1998).

(5) Neutron source: ¹³CO reaction is the major neutron source of s-process nucleosynthesis, which is released in radiative conditions at during the interpulse period, hence gives rise to an efficient s-processing that depends on the ¹³C profile in the ¹³C pocket, q layer. The layer is divided into three zones, in which the neutron exposures change with the various interpulse periods (Gallino et al. 1998, see their Fig. 6). Then the products of nucleosynthesis are ingested into the convective thermal pulse, and mixed with s-process material already from the previous pulses, and with the H-burning ashes from the below H shell. A second small neutron burst from the ²²Ne source operates during convective pulses due to the high neutron density and high temperature. The ²²Ne source to the total neutron irradiation is small, ranging from 0.002 mb^-1 at the 9th pulse to 0.05 mb^-1 in the latest pulses, where the average effective temperature is 23 keV.

(6) Light neutron poisons: Some isotopes can consume neutron produced by the neutron source isotopes, ¹³C or ²²Ne, through some reactions. They are named as neutron poisons. For example, the isotope ¹⁴N through its resonant channel ¹⁴NC, ²⁶Al in its long-lived ground state synthesized by the reaction ²⁵MgAl^g (Gallino et al. 1998 and references therein). However, in the previous convective pulses, ¹⁴N is destroyed in the reaction chain leading to ²²Ne production, and a large fraction of ²⁶Al has undergone substantial depletion by neutron capture. The above discussion makes it clear that compared to a convective burning scenario, the radiative s-process is much less affected by the filtering effect of light neutron poisons (Gallino et al. 1998). So in the radiative s-process nucleosynthesis calculation, we neglect the influence of these light neutron poisons (the slightly effects on nucleosynthesis calculation caused by this will be further discussed in Sect. 4.1).

(7) Value of C/O: At the beginning of the AGB phase, the C/O on the surface of a low mass stars is approximately 0.31, and lower than the initial (solar) value (approximately 0.45) because it has been modified by the first dredge-up process, which reduces the ¹²C abundance of the surface by about 30%. With the third dredge-up process, ¹²C of He-intershell is mixed into the stellar surface with the mixing of s-process material. So the C/O ratio on the surface increases gradually. In calculation, we adopt the C/O ratio from Table 4 of Straniero et al. (1997).

3.2. The heavy-element overabundances of barium stars

3.2.1. The angular momentum conservation model of wind accretion

For the binary system, the two components (an intrinsic AGB star, the present white dwarf, with mass , and a main sequence star, the present barium star, with mass ) rotating around the mass core C, so the total angular momentum is conservative in the mass core reference frame. If the two components exchange material through wind accretion, the angular momentum conservation of total system is showed by:

where µ is reduced mass, and r is the distance from to . , are the distances from , to the mass core C respectively. (=, where P=2 is orbital period) is angular velocity. v is an additional effective velocity defined through the angular momentum variation in the direction of orbital motion of component 2. The first term on the right side of the equal-sign is the angular momentum lost by the escaping material, and the second term is the additional angular momentum lost by the escaping material.

Using the similar method to that adopted by Huang (1956), Boffin & Jorissen (1988) and Theuns et al. (1996), considering the angular momentum conservation of total system and not neglecting the square and higher power terms of eccentricity, we can obtain the change equations of the orbital elements:

where A is the semi-major axis of the relative orbit of component 2 around 1, and e is the eccentricity (more details can be found in Appendix and Liu et al. 2000). Here, we take v=0 (Boffin & Jorissen 1988).

For the mass accreted by the barium star, we use the Bondi-Hoyle (hereafter B-H) accretion rate (Bondi & Hoyle 1944; Theuns et al. 1996; Jorissen et al. 1998):

where is a constant expressing the accretion efficiency. Theuns et al. (1996) indicated that taking between 0.5 and 1, as suggested by the numerical simulations of Ruffert & Arnett (1994), the actual accretion rate deduced from the smoothed particle hydrodynamics (hereafter SPH) simulation in the case was thus about 10 times smaller than that predicted by the B-H formula. Here we take =1 for the B-H formula. In fact, Boffin & Zacs (1994) suggested that the actual accretion rate is between 0.1 and 1 times of B-H rate. We take 0.15 times of the B-H rate as the actual accretion rate for our standard case. is the wind velocity, and is the orbital mean velocity. After fixing the initial conditions, for the mass , ejected at each pulse by the primary star, we can solve the Eqs. (4)-(6) for , the accreted mass by the secondary star. And then, the heavy-element abundances on the surface of barium stars can be calculated.

3.2.2. The heavy-element overabundances of barium stars

The calculation is completed by two separated steps. Firstly, adopting the theory of s-process nucleosynthesis and the latest TP-AGB model, we calculate the degree of the overabundances of the intrinsic AGB star (the present white dwarf) at each ejection. Then, combining the accreting matter predicted by the model of wind accretion on successive occasions and mixing, we calculate the heavy-element overabundances of the barium star. The overabundance factor for nuclide i on the secondary star, , is given by

where is the mass of the outer envelope of barium star, is the mass accreted by barium star during the period of the n-th ejection of the intrinsic star, and is the overabundance factor of the nuclide i of the intrinsic AGB star on that occasion. m is the total ejecting number undergone by the intrinsic AGB star. is the overabundance factor of nuclide i of barium star before the mass accretion. In the above formula we have assumed that the accreting matter has already completely mixed with the outer convective envelope of barium star.

We take as standard case: =3.0, =1.3, =15 and 0.15 times of the Bondi-Hoyle's accretion rate.

© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
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