Astron. Astrophys. 319, 511-514 (1997)

2. Computational details

Theoretical populations of stars were derived using the technique of population simulation. Stars are selected according to the probability that at present they are in the appropriate stage. Briefly, a large number of single and binary star models were generated for constant stellar birth rate history and Salpeter initial mass function. The procedure is explained in more detail by Frantsman (1992). Here only some aspects will be considered.

Most stellar characteristics in the AGB phase can be approximated by analytic formulae derived from precise stellar model calculations. The evolution of stars in the TP-AGB phase was followed using the formulas by Iben and Truran (1978) and Renzini and Voli (1981), and on the E-AGB phase - by Iben and Renzini (1984). It was assumed that at each thermal pulse, an amount of mass is added to the envelope, were is the core mass growth during the preceeding interpulse period and is the dredge-up efficiency. In accordance with Iben's (1983) results, the value 0.1 for the coefficient was adopted for stars with low-mass cores.

Special attention must be paid to the problem of mass loss by stars on the AGB. In our calculations the mass loss was represented by Reimers (1975) law: where denote the star's luminosity, radius and mass in solar units, and is a dimensionless quantity usually taken to be of the order of unity. The observations, however, suggest that apart from the conventional stellar wind, some other mechanism (apart from planetary nebulae ejection) also operates during the AGB phase, substantially increasing the mass loss (Frantsman 1986, 1988, 1989). In our calculations, an abrupt tenfold jump in the mass loss rate for the AGB stars reaching a certain luminosity is suggested ( for and =10 for ), because a quantitative agreement of theoretical investigations with observations would require a rapid mass loss with increasing luminosity on the AGB. This assumption is somewhat supported by IRAS observations of the LMC (Reid et al. 1990). According to their interpretation, the observations suggest that the AGB stars pass through two stages with different mass loss intensity, and that the mass loss may be enhanced for AGB stars with -5:m5.

The analytical expressions to trace the evolution of stars were taken from Iben and Truran (1978) and Renzini and Voli (1981), instead of more recent papers, as for example Groenewegen and de Jong (1993), because, in my opinion, the former better coincide with observations. Fig. 1 represents the initial-final mass relations in accordance with the results of Table 5 of Groenewegen and de Jong (1993), and with ours, under the assumptions mentioned above. The values for the most luminous AGB stars in Magellanic Cloud (MC) clusters are also shown. The carbon-oxygen core mass was taken to be the final mass, as this will be very close to the mass of the remnant white dwarf. These masses were obtained from the AGB star luminosities according to the relations from Boothroyd and Sackmann (1988) and Groenewegen and de Jong (1993). The initial masses come from cluster ages obtained by classical methods; the paper by Frantsman (1988) reviews the references, whereas the age of cluster NGC 1718 is given by Elson and Fall (1988). The relation used between the age and initial mass was from Iben and Laughlin (1989). Our calculations better fit the observations than the calculations by Groenewegen and de Jong (1993), where the final masses for most stars are too large. The same conclusion can be made considering the age-luminosity relation for the most luminous TP-AGB stars in the MC clusters (Fig. 2).

Fig. 1. The initial-final mass lelations. The solid line indicates our results of AGB evolution calculations based on the expressions from Iben and Truran (1978) and Renzini and Voli (1981), and assuming the coefficient in Reimers (1975) mass loss intensity law as for and for . The dashed line indicates the results of the calculations of Groenewegen and de Jong (1993). Dots represent the initial-final mass relations for stars in the Magellanic Cloud clusters (see text for details)

Fig. 2. The age-luminosity relations for the upper parts of the AGB. The difference between solid and dashed lines is described in Fig. 1. Dots represent the results for the Magellanic Cloud clusters: t - the age of the cluster, - luminosity of the most luminous star in the cluster

It was assumed that 50% of the matter which is lost from the Roche-lobe-filling component is transferred to its companion and that 50% of all stars are close binaries. A large number of single and binary star models were generated, and for every system the evolution of the components was traced (as well as for every single star). It was assumed that both components of binary system were formed simultaneously and had a Salpeter initial mass distribution. In the course of evolution (depending on the semi-major axes), in some systems a primary component fills the Roche lobe being in the TP-AGB stage, but the secondary at the same time can be a main sequence star, subgiant, giant or E-AGB star. The distribution of binary stars over the primordial semimajor axes A was taken as , and the distribution over the mass ratio ( and are masses of the primary and secondary components) was taken as for 0.1 1 (Popova et al. 1982). The computational time step was taken as 10⁴ yr.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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