Astron. Astrophys. 363, 970-983 (2000)

2. Evolutionary synthesis model

2.1. Method

The starting point for our modelling effort is the evolutionary synthesis code of Cerviño et al. (2000a), which predicts the time-dependent multi-wavelength energy distribution of a population of discrete stars from radio wavelengths up to the X-ray domain. For this work, we have updated the atmosphere models for massive stars using the CoStar models (Schaerer & de Koter 1997), and we included atmosphere models for the Wolf-Rayet (WR) phase from Schmutz et al. (1992) following the prescriptions of Schaerer & Vacca (1998). In order to predict gamma-ray luminosities, we have included chemical yields for the radioactive isotopes ²⁶Al and ⁶⁰Fe that may either be produced during hydrostatic nucleosynthesis in the interiors of massive stars, or during explosive nucleosynthesis in supernova explosions (cf. Sect. 2.3).

The calculations have been done for two different star formation laws in order to explore the extreme cases of an instantaneous burst (IB) and of a constant star formation rate (CSFR). For the IB model, an initial population of coevally formed stars has been created based on a Monte Carlo method. Using a power-law initial mass function (IMF) of slope as probability density function ¹, we randomly created initial stellar masses within the interval 2-120 until the total number of stars reaches a predefined limit. The evolution of each star is then calculated using the Geneva evolutionary tracks (see below) in time steps of years up to an age of typically 50 Myr. At each time step the spectral energy distribution and the ejected ²⁶Al and ⁶⁰Fe yields are computed for each individual star. The evolution of spectral types is also followed in order to predict the number of O and WR stars in the population. Stars that end their lives during a time step are counted as supernova explosions (as far as they are more massive than 8 ), and are removed from the population for the next time step. Summing the contributions from all individual star results then in predictions for the entire population.

2.2. Evolutionary tracks

The evolution of the stars in the population is followed using the non-rotating stellar tracks from Meynet et al. (1997) (hereafter MAPP97) for stars with initial mass , Meynet et al. (1994) for , and Schaller et al. (1992) otherwise. Solar metallicity tracks are used for this work since we are interested in predictions for star clusters and OB associations in the solar neighbourhood, but in future we plan to extend the calculations also to other metallicities. The possible alterations when rotation is taken into account in the stellar models are briefly discussed in Sect. 4. Rotating stellar models will be included when complete tracks covering all relevant evolutionary phases will become available.

The stellar tracks we used have been calculated for enhanced mass loss during the massive star evolution until the end of the WNL phase. This prescription leads to an improved agreement between predictions and several observed WR properties; in particular, these models can account for the variation of the number ratio of WR to O type stars as a function of the metallicity in zones of constant star formation rate (Maeder & Meynet 1994). The relative populations of WN and WC stars observed in young starburst regions are also better reproduced when models with high mass loss rates are used (Meynet 1995; Schaerer et al 1999).

The models we are using predict a lower initial mass limit of 25 for the formation of WR stars. Uncertainties due to this mass limit will be discussed in Sect. 4. In order to avoid numerical inconsistencies and an unrealistic behaviour of interpolated tracks around the WR mass limit, we have constructed an artificial track at 25.01 going through the WR phases. This track together with the published 25 track, which does not enter the WR phase, allows smooth interpolations in this mass range.

2.3. Chemical yields

The prediction of chemical yields for massive stars depends critically on the assumed stellar physics, such as the treatment of convection, rotation, mass-loss, and the final supernova explosion. Being aware of these uncertainties, we do not intend to predict the yields of ²⁶Al and ⁶⁰Fe to better than within a factor of 2 or so (e.g. Prantzos & Diehl 1996; Woosley & Heger 1999), but we merely want to identify the main characteristics of nucleosynthesis in a massive star population. However, by comparing the models to real massive star populations, one can inverse the problem and try to learn something about massive star nucleosynthesis, and possibly better constrain the theoretical models. In this sense, the employed nucleosynthesis yields can be seen as a hypothesis, which can be verified by comparison to observations.

Two different sites of nucleosynthesis must be taken into account for the prediction of ²⁶Al and ⁶⁰Fe yields from a massive star association. First, the H-burning in the core of the stars may produce appreciable amounts of ²⁶Al that may appear at the stellar surface as an effect of both internal mixing and removal of the external layers by stellar winds. This ²⁶Al can then be ejected into the interstellar medium by the stellar winds. Second, when the star explodes in a supernova event, both ²⁶Al produced during the post H-burning phases and that synthesised at the time of the explosion are then expulsed into the interstellar medium.

To obtain the yields from the population synthesis code, one needs a series of different initial mass stellar models computed with the same physical ingredients. Since the Geneva tracks stop before the presupernova stage and give no predictions concerning the explosive nucleosynthesis, we must complement these data with the yields of ²⁶Al and ⁶⁰Fe ejected during the supernova event.

2.3.1. Stellar winds

Recent calculations of hydrostatic ²⁶Al nucleosynthesis in mass losing stars have been undertaken by Langer et al. (1995) and MAPP97 for non-rotating single massive stars. Despite the different assumptions that both groups made about mass loss and convection, both calculations lead to comparable results (MAPP97). We here use the MAPP97 calculations, which have the advantage of being consistent with the adopted evolutionary tracks which have extensively been compared to observations (see Sect. 2.2). The dependence of the total mass of ²⁶Al ejected by WR stellar winds as a function of the initial mass is shown in the left panel of Fig. 1. We have assumed no ²⁶Al ejection by stellar winds for the 20  stellar model. Log-linear interpolation is used to determine the yields at intermediate masses.

Fig. 1. 26Al yields used in this work. Left panel: 26Al yields as function of initial stellar mass for Wolf-Rayet stars (Meynet et al. 1997, MAPP97, triangles) and type II supernovae (Woosley & Weaver 1995, WW95: diamonds). The inset shows 26Al yields for Helium stars as function of the initial He mass (Woosley et al. 1995, WLW95). Right panel: 26Al yields as function of initial stellar mass after combining the nucleosynthesis models with the evolutionary tracks (see text). For type Ib/c supernovae, 3 link parameters (: thin solid line, : dashed-dotted, : dashed) have been explored, which lead to comparable 26Al yields. The resulting total yield, obtained using , is shown by the thick solid line

Arnould et al. (1997) suggest that some amount of ⁶⁰Fe could be ejected by WR stellar winds. Typically, for a 60 star they find of ⁶⁰Fe ejected during the WR phase, which is well below the quantity of ⁶⁰Fe expelled in the final SN explosion (cf. Fig. 2). The stellar wind contribution of ⁶⁰Fe can therefore safely be neglected for our purposes.

Fig. 2. 60Fe yields used in this work. Left panel: 60Fe yields as function of initial stellar mass for type II supernovae (Woosley & Weaver 1995, WW95). The inset shows 60Fe yields for Helium stars as function of the initial He mass (Woosley, Langer & Weaver 1995, WLW95). Right panel: 60Fe yields as function of initial stellar mass after combining the nucleosynthesis models with the evolutionary tracks (see text). Same symbols as in Fig. 1. For , the final yield corresponds to the yield using as a linking parameter

2.3.2. Type II supernova nucleosynthesis

At solar metallicity, stars in the mass range will end their lives by core collapse, giving rise to type II supernova (SN II) explosions. Explosive ²⁶Al and ⁶⁰Fe nucleosynthesis during these events has been calculated by Woosley & Weaver (1995; hereafter WW95), Thielemann et al. (1996), and Limongi et al. (2000). Their models differ in the treatment of the pre-supernova evolution, the prescription of convection, the employed nuclear reaction networks, and the assumed explosion mechanism. For example, while WW95 included hydrostatic ²⁶Al production in the pre-supernova phase in their models, Thielemann et al. (1996) only predict explosive nucleosynthesis yields in their models. Additionally, WW95 added neutrino driven spallation (the so called -process) in their reaction network while the others ignore this channel (the -process may enhance ²⁶Al production due to additional release of protons).

Of all these models, WW95 predict the highest ²⁶Al yields; for example Thielemann et al. (1996) obtain yields that are almost a factor of 10 lower. We here adopt the WW95 yields for ²⁶Al and ⁶⁰Fe, whose dependence on the initial stellar mass are shown in the left panels of Fig. 1 and Fig. 2 respectively. Note that the WW95 yields do not reach down to 8 , the assumed initial mass limit for SN II. We assume no ²⁶Al production for 7  and perform a log-linear interpolation in the mass interval between 8 and 11 . Note that in the case of ⁶⁰Fe the choice of the inferior mass limit for stars undergoing core-collapse has an important impact on the results since the stars in this mass range may have a considerable contribution.

In order to assign a supernova model (calculated from stellar models neglecting mass loss and with different limits of the convective cores) to the adopted stellar models, we use , the mass of the Carbon-Oxygen core at the end of C-burning, as linking parameter, as suggested by Maeder (1992). This procedure is based on the hypothesis that the relation between and the explosive nucleosynthetic yields does not much depend on the particular set of stellar models. In particular for our case this should be a reasonable assumption since at the time of the supernova explosion the main regions of ²⁶Al and ⁶⁰Fe production are inside the CO core (cf. WW95). from the evolutionary tracks was estimated from the fraction of the convective core before the end of He burning ². Although, for the tracks in common with Maeder (1992), the derived values are somewhat lower than the ones tabulated by Maeder (1992), the yields are only slightly modified. For the WW95 models we use the values from Portinari et al. (1998) calculated by subtracting the amount of hydrogen and helium in the WW95 tables from the initial mass. The resulting ²⁶Al and ⁶⁰Fe yields are shown in the right panels of Fig. 1 and Fig. 2,

2.3.3. Type Ib/c supernova nucleosynthesis

For stars that go through the Wolf-Rayet phase, the above approach of estimating the explosive nucleosynthesis yields is not longer valid. The mass-loss will considerably modify the structure of the star prior to explosion, leading eventually to a type Ib or type Ic supernova explosion at the end of its lifetime. After the evaporation of the hydrogen envelope, such an object may closely resemble a Helium star.

The only computation of explosive nucleosynthesis yields for ²⁶ Al and ⁶⁰Fe in such events comes from Woosley, Langer & Weaver (1995; hereafter WLW95) who calculated the explosion of mass losing Helium stars of initial He masses between 4-20 . Again, the models of WLW95 have to be connected to the evolutionary tracks in our evolutionary synthesis model. In principle there are three different ways for such a link to be done. First, we could use the final mass of the MAPP97 evolutionary tracks and link them to the final masses of the WLW95 models. Second, we could use the mass of the He core at the beginning of core He burning and connect them with the initial masses of the WLW95 models. And third, as for SN of type II, could be used.

The three possible link parameters , and derived from the evolutionary tracks are shown as a function of the initial stellar mass in Fig. 3. Since and are not all directly available from the tracks they were estimated from the mass fraction of the convective core close to the beginning and end of He-burning respectively ³. The values estimated in this manner are found to be 20-40 % lower than from the stellar structure models (Foellmi 1997). As shown by Fig. 3 not all mass ranges overlap with available SN Ib models of WLW95. Some link parameters are therefore of limited practical use.

Fig. 3. , , and mass versus initial stellar mass as determined from the evolutionary tracks. The mass ranges in these three parameters covered by the nucleosynthesis models of WLW95 are shown as vertical lines in the left.

The ²⁶Al and ⁶⁰Fe yield resulting from the use of the different link parameters (excluding extrapolations outside the range covered by the WLW95 models) is shown in the right panels of Fig. 1 and Fig. 2 (values at 25 ). Given the physical and numerical uncertainties in the link between the hydrostatic stellar models and the SN Ib calculations the variations are considered to be small. For masses 70 , the link using provides a somewhat lower ²⁶Al yield than using . However, since these stars are relatively rare and since stellar wind ejection exceeds the SN Ib production by up to an order of magnitude, the precise explosive yield in this domain is not crucial. For consistency with the type II supernova yields, will thus be used for all tracks as the link parameter in the remainder of this work.

Note also that Knödlseder (1999), in his estimation of the global galactic ²⁶Al production rate, used a mass independent, constant SN Ib/c yield of . This prescription is in good agreement with our more refined treatment, which predicts yields between .

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
helpdesk@link.springer.de