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

3. Model predictions

We will now present the temporal evolution of some of the key predictions of our model, such as the supernova rate, the ionising flux, and the ²⁶Al and ⁶⁰Fe nucleosynthesis yields. As indicated earlier (Sect. 2.1) we consider two different star formation histories: a coeval population (instantaneous burst: IB) and a constant star formation rate (CSFR). In the present section an analytic description of the IMF is used. Realistic populations of OB associations and young open clusters, with a limited number of member stars, will be discussed in Sect. 5. We adopt a Salpeter IMF () over the interval 2-120 , variations of the IMF slope will be discussed in Sect. 3.6. Recall that in both the IB and CSFR cases our normalisation yields absolute quantities given per mass of stars formed (IB case), and star formation rate, /yr (CSFR case). All other predictions shown here, referring to relative quantities, are not affected by the adopted normalisation.

3.1. Supernova rates

The predicted supernova rates from our models are shown in Fig. 4. For the IB law, the supernova activity starts with a sharp peak around Myr which then soon turns over into a smoothly declining activity, situated around 1 SN per Gyr and . The peak is due to the fact that stars within the mass interval 60-120 all have about the same lifetime (3.97, 3.95, and 4.05 Myr for a 120, 85, and 60 star, respectively), hence the stars within this mass range explode at almost the same moment. The supernova activity ends around Myr, when all stars more massive than 8 have vanished.

Fig. 4. Predicted temporal evolution of the supernova rate for the IB (solid) and CSFR (dashed) star formation laws.

For the CSFR, the onset of the supernova activity is much more smooth, turning quickly into an almost constant rate of SN per Gyr and /Myr ⁴.

3.2. Ionising flux

The evolution of the ionising flux , defined as the number of photons emitted per second with wavelengths shorter than 912 Å, is shown in Fig. 5. After the onset of star formation decreases with age. In the case of an IB, the decline is very rapid, reducing during the first 12 Myr the ionising flux by a factor of . This comes from the fact that the bulk of ionising flux is provided by stars more massive than , which disappear within only a few million years after their formation. In the case of a CSFR new massive stars constantly replenish the loss of ionising photons from stars which disappear. Since the ionising flux increases strongly with mass, reaches equilibrium more rapidly than the SN rate.

Fig. 5. Predicted temporal evolution of the ionising flux for the IB (solid) and CSFR (dashed) star formation laws.

3.3. ²⁶Al and ⁶⁰Fe ejection rates

The ²⁶Al and ⁶⁰Fe ejection rates, and respectively, defined as the mass of radio-isotopes ejected per Myr, and normalised to the total mass converted into stars, are shown in Fig. 6 for IB models. Several ejection peaks are seen in the evolution of the ²⁶Al rates. The first peak between 2-3 Myr, which presents the maximum ejection rate during the evolution, comes from the onset of strong winds for the most massive stars in the population. The next peak at Myr is due to the almost simultaneous explosion of all stars in the mass interval 60-120 as type Ib/c supernovae (see Sect. 3.1 and Fig. 4). Around Myr type Ib/c supernovae start to dominate ²⁶Al production, until at 7 Myr the first type II supernovae begin to explode. The enhanced ²⁶Al production of type II SN with respect to type Ib/c leads then to the next peak between 7-8 Myr. From this age on, the ²⁶Al ejection rate is dominated by type II events. The time-dependence of the ejection rate then reflects the mass dependence of type II supernova yields (cf. Fig. 1) and the slow decline of the supernova rate (cf. Fig. 4).

Fig. 6. Temporal evolution of the predicted 26Al (left panel) and 60Fe (right panel) ejection rates for instantaneous burst models. The immediate total rate is shown by the solid line. Contributions from hydrostatic nucleosynthesis (dotted line) and from explosive nucleosynthesis (dashed line) are shown. The dashed-dotted line shows the emissivity as defined by Eq. (1) in the text.

For the ⁶⁰Fe ejection rate, a complex structure is found, where the first peak again reflects the burst of supernova explosions around Myr, and the remaining peaks directly reflect the dependency of the ⁶⁰Fe yield on initial stellar mass (cf. Fig. 2). Hence the temporal structure of the ⁶⁰Fe rate depends mainly on the details of the explosive nucleosynthesis models, and following the discussion about the uncertainties of these models (Sects. 2.3, 4), the structure should not be regarded as a physical prediction of our model. In particular, the broad bump between 13-18 Myr mainly reflects the single high-yield point in the WW95 models at , and modifications in the yields due to changes in the assumptions about the stellar physics in the nucleosynthesis models (e.g. Woosley & Heger 1999) can easily shift or remove this bump.

Emissivities (i.e. decay rates) , in units of per Myr, and normalised to the total mass converted into stars, have been obtained by integrating the ejection rates over the past history including the radioactive decay, using

where is the mean life of the radioactive isotope ( Myr for ²⁶Al and Myr for ⁶⁰Fe). As seen in Fig. 6 the emissivities are much smoother than the ejection rates, which is an obvious result of the convolution operation given by Eq. 1. For ²⁶Al, the emissivity rises sharply between 1-3 Myr, which is attributed to stellar wind ejecta from massive stars. From Myr on, the emissivity decreases continuously with a secondary maximum around 7-8 Myr due to the onset of type II supernovae mass ejections. The trend of decreasing ²⁶Al rate with increasing age should be a generic feature for ²⁶Al nucleosynthesis in massive star associations, at least for a Salpeter IMF, independently of the uncertainties in the nucleosynthesis models. The reason is that hydrostatic ²⁶Al production, which dominates ²⁶Al nucleosynthesis in WR stars and SN II, decreases generally with decreasing initial mass since the number of seed nuclei becomes smaller. Also, the SN rate, which defines the number of ejection events within a time interval, decreases with time (cf. Fig. 4). However, the level of the first maximum (i.e. the WR-peak) with respect to the second maximum (i.e. the SN-peak) may depend on details of the nucleosynthesis calculations. Also, the age at which the second SN-peak (due to SN II) occurs depends on the exact mass limit of WR star formation, assumed here to be 25 . E.g. lower values of should shift the SN II peak to older ages. A similar temporal behaviour of ²⁶Al is found in the models of Plüschke et al. (2000).

For ⁶⁰Fe, the emissivity is composed of two broad bumps (at and Myr) that are separated by a local minimum around Myr. After the second bump, the emissivity decreases smoothly, followed by the exponential decline after Myr. Overall, the ⁶⁰Fe production stays almost constant between 5-18 Myr, followed by a slow decline, and due to the uncertainties in the nucleosynthesis calculations, we should only retain this behaviour as characteristic. Given our detailed treatment of yields from SN II and SN Ib, a larger ⁶⁰Fe yield is obtained when compared to the models of Plüschke et al. (2000). This also affects the predicted ⁶⁰Fe/²⁶Al ratio.

3.4. ⁶⁰Fe/²⁶Al emissivity ratio

Part of the ²⁶Al and ⁶⁰Fe are co-produced in the same regions within type II supernovae (Timmes et al. 1995), hence the observation of gamma-ray lines from both isotopes can be used as powerful diagnostics tool of nucleosynthesis conditions in such events. For this reason, we investigate also the time-dependency of the ratio R between ⁶⁰Fe and ²⁶Al emissivities defined as . The predicted dependency is shown in Fig. 7 for the case of an instantaneous burst, and for a continuous star formation rate.

Fig. 7. 60Fe over 26Al emissivity ratio R for the IB (solid) and the CSFR model (dashed). The steady state ratio amounts to 0.6.

For the IB model, R rises steadily as function of time. This is due to the fact that ²⁶Al ejection decreases with increasing age, while the ⁶⁰Fe yields stay roughly constant during the active phase of the association. After the last supernova exploded, at an age around Myr, the accumulated yields decay exponentially, leading to an exponential rise in R with a time scale of

( Myr and Myr are the mean lifetimes of ²⁶Al and ⁶⁰Fe, respectively). Note that for ages younger than Myr, copious ²⁶Al production may appear in an association while no ⁶⁰Fe has been synthesised yet. This is due to the fact that no appreciable amounts of ⁶⁰Fe are supposed to be ejected by stellar winds, and ⁶⁰Fe appears only when the first supernovae begin to explode. Again, this should be a generic feature of the time-evolution of a coevally formed population that should be independent of details in the nucleosynthesis calculations.

For a constant star formation rate, R rises more smoothly and soon turns into its steady state value around 0.6. Our result is slightly in excess of the calculations of Timmes et al. (1995) who inferred a value of from a chemical evolution calculation for the Galaxy, assuming only SN II nucleosynthesis without mass-loss in the initial mass range 11-40   and taking the metallicity gradient of the Galaxy into account. Adopting similar assumptions ⁵ we obtain a ratio of , which is much closer to their findings. The main difference between the Timmes et al. (1995) and our work lies in the treatment of mass loss during the hydrostatic burning phases and its effect on the presupernova structure, which leads to a reduction of ²⁶Al by about 30%, while the ⁶⁰Fe nucleosynthesis remains almost the same.

3.5. Equivalent O7 V star yields

COMPTEL observations of the 1.809 MeV gamma-ray line suggest that the ²⁶Al flux is proportional to the number of ionising photons (Knödlseder et al. 1999). In order to express the proportionality factor in convenient units, Knödlseder (1999) introduced the "equivalent O7 V star ²⁶Al yield", , as the mass of ²⁶Al that is produced by a star of spectral type O7 V, assuming that such a star has an ionising flux of = 49.05 ph s^-1. This terminology follows closely the one employed for the analysis of starburst galaxies, where the strength of the ionising flux is often expressed in terms of equivalent stars of a given subtype (e.g. Vacca 1994). We extend here the definition also to ⁶⁰Fe, where in analogy is the mass of ⁶⁰Fe produced by an O7 V star.

The predicted equivalent O7 V star yields, calculated using

are shown in Fig. 8 for the IB and the CSFR model. Apparently, is an extremely sensitive age indicator for a coevally formed population (this would be also true for , yet the ⁶⁰Fe lines have not been detected so far).

Fig. 8. Equivalent O7 V star 26Al yield (left) and 60Fe yield (right) as predicted by the IB (solid) and the CSFR (dashed) model.

Within 10 Myr, varies by more than 3 orders of magnitude. This strong variation is mainly due to the rapid evolution of the ionising flux which drops considerably as soon as the most massive stars in the association vanished (see Sect. 3.2). In comparison, the most common age-indicator in H II regions, the H equivalent width, varies only within 3 orders of magnitude within 20 Myr (see Cerviño & Mas-Hesse 1994 for more details). A combination of radio observations providing ionising fluxes and 1.809 MeV gamma-ray observations should allow to obtain age estimates. In particular, although the strong time-variation is driven by the rapid drop in ionising flux, adding the gamma-ray observations provides a convenient normalisation, making the age estimate independent of distance, population richness, interstellar extinction, and IMF slope (see also Sect. 3.6).

The evolution of for the IB model can be split into four phases:

1. the stellar wind phase, lasting from the star formation burst up to 3 Myr, and which is characterised by a steep rise of ,

2. the type Ib/c supernova phase, from 3-7 Myr, showing a flattening in the slope, which comes from a slight decline in ²⁶Al production together with a rapid decline in the ionising flux,

3. the type II supernova phase, from 7-37 Myr, starting with a step around 7 Myr due to the most massive SN II, followed by a tail of positive slope since the ionising flux drops quicker than ²⁶ Al production, and

4. the decay phase, after 37 Myr, which is dominated by the exponential decay of ²⁶Al.

While this general picture should not depend on details of the nucleosynthesis and atmosphere models, the exact slopes and time intervals may well change for different input physics. In particular, phase 3 could already start after 5 Myr if the minimum mass required to form a Wolf-Rayet star would be as high as 40 . Fig. 8 also illustrates that the equivalent ²⁶Al star yield is an excellent discriminator between O star nucleosynthesis (i.e. hydrostatic nucleosynthesis in WR stars and explosive nucleosynthesis in type Ib/c SNe) and SN II nucleosynthesis. While phase 1 and 2 are characterised by low values, ranging from zero to , SN II phase is characterised by high equivalent yields well above . Due to the order of magnitude difference, it should be relatively easy to use to discriminate between both contributions.

Although the evolution of for the IB model follows roughly that of , we cannot easily identify distinct phases as for ²⁶Al. Due to the complex behaviour of the ⁶⁰Fe yields, a lot of structure is found in the evolution of the ⁶⁰Fe ejection rates, but no simple characteristic trend. Hence, to first order, the evolution of mainly reflects the fast decline of the ionising flux with increasing age.

The models calculated for a constant star formation rate allow us to predict steady state equivalent yields, which are reached after Myr (cf. Fig. 8). The resulting steady state values, split into contributions from individual source types, are given in Table 1. In particular, stellar wind contributions have been divided into yields ejected prior to (MS-winds) and during (WR-winds) the Wolf-Rayet phase, respectively. Apparently, of the hydrostatically produced ²⁶Al ejected by stellar winds comes from before the WR phase, while the rest is ejected when the Hydrogen envelope gets entirely lost in a Wolf-Rayet phase. As already pointed out by MAPP97 and Knödlseder (1999), stellar wind ejection from massive stars provide an important ( 42 %) contribution to the global ²⁶Al production. Type II supernovae contribute a similar amount, while the rest originates from SN Ib/c explosions. The exact repartition on the different source classes depends, of course, on the nucleosynthesis models, but also on the assumed mass limit for WR star formation, the slope of the IMF, and finally the metallicity (Knödlseder 1999).

Table 1. Steady-state predictions of the equivalent O7 V star yields.

Our model predicts a steady-state equivalent O7 V star ²⁶Al yield of , which is lower than the observed value integrated over the whole Galaxy of (Knödlseder 1999). In view of the uncertainties involved in the nucleosynthesis calculations, the similarity between model and observation is however encouraging. In addition, our models were calculated for solar metallicity only, whereas the gamma-ray observations average over the entire Galaxy, which shows an average metallicity of roughly twice the solar value (Prantzos & Diehl 1996). Higher metallicities potentially increase the ²⁶Al production by Wolf-Rayet stars, due to an increase in mass-loss and the amount of seed nuclei available for ²⁶Al synthesis (e.g. MAPP97). Hence, including metallicity effects in our calculations is expected to raise the estimate, bringing it even closer to the observed value.

Using the estimated galactic Lyman continuum luminosity of photons s^-1 (Bennett et al. 1994), the number of equivalent O7 V stars can be estimated to 31 194, and we can predict galactic nucleosynthesis yields from our CSFR model. A similar approach has been followed by Knödlseder (1999) using a time-independent steady-state model for the Galaxy. In Table 2 we compare his findings for solar metallicity and Salpeter IMF with mass limits 1-120 , to our model (Salpeter IMF with mass limits 2-120 ) and fitting the ionising flux to the observed value. Overall, the agreement between the models is quite satisfactory. Our models predict a total Galactic ⁶⁰Fe mass of 1.7 , which due to cancellation of various differences, turns out to be very similar to the value of Timmes & Woosley (1997).

Table 2. Galactic yield predictions assuming solar metallicity derived in this work, and given by Knödlseder (1999). The star formation rate (SFR) is quoted for the mass interval 1-120  in Knödlseder (1999) work.

3.6. Dependence on IMF slope

No general consensus exists about the slope of the IMF in young massive star associations and related objects (see reviews in Gilmore & Howell 1998). For example from an analysis of young open clusters and OB association in the Milky Way, Massey et al. (1995) derive an average slope of for stars with masses . For O stars within 2.5 kpc from the Sun Garmany et al. (1982) find . Based on NIR photometry of the massive Cyg OB2 association, Knödlseder (2000) found a comparable slope of . Finally, in his most recent revision Kroupa (2000), obtains for stars with masses , taking the scatter introduced by Poisson noise and the dynamical evolution of star clusters into account.

Throughout this work a Salpeter IMF slope () has been used for our "standard" models. The dependence of our results on are illustrated subsequently.

Fig. 9 shows the time-dependent ²⁶Al emissivity for assuming an IMF slope of , -1.35, and -2.0. All three curves have been normalised to the mass transformed into stars in the mass range 2-120 Obviously, the structure in the time-evolution remains similar, but the importance of stellar wind ejecta with respect to supernova ejecta depends strongly on . For the stellar-wind ²⁶Al emissivity peak (at Myr) is almost one magnitude larger than for the Salpeter law, leading to a burst-like lightcurve that is dominated by stellar wind products. In contrast, for the stellar-wind emissivity is of the same level as the type II supernova emissivity, leading to an almost 10 Myrs lasting plateau in the lightcurve.

Fig. 9. 26Al emissivity for three different IMF slopes.

Interestingly the emissivity ratio R and the equivalent O7 V star ²⁶Al yield depend very little on the IMF slope, as shown in Fig. 10. This is due to the fact that both the nucleosynthetic yield and the ionising flux show a similar dependence with initial mass. This finding indicates that the equivalent O7 V star ²⁶Al yield should be fairly reliable age indicator for young massive star associations. From Fig. 10 we estimate a typical age uncertainty of Myr due to IMF variations, which is of the same order as typical uncertainties obtained for massive star associations by isochrone fitting. Also, the IMF variations are smaller than the dispersion introduced by statistical fluctuations in a finite sample, as we will demonstrate for realistic populations in Sect. 5 (cf. Fig. 11 right panel).

Fig. 10. emissivity ratio R (left) and equivalent O7 V star 26Al yield (right) for three different IMF slopes.

Fig. 11. Time-dependent probability density functions for the 26Al emissivity (left) and (right). A logarithmic greyscale was chosen to display also the wings of the PDFs.

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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