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

5. Realistic populations

The model predictions presented in Sect. 3 have been calculated for an infinitely rich association of stars in order to avoid effects due to small number statistics, in particular at the high-mass end of the IMF. In reality, however, galactic associations or open clusters have total stellar masses between a few 100-1000 (Bruch & Sanders 1983), limiting the number of O stars, i.e. stars with initial masses above , to only a few objects. However, since these massive stars provide a considerable fraction of the nucleosynthesis yields and ionising power of the association, the early evolution of the system will depend rather sensitively on the actual distribution of stellar masses. In the following we will describe a Bayesian method that allows to quantify the uncertainties introduced by a finite sample in our predictions.

5.1. Probability density functions

Suppose we want to predict the gamma-ray luminosity and ionising flux from an association of age Myr, which today contains 100 stars within an initial mass range of 8-25 . At 5 Myr, stars above already exploded as supernovae, but the population within 8-25 should still be relatively unaffected by evolution. A possible initial population may then be estimated by randomly selecting stellar masses from a power-law IMF of slope until the number of stars with masses between 8-25 amounts to 100 (cf. Sect. 2.1). This population is then statistically identical to the initial population of the observed association, although it may differ in the exact distribution of stellar masses. Repeating the sampling provides several possible initial realisations, and the statistical variations in the number and distribution of stars among these samples reflects then the ignorance about the precise initial conditions in the association.

The calculation of evolutionary synthesis models for each of the samples provides then time-dependent model predictions, and the variations among the predictions reflect the uncertainty about the precise distribution of the initial stellar population. If the number of random samples and corresponding evolution synthesis models is sufficiently high the probability of observing at an age t the value x for quantity X can be reasonably well approximated by

where is the number of samples for which X was comprised between x and , and is the total number of samples. is called the probability density function (PDF) of X at age t.

For illustration, time-dependent probability density functions for the ²⁶Al emissivity and are shown for our example association in Fig. 11. The PDFs have been computed from 1000 Monte Carlo samples assuming an IMF slope of . During the early evolution ( Myrs), both quantities are subject to considerable uncertainties which can be understood as the combined effect of comparable lifetime, ²⁶Al yield mass dependence, and small number statistics at the high-mass end. As mentioned earlier, stars with initial mass between 60-120 evolve on comparable timescales, providing contemporaneous ²⁶Al ejection in this mass range. ²⁶ Al yields, however, vary by almost one magnitude within this mass range, making the resulting ²⁶Al ejection rates crucially dependent on the particular spectrum of initial masses. Consequently, the large statistical uncertainties in the mass spectrum at the high-mass end translates into a large uncertainty in the ²⁶Al emissivity and during the early evolution of the population.

During the subsequent evolution, the relative uncertainty in the derived quantities is roughly constant, with a minimum around 5-10 Myr followed by a slight rise of the uncertainty with time. To understand this behaviour, note that the relative uncertainty in the number n of stars that contribute to ²⁶Al production within a time step is given by , hence the uncertainty increases with decreasing n. Indeed, for ages Myr the ²⁶Al production is only due to SN explosions, and the slight decrease in the SN rate with time (Fig. 4) is at the origin of the uncertainty increase. Note that this feature depends on the slope that is chosen for the stellar population: with a steeper IMF the decline in the supernova rate would have been reduced (or may even turn into an increase), and consequently the uncertainty increase would become negligible (or might even turn into a decrease of the uncertainties).

5.2. Inclusion of uncertainties

In the above example we assumed that the number of stars within a given mass interval has been determined precisely, and that the slope of the association is known. Both assumptions may not be valid in a realistic case. The determination of the association richness is certainly subject to some uncertainty, due to membership ambiguities, stellar confusion, or invisible members hidden by interstellar obscuration.

Uncertainties in both quantities can be easily included in the analysis by computing the conditional probability density function for a sufficiently fine grid of stellar richness n and IMF slope . The time-dependent association PDF is then calculated using

where and are prior probability density functions quantifying the uncertainties in n and  ⁶. Typically, the prior PDF could be a Gaussian if the uncertainty is symmetric around a mean value, or a bounded constant if the uncertainty is specified as lower and upper limits.

5.3. Inclusion of prior knowledge

Finally, to predict the characteristics of an association, the information about its actual age and its distance should be included in the analysis. In the above example, the age was estimated to Myr, which again can be expressed by a prior PDF . Marginalisation leads then to an age independent estimate

Eq. 6 can be seen as a smoothing operation applied to the time-dependent PDF over a limited age-window, specified by the width of the prior PDF. Since our models are calculated for instantaneous star formation, Eq. 6 can also be interpreted as extending the star formation to a period which is defined by the width of the prior PDF. In this case, is the star formation law, and the age uncertainty is interpreted as an uncertainty in the star formation history.

Similarly, if the distance s of the association is known (to some uncertainty - of course), yields and ionising flux can be converted to gamma-ray and radio fluxes, using

where is the prior PDF, and is the distance dependent flux PDF.

Thus, the final outcome of the evolutionary synthesis model is not a single value, but a probability density function, also called the posterior probability density function, that reflects all uncertainties related to the investigated association. Two examples of such posterior PDFs are shown in Fig. 12 for the 1.809 MeV gamma-ray flux and the equivalent O7 V star ²⁶Al yield . The first example (solid lines) has been obtained by assuming an age uncertainty of Myr, expressed by a Gaussian prior PDF with mean of 5 Myr and standard deviation of 1 Myr. The second example (dashed lines) has been derived for an age uncertainty of 5-15 Myr, described by a constant bounded prior PDF which is non-zero for the interval 5-15 Myr. To obtain the 1.809 MeV flux from the ²⁶Al emissivity a distance of kpc has been assumed, implemented as a prior PDF of Gaussian shape with mean of 1 kpc and standard deviation of 0.2 kpc. Note that is not subject to any distance uncertainty since it is defined as a flux (or yield) ratio for which the actual distance cancels out.

Fig. 12. Posterior PDFs for the 1.809 MeV gamma-ray flux from the decay of radioactive 26Al (left) and the equivalent O7 V star 26Al yield (right). The solid lines show the results for an age of Myr, the dashed line has been calculated for an age uncertainty of 5-15 Myr. A distance of kpc has been assumed for the association.

In many cases, the resulting posterior PDF has approximately a Gaussian shape, but the example of using an age uncertainty of 5-15 Myr illustrates that the distribution may be much more complex. The posterior PDFs may then be used to address various questions about the investigated association. For example, the probability P that the 1.809 MeV flux is comprised in the interval is simply derived by integrating

Or, inverting the problem, one may derive the flux interval that contains of the probability distribution by solving

and

(this definition is similar to the errors often quoted in classical statistics).

The posterior PDFs may also be used to make predictions about the detectibility of an association by a gamma-ray instrument or radio telescope. If the instrument sensitivity is given as smallest detectible flux limit , the probability of detecting the association with the instrument is given by

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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