Astron. Astrophys. 319, 487-497 (1997)

2. Luminosity functions: calculated and observed

To compare theoretical luminosity functions with the observed ones of different globular clusters we need isochrones for various ages, chemical compositions and initial mass functions (IMF). With these quantities we obtain the number of stars in any given interval of visual magnitude by

For the IMF we use the classical power law form:

with m being the stellar mass. To find the best agreement with observational results we tried to fit the data of each cluster with several luminosity functions for different ages, metallicities and exponents for the power law of the IMF (we recall that the IMF affects the MS only); these parameters are either taken from the literature (metallicity, distance modulus) or, in the case of cluster age, have been determined with our own isochrones. The best fit for each cluster will be shown in the next section. All LFs are normalized to the total number of stars on the RGB. We have checked that the normalization does not depend on the brightness range used. As will be demonstrated in Sect. 4, the RGB-part of the LF is not at all influenced by any assumption we have tested, including that of an isothermal main-sequence core. In contrast, the main-sequence part depends strongly on the IMF and - though less - on the helium content. It is therefore natural to prefer the RGB for the normalization of the LFs. Since the distance modulus is not known exactly, the LFs might be shifted by up to 01, which certainly is within the errors. This way, no unique "best fit" is possible, but a small uncertainty in metallicity or age remains, because small changes in these quantities do not change the shape of the LF.

The helium mass fractions of all isochrones is or . This small difference does not influence the LFs (Sect. 4.3 and Ratcliff 1987) The range of age and metallicity covered by the adopted isochrones are 10 20 Gyr and , resp. We only used solar metal ratios within Z. Salaris et al. (1993) have demonstrated that for metal poor stars only the total or global metallicity is important for evolution, isochrones and therefore LFs. This has been confirmed by Salaris et al. (1996). We thus can savely ignore -element metal ratios, as long as the global or total metallicity is correct.

All evolutionary calculations have been made with the Frascati Raphson Newton Evolutionary Code (FRANEC) whose general features and physical inputs have already been described in previous papers (see e.g. Chieffi & Straniero 1989). For all metallicities except we adopted the isochrones of Chieffi & Straniero (1989) and Straniero and Chieffi (1991) with radiative opacity coefficients from the Los Alamos opacity library (Huebner et al. 1977; Ross & Aller 1976 solar metal ratios), combined with the Cox & Tabor (1976) opacities in the low temperature region (below ). For we built isochrones with the latest OPAL opacity tables (Rogers, private communication, and Rogers & Iglesias 1992; Grevesse & Noels 1993 solar metal ratios) combined with the molecular opacities of Alexander & Fergusson (1994). The different choices for the opacity tables is not relevant because, as we will discuss in Sect. 4, the LFs are almost completely unaffected by the adopted opacity coefficients. For the equation of state (EOS) we considered two separate regions: an high-temperature region (T 10⁶ K), where matter can be assumed to be completely ionized and where we adopted the EOS of Straniero (1988) and a low temperature region (T 10⁶ K) where partial ionization takes place. In this last region the thermodynamical properties of partially ionized matter are derived from the Saha equation as described in Chieffi & Straniero (1989); the pressure ionization is included according to the method described by Ratcliff (1987). The colour transformation of Kurucz (1992) was used to transform from the theoretical temperatures and luminosities to colours and visual brightness.

For a comprehensive discussion about the comparison between theoretical and observational luminosity functions, we refer to Ratcliff (1987) and references therein. Here we just wish to note that, despite of some evident advantages of LFs such as the fact that they are almost independent of the unknown details of model envelope structure, the most constraining disadvantage of this method is the requirement of a complete count of stars down to very faint magnitudes (). This is the main reason why luminosity functions are not used very frequently. During the last years, the situation has been improved with the availabily of CCDs and related software packages; however, it still is not possible to claim that the main problems have been solved completely.

The difficulties in building observational luminosity functions include: the problem of lack of completeness at low magnitudes (even if modern techniques are available to make a quantitative evaluation of the completeness, see e.g. Bolte 1989), the proper normalization between various data sets to build the total luminosity function of a cluster, the removal of background and foreground objects, crowding, and possible systematic errors which could occur during the process of data reduction. It is also important to mention statistical noise: to find all possible features in the subgiant region one needs bins as narrow as 02 with a sufficiently large number of stars such that the stochastic variations become smaller than about 10% (see e.g. Chieffi & Gratton 1986). At present, at least to our knowledge, there are few cluster data available which fulfill these requirements. Usually observational data are presented with the statistical error only, but the real errors could be higher. In all cases we are using data already prepared for LFs, i.e. we use the number of stars in brightness bins, where some corrections for completeness had been applied by the observers.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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