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4 Discussion

4.1 The RGB and other caveats

The examples shown thus far are highly idealized cases. The colours of the stars from a synthetic populations are as reliable as the colours of the isochrones. Recent analysis of globular clusters (see Reid 1997 or Gratton et al. 1997 and references cited in those papers) indicate that a good fit for the main sequence stars does not necessarily imply a good agreement with the stars on the red giant branch. The discrepancy partly originates from the uncertainties in the colour transformations for the late type stars from the theoretical to the observational plane. Another cause is likely related to the mixing length parameter used in the calculations of the evolutionary tracks. One therefore has to be cautious in the analysis of selected regions in the CMDs with old stellar populations.

A nice aspect of the diagnostic diagram is that the residuals clearly indicate how large the deviations are. In addition, uncertainties in the treatment of particular evolutionary phases might show up in the diagnostics. However, this can only be properly evaluated through a massive study, where one searches for systematic clumping of the residuals in particular regions of the CMDs. The identification of systematic clumps can then be used to improve the description of a particular evolutionary phase in the computation of new evolutionary tracks. In contrast to Dolphin (1997) it is argued that one should avoid the introduction of factors to reduce the weight of these particular phases in the fitting procedure.

All these caveats are however not related to the general validity of the method presented in Sect. 2. They will depend on the actual implementation of an automated fitting method and they will become important when a comparison with real data is made. However, a thorough discussion of problems associated with the implementation of an automated fitting method or a comparison with real data is beyond the scope of this paper.

4.2 Combining the diagnostics

It will be quite rare that one is going to deal with one of the idealized diagnostic diagrams from Figs. 3a-i. More likely the resulting diagnostic diagram is a combination of these diagrams, indicating that a number of parameters ought to be modified. One has to remain careful, because some effects might partly cancel each other out, like for example age and distance. A different distance for the stellar aggregate induces a change of the best-fitting age of the stellar population (Gratton et al. 1997). However, in a CMD the distribution of the stars is not exactly the same for populations with a different age. The subtle differences might not cancel out through variation of the distance. The resulting diagnostic diagram might therefore indicate that yet another parameter ought to be optimized - such as the star formation rate - and in the end indicate that an acceptable fit has been obtained, while in reality one is dealing with an artifact. However, one of the major problems in (V, V-I) CMDs remains the similarity in the behaviour of the extinction, small age differences, metallicity and the star formation rate. The results therefore might not always be as reliable as they are presented. A heuristic search for the optimum fit obtained with a genetic algorithm might properly disentangle the information for these parameters, but it would be more convenient to avoid this degeneracy and use photometry from additional passbands in which this degeneracy does not occur.

4.3 Unmatched evolutionary phases

In general, one will not always find for large amplitude variable stars a synthetic counterpart within the error ellipse. Those stars give rise to a small bias in the Poisson merit function. But, the total number of large amplitude, variable stars in any field is expected to be considerably smaller than $\sqrt{N}$. Therefore, one can ignore in first approximation any bias in the results due to variable stars.

Some fast evolutionary phases are not necessarily well described by theory or even not well covered by the small number of stars observed. This may lead to the presence of systematic clumps in the CMDs of the residuals from a massive study. The information obtained from these clumps can be used to compute new tracks and isochrones. In many cases however the number of stars present in these clumps is expected to be smaller than $\sqrt{N}$. It is therefore expected that unmatched evolutionary phases in general will not affect significantly the search for an optimum fit to the data.

As an aside, Gallart (1998) demonstrates that models are quite capable to predict subtle details in the observations, despite the fact that some evolutionary phases are not fully understood.

4.4 Galactic structure

In galactic structure studies the stars are distributed along the line of sight. The diagnostics procedure outlined here is also useful for these type of studies, in which the observed stars are a complex mixture of different stellar populations. Instead of applying a tedious scheme to deconvolve this mixture in its individual components, it is more liable to construct a synthetic mixture and compare this directly with the observations. The diagnostics will provide in the first place information about the galactic structure along the line of sight. Once this has been established, one can explore in more detail the initial mass function and the star formation history of the different stellar populations. It should be possible to obtain some feedback for the input stellar library with an improved calibration of the galactic model, and to obtain on the long run in a self-consistent way indications about the adopted solar abundance partition or enrichment law $\Delta {Y}/\Delta {Z}$ (see Chiosi 1996 and references cited therein).

4.5 Open & globular clusters and the Local Group

In the studies of open clusters, metal-rich globular cluster and galaxies with resolved stellar populations from the Local Group a considerable amount of fore- and background stars can be present. It is not easy to take this contribution into account, because it is sometimes not clear if a particular feature is due to stellar evolution and intrinsic to the aggregate[*]. One can clean in a statistical sense the galactic contribution from a neighbouring field. But this is only possible if the extinction and photometric errors are comparable. Ng et al. (1996c,d) used a galactic model to account for the contribution of the fore- and background stars. An unambiguous determination of the age was hampered by the large metallicity range and partly by the estimated amount of differential extinction. In this or other cases the diagnostics scheme as provided in this paper might contribute to a significantly deeper analysis.

4.6 The stellar luminosity function

Figure 4 displays the luminosity function of the original population, together with another realization of this population (case a), a population with a modification in the extinction (case c) and age (case f), and one for which the IMF slope was modified (case h). One can easily verify in Fig. 4 that the differences between the various populations for the majority of the magnitude bins are relatively small with respect to the generally adopted Poisson error bars. Only case f is significantly different, due to the large age difference adopted.

\psfig {,height=6.35cm,width=8.5cm}
}}\end{figure} Figure 4: Luminosity function for the cases a, c, f and h (Table 1). The open circles are obtained from the original input data set. The uncertainty per $0\hbox{$.\!\!^{\rm m}$}2$ bin displayed are Poisson error bars
For a large number of bins (models c and h) the number of stars is not exactly within the 1$\sigma$ uncertainty in case of Poisson errors, but they roughly are within 2$\sigma$.This is a first indication that one did not yet obtain an optimum solution. The diagnostic diagrams, Figs. 3c and 3h, indicate clearly that this is indeed the case. A comparison with model a further indicates that the uncertainty in the number of stars in each bin is slightly over-estimated with Poisson error bars. This is mainly because the bins are not independent from one another. An acceptable solution - like model a - should go through almost every observed point in Fig. 4. About $\sqrt{N_{\rm bins}}$are expected to deviate from this expectation. In case of model a about 5 points might not show a close match in Fig. 4. A close inspection shows that this is indeed the case.

The results indicate that the luminosity functions of various realizations - such as cases a, c, and h - might apparently not be so significantly different from one another and might therefore all be acceptable solutions for the test population. The values for the merit function and the diagnostic diagrams (Figs. 3a, 3c and 3h) however strongly indicate that only model a is acceptable. The method presented in Sect. 2, together with the diagnostic diagram, is more powerful than an analysis in which different model luminosity functions are compared. The global merit function, together with the diagnostic diagrams, gives a better discrimination between different models. The crucial point lies in the application of the Poisson statistics. As outlined in Sect. 2, the only independent quantity to which the Poisson statistics can be applied is the total number of stars brighter than a specific limiting magnitude [*]. Suffice to mention that Table 1 and Figs. 3a, 3c, and 3h show that significant differences are present between cases c and h with respect to case a.

4.7 Future work

Firstly, an automatic procedure should be developed based on the merit functions and the diagnostic diagrams. A search with a genetic algorithm appears to be a promising approach. As a first test one should apply this program to a synthetic dataset, such as the test population used in this paper. The use of real datasets should be avoided initially, because unforeseen problems - which are not associated with the validity of the method - might arise with real data sets. In particular, problems related with relative fast phases of stellar evolution or the colour transformation from the theoretical to the observational plane, see for example Sects. 4.1 and 4.3.

Secondly, a comparison should be made between the results from an automated search program and the results obtained from the isochrone fitting technique. A study of old open clusters in which these techniques are applied is underway (Carraro et al., in preparation). The purpose is to determine if the age of the oldest open cluster Berkeley 17 (Phelps 1997) is as old as the globular clusters or if it has an age comparable to or slightly older than the old clusters in the sample defined by Carraro et al. (1998). The next step is to apply this method to the resolved, multiple stellar populations of dwarf galaxies. However, with respect to the clusters the results might not be as reliable.

Finally, it is intended to improve through (self-) calibration from studies of essentially single stellar populations, like open & globular clusters, the library of stellar evolutionary tracks and isochrones.

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