A Method for Determining Mean Ages and Metal Abundances of Stellar Populations

The understanding acquired in the last Section of the response of line indices to variations of age and elemental abundances can be used to establish a method to estimate both the age and abundance pattern of stars in clusters and galaxies, from the interpretation of Lick indices measured in their integrated spectra.

Initially let us suppose that measurements for all the line indices modelled in this work are available for a given stellar system. The method consists in constraining first the most influential parameters (i.e., those which affect the largest number of observables) and then descend hierarchically towards constraining less influential parameters. Inspection of Figures 14 through 17 tells us that the parameters that affect the largest number of Lick indices are age and metallicity, which in our models is cast in terms of [Fe/H]. Virtually all the Lick indices are affected by age and [Fe/H] variations, to various degrees. Therefore, the starting point of the method is the determination of age and [Fe/H]. According to our conclusions from Section 4.3.2, the best way of estimating age and [Fe/H] is by comparing data with models in a diagram involving an iron index (preferably Fe5270 or Fe5335, or some combination of these) and $H\beta$. Therefore, we assume that the Fe and $H\beta$ indices are only sensitive to [Fe/H] and age and estimate those parameters on the basis of solar-scaled models (the choice of models here is unimportant, provided our assumption that these indices are unaffected by abundance ratios is approximately correct). The second most influential parameter is the abundance of carbon, which affects a large number of line indices, though not all of them (for instance, $H\beta$, Fe5270, and Fe5335 are essentially not affected by carbon abundances), via the contamination of index pseudo-continua and passbands by lines due to CN, CH, or C$_2$, which pervade the spectral region under study. Of all the indices modelled in this paper, the C$_2$4668 index is best suited for carbon abundance determinations, so the next step in our method consists of searching the [C/Fe] value that best matches the $C_2$4668 index for the same age and [Fe/H] as estimated from the analysis of $H\beta$ and Fe indices. Once [C/Fe] is estimated, the next step consists of determining [N/Fe], as the abundance of nitrogen affects a large number of indices, because of its influence on the strength of CN lines. The best indicator of nitrogen abundances are the CN bands themselves, so the next step in the procedure consists of searching the [N/Fe] value for which the CN$_1$ and/or CN$_2$ indices are matched for the same age, [Fe/H], and [C/Fe] that match the measurements of $H\beta$, C$_2$4668 and Fe indices. The remaining parameters in the hierarchical sequence would be [Mg/Fe] and [Ca/Fe], as they influence only a very few line indices, such as Mg $b$, Mg$_2$, and Ca4227. Therefore, the final step of our procedure consists of estimating [Mg/Fe] and [Ca/Fe] by searching the values that match Mg $b$/Mg$_2$ and Ca4227, respectively, for the same age, [Fe/H], [C/Fe], and [N/Fe] as estimated from the match to all the previous indices. Once the latter is achieved, a first estimate of age, [Fe/H], [C/Fe], [N/Fe], [Mg/Fe], and [Ca/Fe] has been reached. The process now needs to be iterated, given that we initially supposed that $H\beta$ and the Fe index/indices of choice were essentially independent of any parameters other than age and [Fe/H], which is not entirely correct. Experience shows that, for most applications and depending on the degree of internal consistency aimed, one iteration is good enough.