Spectroscopic Abundance-Ratio Effects

In the previous section we studied how the adoption of $\alpha$-enhanced isochrones affects our model predictions. Here we show how the sensitivity of line indices to the various elemental abundances alters the model predictions. The method, first proposed by Trager et al. (2000) and further developed by Thomas et al. (2003a) and Korn et al. (2005), is based on estimates of how line indices in the spectra of three types of stars with relevant atmospheric parameters (a turnoff, a giant, and a lower main sequence star) change as a function of variations of the abundances of individual elements. Briefly, the method goes as follows. Initially, synthetic spectra are computed on the basis of model photospheres, for the three stellar types above, assuming a solar-scaled abundance pattern. For each stellar type, a new spectrum is then computed by varying the abundance of a single element. The impact of this sole elemental abundance on all spectral indices is assessed by comparing line indices measured in both the solar-scaled synthetic spectrum and that computed with the altered abundance pattern. The procedure is repeated for all the chemical abundances of relevance in the spectral region of interest and the final results are summarized in the form of sensitivity tables, such as those provided by Tripicco & Bell (1995) and more recently updated by Korn et al. (2005) and Serven, Worthey & Briley (2005). The sensitivity tables are used to estimate incremental changes in absorption line indices as a function of stellar parameters, which are integrated along the theoretical isochrone in order to produce integrated line strength predictions for any desired abundance pattern.

In this section we discuss the behavior of all the line indices studied in this paper as a function of the most important elements affecting their strengths in the range of stellar population parameters considered. In order to study this effect in isolation from that discussed in the previous Section, we perform calculations based on the same set of isochrones (Padova, solar-scaled), but varying the line indices according to their sensitivity to abundance ratio variations. The sensitivity tables used are those by Korn et al. (2005).

Solar-scaled vs. Base Models

The starting point for this discussion are the base models, which are summarized in Table 24. Models for any abundance pattern can be calculated relative to the base models in a differential fashion. As a first step, we generate solar-scaled models, for which [X$_i$/Fe] = 0 for all elements X$_i$, and at all values of [Fe/H]. These models are generated by correcting the base models from the abundance pattern listed in Table 6 to a solar-scaled abundance pattern. Because metallicity in our models is cast in terms of [Fe/H], this parameter is kept fixed whenever we calculate a model with a new abundance pattern. For instance, according to Table 6, if one wants to compute solar-scaled models with [Fe/H]=-0.4, one needs to correct the indices in the [Fe/H]=-0.4 base models for abundance ratio variations of $\Delta$ [O/Fe]=-0.2, $\Delta$ [Mg/Fe]=-0.13, $\Delta$ [Ca/Fe]=-0.06, and $\Delta$ [Ti/Fe]=-0.11. Those are the elemental abundance variations needed to bring the abundance ratios in Table 6 to [X$_i$/Fe]=0 (no need to perform any correction for variations of [C/Fe] and [N/Fe], as these ratios are solar in the spectral library for all values of [Fe/H]). As a result, two models with same [Fe/H] and different abundance patterns have different total abundances [Z/H]. This is different from the procedure followed by Trager et al. (2000) and Thomas et al. (2003a), who cast their models in terms of [Z/H], so that their $\alpha$-enhanced models have lower [Fe/H] for fixed [Z/H].

The result is illustrated in Figure 13, where base and solar-scaled models (dark and gray lines, respectively) are compared in four representative index-index planes. Because the abundance pattern characteristic of the base models only differs importantly from solar-scaled for [Fe/H]$\leq$-0.4, we only expect to find differences in this iron abundance interval. In the upper left panel, the two sets of models are compared in the Fe5270-$H\beta$ plane, where there is hardly any difference between solar-scaled and base models. The reason is that both indices are very little affected by variations of the abundances of any elements other than Fe. The only exception happens at [Fe/H]=-1.3, where the base models have slightly stronger Fe5270, due to the presence of titanium and calcium lines in the index passband. Because [Fe/H] is kept fixed when switching from base to solar-scaled models, the two model sets are virtually identical in the Fe5270-$H\beta$ plane. The same is not the case in the upper-right panel, where models are compared in the G4300-$H\beta$ plane, where the G4300 index appears to display a complex behavior as a function of [Fe/H]. As expected, the index is unchanged for models with [Fe/H]=0 and +0.2, but it becomes stronger in solar-scaled models for lower [Fe/H] values (same-[Fe/H] models are indicated with arrows, for clarity). The G4300 index is mostly a carbon abundance indicator, because of the presence of a strong vibrational band of the CH molecule in the index passband. However, Table 6 tells us that [C/Fe] is solar in the base models, so that it is the same for all [Fe/H] in both sets of models. However, the G4300 index is also affected, in an indirect way, by oxygen abundance variations, in the sense that the index tends to be weaker for higher oxygen abundances. Because the abundance of oxygen is lower in the solar-scaled than in the base models, the former have stronger G4300 for fixed [Fe/H]. One can also conclude from this plot that the G4300 index is very strongly sensitive to age, which can be seen by the fact that the model grids are very far from orthogonal. On the other hand its sensitivity to metallicity and/or carbon abundance is relatively weak, as can be seen by the fact that model lines for different values of [Fe/H] are packed very close together, especially for [Fe/H] $\geq$ -0.7. As a result, the G4300 index is a less than ideal carbon abundance indicator, which is the reason why we decided to include the C$_2$4668 index in our models. A more detailed discussion of this issue will be presented in Graves & Schiavon (2006, in preparation).

In the lower left panel, models are compared in the Fe5270-$H\delta _F$ plane. It can be seen that $H\delta _F$ is slightly weaker in solar-scaled models for [Fe/H]$\leq$-0.4. This is because the passband of the index includes lines due to calcium and magnesium, whose abundances are higher in the base models at these lower values of [Fe/H]. Finally, in the lower-right panel, models are compared in the Ca4227-$H\delta _F$ plane, which allows us to investigate the behavior of the Ca4227 index as a function of abundance ratios. As expected, base models have stronger Ca4227 at [Fe/H]$\leq$-0.4 than solar-scaled models, given that the latter have lower calcium abundances. The effect is in fact enhanced by the strengthening of CN bands in the solar-scaled models, due to their lower oxygen abundances, for reasons that will be discussed in the next section.

Effects of Carbon Enhancement

The blue spectra of G and K-type stars is pervaded by weak to moderately strong absorption lines due to the CN and CH molecules. That is in fact one of the most important challenges for the reliable modeling of the spectra of old/intermediate-age stellar populations in the blue (e.g., Vazdekis 1999, Paper I, Prochaska, Rose & Schiavon 2005, Prochaska et al. 2006). For that reason, it is important to investigate the effects of variations of the abundances of carbon and nitrogen on blue spectral indices.

In Figures 14 and 15 we compare solar-scaled models with models where only the abundance of carbon is enhanced by +0.3 dex. In the upper left panel of Figure 14, the models are compared in the Fe5270-$H\beta$ plane, where it can be seen that the two indices are essentially insensitive to carbon abundance variations. In the upper right panel, on the other hand, one can see that the C$_2$4668 index is extremely sensitive to carbon abundance variations, indeed, far more sensitive than the G4300 index. This is not unexpected, given that the concentration of the C$_2$ molecule in stellar atmospheres depends quadratically on the abundance of carbon, while that of CH depends only linearly on that parameter. In the lower left panel, one can see that $H\delta _F$ is very little affected by carbon abundance variations, in spite of the fact that the index is immersed in a thick forest of CN lines. This has been discussed in Paper I, where it was shown that the small sensitivity of $H\delta _F$ to carbon abundance variations was due to the the fact that the effect of CN lines in both the index passband and pseudo-continuum regions is partially cancelled. For a detailed discussion of the impact of CN lines on $H\delta$ measurements, see Prochaska et al. (2006). In the lower right panel of Figure 14 the models are compared in the Ca4227-$H\delta _F$ plane, where it can be seen that the Ca4227 index is strongly sensitive to carbon abundance variations, in the sense that the index becomes substantially weaker for increasing carbon abundances. This effect has been studied in detail by Prochaska et al. (2005) and it is due to the contamination of the blue pseudo-continuum of the index by a CN band-head. An increase in carbon abundance leads to a depression of the blue continuum and, consequently, to an artificially lower Ca4227 index. Prochaska et al. (2005) defined a new index, Ca4227$_r$, which is far less affected by CN contamination. Analyzing a large sample of nearby early-type galaxies, they showed that this new index presents a correlation with velocity dispersion ($\sigma$) which is much stronger than that of the Lick Ca4227 index, resembling the behavior of other $\alpha$-elements, such as magnesium. In subsequent sections, we will show that when the effect of CN lines is accounted for, one can extract reliable calcium abundances from measurements of the Lick Ca4227 index.

Figure 15 illustrates the effect of carbon abundances on another set of relevant Lick indices. In the upper left panel, one can see that, like Fe5270, the Fe5335 index is unchanged when carbon is varied. The same is true of the Fe5015 index (not shown). In the upper right panel, on the other hand, one can see that the CN$_2$ index is very sensitive to carbon abundance, as expected. It is also sensitive to nitrogen abundance, to a lesser extent. In the lower left panel, the models are compared in the Fe5335-$H\gamma _F$ plane, where it can be seen that this Balmer line is somewhat affected by carbon, in the sense that $H\gamma _F$ is weaker for higher carbon abundances. This is due to contamination of the index pseudo-continuum by CH lines. Finally, in the lower right panel, one can see that the Fe4383 index is only very mildly affected by carbon abundances, in the sense that the index becomes stronger for higher carbon, due to contamination of the index passband by CH lines.

Nitrogen abundance variations affect CN lines and therefore indirectly affect a number of indices, most notably $H\delta _F$, $H\delta _A$, and Ca4227. Since the variation of CN lines as a function of carbon abundances and their impact on line indices has been discussed here and since CN lines respond similarly to carbon and nitrogen (see Korn et al. 2005), there is no need to show model variations as a function of nitrogen here.

In summary, we find that, among the Balmer line indices, the only ones that are affected by the abundance of carbon are the $H\gamma _F$ and $H\gamma _A$ indices. Application of a solar-scaled model to $H\gamma$ measurements taken in the spectrum of a carbon-enhanced stellar population would lead to an age overestimate. For a stellar population with [C/Fe]=+0.3 the effect would be of the order of $\sim$ 5 (1) Gyr for ages of $\sim$ 10 (2) Gyr. Amongst the metal line indices, the ones that are the most sensitive to carbon are C$_2$4668, G4300, CN$_1$, CN$_2$, and Ca4227 (the latter due to a spurious contamination of the index blue pseudo-continuum). The cleanest carbon abundance indicator studied in this work is C$_2$4668, as it is solely dependent on the abundances of carbon and iron, and (very weakly) on age. The CN indices are also very strongly dependent on carbon, but are also sensitive to nitrogen, as expected. The iron indices Fe5270, Fe5335, and Fe5015 are free of the influence of carbon abundance variations, while the Fe4383 index is slightly affected by them.

Effects of $\alpha$-Element Enhancement

The abundance of $\alpha$ elements relative to that of iron provides some of the most fundamental clues available on the history of star formation and chemical enrichment of galaxies (e.g., Matteucci & Tornambé 1987, Wheeler et al. 1989, Peletier 1989, Worthey et al. 1992, Edvardsson et al. 1993, McWilliam 1997, Worthey 1998, Trager et al. 2000). Therefore, it is crucially important to understand how indices respond to variations of $\alpha$-element abundances, so that the latter can be estimated from index measurements taken in the integrated spectra of galaxies. In Figures 16 and 17 we contrast solar-scaled and $\alpha$-enhanced models in a number of index-index diagrams. The $\alpha$-enhanced models are computed by increasing by +0.3 dex the abundances of oxygen, magnesium, calcium, sodium, silicon, and titanium. We emphasize again that these computations do not take into account the effect of oxygen abundances in the stellar interiors (see Section 4.3.1), but only their spectroscopic effect, which is mostly due to changes in the strengths of carbon-based molecules, due to the impact of oxygen abundances on the concentration of free carbon atoms via the dissociation equilibrium of the CO molecule. As another caveat, we note that indices capable of constraining the abundances of sodium, silicon, or titanium are not modelled in this work. However, these abundances only have a small impact on the strengths of some of the indices modelled here (see Korn et al. 2005 for details), so we choose to force these abundances to track those of the other $\alpha$-elements.

In the upper left panels of Figures 16 and 17, the models are compared in the Fe5270-$H\beta$ and Fe5335-$H\beta$ planes, respectively. In these plots it can be seen that $H\beta$ is free of any influence due to the enhancement of $\alpha$-element abundances. This result, combined with that of Figure 14 confirms the finding by other authors (e.g., Korn et al. 2005) that $H\beta$ is the cleanest age indicator within the Lick index family, given its very low dependence on metallicity, and its virtual independence on any abundance-ratio effects. These figures also show that Fe5270, Fe5335, and, to a lesser extent, Fe4383 (lower right panel of Figure 17) are only very mildly sensitive to $\alpha$-enhancement. This result, combined with those of Figures 14 and 15, implies that the Fe5270 and Fe5335 indices are essentially only dependent on iron abundance and (mildly) on age. Therefore, they are all very reliable [Fe/H] indicators. The case of Fe4383 is interesting. According to the Korn et al. (2005) tables, this index is mostly affected by the abundance of iron, followed by that of magnesium, in the sense that the index becomes weaker when [Mg/Fe] increases. Supposedly, this is due to the presence of magnesium lines in one of the index pseudo-continuum windows. However, inspection of the Moore, Minnaert & Houtgast (1966) table of absorption lines identified in the solar spectrum reveals no such lines. Also according to Korn et al. (2005), the Fe5015 index (not shown) is very strongly sensitive to titanium and magnesium, being stronger (weaker) for higher (lower) [Ti/Fe] ([Mg/Fe]). Presumably this is due to the presence of a large number of MgH lines in the index red pseudo-continuum, and strong TiI lines in the index passband (Moore et al. 1966). Therefore, we caution against using this index in a situation where the abundances of magnesium and titanium are unconstrained.

In the upper right panels of Figures 16 and 17, the models are compared in the Mg $b$-$H\beta$ and Mg$_2$-$H\beta$ planes, respectively. One can see that the Mg $b$ and Mg$_2$ indices are very strongly sensitive to [Mg/Fe], as expected, with Mg $b$ being a little more sensitive than Mg$_2$. For this reason, these indices have been used in the literature as the chief $\alpha$-enhancement indicators. For reasons that will be discussed in Section 5.2.2, we will adopt Mg $b$ as our main indicator of magnesium abundance.

The lower left panels of Figures 16 and 17 illustrate how the Balmer line indices $H\delta _F$ and $H\gamma _F$, respectively, respond to variations of $\alpha$-element abundances. In both cases, a mild response to $\alpha$-enhancement is seen, in the sense that the indices tend to be stronger for higher $\alpha$-element abundances, so that non-consideration of $\alpha$-enhancement effects could lead to age underestimates. The effect would be of the order of 4-5 Gyr for old ages and solar metallicity, and about 1 Gyr for ages around 2 Gyr, for a stellar population with [$\alpha$/Fe] $\sim$ +0.3. While in the case of the $H\delta$ indices this is, according to Korn et al. (2005), due to contamination of the index passbands by magnesium and silicon lines (though we failed to find any of the former in the Moore et al. 1966 catalogue), in the case of $H\gamma$ it is because of the weakening of CH lines in the index pseudo-continuum, due to enhanced oxygen abundances.

The lower right panel of Figure 16 shows a comparison of solar-scaled and $\alpha$-enhanced models in the Ca4227-$H\delta _F$ plane. This plot shows that the Ca4227 is very strongly sensitive to [Ca/Fe], so that it will be used here as our chief indicator of calcium abundances. However, as we pointed out in the previous subsection, this index is also very heavily influenced by carbon abundance, which needs to be accounted for if one wants to use the Ca4227 index for calcium abundance determinations. Finally, the lower right panel of Figure 17 compares solar-scaled and $\alpha$-enhanced models in the Fe4383-$H\gamma _F$ plane. This plot suggests that Fe4383 is mildly sensitive to $\alpha$-enhancement, in the sense that it is weaker in $\alpha$-enhanced models. This results from contamination of one of the index's pseudo-continua by Mg lines.

The main conclusions of our investigation of the effects of $\alpha$-enhancement on the Lick indices studied in this work can be summarized as follows. Amongst the Balmer line indices, $H\beta$ is the only one that is not affected by $\alpha$-enhancement, or any other abundance-ratio effects. $H\gamma _F$, $H\gamma _A$, $H\delta _F$, and $H\delta _A$ are similarly affected in the sense that they are stronger in spectra of $\alpha$-enhanced stellar populations, for fixed age and [Fe/H]. Amongst the metal lines, the Fe5270, and Fe5335 indices are essentially insensitive to $\alpha$-enhancement, Fe4383 is only mildly affected by it, and Fe5015 is strongly affected by it. This is good news, telling us that a safe estimate of [Fe/H] is warranted. Our main indicators of $\alpha$-element abundances are Mg $b$, Mg$_2$, and Ca4227. While Mg $b$ is our cleanest indicator of an $\alpha$-element abundance (magnesium, in this case), Ca4227 can be used to estimate calcium, provided that carbon and nitrogen abundances are known, so that the CN effect on Ca4227 can be accounted for. The abundances of carbon and nitrogen can be inferred from the combined use of either G4300 or C$_2$4668 (both sensitive to carbon only, but C$_2$4668 is preferable, see Graves & Schiavon 2006, in preparation, for a discussion) and CN$_1$ and CN$_2$ indices (sensitive to carbon and nitrogen).