The Balmer line series provides the best spectroscopic age indicator among the set of Lick line indices. The Lick system defines five indices (Hβ, HγA, HδA, HγF, and HδF) for three Balmer lines (Worthey 1994; Worthey & Ottaviani 1997). Figure 5.2 shows the passband definitions for all Balmer indices. In combination with a metallicity diagnostic, these higher-order Balmer line indices are widely used to determine (luminosity-weighted) ages and metallicities of galaxies (e.g. Trager et al. 1998, 2000a,b; Kuntschner 2000; Poggianti et al. 2001; Moore et al. 2002; Kuntschner et al. 2002a; Thomas et al. 2003a).
However, different types of diagnostic plots employing different Balmer-line in- dices as age indicators are used throughout the literature. Although the age predict- ing power of an arbitrarily chosen diagnostic plot (most common versions include the Hβ and Mg2 or hFeiindices) might yield accurate-enough results for a specific scientific goal (e.g. the mean age difference between two different galaxy samples), the choice of a specific diagnostic plot is still subject to observational constraints and personal assessment and makes comparisons between studies difficult. As a consequence, most authors use several diagnostic plots with different Balmer line indices and assign equal importance to the results derived from each of those.
In the following we provide a recipe to define a quantity from which the relatively best Balmer-line age indicator can be determined. This quantity takes into account the quality of a given data set and the diagnostic power of theoretical predictions from which one intends to derive the age and metallicity.
In particular, the age sensitivity of an index is a function of the following pa- rameters:
• η: mean error of the data
• ζ: transformation accuracy to the Lick system
• γ: mean error of the original Lick spectra
• δ: accuracy of the Lick fitting functions (Worthey 1994; Worthey & Ottaviani 1997)
• DZ: index range covering all ages at a given metallicity, hereafter termed the dynamic range
• SZ,t: degeneracy parameter, which quantifies the sensitivity to age and metal-
licity at a given metallicity and age (i.e. the impact of the age-metallicity degeneracy)
The according numerical values for each parameter are given in Table 5.1 for each Balmer index. It is worth noting that some of these values are only valid for our data quality in combination with the SSP models of Maraston (2003). For different data and SSP models,η,DZ, andSZ,tare subject to change. To quantify the most age-sensitive and least metallicity-sensitive Balmer index, we define the quantity
R= p DZ· SZ,t
η2+ζ2+γ2+δ2 (5.1)
where the degeneracy parameter, SZ,t, is defined as
SZ,t(I) = ∂I ∂t ¯ ¯ ¯ ¯Z,t · µ∂I ∂Z ¶−1¯¯ ¯ ¯ ¯ t,Z . (5.2)
Figure 5.2: Passband definitions for Balmer-line, Mgb, Mg2, Fe5270, and Fe5335 Lick indices with their feature and adjacent continuum passbands. The overplot- ted spectrum is a high-S/N spectrum of the Galactic globular cluster NGC 6284 (Chapter 2). The resolution is∼7 ˚A and was left untouched to keep satellite lines visible. Line data were taken from Reader & Corliss (1981). Note the large amount of satellite lines which are included in within the passband definitions.
the Age-Metallicit y Degeneracy of Diagnostic Plots 117
in units of ˚A. Columns 9-12 are given in dex/Gyr while the unit ofRin the last column is ˚A·dex/Gyr. index η ζ γ δ D−1.35 D0.0 hDZi S−1.35,3 S−1.35,13 S0.0,3 S0.0,13 hSZ,ti R 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Hβ 0.20 0.232 0.22 1.30 2.55 2.68 2.62 0.389 0.146 0.442 0.120 0.274 0.530 HγA 0.28 0.722 0.48 1.78 7.75 11.01 9.38 0.284 0.068 0.262 0.049 0.166 0.779 HγF 0.28 0.448 0.33 1.34 4.55 5.83 5.19 0.291 0.072 0.123 0.058 0.136 0.478 HδA 0.27 1.043 0.64 1.27 5.73 9.11 7.42 0.262 0.047 0.233 0.047 0.147 0.611 HδF 0.28 0.790 0.40 1.18 3.83 4.47 4.15 0.284 0.054 0.122 0.058 0.130 0.359
two different ages 3 and 13 Gyrs. SZ,t is the ratio of age and metallicity partial derivatives at a given metallicity Z and aget (see Eqn. 5.2). In other words, SZ,t is a measure of the age-metallicity degeneracy and is maximal for indices which are very sensitive to age and least sensitive to metallicity, at the same time.
The highestRindicates the best age indicator with least age-metallicity degen- eracy. In Table 5.1 we provide values forDZ at two different metallicities and for
SZ,t at four age-metallicity combinations for each Balmer line index. Since SSP models do not provide continuous but discrete predictions the partial derivatives are substituted by difference ratios, i.e. ∂I/∂Z→ ∆I/∆Z and ∂I/∂t→ ∆I/∆t. The quotients are determined by linear interpolation of the SSP models.
We determine the relatively best age indicator from the set of five Lick Balmer indices by combining the mean dynamic rangehDZ,ti, the mean age-metallicity sen- sitivityhSZ,ti, and the total index uncertainty which is the denominator in equation 5.1. The final meanRis documented in the last column of Table 5.1. We find that the relatively best age diagnostic for our data is the HγA index followed by the indices HδA and Hβ. HγF and HδF have the smallestRvalues and are considered
as not reliable age indicators.
It is instructive to see that despite the relatively large age-metallicity degener- acy of the HγA index, the most accurate age predictions can be derived with this index. This fact is basically due to the large dynamic range of HγA compared to its mean measurement uncertainty. Hβ, on the other hand, has a relatively large total uncertainty and the measurements will therefore be more scattered over the diagnostic plot’s parameter range. In general, the higher-order Balmer lines require less S/N to guarantee a similar total index accuracy as Hβ. If our data set would be infinitely accurate (i.e. η= 0 in Eq. 5.1), the order ofRfrom the best to worst Balmer index would remain unchanged. This order is predominantly governed by uncertainties in the fitting functions of the respective index. To vary this order the mean measurement uncertainties have to be very discrepant and the SSP-model predictions have to deviate significantly from the model used here. It is expected that the relative accuracy of Balmer index measurements is comparable between different data sets as they are usually derived from one optical spectrum. The
relative age scale of SSP models appears to be quite stable from model to model. This scale is used in our above prescription. It can therefore be expected that no large fluctuation inR will arise from the use of different SSP model predictions1. Henceforth, we use the HγA index as our most reliable age-indicator.