3. Line-strength indices of SSPs
Most sudies of evolutionary population synthesis (EPS) are based on low resolution stellar spectra, see for instance Bressan et al. (1994, 1996) who made use of the Kurucz (1992) library. This fact prevents the direct measure of line-strength indices on the integrated spectral energy distribution (ISED) of stellar aggregates of any complexity, e.g. SSPs and galaxy models. The problem is solved by making use of the so-called fitting functions that express the intensity of the indices as a function of three basic stellar parameters, namely effective temperature, gravity and metal content (see Worthey 1992, Worthey et al. 1994). Knowing these quantitities for each single star in the stellar mix, the integrated line strength indices can be easily calculated both for SSPs and galaxy models in a fully consistent manner with their associated ISEDs, see Bressan et al. (1996) for all details. It goes without saying that this is possible only if the fitting functions are known. Unfortunately, there are indices, for instance the blue ones, for which fitting functions are not available. Therefore one is forced to make use of spectra with the required degree of resolution. This is the subject of the coming section.
3.1. Using empirical medium-resolution spectra
In general, the spectral energy distribution , flux (in suitable units) as a function of the wavelength, of a SSP of any age t and metallicity Z is defined by
where the product between the initial mass function (IMF) and the monochromatic flux, of a star of initial mass M, metallicity Z and age t, is integrated along the isochrone associated with the SSP in question. The integration is performed from , the minimum mass of objects that can become stars, up to , which is the maximum mass of the stars in the most evolved phase still contributing to the total luminosity (see for details Bressan et al. 1994, 1996).
The function indicates the adopted spectrum of the constituent stars of mass M, age t and metallicity Z, i.e. the adopted spectral library as function of effective temperature, gravity and metal content (a trivial re-normalization of the spectra is required to scale the flux emitted by a star according to its total bolometric luminosity). This library has of course its own resolution which determines the final result.
For the purposes of this study, we have adopted the empirical library of stellar spectra by J84, which contains fluxes in units of Å with a mean resolution of 4.5 Å (FWHM) for stars of different spectral type and luminosity class. The spectra cover the range of wavelength 3500 Å - 7500 Å and refer to stars with solar composition.
To incorporate the J84 library in our population synthesis algorithm, we proceed as follows:
(ii) As the J84 spectra cover only a limited spectral range, we have extended them both at the short and long wavelength side by means of the Kurucz (1992) spectra of the same composition, gravity and effective temperature. The Kurucz spectra have been cut at 3510 Å and 7427 Å and after suitable re-scaling, patched with the J84 spectra.
(iii) The spectral ranges used to calculate the scaling factors are [3510 Å-3550 Å] and [7390 Å-7427.2 Å] at the blue and red side of the spectrum, respectively. From a theoretical point of view, the blue and red scaling factors should be the same. Actually, we have detected differences between the blue and red scaling factors of about 26%. Various reasons concur to build up this difference: (a) inaccuracy of the synthetic spectra; (b) errors in the flux calibration by J84; (c) some mismatch in the correspondence between Kurucz (1992) and J84 spectra via the correspondence in gravity and effective temperature.
The monocromatic flux of individual stars in Eq. (1) now reduces to because the J84 library is only for solar composition. However, the dependence on the metallicity (chemical composition) is not completely wiped out, because it still remains via the metal dependence of the isochrones along which relation (1) is integrated. Though reduced, the effect of different metallicities (chemical compositions) on the final results can be noticed.
In order to compare indices derived from direct measurements on the spectra with those obtained from fitting functions, we must now degrade the spectra of our SSP and/or model galaxies (see below) to the resolution of 8.2 Å (FWHM) typical of the Lick-IDS spectra. To this purpose we convolve our spectra with a Gaussian distribution with = 2.92 Å.
Using the degraded spectra we calculate the line-strength indices by means of one of the relations below
In several cases, the results we have obtained do not perfectly agree with the standard Lick standard system, but a small off-set is present. Determining this off-set is not possible because there is no sample of stars or spectra in common between the Lick and the J84 library.
Finally, we have calculated indices for three large grids of SSPs, i.e. at varying metallicity (Z=0.004, Z=0.02, Z=0.05), each grid containing 30 values of the age in the interval 0.05 to 19 Gyr.
3.2. Comparing stellar indices
In this section we check whether stellar indices derived directly from the stellar spectra of the J84 library (therein-after the Jacobi indices) are fully consistent with the same indices but obtained from fitting functions. The comparison is made for all the stars of the J84 catalog.
In Fig. 1 we display the comparison between the Jacobi indices and those obtained from the Lick system fitting functions (Worthey 1992, Worthey et al. 1994) as, however, modified by Longhetti et al. (1998a,c) in the high temperature regime.
Similar comparison has been made using the fitting function of Buzzoni et al. (1992; 1994) for Mg2, Fe5270 and H (open squares) and those of Idiart & de Freitas Pacheco (1995) for Mg2, Mgb and H. The results are also shown in Fig. 1, where the open squares are for Buzzoni et al. (1992, 1994) and open triangles for Idiart & de Freitas Pacheco (1995). Considering all the uncertainties affecting the whole procedure, the agreement we get is remarkable.
3.3. Comparing SSP indices
In this section firstly we examine the red indices for SSPs and compare them with their counterparts based on fitting functions. Secondly we look at the blue indices for SSPs. In such a case, no comparison with fitting function analogs is obviously possible.
In Fig. 2 we show the comparison of SSP red indices with solar metallicity (Z=0.02) obtained from the two methods (fitting functions and direct measurement). Along each curve the symbols change according to the age of the SSP: full squares for 1019 Gyr, open circles for 19 Gyr, full circles for 0.20.9 Gyr and open squares for 0.05 0.1.
The data displayed in Fig. 2 show that both methods yield consistent results. Although almost all red indices strongly depend on the metallicity, i.e. their values significantly change for small variations in the metal content, this effect is not shown here as we are more interested now in the mutual consistency of the two methods rather than the absolute values of the indices as function of age and metallicity.
Particularly remarkable is the agreement for those indices that are commonly considered as valuable age indicators, such as H and G4300.
Specifically, the values of H derived from the two methods at ages older than 1 Gyr show an average difference of with a root mean difference of . For G4300 we find that they differ by with a root mean difference passing from one method to the other. Also in this case, the average difference gets smaller for ages older than 1 Gyr (mean difference , root mean difference ).
A systematic comparison between the Jacobi and Lick indices is presented in Table 1 for all indices in common.
Table 1. Comparison of red indices derived from different methods. The symbols ff and J84 stand for fitting functions and Jacobi et al. (1984) library. See the text for more details. The SSP indices are given at three ages (in Gyr) as indicated.
Finally, in the three panels of Fig. 3 we plot the blue indices 4000, H+K(CaII) and H/FeI as a function of the age for SSP with different metal content. The effect of this is shown by using different symbols. The same data are also listed in Table 2 (available in electronic form only) for the sake of general use. The following remarks can be made:
(a) The indices H+K(CaII) and H/FeI at ages older than Gyr tend to flatten out or, at ages older than about 10 Gyr, even to reverse the trend. They somehow loose ability in deriving the age. The same is true for ages younger than 0.3 Gyr. Therefore, there is only a limited range in age in which these indices vary monotonically with time, i.e. 0.3 to about 3 Gyr. However the effect of metallicity is not neglible bringing an uncertainty on the age of about
(b) The index 4000 works much better than the previous ones, because it is monotonic with the age up to ages older than 10 Gyr. The uncertainty brought about by the metal parameter is however still large ().
© European Southern Observatory (ESO) 1999
Online publication: April 19, 1999