1 Introduction

The Ca II triplet in absorption, at $\lambda\lambda$ 8498, 8542, 8662 Å, is the strongest feature in the near-infrared spectrum of late-type stars and normal galaxies. Pioneer work by Jones et al. (1984) was followed by those of Díaz et al. (1989, hereafter DTT89), Zhou (1991, hereafter Z91) and Mallik (1994), who studied the behaviour of these features in several stellar libraries, as a function of the atmospheric stellar parameters: effective temperature, $T_{\rm eff}$, surface gravity, logg, and iron abundance, [Fe/H].

DTT89 observed a sample of 106 late-type stars (up to K5III) in the near-IR, providing the first homogeneous atlas including CaT. They defined standard spectral windows, free of TiO contamination, to locate the continuum and to measure the index. They concluded that the equivalent width of the two main lines ($\lambda\lambda$ 8542, 8662 Å) of the Ca II triplet, EW(CaT), increases with increasing metallicity and decreasing stellar surface gravity of the star and, at high metallicity ($Z \geq 0.5$$Z_{\odot}$), the surface gravity is the dominant parameter, with values of EW(CaT) larger than 9 Å found only in RSG. This behaviour was later confirmed by Z91 and Mallik (1994).

Zhou reached the same conclusions as DTT89 but adopting slightly different spectral windows for the index definition. The analysis of the values of the EW(CaT) for the stars in common allow us to combine both atlases. In particular, Z91 includes M-late giant stars, not present in DTT's library. For these M giants, it appears that the correlation between EW(CaT) and $T_{\rm eff}$ is stronger than that between EW(CaT) and log g. Z91 shows that in M giants EW(CaT) reaches values lower than the ones predicted only on the basis of the EW-logg calibration previously found by DTT89.

Mallik (1994) observed 91 late-type stars, confirming the results of DTT89 and Z91. The lack of the measurement of the 8662 Å line for a large part of his sample and the different continuum band-passes and spectral resolution, make very difficult the comparison between Mallik's atlas and those of DTT89 and Z91. Mallik's sample does not include stars cooler than M1, and therefore no conclusions about the values of the index for extremely cool stars can be achieved. However, for the coolest stars in this atlas the values of EW(CaT) are lower for lower $T_{\rm eff}$, confirming the $T_{\rm eff}$-dependence of the index for the coolest stars.

Recently, Idiart et al. (1997) have published CaT indices for a sample of 55 stars. Their sample do not include cool M stars neither metal-rich supergiant stars. The cool late-type stars are however included in their calibration since they use those from Z91 (converting the values of CaT given by Z91 to their own system). The definition of their indices is different from the one assumed in this work (common in DTT89 and Z91): they used different continuum band-passes and the three CaT lines (instead of the two strongest ones), being the comparison meaningless. Nevertheless, they confirm the strong dependence on metallicity. With respect to the $T_{\rm eff}$- dependence, they find that the strength of CaT increases from F2 to K5 stars.

From the theoretical point of view, Smith & Drake (1987, 1990) and Erdelyi-Mendes & Barbuy (1991) computed the intensities of CaT lines, as a function of the atmospheric parameters ($T_{\rm eff}$, logg and metallicity). These last authors found that the computed intensity of CaT lines increases exponentially with metallicity (DTT89 had found a linear relation but in a narrower range of metallicities), showing a stronger dependence on metallicity when gravity is low (giant and supergiant stars). They also found a weak dependence on effective temperature and a modest dependence on gravity.

Finally, Jørgensen et al. (1992; hereafter JCJ92) computed a complete grid of models for Ca II lines as a function of $T_{\rm eff}$, logg and [Ca/H] abundance. They synthesized the equivalent widths of CaT lines. They used the DTT89 index definition and therefore compared these results with the published observational data. They found a good agreement between their calibrations and the observed EW(CaT) compiled by DTT89, reaching basically the same conclusions already pointed out, that can be summarized as follows: (1) in high metallicity systems, the stellar surface gravity is the parameter which controls the strength of the CaT lines; (2) the effect of the abundance is very important for giants and supergiants, with EW(CaT) increasing at increasing metallicity, but not for dwarfs; (3) at lower metallicity the effect of the effective temperature is in competition with that of the gravity.

In the present work we compute stellar population synthesis models for the sum of the equivalent widths of the two strongest lines ($\lambda\lambda$8542, 8662 Å) of the CaT. The age considered ranges from 1 Myr to 13 Gyr, and the metallicity from 0.2 $Z_{\odot}$ to 2.5 $Z_{\odot}$. Section 2 describes the main aspects of the evolution related to the appearance of cool stars on the basis on the Padova evolutionary tracks (2.1), and the computed Spectral Energy Distributions, SEDs, in which both, stellar (2.2) and nebular (2.3) contributions have been included. Section 3 is devoted to the CaT synthesis. Two grids of models have been computed: grid I which is based on the theoretical fitting functions of EW(CaT) (Sect. 3.1), and grid II, based on empirical fitting functions derived from the above stellar atlases (Sect. 3.2).

In addition to the models described in Sect. 3, several composite-population models have been computed with different mass percentages of young (2.5 - 5 Myr, able to ionize), intermediate (8- 12 Myr, rich in RSG) and very old (10 Gyr) populations. These models are described in Sect. 4.1 and are meant to constitute a reference frame for the interpretation of the observations of CaT in star-forming regions at different scales (from pure HII regions to Starburst galaxies or even Active Galactic Nuclei, AGN). Section 4.2 discusses the implications of the use of CaT as a metallicity indicator in elliptical galaxies. Finally, Sect. 5 summarizes the conclusions.