3 Calcium triplet synthesis

We calculate the integrated equivalent widths for the CaT lines by combining the individual stars in each evolutionary stage, according to the theoretical isochrone. To this purpose let I_j be the intensity in absorption of the two lines of CaT for each star, j, found in the HR diagram of an SSP:

I_j = f_j EW_j

where f_j is the corresponding flux at the wavelength of 8600 Å for the star in the HR diagram. This quantity is obtained by a linear interpolation between the two central values of the continuum band-passes as defined by DTT89. The fluxes come from a suitable stellar atmosphere model and have been scaled to the luminosity of the corresponding theoretical star in the HR diagram. EW_j is the equivalent width of CaT for a star in the evolutionary stage j, that we assume is known. If N_j describes the number of stars in the evolutionary stage j and N is the total number of points in the HR Diagram, the synthesized equivalent width of CaT for an SSP at a given epoch is:

$$EW {\rm (CaT, SSP)} = \frac{\sum_{j=1}^{N} I_{j}N_{j}} {\sum_{j=1}^{N} f_{j}N_{j} + f_{\rm neb}}$$ (2)

where $f_{\rm neb}$ is the nebular continuum at 8600 Å corresponding to the SSP. In the following both theoretical (grid I) and empirical (grid II) fitting functions have been used to obtain the index as a function of the stellar physical parameters: $T_{\rm eff}$, logg and abundance. The theoretical stellar grid of EW(CaT) is from JCJ92, while the empirical library is from DTT89 plus the M type stars from Z91's atlas. We consider EW(CaT) to be zero for stars hotter than 6700 K which is the observational limit of DTT89's atlas.

3.1 Grid I: Theoretical fitting functions

JCJ92 computed a complete grid of NLTE models for the equivalent widths of CaT lines from stars with $T_{\rm eff}$ ranging between 4000 and 6600 K, logg between 0.00 and 4.00, and calcium abundances between 0.1 and 1.6 solar. From their models, the following fitting functions can be used to calculate the theoretical value of EW(CaT) as a function of $T_{\rm eff}$, logg, and calcium abundance, [Ca/H] = -1.0, -0.5, 0.0 and +0.2 (Eqs. (3), (4), (5) and (6) respectively).

$$\begin{eqnarray} EW_{-1.0} &=& -5.03 \!-\! 0.136\, {\rm log}g \!+\! 0.304\, {\rm... ...\,10^{-7} T_{\rm eff}^{2} - 3.14\,10^{-4}{\rm log}g\, T_{\rm eff}\end{eqnarray}$$ (3)

$$\begin{eqnarray} EW_{-0.5} &=& -10.28 \!-\! 1.83\, {\rm log}g \!+\! 0.493\, {\rm... ...\,10^{-7} T_{\rm eff}^{2} - 2.20\,10^{-4}{\rm log}g\, T_{\rm eff}\end{eqnarray}$$ (4)

$$\begin{eqnarray} EW_{+0.0} &=& -14.25 \!-\! 5.00\,{\rm log}g \!+\! 0.703\,{\rm l... ...,10^{-2}T_{\rm eff}\nonumber \\ && - 1.09\,10^{-6} T_{\rm eff}^{2}\end{eqnarray}$$ (5)

$$\begin{eqnarray} EW_{+0.2} &=& -16.00 \!-\! 5.88\, {\rm log}g \!+\! 0.811\, {\rm... ...10^{-2}T_{\rm eff} \nonumber \\ && - 1.27\,10^{-6} T_{\rm eff}^{2}\end{eqnarray}$$ (6)

where $EW_{\rm [Ca/H]}$ indicates the value, in Å, of CaT index (sum of the equivalent widths from the two strongest lines, at 8542, 8662 Å), computed with the continuum band-passes located as in DTT89. [Ca/H] means the calcium abundance with respect to the solar value. Conversion between the metallicity Z of the isochrones and the [Ca/H] index of the fitting functions is made adopting Z=0.02 for $\rm [Ca/H] = 0$ and by linearly scaling the index for other metallicies. JCJ92 do not compute theoretical EW(CaT) for metallicities higher than 1.6 solar. We have assumed the use of Eq. (6) for our calculations at Z=0.05 (2.5 $Z_{\odot}$), and therefore the values of EW(CaT) could be understimated. For metallicities lower than solar a linear interpolation between the values of the indices given by the above expressions has been done.

  

Figure 3: Comparison between data and models of EW(CaT) in stars. Panel a) shows the EW(CaT) as a function of the gravity. Open circles represent data from DTT89. Solid lines correspond to JCJ92's fitting functions for the values of the effective temperature labelled in the figure. The dotted line is our fit to DTT89's data, which has been used in the models (grid II). Panel b) shows the EW(CaT) as a function of the effective temperature for the coolest stars. For stars cooler than 4000 K, we have extrapolated the expressions given by JCJ92 for stars with $T_{\rm eff}$ between 4000 and 6000 K. Open circles are the data from Z91

3.2 Grid II: Empirical fitting functions

In the second grid of models we made use of the observational data collected by DTT89 complemented by data of M-late type stars from Z91. DTT89 provide the following relation between EW(CaT), gravity and stellar abundance as measured by [Fe/H]:  

$$EW{\rm (CaT)} = 10.21 - 0.95\, {\rm log}g + 2.18\, {\rm [Fe/H]}.$$ (7)

This relation has been adopted for the metallicities Z=0.004 and Z=0.008, assuming [Fe/H] = 0 for Z=0.02.

For models with Z=0.02 and Z=0.05, we have fitted the observational data of the EW(CaT) as a function of the gravity, following Eqs. (8) and (9).

$$EW{\rm (CaT)} = 13.76 - 2.97\, {\rm log}g ;\quad {\rm if}~~{\rm log}g < 2$$ (8)

$$EW{\rm (CaT)} = 9.51 - 0.78\, {\rm log}g ;\quad {\rm if}~~{\rm log}g \geq\ 2.$$ (9)

The above relations are shown in Fig. 3a together with the theoretical calibrations given by JCJ92 for different effective temperatures.

As already anticipated, for M-late stars we adopted the data by Z91, since these stars were not included in DTT89's library. The data given by Z91 have been converted to DTT89's system through the following relation, which has been obtained by fitting a linear regression to 20 common stars in Z91 and DTT89:

$$\begin{eqnarray} EW{\rm (DTT89)} &=& (0.87 \pm 0.07)EW(Z91) \nonumber \\ &&+ (0.70 \pm 0.58).\end{eqnarray}$$ (10)

The resulting final expression adopted for M-late stars ($T_{\rm eff}$ $\leq$ 4000 K and log$g \geq 3.00$) is:

$$\begin{eqnarray} EW{\rm (CaT)} &=& (6.06 \pm\ 1.51)\, 10^{-3}T_{\rm eff} \nonumber \\ && - (14.19 \pm\ 5.31).\end{eqnarray}$$ (11)

Figure 3b shows Z91's data for M-type stars and JCJ92' models, as a function of the effective temperature. Different curves correspond to models with different gravity as indicated in the plot.

With the above fitting functions we computed the synthetic equivalent widths for the two main lines of CaT at $\lambda \lambda\ 8542, 8662$ Å at the four selected metallicities: 0.2 $Z_{\odot}$, 0.4 $Z_{\odot}$, $Z_{\odot}$ and 2.5 $Z_{\odot}$. The results are shown in Fig. 4 and the values are given in Tables 3, 4 (grid I), 5 and 6 (grid II). For each table, Col. (1) lists the logarithm of the age of the SSP (in yr), Col. (2) the continuum luminosity (in units of ${L}_{\odot}$) from the SSP (nebular emission not included), taking an average value in the DTT89's spectral band-passes; Col. (3) the luminosity, in units of ${L}_{\odot}$, absorbed in the Ca II lines at 8542 and 8662 Å by the stars of the SSP, and Col. (4) the equivalent width of CaT, in Å, computed as the ratio between Col. (3) and the total continuum luminosity (in which both the stellar and the nebular contribution are taken into account). Columns (5), (6) and (7) are the same of (2), (3) and (4) respectively, but for a different metallicity.

  

Figure 4: Computed models for the CaT index as a function of the age of the SSP in a logarithmic scale. Panels a), b), c), and d) display the results for metallicities 2.5 $Z_{\odot}$, $Z_{\odot}$, 0.4 $Z_{\odot}$, and 0.2 $Z_{\odot}$ respectively. Solid points correspond to grid I, and therefore based on JCJ92's theoretical calibrations for EW(CaT); and open circles correspond to grid II, based on data from DTT89 and Z91