2.3.3 Tripicco & Bell (1995)

Tripicco & Bell (1995) present theoretically calculated line indices for single stars (not entire populations). The line indices are measured from synthetic spectra based on model atmospheres. This is done for $(T_{\rm eff}, \log g)$ points along a 5 Gyr solar abundance isochrone. The microturbulence is calculated from the luminosity.

The dependence of each index on the abundances of individual elements is determined in the following way. The abundance of a single metal (C, N, O, Mg, Fe, Ca, Na, Si, Cr, and Ti) is increased from 0.0 to 0.3 dex, while keeping the other metals at 0.0 dex. Then the line indices are calculated as described above, still for points along the said solar abundance isochrone. As found by Weiss et al. (1995), the error made by using a solar abundance isochrone instead of an isochrone calculated for the appropriate non-solar abundance ratios is probably small. Tripicco & Bell also try increasing all the metals from 0.0 to 0.3 dex.

Table  lists their results for the indices ${ {\rm Mg}_2}$ and $\log { <{\rm Fe}>}$ for increasing the abundances of C, Mg, and Fe, and the total metallicity. The elements not listed (N, O, Ca, Na, Si, Cr, and Ti) have smaller effects than the above. Note that the table show the result for two individual stars, and not for an entire single-age single-metallicity population. The two types of stars shown, a turnoff star and a cool giant, are expected to be the dominant sources of light in a typical E and S0 galaxy.

It is seen, that ${ {\rm Mg}_2}$ depends mostly on the magnesium abundance, but that it is also sensitive to the carbon abundance as well as the total metallicity. $\log { <{\rm Fe}>}$ depends just as much on the total metallicity as on the iron abundance.

Also shown in the table is ${\widehat{{ {\rm Mg}_2}}}$, which is calculated from Mgb as ${\widehat{{ {\rm Mg}_2}}}= 0.638 \log {\rm Mgb} - 0.133$. The relation ${\widehat{{ {\rm Mg}_2}}}= { {\rm Mg}_2}$ was established by J97, based on 161 E and S0 galaxies. We use ${\widehat{{ {\rm Mg}_2}}}$ as a substitute for ${ {\rm Mg}_2}$ for the spectra where Mgb but not ${ {\rm Mg}_2}$ could be measured, cf. Sect. . It is seen, that for these models the two values do not quite agree, and the the dependence on abundance changes is not always the same. However, since J97 found the above relation to have no significant intrinsic scatter, either (a) the differences cancel out when an entire population is considered; (b) they become smaller for the abundance ratios and metallicities actually found in E and S0 galaxies; or (c) the model is somewhat off. It is worth noting that the ${ {\rm Mg}_2}$ values of J97 are in the range 0.13-0.34, so the turnoff star with ${ {\rm Mg}_2}=0.07$ and the cool giant with ${ {\rm Mg}_2}=0.36$ are both outside this range. On the other hand, the ${\widehat{{ {\rm Mg}_2}}}$ values are in both cases lower by the same large amount. This indicates, that even for a composite stellar population, the model ${\widehat{{ {\rm Mg}_2}}}$ values will be lower than the model ${ {\rm Mg}_2}$ values. This is actually the case for e.g. the Vazdekis et al. (1996) models, although the discrepancy is smaller. For the bimodal $\mu = 1.35$ IMF model, ( ${ {\rm Mg}_2}- {\widehat{{ {\rm Mg}_2}}}$) is in the range 0.00-0.09, and typically 0.02.

 

Table: Spectral Line Index Response to Abundance Changes

Turnoff star ( $T_{\rm eff} = 6200 \, {\rm K}, \log g = 4.1)$

Index	Value	C	Mg	Fe	[M/H]
${ {\rm Mg}_2}$	0.07	0.01	0.02	-0.00	0.01
${\widehat{{ {\rm Mg}_2}}}$	-0.07	-0.03	0.10	-0.02	0.02
$\log { <{\rm Fe}>}$	0.05	0.01	-0.02	0.08	0.09

Cool giant ( $T_{\rm eff} = 4255 \, {\rm K}, \log g = 1.9)$

Index	Value	C	Mg	Fe	[M/H]
${ {\rm Mg}_2}$	0.36	0.04	0.08	-0.02	0.05
${\widehat{{ {\rm Mg}_2}}}$	0.23	-0.07	0.09	-0.02	0.02
$\log { <{\rm Fe}>}$	0.60	0.01	-0.02	0.06	0.05

 

Notes: ${\widehat{{ {\rm Mg}_2}}}= 0.638 \log {\rm Mgb} - 0.133$ (J97). ${ <{\rm Fe}>}= ({\rm Fe5270} + {\rm Fe5335})/2$. The column `Value' lists the value of the given index for a model with solar abundances. The columns `C', `Mg', `Fe', and `[M/H]' list the absolute amount the given index changes when the abundance of C, Mg, Fe, and all metals are increased from 0.0 to 0.3 dex, respectively. From Tripicco & Bell (1995).