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Astron. Astrophys. 335, 855-866 (1998)

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4. Model results

In this section we will show the results of several models and we will convert, by means of the available calibrations, the average stellar abundances of Mg and Fe, [FORMULA] and [FORMULA], as predicted for elliptical galaxies of different mass, into the metallicity indices [FORMULA] and [FORMULA], respectively.

Let us discuss first the metallicity calibrations. Relations linking the strength of metallicity indices to real abundances can be either empirical or theoretical. In the past few years several attempts have been made to calibrate the strength of [FORMULA] against [Fe/H] which has always been considered as the measure of the "metallicity" in stars. It should be said that this is not entirely correct since we know that Mg does not evolve in lockstep with iron in the solar neighborhood nor in elliptical galaxies, due to the different timescales of production of these two elements. It would be much better to calibrate [FORMULA] versus [Mg/H] in order to avoid confusion. Calibrations of [FORMULA] versus [FORMULA] are from Mould (1978), Burstein (1979), Peletier (1989), Buzzoni et al. (1992), Worthey et al. (1992). In all of these calibrations the ratio [Mg/Fe] is assumed to be solar, at variance with the indication arising from population synthesis models showing an overabundance of Mg relative to iron in the nuclei of giant ellipticals (Faber et al. 1992; Worthey et al. 1992; Davies et al. 1993; Weiss et al. 1995).

More recently, Barbuy (1994), Borges et al. (1995) and Tantalo et al. (1998) took into account non-solar ratios of [Mg/Fe] in their calibrations. In addition, some of them (Borges et al. 1995; Tantalo et al. 1998) produced synthetic [FORMULA] indices thus allowing us to calibrate [Fe/H] also against [FORMULA]. This allows us to transform [Fe/H] in to [FORMULA] and [FORMULA], although many uncertainties are involved in this exercise, mainly because, in this way, the derivations of [FORMULA] and [FORMULA] are not independent.

We run several models, in particular: Model I, which is the classic wind model, as described in MG95, with a Salpeter (1955) IMF (namely an IMF with power index x=1.35 over a mass range [FORMULA]); Model II, which is the classic wind model with the Arimoto and Yoshii (1987) IMF (namely an IMF with power index x=0.95 over the same mass range of the Salpeter one); Model III, which is the equivalent of the inverse wind model, as described in Matteucci (1994) with the Arimoto and Yoshii (1987) IMF; Model IV which is the equivalent of the model with variable IMF, as described in Matteucci (1994), which assumes that ellipticals of smaller mass have a steeper IMF than the more massive ones. It is worth noting that the slope of the IMF is kept constant inside a galaxy. In particular, we vary the slope of the IMF from the Salpeter one to the Arimoto and Yoshii one passing from a galaxy with initial luminous mass of [FORMULA] to a galaxy with [FORMULA]. This particular assumption can reproduce the observed tilt of the fundamental plane seen edge-on, namely the increase of M/L versus L as observed by Bender et al. (1992).

Model V assumes a time-variable IMF as suggested by Padoan et al. (1997) and will be discussed in a forthcoming paper. In this formulation the IMF slope varies as a function of gas density and gas velocity dispersion, favoring the formation of massive stars at early epochs.

Model VI assumes a constant IMF with a slope x=0.8 and a star formation efficiency which varies more strongly with the luminous mass than in Model III. The slope x=0.8 is the limiting slope that we can accept to obtain a realistic [FORMULA] ratio for ellipticals, as discussed in Padovani and Matteucci (1993).

The model parameters are described in Tables 1-6 where we list the luminous masses in column 1, the star formation efficiency (in units of [FORMULA]) in column 2, the effective radius (in units of Kpc) in column 3, the time for the occurrence of the galactic wind (in Gyr) in column 4 and the final galactic luminous mass in column 5. For model IV is shown also the slope of the IMF in column 6.


[TABLE]

Table 1. Model I-classic wind, x=1.35



[TABLE]

Table 2. Model II- classic wind, x=0.95



[TABLE]

Table 3. Model III- inverse wind, x=0.95



[TABLE]

Table 4. Model IV-classic wind, variable IMF



[TABLE]

Table 5. Model V- classic wind, time variable IMF



[TABLE]

Table 6. Model VI- inverse wind, x=0.8


We then calculate the average [FORMULA] and [FORMULA] for the stellar component of ellipticals by using Eq. (6) and finally we transform these abundances to observed [FORMULA] and [FORMULA] line indices. In Tables 7-12 we show the results for different models and for the calibration of Tantalo et al. (1998). In particular, in Tables 7-12 we show the luminous mass in column 1, the [FORMULA] in the second column, the [FORMULA] in column 3 and in column 4 and 5 the [FORMULA] and the [FORMULA] indices, respectively. As already said, only the calibrations of Tantalo et al. (1998) and Borges et al. (1995) allow us to transform [Fe/H] into [FORMULA] and therefore to compare model results with the data showing the behavior of [FORMULA] vs. [FORMULA] among nuclei of giant ellipticals. In particular, starting from the synthetic indices of Tantalo et al. (1998) calculated for a fixed [Mg/Fe] and a fixed [Fe/H] we derived calibration relationships of the type:

[EQUATION]

[EQUATION]

which allowed us to derive the indices for any [Fe/H] and [Mg/Fe]. The calibrations we have adopted are:

[EQUATION]

[EQUATION]


[TABLE]

Table 7. Model I- classic wind, x=1.35



[TABLE]

Table 8. Model II- classic wind, x=0.95



[TABLE]

Table 9. Model III- inverse wind, x=0.95



[TABLE]

Table 10. Model IV- classic wind, variable IMF



[TABLE]

Table 11. Model V- classic wind, time variable IMF



[TABLE]

Table 12. Model VI- inverse wind, x=0.8


In Fig. 1 we show the metallicity indices obtained by means of the already mentioned calibrations compared with the data (Gonzalez 1993; Worthey et al. 1992; Carollo and Danziger 1994a,b). As one can easily see the data present a large spread, mostly due to the uncertainties in deriving the [FORMULA] indices. In particular, in Fig. 1a we show the observed and predicted behavior of [FORMULA] vs. [FORMULA] when Model I is adopted. The bestfit to the data implies the following relation, [FORMULA], and is indicated in the figure. However, the spread in the data is large and this prevents us from drawing strong conclusions about a possible trend. The dotted lines in Fig. 1a represent the predictions of Model I obtained by means of the calibrations described before and adopting the same [Mg/Fe] ratio as predicted by the models, as one can see in Table 7. The agreement with the observed trend is not so good, showing that the slope of the predicted relation is steeper than that shown by the data and that the predicted [FORMULA] values do not cover the entire range in [FORMULA]. This is mostly due to the assumed IMF since Model II, which assumes a flatter IMF, predicts values for [FORMULA] which cover the whole range (see Fig. 2a).

[FIGURE] Fig. 1. a (left panel) Predicted and observed metallicity indices. The dotted line and open squares represent the [FORMULA] versus [FORMULA] predicted by Model I for galaxies of different masses, as indicated in Table 7. The dashed line and stars represent the real abundances of Fe and Mg as predicted by Model I for galaxies of different masses and arbitrarily translated in the [FORMULA] vs. [FORMULA] diagram. The error bars referring to the different data samples are also shown. b (right panel) Predicted and observed [FORMULA] versus mass relation. The predictions are from Model I.

In Fig. 1b we show the predicted and observed mass-metallicity ([FORMULA]) relationship. The data are from Carollo et al. (1993). The best-fit to these data indicate [FORMULA], where [FORMULA] is the total galactic mass (dark+luminous). The classic wind model recovers the slope of the [FORMULA] - mass relation, but with a zeropoint offset of [FORMULA] with respect to the observed distribution.

On the other hand, the classic wind model with the Arimoto and Yoshii (1987) IMF (Model II), as shown in Fig. 2b, predicts a slope much steeper than the observed one, although it agrees better than Model I with the [FORMULA] vs. [FORMULA] relation shown in Fig. 2a. It is worth noting that the Arimoto and Yoshii (1987) IMF well reproduces the abundances in the intergalactic medium (MG95; Gibson 1997; Gibson and Matteucci 1997). It is worth noting that in Figs. 1a and 2a and in all the others we show also the relation between real abundances predicted by our models. The relation between [FORMULA] and [FORMULA], arbitrarily translated in the plot of [FORMULA] versus [FORMULA], is indicated by the dashed lines. This is done only with the purpose of comparing the slope of the relation between real abundances with that of the relation between indices and they are very similar, indicating that the adopted calibration does not modify the predicted relation between Mg and Fe abundances. One of the main reasons for that is the adopted calibration which accounts for the right [FORMULA] ratio for each galaxy.

[FIGURE] Fig. 2. a The same as Fig. 1a but relative to the predictions of Model II. b The same as Fig. 1b but relative to the predictions of Model II.

In Fig. 3a,b we show the predictions of the inverse wind model of Matteucci (1994) which predicts a stellar [FORMULA] increasing with galactic mass. The slope of the [FORMULA] versus [FORMULA] relation is in better agreement than in the previous models, and the slope of the [FORMULA] vs. mass relation is also acceptable although the absolute values of the indices are too high.

[FIGURE] Fig. 3. a The same as Fig. 1a but for the predictions of Model III. b The same as Fig. 1b but for the predictions of Model III.

In Fig. 4a,b we show the results of Model IV with a variable IMF from galaxy to galaxy, which also predicts increasing [FORMULA] ratios with galactic mass. The agreement with the [FORMULA] vs. [FORMULA] data is marginally acceptable, but the slope of the mass-metallicity relation is too steep and the predicted absolute values of [FORMULA] are too low. The low absolute values of [FORMULA] are due to the fact that we used slopes steeper than the Salpeter (1955) one for the less massive galaxies and such slopes are not suitable for elliptical galaxies (see MG95) since they predict too low metallicities. However, other numerical experiments, where we used a variable IMF but with flatter slopes for each galactic mass (from x=1.4 in low mass galaxies to x=0.8 in high mass galaxies), have shown that there is a negligible difference in the predicted [FORMULA] vs. [FORMULA] relation while the mass-metallicity relation gets worse.

[FIGURE] Fig. 4. a The same as Fig. 1a but for the predictions of Model IV. b The same as Fig. 1b but for the predictions of Model IV.

In Fig. 5a,b we show the predictions of Model V calculated with the time-variable IMF as suggested by Padoan et al. (1997) and adapted to elliptical galaxies. The slope of this IMF is decreasing with time thus favoring massive stars at early epochs. A similar although more complex formulation of the Padoan et al. (1997) IMF has been recently adopted by Chiosi et al. (1998). The model behaves like a classic wind model, in the sense that the galactic wind occurs first in small galaxies and later in the more massive ones. Concerning the predicted indices, Fig. 5a shows that the [FORMULA] decreases for massive objects, due to the fact that the IMF in these galaxies is less biased towards massive stars than in smaller systems. This is, in turn, due to the fact that the slope of the IMF is inversely proportional to the gas density which is lower in more massive objects. This model predicts a sort of bimodal behavior for the [FORMULA] vs. mass relation and it does not fit the data better than the other models.

[FIGURE] Fig. 5. a The same as Fig. 1a but for the predictions of Model V. b The same as Fig. 1b but for the predictions of Model V.

Finally, in Fig. 6a,b the predictions of Model VI are shown. At variance with all the previous models, Model VI seems to reproduce well the observed slope of the [FORMULA] vs. [FORMULA] relation as well as the [FORMULA]- mass relation. The main problem with this model is the fact that the predicted ranges of [FORMULA] and [FORMULA] are too narrow compared to the observations, especially the range of [FORMULA]. Another potential problem of this model is the predicted [FORMULA] ratio which is [FORMULA] for each galaxy mass. This is a high value for ellipticals unless one believes in a Hubble constant [FORMULA], as discussed in Padovani and Matteucci (1993).

[FIGURE] Fig. 6. a The same as Fig. 1a but for the predictions of Model VI. b The same as Fig. 1b but for the predictions of Model VI.

In Figs. 7 and 8 we show the plot of the mass-metallicity ([FORMULA]) relation as predicted by Model I and Model II, obtained under different assumptions about the calibrating formula. As one can see, some of the calibrations give similar results such as those of Worthey et al. (1992), Casuso et al. (1996) and Buzzoni et al. (1992). These calibrations have in common the use of a solar ratio for [Mg/Fe] ([Mg/Fe]=0). On the other hand, the values of the indices obtained by using the calibrations of Barbuy (1994) and Tantalo et al. (1998) which adopt non-solar ratios, differ from the others and between themselves. It is worth noting that the use of different calibrations may lead even to different slopes for the [FORMULA]-[FORMULA] relationship.

[FIGURE] Fig. 7. [FORMULA]-mass relationships as predicted by Model I by using different calibrations, as indicated in Fig. 8. The best-fit to the data is also shown.

[FIGURE] Fig. 8. [FORMULA]-mass relationships as predicted by Model II by using different calibrations, as indicated in the figure. The best-fit to the data is also shown.

One criticism that could in principle be moved to the results discussed before is that we adopted mass-averaged metallicities and not luminosity-averaged metallicities, as it should be the case. In Fig. 9 we show the indices obtained by the luminosity- and mass-averaged metallicities calculated for the results of Model II, when the calibration of Tantalo et al. (1998) is applied. The luminosity-averaged metallicities, computed with the photometric model of Gibson (1997), are systematically slightly lower than the others and the difference is larger for smaller galaxies, as expected. However, the slope is the same in the two cases, showing that the use of mass-averaged metallicity for this kind of analysis is quite justified.

[FIGURE] Fig. 9. Predicted and observed [FORMULA] versus [FORMULA]. The predicted indices, relative to Model II, are calculated by averaging on the mass (stars) and on the visual luminosities (squares).

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© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998
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