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A different pattern for the [alpha/Fe] vs. [Fe/H] relation compared to the solar vicinity is observed in dwarf spheroidal galaxies (dSphs) of the Local Group, as shown in Figure 43, and this can be easily interpreted in the framework of the time-delay model coupled with different star formation histories.

Figure 43

Figure 43. Observed [alpha/Fe] vs. [Fe/H] in the Milky Way (small points) and in dSphs (points with error bars). Figure from Shetrone et al. (2001).

Before interpreting the [alpha/Fe] diagram, we recall the current ideas about the formation of the dSphs.

6.1. How do dSphs form?

Cold dark matter (CDM) models for galaxy formation predict that the dSphs, systems with luminous masses of the order of 107 Modot, are the first objects to form stars and that all stars in these systems should form on a timescale < 1 Gyr, since the heating and gas loss, due to reionization, must have halted the SF soon. However, observationally all dSph satellites of the Milky Way contain old stars indistinguishable from those of Galactic globular clusters and they seem to have experienced SF for long periods (> 2 Gyr, Grebel & Gallagher 2004). The histories of SF for these galaxies are generally derived from the color-magnitude diagram (e. g. Mateo 1998). By looking at the [alpha/Fe] vs. [Fe/H] relations for dSphs, as shown in Figure 43, one can immediately suggest, on the basis of the time-delay model, that their evolution should have been characterized by a slow and protracted SF, at variance with the suggestion of a fast episode of SF truncated by the heating due to reionization.

Dark matter in dSphs

The dSph satellites of the Milky Way are considered the smallest dark matter dominated systems in the universe. In the past years there have been a few attempts at deriving the amount of dark matter in dSphs, in particular by measuring the mass to light ratios versus magnitude for these galaxies (e.g. Mateo 1998; Gilmore et al. 2007). Gilmore et al. (2007) suggested that the dSphs have a shallow central dark matter distribution and no galaxy is found with a dark mass halo less massive than 5 × 107 Modot, as shown in Figure 44.

Figure 44

Figure 44. Dark matter in dSphs: mass to light ratios versus absolute V magnitude for some Local Group dSphs. The solid curve shows the relation expected if all the dSphs contain about 4 × 107 Modot of dark matter interior to their stellar distributions. Figure from Gilmore et al. (2007).

Observations of dSphs

In recent years there has been a fast development in the field of chemical evolution of dSphs of the Local Group due to the increasing amount of data on chemical abundances derived from high resolution spectra (e.g. Smecker-Hane & McWilliam, 1999; Bonifacio el al. 2000, 2004; Shetrone et al. 2001, 2003; Tolstoy et al. 2004; Bonifacio et al. 2004; Venn et al. 2004; Sadakane et al. 2004; Fulbright et al. 2004; McWilliam & Smecker-Hane 2005a, b; Monaco et al. 2005; Geisler et al. 2005). The abundances of alpha-elements (O, Mg, Ca, Si,) plus the abundances of s- and r- process elements (Ba, Y, Sr, La and Eu) were measured with unprecedented accuracy. Besides these high-resolution studies we recall also the measure of the metallicities of many red giant stars in several dSphs by Tolstoy et al. (2004), Koch et al. (2006), Helmi et al. (2006) and Battaglia et al. (2006) obtained from the low-resolution Ca triplet. An interesting result of Helmi et al. (2006) is that they did not find stars with [Fe/H] < -3.0 dex and that the metallicity distribution of the stars in dSphs is different from that of halo stars in the Milky Way. Other important information, as already mentioned, comes from the photometry of dSphs of the Local Group and in particular from the color-magnitude diagrams. From these diagrams one can infer the history of SF of these galaxies. We recall the studies of Hernandez et al. (2000), Dolphin (2002), Bellazzini et al. (2002), Rizzi et al. (2003), Monelli et al. (2003), Dolphin et al. (2005). The color-magnitude diagrams seem to indicate that the majority of dSphs had one rather long episode of SF with the exception of Carina for which four episodes of SF have been suggested (Rizzi et al. 2003).

Chemical evolution of dSphs

Several papers have appeared in the last few years concerning the chemical evolution of dSphs. For example Carigi et al. (2002) computed models for the chemical evolution of four dSphs by adopting the SF histories derived, from color-magnitude diagrams, by Hernandez et al. (2000). In their model they assumed gas infall and computed the gas thermal energy heated by SNe in order to study galactic winds. In fact, the dSphs must have lost their gas in one way or another (galactic winds and/or ram pressure stripping) since they appear completely without gas. They assumed that the wind is sudden and devoids the galaxy of gas instantaneously. The adopted IMF is the Kroupa et al. (1993) IMF as in the solar vicinity. They predicted a too high metallicity for Ursa Minor and did not match the correct slope for the observed [alpha/Fe] ratios, unless the history of SF in this galaxy was assumed to be different than suggested by its color-magnitude diagram, as shown in Figure 45.

Figure 45a
Figure 45b

Figure 45. Observed and predicted [O/Fe] vs. [Fe/H] relation for the galaxy Ursa Minor. Models and Figures by Carigi et al. (2002). The upper Figure presents models assuming only one burst of SF at high redshift, as suggested by the color-magnitude diagram of this galaxy. The lower Figure shows the predictions of a model with two separate bursts which seems to fit better the data.

Then, Ikuta & Arimoto (2002) proposed a closed box model (no infall nor outflow) for dSphs. In this Simple Model they had to assume some external cause to stop star formation, such as ram pressure stripping. They tested different IMFs and suggested that these galaxies had suffered very low SFRs (1-5% of that in the solar neighbourhood) and that the SF had a long duration (> 3.9-6.5 Gyr). In Figure 46 are shown their predictions for [Mg/Fe] in dSphs. Also here, the predicted slope of the [Mg/Fe] ratio is flatter than observed.

Figure 46

Figure 46. Observed and predicted [Mg/Fe] vs. [Fe/H] relation for the dSph Draco, Ursa Minor and Sextans. The different curves refer to different SF efficiencies (epsilonSF) expressed in Gyr-1, which are equivalent to the quantity nu. Figure from Ikuta & Arimoto (2002).

More recently, Fenner et al. (2006) suggested a model with a galactic wind for Sculptor: they indicated an efficiency of SF of 0.05 Gyr-1. They concluded, from the study of the [Ba/Y] ratio, that chemical evolution in dSphs is inconsistent with the SF being truncated after reionization (at redshift z = 8). In fact, the high value of this ratio measured in stars indicates strong s-process production from low mass stars which have very long lifetimes.

The results of Lanfranchi & Matteucci

Lanfranchi & Matteucci (2003; 2004 hereafter LM04) developed models for dSphs of the Local Group. First they tested a "standard model" devised for describing an average dSph galaxy. This model was based on the following assumptions:

In Figure 47 we show the [alpha/Fe] ratios for different alpha-elements and for different efficiencies of SF, as predicted by the standard model of Lanfranchi & Matteucci (2003).

Figure 47

Figure 47. Observed and predicted [alpha/Fe] versus [Fe/H]. The different lines refer to the "standard model" with different SF efficiency nu going from 1 (dashed-dotted lines) to 0.01 Gyr-1 (continuous lines). The points represent stars in different dSphs: Sagittarius (open triangles), Draco (filled hexagons), Carina (filled circles), Ursa Minor (open hexagons), Sculptor (open circles), Sextans (filled triangles), Leo I (open squares) and Fornax (filled squares). Figure and references from Lanfranchi & Matteucci (2003).

As one can see, the [alpha/Fe] ratios show a clear change in slope followed by a steep decline, in agreement with the data. The change in slope corresponds to the occurrence of the galactic wind which starts emptying the galaxy of gas. In such a situation the SF starts to decrease as does therefore the production of the alpha-elements from massive stars, whereas Fe continues to be produced since its progenitors have long lifetimes. This produces the steep slope: the low SF efficiency and the wind, which decreases furtherly the SF. In this situation, the time-delay model predicts an earlier and steeper decline of the [alpha/Fe] ratios, as we have already discussed.

In LM04, the histories of star formation of specific galaxie, as suggested by their color-magnitude diagrams, were taken into account, and they developed models for six dSphs: Carina, Ursa Minor, Sculptor, Draco, Sextans and Sagittarius. In Table 1 we show the assumed SF formation histories and model parameters. In particular, in column 1 are the galaxy names, in column 2 are the initial baryonic masses, in column 3 are the SF efficiencies, in column 4 are the wind parameters, in column 5 are the numbers of SF episodes, in column 6 are the times at which the SF episodes start, in column 7 are the durations, in Gyr, of the SF episodes and in column 8 the assumed IMF.

Table 1. Models for dSph galaxies. Mtotinitial is the baryonic initial mass of the galaxy, nu is the star-formation efficiency, lambda is the wind efficiency, and n, t and d are the number, time of occurrence and duration of the SF episodes, respectively.

galaxy Mtotinitial (Modot) nu(Gyr-1) lambda n t(Gyr) d(Gyr) IMF

Sextan 5 * 108 0.01-0.3 9-13 1 0 8 Salpeter
Sculptor 5 * 108 0.05-0.5 11-15 1 0 7 Salpeter
Sagittarius 5 * 108 1.0-5.0 9-13 1 0 13 Salpeter
Draco 5 * 108 0.005-0.1 6-10 1 6 4 Salpeter
Ursa Minor 5 * 108 0.05-0.5 8-12 1 0 3 Salpeter
Carina 5 * 108 0.02-0.4 7-11 2 6/10 3/3 Salpeter

In Figures 48, 49, 50, 51 and 52 we show the predictions for specific dSphs by LM04. As one can see, the [alpha/Fe] data in the dSphs are well reproduced and in particular the steep decline of the [alpha/Fe] ratio is well reproduced. This steep decline is due again to the low efficiency SFR, a feature common also to the other models, coupled with a strong and continuous galactic wind which gradually empties the galaxies of gas. In the previous models either the galactic wind was not present or it was assumed instantaneous or not as strong as in LM04, thus predicting a flatter slope for the descent of the [alpha/Fe] vs. [Fe/H], as we have already seen.

Figure 48

Figure 48. Observed and predicted [alpha/Fe] vs. [Fe/H] relation for the galaxy Carina. The different lines represent models with different SF efficiency. The continuous line represents the best model and corresponds to the efficiency nu = 0.1 Gyr-1. Figure from LM04.

Figure 49

Figure 49. Observed and predicted [alpha/Fe] vs. [Fe/H] relation for the galaxy Sculptor. The different lines represent models with different SF efficiencies (nu = 0.05, 0.2, 0.5 Gyr-1). The continuous line represents the best model (nu = 0.2 Gyr-1). Figure from LM04.

Figure 50

Figure 50. Observed and predicted [alpha/Fe] vs. [Fe/H] relation for the galaxy Ursa Minor. The different lines represent models with different SF efficiencies. The continuous line represents the best model (nu = 0.2 Gyr-1). Figure from LM04.

Figure 51

Figure 51. Observed and predicted [alpha/Fe] vs. [Fe/H] relation for the galaxy Draco. The different lines represent models with different SF efficiencies. The continuous line represents the best model (nu = 0.03 Gyr-1). Figure from LM04.

Figure 52

Figure 52. Observed and predicted [alpha/Fe] vs. [Fe/H] relation for the galaxy Sagittarius. The different lines represent models with different SF efficiencies. The continuous line represents the best model(nu = 3.0 Gyr-1) . Here the efficiency of SF is relatively high but the strong galactic wind makes it effectively much lower. Figure from LM04.

Lanfranchi et al. (2006) computed also the expected abundances of s- and r- process elements in dSphs, by adopting the same nucleosynthesis prescriptions used for the chemical evolution of the Milky Way. In particular, they adopted the prescriptions of Cescutti et al. (2006) for Ba, Y, La, Sr and Eu: Ba, Sr , La and Y are mainly s-process elements produced on long timescales by low mass stars (1-3 Modot), but they have also a small r-process component originating in stars in the mass range 12-30 Modot. The Eu instead is considered as a pure r-process element produced in the stellar mass range 12-30 Modot.

In Figures 53, 54, 55 and 56 we show the predictions of Lanfranchi et al. (2006) for s-and r-process elements in dSphs compared with the available data. Also in this case the agreement looks good, although more data are necessary before drawing firm conclusions. The general tendency for the alpha-elements in dSphs is to be less overabundant relative to Fe and the Sun than the stars of the solar vicinity with the same [Fe/H]. This is due to the lower SFR in dSphs (the effect is increased by the galactic wind) which acts to shift the curve [alpha/Fe] vs. [Fe/H] for the solar vicinity towards left in the diagram, whereas a stronger SF than in the solar neighbourhood moves the solar vicinity curve toward right in the diagram (see Figure 33). The same holds for s- and r- process elements: in this case, since [s/Fe] vs. [Fe/H] first increases sharply at low metallicities and then it flattens at higher ones (the opposite of what happens for the alpha-elements), the dSphs show a higher [s/Fe] than the stars in the solar vicinity at the same [Fe/H]. This shows again the effect of the time-delay model.

Figure 53

Figure 53. Predicted and observed [s,r/Fe] vs. [Fe/H] for the galaxy Carina. The model is from Lanfranchi, Matteucci & Cescutti (2006a), where the references for the data can be found.

Figure 54

Figure 54. Predicted and observed [s,r/Fe] vs. [Fe/H] for the galaxy Sculptor. The model is from Lanfranchi, Matteucci & Cescutti (2006a), where the references for the the data can be found.

Figure 55

Figure 55. Predicted and observed [Ba/Eu] vs. [Fe/H] for the galaxy Sculptor. The model is from Lanfranchi, Matteucci & Cescutti (2006a), the data and the figure are from Geisler et al. (2007).

Figure 56

Figure 56. Predicted and observed [s,r/Fe] vs. [Fe/H] for the galaxy Ursa Minor. The model is from Lanfranchi, Matteucci & Cescutti (2006a), where the references for the the data can be found.

The differences between the [s/Fe] vs. [Fe/H] in dSphs and the Milky Way are shown in Figures 57, where data and predictions of Ba and Eu for Sculptor are compared with data and predictions of Ba and Eu in the solar vicinity.

Figure 57a
Figure 57b

Figure 57. Comparison between data and predictions for Ba and Eu in Sculptor and in the Milky Way. Model for the Milky Way: continuous lines. Model for Sculptor: dashed lines. Data for the Milky Way: triangles. Data for Sculptor: full squares. Models from Lanfranchi & al. (2007) where the references to the data can be found.

Finally, another important constraint for models of galactic chemical evolution is represented by the stellar metallicity distribution. In Figures 58 we show the predictions for the stellar metallicity distribution of Carina compared with the observed one and the agreement is very good. The observed distribution is from Koch et al. (2006) who measured the metallicity of 437 giants in Carina by means of Ca triplet and then transformed it into [Fe/H] through a suitable calibration. In Figures 58 we also show the comparison between the stellar metallicity distribution in Carina and the G-dwarf metallicity distribution in the solar vicinity. As one can see, the Carina distribution lies in a range of smaller metallicities due to the lower efficiency of SF assumed for this galaxy.

Figure 58a
Figure 58b

Figure 58. Upper panel: stellar metallicity distribution for Carina. Model from Lanfranchi et al. (2006b). The assumed SF efficiency is nu = 0.15 Gyr-1 and the wind efficiency is lambda = 5. Two different histories of SF have been tested here: the one of Dolphin (2002) (continuous line) and that of Rizzi et al. (2003) (long dashed line), but this does not produce important differences in the results. The main difference between the two histories of SF is the number of bursts (3 in Dolphin and 4 in Rizzi et al.). Data from Koch et al. (2006). Lower panel: predicted stellar metallicity distribution for Carina compared with the predicted G-dwarf metallicity distribution in the solar neighbourhood (dashed line).

To this purpose, owing a word of caution is appropriate: in fact, the Ca triplet in principle traces the abundance of Ca and not that of Fe and we know that Ca and Fe evolve in a different way since Ca is mainly produced in Type II SNe, whereas Fe is produced mainly in Type Ia SNe. This different evolution of Ca and Fe leads, in the Koch paper, to an uncertainty of 0.2 dex. Besides that, the globular clusters which serve as calibrators for obtaining [Fe/H] lie in the range -2.0 - -1.0 dex, whereas Koch's data extend down to lower metallicities.

The good fit of the stellar metallicity distribution indicates that both the assumed history of SF and the IMF are close to reality.

LM04 predicted the stellar metallicity distribution for all the six dSphs and while for Carina the agreement is quite good for Sculptor they cannot reproduce the bimodal stellar distribution recently suggested by Tolstoy et al. (2004) and shown in Figure 59, since their model is a one-zone model. In Figure 60 are shown the predictions of LM04 for the Sculptor galaxy: it is clear from this Figure that to reproduce the two different stellar populations, one has to assume a multizone model possibly with different efficiencies of SF. Kawata et al. (2006) explained the bimodality of stellar populations in Sculptor as a consequence of dissipative collapse which produces higher metallicities at the center of the galaxy.

Figure 59

Figure 59. Observed stellar metallicity distribution for Sculptor. Data and figure are from Tolstoy et al. (2004): all the stars of Sculptor are indicated by the dotted line. The central stars are those indicated by the lower histogram with continuous line, whereas the stars beyond R > 0.2 Kpc are indicated by the upper histogram with continuous line.

Figure 60

Figure 60. Observed and predicted stellar metallicity distribution for Sculptor. The data (the dotted line of Figure 59) are represented by the histogram (long dashed). The models are from LM04: the solid line represents the best model, whereas the dashed lines represent models with higher (the curve on the right) or lower (the curve on the left) SF efficiency.

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