6. Conclusions and discussion
We have developed a stochastic model of the early chemical enrichment of the halo ISM. The aim of the model is to understand the scatter in the [El/Fe] ratios of observed stars at very low metallicities and the transition to the smaller scatter seen at higher metallicities. We computed the evolution of the abundances of the -elements O, Mg, Si and Ca, the iron-peak elements Cr, Mn, Fe and Ni and the r-process element Eu and investigated the mixing of the halo ISM.
We divide the enrichment history of the halo ISM into different phases. At metallicities [Fe/H] , the ISM is not mixed and is dominated by local abundance inhomogeneities, which are caused by individual Type II SNe of different progenitor masses. The second phase at [Fe/H] is defined by a well mixed ISM which shows IMF averaged abundances and is too metal-rich to be dominated by single SN events. A continuous transition from the first to the second phase occurs between [Fe/H] . The onset of SN Ia events marks the beginning of a third phase in the enrichment of the ISM.
The different enrichment phases of the halo ISM can be distinguished in every considered element, after normalising to the stellar yields. On the other hand, Fig. 2 shows that some stellar yields reproduce the IMF averaged element-to-iron ratio [El/Fe] well, but fail to reproduce the abundance scatter observed in metal-poor halo stars. Especially the stellar yields of the -elements O and Mg predict stars with a low element-to-iron ratio ([O/Fe] or [Mg/Fe] ), which are not observed. Typically, metal-poor halo stars show an overabundance of -elements of about [/Fe] to 0.4, as can be seen in Fig. 2. This is especially troublesome, since an attempt to solve this problem would either require a change in the iron yields of the 13 and 15 models of up to a factor six, which would raise every other mean element abundance by about 0.3 dex (a factor of two), or a change in the stellar yields of oxygen and magnesium, which are produced mainly during the hydrostatic burning phases.
Recent observations of metal-poor stars show a decrease of the [Cr/Fe] and [Mn/Fe] ratios for lower metallicities. It is not possible to explain these trends with the metallicity independent stellar yields we have used. Also, the stellar yields of the iron-peak element Ni predict a scatter in [Ni/Fe] which is much too small compared to the observational data. These problems argue strongly for a revision of the theoretical nucleosynthesis models and their extension to lower metallicities.
The unfortunate situation is, however, that there exists no theoretical foundation to do so for Fe-group yields as long as the supernova explosion mechanism is not understood. Thielemann et al. (1996) discussed in detail uncertainties of Fe-group yields due to the choice of mass cut, explosion energy and entropy, as well as the delay time between collapse and explosion, affecting also the neutron-richness of matter. Multidimensional aspects might add further degrees of freedom (Nagataki et al. 1997, 1998). Thus, there is an understanding of dependences, coincidences of abundance features etc., but at present only observational information combined with galactic evolution modelling like in the present paper or, e.g., Tsujimoto & Shigeyama (1998) and Nakamura et al. (1999) can try to provide sufficient constraints for a further understanding of supernova nucleosynthesis.
The advancing enrichment process of the halo ISM can be characterized by the pollution factor , defined as the ratio of the mass polluted by preceding SNe and the ISM mass , all in a unit volume. The enrichment history of the halo ISM now is mainly determined by the mixing efficiency which in turn is fixed by the ratio of the mass of swept-up material in a SN event and . The more mass a SN sweeps up, the less SN events are needed to reach a certain value of , making the mixing more efficient. also determines the average metallicity [Fe/H] of the halo ISM for a given pollution factor. A larger swept-up mass leads to a lower mean ISM metallicity and vice versa. Therefore, the metallicity where the transition from one enrichment stage to the next occurs, depends only on the mixing efficiency and not on the SFR. The SFR is only important if one is interested in the elapsed time that is needed to reach a certain mean metallicity of the halo ISM.
Figs. 4 and 5 support our adopted value of of swept up material. If the swept up mass is higher, the mixing would be more efficient, resulting in an IMF averaged chemical abundance pattern at lower metallicities. This would produce a narrower peak in Fig. 4 and a steeper slope in Fig. 5. Moreover, the whole curve in Fig. 5 would be shifted towards lower metallicities. On the other hand, a smaller mixing mass would reduce the efficiency of the enrichment of the ISM, which could be seen in a broader distribution at higher metallicities in Fig. 4 and a shallower slope in Fig. 5.
Standard 1-zone chemical evolution models predict a well defined age-metallicity relation, based on the assumption that the ISM is well mixed at all times. In our stochastic model the chemical inhomogeneity of the halo ISM and therefore the scatter in metallicity at any time is much too high to reasonably establish such a relation. Nevertheless, the steep rise with large scatter seen in Fig. 6 at very early times marks the chemically inhomogeneous enrichment phase, while the slow increase later-on reflects the well mixed, metal-rich ISM.
The results of our model also quantify the problem of the missing ultra metal-poor stars. From it we have deduced the expected number of ultra metal-poor stars (with [Fe/H] ) which should have been observed, normalized to the number of halo stars in the combined high-resolution sample with metallicities in the range [Fe/H] . We expect about ultra metal-poor stars whereas none was found to date. It is possible that Population III stars have caused a pre-enrichment of the ISM to [Fe/H] and already have disappeared before the onset of the formation of the Galaxy.
© European Southern Observatory (ESO) 2000
Online publication: April 17, 2000