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Astron. Astrophys. 356, 873-887 (2000)

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5. Results

5.1. Chemical mixing of the halo ISM

Starting with an ISM of primordial abundances, a first generation of ultra metal-poor stars is formed in the model. After the first high-mass stars exploded as SNe of Type II, the halo ISM is dominated by local inhomogeneities, since the SN events are spatially well separated and no mixing has yet occurred. In a very metal-poor medium, a single SN event heavily influences its surroundings, so that its remnant shows the element abundance pattern produced by that particular core-collapse SN. Stars born out of this enriched material therefore inherit the same abundances. Since SNe of different progenitor masses have different stellar yields, stars formed out of an incompletely mixed ISM show a great diversity in their element abundances.

Fig. 1 shows a cut through the computed volume, giving the density distribution of the halo ISM at four different times, ranging from the unmixed to the well mixed stage. Each panel shows a different mean metallicity and has a lateral length of 2.5 kpc. The density of the ISM ranges from [FORMULA] in the inner part of a remnant to [FORMULA] in the densest clouds, which is about the gas density of the solar neighbourhood (Binney & Tremaine 1987).

[FIGURE] Fig. 1. Cut through the computed volume, showing the density distribution of the halo ISM during the transition from the unmixed to the well mixed stage (see text for details).

Upper left: After 82 Myr (see below for the scaling of the time units with the assumed SFR), the mean metallicity of the halo ISM is [Fe/H] [FORMULA]. Most of the volume was not yet affected by SN events, which can be seen as bright patches in the otherwise homogeneously distributed ISM. In the inner part of the remnants most of the gas has been swept up and condensed in thin shells, which show up as dark, ring-like structures. The regions influenced are well separated and no or only slight mixing on a local scale has taken place. The halo ISM has to be considered unmixed and is dominated by local inhomogeneities. Stars forming in the neighbourhood of such inhomogeneities show an abundance pattern which is determined by the ejecta of a single SN or a mixture of at most two to three SN events.

Upper right: Beginning of the transition from the unmixed to the well mixed halo ISM, after 170 Myr at a mean metallicity of [Fe/H] [FORMULA]. The separation of single SN remnants has become smaller and a higher number of remnants may overlap. The abundance pattern in overlapping shells still shows a great diversity but is closer to the IMF averaged element abundance than at the completely unmixed stage.

Lower left: After 430 Myr, at a mean metallicity of [Fe/H] [FORMULA], enough SN events have occurred to pollute the mass of the whole ISM twice. Nevertheless there are still cells which were not influenced by a SN, showing therefore primordial abundances.

Lower right: After 1430 Myr, a mean metallicity of [Fe/H] [FORMULA] is reached. At this time, no patches with primordial abundances exist and the ISM starts to be chemically well mixed. The once homogeneous, primordial medium has now a lumpy structure with large density fluctuations and shows mainly an IMF averaged abundance pattern.

Our calculations confirm that the incomplete mixing of the ISM during the halo formation plays a significant rôle in the early enrichment of the metal-poor gas. This can be seen in the [El/Fe] ratios of the considered elements, shown in Fig. 2. The small, filled squares show the [El/Fe] ratio of single model-stars. For comparison, observed metal-poor stars are represented by open squares. In the case of O, Cr and Mn, observations taken before 1995 were marked with open triangles. This was done to highlight possible trends in the [El/Fe] ratio of these elements. If multiple observations of a metal-poor star occurred and the abundances had to be averaged, open diamonds were used (see Sect. 4 for details).

[FIGURE] Fig. 2. Element-to-iron ratio [El/Fe] of O, Mg, Si, Ca, Cr, Mn, Ni and Eu. Open circles depict [El/Fe] ratios of SN II models of the given progenitor mass. Small filled squares represent model stars, open symbols show observed stars.

The numbered circles show the [El/Fe] and [Fe/H] ratios of a single SN II with the indicated mass. The [Fe/H] ratio in the shell of the SN remnants are determined by the mass of the exploding star and the [FORMULA] of gas that are swept up by the shock front. This ratio represents the maximal abundance a single SN II can produce and determines the position of the open circles on the [Fe/H] axis in Fig. 2. If the swept up material subsequently mixes with the surrounding medium, this abundance decreases and stars with lower [Fe/H] abundance can be formed. On the other hand, the [El/Fe] ratio is determined solely by the stellar yields of a SN II and is independent of the mixing mass. For the uncertainties of Fe-ejecta and [El/Fe] ratios in individual SNe II see the discussion in Sect. 3.

We divide the chemical enrichment process in the early evolution of the halo ISM into different enrichment stages: at metallicities [Fe/H] [FORMULA], the ISM is completely unmixed and dominated by local inhomogeneities, originating from SN II events. At about [Fe/H] [FORMULA] the halo ISM shows an IMF averaged abundance pattern and has to be considered chemically well mixed. The continuous transition between these two phases is marked by incomplete mixing which gradually becomes better, as more and more high-mass stars explode as SN II.

The different phases in the enrichment of the halo ISM seen in Fig. 2 can be distinguished in all [El/Fe] - [Fe/H] plots. At very low metallicities ([Fe/H] [FORMULA]), only a few stars exist, showing a considerable spread in their [El/Fe] ratios, ranging from 0.5 dex in the case of Ni to more than 2 dex in the cases of O, Mg and Eu. At this stage, the scatter of the model stars is given by the spread in metallicities of the SN models. At the end of this early phase, less than 0.5% of the total halo ISM mass has been transformed into stars.

At a metallicity of [FORMULA] [Fe/H] [FORMULA], the SN remnants start to overlap and a first, incomplete mixing occurs. New stars form out of material which was influenced by several SNe of different masses. Therefore, they do not show the typical abundance pattern of a single SN, but show an average of the SNe which contributed to the enrichment of the local ISM. The spread in the metallicities gradually decreases from [Fe/H] [FORMULA] to -2.0, reflecting the ongoing mixing process as more and more SNe pollute the ISM. At the beginning of the well-mixed phase, star-formation has consumed about 2.5% of the total halo ISM mass.

This late phase is characterized by a well mixed ISM and begins at [Fe/H] [FORMULA], where the abundance scatter in the model is reduced to a third of its initial value. At this stage, the whole volume considered was influenced several times by SN events. This leads to an IMF averaged [El/Fe] ratio in the ISM which is the same as predicted by simple 1-zone models and which is observed in stars with metallicity [Fe/H] [FORMULA]. Even in an enriched medium, however, a SN event will still have an influence on its neighbourhood, although the change in the abundance pattern will not be as prominent as in a very metal-poor medium. This explains the fact, that even in the well-mixed case, the [El/Fe] ratios show a certain dispersion. At [Fe/H] [FORMULA], about 8% of the total halo ISM mass has been used to form stars.

The onset of SNe of Type Ia marks the beginning of a third phase in the chemical enrichment history of the galaxy, but we will not consider this phase any further. Note, that the mean [El/Fe] ratios at [Fe/H] [FORMULA] in our model do not have to be equal to zero (by definition the solar metallicity), since we have neglected the influence of SN Ia events.

To quantify the enrichment of the ISM we introduce the concept of the polluted mass [FORMULA] in a unit volume at time [FORMULA]. It is defined as the total mass which gets polluted by [FORMULA] isolated SNe, where [FORMULA] is the number of SNe in the unit volume that occurred during the elapsed time [FORMULA]. For a constant mixing-mass [FORMULA] swept up by a SN event, [FORMULA]. Since the polluted mass is directly proportional to the number of SNe, it can become larger than the total ISM mass in the unit volume, [FORMULA]. Furthermore, the pollution factor [FORMULA] is defined as the ratio [FORMULA] and only depends on [FORMULA], for fixed [FORMULA] and [FORMULA]. When [FORMULA], enough SNe have contributed to the chemical enrichment to theoretically pollute the entire ISM in the unit volume, even though there still may be patches of material which were not yet affected by any SN. A higher pollution factor results in a better mixing of the halo ISM, decreasing the local abundance differences and the amount of material with primordial abundances. Therefore, the ratio [FORMULA] determines the mixing efficiency in our model, i.e. how many SNe in the unit volume are needed to reach a certain value of [FORMULA]. Given [FORMULA], [FORMULA] and the mean, IMF integrated iron yield [FORMULA] of a typical SN II, the mean metallicity of the ISM then is determined by

[EQUATION]

where C is the solar iron abundance.

The local inhomogeneities of the ISM begin to disappear when most of the gaseous SN remnants start to overlap. This is the case when more or less every cloud in the halo was influenced at least once by a SN event, i.e. the pollution factor is about equal to one. With the adopted mixing mass of [FORMULA] this is the case at [Fe/H] [FORMULA]. This metallicity gives an upper limit for the end of the early phase and the beginning of the transition to the second, well-mixed enrichment phase.

Table 2 shows the pollution factor needed to reach the mean metallicities shown in the panels of Fig. 1. Also shown are the corresponding SN frequency [FORMULA] and elapsed time [FORMULA], which depend on our model parameters. Here, the SN frequency is defined as the number of SNe per kpc3 and depends on the total ISM mass in the unit volume, whereas the elapsed time scales with the average SFR as


[TABLE]

Table 2. Pollution factor and SN Type II frequency.


[EQUATION]

where [FORMULA] is the mean SFR in the unit volume in our model. Note that the evolution of the abundance ratios as a function of [Fe/H] is independent of the star formation timescale and the SFR specified in the model.

5.2. Comparison with observations

The scatter in the [El/Fe] ratios of the model stars as a function of [Fe/H] shows the same general trend for every element considered, independent of the individual stellar yields. The inhomogeneous mixing of the very metal-poor halo ISM at [Fe/H] [FORMULA] leads to a scatter in the [El/Fe] ratios of up to 1 dex. This scatter continuously decreases for higher metallicities, reflecting the ongoing mixing of the ISM. At [Fe/H] [FORMULA] the model stars show an IMF averaged abundance pattern with an intrinsic scatter of about 0.1 to 0.2 dex. This behaviour matches the general trend of the observations well, as can be seen in Fig. 2. The observations also show a large scatter at low metallicities which again decreases for higher [Fe/H], with some exceptions, however: the iron-group elements Cr and Mn show a strong decrease in the [Cr/Fe] and [Mn/Fe] ratio for lower metallicities. This behaviour can not be reproduced with our adopted metallicity-independent stellar yields and the progenitor-independent mixing mass.

Compared to the observations, the distribution of [Ni/Fe] ratios of the model stars in Fig. 2 shows a scatter that is much too small. This is most likely due to the choice of mass cuts in the SNe II models, which have been set with the aim to reproduce the average solar [Ni/Fe] ratio. Thielemann et al. (1996) discuss in detail that large variations can easily occur. See also the discussion in Sect. 3. We therefore now want to investigate whether the sequence of enrichment stages seen in our model is similar to the observed evolution of abundance ratios even in cases when the employed yields may be incorrect.

To this end we have normalized the scatter in [Ni/Fe] of the model-stars at low [Fe/H] to unity, and have similarly renormalized the range of values for the observed stellar [Ni/Fe] ratios to one. The mean values of both distributions were left unchanged. The resulting renormalized distributions are shown in Fig. 3. The remarkably good agreement of both distributions after this procedure indicates that the enrichment history of the halo ISM implied by the model is consistent with the data, even though the employed Ni yields are not. Based on similar comparisons, we conclude that the abundance ratio data of most elements except Mn and Cr are consistent with the predicted enrichment history, and the scatter plots in Fig. 2 can thus be used to compare the range of the theoretically predicted nucleosynthesis yields with observations.

[FIGURE] Fig. 3. Normalized nickel-to-iron ratio [Ni/Fe] as function of metallicity [Fe/H]. Symbols are the same as in Fig. 2. The scatter of the model-stars and the halo stars was normalized to unity to highlight the enrichment phases of the halo ISM.

To describe the transition from the metal-poor, unmixed to the enriched, well mixed ISM more quantitatively, the relative frequency of stars at a given [El/Fe] ratio has been analysed for the different enrichment phases. In the case of silicon, this detailed enrichment history is shown in Fig. 4. The different enrichment phases from [Fe/H] [FORMULA], [FORMULA] [Fe/H] [FORMULA] and [Fe/H] [FORMULA] are given in the panels from top to bottom. The solid line shows the relative frequency of observed halo stars per [Si/Fe] bin for each enrichment phase and the dashed line the relative frequency of computed model stars per bin. To account for the effect of observational errors on our data, we added a random, normally distributed error with standard deviation 0.1 dex in the [El/Fe] and [Fe/H] ratios to the model stars. The bin size in the [Si/Fe] ratio is 0.1 dex for observed and computed stars, while the position of the histogram for the model stars is shifted by 0.01 dex to the left for better visibility. The total number of stars included in the plot is given in the upper left corner of each panel, where [FORMULA] and [FORMULA] are the number of observed stars and of model stars, respectively.

[FIGURE] Fig. 4. Relative frequency of stars normalized to unity in [Si/Fe] bins for three different metallicity ranges (see text). The solid line shows observational data, the dashed line the model stars. The number of included stars is given in the upper left corner.

In the upper panel, the distributions of both the 22 observed and the 4226 model stars show a spread in the [Si/Fe] ratio of more than one dex. The distribution of the model stars shows two wide, protruding wings and a faint peak at [Si/Fe] [FORMULA]. The "right" wing shows a shallow rise from [Si/Fe] [FORMULA] to the peak. The "left" wing is not as extended and shows a rather steep cutoff at [Si/Fe] [FORMULA]. This asymmetry is due to the nucleosynthesis models of core-collapse SNe, which show a more or less constant value of [Si/Fe] [FORMULA] for progenitor masses in the range of [FORMULA], as can be seen in Fig. 2. The distribution of the halo stars peaks at the same location as the model stars but extends only down to [Si/Fe] [FORMULA]. We attribute this to the poor statistic of the data set, since this gap is filled in the middle panel.

The middle panel of Fig. 4 shows the same distribution for the intermediate mixing stage of the ISM. The distribution of the model stars now has smaller wings, and peaks at [Si/Fe] [FORMULA]. It is still broader than 1 dex, but the majority of the stars fall near the IMF averaged [Si/Fe] ratio. The prominent peak is caused by the already well-mixed regions, whereas the broad distribution shows that the halo ISM is still chemically inhomogeneous. The peak of the observational sample has shifted by about 0.2 dex to the right and lies now at [Si/Fe] [FORMULA]. Compared to the prediction of the model, the relative frequency of the halo stars is too high in the wings of the distribution and too low to the left of the peak.

The lower panel shows the late stage, where the halo ISM is well mixed. The broad wings have completely disappeared and only the very prominent peak at the IMF averaged value remains. The distributions of the 11 observed stars and the 370 000 model stars are in good agreement. At this metallicity no SN of Type Ia should have polluted the by now well mixed ISM and the metal abundance is high enough to restrict the impact of single SN II events on the ISM.

The most prominent feature which characterizes the different enrichment phases, is the intrinsic scatter in the abundances of metal-poor stars. This can be seen in Fig. 5, which shows the standard deviation of [Si/Fe] as a function of metallicity [Fe/H] for the model and the halo stars. The bin size used to compute the standard deviation was 0.1 dex in metallicity. The solid line shows the scatter of the unmodified model stars. The influence of observational errors on our data was simulated by adding a random, normally distributed error with standard deviation 0.1 and 0.2 dex in both [Si/Fe] and [Fe/H]. The resulting scatter in dependence of metallicity is given by the dashed and dotted lines. In the range of [FORMULA] [Fe/H] [FORMULA] the scatter has a more or less constant value of approximately 0.4 dex. It declines rather steeply in the range [FORMULA] [Fe/H] [FORMULA] and levels off again at metallicities higher than -2.0, depending on the assumed observational errors of 0.0, 0.1 or 0.2 dex. These curves show that for errors in this range the scatter at low metallicities is dominated by the intrinsic differences in the element abundances of single stars.

[FIGURE] Fig. 5. Scatter in [Si/Fe] of the model and observed stars. The solid line gives the scatter of the model stars, the dashed and dotted lines show the scatter of the model stars, folded with an error of 0.1 and 0.2 dex. Filled squares give the standard deviation of the observed stars (see text).

For comparison, the scatter in the [Si/Fe] ratio of observed halo stars is represented by filled squares. The observations were binned with a bin size of 0.5 dex to compute the standard deviations. To estimate the reliability of their scatter in [Si/Fe], we built several new data sets by adding a normally distributed random error with standard deviation 0.1 dex to the [Si/Fe] ratio and the metallicity of the stars. For each new data set, the standard deviation in the different bins was computed. The standard deviation for the results from these artificial data sets is given in the plot as 1-[FORMULA] error-bars. The scatter of the observed abundance ratios shows nicely the features already seen in the curves for the model stars. At the first stage of the enrichment, it is approximately constant, followed by a steady decline in the intermediate mixing phase. At higher metallicities, the scatter levels off again.

Since the scatter in [Si/Fe] at [Fe/H] [FORMULA] is about 0.4 dex the observational errors have little influence on the analysis at these low metallicities, unless unknown systematic or confusion errors were large enough to inflate the scatter at [Fe/H] [FORMULA] to also about 0.4 dex. On the other hand, observational errors do dominate the scatter at metallicities [Fe/H] [FORMULA], when the halo ISM is well mixed and the intrinsic scatter of the stars is negligible compared to the observational errors.

5.3. Individual elements and nucleosynthesis

Oxygen & Magnesium: As expected, the IMF averaged [El/Fe] ratios for O and Mg reproduce the mean abundance of the observed metal-poor stars nicely. The [O/Fe] ratio seems to be slightly too low, whereas [Mg/Fe] is slightly too high, but both deviations are smaller than 0.1 dex. No trend in the observational data of Mg can be seen and a trend in O only becomes visible if the observations of Israelian et al. (1998) are considered.

An important fact is that the scatter in the data, although increasing at lower metallicities, does not match the large scatter of more than two dex predicted by the stellar yields. Since no other mixing effects than the overlapping of SN remnants are included in our model, the expected scatter is determined by the nucleosynthesis yields. If gas flows and the random motion of stars in the halo accelerated the chemical mixing, a smaller scatter in the model data would be expected. Even then, the fact that the observed stars only show [O/Fe] or [Mg/Fe] ratios corresponding to the stellar yields produced by [FORMULA] SNe would remain unexplained.

If we assume a top-heavy IMF which favoured high-mass SNe, this problem could be solved. However, the abundance pattern of the other elements should then also reflect this, which is not the case. This leaves us with two explanations: Either the stellar yields of O and Mg or of Fe are incorrect (or both).

Since the exact location of the mass cut is not known, the actual Fe yields are not very well determined (Woosley & Weaver 1995; Thielemann et al. 1996) and direct observational information which links a progenitor to an ejected Fe mass is very limited, maybe with the exception of SN 1987A and 1993J. Otherwise only the IMF integrated Fe-yields are constrained, but not necessarily their progenitor mass dependence (declining, rising, or with a maximum, see Nakamura et al. 1999 and Sect. 3). In order to attain a fit to the observational data within our evolution model we would need to decrease the Fe yields of the 13 or [FORMULA] stars by a factor of six. Without adjusting Fe-yields of the more massive stars, and assuming a standard Salpeter IMF, this would increase the IMF averaged [O/Fe] and [Mg/Fe] ratio by about 0.3 dex (a factor of two) and would therefore result in a much too high IMF averaged value. Equally, every other abundance ratio would be affected by this change.

On the other hand, the stellar yields of the [FORMULA]-elements O and Mg could be too low for the 13 and [FORMULA] progenitors. The abundances of O and Mg are mainly determined in the hydrostatic burning phases and do not depend heavily on the explosion mechanism. Changing the stellar yields of O and Mg for the 13 and [FORMULA] progenitor stars would therefore require to adjust the existing stellar models, which suffer from uncertainties in the theory of convection and the treatment of rotation.

Silicon: The IMF averaged [Si/Fe] ratio and the scatter predicted by the stellar yields fit the observations perfectly. The decrease in the scatter for higher metallicities and therefore the different enrichment phases of the halo ISM are clearly visible. At [Fe/H] [FORMULA], the scatter in the model points reflects the scatter predicted by the stellar yields. During the transition from the unmixed to the well mixed ISM, in the range [FORMULA] [Fe/H] [FORMULA], the scatter decreases steadily and leads to the IMF averaged [Si/Fe] ratio, which is reached at [Fe/H] [FORMULA].

Calcium: The IMF averaged [Ca/Fe] ratio is about 0.2 dex lower than the observed mean for metal-poor stars. If the model data is shifted by this value to reproduce the mean of the observational data, the scatter for very metal-poor stars and the transition to the less metal-poor stars fits the data well. Note that, contrary to the other [FORMULA]-elements, the stellar yields of Ca are no longer approximately proportional to the mass of the progenitor. This behaviour becomes more pronounced in the cases of Cr, Mn and Ni.

Chromium & Manganese: The iron-peak elements Cr and Mn are both produced mainly during explosive silicon burning and show an almost identical, complicated dependence on the mass of the progenitor. The IMF averaged [El/Fe] ratio reproduces the observations of the less metal-poor stars well. The scatter predicted by the stellar yields is only about 0.8 dex and is not as large as for the [FORMULA]-elements.

A notable feature of the observations is the decrease of the [Cr/Fe] and [Mn/Fe] ratios for lower metallicities, seen in the newer data. If these trends are real, they can not be reproduced by metallicity-independent yields unless one assumes a progenitor mass dependent amount of mixing with the interstellar medium (Nakamura et al. 1999).

The upper limits of the [El/Fe] ratios of Cr and Mn are given by the stellar yields of a [FORMULA] SN and correspond to the highest [Cr/Fe] and [Mn/Fe] ratios seen in metal-poor stars, with the exception of the binary CS 22876-032 which shows an unusual high [Mn/Fe] ratio of 0.29 dex. Recent high-signal-to-noise, high-resolution data result in a lower [Mn/Fe] value for this star, placing it near the IMF averaged value (S. G. Ryan, private communication). On the other hand, the lower limits of the observed [Cr/Fe] and [Mn/Fe] ratios do not correspond at all with those given by the stellar yields, which show a ratio which is too high by up to 0.5 dex in the Thielemann et al. (1996) yields. Only a different choice of mass cuts as a function of progenitor mass (Nakamura et al. 1999) or as a function of metallicity would be able to rectify this.

Nickel: The stellar yields of the most important iron-peak element besides Fe completely fail to reproduce the observations. The scatter of the stellar yields is only about 0.5 dex compared to about 1.2 dex seen in the observational data. Compared to the mean of the observations, the IMF averaged abundance of the model-stars is about 0.1 dex to high. The small scatter in the [Ni/Fe] ratio originates from an almost constant ratio of Ni and Fe yields, which means that in the models Ni is produced more or less proportional to iron. To reproduce the scatter seen in the observations, Ni would have to depend on the progenitor mass differently from Fe. This is a shortfall of the employed yields, adjusted to reproduce the average solar [Ni/Fe] ratio with the choice of their mass cuts. Varying neutron excess in the yields, however, can change the [Ni/Fe] ratio drastically (Thielemann et al. 1996and Sect. 3).

Europium: The r-process element Eu reproduces the scatter and the mean of the observational data quite well. But compared to the [FORMULA]-elements O, Mg, Si and Ca its behaviour is very different. The highest [Eu/Fe] ratio is produced by low mass SN and the lowest ratio by high mass SN, as required when one constructs yields under the assumption that the r-process originated from SNe II (see Sect. 3). This is exactly the opposite to what is found for the [FORMULA]-elements. Therefore, a top-heavy IMF would lead to a steadily increasing [Eu/Fe] ratio, since the most massive SNe will explode first (cf. Oxygen & Magnesium).

5.4. Age-metallicity relation

Common 1-zone chemical evolution models are based on the assumption that the system is well mixed at all times. An important consequence of this assumption is a monotonically increasing metallicity, which leads to a well defined age-metallicity relation. Therefore, it is in principle possible to deduce the age of a star if its metallicity is known. This basic assumption of 1-zone chemical evolution models was dropped in our stochastic approach. Therefore, it is not surprising that the well defined age-metallicity relation has to be replaced by a statistical relation. In Fig. 6, the metallicity [Fe/H] of model stars is plotted against the time of their formation. Model stars are represented by small filled squares. For comparison the white line visible in the middle of the black strip shows the mean age-metallicity relation, corresponding to the relation given by a 1-zone model. As can be seen, there is no clear age-metallicity relation at any time. Stars which were formed in the first 500 million years show a metallicity ranging from [Fe/H] [FORMULA] up to [Fe/H] [FORMULA] and in one extreme case up to [Fe/H] [FORMULA]. On the other hand, stars with metallicity [Fe/H] [FORMULA] could have formed at any time in the first [FORMULA] years. Taking these huge uncertainties into account, it is no longer possible to speak of a well-defined age-metallicity relation.

[FIGURE] Fig. 6. Metallicity [Fe/H] vs. age of single model stars.

While the mean [Fe/H] abundance increases about linearly with time, the scatter in Fig. 6 again reflects the different enrichment phases of the ISM. The steep rise at early times marks the metal-poor and chemically inhomogeneous stage of the halo ISM. Contrary to this first phase, where SN events dominated the metal-poor ISM locally and the enrichment of isolated clouds could be very efficient, the late stage is characterized by a well mixed ISM and therefore an inefficient enrichment, which is reflected by the slow increase of the age-metallicity "relation" at later times.

5.5. Ultra metal-poor stars

From our calculations, we can deduce the number of metal-poor stars which we expect to observe in different metallicity bins. The normalized distribution is shown in Fig. 7 for bin sizes of 0.2 dex and 1.0 dex. The number distribution is approximately a power law with slope 0.7 for [Fe/H] [FORMULA] and slope 0.9 for [Fe/H] [FORMULA]. Also plotted are the observed data from a homogeneous intermediate resolution sample of Ryan & Norris (1991) and the combined high resolution data from Table 1, both rebinned to the large bin size (see Tables 3 and 4 for the numerical values). It is possible that the high resolution sample is incomplete in the range [FORMULA], whereas the intermediate resolution sample might have a shortage of stars with [Fe/H] [FORMULA]. Table 3 also lists the number of model stars expected in the three 1.0 dex bins, normalized such that the number of model stars in the range [FORMULA] [Fe/H] [FORMULA] is equal to the number of observed halo stars in the high-resolution sample in this metallicity range. As can be seen, in this case we expect [FORMULA] model stars with [FORMULA] [Fe/H] [FORMULA] while the high resolution sample contains none. If the ratio of stars in these two metallicity bins for this admittedly inhomogeneous sample is representative for the Galactic halo stars, this would suggest a genuine shortage of the most metal-poor stars. In this case a possible solution could be the pre-enrichment of the halo ISM by population III stars. It is conceivable that these already produced an iron abundance of [Fe/H] [FORMULA] before they disappeared, leaving only a pre-enriched ISM.

[FIGURE] Fig. 7. Metallicity distribution in the model at the end of the calculation ([Fe/H][FORMULA]). The number of stars with metallicity [Fe/H] is shown (a) in bins of 0.2 dex (b) in bins of 1.0 dex. In both cases the total number of stars is normalized to one. The solid squares show the intermediate resolution data of Ryan & Norris (1991), the solid triangles the combined high resolution data from Table 1, both binned with bin size 1.0 dex and similarly normalized. There are no observed stars with [Fe/H] [FORMULA]. See Tables 3, 4 and text.


[TABLE]

Table 3. Top: Relative frequency of stars in the homogeneous intermediate resolution survey of Ryan & Norris (1991), the combined high resolution data from Table 1 and our model, binned with binsize 1 dex.Bottom: Absolute numbers. The last row gives the number of stars per bin which we expect to be present, if our model gives a fair representation of the halo metallicity distribution. The number of model-stars is normalized to the number of stars in the range [FORMULA] [Fe/H] [FORMULA] in the high resolution sample. No star was detected with confirmed [Fe/H] [FORMULA], in contrast to the [FORMULA] stars predicted by the model.



[TABLE]

Table 4. Relative frequency of model stars, binned with binsize 0.2 dex.


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Online publication: April 17, 2000
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