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

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2. The model

Very metal-poor halo stars show a great diversity in their element abundances and therefore a scatter in their element-to-iron ratios [El/Fe] of order 1 dex. This scatter gradually decreases at higher metallicities until a mean element abundance is reached which corresponds to the [El/Fe] ratio of the stellar yields integrated over the initial mass function (IMF). The aim of our stochastic halo formation model is to understand the trends seen in the observations and to investigate how the metal-poor interstellar medium (ISM) in the halo evolves chemically.

Our fully 3D-code, contrary to 1-zone chemical evolution models, enables us to resolve local inhomogeneities in the ISM with a spatial resolution of 50 pc. All in all, we model a volume of (2.5 kpc)3, divided into [FORMULA] cells. Every cell of our grid contains detailed information about the enclosed ISM and the mass distribution of stars. We consider simultaneously the evolution of nine elements, the [FORMULA]-elements O, Mg, Si and Ca, the iron-peak elements Cr, Mn, Fe and Ni and the r-process element Eu.

Our initial conditions assume a halo ISM consisting of a homogeneously distributed single gas phase with primordial abundances and a density of 0.25 particles per cm3, which gives a total mass of about [FORMULA] in a volume of (2.5 kpc)3. We adopt a constant time-step of [FORMULA] years since it has to be longer than the dynamical evolution of a supernova (SN) remnant and shorter than the lifetime of the most massive stars. At each time-step 20 000 cells are chosen randomly and independently of each other and of the state of the enclosed ISM. Each selected cell may create a star with a probability proportional to the square of the local ISM density (Larson 1988). The number of stars formed per time-step is the product of the number of cells tested with the probability of star formation in each cell. Various combinations of these parameters are possible to achieve a given SFR; the choice of 20 000 cells proved computationally convenient. The absolute value of the SFR influences the time-scale of the enrichment process (cf. Sect. 5.1), but not the evolution of [El/Fe]-ratios as function of [Fe/H]. Therefore, the main results of this paper are insensitive to the values of these parameters.

The mass of a newly formed star is chosen randomly from a Salpeter IMF. The lower and upper mass limits of the IMF are taken to be [FORMULA] and [FORMULA], respectively. About 5000 stars are formed on average during each step. Newly born stars inherit the abundance pattern of the ISM out of which they form, carrying therefore information about the state of the ISM at the place and time of their birth. To determine the lifetime of a star an approximation to the metallicity dependent mass-lifetime relation of the Geneva Stellar Evolution and Nucleosynthesis Group (cf. Schaller et al. 1992; Schaerer et al. 1993a; Schaerer et al. 1993b; Charbonnel et al. 1993) is used, given by


where T is the lifetime in units of [FORMULA] yr, Z the metallicity in units of solar metallicity [FORMULA] and M the mass in units of solar masses [FORMULA].

Stars in a range of [FORMULA] will explode as core-collapse supernovae (SNe II), resulting in an enrichment of the neighbouring ISM. Stellar yields are taken from Thielemann et al. (1996) and Nomoto et al. (1997) for all elements except Eu. Since there are no theoretical predictions of stellar Eu yields, we use the indirectly deduced yields of Tsujimoto & Shigeyama (1998) which assume that r-process elements originate from SNe II (see the discussion of stellar yields in Sect. 3). We linearly interpolate the stellar yields given in these papers, since we use a finer mass-grid in our simulation. For SNe with masses below [FORMULA] stellar yields are not available. Since the nucleosynthesis models show declining yields towards low progenitor masses, we have for the interpolation arbitrarily set the yields of a [FORMULA] SN to one thousandth of those of a [FORMULA] SN. The interpolation gives IMF averaged values of the [El/Fe] ratios, which are in good agreement with the observed mean values of metal-poor stars in all elements except Ca, which shows a [Ca/Fe] ratio that is about 0.3 dex lower than the observed mean. We do not include supernovae (SNe) of Type Ia, since we are only interested in the very early enrichment of the halo ISM, which is dominated by SNe of Type II.

Intermediate mass stars will evolve to planetary nebulae, returning only slightly enriched material in the course of their evolution. This locally influences the enrichment pattern of the gas, since metal-poor material is returned into the evolved and enriched ISM. It will not change the element abundances [El/H] significantly, but can affect the local element-to-iron ratios [El/Fe] considerably. Low mass stars do not evolve significantly during the considered time. In our model, they serve to lock up part of the gas mass, affecting therefore the local element abundances [El/H] in the ISM.

Since the explosion energy of a core-collapse supernova (SN II) depends only slightly on the mass of its progenitor (Woosley & Weaver 1995; Thielemann et al. 1996), every SN II sweeps up a constant mass of about [FORMULA] of gas (Ryan et al. 1996; Shigeyama & Tsujimoto 1998). In our model the radius of the SN remnant then is computed from the local density of the ISM and lies typically between 100 pc and 200 pc. The ejecta of the SN II and all the swept up, enriched material are condensed in a spherical shell which is assumed to be chemically well mixed. The material in the shell subsequently mixes with the ISM of the cells where the expansion of the remnant stopped. The interior of the remnant, where all the material was swept up, is assumed to be filled with about [FORMULA] of dilute gas from the SN event with the corresponding metal abundances. This gas is unable to form stars until it is swept up by another SN event and mixed with the surrounding ISM. Thus this material contributes to the enrichment only after some delay.

The star formation rate of cells influenced by the remnant will rise, since their density is higher than the average density of a cell and the probability to form a star is assumed to be proportional to the square of the local density. It is still possible to form stars in the field, but cells that are influenced by a SN remnant are favoured. Stars which form out of material enriched by a single SN inherit its abundances and therefore show an abundance pattern which is characteristic for this particular progenitor mass. The most metal-poor stars that form out of material which was enriched by only one SN would therefore allow to reconstruct the stellar yields of single core-collapse SN, if the average swept up mass and the absolute yield of one element were known (Shigeyama & Tsujimoto 1998).

The SN remnant expansion is the only dynamical process taken into account in our model. Therefore, this model has the least possible mixing of the ISM. This is the opposite limit as in the case of closed box models, which assume a complete mixing of the ISM at all times and are therefore not able to explain the scatter seen at low metallicities. We continue our calculation up to an averaged iron abundance of [Fe/H] [FORMULA]. At this metallicity SN events of Type Ia, which we have not included in our model, would start to influence the ISM.

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© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000