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Astron. Astrophys. 341, 709-724 (1999)

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2. Chemically consistent galaxy evolution models

Our galaxy evolution model has been described in detail earlier by Fritze - von Alvensleben 1989, Krüger et al. 1991, Fritze - von Alvensleben & Gerhard 1994 and Lindner et al. 1996. In the following brief outline we concentrate on the chemical evolution, especially on our new concept of chemical consistency which is a considerable step towards a more realistic galaxy modeling. Chemically consistent models account for the increasing initial stellar metallicities of successive generations of stars and use several sets of stellar evolutionary tracks, stellar lifetimes and yields, color calibrations and spectra appropriate for the different metallicity subpopulations present in any type of galaxy. While single burst stellar populations, like star clusters, are well described by one common (age and) metallicity for all stars, the stars in any system with an extended or more complex star formation history (SFH ) have a dispersion not only in age but also in initial metallicity.

Our chemically consistent models describe the first stars forming in a (proto-) galaxy using the lowest metallicity stellar tracks, lifetimes, yields, etc. and consistently use input data bases for higher metallicity as the ISM abundance increases. Our models, however, do not include any dynamical aspects, we have to assume that the gas is always well mixed (cf. Sect. 2.2).

Chemically consistent models can now be developed because sufficiently complete and homogeneous sets of physical input data are becoming available for a range of metallicities, including metallicity dependent stellar yields. With our new model approach and extensive input data bases we calculate in detail the time and redshift evolution of abundances for a large number of different elements including SNI contributions from carbon deflagration white dwarf binaries.

Chemically consistent models similar in principle to the ones presented here were used by Timmes et al. (1995) using the same Woosley & Weaver (1995) yields for massive stars but older yields (Renzini & Voli (1981) for stars with [FORMULA]. Recent chemically consistent models by Portinari et al. (1998) use the Padova set of stellar input physics. Both approaches aim at describing the Galactic enrichment history by comparing to observed stellar abundance patterns.

2.1. General description of models

Starting from an initial gas cloud of mass [FORMULA] stars are formed continuously in a 1-zone model according to a given star formation law. The distribution of the total astrated mass to discrete stellar masses in the range 0.15 [FORMULA]... 40 [FORMULA] is described by a Scalo 1986 initial mass function (IMF) as specified in Eq. (1).

[EQUATION]

with normalization factors [FORMULA] and [FORMULA], slopes [FORMULA], [FORMULA], and [FORMULA] and mass limits [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. The IMF is normalized to [FORMULA] to account for a fraction of astrated material stored away from enrichment and recycling processes in substellar objects [FORMULA] (Bahcall et al. 1985). This normalization at the same time brings our model M/L values after a Hubble time into good agreement with observed M/L values for the respective galaxy types. The influence on our results from the use of this Scalo IMF as compared to a single Salpeter slope [FORMULA] is discussed in Sect. 2.3.5.

For galaxies of various spectral types, we use different parametrisations of their star formation rates (SFR) following Sandage's (1986) semi-empirical determinations. SFRs [FORMULA] in spiral galaxies are assumed to be linear functions of the gas content [FORMULA] with characteristic time scales [FORMULA] ranging from about 2 to 10 Gyr for Sa, Sb, and Sc, respectively, and a constant rate for Sd galaxies (cf. Eq. 2).

[EQUATION]

The total mass is assumed to be constant ([FORMULA]), since we restrict ourselves to closed box models. All masses are given in solar units ([FORMULA]). The characteristic timescale for SF [FORMULA] is defined by [FORMULA]. Gas recycling due to stellar winds, supernovae and planetary nebula is included consistently accounting for the finite stellar lifetimes [FORMULA] of stars of mass m, i. e., no Instantaneous Recycling Approximation is used (cf. Sect. 2.2).

Since dynamical effects are not included in our models, we cannot account for the internal structure or gradients in spiral galaxies or DLA absorbers. Our closed box models do not allow for galactic winds which clearly are important for dwarf galaxies but presumably not for spiral galaxies or their massive DLA progenitors (cf. A. Wolfe (1995), Wolfe & Prochaska (1997), and references therein).

For these SFRs our chemically consistent spectrophotometric models (cf. Möller et al. 1996, Fritze - v. Alvensleben et al. 1996) give detailed agreement, not only with average broad band colors observed for the respective galaxy types from U through K, but also with detailed emission and absorption features of the template spectra from Kennicutt's (1992) atlas. They also give agreement with the observed redshift evolution of galaxy colors for the respective types at least up to [FORMULA], i.e. over roughly a third of the Hubble time and with observations of optically identified QSO absorbers (Lindner et al. 1996).

Although our simple 1-zone models are not able to account for any abundance gradients along (proto-)galactic disks, the range of observed average HII region abundances [FORMULA] for nearby spiral types Sa through Sd as given by Zaritsky et al. (1994) and Oey & Kennicutt (1993) is well covered by our Sa through Sd models at [FORMULA] (cf. Fig. 6 in Sect. 2.3.5). Of particular interest for the comparison with DLA abundances which - most probably - probe the outer regions of the absorbing galaxies are the HII region abundances measured by Ferguson et al. (1998). For three late type spirals with large HI-to-optical sizes they find [FORMULA] solar at 1.5 - 2 optical radii. For their galaxies this corresponds to off-center distances between 12 and 42 kpc where the HI column density still is several [FORMULA]. Impact parameter found for optically identified DLA absorber galaxies typically are in the range 10 - 20 kpc.

2.2. Input physics

Besides the two basic parameters of our models, IMF and SFR, stellar yields [FORMULA] for elements i, stellar remnant masses [FORMULA], and stellar lifetimes [FORMULA] are required as input physics for our models. These data are needed to calculate the total mass E(t) ejected by stars above the turn-off mass [FORMULA]

[EQUATION]

that, together with the SFR, determines the time evolution of the gas mass in our closed box models:

[EQUATION]

Our models only aim at describing average gas phase abundances without accounting for the multiphase nature of the ISM, they assume perfect and instantaneous mixing of the material rejected by the stars. For a simple 1-zone model, the abundance [FORMULA] for each element i is calculated from Eq. (3c)

[EQUATION]

with

[EQUATION]

We follow the formalism outlined by Matteucci & Greggio (1986) and Matteucci & Tornambè (1987) to split up the IMF in the mass range between 3 and 16 [FORMULA] into some fraction A of binary stars that give rise to SNIa in the carbon deflagration white dwarf binary scenario and a fraction [FORMULA] of single stars. The term

[EQUATION]

describes the ejection contribution to an element i from single low mass stars,

[EQUATION]

the respective contribution of type Ia SNe in binary systems with binary mass [FORMULA]. After its lifetime the primary star [FORMULA] is assumed to transform into a white dwarf, ejecting its envelope into the ISM. Later, the secondary [FORMULA] evolves into a red giant, fills its Roche lobe and all the material is assumed to flow onto the primary which eventually reaches the Chandrasekhar limit and gives rise to a carbon deflagration SNIa event. The parameter A gives the fraction of stars in the mass range 1.5 - 8 [FORMULA] that finally give rise to a SNIa event, [FORMULA] and [FORMULA] is - in analogy to the turn-off mass for single stars - the smallest mass fraction that contributes to SNIa at time t. [FORMULA] is an assumed distribution function for binary relative masses. As proposed by Matteucci & Greggio, the parameter A is fixed by the requirement that for a Milky Way model Sbc the resulting SNIa rate at [FORMULA] Gyr is equal to the observed one (Cappellaro et al. 1997). This gives us [FORMULA] which we use for all galaxy types. The enrichment contribution of single stars in this mass range is given by

[EQUATION]

and the SNII contributions from stars above 8 [FORMULA] are described by

[EQUATION]

2.2.1. Cosmological model

To compare the results from our galaxy evolution calculations with the observed element abundances in DLA systems redshift dependent values are needed. To convert any evolution in time to a redshift evolution we adopt a Friedmann-Lemaître model with vanishing cosmological constant ([FORMULA]) and cosmological parameters [FORMULA] and [FORMULA]. As redshift of galaxy formation we chose [FORMULA] = 5. These parameters are in conformity with our spectrophotometric models (cf. Möller et al. 1996, Fritze - v. Alvensleben et al. 1996) and are used throughout the paper. The relation between redshift and time is then calculated directly obtained via (Eq. 4) for the Hubble-time [FORMULA].

[EQUATION]

2.2.2. Supernova yields

Supernova explosions of type II (SN II), i.e. stars heavier than about ten solar masses, are the most productive suppliers of heavy elements to the interstellar medium (ISM). SN type I explosions also supply a considerable contribution to the ISM metallicity (see Nomoto et al. 1997) in case of some elements, i.e. Fe, Ni, Cr and Mn (cf. last row of Table 5); whereas single stars of intermediate and small mass ([FORMULA]) supply considerable contributions only to the abundances of elements C, N and O (cf. Table 6).

Woosley & Weaver (1995) have calculated nucleosynthetic yields of about 144 isotopes from altogether 32 elements (H, He, Li, Be, B, C, N, O, F, Ne, Na, Mg, Al, Si, P, S, Cl, Ar, K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Ge) ejected from SN II explosions of progenitor stars with 12, 13, 15, 18, 20, 22, 25, 30, 35 and 40 solar masses. They considered five different initial metallicities (Z = [FORMULA], Z = 0.1 [FORMULA], Z = 0.01 [FORMULA], Z = [FORMULA] [FORMULA] and Z = 0) and three models with different explosion energies (labeled A, B and C) for very massive stars. For our models we use their SNII yields for 16 of the most abundant elements (H, He, C, N, O, Mg, Al, Si, S, Ar, Ca, Cr, Mn, Fe, Ni and Zn). Ejecta (given in solar mass units [FORMULA]) are listed in Table 5 for all five initial metallicities summed over all isotopes for each element. In the last column of this table we see that differences [FORMULA] between the total ejection of all elements ([FORMULA]) and the sum of ejecta from our 16 selected elements is negligible (between 0.3% and 4.1%).

Ejected masses of all isotopes of one element are added because we are only interested in total element abundances. Woosley & Weaver (1995) did not take into consideration any radioactive decay of isotopes after their production in the SN II explosion. In Table 1 we list those radioactive decays of isotopes which considerably contribute to the abundances of Fe, Cr and Mn. All contributions of other isotopes are negligible for our purpose.


[TABLE]

Table 1. Radioactive decays relevant for our yields


Rows containing results from model B and C are marked with * and [FORMULA] attached to the stellar mass in the first column of Table 5. We see that for larger explosion energy the ejected mass of heavy elements increases. ".0000" in Table 5 indicate that respective values are smaller than [FORMULA].

For an overview of the influence of different initial metallicity we list in Table 5 as an example the SN II yields for a "typical" (i.e. 25 [FORMULA]) star (model A). Ejecta for Z = 0 are clearly smaller than for other metallicities. On the whole, Z=0 yields differ drastically from those for [FORMULA] but these exceptional data do not affect our results because after the first few timesteps our program switches to higher metallicity input data. However, yields for other metallicities likewise do not show any general trend, neither with respect to initial stellar mass for fixed metallicity nor with respect to initial metallicity for fixed stellar mass, and this applies to all elements considered. In case of iron and carbon this is illustrated in Fig. 1 and Fig. 2.

[FIGURE] Fig. 1. Iron yields are given in units of stellar mass m_star as a function of stellar mass m_star (given in units of solar mass M_sun).

[FIGURE] Fig. 2. Carbon yields (given in units of stellar mass m_star) as a function of stellar mass m_star (given in units of solar mass M_sun).

In Fig. 1 we present stellar iron yields for the five initial metallicities calculated by Woosley & Weaver 1995. Yields are given as a fraction of the total stellar mass [FORMULA] and stellar masses are given in solar units [FORMULA]. Lines do split at [FORMULA] and 35[FORMULA] indicating different explosion models. In case of solar metallicity (Z = [FORMULA]) the separate lines for model A, B and C are indicated for stellar mass larger than 25 [FORMULA]. We see that generally more mass is ejected for larger explosion energies. However, no distinct trend neither with increasing stellar mass nor with increasing metallicity can be found.

We use element yields from SNIa calculated from Nomoto's deflagration model W7 (Nomoto et al. 1997), which are presented in the last row of Table 5. These SNIa yields are available for solar metallicity only. However, no important metallicity dependence is expected for SNIa yields.

2.2.3. Yields from intermediate and low mass stars

Intermediate mass stars ([FORMULA]) contribute little to the total metal enrichment of the ISM, they are, however, important for elements C, N and O. Stars of mass less than [FORMULA] do not contribute any metals to the ISM. We use up-to-date stellar yields for three different initial metallicities (Z = [FORMULA], Z = 0.2 [FORMULA] and Z = 0.05 [FORMULA]) calculated by van den Hoek & Groenewegen 1997. The data are listed in Table 6 for stellar masses ranging from 0.9 [FORMULA] to 8 [FORMULA]. Negative values indicate consumption instead of ejection of mass and "0.000" indicate values smaller than [FORMULA]. Yields of the same element but for different metallicities are arranged in neighboring columns of the table to make comparison easy. In case of elements C, N and O differences are quite significant but no trend can be found.

In Fig. 2 we present stellar carbon yields in units of stellar mass [FORMULA] for five initial metallicities calculated by Woosley & Weaver (1995). For stellar masses less than 8 [FORMULA] the plot contains the results calculated by van den Hoek & Groenewegen (1997). Their initial metallicities differ from those used by Woosley & Weaver and we combine those metallicities which are next to each other as listed in Table 2.


[TABLE]

Table 2. Metallicities combined from Woosley & Weaver (1995) for high mass stars and van den Hoek & Groenewegen (1997) for intermediate mass stars


As for iron in Fig. 1 carbon lines do split at [FORMULA] and [FORMULA] indicating different explosion models and clearly more carbon is ejected for larger explosion energy. Likewise no distinct trend with increasing stellar mass or increasing metallicity is visible. These findings are typical not only in case of iron and carbon but for almost all elements under investigation.

2.2.4. Stellar remnants and lifetimes

Neither Woosley & Weaver (1995) nor van den Hoek & Groenewegen (1997) report any stellar lifetimes [FORMULA]. Hence we adopt them from the stellar evolutionary tracks calculated by the Geneva group who gives lifetimes for two different initial metallicities Z = [FORMULA] and Z = 0.05 [FORMULA], listed in Table 7. For stellar masses less than 0.9 [FORMULA] there is no difference for lifetimes between the different metallicities and lifetimes are equal to or larger than the Hubble time, anyway. Hence, only data for [FORMULA] 0.9 [FORMULA] are of interest. To coordinate with yields for different initial metallicities we use [FORMULA] in case of Z = [FORMULA] and [FORMULA] for all other metallicities.

Masses of stellar remnants have been calculated by Woosley & Weaver (1995) for massive stars and by van den Hoek & Groenewegen (1997) for intermediate mass stars. Their results are reported in Table 8 and Table 9, respectively.

2.3. Discussion of models

Now the influence of different initial metallicities and explosion energies (described in the previous Sect. 2.2) on the results of our chemical evolution models will briefly be discussed.

2.3.1. Chemically consistent models

During the evolution of any galaxy the ISM is continuously enriched with metals. Hence it is reasonable to assume that stars which are formed in early phases are very poor in metals and consequently we need to use input data (yields, remnants and lifetimes) of very low metallicity in the beginning of the galaxy evolution. With increasing time the metal content of the ISM is growing and after each time step the actual metallicity is determined to select the appropriate input data. In earlier evolution models only solar metallicity data have been available. Chemically consistent models take into account increasing metal enrichment of the ISM from which successive generations of stars are born and hence are more realistic than models using solar data. A comparison of chemically consistent calculations only with results from models using solar metallicity exclusively for Sa, Sb, Sc and Sd galaxies is shown in Fig. 3. We see that chemically consistent models in general produce less metals than calculations with solar input data because metallicity dependent stellar yields are smaller than their solar counterparts. In the following we will omit curves of Sb and Sc galaxies because they lie between those of Sa and Sd galaxies.

[FIGURE] Fig. 3. Time evolution of global metallicity for Sa, Sb, Sc and Sd galaxies: comparison of results calculated with chemical consistent models vs. models using exclusively solar values.

The average ISM metallicity of our Sb model after a Hubble time is seen to be about 2/3 [FORMULA] in good agreement with recent ISM abundance determinations (cf. Cardelli & Meyer 1997, Sofia et al. 1997, and references therein), H II region abundances (e.g. Vilchez & Esteban 1996) for the solar neighborhood and with B-star abundances (e.g. Kilian et al. 1994, Kilian-Montenbruch et al. 1994).

2.3.2. The influence of stellar yields

As discussed in Sect. 2.2.2 Woosley & Weaver's yields show no clear trends neither with stellar mass at fixed metallicity Z nor with Z at fixed stellar mass. In particular yields for Z = 0 differ drastically from those with [FORMULA].

We decided to use Woosley & Weaver's data because they give yields for five different metallicities which is important for our concept of chemical consistent models. It should be mentioned that there are yield data from other authors.

Thielemann et al. (1996) published SN II yields for solar initial metallicity which to some extent differ from those of Woosley & Weaver. For a detailed investigation of the effects of these differences we refer the reader to D. Thomas et al. (1998). Portinari et al. (1998) take mass loss by stellar winds into account and give stellar yields for a few elements for five initial metallicities.

The impact of yield uncertainties on our results is hard to quantify. Even significant changes for a star of given mass and metallicity, however, do hardly affect the global evolution due to the smoothing power of the IMF.

From a comparison of the stellar yields given by various authors we conclude that while yield differences may have strong impact on abundance ratios of certain elements - which we do not attempt to interpret - they will not strongly affect the abundance evolution and hence our conclusions.

Yield uncertainties may slightly change our enrichment calculations for some elements, e.g. for Fe (or for typical wind elements C, N, O, which, however, we do not discuss since there are very few precise DLA data), but certainly not to the extent as to affect our conclusions which are based on a series of elements for many of which stellar yields are not controversial.

2.3.3. Different explosion energies for SN II

A comparison of results from chemically consistent evolution models for Sa and Sd galaxies using different explosion energies for SN II yields calculated by Woosley & Weaver (1995) is shown in Fig. 4. We see that the curves representing the time evolution of the metal content of the ISM are roughly similar for the three SN II models (named A, B and C by Woosley & Weaver) but they are shifted to larger abundance in case of larger explosion energy. Curves for model B always lie between those of model A and C and will be omitted in the following studies.

[FIGURE] Fig. 4. Time evolution of global metallicity for Sa and Sd galaxies: comparison of results calculated with chemical consistent models using different energies for SN II explosions: models A, B and C from Woosley & Weaver (1995)

2.3.4. Evolution of selected element abundances

As another improvement of our chemical evolution models we can calculate abundances [X/H] (cf. Eq. 5 in Sect. 3) of a great variety of elements because appropriate input data are now available (as was pointed out in Sect. 2.2). In Fig. 5 we present abundances [X/H] for elements X = C, N, O, Mg, Al, Si, S, Ar, Ca, Cr, Mn, Fe, Ni and Zn in Sa galaxies.

[FIGURE] Fig. 5. redshift evolution for various element abundances [X/H] in Sa galaxies. Elements X are listed in the legend.

At first sight all curves nearly have a similar shape (abundances increasing from high to low redshift) indicating that the enrichment history for all elements is roughly the same. But there are important differences in detail. Absolute element abundance values are very different (bearing in mind the log scale). Furthermore the gradients of the curves differ significantly in some parts reflecting the different production histories of various elements (i.e. SNII-, SNI- and intermediate mass star-products).

2.3.5. Influence of IMF and upper mass limit

Generally two different initial mass functions (IMF) are in use. Scalo's (1986) IMF is described in Sect. 2.1 Eq. (1). Applying the same exponent [FORMULA] for the whole mass range [FORMULA] we recover Salpeter's (1955) IMF. Improved data, especially metallicity dependent yields by Woosley & Weaver (1995) are solely available up to [FORMULA][FORMULA]. To study the influence of the IMF and the upper mass limit [FORMULA] on the results we therefore refer to earlier calculations (cf. Fritze 1995a and Fritze et al. 1995b) using only solar metallicity yields.

Fig. 6 presents the time evolution of the global metallicity Z in Sa and Sd galaxies using Scalo and Salpeter IMFs and two different upper mass limits ([FORMULA] = 40 and 85 [FORMULA]). The range of metallicities observed in HII regions of nearby Sa to Sd galaxies (today = 15 Gyrs) by Oey & Kennicutt (1993) and Zaritsky et al. (1994) is indicated at the right edge of Fig. 6. We chose a Scalo IMF with [FORMULA] (as described in Sect. 2.1) throughout the paper.

[FIGURE] Fig. 6. Comparison of global metallicity Z for Sa and Sd galaxies using Salpeter and Scalo IMF with upper mass limit m_up = 85 and 40 solar masses, respectively. Metallicities of HII regions in nearby spiral galaxies are taken from Zaritsky et al. (1994) and Oey & Kennicutt (1993).

2.3.6. Comparison with H II region abundances of nearby spirals

Our simple one-zone models are assumed to give average ISM abundances. We therefore chose to compare them to observed nearby spiral H II region abundances as measured (or extrapolated from observed gradients) at [FORMULA]. This is what Oey & Kennicutt (1993) call characteristic abundances . The range they give for Sa - Sb galaxies is [FORMULA][FORMULA] - [FORMULA][FORMULA] (cf. their Table 4). For Sbc through Sd spiral galaxies (Zaritsky et al. 1994) this characteristic abundance range extends downwards with average characteristic abundances (at 1 [FORMULA]) for Sd galaxies of about 0.5 [FORMULA].


[TABLE]

Table 3. Published element abundance measurements in DLA systems for Al, Ni, S, Fe, Si, Mn, Cr and Zn. Reference numbers for f-values (column reff) and solar abundances (column [FORMULA]) are given in the footnotes.
a) References on solar element abundances:
1: Aller, 1987; 2: Anders, Grevesse, 1989; 3: de Boer, Jura, Shull, 1987; 4: Grevesse, Anders, 1989; 5: Grevesse, Noels, 1993; 6: Hannaford et al., 1992; 7: Withbroe, 1971;
b) References on oscillator strength :
1: Bergeson, Lawler, 1993; 2: Kurucz, Peytremann, 1975; 3: Morton, Smith, 1973; 4: Morton, 1978; 5: Morton, York, Jenkins, 1988; 6: Morton, 1991; 7: Spitzer, Fitzpatrick, 1993; 8: Tripp, Lu, Savage, 1996



[TABLE]

Table 4. DLA systems with element abundances in excess of our Sa model are indicated with "X", and "x" for less reliable data. "O" indicates conformity of observations with models and for "-" no observations are available.)


The compilation of Ferguson et al. (1998) confirms the radial gradients in galaxies out to large radii. In some cases they even find stronger gradients than those derived from the inner regions. Starting from observed spiral ISM abundances and abundance gradients and using a geometrical model Phillipps & Edmunds (1996) find that the average abundance encountered along an arbitrary line of sight through a present day spiral galaxy should be of the order [FORMULA][FORMULA].

It should be noted that oxygen abundances in H II regions may already be locally enhanced with respect to average ISM abundances as soon as the first supernovae explode among the stars that ionize the gas. We therefore decided to transform the observed H II region abundances [O/H] to a global metallicity Z for the comparison with our model results in Fig. 6.

2.3.7. Connection with DLA galaxies

It should be mentioned that for our 1-zone models it would not matter if at the highest redshifts the proto galaxies were not really assembled yet in one coherent structure but rather consisted of a set of subgalactic fragments that imprint their relative velocity differences on the structure of the DLA line profile. In this case, our model could be interpreted as describing the global SF and enrichment history of all the bits and pieces that are bound to later assemble into one present-day galaxy.

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Online publication: December 16, 1998
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