Astron. Astrophys. 338, 413-434 (1998)

3. Observational constraints from the Galactic disk

The whole set of observational constraints confronted with the model predictions is listed in Tables 1 and 2. Some of these constraints are discussed below in more details: the abundance distribution of G dwarfs, the age-metallicity relation, the - relation, and the radial abundance gradients. The radial density distributions have been discussed in more detail by Prantzos & Aubert (1995). A "minimal set" of observables (cf. Prantzos & Aubert 1995) is shown in Figs. 5 and 6.

Table 1. Set of observational constraints from the local Galactic disk.

Table 2. Set of global observational constraints from the Galactic disk.

3.1. Abundance distribution of the nearby long-lived stars

The abundance distribution function (ADF) of the long-lived stars is one of the most severe local constraints for chemical evolution models of the Galactic disk (van den Bergh 1962; Schmidt 1963; Pagel & Patchett 1975; Pagel 1989). The observational data have become significantly improved by the work of Wyse & Gilmore (1995) and Rocha-Pinto and Maciel (1996), respectively. These new data do not relaxe the classical "G dwarf problem", i.e. the prediction of too a high number of low-metallicity long-lived stars by simple evolution models. It seems unlikely that the G dwarf problem is simply due to selection effects in the stellar sample or of metallicity-dependent stellar lifetimes (Rocha-Pinto & Maciel 1997; Meusinger & Stecklum 1992).

The ADF presented by Rocha-Pinto & Maciel (1996) shows a pronounced peak around . On the other hand, Wyse & Gilmore (1995) emphasized that their -distribution does not differ radically from earlier determinations. Differences between these both empirical ADFs may partly be due to different corrections for observational uncertainties. Wyse & Gilmore directly integrated over the solar cylinder by combining a local stellar sample with a distant one and deconvolved the abundance distribution into thin disk and thick disk components. However, as the peak is present also in the uncorrected data by Rocha-Pinto & Maciel, its different strength is perhaps due to different stellar samples rather than due to different data corrections.

We compare the model -distribution of long-lived stars with the observed ADF from Rocha-Pinto & Maciel, as well as with the combined thin+thick disk distribution from Wyse & Gilmore. In the latter case, we take into account that the models do not distinguish between thick disk and thin disk, and that these both components may be the product of a uniform formation process (e.g., Majewski 1993; Wyse & Gilmore 1995).

3.2. The age-metallicity relation (AMR) of the local disk

Empirical studies on ages and metallicities of stars in the solar neighbourhood have established the idea that the heavy element abundances in the ISM, averaged over suitable temporal and spatial regions, have slightly increased in the course of the evolution of the local Galactic disk (Mayor 1976; Twarog 1980; Carlberg et al. 1985; Schuster & Nissen 1989; Meusinger et al. 1991; Edvardsson et al. 1993). On the basis of Strömgren photometric data for nearly 5 000 disk dwarfs, Twarog (1980) created a well-defined sample of about 900 F and G dwarfs to derive the AMR. A re-analysis of this photometric data base was presented by Meusinger et al. (1991) using revised calibrations of the photometric indices, updated stellar evolution models, and a more elaborated method of age determination. Nevertheless, the resulting AMR was quite similar to that found by Twarog. Edvardsson et al. (1993) presented very precise abundance data for nearly 200 nearby F and G dwarfs which provide a very suited data base for studying chemical properties of the Galactic disk. The AMR derived for the stars representing the solar cylinder ring (7.7 kpc kpc, kpc, ) is in good agreement with the previously derived AMRs for the solar neighbourhood, too. The data point for the oldest stars may be slightly biased toward lower matallicities (Edvardsson et al. 1993).

In the present paper, we will confront our model AMR with the AMRs from Edvardsson et al. and Meusinger et al. (Figs. 5 to 8). The age bin for the oldest stars in the latter AMR is much in excess of 12 Gyr. To maintain consisteny with the adopted disk age we decided to scale down the ages in this AMR by a factor 0.8. This may be an oversimplification. However, simulations have shown (Meusinger 1994) that a disk age of about 15 Gyr may be pretended by the age distribution of the stellar AMR sample even if the true age is only about 12 Gyr, simply due to the uncertainties of age determination.

3.3. Abundance scatter

It is a well-known property of the AMR to show a considerable abundance scatter among stars born at the same time at the same Galactocentric distance. Twarog (1980) found a dispersion of dex, nearly in agreement with the expected uncertainty of the abundance determination. A somewhat larger dispersion was present in the revised AMR by Meusinger et al. (1991), ranging from 0.13 dex for stars younger than 2 Gyr to 0.24 dex for the oldest stars. Large abundance variations are also indicated by similarly aged open clusters after correction for the radial abundance gradients across the Galactic plane (e.g., Carraro & Chiosi 1994; Piatti et al. 1995). The accurate data presented by Edvardsson et al. (1993) have clearly demonstrated that the observed abundance scatter at fixed age is much larger than that expected from observational uncertainties assumed as about 0.05 dex. After correction for the kinematic evolution perpendicular to the Galactic plane seems to increase with age from about 0.15 to about 0.25 (Wielen et al. 1996).

The relation between metallicity dispersion and age has been interpreted by Wielen et al. (1996) as confirmation of the hypothesis of stellar orbital diffusion, predicted already by Wielen (1977). Van den Hoek & de Jong (1997) have argued that diffusion of stellar orbits is probably insufficient to explain the observed abundance scatter. Alternative ideas include self-enrichment in regions of sequential star formation (Edmunds 1975), irregular infall of unenriched gas onto the disk (Pilyugin & Edmunds 1996), or a combination of both (van den Hoek & de Jong 1997).

In the present paper, we do not follow in detail the possible processes causing the abundance inhomogeneity in the Galactic disk. However, we have to take into account the abundance scatter when comparing the model ADF with the empirical G dwarf distribution: Following Pagel (1989), Rocha-Pinto & Maciel (1996) have deconvolved the observed ADF with a Gaussian of dispersion to correct for the uncertainties in the abundance determination. In order to account for a further "cosmic" scatter, expressed by , we convolve the model ADF with a Gaussian with the dispersion , where is the age. Because the effect of erroneous age determinations on the abundance scatter at fixed age is difficult to estimate, we will alternatively consider the case . Malinie et al. (1993) have shown that the fit to the empirical ADF is improved in a model of inhomogeneous chemical evolution, especially at the high metallicity end of the ADF.

3.4. relation

The dependence of the abundance ratio for elements on is generally interpreted as due to different lifetimes of the main producers of the different elements. Especially, the relation is explained by the long lifetimes of progenitors of SNe I a responsible for iron enrichment compared to those of SNe II driving the oxygen evolution (e.g., Matteucci & Francois 1989). The observed ratio for can be explained as exlusively originating from SN II enrichment, while the slope of the - relation for (Edvardsson et al. 1993) depends on the ratio of the SN II to SN I a rates, i.e. on the slope of the IMF for .

Therefore, we determined the IMF index by making the demand to reproduce the observed slope for . On the other hand, the maximum stellar mass, , contributing to the ISM enrichment via SNe II , i.e. the mass limit beyond which stars end as black holes without ejecting processed matter into the ISM, has mainly the effect of shifting the - relation along the axis (cf. Tsujimoto et al. 1995 a). Once is specified, we found by fitting at . For the SN II element production rates from Tsujimoto et al. (1995 b) we find a good agreement with the observed - relation assuming and (Fig. 2). The use of the metallicity-dependent SN II yields from Woosley & Weaver (1995), however, does not result in a flattening of at low metallicities. For the search of best-fit models (Sect. 4.2) we restrict our analysis to the data from Tsujimoto et al. (1995 b). In a more detailed investigation of a similar approach, Tsujimoto et al. (1997) found and .

Fig. 2a and b. Top : The slope of the relation in dependence on the slope, , of the IMF in the range . Bottom : The element ratio at as a function of the upper stellar mass contributing to the ISM enrichment via SN II for . The solid curves are for the SN II yields from Tsujimoto et al. (1995 b), the dashed curves for the yields from Woosley & Weaver (1995).

3.5. Radial abundance gradients (RAGs)

Although there are strong indications for the existence of overall radial heavy element abundance gradients in our Galaxy, the values for the gradients are still a matter of debate. From a comprehensive analysis of H II regions, Shaver et al. (1983) found a strong RAG of dex kpc^-1 (for kpc). Over the last years, abundances have been estimated for distant B stars and H II regions by several groups. The results are listed in Table 3, together with RAGs derived from planetary nebulae, disk globular clusters, field stars, and photometric abundances in open clusters. The RAG quoted there for the study by Shaver et al. was obtained from the well-determined abundances (their Eq. 13 a) after scale change to kpc. The RAG from Smartt et al. (1997) is estimated from the average for four blue supergiants at kpc derived by Smartt et al. compared with solar abundance at . In the most studies quoted in Table  3, kpc was adopted. Pasquali & Perinotto (1993) and Edvardsson et al. (1993), respectively, used slightly different values for which do not influence the estimated RAGs. The RAG for the K giants studied by Lewis & Freeman (1989) was corrected by Sommer-Larsen & Yoshii (1989) for the vertical abundance gradient. The corresponding value quoted in Table  3 was derived from these corrected data for the R range given in the Table. The gradients for the F and G dwarfs have been calculated from the abundance data and the mean Galactocentric distances, , given by Edvardsson et al. (1993; their Tables 11 and 12).

Kaufer et al. (1994) and Vilchez & Esteban (1996) have pointed out that the RAGs are becoming shallower at larger R and are vanishing in the outermost part of the disk. The assumption of a linear RAG over the whole disk may be too simple. Therefore, we have recalculated the RAG for the compact H II regions from Table 3 in Afflerbach et al. (1997) using only objects with kpc, in order to have a comparable R range.

The results in Table  3 confirm the existence of a radial gradient, partly however with a significantly lower slope than found by Shaver et al. (1983). As a compromise for all data in Table 3, a RAG of about dex kpc^-1 seems representative for the present-day ISM in the range kpc. Kennicutt & Garnett (1996) found dex kpc^-1 for the disk of M 101 at kpc.

Table 3. Radial abundance gradients in the Galaktic disk.

There are weak indications for a steepening of RAGs during the evolution of the disk (Table 3). Maciel & Köppen (1994) discussed a possible age-dependence in the RAGs from planetary nebulae data, although no individual age determinations are available. The mean ages listed in Table  3 were taken from the original papers for the most object groups. For the disk globulars we simply adopted the limits of globular cluster ages derived by Jimenez et al. (1996). The mean age of the K giants from Lewis & Freeman (1989) was roughly estimated from the mean and the mean kinematic data using the AMR from Sect. 3.2 and the age-velocity dispersion relation from Wielen (1977). The nearby F and G dwarfs studied by Edvardsson et al. (1993) are also well-suited for studying the evolution of the RAGs because individual age data are available. For the 93 thin disk stars with ages Gyr we find dex kpc^-1, in contrast to dex kpc^-1 for the 43 stars with Gyr. Vanishing RAGs are also indicated for field K giants (Lewis & Freeman 1989) and disk globular clusters (Alfaro et al. 1993).

For typical conditions in galaxy disks, the RAGs are, as a first approximation, not influenced by the diffusion of stellar orbits (Wielen et al. 1996). The alternative explanation for the small RAGs indicated by the older disk objects is that RAGs are generated in the course of the chemical evolution of the disk. However, one has to take into account the following: for the old stars abundances are related to iron-group elements whereas young stars are related to oxygen. The empirical relation with is applicable for the nearby disk stars only, but not for the transformation between iron and oxygen RAGs. There is an interesting feature in the Edvardsson et al. data (cf. Nissen 1995): for stars formed in the inner disk have a larger than stars from the outer disk. With a formal gradient dex kpc^-1 implicates that the small gradient dex kpc^-1 found for the old stellar population is nearly in agreement with the oxygen RAG. Nevertheless, there remains a difference between the gradient for the oldest and the younger stars, as indicated e.g. by the data from Edvardsson et al. (1993).

© European Southern Observatory (ESO) 1998

Online publication: September 14, 1998
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