Astron. Astrophys. 362, 921-936 (2000)

2. Model

The adopted model for the chemical evolution of the Milky Way disk is described in detail in BP99. In Sect. 2.1 we briefly recall the main features of the model. In Sect. 2.2 we present in some more detail the only novel ingredient with respect to BP99, namely the metallicity dependent yields of Woosley & Weaver (1995, hereafter WW95) for intermediate mass elements and of Maeder (1992, hereafter M92) for He, C, N, O (in BP99 only yields for stars of solar metallicity are used).

2.1. Description of the model

The galactic disk is considered as an ensemble of concentric, independently evolving rings, progressively built up by infall of primordial composition. The assumption of infall is traditionally based upon the need to explain the locally observed metallicity distribution of long-lived stars, which cannot be explained by the simple "closed-box" model (leading to the well-known "G-dwarf problem"). However, the recent work of Blitz et al. (1999) gives observational support to this idea, showing that the Milky Way and M31 are currently accreting substantial amounts of gas ( 1 /yr) in the form of high velocity clouds of low metallicity.

The infall rate is assumed to be exponentially decreasing in time, i.e.

with a characteristic timescale ) = 7 Gyr in the solar neighborhood ( = 8kpc), in order to reproduce the local G-dwarf metallicity distribution. is assumed to increase outwards, from (R = 2 kpc) = 1 Gyr to (R = 17 kpc) = 12 Gyr. This radial dependence of the timescale of the infall rate is simulating the inside-out formation of galactic disks and, combined with the adopted SFR (Eq. 3) allows to reproduce the observed current profiles of gas, oxygen abundance and SFR in the Milky Way disk (see BP99 and Sects. 2.3 and 4 below). The coefficient is obtained by the requirement that at time T = 13.5 Gyr the current mass profile of the disk is obtained, i.e.

with and a scalelength = 2.6 kpc for the Milky Way disk.

The chemical evolution of each zone is followed by solving the appropriate set of integro-differential equations, without the Instantaneous Recycling Approximation. The adopted stellar Initial Mass Function (IMF) is a multi-slope power-law between 0.1 and 100 from the work of Kroupa et al. (1993), leading to a Return Fraction R = 0.32.

The star formation rate (SFR) is locally given by a Schmidt-type law, i.e. proportional to some power of the gas surface density : , according to the observations of Kennicutt (1998). It varies with galactocentric radius R, as:

where is the circular velocity at radius R. This radial dependence of the SFR is suggested by the theory of star formation induced by density waves in spiral galaxies (e.g. Wyse & Silk 1989). Since in the largest part of the disk, this is equivalent to . The efficiency of the SFR in Eq. 3 is fixed by the requirement that the local gas fraction 0.2, is reproduced at T = 13.5 Gyr.

We assume that the "rings" of the disk are evolving independently from one another. This (over)simplification ignores in general the possibility of radial inflows in gaseous disks, resulting e.g. by viscosity or by infalling gas with specific angular momentum different from the one of the underlying disk; in both cases, the resulting redistribution of angular momentum leads to radial mass flows. The magnitude of the effect is difficult to evaluate, because of our poor understanding of viscosity and our ignorance of the kinematics of the infalling gas. Models with radial inflows have been explored in the past (Mayor & Vigroux 1981; Lacey & Fall 1985; Clarke 1989; Chamcham & Tayler 1994). It turns out that for some combinations of the parameters of infall, radial inflow and SFR, acceptable solutions are obtained, i.e. the current radial profiles of various quantities are successfully reproduced (see, e.g. Portinari & Chiosi 2000 for a recent overview of the problem). However, at the present stage of our knowledge introduction of radial inflows in the models would imply more free parameters than observables. For simplicity reasons we stick to the model of "independently evolving rings" for the Milky Way disk.

2.2. Yields of massive stars

An important ingredient in our study of abundance gradients is the stellar yields of various elements. Most of the intermediate mass elements studied here are produced by massive stars, with the exception of some CNO isotopes that are also produced by intermediate mass stars. We consider no yields from intermediate mass stars in this work ; in the line of Goswami & Prantzos (2000), concerning the evolution of the halo+local disk, our explicit purpose is to check to what extent massive stars can account for observations of intermediate mass elements and for which elements the contribution of intermediate mass stars is mandatory.

We use the metallicity dependent yields of WW95, which are given for stars of mass M = 12, 13, 15, 18, 20, 22, 25, 30, 35 and 40 and metallicities Z/ = 0, 10^-4, 10^-2, 10^-1 and 1. In Fig. 1 we present the WW95 yields, folded with the Kroupa et al. (1993) IMF. They are presented as overproduction factors , i.e. the yields (ejected mass of a given element) are divided by the mass of that element initially present in the part of the star that is finally ejected:

where: is the IMF, and the lower and upper mass limits of the stellar models (12 and 40 , respectively). are the individual stellar yields and the mass of the stellar remnant. Adopting in Eq. (4) creates a slight inconsistency with the definition of the overpoduction factor given above, but it allows to visualize the effects of metallicity in the yields of secondary and odd-Z elements.

Fig. 1. Average overproduction factors (over a Kroupa et al. (1993) IMF, see Eq. 4) of the yields of Woosley & Weaver (1995) for 3 different initial stellar metallicities, covering reasonably well the metallicity evolution of the Milky Way disk. The solid horizontal line is placed at and the two dotted horizontal lines at half and twice that value, respectively. Most of the intermediate mass elements are nicely co-produced (within a factor of two). The "odd-even effect" is clearly seen (e.g. in the cases of Na, Al, P, etc.); notice the small (but significant) metallicity dependence of the Ne and Mg yields. N behaves as a pure "secondary" element. The elements He, C, N, Li and Be obviously require another production site.

As can be seen from Fig. 1: i) most of the intermediate mass elements are nicely co-produced (within a factor of 2) by solar metallicity stars; ii) the "odd-even effect", favoring the production of odd-nuclei at high metallicities, is clearly present; iii) the yields of Ne and Mg show, curiously, some dependence on metallicity (not as large as the one of the odd-elements Na and Al, but still enough to lead to some interesting abundance patterns, as we shall see in Sect. 3); iv) He, C, N, Sc, V and Ti are underproduced relative to Oxygen. He, C and N clearly require another source (intermediate mass stars and/or Wolf-Rayet stars, see Prantzos et al. 1994 and Sect. 4.2), while the situation is less clear for the elements, Sc, V and Ti (see Goswami & Prantzos 2000).

The calculations of WW95 did not consider any mass loss during stellar evolution. Thus, they probably underestimated the yields of several elements that are expelled by the intense winds of massive stars, i.e. He, N and C. The effect of stellar winds is stronger when the stellar metallicity is larger. Maeder (1992) found that stars with 30 and Z0.1 eject considerably larger amounts of He, N and C during their lifetime than their lower metallicity counterparts; for that reason, less matter is left in the He-core to be processed into Oxygen. At lower metallicities, the effect of stellar winds is negligible and stars of all masses evolve almost at constant mass. Elements heavier than Oxygen are produced in the subsequent, very rapid, stages of stellar evolution (after core He exhaustion) and their yields are not directly affected by the intensity of the mass loss. However, the structure of the stellar core may be affected by the loss of mass and this may also affect the final yields of heavy elements (e.g. Woosley et al. 1993).

In Fig. 2 we present the yields of M92 for He, N, C and O as a function of stellar mass; they are given for two metallicities (Z/ = 0.05 and 1, respectively) and are compared to the corresponding yields of WW95. The aforementionned effect of metallicity-dependent stellar winds on the yields of stars with M30 is clearly seen. In Sect. 3.4 we shall explore the effect of those yields on the abundance gradients in the disk.

Fig. 2. Massive star yields of He, C, N and O for different initial metallicities, according to WW95 (filled triangles ) and M92 (open squares ); Solid curves : Z = , dotted curves : Z = 0.1 (for WW95), dashed curves: Z = 0.05 (for M92). There is good general agreement between the two calculations for stars with initial metallicity Z = . He, C and O are primary elements, i.e. their yields are independent of the initial stellar metallicity; the differences between the two sets of M92 yields for stars with M30 are due to the metallicity dependent stellar winds. Nitrogen is produced as a secondary element in both WW95 and M92.

To account for the additional source of Fe-peak elements, required to explain the observed decline of O/Fe abundance ratio in the disk (e.g. Goswami & Prantzos 2000), we utilise the recent yields of SNIa from the exploding Chandrashekhar-mass CO white dwarf models W7 and W70 of Iwamoto et al. (1999). These are updated versions of the original W7 model of Thielemann et al. (1986), calculated for metallicities Z = (W7) and Z = 0 (W70), respectively. In this model, the deflagration is starting in the centre of an accreting white dwarf, burns half of the stellar material in Nuclear Statistical Equilibrium and produces 0.7 of ⁵⁶Fe (in the form of ⁵⁶Ni). These SNIa models lead to an oveproduction of Ni, but the evolution of this element will not be considered here.

It should be emphasised that the evolution of the SNIa rate is not well known, and hardly constrained by observations. For the purpose of this work, we shall adopt the formalism of Matteucci & Greggio (1986) for the rate of SNIa, adjusting it as to have them appearing locally after the first Gyr, i.e. at a time when [Fe/H] -1 in the solar neighborhood. However, use of that same formalism along the disk will lead to different O/Fe abundance ratios at T = 13.5 Gyr as we shall see in Sect. 3, i.e. the final Fe gradient will be different from the one of oxygen (see also Prantzos & Aubert 1995).

2.3. Results for the Milky Way disk

As described in detail in BP99, the simple model presented in Sect. 2.1 can readily account for the evolution of the solar neighborhood, reproducing quite successfully the main observational constraints (age-metallicity relationship, metallicity distribution of long-lived F-stars, current local surface densities of stars, gas, star formation and supernova rates). Also, in Goswami & Prantzos (2000) it is shown that the use of the WW95 yields for massive stars and of the Iwamoto et al. (1999) yields for SNIa leads to a successful agreement between the gaseous composition of the model at an age of 9 Gyr and the observed solar one. A few exceptions concern the elements He, C and N (which are underproduced) and Ni (which is overproduced, because of the adopted SNIa yields).

The model predictions for the disk are equally successful, at least to a first order. Indeed, the adopted combination of SFR (Eq. 3) and infall rate (Eq. 1) leads to final profiles of gas and SFR that are in fair agreeement with the observed ones, as can be seen in Fig. 3. The stellar profile is also in agreement with observations, but it is essentially determined by the boundary condition of Eq. 2. However, the adopted inside-out formation scheme of the disk leads naturally to different scalelengths in the B-band (reflecting mostly the SFR profile in the past 1 Gyr) and the K-band (reflecting the total stellar population, cumulated over T = 13.5 Gyr). As shown in BP99, the corresponding scale-lengths ( 4 kpc in the B-band and 2.6 kpc in the K-band, respectively) are in fair agreement with observations. Moreover, the model also reproduces reasonably well the total current SFR and supernova rates as well as the total luminosities in various wavelenght bands. This is a rather encouraging success, since the number of the constraints is much larger than the number of the parameters. The success of this simple model encourages us to use it for a thorough study of the various abundance gradients in the Milky Way disk.

Fig. 3. Results of the chemical evolution model for the Milky Way disk and comparison to observations. In all panels, results are shown at three different epochs (dotted curves: 1 Gyr, dashed curves: 5 Gyr, solid curves: 13.5 Gyr). The latter are to be compared to observations of present-day profiles in the Milky Way disk, shown as shaded regions in first three panels and within error bars in the lower panel. In all panels, the error bar at = 8 kpc indicates observed quantities in the solar neighborhood. References for data are given in BP99.

© European Southern Observatory (ESO) 2000

Online publication: October 30, 2000
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