A&A 430, 491–505 (2005) Astronomy DOI: 10.1051/0004-6361:20048222 & c ESO 2005 Astrophysics Quantifying the uncertainties of chemical evolution studies I. Stellar lifetimes and initial mass function D. Romano1, C. Chiappini2, F. Matteucci3,andM.Tosi1 1 INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, 40127 Bologna, Italy e-mail: [donatella.romano;monica.tosi]@bo.astro.it 2 INAF - Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, 34131 Trieste, Italy e-mail: [email protected] 3 Dipartimento di Astronomia, Università di Trieste, via G.B. Tiepolo 11, 34131 Trieste, Italy e-mail: [email protected] Received 4 May 2004 / Accepted 9 September 2004 Abstract. Stellar lifetimes and initial mass function are basic ingredients of chemical evolution models, for which different recipes can be found in the literature. In this paper, we quantify the effects on chemical evolution studies of the uncertainties in these two parameters. We concentrate on chemical evolution models for the Milky Way, because of the large number of good observational constraints. Such chemical evolution models have already ruled out significant temporal variations for the stellar initial mass function in our own Galaxy, with the exception perhaps of the very early phases of its evolution. Therefore, here we assume a Galactic initial mass function constant in time. Through an accurate comparison of model predictions for the Milky Way with carefully selected data sets, it is shown that specific prescriptions for the initial mass function in particular mass ranges should be rejected. As far as the stellar lifetimes are concerned, the major differences among existing prescriptions are found in the range of very low-mass stars. Because of this, the model predictions differ widely for those elements which are produced mostly by very long-lived objects, as for instance 3He and 7Li. However, we conclude that model predictions of several important observed quantities, constraining the plausible Galactic formation scenarios, are fairly robust with respect to changes in both the stellar mass spectrum and lifetimes. For instance, the metallicity distribution of low-mass stars is nearly unaffected by these changes, since its shape is dictated mostly by the time scale for thin-disk formation. Key words. Galaxy: abundances – Galaxy: evolution – Galaxy: formation – stars: luminosity function, mass function – stars: fundamental parameters 1. Introduction in the literature, abundances and abundance ratios play a major rôle as cosmic clocks and give hints on the time scales of struc- The formation and evolution of galaxies is one of the outstand- ture formation and evolution (Wheeler et al. 1989; Matteucci ing problems of astrophysics. In the last decade, a great deal of & François 1992; Matteucci 2001). observational work has shed light on the production and distri- bution of chemical elements inside the Galaxy (e.g. Edvardsson In order to build up a chemical evolution model it is neces- et al. 1993; Cayrel 1996; Nissen & Schuster 1997; Gratton et al. sary to define the initial conditions and the basic physical laws 2000; Chen et al. 2003; Gratton et al. 2003; Ivans et al. 2003; governing the evolution of the system during the whole galac- Reddy et al. 2003; Zoccali et al. 2003; Akerman et al. 2004), of- tic lifetime. In short, one needs to specify whether the system ten leading to an evolutionary scenario much more complicated is closed or open (whether any inflow/outflow of gas occurs), than assumed in many models. Even more recently, abundance the chemical composition of the gas from which the computa- data have accumulated for external galaxies at both low and tion starts and that of any infalling material, the stellar birthrate high redshift, thus providing precious information on the chem- function and stellar evolution and nucleosynthesis. In particu- ical evolution of different types of galaxies and on the early lar, the stellar birthrate is expressed as the product of two inde- stages of galaxy evolution (e.g. Pettini 2001; Centurión et al. pendent functions, the star formation rate (SFR) and the stel- 2003; Dietrich et al. 2003a,b; Prochaska et al. 2003; Tolstoy lar initial mass function (IMF). The first is generally expressed et al. 2003; Dessauges-Zavadsky et al. 2004; D’Odorico et al. as a function of time only, while the second, which describes 2004). In this framework, galactic chemical evolution models the stellar mass distribution at birth, is likely to be universal can be regarded as useful tools to discriminate among different (Kroupa 2002; but see e.g. Jeffries et al. 2004) and not vary as scenarios of galaxy formation. In fact, as stressed many times a function of time (Chiappini et al. 2000). Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20048222 492 D. Romano et al.: Quantifying the uncertainties of chemical evolution studies. I. −1.35 The free parameters that one introduces in the model sim- if using the IMF by mass, ϕSalpeter(m) ∝ m . In what follows ply reflect our poor understanding of the basic physical pro- we always use the IMF by mass. cesses governing the formation and evolution of galactic struc- More realistic, multi-slope expressions give a better de- tures. However, having a number of observational constraints scription of the luminosity function of main sequence stars formally larger than that of the free parameters allows us to re- in the solar neighbourhood that is actually observed (Tinsley strict the range of variation of the parameters themselves and 1980; Scalo 1986; Kroupa et al. 1993; Scalo 1998; see the orig- gain useful insight into the mechanisms of galaxy formation inal publications for details): and evolution (Matteucci 2001). This is the case for our own −1.0 A m if m < 2 M Galaxy and will soon become the standard for an increasing Tinsley −1.3 ϕ (m) = B m if 2 < m/M < 10 number of external galaxies, thanks to the capabilities of mod- Tinsley Tinsley −2.3 ern telescopes and instrumentation. CTinsley m if m > 10 M, In an epoch where the uncertainties in the data have be- A . , B . , C . come really small, it is worth trying to consider “theoretical Tinsley 0 21 Tinsley 0 26 Tinsley 2 6; error bars” as well. An attempt to do so has recently been made −1.35 A m if m < 2 M by Romano et al. (2003), who compare the evolution of light el- ϕ = Scalo 86 Scalo 86(m) −1.70 ements predicted by two independent models of chemical evo- BScalo 86 m if m > 2 M, lution for the Milky Way and try to ascertain the origin of the A . , B . differences in the model predictions. In the present paper we in- Scalo 86 0 19 Scalo 86 0 24 tend to assess the uncertainties in the model predictions which (notice that we adopt a simplified two-slope approximation to arise when exploring different prescriptions for the IMF and the the actual Scalo 1986 formula, similarly to what is done in stellar lifetimes in a model for the Milky Way. To this purpose Matteucci & François 1989); we adopt a chemical evolution model which has been proven −0.3 to successfully reproduce the main observational features of A m if m < 0.5 M Kroupa −1.2 the solar neighbourhood (Chiappini et al. 1997) and change the ϕ (m) = B m if 0.5 < m/M < 1 Kroupa Kroupa assumptions on the stellar IMF. As a result we set up a range −1.7 C m if m > 1 M, of possible variations for several predicted quantities. Then we Kroupa repeat the same analysis by changing the prescriptions for the AKroupa 0.58, BKroupa = CKroupa 0.31; stellar lifetimes. −0.2 The paper is organized as follows. In Sect. 2 we review the A m if m < 1 M Scalo 98 −1.7 general features of the IMF which have emerged over the years, ϕ (m) = B m if 1 < m/M < 10 Scalo 98 Scalo 98 and emphasize differences and similarities among the various −1.3 C m if m > 10 M, IMFs adopted in this work. In Sect. 3 we discuss the prescrip- Scalo 98 tions for the stellar lifetimes. In Sect. 4 we describe the chemi- AScalo 98 = BScalo 98 0.39, CScalo 98 0.16. All of them are cal evolution model for the solar vicinity. In Sect. 5 we present considered in the present work. The normalization is always model results. Finally, Sect. 6 is devoted to a critical discussion performed in the mass range 0.1–100 M. of the problem and some conclusions are drawn. Notice that More recently a lognormal form has been suggested for the the paper is structured in such a way that it is easy to concen- low-mass part of the IMF (m ≤ 1 M), eventually extending trate on only one among the proposed topics, while skipping into the substellar regime (Chabrier 2003): the others, if one wishes to do so. 2 2 −(log m−log mc) /2 σ AChabrier e if m ≤ 1 M 2. The stellar initial mass function ϕ = Chabrier(m) −(1.3±0.3) BChabrier m 2.1. The adopted parametrizations if m > 1 M. The most widely used functional form for the IMF is an exten- According to Chabrier (2003) the IMF depends weakly on the sion of that proposed by Salpeter (1955) to the whole stellar environment, except perhaps for early star formation condi- mass range: tions. Values of mc = 0.079 M and σ = 0.69 well charac- −(1+x) terize the IMF for single objects belonging to the Milky Way φSalpeter(m) = ASalpeter m , disk.