The chemical evolution of a Milky Way-like galaxy: the importance of a cosmologically motivated infall law
aa r X i v : . [ a s t r o - ph ] F e b Astronomy&Astrophysicsmanuscript no. articolo9˙AA˙revised˙3 c (cid:13)
ESO 2018November 15, 2018
The chemical evolution of a Milky Way-like galaxy: theimportance of a cosmologically motivated infall law
E. Colavitti ,⋆ , F. Matteucci , , and G. Murante Dipartimento di Astronomia, Universit´a di Trieste, Via G. B. Tiepolo 11, I-34143 Trieste (TS), Italy I.N.A.F. Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, I-34143 Trieste (TS), Italy I.N.A.F. Osservatorio Astronomico di Torino, Strada Osservatorio 20, I-10025 Pino Torinese (TO), ItalyReceived xxxx / Accepted xxxx
ABSTRACT
Aims.
We aim at finding a cosmologically motivated infall law to understand if the Λ CDM cosmology can reproduce the main chemicalcharacteristics of a Milky Way-like spiral galaxy.
Methods.
In this work we test several di ff erent gas infall laws, starting from that suggested in the two-infall model for the chemical evolutionof the Milky Way by Chiappini et al., but focusing on laws derived from cosmological simulations which follows a concordance Λ CDMcosmology. By means of a detailed chemical evolution model for the solar vicinity, we study the e ff ects of the di ff erent gas infall laws on theabundance patterns and the G-dwarf metallicity distribution. Results.
The cosmological gas infall law, derived from dark matter halos having properties compatible with the formation of a disk galaxylike the Milky Way, and assuming that the baryons assemble like dark matter, resembles the infall law suggested by the two-infall model. Inparticular, it predicts two main gas accretion episodes. Minor infall episodes are predicted to have followed the second main one but they areof small significance compared to the previous two. By means of this cosmologically motivated infall law, we study the star formation rate, theSNIa and SNII rate, the total amount of gas and stars in the solar neighbourhood and the behaviour of several chemical abundances. We findthat the results of the two-infall model are fully compatible with the evolution of the Milky Way with cosmological accretion laws. We derivethat the timescale for the formation of the stellar halo and the thick disk must have not been longer than 2 Gyr, whereas the disk in the solarvicinity assembled on a much longer timescale ( ∼ Conclusions.
A gas assembly history derived from a DM halo, compatible with the formation of a late-type galaxy from the morphologicalpoint of view, can produce chemical properties in agreement with the available observations.
Key words.
Galaxy: evolution, Galaxy: formation, Galaxy: disk, Galaxy: abundances
1. Introduction
In many models of the chemical evolution of the Milky Way gasinfall has been invoked to explain the formation of the Galacticdisk (e.g. Chiosi (1980), Matteucci & Franc¸ois (1989); Lacey& Fall (1985); Chiappini et al. (1997); Boissier & Prantzos(2000) among others). Originally, the gas infall was introducedas a possible solution to the G-dwarf problem (Pagel 1989). Ingeneral terms the gas infall rate gives the law for the assemblyof baryons in a galaxy. However, in the majority of the chem-ical evolution papers existing in the literature, the gas infalllaw has been treated as a free parameter with no connectionto a galaxy’s cosmological context. In other words, in most ofthe cases the assumed infall law is independent of the detailsof the galactic dark matter (DM) halo’s assembly which, in-stead, should have a dominant e ff ect on it. On the other hand,the infall law is clearly very important in determining the maincharacteristics of a galaxy. In this paper we aim at studying the ⋆ email to: [email protected] infall law which descends directly from the DM halo and itsassembly.In this way, we will have an infall law for the gas whichis related to cosmology and does not contain free parameters.Once achieved that, we will test this cosmological infall lawin a detailed model of chemical evolution of the Milky Waywhich follows the evolution of many chemical species by tak-ing into account the stellar lifetimes, detailed nucleosynthesisprescriptions and supernova (type II, Ib / c and Ia) rates. Severalauthors have tried before us to build a model for the evolu-tion of the disk galaxies in a cosmological context, but noneof these considered the chemical evolution in such a detail asour model. Chemo-dynamical models for the Milky Way wereproposed by Theis, Burkert & Hensler (1992), where the evo-lution of massive spherical galaxies was calculated by a multi-component hydrodynamical approach but with no cosmologi-cal context. In Raiteri, Villata & Navarro (1996), instead, N-body / hydrodynamical cosmological simulations were used toinvestigate the chemical evolution of the Galaxy by assuming E. Colavitti et al.: A cosmological infall law for the Milky Way that it formed by the collapse of a rotating cloud of gas anddark matter. However, their chemical analysis, although de-tailed, was limited to only oxygen and iron. Another importantchemodynamical paper appeared in Samland, Hensler & Theis(1997), in which they presented their two-dimensional chemo-dynamical code CoDEx. Their model contains nucleosynthesisfrom supernovae of type I and II and some chemical evolu-tion, but no cosmological context was assumed. More recently,Abadi et al. (2003) presented simulations of galaxy formationin a Λ cold dark matter universe ( Λ CDM) and studied the dy-namical and photometric properties of disk galaxies, but nochemical evolution was included.Robertson et al. (2005) adopted the hierarchical scenariofor galaxy formation to see if in this context they could re-produce the rich data set of stellar abundances in the galactichalo and Local Group dwarf galaxies. They used an analyticalexpression for the growth of DM halos in a Λ CDM cosmol-ogy. Their baryonic infall law is proportional to the DM one.The hierarchical formation scenario, when applied to the stellarhalo of the Milky Way, suggests that it formed through accre-tion and disruption of dwarf galaxies. They concluded that themajority of stars in the stellar halo were formed within a rel-atively massive dwarf irregular sized dark matter halos, whichwere accreted and disaggregated ∼
10 Gyr ago. In their sce-nario, these systems had rapid star formation histories and wereenriched primarily by supernovae (SNe) of type II. They alsosuggested that the still existing dwarf irregular galaxies formedstars more gradually and they underwent both SNIa and II en-richment. On the other hand, dwarf spheroidal galaxies shouldbe systems where the abundances are determined by galacticwinds. In summary, the paper dealt mostly with the compar-ison between the [ α / Fe] ratios in the galactic halo and dwarfgalaxies.Also Naab & Ostriker (2006) studied the metallicity andphotometric evolution of a generic disk galaxy, by assumingthat it forms through mergers of dark matter halos. They took apoint of view similar to that of the present paper: in particular,they derived a cosmological infall law and concluded that theinfall rate should have been almost constant during the lifetimeof the disk. No detailed chemical evolution was followed andno consideration was given to the formation of the stellar halo.Finally, another paper dealing with chemical evolution in acosmological context is that by Nagashima & Okamoto (2006).The authors investigated the chemical evolution in Milky Way-like galaxies based on the CDM model in which cosmic struc-tures form via hierarchical merging. They adopted a semi-analytical model for galaxy formation where the chemical en-richment due to both SNeIa and SNeII was considered. Theysuggested that the so-called G-dwarf metallicity problem canbe fully resolved by the hierarchical formation of galaxies. Infact, the infall term introduced by the traditional monolithiccollapse models to solve the G-dwarf problem can be explainedby some physical processes such as injection of gas and metalsinto hot gas due to SNe. The model, however, was not testedon large number of chemical elements but was limited to the[O / Fe] vs. [Fe / H] plot.In this paper we will first study the e ff ect of di ff erent gasinfall laws taken from the literature and compare the chemical results with those of Franc¸ois et al.’s (2004) model, which isbased on the two-infall model of Chiappini et al. (1997). In thetwo-infall model it is assumed that the halo and the thick diskformed by means of a first infall episode on a timescale notlonger than 2 Gyr, whereas the thin disk should have formedby means of an independent second infall episode lasting muchlonger. In particular, the timescale for the formation of the solarvicinity was 7 Gyr, as suggested by the G-dwarf metallicity dis-tribution, while the internal parts of the thin disk formed fasterand the outermost regions are still forming now. This scenariohas proven to be very successful in reproducing the majorityof the properties of the solar vicinity and the whole disk andit was adopted by the majority of the chemical evolution mod-els of the Milky Way. In Franc¸ois et al. (2004) the evolution of35 chemical species including C, N, O, α -elements (Ne, Mg,Si, Ca, Ti), Fe-peak elements plus light elements such as D,He and Li is followed in detail. In this paper we run a cos-mological simulation to find a suitable Dark Matter (DM) halofor a Milky Way-like galaxy by adopting GADGET2 (Springel2005) and to obtain the infall law for the gas. In particular, wederive the law for the accretion of the DM halo by assumingthat the same law is followed by the assembling baryons. Onceobtained, this law is tested in the chemical evolution model tosee if it is consistent with the two-infall or other scenarios. Inorder to do that we calculate in detail the evolution of the abun-dances of several chemical elements, the SN rates and all thephysical quantities relevant to the evolution of the solar vicin-ity. Therefore, we start from a di ff erent approach relative toall the previous hierarchical models for the formation of theMilky Way (but see Sommer-Larsen et al. 1999). The reasonfor considering only gas accretion and not dwarf galaxies, asin other papers, is suggested by the di ff erent chemical histo-ries observed in dwarf galaxies relative to the Milky Way (e.g.Lanfranchi & Matteucci, 2004).The paper is organized as follows: in section 2 we showthe nucleosynthesis prescriptions adopted. Section 3 presentsa brief description of the model by Chiappini et al. (1997). Insection 4 we describe the cosmological simulation, done usingthe simulator Gadget2. Section 5 describes the adopted infalllaws. In section 6 we present the results obtained, comparingthe models predictions with the observed properties. Finallysection 7 presents the conclusions.
2. Nucleosynthesis prescriptions
One of the most important ingredients for chemical evolutionmodels is represented by the nucleosynthesis prescriptions andconsequently by the stellar yields.The single stars in the mass range 0 . M ⊙ ≤ M ≤ M ⊙ (low and intermediate-mass stars) contribute to the Galactic en-richment through planetary nebula ejection and quiescent massloss. They enrich the interstellar medium mainly in He, C, Nand heavy s-process elements (e.g. Cescutti et al. 2006). Weadopt here the stellar yields for low and intermediate mass starsof van den Hoek & Groenewegen (1997) computed as functionsof stellar metallicity, their case with variable mass loss. Thesestars are also the progenitor of Type Ia supernovae (SNe), ifthey are in binary systems, which originate from carbon de- . Colavitti et al.: A cosmological infall law for the Milky Way 3 flagration of C-O white dwarfs. We adopt in this paper thesingle-degenerate progenitor scenario (Whelan & Iben, 1973;Han & Podsiadlowski 2004). Type Ia SNe contribute a substan-tial amount of Fe ( ∼ . M ⊙ per event) and Fe-peak elementsas well as non negligible quantities of Si and S. They also pro-duce other elements, such as O, C, Ne, Ca, Mg and Ni, but invery small amounts compared to Type II SNe. We assume thestellar yields for Type Ia SNe from Iwamoto et al. (1999).Massive stars (8 M ⊙ < M ≤ M ⊙ ) are the pro-genitor of core-collapse SNe which can be either Type II SNeor Type Ib / c SNe. These latter can arise from binary systemsor Wolf-Rayet stars whereas Type II SNe originate from themassive stars in the lower mass range. Type II SNe mainly pro-duce the so called α -elements, such as O, Mg, Ne, Ca, S andSi and Ti, but also some Fe and Fe-peak elements although insmaller amounts than Type Ia SNe. We adopt here the stellaryields for massive stars by Woosley & Weaver (1995) with thesuggested modifications of Franc¸ois et al. (2004). However, themost important modifications concern some Fe-peak elements,except Fe itself, whereas for the α -elements, with the exceptionof Mg which has been increased relative to the original yields,the yields are substantially unmodified. The modifications ofthe yields in Franc¸ois et al. (2004) were required to fit at bestand at the same time the [element / Fe] versus [Fe / H] patternsand the solar absolute abundances. We keep the same prescrip-tions here with the purpose of testing the infall laws withoutchanging the other model parameters.Finally, we start with primordial gas and the assumed pri-mordial abundances of D and He we have chosen: 3 . · − and 2 . · − , respectively. The reference solar abundancesare those by Asplund et al. (2005).
3. The model by Chiappini et al. (1997)
Prior to the two-infall model of Chiappini et al. (1997), dif-ferent models assuming gas accretion onto the galactic diskhad been constructed. For example, dynamical models, suchas the one of Larson (1976), viscous models (Lacey & Fall1985; Sommer-Larsen & Yoshii 1989, 1990; Tsujimoto et al.1995), inhomogeneous models (Malinie et al. 1993), detailedchemical evolution models (Matteucci & Greggio 1986; Tosi1988; Matteucci & Franc¸ois 1989; Pagel 1989; Matteucci &Franc¸ois 1992; Carigi 1994; Giovagnoli & Tosi 1995; Ferriniet al. 1994; Pardi & Ferrini 1994; Pardi, Ferrini & Matteucci1995; Prantzos & Aubert 1995; Timmes, Woosley & Weaver1995) and chemodynamic models (Samland & Hensler 1996,Burkert, Truran & Hensler 1992). The model by Chiappini etal. (1997) was the first in which two main infall episodes for theformation of the Galactic components were suggested. In par-ticular, they assumed that the first infall episode was responsi-ble for the formation of the halo and thick-disk stars that origi-nated from a fast dissipative collapse. The second infall episodeformed the thin-disk component, with a timescale much longerthan that of the thick-disk formation. The authors included inthe model also a threshold in the gas density, below which thestar formation process stops. The existence of such a thresholdvalue is suggested by observations relative to the star formationin external disk galaxies (Kennicutt 1998, but see Boissier et al. 2006). The physical reason for a threshold in the star forma-tion is related to the gravitational stability, according to which,below a critical density, the gas is stable against density con-densations and, consequently, the star formation is suppressed.In the two-infall model the halo- thick disk and the thin diskevolutions occur at di ff erent rates, mostly as a result of di ff er-ent accretion rates. With these precise prescriptions it is possi-ble to reproduce the majority of the observed properties of theMilky Way and this shows how important is the choice of theaccretion law for the gas coupled with the star formation ratein the Galaxy evolution.In the model by Chiappini et al. (1997) the Galactic diskis approximated by a series of concentric annuli, 2 kpc wide,without exchange of matter between them. The basic equationsare the same as in Matteucci & Franc¸ois (1989). The two maindi ff erences between the model by Chiappini et al. (1997) andMatteucci & Franc¸ois (1989) are the rate of mass accretion andthe rate of star formation. Moreover, in the model by Chiappiniet al. (1997) the material accreted by the Galactic thin diskcomes mainly from extragalactic sources. These extragalacticsources could include, for instance, the Magellanic Stream ora major accretion episode (see Beers & Sommer-Larsen 1995and references therein). The two models have in common the“inside-out” formation of the thin disk, in the sense that bothassume that the timescale for the disk formation increase withgalactocentric distance (see section 5). This choice was dictatedby the necessity of reproducing the abundance gradients alongthe Galactic disk.The SFR is a Schmidt (1955) law with a dependence on thesurface gas density ( k = .
5, see Kennicutt 1998) and also onthe total surface mass density (see Dopita & Ryder 1994). Inparticular, the SFR is based on the law originally suggested byTalbot & Arnett (1975) and then adopted by Chiosi(1980): ψ ( r , t ) = ν Σ ( r , t ) Σ gas ( r , t ) Σ ( r ⊙ , t ) ! ( k − Σ gas ( r , t ) k (1)where the constant ν is a sort of e ffi ciency of the starformation process and is expressed in Gyr − : in particular, ν = Gyr − for the halo and 1 Gyr − for the disk ( t ≥ Gyr ).The total surface mass density is represented by Σ ( r , t ), whereas Σ ( r ⊙ , t ) is the total surface mass density at the solar position, as-sumed to be r ⊙ = Σ gas ( r , t )represents the surface gas density and t represents the time.These choices of values for the parameters allow the model tofit very well the observational constraints, in particular in thesolar vicinity. A threshold gas density for the star formationin the disk of 7 M ⊙ pc − is adopted in all the models presentedhere.The IMF is that of Scalo (1986) normalized over a massrange of 0.1-100 M ⊙ and it is assumed to be constant in spaceand time.
4. The cosmological simulation
The main aim of our work is to follow the chemical evolutionof spiral galaxies in a cosmological context. To this aim, werun a dark matter-only cosmological simulation, using the pub-lic tree-code GADGET2 (Springel 2005), in order to produce
E. Colavitti et al.: A cosmological infall law for the Milky Way and study dark matter halos in which a spiral galaxies can form.Our simulated box has a side of 24 h − Mpc. We used 256 par-ticles. We adopted the standard cosmological parameters fromWMAP 3-years (Spergel et al. 2007), namely Ω = . Ω λ = .
725 and Ω b = . . · h − M ⊙ and the Plummer-equivalent soften-ing length is set to 3.75 h − comoving kpc till redshift z = . h − physical kpc since z =
2. We use the public packageGRAFIC (Bertschinger 1995) to set up our initial conditions.The simulation started at redshift z =
20 and 28 outputs wereproduced. We have chosen to use a quite large spread in theredshifts at the beginning, while in the last part of the simula-tion, where a small change in the redshift corresponds a largechange in time, the redshifts are closer. We checked that the fi-nal mass function of DM halos and the power spectrum are inagreement with theoretical expectations.We identified DM halos at redshift z = l = . δ ≈
100 times the critical density,with δ given by the cosmological parameter as in Navarro &Steinmetz (2000).We then built the mass accretion history of our halos. Toachieve this goal, we analysed 28 outputs from redshift z = . z =
0. We identified all DM halos in each snapshot using theprocedure sketched above, except for the fact that we used theredshift-dependent density contrast given by Bryan & Norman(1997) to define the virial radius as a function of z . At any out-put z i + , we found all the progenitors of our halos at redshift z i .We defined a halo at redshift z i + to be a progenitor of one at z i if at least 50% of its particles belong to the candidate o ff spring(see e.g. Kau ff mann 2001, Springel et al. 2001 for a discussionof this threshold). The mass accretion history is defined as themass of the main progenitor of the halo as a function of redshift.Having the mass accretion histories, we were able to identifythe redshift of formation (defined as the epoch at which half ofthe mass of the forming halos were accreted) and the redshift atwhich each halo experienced its last major merger (defined asan increase of at least 25% of its mass with respect to the massof its main progenitor at the previous redshift). To identify theDM halos which can host a spiral galaxy similar to the MW weused selection criteria based on four di ff erent characteristics ofthe halos: – mass between 5 · M ⊙ and 5 · M ⊙ ; – spin parameter λ > . – redshift of last major merger larger than z = . – redshift of formation larger than z = . . · M ⊙ . This approach is similar to that followed byRobertson et al. (2005) except that we did not make any hy-pothesis on the fraction of cold gas falling into the disk butwe used the observations to fix it. In this way, we obtained thebaryon infall law from the mass accretion history of each halo.Here, we do not make any attempt to model the disk for-mation inside the hierarchically growing DM halo. This is un-doubtedly an over-simplification of the physics involved. Onthe other hand, the issue of disk galaxy formation in hierarchi-cal cosmologies is far from being solved. Any attempt to modelthe formation of the disk should use a number of assumptionswhich are currently under debate. As an example, the structureof the disk is obviously driven by the gas cooling coupled withits angular momentum content. Semi-analitical galaxy forma-tion models (SAMs) usually assume that DM and gas sharethe same specific angular momentum. But this point is verycontroversial (see e.g. D’Onghia & Burkert 2004, D’Onghiaet al. 2006, and references therein). Even direct self consis-tent numerical simulations are not currently able to solve theproblem, which may (Governato et al 2007) or may not (Abadiet al 2003) be simply due to insu ffi cient numerical resolutionand / or an insu ffi ciently detailed treatment of supernovae feed-back. Lacking a widely accepted model for the formation ofthe disk, we prefer to keep our model as simple as possible andto verify if the cosmological growth of the halo is compatiblewith the observational constraints obtained using available dataon the chemical composition of stars and gas in the Milky Way.In particular, we assumed that the derived infall law hasthe same functional form for the whole Milky Way, but thatthe normalization constant is di ff erent for di ff erent Galacticregions. In other words, the normalization constants were ob-tained by reproducing the present time total surface mass den-sity at any specific galactocentric distance (see next section),although here we will focus on the solar neighbourhood, leav-ing to a forthcoming paper a more detailed study of the wholeGalactic disk. Finally, we also considered an arithmetic meanof the infall laws of all four halos, in order to have an “average”cosmological infall law to study. In table 1 we summarize thecharacteristics of the halos. Figure 1 represents our best cos-mological halo (halo 48001) at four di ff erent redshifts (z = = = = . Colavitti et al.: A cosmological infall law for the Milky Way 5 Fig. 1.
This figure represents our best cosmological halo, i.e. halo 48001, at four di ff erent redshifts (z = = = = Table 1.
Characteristics of the chosen DMhalos
Group Mass [10 M ⊙ ] Spin parameter Redshift major merger Redshift of formation48001 90.26 0.045 5.00 1.75 - 1.5052888 465.75 0.059 3.75 1.50 - 1.2556009 90.73 0.049 3.25 2.00 - 1.756460 61.94 0.041 2.50 1.25 - 1.00
5. The infall laws
In testing the accretion laws, we started by adopting the two-infall law model, as suggested by Chiappini et al. (1997). Thislaw presents two distinct peaks. During the first peak the haloand thick disk formed whereas during the second peak the thindisk was assembled. The two accretion events are considered tobe independent from each other and only a very small fractionof the gas lost from the halo was assumed to have fallen ontothe disk. The infall law that we indicate as A ( r , t ) is expressed as: A ( r , t ) = a ( r ) e − t /τ H ( r ) + b ( r ) e − ( t − t max ) /τ D ( r ) [ M ⊙ pc − Gyr − ] (2)where a ( r ) and b ( r ) are two parameters fixed by reproducingthe total present time surface mass density along the Galacticdisk. In particular, in the solar vicinity the total surface massdensity Σ tot = ± M ⊙ pc − (see Boissier & Prantzos 1999). t max = . E. Colavitti et al.: A cosmological infall law for the Milky Way disk, τ H = . τ ( r ) is the timescale for the formation ofthe thin disk and it is a function of the galactocentric distance(formation inside-out, Matteucci and Franc¸ois 1989; Chiappiniet al. 2001). In particular, it is assumed that: τ D = . r ( kpc ) − . Gyr (3)Besides this infall law, we tested other possible laws, such as atime constant infall rate. In particular: A ( r , t ) = .
80 [ M ⊙ pc − Gyr − ] (4)This law is probably not realistic although Naab & Ostriker2006 concluded that an almost constant infall law over the disklifetime was to be preferred. Here we adopted it mainly for thepurpose of comparison with more realistic laws. We adoptedthat particular value of the infall rate in order to reproduce thepresent time infall rate (see Table 3), as well as the present timetotal surface mass density.The third infall law we tested it is a linear infall law, givenby: A ( r , t ) = . − . · t [ M ⊙ pc − Gyr − ] (5)Again, we used this particular expression in order to reproducethe present time Σ tot and infall rate.The fourth adopted infall law is the same as that ofChiappini et al. (1997) but with pre-enriched infalling gas. Themetallicity of the infalling gas which forms the disk was as-sumed to be 10 times lower than the present time interstellarmedium (ISM) metallicity while the infalling gas which formsthe halo is still primordial. The assumed chemical compositionof the infalling gas does not assume solar abundance ratios butreflects the composition of the halo-thick disk.Then, we tested the infall laws derived from the cosmo-logical simulations done with GADGET2 (Springel 2005), asdescribed before. In particular, to derive the cosmological infalllaw we proceeded in the following way: A ( r , t ) = a ( r )0 . dM DM dt [ M ⊙ pc − Gyr − ] (6)where 0.19 is the cosmological baryonic fraction and a ( r ) is anormalization constant fixed to reproduce the present time totalsurface mass density along the disk, in analogy with eq. (2). Forthe solar ring a ( r ) = Σ ( r ⊙ , t G ) M Gal , with M Gal = . M DM being thebaryonic mass of our Galaxy and t G the Galactic lifetime. Infigure 2 we show the values of a ( r ) versus the galactocentricdistance.One infall law is given by the arithmetic average of the in-fall laws derived for the four halos and the last infall law is thatsuggested by Naab & Ostriker (2006). In Table 2 we show themodel parameters. The di ff erent models are identified mainlyby their infall histories.Our infall laws for the solar region (8 kpc from the Galacticcenter) are shown in figure 3, whereas in figure 4 we show theincrease in time of the total surface mass density obtained bythe mass accretion history of the simulated halos. It is worthnoting that the infall law derived for the best halo selected as Fig. 2. a ( r ) vs radius. This normalization constant is fixed toreproduce the present time total surface mass density along thedisk (see eq. 6).representative of the Milky Way halo is very similar to the two-infall law by Chiappini et al. (1997).We selected our best halo by choosing the one which hasa very high redshift of last major merger. This is to ensure theright spin parameter for a Milky Way-like galaxy. The assem-bly history of this particular halo presents two distinct accretionpeaks which produce an infall law very similar to the two-infallmodel by Chiappini et al. (1997). The only di ff erence with thetwo-infall model is that in this case the two peaks are placedat a lower redshifts. After the two main peaks there are otherssmaller peaks. The remarkable fact is that all models predicta present time infall rate which is in good agreement with theobserved one, as quoted by Naab & Ostriker (2006). So we cansay that according to the infall laws derived from cosmologicalsimulations the Galaxy had some large infall episodes at highredshift, followed by smaller ones.In figure 4 we present the total surface mass density Σ tot ,expressed as M ⊙ pc − , as a function of time for all the models.Once again Models 1 and 4 (two-infall model with primordialand enriched infall, respectively) have the same Σ tot . The linearmodel predicts the larger final amount of matter, equal to 51.88 M ⊙ pc − . The constant model has a linear growth (in this case Σ tot is the integral of a constant infall law) and produces 49.98 M ⊙ pc − . Model 10, i.e. the model by Naab & Ostriker (2006),is the only one which starts to increase the amount of mattervery slowly (in the solar neighbourhood). After 5 Gyr from theBig Bang it only has reached 6.00 M ⊙ pc − . The cosmologicalmodels produce results which are quite similar to the two-infallmodel. At the beginning their growth is slower but after ∼
3- 3.5 Gyr their Σ tot increases with a steeper slope, due to thepeaks in the infall law.In figure 5 we show the infall law derived from Model 5(our best halo) for three galactocentric distances (4, 8 and 14kpc). As one can see the accretion histories are di ff erent at dif- . Colavitti et al.: A cosmological infall law for the Milky Way 7 ferent galactocentric distances, although no assumptions arepresent about the timescales of disk formation at any radius.This particular behaviour of the infall law with radius needs tobe tested and this will be the subject of a forthcoming paper.At the moment we only checked the gradient of O along thegalactic disk which is predicted to be very similar to the oneobtained with the two-infall model for R G > R G ≤
6. Results
In this section we present the chemical evolution results.Some results are shown in Tables 3, 4 and 5. In particular, inTable 3 we show the predicted present star formation rates, thepresent infall and the present SNIa and SNII rates, comparedwith the corresponding observational values. In Table 4 we plotthe total amount of gas and stars, the Σ gas / Σ tot and the totalsurface mass density in a ring of 2 kpc centered at the Sun’sgalactocentric distance (8 kpc). Finally, Table 5 presents thepredicted solar absolute abundances by mass for Fe, C, Mg, N,O and Si, namely the abundances in the ISM at the time of birthof the solar system 4.5 Gyr ago, compared with the observedones by Asplund et al. (2005).Figure 6 shows the star formation rate as a function of cos-mic time for all the models. At high redshift there is a gap in theSFR for some of the models. This gap is due to the adoption ofa threshold in the surface gas density, below which star forma-tion does not occur. In all models we have adopted a thresholdwhich is equal to 7.0 M ⊙ pc − during the formation of the thindisk. Model 10 instead, adopting the infall law suggested byNaab & Ostriker (2006), has a star formation threshold equalto 7.0 M ⊙ pc − both for the halo and the disk.From figure 6 we deduce that the constant infall model pre-dicts a growing star formation rate at low redshifts, a trendwhich is not predicted by the other laws. On the other hand,the cosmological best model (Model 5) predicts a very impor-tant peak between 3 and 6 Gyr, which should correspond tothe formation of the bulk of stars in the thin disk. This peak isdirectly connected to the trend of the infall law. After 10 Gyrfrom the Big Bang the threshold is easily reached in most of themodels, thus causing the SFR to have an oscillating behaviour.In figure 7 we present the SNIa rates for all the models.The cosmological law of Model 5 predicts a peak for the SNIarate at about 6 Gyr. This is due to the fact that the SFR in thismodel has an important peak at about 5 Gyr. Thanks to thispeak, many stars form and many SNIa explode after a delay ofabout 1 Gyr. All the models predict a SNIa rate between 0.003and 0.004 SNe pc − Gyr − , in good agreement with the valuegiven by Boissier & Prantzos (1999), i.e. 0.0042 ± / H] as a function oftime for all models. It is important to note that the model with a constant infall law (Model 2) and Model 10 never reach the so-lar abundance. The reason is that in both models the infall rateduring the whole galactic lifetime is probably overestimated.In the model by Chiappini et al. (1997) (our Model 1) [Fe / H]reaches a local peak at 1 Gyr, then decreases slightly to increaseagain. The little depression in [Fe / H] is due to the predicted gapin the SFR just before the formation of the thin disk. In fact, thesecond infall episode coupled with the halt in the SF producesa decrease of [Fe / H]. We can see the same behaviour in the cos-mological models. In particular in Model 5 the peak is followedby a deeper depression of [Fe / H] and this is due to the longergap in the SFR predicted by this model (1-2 Gyr) as opposedto that predicted by Model 1 which is less than 1 Gyr. This isan important prediction and it can be tested via chemical abun-dances. In fact, both Gratton et al. (1996) and Furhmann (1998)detected such an e ff ect in the [Fe / O] vs. [O / H] and [Fe / Mg] vs.[Mg / H], respectively.A very important constraint for the chemical evolution ofthe galaxies is represented by the G-dwarf metallicity distribu-tion. This is the relative number of G-dwarf stars as a functionof [Fe / H]. We have used the data from Rocha-Pinto & Maciel(1996), Kotoneva (2002), Jorgensen (2000) and Wyse (1995).Our predicted metallicity distributions are shown in figure 9.From this figure, it is clear that Model 10 predicts insu ffi cienthigh metallicity stars. On the other hand, some of the cosmo-logical models such as Model 7 and Model 8 predict too manymetal-poor stars. Our best cosmological model, i.e. Model 5,shows a bimodal metallicity distribution, which is clearly atodds with the data.The last constraint we study concerns the chemical abun-dances of several elements, such as O, Mg, Si, N and C. Infigure 10 the [O / Fe] as a function of [Fe / H] can be seen. Here,the range of [Fe / H] has been restricted to − . + . / H] down to − . α -elements are no longer produced whereas Fe continues to beproduced. This induces the [O / Fe] to decrease and also the[Fe / H] ratio to decrease to a lesser extent, because of the accre-tion of primordial gas. Then when SF starts again the [O / Fe]increases again. This loop is very prominant in some modelsand not in agreement with the data, although some spread ispresent. It is interesting to note that Model 4, which is the sameas Chiappini et al’s model but with the pre-enriched gas, is ac-ceptable. This is due to the fact that the metallicity of the pre-enriched infalling gas is not so di ff erent from the metallicity ofthe primordial infalling gas.Figures 12 and 13 present the [Mg / Fe] and the [Si / Fe] asa function of [Fe / H]. The data in figures 10, 11, 12 and 13 arefrom Cayrel et al. (2004) for the very metal poor stars and fromthe compilation of Franc¸ois et al. (2004) for all the others. Onceagain all the considerations made above for [O / Fe] are valid forthese other α -elements. E. Colavitti et al.: A cosmological infall law for the Milky Way
Table 2.
Model parameters. In the first column there is the number of the model, inthe second one the adopted infall law, in the third the time scale for the halo, in thefourth that for the disk and in the fifth the type of infalling gas. All the models adopt athreshold gas density for star formation in the disk of 7 M ⊙ pc − . The model by Naab& Ostriker (2006) is the only one which has the threshold also during the formationof the halo. Note that our best cosmological model is Model 5. Model Infall law τ halo τ disk Gas t = M ⊙ pc − Gyr − ] [Gyr] [Gyr]1 Two-infall law 0.8 7 Primordial2 3 .
80 0.8 7 Primordial3 6 . − . · T / Z today )5 Group 48001 - - Primordial6 Group 52888 - - Primordial7 Group 56009 - - Primordial8 Group 6460 - - Primordial9 Mean - - Primordial10 Naab & Ostriker - - Primordial Table 3.
Present time values for all the models and observed values as reported inBoissier & Prantzos (1999) and Chiappini et al. (2001).
Model SFR Infall SNII rate SNIa rate[ M ⊙ pc − Gyr − ] [ M ⊙ pc − Gyr − ] [ pc − Gyr − ] [ pc − Gyr − ]1 2.66 1.100 0.00900 0.003302 4.55 3.800 0.01928 0.004113 2.81 1.320 0.01194 0.003914 2.66 1.100 0.00900 0.003325 2.65 0.528 0.00584 0.003666 2.69 2.273 0.01140 0.003477 2.65 0.126 0.00229 0.003668 4.01 0.998 0.01712 0.004129 2.69 0.979 0.01147 0.0038110 4.72 3.406 0.02002 0.00397Boissier & Prantzos (1999) 2-5 1.0-3.3 0.02 0.0042 ± Table 4.
Present time values for all the models and observed values as reported inBoissier & Prantzos (1999) and Chiappini et al. (2001)
Model Gas Stars Σ gas Σ tot Total[ M ⊙ pc − ] [ M ⊙ pc − ] [ M ⊙ pc − ]1 7.00 35.24 0.1444 48.462 10.11 35.09 0.2024 49.983 7.42 38.66 0.1431 51.884 7.00 35.24 0.1444 48.465 6.99 36.13 0.1439 48.556 7.06 35.60 0.1455 48.537 7.00 36.69 0.1442 48.568 9.21 34.75 0.2056 48.559 7.06 36.32 0.1455 48.5510 10.29 34.29 0.2099 49.04Boissier & Prantzos (1999) 13 ± ± ± Table 5.
Predicted and observed solar abundances by mass (after 8.64 Gyr from theBig Bang)
Model Fe C Mg N O Si1 0.162E-02 0.156E-02 0.774E-03 0.121E-02 0.592E-02 0.980E-032 0.987E-03 0.119E-02 0.585E-03 0.932E-03 0.461E-02 0.665E-033 0.149E-02 0.135E-02 0.778E-03 0.105E-02 0.602E-02 0.940E-034 0.111E-02 0.157E-02 0.691E-03 0.121E-02 0.547E-02 0.771E-035 0.169E-02 0.199E-02 0.797E-03 0.142E-02 0.608E-02 0.102E-026 0.917E-03 0.140E-02 0.604E-03 0.105E-02 0.483E-02 0.653E-037 0.107E-02 0.168E-02 0.701E-03 0.124E-02 0.559E-02 0.761E-038 0.126E-02 0.212E-02 0.796E-03 0.144E-02 0.635E-02 0.879E-039 0.111E-02 0.173E-02 0.716E-03 0.127E-02 0.570E-02 0.783E-0310 0.531E-03 0.102E-02 0.439E-03 0.784E-03 0.362E-02 0.432E-03Asplund & al. (2005) 0.116E-02 0.217E-02 0.601E-03 0.623E-03 0.540E-02 0.669E-03
Other two important elements are C and N. Figures 14 and15 show the behaviour of [C / Fe] and [N / Fe] as a function of[Fe / H]. The data in figure 14 are from Spite et al. (2005) (ma-genta points), Carbon et al. (1987) (red points), Clegg, Lambert& Tomkin (1981) (cyan points), Laird (1985) (black points) andTomkin et al. (1995) (green points). Figure 15 presents the datafrom Spite et al. (2005) (magenta points), Israelian et al. (2004)(blue points), Carbon et al. (1987) (red points), Clegg, Lambert& Tomkin (1981) (cyan points) and Laird (1985) (black points).From figure 14 it can be seen once again that the cosmolog-ical models are very similar to the model by Chiappini et al.(1997). The predicted curves are di ff erent only for values of[Fe / H] higher than − . / Fe]. In both cases, cosmological models have a particularbehaviour at high metallicities. This behaviour is common toall the elements analysed and is due to the gap in the SFR atabout 1 Gyr, as discussed before. In the cosmological modelsthis e ff ect is larger because of the longer duration of the gap.However in the case of [C / Fe] and [N / Fe] we cannot draw anyfirm conclusion because of the large spread in the data. Finally,in figure 16 we show the O abundance gradient as predictedby Model 1 and Model 5, compared with a compilation of dataincluding Cepheids (see Cescutti et al. 2007). As one can see,the O gradient predicted by Model 5 flattens for r < r ≥ ff ect is predominating over theincrease of the timescale for disk formation. This deserves amore detailed study which we plan to do in a more detail thedisk evolution in a cosmological context in a forthcoming pa-per.Figures 17, 18 and 19 present the results obtained by usinga di ff erent infall law, derived from the cosmological simulationbut selecting di ff erent parameters. In this case we selected a halo which is not expected to produce a spiral galaxy, so welooked for a spin parameter lower than 0.04, a redshift of lastmajor merger lower than 2.5 and a redshift of formation lowerthan 1.0. Such a halo is perhaps more appropriate for an ellip-tical or S0 galaxy. We found a halo with the following charac-teristics: – mass = . · M ⊙ – λ = . – redshift of major merger = – redshift of formation = ff erent. In particular, it has amajor peak at a redshift of about 0.3. This produces a peakat the same redshift in the star formation rate and, of course,in the SNII rate. Moreover, there is a strong depression in the[Fe / H] ratio between 1.8 and 3 Gyr from the beginning of thesimulation, di ffi cult to reconcile with observations.In figures 18 and 19 we show the results for the [O / Fe] andfor the G-dwarf metallicity distribution. The main di ff erencebetween this halo and Models 1 and 5 is that the loop placedat [Fe / H] ∼ − . / Fe]at low [Fe / H], which is not observed in Galactic stars. As faras the G-dwarf metallicity distribution is concerned, the haloforms too many stars with low metallicity as a consequence ofthe deep depression in the [Fe / H] ratio (see the plot on the bot-tom right part of figure 17), again not in agreement with thedata, and resembles an early-type galaxy. This example con-firms the importance of the cosmological assembly history ofthe DM halo in determining not only the morphological param-eters of the galaxy it hosts, but also its chemical properties.
7. Conclusions
We have tested di ff erent gas infall laws for models of the for-mation of the Milky Way and especially cosmologically de-rived infall laws, obtained by means of cosmological simula-tions for the formation of the DM halo of the Milky Way. Inparticular, we assumed that the accretion law for the DM haloholds also for the baryonic matter. We found four di ff erent DM Fig. 3.
Infall vs time. Upper left panel: red solid line is the two-infall model (Model 1); black dashed line is the cosmologicalmean model (Model 9); green dotted line is the model by Naab & Ostriker (2006) (Model 10). Upper right panel: magentasolid line is the constant infall model (Model 2); blue dashed line is the linear infall model (Model 3); cyan dotted line is thepre-enriched model ( Z in f = / Z today , Model 4). Bottom left panel: black solid line is Model 5; magenta dashed line is Model6. Bottom right panel: blue solid line is Model 7; cyan dashed line is Model 8. In the bottom left panel the black solid arrowrepresents the redshift of last major merger for Model 5, the magenta dotted arrow the redshift of last major merger for Model 6,the black solid interval the redshift of formation for Model 5 and the magenta dotted interval the redshift of formation for Model6. In the bottom right panel the blue solid arrow represents the redshift of last major merger for Model 7, the cyan dotted arrowthe redshift of last major merger for Model 8, the blue solid interval the redshift of formation for Model 7 and the cyan dottedinterval the redshift of formation for Model 8.halos with properties compatible with a disk galaxy, with one inparticular seeming better than the others. All these infall lawswere then compared with the one proposed by Chiappini et al.(1997), called two-infall law, which predicts that there weretwo main accretion episodes which formed the halo-bulge-thick disk and the thin disk, respectively. We found that ourbest cosmological infall law is very similar to the two-infallone, which has already proven to be able to reproduce the ma-jority of the chemical properties of the Milky Way in the solarneighbourhood. Our cosmological infall laws have been testedin a detailed chemical evolution model for the Milky Way, fol- lowing the evolution of several chemical elements by takinginto account stellar lifetimes, SN progenitors and stellar nucle-osynthesis.Our main conclusions can be summarized as follows: – A model with constant infall predicts a present day infallrate and SFR larger than all the other models. Moreover,it is the only model which produces an unrealistically in-creasing SFR during the last billion years. This is probablyan unrealistic law, and we only used for a purpose of com-parison with other infall laws. . Colavitti et al.: A cosmological infall law for the Milky Way 11
Fig. 4. Σ tot vs time. Upper left panel: red solid line is the two-infall model (Model 1); black dashed line is the cosmological meanmodel (Model 9); green dotted line is the model by Naab & Ostriker (2006) (Model 10). Upper right panel: magenta solid lineis the constant infall model (Model 2); blue dashed line is the linear infall model (Model 3); cyan dotted line is the pre-enrichedmodel ( Z in f = / Z today , Model 4). Bottom left panel: black solid line is Model 5; magenta dashed line is Model 6. Bottomright panel: blue solid line is Model 7; cyan dashed line is Model 8. – The linear model predicts the largest amount of starspresently in the solar neighbourhood but it seems to repro-duce reasonably well all the other observables. However,this model does not describe the evolution of our Galaxy aswell as an exponential law does. – The model adopting the two-infall law but where the gas isassumed to be pre-enriched during the formation of the diskat the level of 1 /
10 of solar well reproduces the G-dwarfmetallicity distribution, as expected. – The cosmological laws, and in particular our preferred bestfit, seem to fit well all the data. This law predicts two mainaccretion episodes which can be identified with the forma-tion of halo-thick disk and thin disk , respectively, very sim-ilar to the two-infall law. Moreover, there seems to be a gapof 1-2 Gyr in the SFR between the two episodes, largerthan predicted by Chiappini et al. (1997) ( < / O] vs. [O / H] (Gratton et al. 1996) and at [Fe / Mg]vs. [Fe / H] (Fuhrmann 1998), although new data are nec-essary to draw firm conclusions. The model including thiscosmological infall law can well reproduce most of the ob-servational constraints. It predicts for the G-dwarf metallic-ity distribution, in the solar vicinity, two di ff erent peaks: wespeculate that the first peak represents the stars of the haloand thick disk while the second peak represents the stars ofthe thin disk. The same metallicity distribution computedfor the central region should include also the bulge stars.The predicted timescales for the formation of the halo-thickdisk and the thin disk, respectively, are in excellent agree-ment with those suggested by Chiappini et al. In particu- Fig. 5.
This figure represents the infall law our best cosmological halo, i.e. halo 48001, at four di ff erent radius (4 kpc: blue dottedline; 8 kpc: red solid line; 14 kpc: green dashed line).lar, the halo-thick disk must have formed on a timescalenot longer than 1-2 Gyr whereas the thin disk in the solarvicinity took at least 6 Gyr to assemble 60% of its mass. Asa consequence of the gap between the halo-thick disk andthe thin disk, we predict that the thin disk is at least 2 Gyryounger than the halo. – The other cosmological infall laws are characterized byseveral minor accretion events after the two main ones andpredict larger gaps in the SFR which are not observed in the[Fe / O] vs. [O / H] and [Fe / Mg] vs. [Mg / H] which indicate agap not larger than 1-2 Gyr. – A model adopting a cosmologically inferred infall law byNaab & Ostriker (2006) presents a behaviour very similarto the constant infall law and predicts too low metallici-ties at the Sun age and at the present time. Moreover, thismodel predicts a too small number of G-dwarf with highmetallicity. In their paper they present a G-dwarf metallic-ity distribution but as a function of Z which represents Oand not Fe as in the observations. – Our results strongly depend on what criteria were used toselect the dark matter halo from the cosmological simula-tions. If they are not suitable for forming a spiral galaxyit is possible to see that the results are not in good agree-ment with the observations. We prove it by using a DM halowith dynamical parameters compatible with an early-typegalaxy. – Our results can be compared with the work of Robertsonet al. (2005), in which the authors studied the chemical en-richment of the stellar halo of the Milky Way, using the pre-scriptions of the hierarchical scenario. They supposed thatmost of the mass in the MW halo was acquired via mergerswith massive dIrr-type DM halos, occurred at a look-backtime of ∼
10 Gyr. They used three examples of mass accre-tion history, supposing that the cumulative mass accretionin individual DM halos can be well described by a an ana-lytical function obtained by Wechsler et al (2002). . Colavitti et al.: A cosmological infall law for the Milky Way 13
Fig. 6.
SFR vs time. Upper left panel: red solid line is the two-infall model (Model 1); black dashed line is the cosmological meanmodel (Model 9); green dotted line is the model by Naab & Ostriker (2006) (Model 10). Upper right panel: magenta solid lineis the constant infall model (Model 2); blue dashed line is the linear infall model (Model 3); cyan dotted line is the pre-enrichedmodel ( Z in f = / Z today , Model 4). Bottom left panel: black solid line is Model 5; magenta dashed line is Model 6. Bottomright panel: blue solid line is Model 7; cyan dashed line is Model 8.Moreover, they assumed that the cold gas inflow rate tracksthe DM accretion rate and that the fraction of cold gas isequal to 2%.In order to build the stellar halo of the Milky Way theyused a dIrr-type dark matter halo with a virial mass M =
6x 10 M ⊙ , accreted 9 Gyr ago, following their assumed ac-cretion law. In this case the time available for the star for-mation and the consequent chemical enrichment is only ∼ – In the future we plan to extend the current work, and in par-ticular our cosmologically derived baryonic infall laws, tothe study of the chemical properties of the whole disk. Aswe have already shown in this paper, by normalizing the in-fall law to the present time total surface mass density alongthe disk, we obtain di ff erent timescales for the assembly ofthe disk as a function of galactocentric distance, althoughthe inside-out e ff ect is not as marked as in the Matteucci &Franc¸ois (1989) and Chiappini et al. (2001) models.The fact that all our four suitable DM halos show an ac-cretion law which resembles that used in the two-infall modelcould be linked to the way in which such halos assemble.Indeed, they have their last major merger at high redshift, largerthan z = .
5, by selection and they reach a mass larger than
Fig. 7.
SNIa rate vs time. Upper left panel: red solid line is the two-infall model (Model 1); black dashed line is the cosmologicalmean model (Model 9); green dotted line is the model by Naab & Ostriker (2006) (Model 10). Upper right panel: magenta solidline is the constant infall model (Model 2); blue dashed line is the linear infall model (Model 3); cyan dotted line is the pre-enriched model ( Z in f = / Z today , Model 4). Bottom left panel: black solid line is Model 5; magenta dashed line is Model 6.Bottom right panel: blue solid line is Model 7; cyan dashed line is Model 8.50% of their final one at lower redshift. As a consequence ofour requirement not to have late major mergers, such late as-sembly happens via accretion of material from the field, namelyfilaments, or via minor mergers. These two epochs of impor-tant accretion qualitatively corresponds to the two peaks usedin the two-infall model and give it a cosmological motivation.Obviously the details of the late accretion episode will dependon the dynamical history of the single DM halo, and will gen-erate di ff erences in the chemical patterns of individual late-type galaxies without destroying their overall properties. Onthe other hand, halos should acquire their angular momentumthanks to the cosmological torques acting at high redshifts onthe material (both baryons and dark matter) which will coa-lesce to form them. Such torques will also influence their massaccretion histories. Thus, selecting DM halos with high spinvalues could also result in selecting halos with similar dynam- ical histories. Astrophysical processes acting on baryons, e.g.feedback, should not be able to dramatically alter this scenario.Finally we note that, while in the two-infall model the timing ofthe two episodes is a free parameter, in the cosmological infallscenario the timing is directly given by the gravitational evo-lution of the halos. In this sense, the agreement between suchmodels is not a-priori guaranteed and could be interpreted asan interesting link between the morphological properties of thelate-type galaxies (used to fix our requirement) and their chem-ical properties, via the hierarchical model.
8. Acknowledgments
We thank Gabriele Cescutti and Cristina Chiappini for somehelpful suggestions and Donatella Romano, Patrick Franc¸oisand Silvia Kuna Ballero for their collaboration concerning the . Colavitti et al.: A cosmological infall law for the Milky Way 15 -2-1.5-1-0.500.5 Redshift Redshift0 5 10-2-1.5-1-0.500.5 Time [Gyr] 0 5 10Time [Gyr]
Fig. 8. [Fe / H] vs time. Upper left panel: red solid line is the two-infall model (Model 1); black dashed line is the cosmologicalmean model (Model 9); green dotted line is the model by Naab & Ostriker (2006) (Model 10). Upper right panel: magenta solidline is the constant infall model (Model 2); blue dashed line is the linear infall model (Model 3); cyan dotted line is the pre-enriched model ( Z in f = / Z today , Model 4). Bottom left panel: black solid line is Model 5; magenta dashed line is Model 6.Bottom right panel: blue solid line is Model 7; cyan dashed line is Model 8.data. We also thank the referee Chris Flynn for valuable com-ments. References
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Upper right panel: magenta solidline is the constant infall model (Model 2); blue dashed line is the linear infall model (Model 3); cyan dotted line is the pre-enriched model ( Z in f = / Z today , Model 4). Bottom left panel: black solid line is Model 5; magenta dashed line is Model 6.Bottom right panel: blue solid line is Model 7; cyan dashed line is Model 8. The data are from: Spite et al. (2005) (magenta starswith three arms), Israelian et al. (2004) (blue crosses), Carbon et al. (1987) (red circles), Clegg, Lambert & Tomkin (1981) (cyanstars with five arms) and Laird (1985) (black triangles). . Colavitti et al.: A cosmological infall law for the Milky Way 23 Fig. 16.
Predicted and observed O abundance gradients in the galactocentric distance range 4 - 14 kpc. The continuous line is theprediction of the two-infall model, whereas the dashed line is the prediction of Model 5. The data points are from Cepheids. Thebig squares with error bars represent averages of the points with their errors (see Cescutti et al. 2007 and reference therein).
Fig. 17.
These plots represent the infall law (upper left panel), the star formation rate (upper right panel), the SNIa rate (bottomleft panel) and the [Fe / H] (bottom right panel) as a function of time for the two-infall model (Model 1, red solid line), for Model5 (black dashed line) and for the halo 20912 (blue dotted line). . Colavitti et al.: A cosmological infall law for the Milky Way 25 -2 -1.5 -1 -0.5 0-0.200.20.4 [Fe/H]
Fig. 18.
This plot represents the [O / Fe] as a function of [Fe / H]. The red solid line represents the two-infall model (Model 1), theblack dashed line represents Model 5 and the blue dashed line the halo 20912. -1.5 -1 -0.5 0 0.500.050.10.150.20.25 [Fe/H]