New analytical solutions for chemical evolution models: characterizing the population of star-forming and passive galaxies
aa r X i v : . [ a s t r o - ph . GA ] F e b Astronomy & Astrophysics manuscript no. 29745˙ap c (cid:13)
ESO 2018June 27, 2018
New analytical solutions for chemical evolution models:characterizing the population of star-forming and passive galaxies
E. Spitoni ⋆ , F. Vincenzo , , and F. Matteucci , Dipartimento di Fisica, Sezione di Astronomia, Universit`a di Trieste, via G.B. Tiepolo 11, I-34131, Trieste, Italy I.N.A.F. Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, I-34131, Trieste, ItalyReceived xxxx / Accepted xxxx
ABSTRACT
Context.
Analytical models of chemical evolution, including inflow and outflow of gas, are important tools for studyinghow the metal content in galaxies evolves as a function of time.
Aims.
We present new analytical solutions for the evolution of the gas mass, total mass, and metallicity of a galacticsystem when a decaying exponential infall rate of gas and galactic winds are assumed. We apply our model to characterizea sample of local star-forming and passive galaxies from the Sloan Digital Sky Survey data, with the aim of reproducingtheir observed mass-metallicity relation.
Methods.
We derived how the two populations of star-forming and passive galaxies differ in their particular distributionof ages, formation timescales, infall masses, and mass loading factors.
Results.
We find that the local passive galaxies are, on average, older and assembled on shorter typical timescales thanthe local star-forming galaxies; on the other hand, the star-forming galaxies with higher masses generally show olderages and longer typical formation timescales compared than star-forming galaxies with lower masses. The local star-forming galaxies experience stronger galactic winds than the passive galaxy population. Exploring the effect of assumingdifferent initial mass functions in our model, we show that to reproduce the observed mass-metallicity relation, strongerwinds are requested if the initial mass function is top-heavy. Finally, our analytical models predict the assumed sampleof local galaxies to lie on a tight surface in the 3D space defined by stellar metallicity, star formation rate, and stellarmass, in agreement with the well-known fundamental relation from adopting gas-phase metallicity.
Conclusions.
By using a new analytical model of chemical evolution, we characterize an ensemble of SDSS galaxies interms of their infall timescales, infall masses, and mass loading factors. Local passive galaxies are, on average, older andassembled on shorter typical timescales than the local star-forming galaxies. Moreover, the local star-forming galaxiesshow stronger galactic winds than the passive galaxy population. Finally, we find that the fundamental relation betweenmetallicity, mass, and star formation rate for these local galaxies is still valid when adopting the average galaxy stellarmetallicity.
Key words. galaxies: abundances - galaxies: evolution - galaxies: ISM
1. Introduction
Chemical evolution of galaxies studies how subsequent stel-lar generations and gas flows alter the chemical composi-tion of the galaxy interstellar medium (ISM) to give riseto the present-day observed chemical abundance pattern ofgalaxies. In this respect, the galaxy star formation and gasmass assembly histories play a major role, together with theassumed initial mass function (IMF) and stellar nucleosyn-thetic yields. Chemical evolution is an essential ingredientin the broader framework of galactic archeology, which aimsat constraining and recovering the formation and evolutionof galaxies, starting from the observed chemical, dynami-cal, and photometric properties of their member stars atthe present time.Simple analytical models of chemical evolution have pre-viously enabled deriving analytical functions for the metal-licity evolution of a stellar system. We assume a constantIMF and complete mixing of the various chemical specieswithin the ISM of a galaxy at any time of its evolution.Finally, a further fundamental hypothesis is to retain the ⋆ email to: [email protected] instantaneous recycling approximation (IRA): all the starswith mass m ≥ ⊙ instantaneously die as they form,whereas all the stars with m < ⊙ have infinite life-times. This type of models still represents a useful tool fortracing the metallicity evolution of galaxies, but only whenthe abundance of chemical elements produced on typicalshort timescales is considered. An example of such a chem-ical element is given by oxygen, which incidentally repre-sents the best proxy for the total metallicity of the galaxyISM. Pioneering works in this field are considered those ofSchmidt (1963), Searle & Sargent (1972), Tinsley (1974),and Pagel & Patchett (1975).To build a realistic analytical model for the galaxychemical evolution, gas flows must be included in theset of differential equations to be solved because galax-ies do not evolve as closed boxes. Analytical and semi-analytical solutions for models including infall and out-flow of gas have been known for at least 30 years(see, for example, Chiosi 1980; Clayton & Pantelaki 1986,1993; Pagel & Patchett 1975; Hartwick 1976; Clayton 1988;Twarog 1980; Edmunds 1990; Erb 2008; Recchi & Kroupa2015; Kudritzki et al. 2015). Moreover, solutions have also been found later with radial gas flows, provided a partic-ular gas velocity profile is assumed in the equations (seeLacey & Fall 1985; Edmunds & Greenhow 1995; Martinelli1998; Portinari & Chiosi 2000; Spitoni & Matteucci 2011;Pezzulli & Fraternali 2015). A summary of some of themost frequently used analytical solutions for the metallicityevolution of a system with different prescriptions for inflowsand outflows of gas can be found in Recchi et al. (2008) andSpitoni et al. (2010).Recently, Spitoni (2015) presented an analytical solu-tion for the evolution of the metallicity of a galaxy in pres-ence of environmental effects. In this work, a galaxy suffersthe infall of enriched gas from another evolving galacticsystem, with the metallicity of the infalling material evolv-ing according to the metallicity evolution of the companiongalaxy, hence with chemical abundances variable in time.More recently, Weinberg et al. (2016) showed chemicalanalytical solutions with different prescriptions for the starformation rate (SFR) analyzing constant, exponential, orlinear-exponential star formation histories.To solve the set of differential equations for the gasmass, total mass, and metallicity of the galaxy and henceobtain analytical solutions for these quantities, some ofthe previous works in the literature (see, for example,Matteucci & Chiosi 1983, Matteucci 2012) assumed an in-fall rate of gas that is directly proportional to the SFR overthe entire galaxy evolution. This assumption is not phys-ical, and it represents a strong simplification in analyticalmodels. Nevertheless, by exploring the effects of differentprescriptions for the infall term in the equations, the finalpredicted physical properties of the galaxy have been foundto not deviate substantially from the case of a generic expo-nential infall law (Recchi et al., 2008). The infall of gas thatfollows an exponential law is a fundamental assumptionadopted in most of the detailed numerical chemical evo-lution models in which IRA is relaxed. Chemical evolutionmodels of our Galaxy (Chiappini et al. 1997; Romano et al.2010; Brusadin et al. 2013; Micali et al. 2013) assume thatthe various different stellar components formed throughdifferent separated accretion episodes of gas, with the ac-cretion rate of each episode obeying to a decaying ex-ponential law. Colavitti et al. (2008) reported that thetwo-infall model of Chiappini et al. (1997) is qualitativelyin agreement with results of the GADGET2 (Springel2005) cosmological hydrodynamical simulations when thestandard cosmological parameters from WMAP three-years (Spergel et al. 2007) are assumed, namely Ω =0.275,Ω λ =0.725, and Ω b =0.041.We here present the results of an analytical chemicalevolution model in which a decaying exponential infall rateof gas is assumed as a function of time; we show that an-alytical solutions for the evolution of the galaxy metal-licity, gas mass, and total mass can be found under thisassumption. Furthermore, we apply our model to inves-tigate and explain the observed mass-metallicity relation(MZ relation, hereafter), as derived in a sample of galaxiesfrom the Sloan Digital Sky Survey (SDSS) by Peng et al.(2015). We extend and update the methods and results ob-tained by Spitoni et al. (2010), which reproduced the ob-served MZ relation in 27730 local SDSS star-forming galax-ies (Kewley & Ellison, 2008) with an analytical model ofchemical evolution. In particular, we aim at characterizingthe two distinct MZ relations that Peng et al. (2015) de-rived for the local actively star-forming (we refer to them as star-forming ) galaxies and the passively evolving ( passive )galaxies. A tight relation between the stellar mass and thegas-phase metallicity such as displayed by a large sampleof galaxies like the one provided by the SDSS can be usedto constrain the various fundamental parameters playing arole in any galaxy formation theory: the infall timescale,wind loading factor parameters, infall mass values, and thestar formation efficiencies.On the one hand, the analytical model of Spitoni et al.(2010) assumed both the gas outflow rate and the gas infallrate to be directly proportional to the galaxy SFR; whenthis simplifying assumption is made, the analytical solutionfor the metallicity of the system does not explicitly dependupon the time variable, which indeed turns out to be hiddenin the galaxy gas mass fraction entering in the equations.Furthermore, Kudritzki et al. (2015) assumed constantratios of galactic wind mass-loss and accretion mass gainto SFR in the IRA approximation. They investigated theradially averaged metallicity distribution of the interstellarmedium of a sample of 20 local star-forming disk galaxiesby means of analytical chemical evolution model.On the other hand, in this work, we show that when adecaying exponential infall law is assumed, the analyticalsolution for the evolution of the galaxy metallicity explicitlydepends upon the time variable; this fact will allow us toalso provide an estimate for the age of the galaxies andcharacterize them in terms of their infall timescale.Mannucci et al. (2010) added a further dimension to theobserved MZ relation of local galaxies; in particular, theyfound that local galaxies place themselves on a tight sur-face in the 3D space defined by gas phase metallicity, SFR,and stellar mass, with a small residual dispersion of about0 .
05 dex and hence showing an underlying fundamental re-lation. In this work, we test whether our new analyticalmodel is able to recover a similar fundamental relation forthe local population of passive and star-forming galaxies.Our paper is organized as follows. In Sect. 2, aftersummarizing the basic assumptions of our chemical evolu-tion model, we present the new analytical solutions for thegalaxy metallicity, gas mass, and total mass. In Sect. 3 wediscuss the observed MZ relation in the population of localpassive and star-forming galaxies by Peng et al. (2015) andpresent the method we employ to reproduce it. In Sect. 4we present the methods used to reproduce the MZ relationwith the new analytical solutions. In Sect. 5 we report ourresults, and in Sect. 6, we discuss whether the population ofgalaxies drawn by our new analytical model are predicted tofollow the fundamental relation of Mannucci et al. (2010).Finally, our conclusions are drawn in Sect. 7.
2. Model
In this section we present new analytical solutions for theevolution of the metallicity, gas mass, and total mass ofgalaxies in the framework of simple models of chemical evo-lution where an exponential infall of gas is assumed. Wesummarize the basic assumptions of our model and showthe system of differential equations that are to be solved.We show the new analytical solution and explore the effectof varying the main free parameters of our model.
Passive Galaxies Star-Forming galaxies Passive Galaxies Star-Forming galaxiesRange
75 %
Range
75 %
Range
75 %
Range
75 %
Infall timescale τ [Gyr] 0 . − . ≤ . . − . ≤ . − . ≤ . . − . ≤ . . − . ≤ . − . ≤ . . − . ≤ . − ≤ . λ − . ≤ . − . ≤ .
25 0 . − ≤ . . − . ≤ . M inf [M ⊙ ]) 9 . − . ≤ . . − . ≤ .
43 10 . − . ≤ . . − . ≤ . Table 1.
Principal characteristics of the computed passive and star-forming galaxies. In each column we indicate therange spanned by the values related to the infall timescale τ , the ages of the computed local galaxies, the wind parameter λ, and the infall mass M inf . Moreover, the range of values at which 75% of the galaxies are found is presented. Resultsare also indicated for two different IMFs: a Chabrier (2003) and Salpeter (1955) IMF. The main assumptions of the simple model (Tinsley, 1980)are as follows:1. The IMF is constant in time and space, which meansthat every galaxy stellar generation hosts stars with amass sampling a universal distribution, regardless of theage, metallicity, and birthplace of the stellar generation.2. The gas is well mixed at any time of the galaxy evolution( instantaneous mixing approximation ).3. Stars with mass m ≥ ⊙ die instantaneously, as soonas they form (IRA), while stars with mass m < ⊙ have infinite lifetimes.By making these simplifying assumptions, we can deriveanalytical formulas for the evolution of the main galaxyphysical properties, such as the metallicity Z , SFR, gas,and stellar mass; the two following quantities appear in thevarious differential equations: R = Z ∞ ( m − M R ) φ ( m ) dm, (1)which represents the so-called returned mass fraction,where φ ( m ) is the IMF and M R is the mass of the stel-lar remnant, and y Z = 11 − R Z ∞ m p Z ( m ) φ ( m ) dm, (2)which represents the so-called yield per stellar generation,with p Z ( m ) being the fraction of the newly produced andejected metals by a star of mass m . The values of y Z and R for different IMFs are taken from Vincenzo et al.(2016), who also showed that the effect of metallicity onthese quantities is minor compared to the adopted IMFand set of stellar yields. The IMF is defined in the stellarmass range 0.1- 100 M ⊙ . In particular, we assume aver-age values over metallicity from Table 2 of Vincenzo et al.(2016), corresponding to the compilation of stellar yields ofRomano et al. (2010).In our model we explore the effect of the two followingIMFs: 1. Chabrier (2003, which is similar to Kroupa 2001), forwhich we obtain a return mass fraction R = 0 . y Z = 0 . , anda yield of oxygen per stellar generation y O = 0 . R = 0 . y Z = 0 . , and y O = 0 . ψ ( t ) = S × M gas ( t ) , (3)where M gas ( t ) is the galaxy gas mass at the time t, and S is the so-called star formation efficiency (SFE), a freeparameter of our model that is measured in Gyr − .The infall model is taken from Chiosi (1980) and therate is assumed to obey the following decaying exponentiallaw: I ( t ) = Ae − t/τ , (4)where τ is the so-called infall timescale, which determinesthe typical timescale over which the galaxy is assumed toassemble, and A is a constant constrained by the total infallgas mass ( M inf ) by the following equation: Z t G Ae − t/τ dt = M inf ⇒ A = M inf τ (cid:0) − e − t G /τ (cid:1) . (5)With Eq. (5) we impose that the integral of the infall rateover the entire galaxy lifetime is the total gas infall mass, M inf . We assume the galactic lifetime to be t G = 14 Gyr.In our model, we also take into account outflow gasepisodes in galaxies. The outflow rate is assumed to beproportional to the SFR in the galaxy (see Matteucci 2012;Matteucci & Chiosi 1983): W ( t ) = λψ ( t ) , (6)with the wind parameter λ being a dimensionless quantity.We are aware that in our study galactic winds are treatedin a simple way. On the other hand, Recchi & Hensler(2013) showed that the pure galactic wind mass-loss de-pends on the gas stratification: we can obtain various windstrengths and different mass loading factors for the same SFR. However, Eq. (6) has been used in almost all thechemical evolution models in the literature up to now (seeVincenzo et al. 2014; Recchi et al. 2008; Lanfranchi et al.2008).
The set of differential equations we have to solve to char-acterise the evolution of the galaxy total mass, gas mass,and metallicity is the following: ˙ M tot ( t ) = Ae − t/τ − λψ ( t )˙ M gas ( t ) = − (cid:0) − R (cid:1) ψ ( t ) + Ae − t/τ − λψ ( t )˙ M Z ( t ) = (cid:0) − Z ( t ) + y z (cid:1)(cid:0) − R (cid:1) ψ ( t ) − λZ ( t ) ψ ( t ) + Z inf Ae − t/τ , (7) where M tot and M gas are the total mass and the gas massof the galaxy, respectively; Z = M Z /M gas represents thegas metallicity of the system, and Z inf the metallicity ofthe infalling gas. The total stellar mass of the system canbe retrieved by means of the following formula: M ⋆ ( t ) = M tot ( t ) − M gas ( t ).By recalling that M Z = Z × M gas , if we differentiatethe latter with respect to time and then combine the sec-ond and the third equations in the system of Eq. (7), thetemporal evolution of the gas metallicity, Z , obeys the fol-lowing differential equation:˙ Z ( t ) = y z (cid:0) − R (cid:1) S + A (cid:0) Z inf − Z ( t ) (cid:1) e − t/τ M gas ( t ) . (8)By assuming Z inf = 0, the second term in the right-handside of Eq. (8) is responsible for a dilution of the metalcontent within the galaxy ISM as the infall rate of gas pro-ceeds. In this section, we present the analytical solutions for thesystem of Eq. (7) and show the effects on M gas ( t ), M tot ( t ),and Z ( t ) caused by the variation of the adopted IMF andwind parameter λ .We assume for Eq. (7) the following initial conditions:1. At t = 0, we assume M tot (0) = M gas (0) ≪ M inf . Tointegrate Eq. (7), an initial gas mass different from zerois required; we consider it as negligible with respect tothe infall mass.2. The metallicity of the gas infall is constant: Z inf = 0.3. The metallicity of the galaxy is primordial at the for-mation of the galaxy: Z (0) = Z inf = 0.The solution for the gas mass is M gas ( t ) = e − αt A (cid:2) e − t/τ + αt − (cid:3) τατ − M gas (0) ! , (9)while the total galaxy mass ( M gas ( t ) + M ⋆ ( t )) evolves ac-cording to the following formula: M tot ( t ) = M gas (0) Sα (cid:0) λe − αt + 1 − R (cid:1) ++ Sα (cid:0) − R (cid:1) Aτ − Aτ e − t/τ + Sα λAτ (cid:0) τ αe − t/τ − e − αt (cid:1)(cid:0) ατ − (cid:1) . M / M s un t [Gyr]No Wind M tot M star M gas M / M s un t [Gyr]Wind λ =0.5M tot M star M gas Fig. 1.
Effects of galactic winds. We show in both panelsthe predicted evolution in time of the galaxy gas mass M gas (gray dashed lines), stellar mass M ⋆ (blue dotted lines), andtotal mass M tot = M gas + M ⋆ (red solid lines). The modelassumes an exponential infall law with timescale τ = 2 Gyrand infall mass M inf = 10 M ⊙ , and an SFE S = 1 Gyr − . Left panel : model without galactic winds ( λ = 0); rightpanel : model with wind parameter λ = 0 . Z t [Gyr]Salpeter (1955) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 W i nd pa r a m e t e r λ Z t [Gyr]Chabrier (2003) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 W i nd pa r a m e t e r λ Fig. 2.
Effect of varying the wind parameter λ on the timeevolution of the metallicity Z . The model assumes an expo-nential infall law with timescale τ = 2 Gyr and infall mass M inf = 10 M ⊙ , and an SFE S = 1 Gyr − . In the left panelwe show the results of models with a Salpeter (1955) IMF,while in the right panel a Chabrier (2003) IMF is assumed.Finally, the solution for the galaxy gas-phase metallicity is Z ( t ) = y z S (cid:0) − R (cid:1) ατ − ×× M gas (0) t (cid:0) ατ − (cid:1) + Aτ (cid:2) t − τ (1 + αt ) + τ e αt − t/τ (cid:3) Aτ (cid:0) e αt − t/τ − (cid:1) + M gas (0) (cid:0) ατ − (cid:1) . In these equations, we have defined the parameter α asfollows: α = (cid:0) λ − R (cid:1) S. (10)We note that when we consider the case without infall mass( M inf = 0, and hence A = 0) and without winds ( λ = -1-0.8-0.6-0.4-0.2 0 0.2 9 9.5 10 10.5 11 11.5 l og ( Z * / Z s un ) log(M * /M sun )Observational data by Peng et al. (2015) P a s s i v e g a l a x i e s S t a r- f o r m i n g g a l a x i e s Fig. 3.
Observed uncertainties of the MZ relations for star-forming and passive galaxies by Peng et al. (2015) are re-ported with blue lines. We show with orange lines the third-order polynomial fits for the passive and star-forming galax-ies adopted in this article.0), we recover the closed-box solution. We have with theseassumptions that M tot , cb ( t ) = M gas (0) , (11) M gas , cb ( t ) = M gas (0) e − (1 − R ) St , (12)and Z cb ( t ) = y z S (1 − R ) t. (13) M tot,cb ( t ), M gas,cb ( t ) , and Z cb ( t ) are exactly the solu-tions of the closed-box model reported in Spitoni (2015)for the total mass, gas mass, and metallicity of the gas,respectively.In Fig. 1 we show how M gas , M ⋆ and M tot evolvewhen we assume an infall timescale τ = 2 Gyr, infallmass M inf = 10 M ⊙ , and SFE S = 1 Gyr − . A modelwith wind parameter λ = 0 . t G , both models predict the total galaxy gas massto be zero and hence the stellar mass to approach thegalaxy total mass. In the model without galactic winds, M ⋆ ( t G ) ≈ M tot ( t G ) = M inf , while in the model with out-flow we find that M ⋆ ( t G ) ≈ M tot ( t G ) < M inf , since thegalaxy loses a substantial fraction of its total infall massin the intergalactic medium (IGM) because of the galacticwinds. The shape of the temporal evolution of the galaxySFR can be retrieved from the behavior of M gas ( t ), sincewe assume them to be proportional between each other.In Fig. 2 we show how the gas metallicity Z is pre-dicted to evolve with time when we assume different windparameters, λ . The right panel shows the results of modelswith a Chabrier (2003) IMF, while the left panel showsmodels with a Salpeter (1955) IMF. All the models as-sume M inf = 10 M ⊙ , τ = 2 Gyr , and S = 1 Gyr − .The figure clearly shows that as the wind parameter λ in-creases, the galaxy metallicity is predicted to saturate atprogressively earlier epochs toward always lower metallici-ties. Interestingly, the assumption of a Chabrier (2003) IMFcauses an enhanced chemical enrichment of the galaxy ISM.At any galactic time, the gas metallicity with a Chabrier (2003) IMF is about twice higher than the metallicity witha Salpeter (1955) IMF, which is due to the large portionof massive stars in the Chabrier (2003) IMF. As expected,the choice of the IMF has a strong effect on the metallicityevolution of a galactic system.
3. MZ relation of the SDSS sample of star-formingand passive galaxies
By analyzing a sample of local galaxies from SDSS data,Peng et al. (2015) were able to separate the local popula-tion of actively star-forming and gas-rich galaxies from thepassive and gas-poor galaxies. In Fig. 3 we report the databy Peng et al. (2015), showing that these two populationsof galaxies present distinct relations in the MZ plane, withthe passive galaxies having on average higher stellar metal-licities than the star-forming galaxies.For a given stellar mass, there is a gap in metallicitybetween the two galaxy populations, which is observed todiminish as the stellar mass increases. At M ⋆ ≈ M ⊙ and beyond, star-forming and passive galaxies share almostthe same average stellar metallicities. In Fig. 3 we also showthe fits of the passive and star-forming sequences by meansof third-order polynomial functions that we use in this pa-per.To explain the observed MZ relation of the star-forming and passive galaxy populations in the SDSS data,Peng et al. (2015) suggested that galaxies ceased to accretegas from the outside and kept forming stars only by ex-hausting the remaining available cold gas reservoir withintheir potential well. In this way, as soon as the galaxy isstrangulated and stops accreting gas, the concentration ofmetals in the galaxy can steeply increase, and similarly themetallicity of all the subsequent stellar generations rises.The average time needed for star-forming galaxies to reachthe high-metallicity stripe of passive galaxies in the MZ re-lation is predicted by Peng et al. (2015) to be on the orderof ∼
4. Methods
For the MZ relation, Peng et al. (2015) adopted the averagestellar metallicity, Z ⋆ ( t ); we recover this quantity from ourmodels by averaging the gas metallicity of each galaxy inthe following way: h Z ⋆ ( t ) i = R t dt ′ Z ( t ′ ) ψ ( t ′ ) R t dt ′ ψ ( t ′ ) , (14)where the metallicity of the various galaxy stellar popula-tions is weighted with the total number of stars formed withthat metallicity (the latter quantity is directly proportionalto the SFR). This expression represents the so-called mass-weighted average metallicity of all the stellar populationsever born in the galaxy (see Pagel 1997). -1-0.8-0.6-0.4-0.2 0 0.2 9 9.5 10 10.5 11 11.5 l og ( Z * / Z s un ) log(M * /M sun )Chabrier (2003) IMF 0 1 2 3 4 5 6 7 8 I n f a ll t i m e - sc a l e τ [ G y r ] P a s s i v e g a l a x i e s S t a r- f o r m i n g g a l a x i e s -1-0.8-0.6-0.4-0.2 0 0.2 9 9.5 10 10.5 11 11.5 l og ( Z * / Z s un ) log(M * /M sun )Salpeter (1955) IMF 0 1 2 3 4 5 6 7 8 I n f a ll t i m e - sc a l e τ [ G y r ] P a s s i v e g a l a x i e s S t a r- f o r m i n g g a l a x i e s Fig. 4.
Upper panels : The MZ relation by Peng et al. (2015) (fitted by third-order polynomial functions shown in Fig. 3)for passive and star-forming galaxies. We reproduce it using the new analytical solution presented in this paper and varydifferent parameter models. In this figure we focus on the timescale parameter τ , and the color code indicates different τ values of the computed galaxies that reside along the MZ relations. In the left panel we adopt a Chabrier (2003) IMF,whereas in the right panel the Salpeter (1955) IMF is assumed. Lower panels : The distribution of the predicted passivegalaxies (red histogram) and star-forming galaxies (blue histogram) that reproduced the MZ relation in terms of thetimescale parameter τ with a Chabrier (2003) IMF (left panel) and a Salpeter (1955) IMF (right panel). To define a galaxy as passive, we adopt the following cri-terium introduced by Fossati et al. (2015), which takes intoaccount the galaxy specific star formation rate (sSFR), thatis, the SFR per unit galaxy stellar mass:sSFR pass < bt z , (15)where b is the birthrate parameter b = SF R/ h SF R i asdefined by Sandage (1986), and t z is the age of the Universeat redshift z . As in Fossati et al. (2015), we assume thevalue proposed by Franx et al. (2008): b = 0 . h z i = 0 .
05, thepassive galaxy population is characterized by sSFR pass < . × − yr − . For star-forming galaxies, we adopt the following scaling re-lation between µ ⋆ and M ⋆ found by Boselli et al. (2014) for a sample of the Herschel Reference Survey SDSS galaxies:log( µ ⋆ ) = − .
74 log( M ⋆ /M ⊙ ) + 7 . , (16)where µ ⋆ is the ratio between M gas and M ⋆ : µ ⋆ = M gas /M ⋆ . Molecular gas masses are estimated fromthe H -band luminosity-dependent conversion factor ofBoselli et al. (2002), while the galaxy stellar massesare derived from the galaxy i -band luminosities by as-suming a Chabrier (2003) IMF and using the g − i color-dependent stellar mass-to-light ratio relation fromZibetti et al. (2009).A further scaling relation found by Boselli et al. (2014)is the following relation between the typical galaxy gas de-pletion timescale, τ gas , and the galaxy stellar mass:log( τ gas ) = − .
73 log( M ⋆ /M ⊙ ) + 16 . , (17)where τ gas = M gas / SFR is defined as the inverse of our SFE,namely τ gas = 1 /S . According to Eq.(17), galaxies withhigher stellar mass would consume their available gas masson progressively shorter typical timescales if only star for-mation activity were taking place in the galaxy; this meansthat larger galaxies are expected to experience, on average,higher SFEs (see Matteucci 2012). We adopt Eq. (17) to -1-0.8-0.6-0.4-0.2 0 0.2 9 9.5 10 10.5 11 11.5 l og ( Z * / Z s un ) log(M * /M sun )Chabrier (2003) IMF 0 2 4 6 8 10 12 14 A ge [ G y r ] P a s s i v e g a l a x i e s S t a r- f o r m i n g g a l a x i e s -1-0.8-0.6-0.4-0.2 0 0.2 9 9.5 10 10.5 11 11.5 l og ( Z * / Z s un ) log(M * /M sun )Salpeter (1955) IMF 0 2 4 6 8 10 12 14 A ge [ G y r ] P a s s i v e g a l a x i e s S t a r- f o r m i n g g a l a x i e s Fig. 5.
Upper panels : As in Fig. 4, but here, the color code indicates different ages of the galaxies that reproduce theMZ relations.
Lower panels : The distribution of the predicted passive galaxies (red histogram) and star-forming galaxies(blue histogram) that reproduced the MZ relation in terms of the age of galaxies. In the left panel we adopt a Chabrier(2003) IMF, whereas in the right panel we show the results assuming a Salpeter (1955) IMF.constrain the galaxy SFE (which is kept fixed during thegalaxy evolution), given an initial value for the galaxy infallmass, M inf . The free parameters entering in the analytical solutions ofour chemical evolution model are the following:1. the infall timescale τ ;2. the infall mass M inf ;3. the wind parameter λ .We created a set of chemical evolution models by varyingthese free parameters with a very fine resolution. The as-sumed infall timescale τ spans the range between 0.1 and 8Gyr, with a resolution of the grid values of ∆ τ =0.05 Gyr.The wind parameter λ is defined between 0 and 10, witha resolution of ∆ λ =0.5. The infall masses are in the rangebetween 10 . and 10 . M ⊙ .The SFE is determined by means of Eq. (17), where M ⋆ is replaced with M inf . We cannot vary the SFE of thegalaxy according to its stellar mass, since the system of Eq.(7) is solved by keeping the SFE constant.The observational constraints that we assume to charac-terize the star-forming and passive populations of galaxiesin the SDSS sample of Peng et al. (2015) are the following: 1. Our models for passive and star-forming galaxies haveaverage stellar metallicities defined such that | log( Z ⋆, mod ) − log( Z ⋆, obs ) | < .
001 dex, where Z ⋆, mod are computed with Eq. (14) and Z ⋆, obs are the stellarmetallicities obtained with the fit reported in Fig. 3 forthe observed values by Peng et al. (2015).2. Our models for star-forming galaxies have | µ ⋆, mod − µ ⋆, obs | < . µ ⋆, obs are computed by meansof Eq. (16).3. Our models for passive galaxies must havesSFR mod , passive < . × − yr − , where sSFR mod isthe predicted specific SFR.4. Our models for star-forming galaxies must havesSFR mod , star − forming > . × − yr − .In this way, we are able to select and distinguish whichchemical evolution model best represents the sample ofPeng et al. (2015). We follow the chemical evolution of ourentire galaxy sample with a fixed time resolution ∆ t =0 . Z inf = 0). Although Lehner et al. (2013)studied the circumgalactic medium of galaxies and recentlydemonstrated that metal-enriched infalls occur, previouspapers (Tosi 1988; Matteucci 2012) have shown that an -1-0.8-0.6-0.4-0.2 0 0.2 9 9.5 10 10.5 11 11.5 l og ( Z * / Z s un ) log(M * /M sun )Chabrier (2003) IMF 0 2 4 6 8 10 w i nd pa r a m e t e r λ P a s s i v e g a l a x i e s S t a r- f o r m i n g g a l a x i e s -1-0.8-0.6-0.4-0.2 0 0.2 9 9.5 10 10.5 11 11.5 l og ( Z * / Z s un ) log(M * /M sun )Salpeter (1955) IMF 0 1 2 3 4 5 6 w i nd pa r a m e t e r λ P a s s i v e g a l a x i e s S t a r- f o r m i n g g a l a x i e s Fig. 6.
Upper panels : As in Fig. 4, but here, the color code indicates different wind parameters λ of the galaxies thatreproduce the MZ relations. Lower panels : The distribution of the predicted passive galaxies (red histogram) and star-forming galaxies (blue histogram) that reproduced the MZ relation in terms of the wind parameter λ of galaxies. In theleft panel we adopt a Chabrier (2003) IMF, whereas in the right panel a Salpeter (1955) IMF is adopted.infall enriched with metallicity Z ≤ Z ⊙ does not pro-duce differences in the evolution of the solar neighborhood.Moreover, assuming an infall with a metallicity higherthan 0.4 Z ⊙ requires very specific situations. Recently,Spitoni et al. (2016) considered the effects of an enrichedinfall of gas with the same chemical abundances as thematter ejected and/or stripped from dwarf satellites of theMilky Way on the chemical evolution of the Galactic halo.We found that α elements are only slightly affected by suchan enriched infall of gas.
5. Results
In this section we present our results concerning the char-acterization of the local passive and star-forming galaxiesof the SDSS sample of Peng et al. (2015), by making useof the new analytical solutions presented in Sect. 2.3 in thepresence of an exponential infall of gas. We remark on thefact that we selected the best chemical evolution modelscharacterizing the SDSS sample of Peng et al. (2015) byimposing the set of constraints presented in the previousSect. 4.3 and by varying all the free parameters simultane-ously.In Table 1 we summarize the main results of our work.We report the range of values spanned by the main freeparameters of our models to reproduce the observed MZ relation of the local star-forming and passive galaxies. Foreach parameter we also report the value within which 75per cent of the galaxies are expected to reside. We showin the table our results for both the Salpeter (1955) andthe Chabrier (2003) IMFs. A fundamental quantity thatwe can predict is the age distribution of the passive andstar-forming galaxy SDSS population, corresponding to thesecond row in the table.We assume that galaxies can form at different times,and by age we mean the galaxy evolutionary time, namelythe difference between the time corresponding to redshift z = h . i and the formation epoch.To better visualize our results, in Figs. 4-7 we show howlocal star-forming and passive galaxies are expected to bedistributed in the MZ relation (upper panels) and in rela-tive number (lower panels) for different values of the infalltimescale, age, wind parameter, and infall mass, respec-tively. In the lower panels of the figures we indicate withN/N tot the ratio between the number of the computed star-forming (or passive) galaxies in the considered bins of infalltimescale, age, wind parameter, and infall mass values overthe total number of the computed star-forming (passive)galaxies. The plots on the left correspond to our resultswith a Chabrier (2003) IMF, while on the right we showour results with a Salpeter (1955) IMF. -1-0.8-0.6-0.4-0.2 0 0.2 9 9.5 10 10.5 11 11.5 l og ( Z * / Z s un ) log(M * /M sun )Chabrier (2003) IMF 10 10.5 11 11.5 12 12.5 l og ( M i n f / M s un ) P a s s i v e g a l a x i e s S t a r- f o r m i n g g a l a x i e s -1-0.8-0.6-0.4-0.2 0 0.2 9 9.5 10 10.5 11 11.5 l og ( Z * / Z s un ) log(M * /M sun )Salpeter (1955) IMF 9.8 10 10.2 10.4 10.6 10.8 11 11.2 11.4 11.6 l og ( M i n f / M s un ) P a s s i v e g a l a x i e s S t a r- f o r m i n g g a l a x i e s Fig. 7.
Upper panels : As in Fig. 4 but here, the color code indicates different infall mass M inf of the galaxies whichreproduce the MZ relations. Lower panels : The distribution of the predicted passive galaxies (red histogram) and star-forming galaxies (blue histogram) which reproduced the MZ relation in terms of infall mass M inf of galaxies. In the leftpanel we adopt a Chabrier (2003) IMF, whereas in the right one the Salpeter (1955) one;Figure 4 shows that our models for the local pas-sive galaxies are characterized by shorter typical formationtimescales than the star-forming galaxies. This is mainlydue to the requirement of very low sSFRs for these galaxiesat the present time. Therefore, they had the time to con-sume or remove most of their total infall gas mass throughthe star formation activity or galactic wind. The short typi-cal formation timescales also enhance the star formation ac-tivity and hence the metal production in the earliest epochsof the galaxy evolution. According to the predictions of ourmodels, local passive galaxies are currently undergoing thedeclining, fading phase of their SFR evolution.In our analytical model of chemical evolution, the IMFonly enters in the calculation of the yield of metals perstellar generation and the returned mass fraction. Top-heavy IMFs determine higher yields of metals per stellargeneration and hence a more effective chemical enrichmentof the galaxy ISM, fixed all the other model parameters.Therefore, passive galaxies with a top-heavy IMF that reachtheir relatively high observed stellar metallicity are on av-erage characterized by longer formation timescales (a lessintensive SFR at early times). We find that when we as-sume the Salpeter (1955) IMF, the distribution of the for-mation timescales of passive galaxies is narrower than whenwe assume a Chabrier (2003) IMF, which contains a largernumber of massive stars. We find that almost 75 per cent of all passive galaxies are expected to assemble on τ ≤ . τ ≤ . * /M sun ) -3-2-1 0 1 2 log(SFR)-1-0.8-0.6-0.4-0.2 0 0.2 l og ( Z * / Z s un ) Chabrier (2003) IMFStar-forming galaxiesPassive galaxies 9 9.5 10 10.5 11 11.5 log(M * /M sun ) -3-2.5-2-1.5-1-0.5 0 0.5 1 log(SFR)-1-0.8-0.6-0.4-0.2 0 0.2 l og ( Z * / Z s un ) Salpeter (1955) IMFStar-forming galaxiesPassive galaxies 9 9.5 10 10.5 11 11.5 log(M * /M sun ) -3-2-1 0 1 2 log(SFR) -1-0.8-0.6-0.4-0.2 0 0.2 l og ( Z * / Z s un ) * /M sun ) -4-3-2-1 0 1 log(SFR) -1-0.8-0.6-0.4-0.2 0 0.2 l og ( Z * / Z s un ) Fig. 8.
Upper panels:
Simulated passive and star-forming galaxies that are able to reproduce the MZ relations of Peng etal. (2015) in the 3D plot to test a more general relation between stellar M ∗ , average stellar metallicity Z, and the SFRusing a Chabrier (2003) (left panel) and a Salpeter IMF (right panel). Lower panels:
The 2D projection in the M ∗ - SFRplane.Considering the difference between the median age val-ues of our computed star-forming and passive galaxies, wefind that our results are in agreement with the data byPeng et al. (2015). With the Chabrier (2003) IMF, we findthat the median stellar ages for star-forming and passivegalaxy are 0 . . . . . . λ , associated with our best modelsfor the star-forming and passive galaxy populations of theSDSS sample of Peng et al. (2015). According to our re-sults, the population of galaxies that suffer more promi-nent wind episodes are the star-forming galaxies. To repro-duce the observed average stellar metallicity of star-forminggalaxies at lower stellar mass, stronger winds are needed.On the other hand, the wind parameter, λ , is predicted todecrease toward star-forming galactic systems with higher stellar mass. In summary, the observed MZ relation canbe well reproduced by assuming a variable mass-loadingfactor, which increases when passing from more massiveto less massive galactic systems. This conclusion has beendiscussed by several authors in the past (see, for example,Spitoni et al. 2010, and references therein). This correlationbetween the mass-loading factor, λ , and the galaxy stellarmass is also valid for passive galaxies, although the galacticwinds for these systems are predicted to be much weakerthan for the star-forming galaxies.The IMF choice can also affect the typical values ofthe wind parameter that best reproduce the observed MZrelation. Our best models for low-mass and metal-poorstar-forming galaxies with a Salpeter (1955) IMF requireweaker galactic winds than similar models with a Chabrier(2003) IMF, which predict a faster and hence more effec-tive chemical enrichment of the galaxy ISM. Table 1 andFig. 6 show that the distribution of the wind parametersfor star-forming galaxies with a Salpeter (1955) IMF spanthe range 0 < ∼ λ < ∼ .
8, and the 75% of star-forming galax-ies are expected to have λ < ∼ .
25, while the models ofstar-forming galaxies with a Chabrier (2003) IMF span therange 0 . < ∼ λ < ∼ .
2, and and the 75% of star-forminggalaxies are expected to have λ < ∼ . -1-0.8-0.6-0.4-0.2 0 0.2 -3 -2 -1 0 1 2 l og ( Z * / Z s un ) log(SFR)Salpeter (1955) IMF 9 9.5 10 10.5 11 11.5 l og ( M * / M s un ) -1-0.8-0.6-0.4-0.2 0 0.2 -3 -2 -1 0 1 2 l og ( Z * / Z s un ) log(SFR)Chabrier (2003) IMF 9 9.5 10 10.5 11 11.5 l og ( M * / M s un ) Fig. 9.
Stellar metallicity as a function of the SFR for thecomputed star-forming and passive galaxies. The color codeindicates the stellar mass. In the upper panel model resultswith a Salpeter (1955) IMF are drawn, and in the lowerpanel a Chabrier (2003) IMF is adopted.In Fig. 7 we show our results for the distribution ofthe infall mass of our best models for the local populationof star-forming and passive galaxies. We find that passivegalaxies are characterized by lower infall masses than thestar-forming galaxies. Passive galaxies in the Local Universemust be characterized by very low sSFRs, which can be ob-tained in the framework of our analytical model with an ex-ponential infall of gas by assuming low infall gas masses to-gether with short typical timescales (as we discussed abovefor the passive population). In this way, nearly most of theinfall gas mass is converted into stars at the later epochsof the galaxy evolution. Typical values of the infall mass ofour best models for the passive and star-forming galaxiescan be found in Table 1.The results for the infall mass distribution of the localpopulation of star-forming and passive galaxies are stronglyaffected by the adopted IMF. Higher infall masses are re-quired with a Chabrier (2003) IMF. We recall here thatwe assumed a primordial infall of gas. Because a top-heavyIMF leads to a more efficient chemical enrichment thana Salpeter (1955) IMF, a stronger dilution of metals andtherefore higher infall masses are required to obtain theobserved MZ relation.
6. Predicted fundamental relation of the localstar-forming and passive galaxies
In this section, we analyze the correlations that local galax-ies are predicted to show by our chemical evolution modelin the 3D space defined by stellar mass, gas-phase metal-licity, and SFR. Mannucci et al. (2010) showed that localgalaxies lie on a tight surface in such a space and named theresulting correlation the “fundamental relation” , implyingthat it is valid at all redshifts.Since we do not possess gas-phase metallicities for thepassive galaxies in the sample of Peng et al. (2015), we usethe average galaxy stellar metallicity instead of the gas-phase metallicity to study the MZ relation. The gas metal-licity is typically higher than the stellar one for evolvedgalactic systems. Our results are shown in Fig. 8, where weshow that local star-forming and passive galaxies lie on atight surface when considering their stellar mass, M ⋆ , mass-weighted stellar metallicity, h Z ⋆ i , and SFR. The projectionof this 3D diagram onto the M ⋆ - Z ⋆ place gives rise to theMZ relation that we discussed in the previous sections. Thefigure shows that the two populations of galaxies are sepa-rated by a discontinuity that is due to the different metal-licity distributions of star-forming and passive objects.The discontinuity is also visible in Fig. 9, where we showthe projection of the predicted fundamental relation ontothe Z ⋆ -SFR plane. In this figure, the color-coding corre-sponds to the galaxy stellar mass. The local galaxy popula-tions are predicted to show a similar trend as the one pre-sented by Mannucci et al. (2010, see the right panel of theirFig. 1), where higher stellar masses are found for galaxieswith higher SFRs and metallicities.
7. Conclusions
We have developed a new analytical model of chemical evo-lution, in which an exponential infall of gas and galac-tic winds are assumed. We applied this model to repro-duce the observed MZ relation for local SDSS galaxies(Peng et al. 2015); in particular, we characterized the pop-ulations of gas-rich and star-forming and passive (gas-poorand quiescent) galaxies, by showing how their ages, forma-tion timescales, mass-loading factors and infall masses mustrelate to each other. Finally, we analyzed the fundamentalrelation for these local galaxies in the 3D space, defined bystellar mass, average stellar metallicity, and SFR. Our mainconclusions can be summarized as follows:1. We assume that all galaxies form by gas accretion withan exponential law. We find that passive galaxies arecharacterized by shorter typical formation timescalesand are older objects than the star-forming galaxies.2. Galactic winds in star-forming galaxies are found onaverage to be stronger than in passive galaxies. The in-tensity of the galactic winds depends on the adoptedIMF, with the higher values of the mass-loading factorcorresponding to top-heavy IMFs. This is a consequenceof the fact that top-heavy IMFs lead to a more efficientmetal enrichment of the galaxy ISM than bottom-heavyIMFs.3. The observed MZ relation of Peng et al. (2015) can bereproduced by our models without invoking any stran-gulation effects. This is because the assumption of anexponential infall of gas, also coupled to galactic winds, naturally reduces the gas accretion after the assumedinfall timescale, hence mimics the effect of strangula-tion. Nevertheless, we conclude that strangulation is notthe main physical mechanism driving the transition ofgalaxies toward the passive evolution.4. We have shown that our models for star-forming andpassive galaxies imply that they obey the so-called fun-damental relation of Mannucci et al. (2010), which a ismore general relation between stellar mass, metallicity,and SFR. The fundamental relation of Mannucci et al.(2010) adopts the gas-phase metallicity, and we find thatit is still valid when adopting the average galaxy stellarmetallicity ( M ⋆ - SFR - Z ⋆ ). Acknowledgments
We thank the anonymous referee for the suggestions thatimproved the paper. The work was supported by PRINMIUR 2010-2011, project “The Chemical and dynamicalEvolution of the Milky Way and Local Group Galaxies”,prot. 2010LY5N2T.
References