APOGEE DR16: a multi-zone chemical evolution model for the Galactic disc based on MCMC methods
E. Spitoni, K. Verma, V. Silva Aguirre, F. Vincenzo, F. Matteucci, B. Vai?ekauskait?, M. Palla, V. Grisoni, F. Calura
AAstronomy & Astrophysics manuscript no. spitoni_MCMC_APOGEEDR16 © ESO 2021January 25, 2021
APOGEE DR16: a multi-zone chemical evolution model for theGalactic disc based on MCMC methods
E. Spitoni (cid:63) , K. Verma , V. Silva Aguirre , F. Vincenzo , , F. Matteucci , , ,B. Vaiˇcekauskait˙e , , M. Palla , , , V. Grisoni , , and F. Calura Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C,Denmark Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 West Woodru ff Avenue, Columbus, OH 43210,USA Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA Dipartimento di Fisica, Sezione di Astronomia, Università 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, Italy I.N.F.N. - Sezione di Trieste, Via Valerio 2, I-34100 Trieste Technology University Dublin, School of Physics and Clinical & Optometric Sciences, Kevin Street, Saint Peter’s, Dublin 2, D08X622, Ireland S.I.S.S.A. - International School for Advanced Studies, via Bonomea 265, I-34136, Trieste, Italy I.N.A.F. - Osservatorio Astronomico di Bologna, Via Gobetti 93 /
3, 40129 Bologna, ItalyReceived xxxx / Accepted xxxx
ABSTRACT
Context.
The analysis of the APOGEE DR16 data suggests the existence of a clear distinction between two sequences of disc stars atdi ff erent Galactocentric distances in the [ α / Fe] vs. [Fe / H] abundance ratio space: the so-called high- α sequence, classically associatedto an old population of stars in the thick disc with high average [ α / Fe], and the low- α sequence, which mostly comprises relativelyyoung stars in the thin disc with low average [ α / Fe].
Aims.
We aim to constrain a multi-zone two-infall chemical evolution model designed for regions at di ff erent Galactocentric distancesusing measured chemical abundances from the APOGEE DR16 sample. Methods.
We perform a Bayesian analysis based on a Markov Chain Monte Carlo method to fit our multi-zone two-infall chemicalevolution model to the APOGEE DR16 data.
Results.
An inside-out formation of the Galaxy disc naturally emerges from the best fit of our two-infall chemical-evolution model toAPOGEE-DR16: inner Galactic regions are assembled on shorter time-scales compared to the external ones. In the outer disc (withradii R > / Fe] vs. [Fe / H] abundance pattern in the low- α sequence. In the inner disc, in the framework of the two-infall model, weconfirm the presence of an enriched gas infall in the low- α phase as suggested by chemo-dynamical models. Our Bayesian analysisof the recent APOGEE DR16 data suggests a significant delay time, ranging from ∼ Conclusions.
Our results propose a clear interpretation of the [Mg / Fe] vs. [Fe / H] relations along the Galactic discs. The signaturesof a delayed gas-rich merger which gives rise to a hiatus in the star formation history of the Galaxy are impressed in the [Mg / Fe]vs. [Fe / H] relation, determining how the low- α stars are distributed in the abundance space at di ff erent Galactocentric distances, inagreement with the finding of recent chemo-dynamical simulations. Key words.
Galaxy: abundances - Galaxy: evolution - ISM: general - methods: statistical
1. Introduction
Our understanding of the formation and evolution of our Galaxydisc is essentially based on the study and interpretation ofsignatures imprinted in resolved stellar populations, such astheir chemical and kinematic properties as traced by large sur-veys and observational campaigns. The current synergy betweenthe Apache Point Observatory Galactic Evolution Experimentproject (APOGEE; Majewski et al. 2017; in particular the latestdata release DR16, Ahumada et al. 2020) and the Gaia mission(DR2; Gaia Collaboration et al. 2018), o ff ers an unparalleled op- (cid:63) email to: [email protected] portunity to simultaneously rely upon accurate spectroscopic andkinematic properties to constrain models of Galactic chemicalevolution.The analysis of the APOGEE DR16 data (Ahumada et al.2020; Queiroz et al. 2020) suggests the existence of a clear dis-tinction between two sequences of disc stars in the [ α / Fe] vs.[Fe / H] abundance ratio space: the so-called high- α and low- α se-quences. This dichotomy in the chemical abundance ratio spacehas been also confirmed by the Gaia-ESO survey (e.g., Recio-Blanco et al. 2014; Rojas-Arriagada et al. 2016, 2017), the AM-BRE project (Mikolaitis et al. 2017), and the GALAH survey(Buder et al. 2019). Article number, page 1 of 15 a r X i v : . [ a s t r o - ph . GA ] J a n & A proofs: manuscript no. spitoni_MCMC_APOGEEDR16
By analyzing the APOKASC (APOGEE + Kepler Astero-seismology Science Consortium) sample for the solar neigh-borhood, Silva Aguirre et al. (2018) pointed out that the twosequences are characterized by two di ff erent ages: the high- α stars have ages of ∼
11 Gyr, while the low- α sequence peaks at ∼ ∼ Gaia colour-magnitude diagrams. By analyzing the HighAccuracy Radial velocity Planet Searcher (HARPS) spectra oflocal solar twin stars, Nissen et al. (2020) found that the age-metallicity distribution has two distinct populations with a clearage dissection. The authors suggested that these two sequencesmay be interpreted as evidence of two episodes of accretion ofgas onto the Galactic disc with quenching of star formation inbetween them, in agreement with the scenario proposed by Spi-toni et al. (2019b, 2020).In a cosmological framework, the existence of a double se-quence was predicted for the first time by Calura & Menci(2009) who, by means of a semi-analytic model based on the ex-tended Press and Schechter (Bond et al. 1991), modelled in post-processing the abundance pattern of a sample of model galaxiesselected as Milky Way analogues. A second accretion phase af-ter a prolonged period with a quenched star formation has beensuggested by the dynamical models of Noguchi (2018) in whicha first infall episode rapidly builds up the high- α sequence, butthen the star formation is starved from the lack of gas supplyfrom the intergalactic medium (IGM) until the shock-heated gasin the Galactic dark matter halo has radiatively cools down andis accreted by the Galaxy giving rise to a delayed second gas in-fall episode. In this framework, Noguchi (2018) found that theSFR of the Galactic disc is characterised by two distinct peaksseparated by ∼ / H]-[ α / Fe] plane may be a consequence ofa significantly lowered gas accretion rate at ages between 6 and9 Gyr. In the framework of cosmological hydrodynamic simula-tions of Milky Way like galaxies, Buck (2020) for example foundthat a dichotomy in the α -sequence is a generic consequence ofa gas-rich merger occurred at a certain epoch in the evolution ofthe Galaxy, which destabilized the gaseous disc at high redshift.The significant delay in the two-infall model of Spitoni et al.(2019b, 2020) has been discussed by Vincenzo et al. (2019)in the context of the stellar system accreted by the Galactichalo, AKA Gaia-Enceladus (Helmi et al. 2018; Koppelman et al.2019). It was proposed that the mechanism which quenched theMilky Way star formation at high redshift was a major mergerevent with a satellite like Enceladus (by heating up the gas inthe dark matter halo). This proposed scenario is in agreementwith the recent Chaplin et al. (2020) study. They constrained themerging time with the observations of the very bright, naked-eye star ν Indi, finding at 68% confidence that the earliest the mergercould have started was 11.6 Gyr ago.As outlined by Hayden et al. (2015) and Queiroz et al.(2020), the two sequences in the [ α / Fe] vs. [Fe / H] abundanceratio relation from APOGEE have di ff erent features and trendsthroughout the Galactic disc. While the low- α sequence is dis-tributed at increasingly lower metallicity towards the outer disc,it is found at super-solar values in the inner disc. Moreover,it is worth noticing that the ratio between the number of low- α and high- α stars increases when moving from the inner tothe outer Galactic disc. Hence, the formation of the low- α se-quence in the entire Galaxy seems to be more complex than asimple sequential process as assumed in the model of Spitoniet al. (2020) for the solar vicinity and may be driven by di ff er-ent physical processes. For instance, in the cosmological simu-lations presented by Agertz et al. (2020), Renaud et al. (2020b)and Renaud et al. (2020a), they concluded that the low- α se-quence has been assembled through di ff erent physical processthat interplay together in the whole disc. Two distinct channelsof gas infall fuel the Galactic disc; a chemically enriched gas ac-cretion event (by outflows from a massive galaxy with ∼ / ff erent one fuels the outer gas disc,which is inclined with respect to the main Galactic plane and hassignificantly poorer chemical content. However, their predictedlow- α sequence is shifted towards larger [ α / Fe] values than theAPOGEE sample by ∼ / Fe]vs. [Fe / H] of APOGEE (Hayden et al. 2015) at di ff erent Galac-tocentric distances. Palla et al. (2020) proposed that a delay of t max = / Fe] vs.[Fe / H] abundance ratio. Khoperskov et al. (2021) found that inthe infalling gas during inner thin disc formation phase is not pri-mordial because the gaseous halo has been significantly pollutedduring the formation of the thick disc, providing a tight connec-tion between chemical abundance patterns in the two Galacticdisc components.In this article, we present a multi-zone two-infall chemicalevolution model with the aim to extend the results of Spitoniet al. (2019b, 2020) for the solar vicinity to the whole disc. Wequantitatively infer the free parameters by fitting the APOGEEDR16 (Ahumada et al. 2020) abundance ratios at di ff erent Galac-tocentric distances using a Bayesian technique based on MarkovChain Monte Carlo (MCMC) methods. The Bayesian analysis isnow being widely used in testing the Galactic chemical evolutionmodels (see e.g. Côté et al. 2017; Rybizki et al. 2017; Philcoxet al. 2018; Frankel et al. 2018; Belfiore et al. 2019; Spitoni et al.2020). In fact, thanks to the wealth of information from largesurveys, large datasets are currently being exploited by means ofstatistic methods to constrain the parameters of Galactic chemi-cal evolution models.The paper is organised as follows. In Section 2, the obser-vational data used in the Bayesian analysis are presented. InSection 3, we present the main characteristics of the multi-zonechemical evolution model adopted in this work, and describe thefitting method. In Section 4, we present our results, and finallyin Section 5, we draw our conclusions. Article number, page 2 of 15pitoni et al.: Multi-zone chemical evolution model based on MCMC methods
Fig. 1.
Observed stellar [Mg / Fe] vs. [Fe / H] abundance ratios from APOGEE DR16 (Ahumada et al. 2020) for three bins of di ff erent Galactocentricdistances. The regions are 4 kpc-wide centered at 4 kpc (left panel), 8 kpc (middle panel) and 12 kpc (right panel), respectively. The contour linesenclose fractions of 0.90, 0.75, 0.60, 0.45, 0.30, 0.20, 0.05 of the total number of observed stars. Details on the data selection are reported in thetext.
2. The APOGEE DR16 sample
Here, following Spitoni et al. (2020), we use a Bayesian frame-work based on MCMC methods to fit a multi-zone Galacticchemical evolution model to the observed chemical abundancesfor Mg and Fe provided by APOGEE DR16 (Ahumada et al.2020) and related Galactocentric distances and vertical heightsabove the Galactic plane, as found by the
Gaia mission (DR2;Gaia Collaboration et al. 2018).Di ff erent methods have been introduced to compute properGalactocentric distances. Luri et al. (2018) highlighted that theestimation of distances from parallaxes has to be addressed asa fully Bayesian inference problem, as shown in Bailer-Joneset al. 2018 for Gaia data. Moreover, in the Bayesian frameworkQueiroz et al. (2018) presented spectro-photometric distancesestimated with the StarHorse tool which, in the case of APOGEEDR16, are derived from a set of photometric bands, APOGEEspectra and
Gaia parallaxes. In this paper, we adopt the Galac-tocentric distances computed by Leung & Bovy (2019) and re-ported in the astroNN catalogue for APOGEE DR16 stars. Toobtain precise distances for distant stars, Leung & Bovy (2019)designed a deep neural network, and trained it using preciselymeasured parallaxes of nearby stars in common between Gaia and APOGEE to determine the spectro-photometric distances forAPOGEE stars. They included a flexible model to calibrate par-allax zero-point biases in
Gaia
DR2 in order to avoid the propa-gation of systematic uncertainties present in the training data setto the inferred distances. On top of the versatility of the neuralnetwork, they employed a robust way of Bayesian deep learningthat takes data uncertainties in the training set into account andalso estimates uncertainties in predictions made with the neuralnetwork using the drop out variational inference. One major lim-itation of the method is the size of the training set it rests upon.Fortunately, the amount of data to train their algorithm will in-crease in the future thanks to new APOGEE and
Gaia data re-leases and other spectroscopic surveys such as GALAH. https: // data.sdss.org / sas / dr16 / apogee / vac / apogee-astronn The Galactocentric positions and velocities used in the com-putation of the orbital properties are calculated assuming that theGalactocentric distance R (cid:12) from the Sun to the Galactic centeris 8.125 kpc (Gravity Collaboration et al. 2018) and located 20.8pc above the Galactic midplane (Bennett & Bovy 2019).Stars that are part of the Galactic disc have been chosen withthe same quality cuts suggested in Weinberg et al. (2019) assum-ing signal-to-noise ratio ( S / N ) >
80, logarithm of surface grav-ity between 1 . < log g < . | z | < α / Fe] vs [Fe / H] abundance patterns atdi ff erent radii for the MCMC fitting, since we adopt a one-zonechemical evolution model for each radial range of Galactocentricdistances, which does not account for the small displacementsobserved in the APOGEE abundance distributions of the high- α and low- α sequences at high | z | (Hayden et al. 2015).The uncertainties in metallicity reported in this compilationcorrespond to the internal precision, which are of the order of ∼ ff ort to better estimate systematic uncertainties,we added, as in Silva Aguirre et al. (2018), the median di ff erencebetween APOGEE results for clusters and the standard literaturevalues as reported in Table 3 of Tayar et al. (2017) ( ∼ .
09 dex)in quadrature.We have split the data into 3 concentric annular Galactic re-gions, each 4 kpc-wide, spanning the range between 2 and 14kpc. In Fig. 1, it is clear that the further out stars (right panel,10-14 kpc region) preferentially populate the low- α sequence inthe [Mg / Fe] vs. [Fe / H] relation, and few stars are located in thehigh- α population. Moreover, the more the regions are external,the more the locus of the low- α sequence is shifted towards lowermetallicity. On the contrary, in the annular region enclosed be-tween 2 and 6 kpc (left panel of Fig. 1), the low- α phase peaksat super-solar metallicity, however a clear bimodality is still evi-dent as highlighted by the isodensity contours.We did not consider innermost regions with R < ff erent Galactic chemical assumptions and prescriptions needto be applied (Matteucci et al. 2019, 2020; Gri ffi th et al. 2020). Article number, page 3 of 15 & A proofs: manuscript no. spitoni_MCMC_APOGEEDR16
The recent analysis of large samples from APOGEE DR16 data(Queiroz et al. 2020; Rojas-Arriagada et al. 2020) suggested thatbulge structure could extend up to Galactocentric distances of3 − . ff ect the region enclosed between 2 and 6 kpc.However, we checked that stars with Galactocentric distancesselected in the range between 2 and 6 kpc give place to an al-most identical distribution in the [Mg / Fe] vs. [Fe / H] abundanceratio space as the ones enclosed in a region between 3 and 6 kpc.The numbers of stars in the considered di ff erent annular re-gions are the following ones: 7440 in the zone centered at 4 kpc,9169 in the one at 8 kpc, and 10081 in the outermost region cen-tered at 12 kpc. We believe that in these three zones the maintrends of the Galactic disc in terms of the [Mg / Fe] vs. [Fe / H]abundance as a function of the Galactocentric distance are im-printed . We have checked that the 4 kpc-wide region centered at16 kpc has only 882 stars, almost all of which are in the low- α sequence.
3. Multi-zone chemical evolution model for theGalactic disc
In this Section we present the main assumptions and character-istics of the multi-zone chemical evolution model which extendsthe previous ones introduced by Spitoni et al. (2019b, 2020) forthe solar neighborhood. After a brief explanation of the reasonsfor neglecting stellar migration e ff ects, we describe the adoptedMCMC methods. We extend the chemical evolution model designed for the solarneighborhood presented by Spitoni et al. (2019b, 2020) to dif-ferent Galactocentric regions centered at 4 kpc, 8 kpc and 12kpc. Retaining the assumption that the Milky Way disc has beenformed by two distinct accretion episodes of gas, we assume thatthe gas infall rate is a function of the Galactic distance R (Chiap-pini et al. 2001; Grisoni et al. 2018; Spitoni et al. 2009, 2019a)and is expressed by the following expression, I i ( t , R ) = X , i ( R ) N ( R ) e − t /τ ( R ) ++ θ ( t − t max , R ) X , i ( R ) N ( R ) e − ( t − t max , R ) /τ ( R ) , (1)where τ ( R ) and τ ( R ) are the time-scales of gas accretion for theformation of the high- α and low- α disc phase, respectively. Thequantity θ in the eq. (1) is the Heaviside step function. X , i ( R )and X , i ( R ) are the abundance by mass of the element i in theinfalling gas for the first and second gas infall, whereas t max , R is the time of the maximum infall rate on the second accretionepisode, i.e. it indicates the delay of the beginning of the sec-ond infall. Spitoni et al. (2019b) underlined the importance of aconsistent delay of t max ∼ R = ffi cients N ( R ) and N ( R ) are obtained by im-posing a fit to the observed current total surface mass density atdi ff erent radii R with the following relations: N ( R ) = σ ( R ) τ ( R ) (cid:0) − e − t G /τ ( R ) (cid:1) , (2) N ( R ) = σ ( R ) τ ( R ) (cid:0) − e − ( t G − t max , R ) /τ ( R ) (cid:1) , (3)where σ ( R ) and σ ( R ) are the present-day total surface massdensity of the high- α and low- α sequence stars, respectively, and t G is the age of the Galaxy.Following Spitoni et al. (2020), we use the value of total sur-face density (sum of high- α and low- α ) in the solar neighbor-hood of 47.1 ± (cid:12) pc − as provided by McKee et al. (2015).In Spitoni et al. (2020) it was assumed that in the solar neigh-borhood the total surface mass densities ( σ tot , (cid:12) = σ , (cid:12) + σ , (cid:12) ) isconstant, as given by McKee et al. (2015). The present-day totalsurface mass density at a certain Galactocentric distance R canbe written as σ tot ( R ) = σ tot , (cid:12) e − ( R − R (cid:12) ) / R d , (4)after having imposed that the total mass declines with the radiusthrough an exponential law and the scale-length of the disc is R d = . ff erentscale-lengths for the thick and thin disc phases were tested andassumed (see their Fig. 6), here we consider the ratio betweenthe surface gas densities as a free parameter of the model.Recalling that σ /σ is the ratio between the low- α andhigh- α , the values of the present-day total surface mass densities σ ( R ) and σ ( R ) to insert in eqs. (3) and (2) are the followingones: σ ( R ) = σ tot ( R )1 + (cid:18) σ σ (cid:12)(cid:12)(cid:12)(cid:12) R (cid:19) − , (5) σ ( R ) = σ tot ( R ) − σ ( R ) . (6)The SFR is expressed as the Kennicutt (1998) law, ψ ( t , R ) ∝ ν ( t , R ) σ g ( t , R ) k , (7)where σ g is the gas surface density and k = . ν ( t , R ) is the star formation e ffi ciency (SFE).Motivated by the theory of star formation induced by spiral den-sity waves in Galactic discs (Wyse & Silk 1989), we considera variable SFE as a function of the Galactocentric distance inthe low- α phase. In several chemical evolution models (Colavittiet al. 2008; Spitoni et al. 2015; Grisoni et al. 2018; Palla et al.2020) it has been claimed that observed abundance gradients inthe Galactic disc may be explained by assuming higher SFE val-ues in the inner regions than in the outer ones (along with the’inside-out’ formation scenario and radial gas flows).Moreover, di ff erent infall episodes in principle could be char-acterized by di ff erent SFEs. In fact, in the classical two-infallmodel (Chiappini et al. 2001; Grisoni et al. 2017, 2019, 2020;Spitoni et al. 2020) the SFEs associated to the high- α and low- α sequences are di ff erent: ν = − and ν = − for thesolar vicinity.We adopt the Scalo (1986) initial stellar mass function(IMF), constant in time and space. Article number, page 4 of 15pitoni et al.: Multi-zone chemical evolution model based on MCMC methods
Although an important ingredient of the Nidever et al. (2014)chemical evolution model to reproduce the APOGEE data wasthe inclusion of Galactic winds proportional to the SFR coupledto a variable loading factor, in this paper we do not consider out-flows. In fact, while studying the Galactic fountains originatedby the explosions of Type II SNe in OB associations, Melioliet al. (2008, 2009) and Spitoni et al. (2008, 2009) found that theejected metals fall back close to the same Galactocentric regionwhere they are delivered and thus do not modify significantly thechemical evolution of the disc as a whole.As in Spitoni et al. (2019b, 2020), we adopt the same nucle-osynthesis prescriptions as proposed by François et al. (2004) forFe, Mg and Si. The authors artificially increased the Mg yieldsfor massive stars from Woosley & Weaver (1995) to reproducethe solar Mg abundance. Mg yields from stars in the range 11-20M (cid:12) have been increased by a factor of 7, whereas yields for starswith mass >
20 M (cid:12) are on average a factor ∼ >
40 M (cid:12) ) are increased by a factor of 2. Con-cerning Type Ia SNe, in order to preserve the observed [Mg / Fe]vs. [Fe / H] pattern, the yields of Iwamoto et al. (1999) for Mgwere increased by a factor of 5.This set of yields has been widely used in the literature (Ces-cutti et al. 2007; Spitoni et al. 2014, 2015, 2017, 2019a; Mottet al. 2013; Vincenzo et al. 2019) and turned out to be able to re-produce the main features of the solar neighbourhood. We adoptthe photospheric values of Asplund et al. (2005) as our solar ref-erence abundances, in order to be consistent with the APOGEEDR16 release.
The existence of stellar radial migration is established beyondany doubts (see, e.g., Roškar et al. 2008; Schönrich & Binney2009; Loebman et al. 2011; Minchev et al. 2012; Kubryk et al.2013). The main physical mechanisms responsible for stellarmigration are churning and blurring (e.g., Sellwood & Binney2002; Minchev et al. 2011), as well as the overlap of the spiraland bar resonances on the disc (Minchev et al. 2011). However,the real impact of radial migration on the chemical evolution ofthe Galactic disc is still under debate.Nidever et al. (2014) and Sharma et al. (2020) presentedchemical evolution models where the dichotomy in the abun-dance space is entirely explainable only in terms of stellar mi-gration. They concluded that the high- α disc has been built bymigrator stars and gas of the thin disc.In particular, the model presented by Sharma et al. (2020)assumed empirical tracks for the evolution of [ α / Fe] and [Fe / H]as a function of time at di ff erent radii. These empirical age-metallicity relations may be inconsistent with their assumed em-pirical relations for the SFR as a function of time at di ff erentradii, which did not require a hiatus in the star formation be-tween the high- α and low- α sequences, as predicted by the two-infall chemical evolution model and chemo-dynamical simula-tions. Finally, we note that stellar migration in Sharma et al.(2020) follows a parametric di ff usion approach (e.g., Schönrich& Binney 2009).On the contrary, the analysis of recent results from chemo-dynamical simulations has raised important doubts about the im-portance of stellar migration in the evolution of chemical abun-dance ratios, such as [ α / Fe] vs. [Fe / H], in the Galactic disc. Forinstance, by means of a self-consistent chemo-dynamical modelfor the Galactic disc evolution, Khoperskov et al. (2021) con- cluded that radial migration has a negligible e ff ect on the [ α / Fe]vs. [Fe / H] distribution over time (the distribution is slightlysmoothed by migrators from the inner and outer disc regions),suggesting the α -dichotomy is strictly linked to di ff erent star for-mation regimes over the Galaxy’s life.Similar results are found by the cosmological simulation ofVincenzo & Kobayashi (2020). The authors concluded that thetwo main gas accretion episodes occurred 0-2 and 5-7 Gyr ago,determinant for the rise of the double sequence in the [ α / Fe] vs.[Fe / H] plot. In their Fig. 13, it is remarkable that the abundancesin stars with ages smaller than 8 Gyr in the solar neighborhoodperfectly trace the gas phase abundances in the same region.They concluded that the signature impressed in the chemicalabundances of the stars may be linked to infalling of primor-dial or poorly enriched gas. Supported by the these results, wecan safely assume that the stellar migration did not alter or af-fect much the [Mg / Fe] vs. [Fe / H] evolution in our analysis whichadopts Galactic annular regions 4 kpc-wide.Although stellar migration has played an important role inGalactic evolution, i.e. in flattening of the radial metallicity pro-files and a ff ecting the [ α / Fe]-age relation of thin-disc stars (Vin-cenzo & Kobayashi 2020), we investigate a complementary sce-nario with respect to that proposed by Sharma et al. (2020), inwhich the radial variation in the [ α / Fe] vs. [Fe / H] abundancehave been entirely caused by the stellar migration.
As in Spitoni et al. (2020), we use a Bayesian analysis based onMarkov Chain Monte Carlo (MCMC) methods to find the best-fitchemical evolution models at di ff erent Galactocentric distances, R . Here, we briefly recall the main assumptions and refer thereader to Spitoni et al. (2020) for a more detailed description ofthe fitting method.At a fixed R , the set of observables is x R = { [Mg / Fe] , [Fe / H] } while the set of model parameters is Θ R = { τ , τ , t max , σ /σ } . The adopted likelihood, L , used to com-pute the posterior probability distribution can be written asln L = − N (cid:88) n = ln (2 π ) d / d (cid:89) j = σ n , j − N (cid:88) n = d (cid:88) j = (cid:32) x n , j − µ n , j σ n , j (cid:33) , (8)where N is the number of stars in a Galactic region correspond-ing to R . The quantities x n , j and σ n , j are respectively the mea-sured value of j th observable and its uncertainty for n th star. µ n , j is the model value of the j th observable for the n th star. As un-derlined in Spitoni et al. (2020), the curve predicted by the two-infall model in the plane [Mg / Fe] vs. [Fe / H] is multi-valued (seetheir Fig. 1). As a result, an observed data point in the [Fe / H]-[ α/ Fe] plane cannot be associated unambiguously to a point onthe curve. To get through this problem they associated a datapoint to the closest value on the curve given a data point x n , j ,defining the following "distance data-model" function D , D n , i ≡ (cid:118)(cid:117)(cid:116) d (cid:88) j = (cid:32) x n , j − µ n , j , i σ n , j (cid:33) (9)where i runs over a set of discrete values on the curve. Hence,the closest point on the curve is µ n , j = µ n , j , i (cid:48) which fulfils thefollowing relation: S n ≡ min i (cid:8) D n , i (cid:9) = (cid:118)(cid:117)(cid:116) d (cid:88) j = (cid:32) x n , j − µ n , j , i (cid:48) σ n , j (cid:33) . (10) Article number, page 5 of 15 & A proofs: manuscript no. spitoni_MCMC_APOGEEDR16
Fig. 2.
Corner plot showing the posterior PDFs of the chemical evo-lution model parameters for the region at 8 kpc. For each parameter,the median and the 16th and 84th percentiles of the posterior PDF areshown with dashed lines above the marginalised PDF. The SFEs arefixed at values of 2 and 1 Gyr − for the high- and low- α phases, respec-tively. Here, we present the priors on Θ R = { τ , τ , t max , σ /σ } .We use uniform priors for all parameters, which are independentof the Galactocentric distances (i.e priors are the same for allmodel runs at di ff erent R ). In the classical two-infall model (Chi-appini et al. 1997; Spitoni et al. 2020), the first gas infall is char-acterized by a short time-scale of accretion in the solar neigh-borhood, fixed at the value of τ = τ = . τ exploring therange 0 < τ < τ , is con-nected to a slower accretion episode. We set a uniform prior on τ exploring the range 0 < τ <
14 Gyr. For the delay t max weset a uniform prior exploring the range 0 < t max <
14 Gyr.Concerning the present-day ratio between the total surfacemass densities σ /σ , we recall that Fuhrmann et al. (2017) de-rived in the solar vicinity a local mass density ratio betweenthe thin and thick disc stars of 5.26, which becomes as low as1.73 after correction for the di ff erence in the scale height. Whilestudying APOGEE stars, Mackereth et al. (2017) found that therelative contribution of low- to high- α is 5.5. In the solar an-nulus, Spitoni et al. (2020) found that the best models span therange between 3 . − .
3. In this work, studying di ff erent Galacticregions, we set this prior in the range, 0 . < σ /σ < ffi ne invariant MCMC ensemble sampler, " emcee : themcmc hammer" code , proposed by Goodman & Weare (2010);Foreman-Mackey et al. (2013) has been used to sample the pos-terior probability distribution. https: // emcee.readthedocs.io / en / stable / ;https: // github.com / dfm / emcee Fig. 3.
Observed [Mg / Fe] vs. [Fe / H] abundance ratios from APOGEEDR16 (Ahumada et al. 2020) (grey points with associated errors) in theGalactocentric region between 6 and 10 kpc compared with the best-fit chemical evolution model (thick curve) in that region. As in Fig. 1,the contour lines enclose fractions of 0.90, 0.75, 0.60, 0.45, 0.30, 0.20,0.05 of the total number of observed stars. The color coding representsthe cumulative number of stars formed during the Galactic evolutionnormalized to the total number N tot . The open circles mark the modelabundance ratios of stellar populations with di ff erent ages. In the insetwe show the surface stellar mass density ∆ M (cid:63) formed in di ff erent agebins as a function of age, where the bin sizes are delimited by the ver-tical dashed lines and correspond to the same age values as indicated inthe [Mg / Fe] vs [Fe / H] plot.
4. Results
Here, we show predictions of chemical evolution models of theGalactic disc computed at di ff erent Galactocentric distances. InSection 4.1 we present and discuss our findings for chemical evo-lution models computed at 8 and 12 kpc, whereas the innermostregion centered at 4 kpc is presented in Section 4.2. In Section4.3 we interpret our results in terms of the "inside-out" formationscenario and we show a comparison between our predictions andsome of the most important observables of the Galactic disc. InSection 4.4 the metallicity and the [Mg / Fe] distributions will bepresented. Finally, in Section 4.5 we compare our findings withthe chemical evolution predictions of the recent study presentedby Palla et al. (2020).
First, in this section we present the results of the best-fit chemi-cal evolution model at 8 kpc, which was obtained by fitting theabundance ratios [Mg / Fe] and [Fe / H] of stars in the APOGEEDR16 sample using our Bayesian technique based on the MCMCmethods.This model assumes infall episodes with primordial chemi-cal composition for both high- α and low- α sequences ( X and X quantities in eq. 1) and di ff erent SFE: ν = − and ν = − (Chiappini et al. 1997; Grisoni et al. 2018; Spi- Article number, page 6 of 15pitoni et al.: Multi-zone chemical evolution model based on MCMC methods
Table 1.
Observed solar chemical abundances compared with modelpredictions.
Abundance
Observations Model log( X / H) +
12 Asplund et al. (2005) R = ± ± ± Θ (cid:12) = { τ , τ , t max , σ /σ } , for our modelat a Galactocentric distance of 8 kpc. We find a significant de-lay in the start of the second gas infall t max = . + . − . Gyr,confirming the previous results of Spitoni et al. (2019b, 2020).The best model predicts a value of 5.635 + . − . for the σ /σ ra-tio, in accordance with the findings of Mackereth et al. (2017);Fuhrmann et al. (2017); Spitoni et al. (2020). Predicted infalltime-scales τ = . + . − . Gyr and τ = . + . − . Gyr areshorter than the ones of Spitoni et al. (2020). We recall that inthat work the model was compared with the APOKASC sam-ple by Silva Aguirre et al. (2018), which consisted of 1180 redgiants in a narrower region in the solar vicinity (2 kpc-wide).The APOKASC sample has measured solar-like oscillations, al-lowing asteroseismic determination of stellar ages. In Spitoniet al. (2020), the determined stellar ages were used as additionalconstraint. Recall that here we adopt di ff erent selection crite-ria for stars following Weinberg et al. (2019) and Vincenzo &Kobayashi (2020).In fact, while the APOKASC sample shows absence of starswith [ α / Fe] < -0.05 dex (see Fig. 1 in Spitoni et al. 2020), thesample adopted here has a significant fraction of low- α stars with[Mg / Fe] < -0.05 dex. Hence, the di ff erences in the best-fit valuesfor the infall time-scales can be attributed to the di ff erences inthe data used in the two studies.In Fig. 3 we compare the best-fit model computed at 8 kpcwith the [Mg / Fe] vs. [Fe / H] abundance ratios. The color codingwith the cumulative number of stars formed during the Galacticevolution shows that the bulk of the stars are formed during thelow- α sequence. Moreover, the surface stellar mass density ∆ M (cid:63) formed in di ff erent age bins as a function of age (see the insetplot of Fig. 3), clearly shows the existence of a low- α and high- α bimodality.For the sake of clarity, in response to a question posed inLian et al. (2020) we underline that this bimodality was alreadypresent in the best fit model of Spitoni et al. (2019b) who didnot show the variation of the stellar mass content along with thechemical evolution tracks in their figures.As illustrated in Spitoni et al. (2020, 2019b), the gas dilutionoriginated by a strong second gas infall is a key process to ex-plain APOKASC abundance ratios with our model. In particular,the second accretion event of pristine gas decreases the metallic-ity of the stellar populations born immediately after keeping aroughly constant [Mg / Fe] ratio since the accretion involves Hand He but basically no metals. When star formation resumes,Type II SNe produce a steep bump in [Mg / Fe], which subse-quently decreases at higher metallicities due to iron from TypeIa SNe (Matteucci et al. 2009; Bonaparte et al. 2013). This se-quence produces a loop feature in the chemical evolution trackof [Mg / Fe] vs. [Fe / H]. We notice from the inset plot of Fig. 3 that
Fig. 4.
Same as Fig. 2, but for the annular region enclosed between10 and 14 kpc. The SFEs are fixed at values of 2 and 0.5 Gyr − for thehigh- α and low- α phases, respectively. Fig. 5.
Same as Fig. 3, but for the observed abundance [Mg / Fe] vs.[Fe / H] ratio for stars with Galactocentric distances between 10 kpc and14 kpc and the corresponding best-fit chemical evolution model. a negligible mass fraction of low- α stars is formed during the di-lution phase of the loop, with ages in the range between 9.7 and9.2 Gyr ( ∆ M (cid:63) = (cid:12) pc − , corresponding to ∼ ∆ M (cid:63) = (cid:12) pc − ( ∼ α sequence stellar mass). Finally, almost the to- Article number, page 7 of 15 & A proofs: manuscript no. spitoni_MCMC_APOGEEDR16
Fig. 6.
Same as Fig. 2, but for the annular region enclosed between 2and 6 kpc. For the second gas accretion episode, an enriched infall with[Fe / H] = -0.5 dex is considered. The SFEs are fixed at values of 3 and1.5 Gyr − for the high- α and low- α phases, respectively. tality of stellar mass is produced in the age interval between 8.7Gyr and 0.2 Gyr, namely ∆ M (cid:63) = (cid:12) pc − , correspondingto ∼ α / Fe] vs. [Fe / H] space,as shown in Spitoni et al. (2020).We stress that, as shown in Spitoni et al. (2019b), it is likelythat the loop feature is hidden inside the observational errors.Using a "synthetic" model (at each Galactic time a random er-ror was assigned to the ages and metallicities of the stellar pop-ulations), Spitoni et al. (2019b) were capable to reproduce thedata spread in the low- α sequence of the APOKASC sample(Silva Aguirre et al. 2018) in the [ α / Fe] vs. [Fe / H] abundanceratio space.In Table 1, we compare the solar abundances of Fe, Mg, andSi predicted by our best fit model in the solar neighborhood withAsplund et al. (2005) photospheric values. In the model, solarabundances are determined from the composition of the ISM atthe time of the formation of the Sun (after 9.5 Gyr from the BigBang). It is evident that our model is able to reproduce well thesolar abundance ratios for these elements.In Figs. 4 and 5 we show the results for the external regioncentered at 12 kpc. For this model we also assume primordialinfall for both the high- and low- α sequences and di ff erent SFEsfor the high- and low- α phases, ν = − and ν = − ,respectively. As we stated in Section 3.1, a lower SFE in outerGalactic regions has been assumed by several chemical evolu-tion models (Colavitti et al. 2008; Spitoni et al. 2015; Grisoniet al. 2018; Palla et al. 2020) in order to reproduce abundancegradients (see Section 4.3). Fig. 7.
Same as Fig. 3, but for stars of APOGEE DR16 (Ahumada et al.2020) sample with Galactocentric distances between 2 and 6 kpc and thecorresponding best fit chemical evolution model. Here, we considereda pre-enriched second gas infall with metallicity, [Fe / H] = -0.5 dex. TheSFEs are fixed at values of 3 and 1.5 Gyr − for the high- and low- α phases, respectively. Comparing the best fit model parameters for the Galacticregion centered at 8 kpc with those at 12 kpc, we note thatlonger time-scales of gas accretion τ ( R ) and τ ( R ) are associ-ated to more external region. Hence, the ‘inside-out’ formationscenario invoked by the classical two-infall model of Chiappiniet al. (2001) in our model is a natural consequence of the fit tothe APOGEE DR16 abundance ratio, as we shall discuss it thor-oughly in Section 4.3. Concerning the evolution of the inner disc, we consider the pres-ence of an enriched gas infall for the low- α phase ( X quantity ineq. 1). In Palla et al. (2020), it was already pointed out that a pri-mordial infall for the inner thin disc cannot explain the observedbehaviour of the [ α / Fe] vs. [Fe / H] in the APOGEE data.Di ff erent physical reasons may be associated with this en-riched gas infall event. For instance, Palla et al. (2020) discussedthat an enriched infall could partly be due to gas lost from the for-mation of the thick disc, Galactic halo or the Galactic bar whichthen gets mixed with a larger amount of infalling primordial gasas proposed by Gilmore & Wyse (1986). In Khoperskov et al.(2021), the infalling gas during inner thin disc formation phaseis not primordial because the gaseous halo has been significantlypolluted during the high- α disc formation, providing a tight con-nection between chemical abundance patterns in the high- α andlow- α discs. Alternatively, as already mentioned in the Introduc-tion, Renaud et al. (2020a) proposed that the Galaxy disc is fu-eled by two distinct gas flows and that the one responsible for theformation of the inner α sequence is enriched by outflows frommassive galaxies (with ∼ / Article number, page 8 of 15pitoni et al.: Multi-zone chemical evolution model based on MCMC methods
Following the best model prescriptions of Palla et al. (2020)for the inner thin disc, we impose that for the second gas infall(low- α ) a chemical enrichment obtained from the model of thehigh- α disc phase corresponding to [Fe / H] = -0.5 dex. In Figs.6 and 7 we present the chemical evolution model predictionsfor the region centered at 4 kpc assuming, for the high- α se-quence, a SFE fixed at the value of ν = − . This highervalue compared to external regions could be motivated by star-burst episodes, as suggested by Agertz et al. (2020); Renaudet al. (2020b,a). The time-scale τ of the second gas accretionepisode and the ratio between low- α and high- α surface massdensity σ /σ are smaller compared to the external parts, in per-fect agreement with the inside-out formation scenario (see Sec-tion 4.3 for a detailed discussion).We also explore the possibility of a SFE fixed at the valueof 2 Gyr − which as the same as in the outer regions. Althoughwe found a similar [Mg / Fe] vs. [Fe / H] relation, an extremelyshort (and maybe unrealistic) infall time-scale of τ = ν ∼ − ) in thehigh- α disc phase provides more reasonable results in the two-infall framework. In Table 2 we summarize the best fit model parameters at dif-ferent Galactocentric distances: the delay t max , surface densityratio σ / σ , infall time-scales τ and τ values. The presence ofa significant delay between the two infall episodes is a robustresult, confirming the previous results of the works focusing onthe solar neighborhood (Spitoni et al. 2019b, 2020).In Fig. 8 we show that our model predictions are in favourof the ’inside-out’ formation scenario: inner Galactic regionsare assembled faster compared to the external one (Matteucci& Francois 1989; Chiappini et al. 2001; Schönrich & McMillan2017; Frankel et al. 2019). In Fig. 1 data show that in the outerregions the locus of the low- α sequence shifts towards lowermetallicity and in Fig. 8 we find that external Galactic regionsare formed on longer accretion time-scales τ , hence the chemi-cal enrichment is weaker and less e ffi cient than the inner Galacticregions, leading to a lower metallicity. We recall that we imposedalso a radial variation of the SFE.Moreover, in Fig. 1 the radial bin between 10 and 14 kpc hasfewer high- α stars and we associate the more prominent low- α sequence to a larger surface density ratio σ / σ compared tothe innermost regions in agreement with Palla et al. (2020). Animportant result of this study is that extending the predictionsfor σ / σ to the whole Galactic disc, we predict a clear and neattrend: the ratio increases with the Galactocentric distance (Fig.8). The Galactic ’inside-out’ formation is well motivated by thedissipative collapse scenario (Larson 1976; Cole et al. 2000) andit has been a widely adopted assumption coupled with a variableSFE and the radial gas flows in order to reproduce the observedabundance gradients (Spitoni et al. 2015; Grisoni et al. 2018;Palla et al. 2020).We have shown that the two-infall model of Spitoni et al.(2019b, 2020) can be extended to the whole disc admitting amore complex nature of the Galactic disc evolution instead ofa simple sequential scenario, with the coexistence of di ff erentphysical processes and di ff erent gas infall enrichment as a func-tion of the Galactocentric distance. Moreover, with a quantitative Fig. 8.
Time-scale τ for the gas accretion in the low- α phase and theratio σ /σ between the low- α and high- α total surface mass densitiesas a function of the Galactocentric distance are drawn with red and bluepoints, respectively. Fig. 9.
Observed and predicted radial [Mg / H] present-day abundancegradient. The prediction of the best-fit models are indicated with bigfilled circles, connected with a yellow line. The observational data arethe Cepheid observations from Luck & Lambert (2011) (blue circles)and Genovali et al. (2015) (blue diamonds with red edges). With theempty circles we report the average values and associated errors of thefull data sample. estimation of the model free parameters using a Bayesian ap-proach, we confirm the results of the chemical evolution modelproposed by Palla et al. (2020), where an enriched gas infall hasbeen invoked to reproduce the chemical evolution of the innerGalactic disc (see Section 4.5 for a detailed comparison withPalla et al. 2020 predictions). The proposed multi-zone chem-ical evolution model based on the MCMC methods to fit theabundance distributions of [Mg / Fe] vs. [Fe / H] ratios from theAPOGEE DR16 sample is also able to reproduce the other im-portant observables of the Galactic disc.In Fig. 9 we compare the observed abundance gradient formagnesium of Luck & Lambert (2011) and Genovali et al.(2015) with the one predicted by our best model computed atthe present-day. To be consistent with these data sets, the modelabundance ratios are referred to the solar value of Grevesse et al.
Article number, page 9 of 15 & A proofs: manuscript no. spitoni_MCMC_APOGEEDR16
Table 2.
In the upper part of the Table we show the chemical composition of the two gas infall episodes ( X and X ) and the star formatione ffi ciencies ( ν and ν ) for the high- α and low- α sequences assumed in our model at various Galactocentric distances. In the lower part of the Tablewe show the accretion time-scales ( τ and τ ), the present-day total surface mass density ratio ( σ / σ ) and delay t max computed for our best fitmodels at 4 kpc, 8 kpc and 12 kpc (see text for model details). In the last column on the right, we also provide the ranges admitted by our studycomputed from our best-model estimates at all Galactocentric radii. Models X Primordial Primordial Primordial X [Fe / H] = -0.5 dex Primordial Primordial ν [Gyr − ] 3.0 2.0 2.0 ν [Gyr − ] 1.5 1.0 0.5 MCMC Results
Range τ [Gyr] 0.115 + . − . + . − . + . − . . − . τ [Gyr] 3.756 + . − . + . − . + . − . . − . σ / σ + . − . + . − . + . − . . − . t max [Gyr] 4.647 + . − . + . − . + . − . . − . Fig. 10.
Observed and predicted radial gas surface density gradient. Theblack dashed curve is the average between the Dame (1993) and Nakan-ishi & Sofue (2003, 2006) data sets. The grey shaded region representsthe typical uncertainty at each radius, for which we adopt either 50%of the average (see Nakanishi & Sofue 2006) or half the di ff erence be-tween the minimum and maximum values in each radial bin (if larger).The big filled circles show the model predictions. (1996). It is clear that our model prediction well reproduces theobserved abundance gradient.It is clear from Fig. 10 that also the observed surface gasdensity profiles are well reproduced by our model. Concerningthe present-day surface stellar mass density (see Fig. 11), thepredicted value at 8 kpc is 34.2 M (cid:12) pc − in very good agree-ment with the observed local value of 33 . ± (cid:12) (McKee et al.2015). Assuming that the stellar surface density decreases expo-nentially outwards with a characteristic length-scale of R (cid:63) = Fig. 11.
Radial stellar surface stellar density profile of our model (bigcircles connected by a yellow line). The observed local stellar density is33 . ± (cid:12) pc − (McKee et al. 2015, small black dot and associated er-ror bar). In our model we assume that the stellar profile decreases expo-nentially outward with a characteristic scale-length of 2.7 kpc (Kubryket al. 2015). The blue and grey shaded areas indicate zones within 1 σ and 2 σ , respectively. by Rana (1991) and the following analytical fit of SN remnantscompilation by Green (2014):SFR( R ) / SFR (cid:12) = (cid:18) RR (cid:19) b e − c (cid:0) R − R R (cid:1) , (11)where R = b = c = . ff erent Galactocentric distances reported in the right panel ofFig. 12, the two star formation phases (high- α and low- α stars)and the hiatus in between are evident. The delay t max betweenthe two infall gas episodes is longer as we move towards theinner region. Such a variation is statistically significant (see Ta-ble 2), and t max values span the range between 3.0 and 4.7 Gyr. Article number, page 10 of 15pitoni et al.: Multi-zone chemical evolution model based on MCMC methods
Fig. 12.
Left panel : Observed and predicted radial SFR density gradientrelative to the solar neighbourhood. The model results are the big solidcircles connected by a solid yellow line. The black line is the analyticalform suggested by Green (2014) for the Milky Way SFR profile. Theopen circles with error bars are observational data from Rana (1991).
Right panel : SFR evolution predicted by our best fit models computedat 4, 8 and 12 kpc. The dark green shaded area indicates the present-daymeasured range in the solar annulus by Prantzos et al. (2018).
Longer cooling timescales due to a more intense star formationactivity and stronger feedback are expected for the innermostGalactic regions at early times. Hence, we propose that in the in-ner (outer) regions the ISM gas needed more (less) time to cooldown in order to begin the SF activity associated with the low- α sequence, leading to larger (smaller) values for t max . This sce-nario is supported by several chemo-dynamical simulations ina cosmological framework where a hot gas phase is already inplace at early times and the halo tends to inhibit gas filamentsto penetrate into the central regions (Kereš et al. 2005; Dekel &Birnboim 2006; Brooks et al. 2009; Fernández et al. 2012; Grandet al. 2018).In Fig. 13, we report the time evolution of the Type Ia SNand Type II SN rates. The present-day Type II SN rate in thewhole Galactic disc predicted by our model is 0.93 / [100 yr], asmaller value (but within 2 σ error) than the observations of Liet al. (2011) which yield a value of 1.54 ± / [100 yr]. Thepredicted present-day Type Ia SN rate in the whole Galactic discis 0.27 / [100 yr], in good agreement with the value provided byCappellaro & Turatto (1997) of 0.30 ± / [100 yr]. In the lowerpanel of Fig. 13 we also show the time evolution of the infallrate. We note that the present day value computed at 8 kpc is1.01 M (cid:12) pc − Gyr − , consistent with the range 0.3 -1.5 M (cid:12) pc − Gyr − suggested by Matteucci (2012).In Fig. 14 the time evolution of the metallicity [M / H] and the[ α / Fe] ratios computed at 4, 8, and 12 kpc are compared with thelocal APOKASC sample by Silva Aguirre et al. (2018). Here, α is computed by means of the sum of the abundances of Mg andSi. The metallicity [M / H] is computed, as in Silva Aguirre et al.(2018), using the following expression introduced by Salariset al. (1993):[M / H] = [Fe / H] + log (cid:16) . × [ α/ Fe ] + . (cid:17) . (12)We combine the abundance ratios [Fe / H] and [ α / Fe] predicted byour model using this formulation to be consistent with the data.We notice that the best fit model at 8 kpc is very similar to thebest model in Spitoni et al. (2019b) constrained by APOKASCabundance ratios and asteroseismic ages. Spitoni et al. (2019b)showed that the steep drop in [M / H] and bump in [ α / Fe] associ-ated with the second accretion episode (not obvious in the obser-vations), are hidden behind the observational uncertainties.The highest metallicity values are reached at any Galactictime by the innermost region. Di ff erent slopes in the [M / H] and
Fig. 13.
Upper panels : Observed and predicted Type II SN rates (left)and Type Ia SN rates (right) as a function of the Galactic age. The SNrates predicted for the whole disc are reported with the black solid lines,and they represent the sum of the contributions from di ff erent Galacticregions indicated with colored dashed lines. The observed present-dayType II SN rate of Li et al. (2011, left panel) for the whole Galaxy isreported with the solid star (1 σ and 2 σ errors are indicated with greyand yellow bars, respectively) whereas the solid star in the right panelstands for Type Ia SN rate of Cappellaro & Turatto (1997) with theassociated 1 σ error bar. Lower panel : Infall rate evolution predicted byour best fit models computed at 4 (red line), 8 (green line) and 12 kpc(light-blue line). The dark green shaded area indicates the present-dayvalues in the solar annulus suggested by Matteucci (2012). [ α / Fe] ratios characterize the evolution of the low- α sequencesat di ff erent Galactcocentric distances. This is due to the interplayof di ff erent best-fit model values for time-scale of accretion τ ,SFEs, and gas infall enrichment in diverse Galactic regions. In Fig. 15, it is shown that the predicted [Fe / H] distribution func-tions at di ff erent Galactocentric distances are generally in agree-ment with the data.To highlight once more the low- α and high- α bimodality, in16 we show the [Mg / Fe] distributions, where we see that theAPOGEE DR16 data in the annular region centered at 8 kpc ex-hibit two neat peaks. Although the best-fit model accounts for theobserved bimodality and the median distribution value is consis-tent with the data, the predicted peaks are shifted towards higher[Mg / Fe] values. In the right panel of Fig. 16, we draw the samedistribution but only for stars with [Fe / H] in the range between-0.2 and 0 dex (see the highlighted region in the enclosed plot).In this case, the predictions are in better agreement with the data.The reason why the full data set seems in contrast with modelis largely due to the large uncertainty in the assumed stellar nu-cleosynthesis yields of Mg from massive stars, which cause the
Article number, page 11 of 15 & A proofs: manuscript no. spitoni_MCMC_APOGEEDR16
Fig. 14.
Evolution of the [M / H] (left panel) and [ α / Fe] (right panel)abundance ratios of our best-fit models computed at 4, 8 and 12 kpccompared with the abundances observed in a stellar sample in the solarannulus (Silva Aguirre et al. 2018). Magenta points depict the high- α population, whereas green points indicate the low- α one. As in Spitoniet al. (2019b, 2020), we have not taken into account young α -rich stars. model to have higher [Mg / Fe] ratios ( ∼ / H]than the observations from APOGEE. Moreover, our assumptionof a bottom-heavy IMF (Scalo 1986) slows down the time evolu-tion of the [Fe / H] abundances in the ISM, creating a bias towardshigher [Mg / Fe] ratios after each infall event. Additionally, sincetwo-infall model is an approximate representation of the truth,significantly larger number of stars in the low-alpha sequencecompared to the high-alpha can compromise the fit for the high- α sequence (as it gets less weight in the optimization processpresented in Section 3.3). A better agreement is achieved for theexternal region centered at 12 kpc as shown in Fig. 17, where theposition of two peaks and the median value of the distribution arein good agreement with the data.In Fig. 18 the observed and predicted [Mg / Fe] distributionfor the region centered at 4 kpc are compared. Once again, wesee that the median value and the location of the two peaks arequite di ff erent from the data. In order to understand this dis-crepancy, we ran a new model with the same best-fit parame-ters as in Table 2 for the 4 kpc case, but with a smaller sur-face mass density σ /σ ratio. In Fig. 19, we show the predicted[Mg / Fe] vs. [Fe / H] relation and the [Mg / Fe] distribution func-tion imposing σ /σ =
1. In the upper panel of Fig. 19, we notethat, as expected, the maximum [Mg / Fe] value reached in thelow- α sequence is smaller compared to the one in Fig. 7 (with σ /σ = .
8) because of the smaller mass associated with thelow- α phase as clearly shown in the inset plots of Figs. 7 and19. In fact, the model with the lowest σ /σ ratio presents thehighest increase of stellar mass ∆ M (cid:63) in the high- α phase.In the lower panel of Fig. 19, we can appreciate that themodel reproduces better the data than the results reported in Fig.18 given also the intrinsic uncertainties due to the Mg stellaryields from massive stars and IMF. In fact, the median valuesof the predicted [Mg / Fe] and observed data are pretty similar. Inconclusion, a smaller σ /σ has a double e ff ect: (i) the peak ofthe [Mg / Fe] abundance associated with the low- α is shifted to-wards smaller values, and (ii) an increase of the number of thehigh- α stars (the second peak in the distribution has a highernumber of stars with larger [Mg / Fe] values than in Fig. 18).
Recently, Palla et al. (2020) presented a revised Galactic chem-ical evolution model for the disc formation based on the two-
Fig. 15.
Metallicity distributions predicted by the best-fit models (col-ored histograms) computed at 4 kpc (upper panel), 8 kpc (middle panel),and 12 kpc (lower panel). The observed APOGEE DR16 distributionsare shown by the black empty histograms. The vertical lines indicatethe median values of each distribution. In each plot, the distributionsare normalised to the corresponding maximum number of stars, N max . infall scenario. In agreement with our findings, their best modelsuggests that a variable SFE should be acting together with the’inside-out’ mechanism for the low- α disc formation. In ad-dition, they claimed that radial gas inflows can help to createan abundance gradient confirming the findings of Spitoni et al.(2015) and Grisoni et al. (2018). Article number, page 12 of 15pitoni et al.: Multi-zone chemical evolution model based on MCMC methods
Fig. 16.
Left panel : [Mg / Fe] distribution predicted by our best-fit model computed at 8 kpc (green histogram) compared with the APOGEE data instars with Galactocentric distances between 6 and 10 kpc. Black and green vertical dashed lines indicate the median values of the data and model,respectively. Distributions are normalised to the corresponding maximum number of stars, N max . Right panel : Same as the left panel, but computedfor a limited metallicity range, − . ≤ [Fe / H] ≤ Fig. 17. [Mg / Fe] distribution predicted by our best-fit model computedat 12 kpc (blue histogram) compared with the APOGEE data in starswith Galactocentric distances between 10 and 14 kpc. Black and bluevertical lines indicate the median values of the data and model, respec-tively.
In order to reproduce the observed [Mg / Fe] vs. [Fe / H] ofAPOGEE (Hayden et al. 2015) at di ff erent Galactocentric dis-tances, Palla et al. (2020) proposed a delay of t max = / Fe] vs. [Fe / H] abundance ratio, in accordance withour study (see Section 4.2).Di ff erences in the chemical evolution tracks in the inner-most disc region are primarily due to the di ff erent stellar nu-cleosynthesis prescriptions. Here, we adopt the yield collectionproposed by François et al. (2004), whereas Palla et al. (2020) Fig. 18. [Mg / Fe] distribution predicted by our best-fit model computedat 4 kpc (red histogram) compared with the APOGEE data in a stel-lar sample with Galactocentric distances between 2 and 6 kpc. Blackand red vertical lines indicate the median values of the data and model,respectively. used those from Romano et al. (2010). Moreover, our results arebased on a Bayesian analysis to fit the latest APOGEE DR16data (Ahumada et al. 2020). Finally, Palla et al. (2020) imposea di ff erent length-scale for the two disc components instead of avariable ratio between the low- α and high- α surface mass den-sities. Notwithstanding all these di ff erences, our model and theone of Palla et al. (2020) share a similar growth of the Galacticdisc following the ’inside-out’ scenario and predict pretty similarmetallicity distribution functions. Article number, page 13 of 15 & A proofs: manuscript no. spitoni_MCMC_APOGEEDR16
Fig. 19.
APOGEE DR16 data in the region 2-6 kpc compared withmodel predictions computed at 4 kpc with the same best-fit parametersas in the first column of Table 2 but with the surface mass density ratio σ /σ fixed at the value of 1. Upper panel : [Mg / Fe] vs. [Fe / H] abun-dance ratios. As in Fig. 1, the contour lines enclose fractions of 0.90,0.75, 0.60, 0.45, 0.30, 0.20, 0.05 of the total number of observed stars.The color coding represents the cumulative number of stars formed dur-ing the Galactic evolution normalized to the total number N tot . The opencircles mark the model abundance ratios of stellar populations with dif-ferent ages. In the inset we show the surface stellar mass density ∆ M (cid:63) formed in di ff erent age bins as a function of age, where the bin sizes aredelimited by the vertical dashed lines and correspond to the same agevalues as indicated in the [Mg / Fe] vs [Fe / H] plot.
Lower panel : [Mg / Fe]distributions. Black and red vertical lines represent the median values ofthe data and model, respectively.
5. Conclusions
We have presented a multi-zone chemical evolution model de-signed for the whole Galactic disc constrained by chemical abun-dance ratios of APOGEE DR16 (Ahumada et al. 2020) data us-ing the Bayesian analysis presented in Spitoni et al. (2020). Inthis study, we have considered four free parameters: accretion time-scales τ and τ , delay t max and present-day surface massdensity ratio σ /σ .Our main conclusions can be summarized as follows.1. The Bayesian analysis based on the recent APOGEE DR16data (Ahumada et al. 2020) suggests the presence of a sig-nificant delay time between the two gas infall episodes forthe thick-disc and thin-disc formation in all analyzed Galac-tocentric regions. We find that the best values for the delaytimes are in the range between 3 and 4.7 Gyr, confirming thefindings of Spitoni et al. (2019b, 2020) for the solar neigh-borhood based on the APOKASC data.2. An inside-out formation of the thin-disc of our Galaxy nat-urally emerges from the best fit of our multi-zone chemical-evolution model to APOGEE-DR16 data: inner Galactic re-gions are assembled on shorter time-scales than externalGalactic zones. Moreover, our best-fit model predicts larger σ /σ (ratio of low- α to high- α surface mass densities) val-ues towards outer Galactic regions (see Fig. 8), in agreementwith the fact that as we move towards external regions, theAPOGEE DR16 data sample presents less and less stars inthe high- α phase compared to the low- α sequence (Queirozet al. 2020).3. In outer disc regions with Galactocentric distances R > / Fe] vs. [Fe / H] abundance pattern in thelow- α phase, extending the findings of the models presentedby Spitoni et al. (2019b, 2020) for the solar neighborhood.4. In the inner disc, for the two-infall model to work, an en-riched gas infall for the formation of low- α sequence starsis required to reproduce the observed data as suggested byPalla et al. (2020). Di ff erent physical explanations could beinvoked: the gas might be enriched with metals from out-flows originated in massive galaxies (Agertz et al. 2020; Re-naud et al. 2020b,a), or it could be due to gas lost from theformation of the thick disc, which then gets mixed with alarger amount of infalling primordial gas (Gilmore & Wyse1986).5. Our model reproduces important observational constraintsfor the chemical evolution of the whole disc reasonably well,such as the present-day profiles of the SFR, the stellar andgas surface densities. Moreover, the predicted abundancegradient is in good agreement with the observations, thanksto the longer time-scales of accretion in the outer regionsand to a variable SFE for the low- α sequence. In the solarneighborhood, the model is able to reproduce the solar pho-tospheric abundance values of Asplund et al. (2005).The above mentioned results suggest that the signatures of adelayed gas-rich merger giving rise to a hiatus in the star forma-tion history are impressed in the [Mg / Fe] vs. [Fe / H] relation anddetermine the distribution of the low- α stars in the abundancespace at di ff erent Galactocentric distances. Acknowledgement
Funding for the Stellar Astrophysics Centre is provided by TheDanish National Research Foundation (Grant agreement no.:DNRF106). E. Spitoni and V. Silva Aguirre acknowledge sup-port from the Independent Research Fund Denmark (Researchgrant 7027-00096B). F. Vincenzo acknowledges the support ofa Fellowship from the Center for Cosmology and AstroParti-cle Physics at The Ohio State University. V. Grisoni acknowl-edges financial support at SISSA from the European Social
Article number, page 14 of 15pitoni et al.: Multi-zone chemical evolution model based on MCMC methods
Fund operational Programme 2014 / ffi / collaboration / a ffi liations / .This work has made use of data from the European SpaceAgency (ESA) mission Gaia ( ), processed by the Gaia
Data Processing and Anal-ysis Consortium (DPAC, ). Funding for the DPAC hasbeen provided by national institutions, in particular the institu-tions participating in the
Gaia
Multilateral Agreement.
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