The formation history of the Milky Way disc with high-resolution cosmological simulations
M. Giammaria, A. Spagna, M. G. Lattanzi, G. Murante, P. Re Fiorentin, M. Valentini
MMNRAS , 1–15 (2020) Preprint 5 February 2021 Compiled using MNRAS L A TEX style file v3.0
The Formation History of the Milky Way disc with high-resolutioncosmological simulations
Marco Giammaria, , ★ Alessandro Spagna, † Mario G. Lattanzi, Giuseppe Murante, Paola Re Fiorentin, Milena Valentini , , INAF - Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025, Pino Torinese, Turin, Italy Department of Physics, University of Turin, Via P. Giuria, 1, I-10125 Turin, Italy Universitäts-Sternwarte München, Fakultät für Physik, LMU Munich, Scheinerstr. 1, 81679 München, Germany Excellence Cluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany INAF - Osservatorio Astronomico di Trieste, via Tiepolo 11, I-34131 Trieste, Italy
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We analyse from an observational perspective the formation history and kinematics of a MilkyWay-like galaxy from a high-resolution zoom-in cosmological simulation that we compareto those of our Galaxy as seen by Gaia DR2 to better understand the origin and evolutionof the Galactic thin and thick discs. The cosmological simulation was carried out with theGADGET-3 TreePM+SPH code using the MUlti Phase Particle Integrator (MUPPI) model.We disentangle the complex overlapping of stellar generations that rises from the top-downand inside-out formation of the galactic disc. We investigate cosmological signatures in thephase-space of mono-age populations and highlight features stemming from past and recentdynamical perturbations. In the simulation, we identify a satellite with a stellar mass of1 . · M (cid:12) , i.e. stellar mass ratio Δ ∼ . 𝑧 ∼ .
6, whichresembles the major merger Gaia-Sausage-Enceladus that produced the Galactic thick disc,i.e. Δ ∼ 𝑧 ∼ . . · M (cid:12) and 5 . · M (cid:12) that are associated to a strong starburst in the Star FormationHistory, which appears fairly similar to that recently found in the Solar Neighbourhood. Ourfindings highlight that detailed studies of coeval stellar populations kinematics, which aremade available by current and future Gaia data releases and in synergy with simulations, arefundamental to unravel the formation and evolution of the Milky Way discs. Key words:
Galaxy: kinematics and dynamics – Galaxy: disc – galaxies: formation – galaxies:evolution – galaxies: star formation – methods: numerical
Many of the chemo-kinematic properties of the Milky Way (MW)and nearby galaxies are related to events that occurred a long timeago. The aim of Local Cosmology is to provide a link betweenthe local scenario at redshift 𝑧 = Λ CDM model pre-dicts that our Galaxy formed through the hierarchical merging ofsubstructures, whose accretion history can be inferred from the ★ E-mail: [email protected] † E-mail: [email protected] chemo-kinematic signatures that we observe today in the stellarpopulations of the Galactic bulge, halo, thick and thin discs. Withthe advent of cosmological hydrodynamical zoom-in simulations ofMW-like galaxies, we are able to analyse in detail the complex inter-play of physical processes (e.g. dynamics of collisionless systems,gas flows, star formation, stellar nucleosynthesis, ...) that generatedour Galaxy, which represents the Rosetta stone of galaxy evolution.Recently, several studies have confirmed that our Galaxy ex-perienced ∼
10 Gyr ago a major merger with a massive dwarfgalaxy, named Gaia-Sausage-Enceladus (GSE, Belokurov et al.2018; Helmi et al. 2018; Di Matteo et al. 2019; Vincenzo et al. 2019;Gallart et al. 2019). According to these studies, the stellar inner halois mainly made up of stellar debris from the accreted satellite andheated disc stars. However, the actual origin of the present thickdisc is not clear and it is a still matter of debate whether it de-rives mainly from the kinematic heating of the proto-disc, or froma starburst triggered by merging events and fed by the infall of a © a r X i v : . [ a s t r o - ph . GA ] F e b Giammaria et al. gas-rich satellite (Brook et al. 2004, 2012). An extensive review ofthe early accretion history of the MW is described by Helmi (2020),while detailed comparisons with the chemo-dynamical propertiesof cosmological simulations are presented by Minchev et al. (2013),Stinson et al. (2013), Buck et al. (2019) and Grand et al. (2020).Other authors have also produced and analysed high resolutionsimulations in order to reconstruct the formation history of our MW;some of these studies are listed in Table 1.Here, we provide a detailed analysis of Aquila-C-4 (AqC4),a hydrodynamical cosmological simulation of a MW-mass galaxybased on the MUPPI algorithm (Murante et al. 2010, 2015). TheAqC series of simulations has been presented by Springel et al.(2008) and used, among other papers, in the Aquila comparisonproject (Scannapieco et al. 2012). The validity of the MUPPI al-gorithm has been confirmed by several studies (e.g. Monaco et al.2012; Goz et al. 2015; Valentini et al. 2017). It consists of an un-constrained simulation, i.e. it is not designed to exactly mimic thedynamic history of our Galaxy as in simulations aimed to reproducethe Local Group environment (see for example Sawala et al. 2016;Carlesi et al. 2016). Thus, according to the precepts of Local Cos-mology, AqC4 results as a typical cosmological product with massand phase-space properties similar to the MW, and with a merginghistory comparable to a disc-like galaxy.In this paper, we aim to compare AqC4, namely a theoretical‘error-free’ catalogue predicted for a Milky Way-like galaxy, to ourGalaxy as seen by Gaia DR2 (Gaia Collaboration et al. 2018a).Therefore, the analysis presented in this work is carried out as ifthe simulation were a real stellar survey of our Galaxy and wefocus on the phase-space properties of stellar particles contained ina simulated region representative of the Solar Neighbourhood. Wecharacterize the spatial and kinematical parameters of the stellardisc and investigate the Star Formation History (SFH) of AqC4 forredshift 0 < 𝑧 (cid:46)
2, in order to identify signatures of past and recentmerging events that can be used to disentangle the accretion historyof our Galaxy.In Sect. 2 we describe the simulation and summarize its mainparameters. In Sect. 3 we characterize the spatial distribution ofparticles, focusing on the radial and vertical structures of the stellardisc, i.e. determining its scale length and scale height. We investigatethe mono-age populations in order to study the flaring of the stellardisc. In Sect. 4 we analyse the kinematic properties of AqC4. Westudy the global rotation curve of AqC4 and we make a directcomparison with recent observational data of the MW as seen byGaia. Then, we focus on the substructures of the stellar disc inthe region defined as the Simulated Solar Ring (hereafter SSR). InSect. 5 we link our previous findings to the formation and evolutionof AqC4; specifically, we extensively compare the Star FormationHistory of the simulation with recent estimates for the MW, andthen we extend our investigation to the accretion history. Finally, inSect. 6 we summarize our findings and discuss our results.
In this Section, we introduce the simulation. The cosmologicalhydrodynamical simulation that we performed has been carriedout with the TreePM+SPH (smoothed particle hydrodynamics)GADGET3 code, a non-public evolution of the GADGET2 code(Springel 2005). As initial conditions (ICs), we adopt the zoomed-in ICs AqC4 introduced by Springel et al. (2008). In our realiza-tion, they describe an isolated DM halo at redshift 𝑧 =
0, with a
Table 1.
Cosmological hydrodynamical simulations of MW-like disc galax-ies. Main properties of recent high resolution zoom-in simulations.
Project Mass particle Softening Reference[M (cid:12) ] [pc]Eris M DM ∼ · 𝜖 ∗ ∼
120 Guedes et al. (2011)M gas ∼ · 𝜖 gas ∼ DM ∼ · 𝜖 ∗ ∼
369 Grand et al. (2017)M gas ∼ · GIZMO M DM ∼ · 𝜖 ∗ ∼
50 Ma et al. (2017)M gas ∼ · 𝜖 gas ∼ DM ∼ · 𝜖 ∗ ∼
740 Nelson et al. (2018)TNG100 M gas ∼ · 𝜖 gas (cid:38) DM ∼ · Mackereth et al. (2019)M gas ∼ · 𝜖 gas (cid:46) DM ∼ · 𝜖 ∗ ∼
273 Buck et al. (2020)(g7.08e11) M gas ∼ · 𝜖 gas ∼ DM ∼ · 𝜖 ∗ ∼
223 This workM gas ∼ · 𝜖 gas ∼ virial mass of 𝑀 vir ∼ . · M (cid:12) within a spherical vol-ume of 𝑅 vir = .
13 kpc. Current estimates of MW virial massare consistent with the range of values 0.5 – 2 · M (cid:12) (Bland-Hawthorn & Gerhard 2016). In particular, studies based on GaiaDR2 give 𝑀 vir ∼ . · M (cid:12) when considering the dynamicsof globular clusters (Posti & Helmi 2019; Watkins et al. 2019), and 𝑀 vir ∼ · M (cid:12) using MW rotation curve (Eilers et al. 2019;Crosta et al. 2020). In what follows, we consider the galactic radius, 𝑅 gal , as 1 /
10 of the virial one .We assume a Λ CDM cosmology with Ω m = . Ω Λ = . Ω baryon = . 𝜎 = . 𝑛 𝑠 =
1, and 𝐻 = ℎ km s − Mpc − =
73 km s − Mpc − where ℎ represents the reduced Hubble constant.The zoomed-in region that we simulate has been extracted from acosmological volume of 100 ( ℎ − Mpc ) of the DM-only, parentsimulation.The Plummer-equivalent softening length for the computationof the gravitational force is 𝜖 Pl = ℎ − pc. DM particles have amass of 2 . · ℎ − M (cid:12) , and the initial mass of gas particles is5 . · ℎ − M (cid:12) .As anticipated in Sect. 1, we adopt a sub-resolution modelcalled MUPPI (MUlti Phase Particle Integrator). Here we outlineits most relevant features, while we refer the reader to Murante et al.(2010, 2015) for a more comprehensive description. In particular,all the parameters of the model which are not explicitly mentionedhere are set to the same values as in Murante et al. (2015). TheMUPPI sub-resolution model describes a multi-phase interstellarmedium (ISM). Among the most important processes implemented,it features star formation and stellar feedback, metal cooling andchemical evolution. It accounts for hot and cold ( 𝑇 =
300 K) gasphases in pressure equilibrium by means of multiphase particles,which are the building blocks of the model. A gas particle enters amulti-phase stage if its density increases above a density threshold The virial quantities are those calculated in a sphere that is centred onthe minimum of the gravitational potential of the halo and that encloses anover-density of 200 times the critical density at present time. We count a total number of DM, gas and stellar particles respectivelyof 5 518 587, 1 348 120, and 6 919 646 at redshift 𝑧 =000
300 K) gasphases in pressure equilibrium by means of multiphase particles,which are the building blocks of the model. A gas particle enters amulti-phase stage if its density increases above a density threshold The virial quantities are those calculated in a sphere that is centred onthe minimum of the gravitational potential of the halo and that encloses anover-density of 200 times the critical density at present time. We count a total number of DM, gas and stellar particles respectivelyof 5 518 587, 1 348 120, and 6 919 646 at redshift 𝑧 =000 , 1–15 (2020) ormation History of the Milky Way disc ( 𝑛 = .
01 cm − ) and its temperature drops below a temperaturethreshold ( 𝑇 = K).A set of ordinary differential equations describes mass andenergy flows among different components within each multiphaseparticle: for instance, radiative cooling moves mass from the hot intothe cold phase, while a tiny fraction of the cold gas evaporates dueto the destruction of molecular clouds. We rely on the phenomeno-logical prescription by Blitz & Rosolowsky (2006) to estimate thefraction of cold gas which is in the molecular phase and which fuelsstar formation. Star formation is then modelled according to thestochastic algorithm introduced by Springel & Hernquist (2003): asa consequence, a multiphase gas particle can generate (up to fourgenerations of) star particles.The MUPPI model features stellar feedback both in thermaland kinetic forms (as described in Murante et al. 2015). Besidesstellar feedback in energy, star formation and evolution also result ina chemical feedback, and galactic outflows generated by supernova(SN) explosions promote metal spread and circulation within thegalaxy (Valentini et al. 2017, 2018).Our model accounts for stellar evolution and chemical enrich-ment following Tornatore et al. (2007), where a thorough descrip-tion can be found. In summary, each star particle is considered tobe a simple stellar population. Assuming an initial mass function(Kroupa et al. 1993), as well as predictions for stellar lifetimes(Padovani & Matteucci 1993) and stellar yields (see Murante et al.2015, for details), we evaluate the number of stars aging and eventu-ally exploding as SNe, and the amount of metals injected in the ISM.Heavy elements released by star particles are distributed to neigh-bouring gas particles. The chemical evolution of 9 metals (C, Ca,O, N, Ne, Mg, S, Si, Fe) synthesized by different sources (namelyasymptotic giant branch stars, SNe Ia and SNe II) is individuallyfollowed. The model also features metallicity-dependent radiativecooling (following Wiersma et al. 2009), and includes the effect ofan ionizing cosmic background (Haardt & Madau 2001).
Fig. 1 shows the face-on and edge-on projected density of stellar(upper panels) and gas (lower panels) particles. The reference sys-tem used is centred in the minimum of the gravitational potentialand rotated to be aligned with the angular momentum vector ofmulti-phase gas and star particles within 8 kpc from the centre,while the 𝑋𝑌 plane is orthogonal to it. This reference system hasbeen adopted for the entire analysis presented in this paper.The presence of a disc structure is clear both in the stellarand gas distribution. This disc dominates the central part of thegalactic volume and is extended until 𝑅 ∼
10 kpc. In the innermostcentral region, a non-axisymmetric bar structure is visible in bothcomponents. The stellar disc is quite symmetric around the 𝑋𝑌 plane which can be defined itself as the galactic plane. Finally, aspiral pattern is visible in the outer part of the disc region, while thegas exhibits a warped shape at its edge.We characterize the main components of AqC4 by means ofthe stellar mass distribution as a function of the orbit circularity ofall star particles within 𝑅 gal for our simulation at redshift 𝑧 = 𝜖 = 𝐽 Z / 𝐽 circ , where 𝐽 Z isthe specific angular momentum in the direction perpendicular tothe disc, and 𝐽 circ is the specific angular momentum of a referencecircular orbit. The results are shown in Fig. 2.The prominent peak at 𝜖 ∼ 𝜖 ∼
0. We compute the ratio of bulge-over-total stellar mass 𝐵 / 𝑇 by doubling the mass of the counter-rotatingstars within 𝑅 gal , under the hypothesis that the bulge is supportedby velocity dispersion and thus has an equal amount of co- andcounter-rotating stars. We highlight that this is a ‘dynamical’ valueof the bulge-to-total stellar mass ratio as defined by Scannapiecoet al. (2010). The resulting ratio is 𝐵 / 𝑇 = .
34, fairly comparablewith the upper limit estimated for the MW, 𝐵 / 𝑇 MW (cid:39) .
15 – 0.33(see e.g. Bland-Hawthorn & Gerhard 2016; Bell et al. 2017, andreferences therein).
It is common to describe the stellar disc of the MW with a double-component exponential decay profile with a radial scale length of ℎ Rt = . ± . ℎ RT = . ± . | 𝑍 | ≤ (cid:12) pc − ) for cylindrical radial bins of Δ 𝑅 = .
25 kpcas shown in Fig. 3. Then, to infer the scale length, we limit the fit inthe disc-dominated region between 2 . ≤ 𝑅 [ kpc ] ≤ 𝜖 = .
163 kpc.We analyse the stellar particles close to the galactic plane, i.e. | 𝑍 | ≤ a radial scale length of ℎ R = . ± .
002 kpc that is smaller by a factor of ∼
21 per cent with respect tothe MW thin disc. Thus, in the following we adopt a proportionallysmaller Simulated Solar Ring (SSR) of 6 ≤ 𝑅 [ kpc ] ≤ 𝑅 (cid:12) = . ± .
033 kpc, as estimated by the Gravity Collaborationet al. (2018).
The vertical mass distribution of the MW disc is modelled as adouble-component exponential function.As reported by Bland-Hawthorn & Gerhard (2016), the thindisc scale height at solar distance from the Galactic Centre is ℎ Zt = ±
50 pc, while for the thick disc is ℎ ZT = ±
180 pc.Moreover, several studies have tried to determine the relative densitynormalization 𝑓 = 𝜌 T / 𝜌 t of the thick disc compared to the thin discwith estimates ranging from 6 to 12 per cent (e.g. Jurić et al. 2008;Just & Jahreiß 2010; Bovy et al. 2015).In order to reduce the contamination from halo stars, we select We set the Credible Interval (CI) as the ± 𝜎 equivalent range betweenthe 16 th and 84 th percentile of the posterior distribution, i.e. the percentageequivalents of the 1 𝜎 range in a Gaussian distribution. From the posteriors,it results very narrow and thinner than the best-fitting line.MNRAS , 1–15 (2020) Giammaria et al.
Figure 1.
Stellar (upper panels) and gas (lower panels) projected density for the AqC4 simulation (face-on and edge-on view on left and right panels,respectively). The 𝑍 -axis of the coordinate system is aligned with the angular momentum vector of multi-phase gas and stars enclosed within 8 kpc from theposition of the minimum of the gravitational potential. The total box size is 50 kpc. all the 320 354 stellar particles in the SSR within | 𝑍 | ≤ 𝜌 ( 𝑍 ) = 𝐴 (cid:32) 𝑒 − | 𝑍 | ℎ + 𝑓 · 𝑒 − | 𝑍 | ℎ (cid:33) , (1)where 𝐴 is the total density normalization, 𝑓 is the relative den-sity normalization, ℎ 𝑖 , for 𝑖 = ,
2, is the scale height of the twocomponents of the disc, and 𝑍 is the vertical coordinate.We estimate the scale heights, ℎ = . ± .
005 kpc that is in good agreement with the MW thin disc, and ℎ = . ± .
02 kpcthat results ∼
50 per cent larger than what it is estimated for the MWthick disc (see Fig. 4). We also found a relative density parameterof 𝑓 = . ± . MNRAS000
50 per cent larger than what it is estimated for the MWthick disc (see Fig. 4). We also found a relative density parameterof 𝑓 = . ± . MNRAS000 , 1–15 (2020) ormation History of the Milky Way disc Figure 2.
Distribution of the stellar mass of AqC4 as a function of the orbitcircularity for all stellar particles within 𝑅 gal at redshift 𝑧 = Figure 3.
Radial density distribution (green dots) of stellar particleswithin | 𝑍 | ≤ Δ 𝑅 = .
25 kpc. Error bars are smaller than the size of data point. Red linerepresents the best fit obtained from the posteriors of the MCMC analysis.The CI is thinner than the best-fitting red line. Vertical black lines show theradial interval adopted for the fit. ponents: (i) the young disc population with stellar ages up to 4 Gyr(blue and orange lines) that shows an exponential decreasing fromthe galactic plane similar to the thin disc described in Sect. 3.2; (ii)an intermediate old disc population (green and red lines, i.e. 4-8 Gyrold), whose vertical distribution clearly shows the superposition ofa thin and a thick component; (iii) a prominent thick disc populationin the age interval 8-10 Gyr (purple line) corresponding to the bulkof the ‘geometric’ thick disc; (iv) the spheroidal oldest population with age greater than 10 Gyr (brown and pink lines) that showswidespread distribution, and includes a large fraction of halo starparticles.We point out that 40 per cent of the sample is made up of youngdisc stars with ages ≤ ≤ Age [ Gyr ] ≤ ≤ Age [ Gyr ] ≤
10, corresponding tothe thick disc . Figure 4.
Vertical density distribution (red dots) of stellar particles in theSSR with | 𝑍 | ≤ Δ 𝑍 = .
25 kpc. Blackline represents the best fit obtained from the posteriors of MCMC analysis.The CI is thinner than the best-fitting black line. Colour lines representmono-age stellar populations divided in bins of 2 Gyr as listed in Table 2.
Table 2.
Age and relative weights of the mono-age populations with respectto the total stellar particles ( 𝑁 ) in the SSR (6 ≤ 𝑅 [ kpc ] ≤
7) within | 𝑍 | ≤ 𝑁 / 𝑁 blue 0-2 0.22orange 2-4 0.18green 4-6 0.12red 6-8 0.10purple 8-10 0.22brown 10-12 0.10pink 12-14 0.07black 0-14 𝑁 =320 354 These results reveal a SFH affected by significant changes overtime, which are discussed in more detail in Sect. 5.1.
Another important spatial feature which stems from the inside-outis the disc flaring (see e.g. Minchev et al. 2017, and referencestherein). Here, we investigate this property by studying the verticaldistribution of AqC4 stellar particles as a function of the cylindricalradial distance for both the whole sample and for the mono-agepopulations listed in Table 2.As shown in Fig. 5, we divided the star particles in rings of Δ 𝑅 = | 𝑍 | ≤ ℎ and ℎ parameters along the disc. Then, we estimate the thicknessof each mono-age population as the height, ℎ 𝑍 , where the densitydistribution decreases by a factor e − with respect to the galacticplane, 𝑍 = ≤ ℎ 𝑍 (cid:46) . 𝑅 (cid:39) MNRAS , 1–15 (2020)
Giammaria et al.
Figure 5.
Variation of stellar height, | 𝑍 | , with 𝑅 . Flaring is present for allmono-age populations (colour code as in Table 2). Solid lines show whenthe density decays by a factor of e − . The ℎ and ℎ scale heights for eachradial bin integrated over all stellar ages (see Table 3) are represented bycircles and diamonds, respectively. intermediate old-thin disc population (4 ≤ Age [ Gyr ] ≤
8) showsa significant flaring already at 𝑅 (cid:39) thick disc population (i.e. 8 ≤ Age [ Gyr ] ≤ ℎ 𝑍 (cid:39) . − . 𝑅 = − ℎ (cid:39) .
33 kpc estimated inSect. 3.2. A radial flaring is already apparent at 𝑅 (cid:39) 𝑧 (cid:39) . Now, we extend the analysis of the galactic structure done in Sect. 3to the kinematics of the stellar populations ‘observed’ at redshift 𝑧 = The galactic Rotation Curve (RC) constitutes one of the key featuresthat characterize disc galaxies like the MW. The significant discrep-ancy existing between the empirical curves observed (e.g. Sofue &Rubin 2001; Eilers et al. 2019; Crosta et al. 2020), and theoretical
Table 3.
The ℎ and ℎ scale heights, and relative density normalization 𝑓 with the 1 𝜎 CI integrated over all the 𝑁 ∗ stellar particles within each radialbin. 𝑅 [kpc] ℎ [kpc] ℎ [kpc] 𝑓 [%] 𝑁 ∗ . + . − . . + . − . . + . − .
767 3113-4 0 . + . − . . + . − . . + . − .
687 8324-5 0 . + . − . . + . − . . + . − .
592 2075-6 0 . + . − . . + . − . . + . − .
436 2706-7 0 . + . − . . + . − . . + . − .
320 3547-8 0 . + . − . . + . − . . + . − .
234 4548-9 0 . + . − . . + . − . . + . − .
171 6749-10 0 . + . − . . + . − . + −
96 15910-11 1 . + . − . . + . − . + −
46 019 circular velocity profiles derived form the baryonic mass distribu-tion (considering test particles that would move in an axisymmetricgravitational potential Φ according to Newtonian theory) representsone of the main line of evidence of the presence of dark matter(DM) in haloes (e.g. Iocco et al. 2015; McMillan 2017; de Salaset al. 2019).Fig. 6 shows the circular velocity 𝑉 𝑐 ( 𝑅 ) = √︁ 𝑅𝜕 Φ / 𝜕𝑅 = √︁ 𝐺 𝑀 ( < 𝑅 )/ 𝑅 for the total mass (black solid line), and for the indi-vidual component of DM (gray dashed), stars (gray dot-dashed), gas(dotted). As expected, after a steep linear increase for 𝑅 ≤ ≤ 𝑅 [ kpc ] ≤
11, reaches its peak 𝑉 𝑐 = . − at 𝑅 = .
25 kpc, and finally has a smooth decrease for 𝑅 ≥
11 kpc inthe halo dominated region. In the central region, for 𝑅 (cid:46) 𝑅 (cid:38) . − , whichis ∼
20 per cent faster than the value 𝑉 𝜙 (cid:39)
234 km s − measured atthe Sun position 𝑅 (cid:12) = .
122 kpc by Crosta et al. (2020). However, ifwe scale the size (with the radial scale length estimated in Sect. 3.1)and the velocity, keeping constant the angular momentum 𝐿 𝑍 at theSun position, we can reasonably overplot the RC of the MW (greenstar symbols in Fig. 6) on that of AqC4.Thus, AqC4 appears to be a disc galaxy fairly similar to the MWwith a ∼
20 per cent faster rotating disc and a shorter pseudo -solarposition location. The re-scaled Gaia data that describe the Galacticdisc kinematics results in good agreement with the rotation curveof AqC4.
In Sect. 3.2, we have modeled the stellar vertical distribution in theSSR with the superposition of a thin disc ( ℎ (cid:39) .
305 kpc) andthick disc ( ℎ (cid:39) .
33 kpc) that includes a residual contamination ofhalo star particles.Here, we focus on the rotation velocity distribution of the stellarparticles within the SSR in order to identify and characterize the
MNRAS000
MNRAS000 , 1–15 (2020) ormation History of the Milky Way disc Figure 6.
Rotation curves for AqC4. Solid line shows the total curve, whiledashed, dot-dashed and dotted lines show the contribution of the DM, stel-lar and gas component, respectively. Red star symbols with correspondinguncertainties represent observational data for the MW from Crosta et al.(2020). Green star symbols are the same Gaia DR2 data scaled as describedin Sect. 4.1. different components. This methodology is usually applied to realstellar surveys in order to select stellar samples belonging to theGalactic populations, e.g. the thin and thick disc, and inner halo(Bond et al. 2010; Spagna et al. 2010). Only a few authors haveapplied such kinematic decomposition to cosmological simulations(see Abadi et al. 2003a,b; Obreja et al. 2018, 2019, and referencestherein), as the high-resolution level required has been only recentlyachieved.We select the 333 616 stellar particles in the SSR with | 𝑍 | ≤ 𝑓 ( 𝑉 𝜙 ) ,of the azimuthal velocity by normalizing to 1 the integral of thetotal sample. We adopt a model based on a Triple Normal Mixturedistribution (TNM), namely 𝑓 ( 𝑉 𝜙 ) = ∑︁ 𝑖 = 𝑤 𝑖 𝑁 (cid:16) 𝑉 𝜙 |(cid:104) 𝑉 𝜙,𝑖 (cid:105) , 𝜎 𝑉 𝜙,𝑖 (cid:17) , (2)where 0 ≤ 𝑤 𝑖 ≤ 𝑖 -th component and 𝑁 is a Normal distribution with mean (cid:104) 𝑉 𝜙, i (cid:105) and standard deviation 𝜎 𝑉 𝜙,𝑖 . The results of the MCMC analysis are listed in Table 4 andvisualised in Fig. 7 (left panel), where the PDF model (red line)appears in good agreement with the observed velocity distribution(blue dots). Further information on the posterior probability of thefitted parameters are reported in Appendix A2.We also tested alternative models with a different number ofcomponents. The results for a double Gaussian distribution such ashalo+disc or disc+disc are not statistically significant and do not pro-vide good curve fitting of the data. A four-component model, aimedto represent additional disc populations, produced non-significantimprovements to the model.We note that the weights of the two disc components listed inTable 4 correspond to the total mass of the young disc (Age ≤ old disc/thick disc (4 ≤ Age [ Gyr ] ≤ old thin disc and the thick disc populations are difficult to distinguish because of their very similarvelocity distributions. Moreover, the difference between the mean
Table 4.
TNM model parameters: 𝑤 𝑖 is the mixture weight of the 𝑖 -thcomponent (i.e. young disc, old disc/thick disc, halo), while (cid:104) 𝑉 𝜙 (cid:105) and 𝜎 𝑉 𝜙 are the corresponding mean and the standard deviation of the Normaldistribution. Note that 0 ≤ 𝑤 𝑖 ≤ (cid:205) 𝑖 𝑤 𝑖 = 𝑖 Component 𝑤 (cid:104) 𝑉 𝜙 (cid:105) 𝜎 𝑉 𝜙 [km s − ] [km s − ]1. young disc 0.389 ± ± ± ± ± ± ± ± ± rotation velocity of the AqC4 discs is Δ 𝑉 𝜙 = . − . (cid:39)
27 km s − . (3)This value is smaller than in the MW, where Δ 𝑉 𝜙, MW = . − . =
38 km s − (Han et al. 2020). Actually, this differencedepends on the fact that in AqC4 the two main disc componentsrepresent the young and the old/thick discs instead of the wholethin disc and the thick disc, as in the MW. Moreover, the high rota-tion velocity of the AqC4 disc is consistent with its more compactstructure with respect to the MW disc (Sect. 3.1).Finally, the halo component shows a small prograde rotation asobserved in the inner halo of the MW, while its velocity dispersion 𝜎 𝑉 𝜙 (cid:39)
160 km s − is about 70 −
100 per cent higher than in theSolar Neighbourhood (Re Fiorentin et al. 2015; Bland-Hawthorn &Gerhard 2016, and references therein).In summary, here we robustly support what is discussed inSect. 3: in the SSR, we can distinguish at least two kinematic disccomponents which rotate faster than what is measured in the So-lar Neighbourhood (as discussed globally in Sect. 4.1), and with asmaller discrepancy between the mean values. These results extendthe spatial characterisation of AqC4 stellar disc beyond the determi-nation of the scale lengths and scale heights, allowing us to properlyinvestigate the complete phase-space of mono-age populations, asusually done in Galactic surveys.
Fig. 8 shows the three median velocity components (cid:16)(cid:101) 𝑉 𝑅 , (cid:101) 𝑉 𝜙 , (cid:101) 𝑉 𝑍 (cid:17) and the dispersions ( 𝜎 𝑉 𝑅 , 𝜎 𝑉 𝜙 , 𝜎 𝑉 𝑍 ) as a function of 𝑅 , computedin radial annular bins of Δ 𝑅 = .
25 kpc, for the mono-age stellardisc populations younger than 10 Gyr and with | 𝑍 | ≤ 𝑅 ≤ ≤ 𝑅 [ kpc ] ≤
11 (white area) the region dominated by the disc,where we have the lowest dispersions and the RC flat regime; (iii)finally, for 𝑅 ≥
11 kpc (grey area) the halo region where the velocitydispersions increase and, conversely, the rotation velocity decreases.In the disc region, the median radial velocity turns out to benegative, i.e. (cid:101) 𝑉 𝑅 = − ÷ − , for almost all mono-age popu-lations (see Fig. 8, top-left panel), meaning that the system is out ofequilibrium in contrast to what observed in the MW. Examining thelast ∼ . MNRAS , 1–15 (2020)
Giammaria et al.
Figure 7.
Left panel : Kinematic decomposition of 𝑉 𝜙 PDF for stellar particles in the SSR (i.e. 6 ≤ 𝑅 [ kpc ] ≤
7) with | 𝑍 | ≤ Δ 𝑉 =
10 km s − are represented by blue dots. Young disc , old disc/thick disc , and halo structure are shown with dashed, dot-dashed and dottedlines, respectively. The reconstructed profile of the PDF using the posteriors estimates of the TNM model is shown by the red line. Right panel : 𝑉 𝜙 distributionfor mono-age stellar particles. Colour code as in Table 2. present time only, since (cid:101) 𝑉 𝑅 ∼ − in the previous snapshotsof the simulation. Unfortunately, the nature of this kinematical sig-nature cannot be easily investigated because it occurred only in thelast snapshot of the simulation. The dynamical perturbation of theAqC4 disc may have been produced by the last satellite detected atredshift 𝑧 = .
014 (see next section for details), whose high veloc-ity and retrograde orbit shows a perigalactic passage in the outerregions of the disc. Further investigations are necessary to confirmthis hypothesis. Despite such zero-point offset in AqC4, we observean ‘U-shape’ with a minimum at about the Sun position fairly simi-lar to the MW, although the Gaia data samples only a portion of theGalactic disc (Fig. 12 in Gaia Collaboration et al. 2018b).The rotation velocity (cid:101) 𝑉 𝜙 is slower for the older stellar genera-tions, as expected from asymmetric drift, as well as to the secularprocesses (e.g. spiral arms perturbations and bar resonances) andto merging events. The presence of significant dynamical disc per-turbations may also explain why the young stellar particles with6 < 𝑅 [ kpc ] <
11 and | 𝑍 | ≤ 𝑉 𝑐 (middle-leftpanel). These stars belong to the most angular-momentum sustainedpopulation and may be accelerated by the momentum of accretedgas. As expected by the azimuthal averaging, the radial gradientof the vertical velocity is almost flat, with a median value (cid:101) 𝑉 𝑍 (cid:39) − and with larger fluctuations for younger populations (Fig. 8,bottom-left panel). Instead, the apparent radial increase of the meanvertical velocity, 𝑉 𝑍 , found in the outer disc of the MW by Poggioet al. (2018, Fig. 3) represents a local signature of the Galacticwarp, due to the limited volume sampled by Gaia DR2 and to thepeculiar Sun position close to the line of nodes.In the disc region, the two velocity dispersion components 𝜎 𝑉 𝑅 and 𝜎 𝑉 𝜙 decrease from 𝑅 (cid:39) ∼ 𝑅 (cid:39) 𝜕𝜎 𝑉 / 𝜕𝑅 and cooler isothermal curves than theolder populations. These results are consistent with the monotonicradial decrease of the velocity dispersions observed in our Galaxy,since the colder longer-scale length young disc is dominant in theouter disc.The behaviour of 𝜎 𝑉 𝑍 in AqC4 is different from the othertwo components and shows a positive radial gradient for mono-age populations between 2 and 8 Gyr old (i.e. the intermediate old disc population), while the youngest and oldest stellar particles arealmost isothermal. We argue that this is a cosmological signature ofa heating process due to mergers that occurred in the last 7 Gyr ofthe simulation as discussed below in Sect. 5.1.Even though our investigation focuses on the disc region, weremark the peculiar velocity patterns within 𝑅 ≤ ( 𝑎 ) the global inward 𝑉 𝑅 systematic motion, ( 𝑏 ) the faster rotation 𝑉 𝜙 > 𝑉 𝑐 of the youngest stars, and ( 𝑐 ) the pecu-liar velocity dispersions shown by mono-age populations. Moreover,the gas accretion in the external region of the disc shown in Fig. 1highlights that the galaxy is still out-of-equilibrium, as discussedin Valentini et al. (2020). All these properties are consistent withthe recent studies based on Gaia DR2 that show how the accretionhistory is the key to understanding the origin of the ancient thickdisc and inner halo (e.g. Stinson et al. 2013; Helmi et al. 2018, andreferences therein), as well as to disentangle the signatures of the“dynamically young and perturbed MW disk” (Antoja et al. 2018). Star Formation History (SFH) does enclose fundamental informa-tion about the origin and evolution of disc-like galaxies and of theMW, as well. In order to understand the phase-space properties ofmono-age populations discussed in Sects. 3.2, 4.2 and 4.3, here weinvestigate the star formation and the accretion history of AqC4.Similarly to Bignone et al. (2019) and Grand et al. (2020), we con-trast the total Star Formation Rate (SFR) with the accretion historyof our simulated galaxy. In addition, we consider the local SFR, asinferred by the stellar particles within the SSR, that we comparewith the observational results recently derived by Mor et al. (2019).
First, we analyse the SFH of the stellar populations in AqC4 byinvestigating the SFR, which results ∼ .
65 M (cid:12) yr − for the whole MNRAS000
65 M (cid:12) yr − for the whole MNRAS000 , 1–15 (2020) ormation History of the Milky Way disc Figure 8.
AqC4 median velocity components (cid:101) 𝑉 𝑅 , (cid:101) 𝑉 𝜙 , (cid:101) 𝑉 𝑍 (left, from top to bottom), and dispersions 𝜎 𝑉 𝑅 , 𝜎 𝑉 𝜙 , 𝜎 𝑉 𝑍 (right, from top to bottom), as afunction of 𝑅 for different mono-age stellar populations within | 𝑍 | ≤ 𝑅 ≤ 𝑅 ≥
11 kpc) stands for halo region. For 3 ≤ 𝑅 [ kpc ] ≤
11 the disc dominates. Error bars are derived via bootstrappingwith 100 re-samples. In panel left-middle we show also the total mass RC (black solid line) for a better comparison. galaxy at redshift 𝑧 =
0. This value is at least 1.5 times higher thanwhat is measured for the MW by Robitaille & Whitney (2010) andLicquia & Newman (2015).Fig. 9 shows the evolution of the SFR per unit of surface,for the star particles inside a spherical volume of 𝑅 gal (red line)and in the SSR with | 𝑍 | ≤ 𝑧 > . 𝑧 ∼ . 𝑧 ∼ . ≈ (cid:12) yr − ,which appears more similar to value of ∼ . (cid:12) yr − found inthe EAGLE simulation studied by Bignone et al. (2019), than to MNRAS , 1–15 (2020) Giammaria et al. the much stronger starburst up to ∼
25 M (cid:12) yr − resulting from theAURIGA simulation analysed by Grand et al. (2020).Meanwhile, the subset of stellar particles within the SSR high-lights an irregular SFH, quite different from the global SFR, thatevidences the complex evolution of the galactic disc. The peak atcosmic times 𝑇 < 𝑧 ∼ . 𝑧 ∼ .
4) evidences the formation ofthe ancient thick disc. We point out the consistency between thesebursts in the star formation at 𝑇 ∼ 𝑇 > (cid:12)
Gyr − pc − for 𝑇 >
11 Gyr, that formed a mas-sive young disc, as already discussed in Sect. 3.2. Despite the largeuncertainties, the SFR observed by Mor et al. (2019) in the SolarNeighbourhood results in very good agreement with our estimatesfor AqC4 in the SSR.The large time-scale (almost 3.5 Gyr) and the large amountof mass that is involved suggest that this event is produced by anexternal factor. Indeed, lower panels of Fig. 1 show a large reservoirof gas both in the disc and falling from the halo. We argue that thelow-redshift merging events shown in the next Section may havecontributed to the recent gas accretion.Although more significant in AqC4, this scenario is consistentwith the results published by Mor et al. (2019) and supports thehypothesis that the increasing SFR in the Solar Neighbourhood ofthe MW may be due to the recent merging event claimed by Lianet al. (2020).
In order to understand the features found in the SFH of AqC4, weselect seven snapshots with redshift 𝑧 = . . . . . .
178 and 0 . 𝑟 sat . A similar procedure is performed to estimate the mass of themain disc-like galaxy, but considering a cylindrical volume definedby the radial disc extension at that epoch, 𝑅 disc , within | 𝑍 | < Δ = 𝑀 sat ∗ / 𝑀 disc ∗ variesfrom ∼ . 𝑧 = . . · M (cid:12) is pretty similar to that estimatedby Mackereth et al. (2019) and Fattahi et al. (2019) for the GSEprogenitor. In fact, these authors claim a mass of a few 10 M (cid:12) ,assuming a 10 M (cid:12) stellar mass for the thick disc present at thetime. We obtain a stellar mass ratio Δ ∼ . ∼ 𝑀 sat ∗ (cid:46) M (cid:12) corresponding to amass ratio Δ ≤ Table 5.
Estimates of the total stellar mass of selected satellite galaxies andthe main galactic disc, with corresponding mass ratios
Δ = 𝑀 sat ∗ / 𝑀 disc ∗ atdifferent redshift 𝑧 . The spherical radius, 𝑟 sat , is centred on satellite galaxies,while the radial extension of the disc, 𝑅 disc , is in the reference frame of themain galaxy. 𝑧 𝑟 sat 𝑀 sat ∗ 𝑅 disc 𝑀 disc ∗ Δ [kpc] [ M (cid:12) ] [kpc] [ M (cid:12) ] [ % ] secondary fluctuations of the SFR in the SSR after redshift 𝑧 ∼ . 𝑀 sat ∗ (cid:39) . · M (cid:12) for the accretedsatellite detected at 𝑧 = . 𝑀 sat ∗ (cid:39) . · M (cid:12) for the one detected at redshift 𝑧 = . old disc population that explains the large scale height of the stellar particleswith age 4-8 Gyr shown in Fig. 5. The presence of merging eventsduring the intermediate phase of the disc formation is consistentwith the late-accretion model proposed by Lian et al. (2020), whosuggest that a recent merger event occurred in the MW at 8 . 𝑧 ∼ . 𝑀 ∗ < M (cid:12) for the gas-rich dwarf galaxyinvolved.Finally, we detect two more satellites at redshift 𝑧 = .
178 and0 .
014 , with M sat ∗ (cid:39) . · M (cid:12) and 2 . · M (cid:12) , respectively,which appear associated to the increasing SFR in the SSR of AqC4during the last 2 Gyr.In summary, the SFH and the accretion history of AqC4 areconsistent with its spatial and kinematic properties described inSects. 3 – 4 and, in particular, they clarify the formation of the pe-culiar features observed in the young disc and old disc populations.Moreover, the overall formation and evolution of AqC4 andthe MW appear fairly similar. Indeed, our findings are consistentwith both the GSE scenario for the origin of the Galactic thick disc (Brook et al. 2004, 2012; Stinson et al. 2013; Helmi et al. 2018;Gallart et al. 2019), and with the recent accretion event proposed toexplain the increasing SFR in the Solar Neighbourhood (Mor et al.2019). We argue that the higher SFR of our simulated galaxy withrespect to the MW depends on both the gas contribution from thelate accreted satellites and the infall of gas previously expelled bythe strong starbursts and supernova explosions that occurred at highredshift. In this work, we analysed the spatial and kinematic properties ofstellar particles of the MW-like galaxy AqC4. Our approach con-sisted in considering such simulation as a real survey of the stellarcontents of our Galaxy. Therefore we implemented methods of in-vestigation usually applied to real stellar catalogues. Our aim wasto identify cosmological signatures enclosed in the mono-age stel-
MNRAS000
MNRAS000 , 1–15 (2020) ormation History of the Milky Way disc Figure 9.
SFR per unit of surface as a function of cosmic time for 𝑟 ≤ . lar populations, defined as coeval particles sub-samples of 2 Gyrage bins, and compare their phase-space properties with the MW.We focused on the Simulated Solar Ring (SSR), an annular regionwithin 6 < 𝑅 [ kpc ] <
7, corresponding approximately to the SolarNeighbourhood, given the ∼
20 per cent smaller radial scale lengthof the thin disc of AqC4 with respect to the MW ( ℎ R (cid:39) . 𝜖 Pl = ℎ − pc.In the SSR, we confirmed the presence of at least two disccomponents with vertical exponential distribution (see Sect. 3.2).We estimated a scale height ℎ ∼ . ℎ ∼ .
33 kpc that is ∼
50 per cent larger with respect to the MW thick disc.The inspection of the vertical distributions of the mono-agestellar particles in the SSR clarified that the thin disc of AqC4 ismainly formed by a young disc population with ℎ Z ∼
250 – 500 pcand age ≤ old disc (age 4-8 Gyr) having intermediate thickness ( ℎ Z ∼
500 –1000 pc), and a prominent thick disc population with age 8-10 Gyrand ℎ Z ∼ (cid:104) 𝑉 𝜙, (cid:105) (cid:39) . − and (cid:104) 𝑉 𝜙, (cid:105) (cid:39) . − . These results are consistent with the morecompact structure of AqC4 with respect to the MW.The median (cid:101) 𝑉 𝑅 and (cid:101) 𝑉 𝜙 components of mono-age stellar parti-cles and their relative dispersions in Fig. 8 revealed that the stellardisc of AqC4 is out of equilibrium and shows the dynamical signa-tures of the perturbations due to both recent mergers, cosmic impactand gas accretion. Indeed, in the disc region 3 ≤ 𝑅 [ kpc ] ≤ (cid:101) 𝑉 𝑅 evidences a systematic inward motion for all the mono-age popu-lations in contrast to what measured in the MW. We suppose thatthe very recent impact of the high-speed, counter-rotating satellitefound at redshift 𝑧 = .
014 (see lower panels of Fig. 11) may haveproduced this dynamical perturbation. On the other hand, younger
MNRAS , 1–15 (2020) Giammaria et al.
Figure 10.
Stellar density maps for the AqC4 simulation at redshift 𝑧 = . .
314 and 1 . stars show an higher median (cid:101) 𝑉 𝜙 than older ones as expected frominside-out formation and secular processes occurred in the disc, butalso rotate faster than the circular velocity, accelerated by accreatedgas. These kinematic features are consistent with the recent studiesbased on Gaia DR2 data that show how the accretion history is thekey to understand the origin of the ancient MW thick disc and innerhalo (e.g. Brook et al. 2004, 2012; Stinson et al. 2013; Helmi et al. 2018; Gallart et al. 2019), as well as to disentangle the signaturesof the “dynamically young and perturbed MW disk” (Antoja et al.2018).We suggest that the prominent thick disc population was gener-ated by the major accreted satellite detected at redshift 𝑧 = .
6. Suchmerging event is associated to the starburst at redshift 1 < 𝑧 < ∼ (cid:12) yr − . Thisscenario is consistent with the results published by Bignone et al. MNRAS000
6. Suchmerging event is associated to the starburst at redshift 1 < 𝑧 < ∼ (cid:12) yr − . Thisscenario is consistent with the results published by Bignone et al. MNRAS000 , 1–15 (2020) ormation History of the Milky Way disc Figure 11.
As Fig. 10 but at redshift 𝑧 = . . .
178 and 0 . , 1–15 (2020) Giammaria et al. (2019) and Grand et al. (2020) who analysed independent cosmo-logical simulations of MW-like galaxies selected from the EAGLEand Auriga projects, respectively. Although such simulations showSFH’s quite different from AqC4, all these studies evidence a sig-nificant increase in the SFR triggered by the galaxy merger andassociated to the thick disk formation.We also compared the age distribution of the disc stellar par-ticles selected within the SSR with the SFH in the Solar Neighbor-hood. At redshift 1 < 𝑧 <
2, we detected an apparent signature of thestarburst above discussed, which matches quite well the high SFR( ∼
10 M (cid:12)
Gyr − pc − ) recently derived at the same epoch by Moret al. (2019). The SFR in the SSR is fairly similar to that derived forthe MW by Mor et al. (2019) till 𝑧 (cid:39) .
2. The comparison with theaccretion history of AqC4 supports the hypothesis that the starburstthat occurred 2 – 4 Gyr ago in the Solar Neighbourhood may be dueto a late merging event, as claimed for the MW at 𝑧 ∼ . (cid:101) 𝑉 𝑅 trends represents an interesting start-ing point for future studies that will have to include chemo-dynamical analysis in order to improve and detail our knowledgeof the MW thin and thick discs origin and evolution, as well as in-vestigations on the in-situ/ex-situ star formation contributions andstellar back-time tracking in order to select halo streams and identifycommon progenitors to study the Galactic halo. In this respect, theuse of constrained simulation (e.g. Carlesi et al. 2016) can be verypromising.There is no doubt that current and future Gaia data releases andthe important synergies with spectroscopic ground surveys such asAPOGEE (Majewski et al. 2017) and GALAH (De Silva et al. 2015)and detailed comparison with simulations will bring tremendousand fundamental contributions to the studies of Galactic Archaeol-ogy. ACKNOWLEDGEMENTS
We would like to thank the anonymous referee and the AssistantEditor for their thoughtful comments and suggestions that helped usimprove our manuscript. We thank also Eloisa Poggio and RonaldDrimmel for the useful discussions.Simulation was carried out using ULISSE at SISSA and Mar-coni at CINECA, Italy (project IsB16_DSKAGN, PI:G. Murante).The post-processing has been performed using the PICO HPC clus-ter at CINECA through our expression of interest. We thank VolkerSpringel for making the GADGET3 code available to us. This re-search made use of python libraries scipy (Virtanen et al. 2020),corner (Foreman-Mackey 2016), and PyMC3 (Salvatier et al. 2016).We are indebted to the Italian Space Agency (ASI) for their con-tinuing support through contract 2018-24-HH.0 to the National In-stitute for Astrophysics (INAF). MV is supported by the ExcellenceCluster ORIGINS, which is funded by the Deutsche Forschungsge-meinschaft (DFG, German Research Foundation) under Germany’sExcellence Strategy - EXC-2094 - 390783311.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable requestto the corresponding author.
REFERENCES
Abadi M. G., Navarro J. F., Steinmetz M., Eke V. R., 2003a, ApJ, 591, 499Abadi M. G., Navarro J. F., Steinmetz M., Eke V. R., 2003b, ApJ, 597, 21Antoja T., et al., 2018, Nature, 561, 360Bell E. F., Monachesi A., Harmsen B., de Jong R. S., Bailin J., Radburn-Smith D. J., D’Souza R., Holwerda B. W., 2017, ApJ, 837, L8Belokurov V., Erkal D., Evans N. W., Koposov S. E., Deason A. J., 2018,MNRAS, 478, 611Bignone L. A., Helmi A., Tissera P. B., 2019, ApJ, 883, L5Bland-Hawthorn J., Gerhard O., 2016, ARA&A, 54, 529Blitz L., Rosolowsky E., 2006, ApJ, 650, 933Bond N. A., et al., 2010, ApJ, 716, 1Bovy J., Bird J. C., García Pérez A. E., Majewski S. R., Nidever D. L.,Zasowski G., 2015, ApJ, 800, 83Bovy J., Rix H.-W., Schlafly E. F., Nidever D. L., Holtzman J. A., ShetroneM., Beers T. C., 2016, ApJ, 823, 30Brook C. B., Kawata D., Gibson B. K., Freeman K. C., 2004, ApJ, 612, 894Brook C. B., et al., 2012, MNRAS, 426, 690Buck T., Macciò A. V., Dutton A. A., Obreja A., Frings J., 2019, MNRAS,483, 1314Buck T., Obreja A., Macciò A. V., Minchev I., Dutton A. A., Ostriker J. P.,2020, MNRAS, 491, 3461Carlesi E., et al., 2016, MNRAS, 458, 900Crosta M., Giammaria M., Lattanzi M. G., Poggio E., 2020, MNRAS, 496,2107De Silva G. M., et al., 2015, MNRAS, 449, 2604Di Matteo P., Haywood M., Lehnert M. D., Katz D., Khoperskov S., SnaithO. N., Gómez A., Robichon N., 2019, A&A, 632, A4Eilers A.-C., Hogg D. W., Rix H.-W., Ness M. K., 2019, ApJ, 871, 120Fattahi A., et al., 2019, MNRAS, 484, 4471Foreman-Mackey D., 2016, The Journal of Open Source Software, 1, 24Freeman K., Bland-Hawthorn J., 2002, ARA&A, 40, 487Gaia Collaboration et al., 2018a, A&A, 616, A1Gaia Collaboration et al., 2018b, A&A, 616, A11Gallart C., Bernard E. J., Brook C. B., Ruiz-Lara T., Cassisi S., Hill V.,Monelli M., 2019, Nature Astronomy, 3, 932González Delgado R. M., et al., 2014, A&A, 562, A47Goz D., Monaco P., Murante G., Curir A., 2015, Monthly Notices of theRoyal Astronomical Society, 447, 1774Grand R. J. J., et al., 2017, MNRAS, 467, 179Grand R. J. J., et al., 2020, MNRAS, 497, 1603Gravity Collaboration et al., 2018, A&A, 615, L15Guedes J., Callegari S., Madau P., Mayer L., 2011, ApJ, 742, 76Haardt F., Madau P., 2001, in Neumann D. M., Tran J. T. V., eds, Clustersof Galaxies and the High Redshift Universe Observed in X-rays. CEA,Saclay, p.64. ( arXiv:astro-ph/0106018 )Han D. R., Lee Y. S., Kim Y. K., Beers T. C., 2020, ApJ, 896, 14Helmi A., 2020, arXiv e-prints, p. arXiv:2002.04340Helmi A., Babusiaux C., Koppelman H. H., Massari D., Veljanoski J., BrownA. G. A., 2018, Nature, 563, 85Iocco F., Pato M., Bertone G., 2015, Nature Physics, 11, 245Jurić M., et al., 2008, ApJ, 673, 864Just A., Jahreiß H., 2010, MNRAS, 402, 461Kroupa P., Tout C. A., Gilmore G., 1993, MNRAS, 262, 545Lian J., et al., 2020, MNRAS, 494, 2561Licquia T. C., Newman J. A., 2015, ApJ, 806, 96Ma X., Hopkins P. F., Wetzel A. R., Kirby E. N., Anglés-Alcázar D., Faucher-Giguère C.-A., Kereš D., Quataert E., 2017, MNRAS, 467, 2430Mackereth J. T., et al., 2019, MNRAS, 482, 3426Majewski S. R., et al., 2017, AJ, 154, 94McMillan P. J., 2017, MNRAS, 465, 76 MNRAS000
Abadi M. G., Navarro J. F., Steinmetz M., Eke V. R., 2003a, ApJ, 591, 499Abadi M. G., Navarro J. F., Steinmetz M., Eke V. R., 2003b, ApJ, 597, 21Antoja T., et al., 2018, Nature, 561, 360Bell E. F., Monachesi A., Harmsen B., de Jong R. S., Bailin J., Radburn-Smith D. J., D’Souza R., Holwerda B. W., 2017, ApJ, 837, L8Belokurov V., Erkal D., Evans N. W., Koposov S. E., Deason A. J., 2018,MNRAS, 478, 611Bignone L. A., Helmi A., Tissera P. B., 2019, ApJ, 883, L5Bland-Hawthorn J., Gerhard O., 2016, ARA&A, 54, 529Blitz L., Rosolowsky E., 2006, ApJ, 650, 933Bond N. A., et al., 2010, ApJ, 716, 1Bovy J., Bird J. C., García Pérez A. E., Majewski S. R., Nidever D. L.,Zasowski G., 2015, ApJ, 800, 83Bovy J., Rix H.-W., Schlafly E. F., Nidever D. L., Holtzman J. A., ShetroneM., Beers T. C., 2016, ApJ, 823, 30Brook C. B., Kawata D., Gibson B. K., Freeman K. C., 2004, ApJ, 612, 894Brook C. B., et al., 2012, MNRAS, 426, 690Buck T., Macciò A. V., Dutton A. A., Obreja A., Frings J., 2019, MNRAS,483, 1314Buck T., Obreja A., Macciò A. V., Minchev I., Dutton A. A., Ostriker J. P.,2020, MNRAS, 491, 3461Carlesi E., et al., 2016, MNRAS, 458, 900Crosta M., Giammaria M., Lattanzi M. G., Poggio E., 2020, MNRAS, 496,2107De Silva G. M., et al., 2015, MNRAS, 449, 2604Di Matteo P., Haywood M., Lehnert M. D., Katz D., Khoperskov S., SnaithO. N., Gómez A., Robichon N., 2019, A&A, 632, A4Eilers A.-C., Hogg D. W., Rix H.-W., Ness M. K., 2019, ApJ, 871, 120Fattahi A., et al., 2019, MNRAS, 484, 4471Foreman-Mackey D., 2016, The Journal of Open Source Software, 1, 24Freeman K., Bland-Hawthorn J., 2002, ARA&A, 40, 487Gaia Collaboration et al., 2018a, A&A, 616, A1Gaia Collaboration et al., 2018b, A&A, 616, A11Gallart C., Bernard E. J., Brook C. B., Ruiz-Lara T., Cassisi S., Hill V.,Monelli M., 2019, Nature Astronomy, 3, 932González Delgado R. M., et al., 2014, A&A, 562, A47Goz D., Monaco P., Murante G., Curir A., 2015, Monthly Notices of theRoyal Astronomical Society, 447, 1774Grand R. J. J., et al., 2017, MNRAS, 467, 179Grand R. J. J., et al., 2020, MNRAS, 497, 1603Gravity Collaboration et al., 2018, A&A, 615, L15Guedes J., Callegari S., Madau P., Mayer L., 2011, ApJ, 742, 76Haardt F., Madau P., 2001, in Neumann D. M., Tran J. T. V., eds, Clustersof Galaxies and the High Redshift Universe Observed in X-rays. CEA,Saclay, p.64. ( arXiv:astro-ph/0106018 )Han D. R., Lee Y. S., Kim Y. K., Beers T. C., 2020, ApJ, 896, 14Helmi A., 2020, arXiv e-prints, p. arXiv:2002.04340Helmi A., Babusiaux C., Koppelman H. H., Massari D., Veljanoski J., BrownA. G. A., 2018, Nature, 563, 85Iocco F., Pato M., Bertone G., 2015, Nature Physics, 11, 245Jurić M., et al., 2008, ApJ, 673, 864Just A., Jahreiß H., 2010, MNRAS, 402, 461Kroupa P., Tout C. A., Gilmore G., 1993, MNRAS, 262, 545Lian J., et al., 2020, MNRAS, 494, 2561Licquia T. C., Newman J. A., 2015, ApJ, 806, 96Ma X., Hopkins P. F., Wetzel A. R., Kirby E. N., Anglés-Alcázar D., Faucher-Giguère C.-A., Kereš D., Quataert E., 2017, MNRAS, 467, 2430Mackereth J. T., et al., 2019, MNRAS, 482, 3426Majewski S. R., et al., 2017, AJ, 154, 94McMillan P. J., 2017, MNRAS, 465, 76 MNRAS000 , 1–15 (2020) ormation History of the Milky Way disc Minchev I., Chiappini C., Martig M., 2013, A&A, 558, A9Minchev I., Steinmetz M., Chiappini C., Martig M., Anders F., MatijevicG., de Jong R. S., 2017, ApJ, 834, 27Monaco P., Murante G., Borgani S., Dolag K., 2012, Monthly Notices of theRoyal Astronomical Society, 421, 2485Mor R., Robin A. C., Figueras F., Roca-Fàbrega S., Luri X., 2019, A&A,624, L1Muñoz-Mateos J. C., Boissier S., Gil de Paz A., Zamorano J., KennicuttR. C. J., Moustakas J., Prantzos N., Gallego J., 2011, ApJ, 731, 10Murante G., Monaco P., Giovalli M., Borgani S., Diaferio A., 2010, MNRAS,405, 1491Murante G., Monaco P., Borgani S., Tornatore L., Dolag K., Goz D., 2015,MNRAS, 447, 178Nelson D., et al., 2018, MNRAS, 475, 624Obreja A., Macciò A. V., Moster B., Dutton A. A., Buck T., Stinson G. S.,Wang L., 2018, MNRAS, 477, 4915Obreja A., et al., 2019, MNRAS, 487, 4424Padovani P., Matteucci F., 1993, ApJ, 416, 26Poggio E., et al., 2018, MNRAS, 481, L21Posti L., Helmi A., 2019, A&A, 621, A56Re Fiorentin P., Lattanzi M. G., Spagna A., Curir A., 2015, AJ, 150, 128Robitaille T. P., Whitney B. A., 2010, ApJ, 710, L11Salvatier J., Wiecki T. V., Fonnesbeck C., 2016, PeerJ Computer Science, 2,e55Sawala T., et al., 2016, MNRAS, 457, 1931Scannapieco C., Gadotti D. A., Jonsson P., White S. D. M., 2010, MNRAS,407, L41Scannapieco C., et al., 2012, Monthly Notices of the Royal AstronomicalSociety, 423, 1726Sofue Y., Rubin V., 2001, ARA&A, 39, 137Spagna A., Lattanzi M. G., Re Fiorentin P., Smart R. L., 2010, A&A, 510,L4Springel V., 2005, MNRAS, 364, 1105Springel V., Hernquist L., 2003, MNRAS, 339, 289Springel V., et al., 2008, Monthly Notices of the Royal Astronomical Society,391, 1685Stinson G. S., et al., 2013, MNRAS, 436, 625Tornatore L., Borgani S., Dolag K., Matteucci F., 2007, MNRAS, 382, 1050Valentini M., Murante G., Borgani S., Monaco P., Bressan A., Beck A. M.,2017, MNRAS, 470, 3167Valentini M., Bressan A., Borgani S., Murante G., Girardi L., Tornatore L.,2018, preprint, ( arXiv:1805.00028 )Valentini M., Borgani S., Bressan A., Murante G., Tornatore L., Monaco P.,2019, MNRAS, 485, 1384Valentini M., et al., 2020, MNRAS, 491, 2779Vincenzo F., Spitoni E., Calura F., Matteucci F., Silva Aguirre V., MiglioA., Cescutti G., 2019, MNRAS, 487, L47Virtanen P., et al., 2020, Nature Methods, 17, 261Watkins L. L., van der Marel R. P., Sohn S. T., Evans N. W., 2019, ApJ, 873,118Wiersma R. P. C., Schaye J., Smith B. D., 2009, MNRAS, 393, 99de Salas P. F., Malhan K., Freese K., Hattori K., Valluri M., 2019, J. Cos-mology Astropart. Phys., 2019, 037
APPENDIX A: POSTERIORS PDFA1 Vertical distribution of stellar disc
Fig. A1 shows the posterior distributions of the parameters accord-ing to Eq. (1) for the vertical distribution of stellar particles. Dashedlines in each histogram refer to the 10 th , 16 th , 50 th (i.e. median),84 th and 90 th percentiles of the relative distribution, while numberson top indicate the medians and the 1 𝜎 CIs. Thick black contoursindicate the 1 and 2 𝜎 CI of the two-dimensional correlations of theposteriors.As for the radial scale length, the uncertainties take into account
Figure A1.
Posterior distributions of the MCMC analysis for the verticalstellar particles distribution in the SSR. The one-dimensional (histogram)posterior distributions for each parameter are shown on the diagonal, whilethe other panels represent the two-dimensional (contours) correlations. the Poissonian statistic within each bin and are smaller than the sizeof single data points in the plot. Consequently, the CIs are verynarrow and we obtain well-peaked posteriors on the parametersestimates.The analysis highlights that ℎ and ℎ are positively correlated,and they are both negatively correlated with the local density nor-malization parameter 𝑓 indicating the intrinsic physical overlappingof the two disc components. The closest correlation is between ℎ and 𝑓 . A2 TNM model
The posterior distributions of the parameters for TNM model areshown in Fig. A2. The means of the posteriors are indicated with ablue square, while dashed black lines and numbers on top of eachhistogram have the same meaning as in Appendix A1. The posteriorsare well approximated by normal distribution, as the means and themedians are similar and the CI are pretty symmetric.The closest correlations are between the arrays of parametersthat describe the two discs components, i.e. the components with 𝑖 = ,
2. This is again a signature of the intrinsic overlap of the stellargenerations that compose the two populations and is similar to whatreported in Appendix A1. The last kinematic component with 𝑖 = This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS , 1–15 (2020) Giammaria et al.
Figure A2.
As in Fig. A1 for TNM model parameters according to Eq. 2. The blue square shows the mean value of each posterior distribution.MNRAS000