The Origin of Elements from Carbon to Uranium

Often GCE model predictions directly come from the input stellar physics and nucleosynthesis yields. Based on recent developments in stellar astrophysics, we summarize the chemical enrichment sources that are chosen to be included in this section.

2.1.1. AGB Stars and Core-collapse Supernovae

Stellar winds—All dying stars return a fraction or all of their envelope mass to the interstellar medium (ISM) by stellar winds. These winds (for massive stars occurring before the final supernova explosions) carry newly processed metals and the unprocessed metals that were trapped inside the star at its formation and return them to the ISM. Usually, both the processed and unprocessed components are included in the nucleosynthesis yield table of AGB stars, while only the former is included for supernova yields (see Equation (9) in K00), and the latter is added in the GCE models using the abundance pattern of the ISM at the time when the stars formed (Equation (8) in K00). The wind mass is given by ${M}_{\mathrm{wind}}={M}_{\mathrm{init}}-{M}_{\mathrm{remnant}}-{{\rm{\Sigma }}}_{i}\,{p}_{{z}_{i}m}$, for the initial mass M_init, the mass of remnant M_remnant, i.e., black hole (BH), neutron star (NS), or WD mass. Nucleosynthesis yields, ${p}_{{z}_{i}m}$, of an element/isotope i are given in the yield tables (see below for more details). For stars with initial masses of 0.7 and 0.9M_⊙, the He core mass is set as M_remnant = 0.459 and 0.473M_⊙, respectively, and ${p}_{{z}_{i}m}=0$, as in K06/K11.

AGB stars—Stars with initial masses between roughly 0.9–8M_⊙ (depending on metallicity) pass through the thermally pulsing AGB phase. The He-burning shell is thermally unstable and can drive mixing of material from the core into the envelope, which has been processed by nuclear reactions. This mixing is known as third dredge-up (TDU), and is responsible for enriching the surface in ¹²C and other products of He-burning, as well as s-process elements. In AGB stars with initial masses ≳4 ${M}_{\odot }$, the base of the convective envelope becomes hot enough to sustain proton-capture nucleosynthesis (hot bottom burning, HBB). HBB can change the surface composition because the entire envelope is exposed to the hot burning region a few thousand times per interpulse period. The CNO cycles operate to convert the freshly synthesized ¹²C into primary¹⁴N, and the NeNa and MgAl chains may also operate to produce ²³Na and Al.

At the deepest extent of each TDU, it is assumed that the bottom of the H-rich convective envelope penetrates into the ¹²C-rich intershell layer resulting in a partial mixing zone (PMZ) leading to the formation of a ¹³C pocket via the ¹²C(p,γ)¹³N(β⁺)¹³C reaction chain. While many physical processes have been proposed, there is still not full agreement on which process(es) drives the mixing. The inclusion of ¹³C pockets in theoretical calculations of AGB stars is still one of the most significant uncertainties affecting predictions of the s-process and in particular the absolute values of the yields (Karakas & Lugaro 2016; Buntain et al. 2017, and references therein). Other major uncertainties come from the rates of the neutron source reactions ¹³C(α, n)¹⁶O and ²²Ne(α, n)²⁵Mg (Bisterzo et al. 2015) and the neutron-capture cross sections of some key isotopes (Cescutti et al. 2018).

In this paper, we take the nucleosynthesis yields including s-process and WD masses primarily from Lugaro et al. (2012) for Z = 0.0001, Fishlock et al. (2014) for Z = 0.001, Karakas et al. (2018) for Z = 0.0028, and Karakas & Lugaro (2016) for Z = 0.007, 0.014, and 0.03. In these post-processing nucleosyntheses, protons are added to the top layers of the He-intershell at the deepest extent of each TDU episode by means of an artificial PMZ. The mass of the PMZ, i.e., how deep it reaches below the base of the convective envelope, is given by a free parameter M_mix as a function of mass and metallicity, as discussed in detail by Karakas & Lugaro (2016). In addition, for this paper we calculated some selected low-mass star models with Z = 0.014, 0.007, and 0.0028 using a smaller PMZ mass; namely these models set the PMZ mass to be 0.001M_⊙ compared to the standard size, 0.002M_⊙, used in Karakas & Lugaro (2016). The adopted PMZ mass of our fiducial model is summarized in Table 1. We also show a GCE model with the original yields of Karakas & Lugaro (2016) for Ba in this paper (Figure 36). Non-time-dependent overshoot, which essentially only affects the depth of the TDU and not the formation of the PMZ, is also included in some models with the parameter N_ov set to 1.0 for the 1.5 and 1.75M_⊙ models of Z = 0.007, to 3.0 and 2.0, respectively, for the 1.5 and 1.75M_⊙ models of Z = 0.014, and to 2.5, 2.0, and 1.0, respectively, for the 2.5, 2.75, and 3.0M_⊙ models of Z = 0.03. For the Z = 0.0028 models, overshoot was included in the 1.15 and 1.25 M_⊙ models where N_ov was set to 1.0 for both cases.

Table 1. Mass of PMZs, M_mix, Adopted for the AGB and Super-AGB Models as a Function of Initial Mass and Metallicity

Z	0.0001	0.001	0.0028	0.007	0.014	0.03
Mmix = 0	⋯	⋯	1M⊙	1–1.25M⊙	1–1.25M⊙	1–2.25M⊙
${M}_{\mathrm{mix}}=2\times {10}^{-3}$	0.9–2.25M⊙	1–2.5M⊙	⋯	⋯	⋯	⋯
Mmix = 1 × 10−3	2.5–3M⊙	2.75M⊙	1.15–2.75M⊙	1.5–3.75M⊙	1.5–4M⊙	2.5–4M⊙
Mmix = 5 × 10−4	⋯	3M⊙	3–3.5M⊙	⋯	⋯	⋯
Mmix = 1 × 10−4	⋯	⋯	3.75–4M⊙	4–4.25M⊙	4.25–5M⊙	4.25–5M⊙
Mmix = 0	3.5–6M⊙	3.25–7M⊙	4.5–7M⊙	4.5–7.5M⊙	5.5–8M⊙	5.5–8M⊙
Z	0.0001	0.001	0.004	0.008	0.02
Mmix = 0	6.5–7.5M⊙	7.5M⊙	7.5–8M⊙	8–8.5M⊙	8.5–9M⊙

Download table as:  ASCIITypeset image

In these yield tables, the mass of each element expelled over the stellar lifetime is listed, which contains the unprocessed metals. The newly produced metal yields, ${p}_{{z}_{i}m}$, are calculated as the difference between the amount of the species in the winds and the initial amount in the envelope of the progenitor star. The initial abundances are set as the scaled proto-solar abundances, which are calculated from Tables 1, 3, and 5 from Asplund et al. (2009); the meteoritic values are chosen if the errors are smaller than the photospheric values, and the proto-solar abundances are used for C, N, O, Ne, Mg, Si, S, Ar, and Fe. Therefore, ${p}_{{z}_{i}m}$ can have negative values especially for H, but after adding the unprocessed metals in the GCE, the mass of each element becomes positive. For Z = 0, the models of Z = 0.0001 are used, although the yields from Campbell & Lattanzio (2008) were adopted in K11. The upper and lower mass ranges of the AGB models can also be found in Table 1 as a function of initial metallicity. It is important to note that these AGB models successfully reproduce both the trend with metallicity observed in a large sample of Ba stars (Cseh et al. 2018), and the heavy element composition of meteoritic stardust silicon carbide (SiC) grains that formed around C-rich AGB stars (Lugaro et al. 2018).

Note that, however, when we discuss the model dependence for the elements up to Zn (namely, Figures 5, 6, 8, and 10), in order to make a fair comparison, we use the same AGB yields as in K11, i.e., the yields of Karakas (2010). There are no differences between this model with the yields of K11 and the fiducial model with s-process yields, except for C and N.

Super-AGB stars—The fate of stars with initial masses between about 8 and 10M_⊙ (at Z = 0.02) is uncertain (Doherty et al. 2017, for a review), and their contributions were not included in K11. The upper limit of AGB stars, ${M}_{\mathrm{up},{\rm{C}}}$, is defined as the minimum mass for carbon ignition, and is estimated to be larger at high metallicity and also for metallicities lower than Z ∼ 10⁻⁴ (Gil-Pons et al. 2007; Siess 2007). Just above ${M}_{\mathrm{up},{\rm{C}}}$, neutrino cooling and contraction leads to the off-center ignition of a C flame, which moves inward but does not propagate to the center. This may form a hybrid C + O+Ne WD (see Section 2.2 of Kobayashi et al. 2015, for more details). These hybrid WDs can be progenitors of a sub-class of SNe Ia, called SNe Iax (Foley et al. 2013), which are expected preferably in dwarf galaxies (Kobayashi et al. 2015; Cescutti & Kobayashi 2017). We take the nucleosynthesis yields of an SN Iax from Fink et al. (2014).

Above this mass range, the off-center C ignition moves inward all the way to the center ($\lesssim 9{M}_{\odot }$), or stars undergo central carbon ignition ($\gtrsim 9{M}_{\odot }$). For both cases, a strongly degenerate O+Ne + Mg core is formed (O + Ne dominant, but Mg is essential for electron capture). If the stellar envelope is lost by winds or binary interaction, an O+Ne+Mg WD may be formed. This upper mass limit is defined as the minimum mass for the Ne ignition, ${M}_{\mathrm{up},\mathrm{Ne}}\sim 9\pm 1{M}_{\odot }$, and is smaller for lower metallicities (e.g., Siess 2007; Doherty et al. 2015).

Stars with ${M}_{\mathrm{up},\mathrm{Ne}}\lt M\lt 10{M}_{\odot }$ may have cores as massive as $\gtrsim 1.35{M}_{\odot }$ and ignite Ne off-center. If Ne burning is not ignited at the center (for a core mass $\lt 1.37{M}_{\odot }$; Nomoto 1984), or if off-center Ne burning does not propagate to the center ($M=8.8{M}_{\odot }$; Jones et al. 2014), it has been believed that such a core eventually undergoes an electron-capture-induced collapse. ECSNe (see Section 2.3.2 of Nomoto et al. 2013, for more details) are one of the candidate r-process sites (Section 1, see Section 2.1.2 for more details). Note that recent 3D simulations showed that the fate of the O+Ne+Mg core may depend on the density, and the explosion may result in thermonuclear disruption leaving behind an O+Ne+Fe WD instead (Jones et al. 2016).

In this paper, the mass ranges of the C+O+Ne WDs, O+Ne+Mg WDs, and ECSNe are taken from Doherty et al. (2015). Note that Ne burning is not followed in Doherty et al. (2015); the lower limit of ECSNe is defined with the temperature ∼1.2 × 10⁹ K, and the upper limit is defined with the core mass $=1.375{M}_{\odot }$ at the end of C burning. These may underestimate the ECSN rate. We also note that these mass ranges are highly affected by convective overshooting, mass-loss, and reaction rates, as well as binary effects, and some of the important physics, such as the URCA process (Jones et al. 2014), are also not included. There is no region where the core mass is larger than the Chandrasekhar (Ch) mass limit in the models considered by Doherty et al. (2015); if there were, the stars could explode as so-called Type 1.5 SNe, although no signature of such supernovae has yet been observed (K06). In previous GCE models (e.g., Cescutti & Chiappini 2014), a much larger mass range was adopted; for example, if $8\mbox{--}10{M}_{\odot }$, the ECSN rate is ∼8–18 times larger than in our models depending on the metallicity.

The nucleosynthesis yields (up to and including Ni) of super-AGB stars are taken from Doherty et al. (2014a, 2014b); the available models are 6.5–7.5M_⊙, 6.5–7.5M_⊙, 6.5–8M_⊙, 6.5–8M_⊙, and 7–9M_⊙, respectively, for Z = 0.0001, 0.001, 0.004, 0.008, and 0.02, and we use these at the masses where Karakas's yields are not available (Table 1). The initial abundances are the scaled solar abundances from Grevesse et al. (1996).

Core-collapse supernovae—Although a few groups have presented multidimensional simulations of exploding 10–25M_⊙ stars (Marek & Janka 2009; Kotake et al. 2012; Bruenn et al. 2013; Burrows 2013), the explosion mechanism of core-collapse (Type II, Ib, and Ic) supernovae is still uncertain; the ejected iron mass in explosion simulations is not as large as observed (Bruenn et al. 2016), and the formation of black holes is also not followed in most cases, except for in Kuroda et al. (2018). Therefore, we use the nucleosynthesis yields from 1D calculations of K06/K11. ⁸

Similar 1D nucleosynthesis yields of massive stars and supernovae have been provided by three different groups (Woosley & Weaver 1995; Nomoto et al. 1997a; Limongi et al. 2003, see Figure 5 of Nomoto et al. 2013 for the comparison) and are constantly being updated (e.g., Kobayashi et al. 2006; Heger & Woosley 2010; Limongi & Chieffi 2018; see also Ritter et al. 2018b). The uncertainties include the reaction rates (namely, of ¹²C(α, γ)¹⁶O), mixing in stellar interiors, rotationally induced mixing processes, and mass loss via stellar winds, which affect the yields of elements/isotopes formed during hydrostatic burning. Furthermore, the most important uncertainty in the yields is associated with the formation of remnants (i.e., NSs or BHs) in massive stars, and different methods have been used to address this problem. The iron mass of Woosley & Weaver (1995) is known to be too large, and is usually reduced by a factor of two or three (e.g., Romano et al. 2010), but this modification causes an inconsistency for the other iron-peak elements, which are formed in the same layer as iron and should be reduced by the same amount. In Nomoto et al. (1997a), the remnant masses were determined from one parameter, the mass cut, which self-consistently determined the yields of iron-peak elements as well. As shown in multidimensional simulations (e.g., Janka 2012; Bruenn et al. 2016), remnant formation is not well described by the mass cut, and the material around the boundary is mixed and partially ejected or falls back onto the remnant. To mimic these phenomena in 1D calculations, the mixing-fallback model was introduced by Umeda & Nomoto (2002).

As in K06, the ejected explosion energy and ⁵⁶Ni mass (which decays to ⁵⁶Fe) are determined to meet an independent observational constraint: the light curves and spectral fitting of individual supernova (Nomoto et al. 2001, 2013). As a result it is found that many core-collapse supernovae (M ≥ 20M_⊙) have an explosion energy that is more than 10 times that of a regular supernova (${E}_{51}\equiv E/{10}^{51}$ erg ≳10), and they produce more iron and α elements. These are called hypernovae (HNe), while all other supernovae with E₅₁ = 1 are referred to as SNe II. The nucleosynthesis yields are provided separately for SNe II and HNe as a function of the progenitor mass (M = 13, 15, 18, 20, 25, 30, and 40M_⊙) and metallicity (Z = 0, 0.001, 0.004, 0.02, and 0.05). As mentioned above, the yield tables provide the amount of processed metals (${p}_{{z}_{i}m}$) in the ejecta (in M_⊙), and the unprocessed metals are added in the GCE. The fraction of HNe at any given time is uncertain and is set as _HN = 0.5 for masses $M\geqslant 20{M}_{\odot }$ following previous works (K06/K11), while a metal-dependent fraction _HN = 0.5, 0.5, 0.4, 0.01, and 0.01 for Z = 0, 0.001, 0.004, 0.02, and 0.05 was introduced in Kobayashi & Nakasato (2011) in order to match the observed rate of broad-line SNe Ibc at the present day (e.g., Podsiadlowski et al. 2004). The metallicity-dependent HN fraction is also tested in this paper.

It is known that multidimensional effects are particularly important for some elements, e.g., Sc, V, Ti, and Co (Maeda & Nomoto 2003; Tominaga 2009). We calculated the K15 GCE model, which is plotted in Sneden et al. (2016), Zhao et al. (2016), and Reggiani et al. (2017), applying constant factors, +1.0, 0.45, 0.3, 0.2, and 0.2 dex for [(Sc, Ti, V, Co, and ⁶⁴Zn)/Fe] yields, respectively, which take the 2D jet effects of HNe into account. We also show this K15 model for some elements in this paper.

Stellar rotation induces mixing of C into the H-burning shell, producing a large amount of primary nitrogen, which is mixed back into the He-burning shell (Meynet & Maeder 2002; Hirschi 2007). For high initial rotational velocities at low metallicity ("spin stars"), this process results in the production of s-process elements, even at low metallicities (Frischknecht et al. 2016; Choplin et al. 2018; Limongi & Chieffi 2018). Chiappini et al. (2006) showed that rotation is necessary to explain the observed N/O–O/H relations with a GCE model, and the same result was shown in Figure 13 of K11. However, using more self-consistent cosmological simulations, Vincenzo & Kobayashi (2018b) reproduced the observed relation not with rotation but with inhomogeneous enrichment from AGB stars. Therefore, we do not include yields from rotating massive stars in this paper. Prantzos et al. (2018) showed a GCE model assuming a metallicity-dependent function of the rotational velocities, and concluded that because of the contribution from rotating massive stars, a further light element primary process (LEPP) is not necessary to explain the elemental abundances with A < 100. In the following sections, we will show that we do not need to include fast rotating stars for these elements since other sources are present in our models.

Failed supernovae—The upper limit of SNe II supernovae, ${M}_{{\rm{u}},2}$, is not well known, owing to uncertainties in the physics of black hole formation, and was set as ${M}_{{\rm{u}},2}=50{M}_{\odot }$ in K06/K11, which is the same for HNe. However, recently it has been questioned if massive SNe II can explode or not, both observationally and theoretically. In searching for the progenitor stars at the locations of nearby SNe II-P, no progenitor stars have been found with initial masses M > 30M_⊙ (Smartt 2009). In multidimensional simulations of supernova explosions, it seems very difficult to explode stars ≳25M_⊙ with the neutrino mechanism (e.g., Janka 2012), and similar results are obtained with parameterized 1D models (Ugliano et al. 2012; Müller et al. 2016). At lower metallicities, since the stellar cores become more compact, they might be even harder to explode. However, the metallicity dependence is probably not very straightforward and may be non-monotonic (Pejcha & Thompson 2015).

Therefore, in this paper, we include new nucleosynthesis yields of "failed" supernovae (C. Kobayashi & N. Tominaga 2020, in preparation) at the massive end of SNe II, while keeping the contributions from HNe. It is assumed that all CO cores fall onto black holes and are not ejected into the ISM, since the timescales of the multidimensional simulations are not long enough to follow this process. The upper mass limit of SNe II, ${M}_{{\rm{u}},2}$, is treated as a free parameter, while the upper mass limit of HNe is the same as the upper limit of the initial mass function (IMF), M_u. In our fiducial model, ${M}_{{\rm{u}},2}=30{M}_{\odot }$ and ${M}_{{\rm{u}}}=50{M}_{\odot }$ are adopted (at $\gt 30{M}_{\odot }$, the yields are interpolated between the values at 30M_⊙ and 0 at 40M_⊙). If we assume all ejecta collapse onto black holes, the evolution of C and N is slightly different, but there is no significant difference in the evolution of heavier elements.

Confusingly, failed supernovae are not related to faint supernovae (Nomoto et al. 2013; Ishigaki et al. 2014, 2018), which are suggested by completely different observational results. At [Fe/H] ≲  −2.5, a large fraction of stars are carbon enhanced relative to iron (CEMP stars, [C/Fe]   >  0.7 in Aoki et al. 2007, but see Beers & Christlieb 2005 for a different definition), increasing the fraction toward lower metallicities (e.g., Placco et al. 2014). CEMP stars with an s-process enrichment (CEMP-s, [Ba/Fe]  >  1) are well explained by the binary mass transfer from AGB stars, while CEMP stars with no s-process enhancement (CEMP-no stars, [Ba/Fe] < 0) are observed to be both single and binaries (Hansen et al. 2016), and several scenarios are suggested (see Nomoto et al. 2013 and the references therein). Faint supernovae are core-collapse supernovae from massive (≳13M_⊙) stars possibly only at Z = 0, with normal or large explosion energies (i.e., faint SNe or faint HNe) that leave relatively large black holes and eject C-rich envelopes. Because of the small ejecta mass, the contribution to GCE is negligible, and thus we do not include the yields of faint supernovae and exclude CEMP stars from most of the figures in this paper.

Pair-instability supernovae—Stars with 100M_⊙ ≲ M ≲ 300M_⊙ encounter the electron–positron pair instability and do not reach the temperature of iron photodisintegration. Pair-instability supernovae (PISNe) are predicted to produce a large amount of metals such as S and Fe (Barkat et al. 1967; Heger & Woosley 2002; Umeda & Nomoto 2002). Despite searching for many years, no conclusive signature of PISNe has been detected in metal-poor stars in the solar neighborhood (Umeda & Nomoto 2002; Cayrel et al. 2004; Keller et al. 2014), the bulge (Howes et al. 2015), or in the metal-poor damped Lyα system (Kobayashi et al. 2010). Therefore, we do not include PISNe in this paper.

Table 2 summarizes the possible necessary conditions for the different types of core-collapse supernovae discussed in this and the next subsection. Note that these are very uncertain, and should be investigated with 3D/GR/MHD simulations of supernova explosions.

Table 2. Mass Ranges of Core-collapse Supernovae Used in Our Fiducial GCE Model, and Necessary Conditions for the Explosions

	Stellar mass (M⊙)	Rotation	Magnetic field
ECSN	∼8.8–9	no	no
SNII/Ibc	10–30	no	no
failed SN	30–50	no	no
HN	20–50	yes	weak?
MRSN	25–50	yes	strong

Note. See the text for the details.

Download table as:  ASCIITypeset image

2.1.2. Sites for Rapid Neutron-capture Processes

2.1.3. SNe Ia

2.2.1. Basic Equations and Constants

The basic equations of chemical evolution are described in K00. The code follows the time evolution of elemental and isotopic abundances in a system where the ISM is instantaneously well mixed (and thus it is called a one-zone model). No instantaneous recycling approximation is adopted, and chemical enrichment sources with long time delays (Section 2.1) are properly included.

We adopt the IMF from Kroupa (2008), which is a power-law mass spectrum $\phi (m)\propto {m}^{-x}$ with three slopes at different mass ranges: x = 1.3 for $0.5{M}_{\odot }\leqslant m\leqslant 50{M}_{\odot }$, x = 0.3 for $0.08{M}_{\odot }\leqslant m\leqslant 0.5{M}_{\odot }$, and x = −0.7 for $0.01{M}_{\odot }\,\leqslant m\,\leqslant 0.08{M}_{\odot }$, which are the same as the canonical stellar IMF in Kroupa (2001) and very similar to the Chabrier (2003) IMF. ¹⁰ The IMF is normalized to unity at $0.01{M}_{\odot }\leqslant m\leqslant {M}_{{\rm{u}}}$, and ${M}_{{\rm{u}}}=50{M}_{\odot }$ is adopted in the fiducial model of this paper. Table 3 shows the IMF-weighted return fractions and yields of core-collapse supernovae for the models we show in the following sections. The net yields are defined as $1/(1-R)y\,=1-2{Z}_{\odot }$ (Tinsley 1980).

Table 3. IMF-weighted Return Fractions and Yields of Core-collapse Supernovae for the Models in Figure 1 for Different Upper Mass Limits of the IMF (M_u) and SNe II (M_u,2)

	Mu $({M}_{\odot })$	${M}_{{\rm{u}},2}$ $({M}_{\odot })$	R(Z = 0)	R(0.02)	y(0)	y(0.02)
K20	50	30	0.43	0.51	0.017	0.015
K20 no failed SNe	40	40	0.43	0.50	0.018	0.015
K20 metal-dependent HN	50	30	0.43	0.50	0.017	0.012
VK18	50	25, Z ≥ Z⊙	0.44	0.50	0.021	0.006
K11	50	50	0.43	0.51	0.021	0.019

Download table as:  ASCIITypeset image

The metallicity-dependent MS lifetimes are taken from Kodama & Arimoto (1997) for $0.6-80{M}_{\odot }$, which are calculated with the stellar evolution code described in Iwamoto & Saio (1999). These are in excellent agreement with the lifetimes in Karakas (2010) for low- and intermediate-mass stars.

We use similar models as K06 for the star formation histories of the solar neighborhood, halo, bulge, and thick disk, but with the Kroupa IMF. The formation of these components is not simple and is affected by dynamical effects such as radial migration and satellite accretion (see Athanassoula & Misiriotis 2002; Zoccali et al. 2017; Barbuy et al. 2018 for the bulge, and Minchev et al. 2012; Grisoni et al. 2017; Spitoni et al. 2019 for the thick disk). In this paper, in order to give rough evolutionary tracks, we adopt one-zone models.

The gas fraction and the metallicity of the system evolve as a function of time as a consequence of star formation, as well as inflow and outflow of matter to/from the outside of the system. The star formation rate (SFR) is assumed to be proportional to the gas fraction; $\tfrac{1}{{\tau }_{{\rm{s}}}}{f}_{{\rm{g}}}$. The infall of primordial gas from the outside of the system is given by the rate $\propto t\exp [-\tfrac{t}{{\tau }_{{\rm{i}}}}]$ for the solar neighborhood (Pagel 1997), and $\tfrac{1}{{\tau }_{{\rm{i}}}}\exp (-\tfrac{t}{{\tau }_{{\rm{i}}}})$ for the other systems. In addition, during the starburst at the early stages of galaxy formation (such as in the bulge) or the star formation in the shallow gravitational potential well (such as in the halo), outflow is an important process. The driving source of the outflow is the feedback from supernovae, and hence the outflow rate is also proportional to the gas fraction; $\tfrac{1}{{\tau }_{{\rm{o}}}}{f}_{{\rm{g}}}$. The outflow also removes some metals with the composition of the average ISM in the system at the time. The outflow gas could later fall onto the disk (so-called "fountain"), but this process is not included in the model. For the bulge and thick disk, star formation may be quenched suddenly by a galactic wind at a given epoch (t = t_w), which is driven by a large number of supernovae or by a central super-massive black hole. The adopted GCE model parameters are summarized in Table 4.

Table 4. Parameters of the GCE Models: Timescales for Infall (τ_i), Star Formation (τ_s), and Outflow (τ_o), and the Galactic Wind Epoch τ_w, All in Gyr

	τi	τs	τo	τw
solar neighborhood	5	4.7	⋯	⋯
halo	⋯	15	1	⋯
bulge	5	0.2	⋯	3
thick disk	5	2.2	⋯	3
halo, outflow	⋯	5	0.3	⋯
bulge, outflow	5	0.1	0.3	⋯

Download table as:  ASCIITypeset image

Compared to K06 and K11, one of the major revisions here is the adopted solar abundances, which are now taken from Tables 1 and 3 from Asplund et al. (2009, hereafter AGSS09) for all elements except for O, Th, and U. For most elements (except for Li and Pb), we use the photospheric value; when that is not available, we adopt the meteoritic values. For O we adopt the oxygen abundance, ${A}_{\odot }({\rm{O}})=8.76\pm 0.02$, from Steffen et al. (2015), which is higher than ${A}_{\odot }({\rm{O}})\,=8.69\pm 0.05$ in AGSS09. Note that in K06 and K11, the solar abundances were taken from Anders & Grevesse (1989, hereafter AG89), where the oxygen abundance was A_⊙(O) =8.93. There is no difference in the solar Fe abundance (A_⊙(Fe) = 7.51 in AG89 and 7.50 in AGSS09). Thus the [O/Fe] ratios in this paper appear to be 1.5 times larger than those in K06 and K11. This difference appears only in the comparison to observational data, and affects the choice of GCE model parameters. For Th and U, we adopt the initial solar system values ${A}_{\odot }(\mathrm{Th})=0.22$ and A_⊙(U) = −0.02 from Lodders (2019), which took into account the radioactive decay of the Th and U isotopes over the past 4.567 Gyr, and we show the model predictions after the long-term decay at each time.

These solar abundances are also applied to the observational data plotted in the figures in the following sections, if necessary. When compared with theoretical predictions, it is better to compare the relative abundances for some cases such as NLTE analysis (e.g., Zhao et al. 2016) and LTE differential analysis (Reggiani et al. 2017). If the solar abundances are not measured by the same analysis and are taken from the literature (e.g., AG89, AGSS09), then constant shifts are applied to re-normalize with our solar abundances (including O of Steffen et al. 2015). For Fe, most of the observational papers used ${A}_{\odot }(\mathrm{Fe})=7.50$ or 7.51, and re-normalization is applied only for the other cases such as with 7.45.

The primordial abundances are also updated from K06/K11, which does not affect the figures showing [X/Fe]. The adopted values are D/H = 2.527 × 10⁻⁵ (Cooke et al. 2018; metal-poor damped Lyα systems), ³He/H = 1.1 × 10⁻⁵ (Bania et al. 2002; Milky Way H ii regions), Y = 0.2449 (Aver et al. 2015; low-metallicity H ii regions, but see Izotov et al. 2014), and theoretical values of ⁶Li/H = 1.27 × 10⁻¹⁴ and ⁷Li/H =5.623 × 10⁻¹⁰ (Pitrou et al. 2018), which is higher than in Sbordone et al. (2010) but is comparable to the value in Meléndez et al. (2010) with a correction of stellar depletion. For ⁹Be and ^10,11B, theoretical values are taken from Coc et al. (2012).

2.2.2. MDFs and Star Formation Histories

The parameters for stellar physics (e.g., the IMF) can be determined from independent observations, while the galactic parameters (e.g., τ_i, τ_s, and τ_o) that describe the formation history of the system have to be determined by comparing GCE model predictions to observations. The MDF is the most important constraint for this purpose. The GCE model parameters chosen to match the observed MDF of each system are summarized in Table 4.

The resultant SFR histories (panel (a)), age–metallicity relations (panel (b)), and MDF (panel (c)) of our solar neighborhood models are shown in Figure 1. In the solar neighborhood, star formation takes place over 13 Gyr; the SFR peaked ∼3 Gyr ago and declined at the age ≳5 Gyr, which is consistent with WD observations (Tremblay et al. 2014). In the recent observational data, there is no tight relation between stellar ages and metallicities (Holmberg et al. 2007; Casagrande et al. 2011). Our model value is slightly lower than the solar ratio ([Fe/H] = 0) at the time of the Sun's formation (4.6 Gyr ago), which implies that the Sun is slightly more metal-rich than the average ISM of the solar neighborhood. The recent MDF (Casagrande et al. 2011) is narrower than in previous works (Edvardsson et al. 1993; Wyse & Gilmore 1995), where thick disk stars were also included. The peak is almost solar but is slightly subsolar, which also means that the Sun is slightly more metal-rich than the average of low-mass stars at present in the solar neighborhood. The K11 model (cyan dotted–dashed lines) was constructed to meet the previous MDFs, while in this paper, the models are updated in order to match the recent MDF as much as possible. Since we do not assume pre-enrichment or unreasonably slow infall, it is difficult to perfectly reproduce the narrow MDF as observed.

Zoom In Zoom Out Reset image size

Figure 2 is the same as Figure 1 but for the bulge, halo, and thick disk models. The GCE model parameters are chosen to match the observed MDFs and are summarized in Table 4. The first four models are the same as in K11, and the second halo model (with stronger outflow) is very similar to the model used in Carlos et al. (2018).

Zoom In Zoom Out Reset image size

In the bulge, the MDF is peaked at super-solar metallicity and has a sharp cut at the metal-rich end. This is well reproduced with rapid star formation (with a short star formation timescale) truncated with a strong outflow or galactic wind. Infall is also required to explain the lack of metal-poor stars. Then the metallicity increases rapidly and reaches solar metallicity only after 1 Gyr, which results in the high [α/Fe] ratios at [Fe/H]  ≳ −1 (Matteucci & Brocato 1990). The first bulge model in this paper (red long-dashed lines) includes infall and winds at 3 Gyr after formation, which results in a peak metallicity of [Fe/H] ∼ +0.3 at 3 Gyr. A much higher efficiency of chemical enrichment, e.g., a flatter IMF is not required, unless the duration is much shorter than 3 Gyr. Note that the 3 Gyr duration is consistent with chemodynamical simulations of Milky Way–like galaxies from cold dark matter initial conditions (KN11). A similar MDF can be produced with the outflow model (olive dotted–short-dashed lines) where the star formation is more gradually suppressed by outflows. In this second bulge model, young and metal-rich stars can be formed, and [Fe/H] increases steadily to +0.3 by present day.

Also for the thick disk (magenta dotted–long-dashed lines), we use the infall+wind model, which is in good agreement with the observed age–metallicity relation (Bensby et al. 2004b). The formation timescale is as short as in the bulge (∼3 Gyr), and the star formation efficiency is smaller than for the bulge but is larger than for the solar neighborhood. Not only the short duration of star formation but also intense star formation are necessary to reproduce the observed [α/Fe]–[Fe/H] relations in the thick disk stars (Bensby et al. 2004a).

In the halo, the MDF has a peak at a much lower metallicity ([Fe/H] ∼ − 1.6; e.g., Chiba & Yoshii 1998) and is distributed over a wider range of metallicities. This can be well reproduced with an outflow model without infall. In the first halo model (green short-dashed lines), the age–metallicity relation is similar to that in the solar neighborhood. However, Carlos et al. (2018) suggested a faster star formation in the halo from the observed Mg isotopic ratios. With a shorter star formation timescale, the metallicity would become too high; for this reason a stronger outflow is adopted in the second halo model (light-blue dotted lines). This model gives an age–metallicity relation similar to that in the thick disk.

Figure 3 shows the evolution of [O/Fe] against [Fe/H] for the solar neighborhood. In the early stages of galaxy formation, only SNe II/HNe contribute and the [O/Fe] ratios form a plateau at a wide range of [Fe/H] ([O/Fe] = 0.62, 0.57, 0.52 at [Fe/H] = − 3, −2, −1.1). The small slope at the low-metallicity end is caused by the mass dependence of the SN II/HN yields. Around [Fe/H] ∼ −1, SNe Ia start to occur, which produce more iron than α elements such as oxygen. This delayed enrichment of SNe Ia causes the decrease in [O/Fe] with increasing [Fe/H] (Matteucci & Greggio 1986). The observational data are the NLTE abundances obtained from the homogeneous analysis of a relatively large sample of high-resolution spectra of nearby stars and of the Sun (Zhao et al. 2016), which reveal the following three features. First, the [O/Fe] plateau value obtained here is ∼0.6, slightly higher than in K06 and K11 (cyan dotted–dashed line). Second, the [O/Fe] plateau continues to [Fe/H]  ∼ −0.8, and then the [O/Fe] ratio sharply decreases. The [Fe/H] at which the [α/Fe] starts to decrease depends on the adopted SN Ia progenitor model, and is determined not by the lifetime but by the metallicity dependence of SN Ia progenitors in our models. It is very difficult to reproduce this rapid evolutionary change without the metallicity effect of SNe Ia (Kobayashi et al. 1998; Kobayashi & Nomoto 2009; Figure 15 of Kobayashi et al. 2020). Third, the abundance ratios approach the solar ratios (i.e., [O/Fe] = [Fe/H] = 0). Our fiducial model (red solid line) can reproduce all of these features very well.

Zoom In Zoom Out Reset image size

With the metal-dependent HN fraction (blue short-dashed line), at Z ≳ Z_⊙, the metal production from core-collapse supernovae is assumed to be very small compared to SNe Ia; the present-day HN fraction is only 1%, and the rest of the massive supernovae are failed supernovae (i.e., no O and Fe production). This results in lower [O/Fe] at [Fe/H] ∼0, which may be more consistent with these observational data. If we simply exclude failed SNe, the predicted [O/Fe] plateau value would become higher than observed. Therefore, in the model without failed SNe, we reduce the upper mass limit of the IMF from 50M_⊙ to 40M_⊙ (green dotted line), so that the model has a [O/Fe] plateau value consistent with the observations. In VK18 (magenta long-dashed line), we assumed failed SNe at mass $\geqslant 25{M}_{\odot }$ and metallicity ≥0.02, where all synthesized O and heavier elements fall back onto a black hole, except for H, He, C, N, and F, which are synthesized in the outermost layers of the SN ejecta.

More parameter studies are shown in Figure 4 for the [O/Fe]–[Fe/H] relation. In the adopted nucleosynthesis yields, at any given mass, [O/Fe] is larger for SNe II than for HNe. Therefore the models with failed SNe at lower progenitor masses give systematically lower [O/Fe] ratios (red solid, green dotted, and blue short-dashed lines). Adopting 30M_⊙ as the upper limit of SNe II provides the best fit to the observations. As mentioned above, without failed SNe, i.e., if SNe II occur up to $50{M}_{\odot }$ as HNe (magenta long-dashed line), the [O/Fe] ratio becomes too high. With changing the IMF upper limit from $50{M}_{\odot }$ to $40{M}_{\odot }$ (cyan dotted–dashed line), both SNe II and HNe occur only up to $40{M}_{\odot }$; then the [O/Fe] ratio is consistent with observations. This conclusion depends on the updated solar oxygen abundance, and was not drawn in K06 or K11.

Zoom In Zoom Out Reset image size

Note that in the NLTE analysis of Zhao et al. (2016), the solar abundances are obtained for each line, and the oxygen solar abundances vary from ${A}_{\odot }({\rm{O}})=8.74$ to 8.82 (and are not shifted; see Section 2.2). Our value of 8.76 (that applied to other observations) lies in the range, but not the AGSS09 value. Since there is a 0.05 dex uncertainty depending on the choice of the solar abundance, the model without failed SNe but with ${M}_{{\rm{u}}}=40{M}_{\odot }$ (green dotted line) and the model with failed SNe at $\gt 25{M}_{\odot }$ (blue short-dashed line) are also acceptable.

Based on our fiducial model (the red solid line in all of the figures), which includes super-AGB stars, Figures 5 and 6 show the evolution of elemental abundance ratios [X/Fe] against [Fe/H] from C to Zn in the solar neighborhood, compared to the other models. In Figures 7–30, we compare the fiducial, K15, and K11 models with more observational data, not only from the NLTE analysis but also from other careful analysis.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The contribution to GCE from AGB stars (green dotted lines in Figure 5) can be seen mainly for C and N, and only slightly for Na, compared with the model that includes supernovae only (blue dashed lines). Hence it does not seem possible to explain the O–Na anticorrelation observed in globular cluster stars with a smooth star formation history as in the solar neighborhood. Although AGB stars produce significant amounts of Mg isotopes (see Section 3.3), their inclusion does not affect the [Mg/Fe]–[Fe/H] relation. The contribution from super-AGB stars (red solid lines) is very small; with super-AGB stars, C abundances slightly decrease, while N abundances slightly increase. It would be very difficult to put a constraint on super-AGB stars from the average evolutionary trends of elemental abundance ratios, but it might be possible to see some signatures of super-AGB stars in the scatters of elemental abundance ratios. With ECSNe (magenta long-dashed lines), Ni, Cu, and Zn are slightly increased. These yields are in reasonable agreement with the high Ni/Fe ratio in the Crab Nebula (Nomoto 1987; Wanajo et al. 2009). No difference is seen with/without SNe Iax (cyan dotted–dashed lines) in the solar neighborhood because of the narrow mass range of hybrid WDs. As noted before (Section 2.1), this mass range depends on convective overshooting, mass-loss, and reaction rates. Even with the wider mass range in Kobayashi et al. (2015; ${\rm{\Delta }}M\sim 1{M}_{\odot }$), however, the SN Iax contribution is negligible in the solar neighborhood; but it can be important at lower metallicities such as in dwarf spheroidal galaxies with stochastic chemical enrichment (Cescutti & Kobayashi 2017).

With failed SNe, our fiducial model (red solid lines in Figure 6) is in good agreement with observations of most of the major elements. Strictly speaking, the predicted Mg, Si, and S abundances are slightly higher, and the Ca, Co, and Ni abundances are slightly lower than in the observations. Compared with the K11 model (cyan dotted–dashed lines), the match is improved for most of elements, except for Ca at [Fe/H] ≲ −1 and Co at all [Fe/H], which implies higher energies or a larger HN contribution. The improvement is due to the inclusion of failed SNe (i.e., the exclusion of massive SNe II) and/or the updated solar abundances (see below for more details). With the metal-dependent HN fraction (blue short-dashed line), Cu and Zn are also underproduced at [Fe/H] ≳ −1 in the model, and this is why we use a constant HN fraction for our fiducial model (Section 2.1). It is possible to keep the agreement without failed SNe (green dotted lines) if we change the upper limit of the IMF (i.e., of both SNe II and HNe) as discussed above. This gives slightly better matching for Al and Cu, but not Na, at [Fe/H] ≳ −1. Finally, the VK18 model (magenta long-dashed lines) gives slightly too large ratios from O to S, relative to Fe, and the N/O ratios in Vincenzo & Kobayashi (2018b) would be ∼0.1 dex larger with the yields in our fiducial model. To put a further constraint on supernova explosions, it is very important to measure elemental abundances with ∼0.1 dex accuracy, not only at low metallicities but also at high metallicities.

Carbon—Half of carbon in the universe is produced by massive stars (>10M_⊙), while the rest is mainly produced by low-mass AGB stars (1–4M_⊙; K11). However, the [C/Fe] ratio is enhanced efficiently by low-mass stars because these stars produce no Fe. In Figure 7, the fiducial model (solid line) reproduces the observed trend slightly better than the K11 model (dashed line). When we include AGB yields (green dotted lines in Figure 5), [C/Fe] increases from [Fe/H] ∼ −1.5, which corresponds to the lifetime of ∼4M_⊙ stars (∼0.1 Gyr). At [Fe/H] ∼ −1, [C/Fe] reaches 0.21 (0.16 with s-process), which is 0.31 dex larger than the case without AGB yields (blue short-dashed lines in Figure 5). The inclusion of super-AGB stars (red solid line in Figure 5) increases [C/Fe] only by 0.004 dex. The peak value of [C/Fe] is in excellent agreement with the measurements from Zhao et al. (2016), which are based on 1D NLTE analysis of C i lines, as well as 1D LTE analysis of molecular CH and C₂ lines.

At lower and higher metallicities than [Fe/H] ∼ −1, however, the predicted [C/Fe] is 0.1–0.2 dex lower than the observations. This is at least partially due to the fact that the AGB contribution appears suddenly in the one-zone models. With inhomogeneous enrichment in chemodynamical simulations (Kobayashi & Nakasato 2011), the [C/Fe] variation would become weaker. In particular, AGB stars can contribute at metallicities below [Fe/H]  ≲ −1.5 when inhomogeneous chemical enrichment is taken into account (Kobayashi 2014; Vincenzo & Kobayashi 2018a).

At [Fe/H] ≳ −1, [C/Fe] shows a decrease in the NLTE observation, which is consistent with the LTE observation from Bensby & Feltzing (2006) with the forbidden [C i] line at 872.7 nm. Our models also show a decrease due to SNe Ia, but it is steeper than that shown by these observations. C yields from AGB could be increased with overshoot (Pignatari et al. 2016), which could also increase s-process yields. The K11 model (dashed line in Figure 7) gives lower [C/Fe] ratios at [Fe/H] ≲ −1 than the fiducial model (solid line), which is due to the adopted higher solar abundance (A_⊙(C) = 8.56 in AG89, instead of 8.43 in AGSS09).

At [Fe/H] ≲ −2.5, the model [C/Fe] is in good agreement with the observations from Spite et al. (2006), although the observational data show a significant scatter. Note that Spite et al. (2006) flagged "mixed" stars, where C is likely to have been transformed into N. These are plotted with smaller symbols in the figures and should be excluded from the comparison. It is known that a significant fraction of extremely metal-poor stars (the CEMP stars) show carbon enrichment, which are also plotted with smaller symbols for the data in Cohen et al. (2013) and Yong et al. (2013) in the figures. One of the scenarios for the CEMP stars is faint supernovae, which are not included in our models (Section 2.1). The model [C/Fe] shows roughly 0, with a very weak increase toward lower [Fe/H], which is consistent with the lower boundary of the plotted [C/Fe] ratios of the unmixed stars in Spite et al. (2006). The weak increase is also in good agreement with recent analysis of 3D/NLTE C and 3D/LTE Fe abundances from Fe ii lines in Amarsi et al. (2019a).

Figure 8 shows the [C/O] ratio against [O/H] for the models in Figure 3. At low metallicity, there is some variation in the plateau values among these models, but all models show a weak increase from [O/H] ∼ −1 to −3. Although Amarsi et al. (2019a) reported that [C/O] rather decreases toward lower [O/H] with their 3D/NLTE analysis, the slope of our models is in good agreement with the plotted observations including those from Amarsi et al. (2019a). At high metallicity, all models predict [C/O] ratios significantly lower than observed, although the model with the metallicity-dependent HN fraction (blue short-dashed line) gives [C/O] ratios closest to the observational data. C yields from AGB stars could be larger with more overshoot, but it is not clear if it would be enough to explain the ∼0.3 dex offset.

Nitrogen—Different from C, N is produced mainly by intermediate-mass AGB stars (4–7M_⊙; K11). Therefore the contribution from AGB stars is seen already from $[\mathrm{Fe}/{\rm{H}}]\,\sim -2.5$ (green dotted lines in Figure 5). At [Fe/H] ∼ −1, [N/Fe] reaches 0.37 (0.39 with s-process), which is 0.94 dex larger than the case without AGB yields (blue dashed lines in Figure 5). With super-AGB stars, the peak [N/Fe] is slightly higher, 0.44 (red solid lines in Figure 5), and the trend agrees very well with the plotted observational data.

At [Fe/H]  ≳ −1, the model [N/Fe] shows a decrease due to SNe Ia. In the observational data, such a decrease is not clearly seen, but at [Fe/H]  ∼ 0, the [N/Fe] ratio is ∼0, which is consistent with our new models with AGB and super-AGB stars. The [N/Fe] ratio at [Fe/H] = 0 is −0.59 without AGB (blue dashed line in Figure 5), and −0.23 in the K11 model (dashed line in Figure 9). Figure 9 shows that the fiducial model (solid line) gives a better match than the K11 model (dashed line) at all metallicity ranges. This difference is caused mainly by the adopted solar abundance (A_⊙(N) = 8.05 in AG89, instead of 7.83 in AGSS09).

Although no difference is seen at [Fe/H]  ≲ −2.5 with and without the AGB yields in these one-zone models (Figure 5), AGB stars can also contribute to N production even at [Fe/H]  ≲ −2.5 when taking inhomogeneous chemical enrichment into account (Kobayashi & Nakasato 2011; Vincenzo & Kobayashi 2018a), and Vincenzo & Kobayashi (2018b) reproduced the observed N/O–O/H relations, not with rotating massive stars, but with the failed SN model of VK18. Figure 10 shows the N/O ratio against oxygen abundance for the solar neighborhood models represented in Figure 3. All models shows a strong increase of N/O ratios toward higher metallicities, and the N/O increases the most in the VK18 model, which allowed for the reproduction the observed N/O–O/H relation of galaxies (Vincenzo & Kobayashi 2018a, 2018b).

α elements—For all α elements (O, Ne, Mg, Si, S, Ar, and Ca), the same trend as in O is present: the plateau caused by SNe II/HNe and the decrease from [Fe/H] ∼ −1 by SNe Ia (Figures 11–16). The [O/Fe] plateau value of the NLTE observation is ∼0.6, and the trend is surprisingly consistent with the LTE analysis in Clegg et al. (1981), ¹¹ Meléndez & Barbuy (2002), Fulbright & Johnson (2003), and Bensby et al. (2003). On the other hand, the observed [Mg/Fe] plateau value may be ∼0.3 dex lower. This observed positive [O/Mg] ratio was discussed in Figure 9 of K06, although the results were not conclusive because of the uncertainty of the observational data. Observationally, the [Mg/Fe] plateau value was reported to be 0.27 (Cayrel et al. 2004), which was due to underestimated equivalent widths of the Mg lines. This was updated by Andrievsky et al. (2010) to 0.31 with LTE, and 0.61 with NLTE analysis for Mg (but not for Fe), which would result in [O/Mg] ∼0. However, the differential analysis of Reggiani et al. (2017), although with LTE, produces very similar results as the NLTE analysis of Zhao et al. (2016), with a low [Mg/Fe] plateau. Bergemann et al. (2017) also obtained ∼0.3 with 1D NLTE, and an even lower value with their $\langle 3{\rm{D}}\rangle$ NLTE analysis. The difference between these two NLTE Mg abundances should be investigated further.

In our fiducial model, the [Mg/Fe] ratios are 0.45, 0.45, and 0.43 at [Fe/H] = −3, −2, and −1.1 (Figure 12), which are only ∼0.15 dex lower than those of [O/Fe]. Figure 13 shows the O/Mg ratio against oxygen abundance in the various GCE models for the solar neighborhood. The predicted [O/Mg] is never higher than 0.25 and is lower than observed at −1.5 ≲  [O/H] ≲ −0.5.

Since the majority of O and Mg are formed during hydrostatic burning of stellar evolution of massive stars, it is not possible to greatly modify the [O/Mg] ratios during supernova explosions. As mentioned in Section 2.1, a different ¹²C(α, γ)¹⁶O reaction rate could change the [O/Mg] ratio during stellar evolution and may explain the large [O/Mg] at the plateau ([O/H] ≲ − 0.5); the core-collapse supernova yields used here were calculated with 1.3 times the value given in Caughlan & Fowler (1988), which is up to a factor of two lower than that calculated by deBoer et al. (2017). At high metallicities, however, it would be difficult to vary [O/Mg] as much as observed. Some of the observational data (open circles and filled triangles) indicate that [O/Mg] may decrease for higher metallicities, which might require a different new physical explanation (K06).

Similar to [Mg/Fe], the observed plateau values of [Si/Fe] and [Ca/Fe] are ∼0.3. In our fiducial model, the [Si/Fe] ratios are ∼0.58, 0.51, and 0.52 at [Fe/H] = −3, −2, and −1.1, which is ∼0.2 dex higher, and the [Ca/Fe] ratios are 0.28, 0.21, and 0.25 at [Fe/H] −3, −2, and −1.1, which is ∼0.1 dex lower than observed. Si and Ca yields are affected by explosive burning, and it is unclear if the ¹²C(α, γ)¹⁶O rate could solve these mismatches as well as for O and Mg. Note that the differential analysis of Reggiani et al. (2017) leads to systematically lower [Si/Fe] ratios, compared to other studies. Their abundances are based on the Si i 390.5 nm line that is blended with CH, and may suffer from NLTE effects in the metal-poor regime (Amarsi & Asplund 2017; up to ∼+ 0.2 dex).

S abundances are difficult to measure in stellar spectra with a significant NLTE effect depending on the lines. The predicted [S/Fe] ratios are 0.52, 0.45, and 0.47 at [Fe/H] −3, −2, and −1.1, which is ∼0.1 dex higher than in Spite et al. (2011) and Nissen et al. (2007), but is in good agreement with Takada-Hidai et al. (2005) at low metallicities. At high metallicities, the K11 model gives a slightly better match with Chen et al. (2002) and also more recent observations by Costa Silva et al. (2020). Note that for S, the solar abundance adopted in the K11 model was higher; A_⊙(S) = 7.27 in AG89, and 7.12 in AGSS09. Ne and Ar also show a similar trend, and the solar abundances are also decreased by 0.16 dex in AGSS09.

Odd-Z elements—The production of odd-Z elements depends on the surplus of neutrons from ²²Ne, which is made during He-burning from ¹⁴N produced in the CNO cycle, and hence the yields depends on the metallicity of the progenitors (see Figure 5 in K06). In GCE models, at [Fe/H] ≲ −1, [(Na, Al, Cu)/Fe] show a decrease toward lower metallicities (Figures 17–20). The observed Na and Al abundances are largely affected by NLTE effects, and our models are in good agreement with the NLTE observations (Andrievsky et al. 2007, 2008; Zhao et al. 2016; Mashonkina et al. 2017) as well as the LTE differential analysis (Reggiani et al. 2017).

For Cu, the LTE data shows a decrease with a large scatter (Primas et al. 2000), and our models reproduce the average trend very well, giving [Cu/Fe] = −0.63 at [Fe/H] = −2.5. Our [Cu/Fe] trend is also quantitatively consistent with that first found by Sneden & Crocker (1988). The recent NLTE analysis by Andrievsky et al. (2018), however, found no such Cu decrease at −4  ≲ [Fe/H]  ≲ −1.5. Another NLTE analysis by Shi et al. (2018) found a decrease very similar to our model. This could be tested with Cu ii lines; Roederer & Barklem (2018) found a very similar decrease at −2.5 ≲ [Fe/H] ≲ −1 with LTE, while Korotin et al. (2018) found a shallower decrease. It is important to obtain NLTE abundances using Cu i and Cu ii lines for a larger sample of metal-poor stars.

At [Fe/H] ≳ −1, Na and Al show a decrease toward higher metallicities owing to the contribution from SNe Ia, which is shallower than the trend for the α elements. With the updated reaction rates, ¹² Na yields from AGB stars were reduced in K11. Nonetheless, in the K11 model (dashed line in Figure 17), Na and Al were overproduced, and this problem is solved in the fiducial model (solid line) of this paper. Although the predicted [Na/Fe] is in excellent agreement, [Al/Fe] may be decreased slightly too much, compared with the NLTE abundances in Zhao et al. (2016).

Similar to Al, Cu was overproduced at [Fe/H] ≳ −1 in the K11 model (dashed line in Figure 20), and this problem is also solved in the fiducial model (solid line). [Cu/Fe] may be slightly too decreased, compared with the NLTE abundances in Zhao et al. (2016). With the s-process (dotted–dashed line), AGB stars produce some Cu, but this gives only a small contribution from [Fe/H]  ∼ −2, and [Cu/Fe] is increased only by 0.03 dex at [Fe/H] = 0 with s-process (Section 3.4).

Recently, P abundances became available with near-UV or IR spectra. The predicted [P/Fe] shows a weak decrease toward lower metallicities due to the same metallicity dependence of the yields, which is in reasonably good agreement with the observations (Figure 19). It would be better if P yields increase toward higher metallicity to reach a peak [P/Fe]  ∼ 0.4 at [Fe/H] ∼ −1 and then sharply decrease due to SNe Ia. Note that the solar P abundance is decreased by 0.16 dex, compared with K11.

Cl, K, Sc, V, and Ti—K, Sc, V, and Ti are known to be underproduced at all metallicity ranges in theoretical models with respect to the observations (K06; Figures 21–23), and it has been shown that some multidimensional effects can increase Sc, V, and Ti abundances, as in the K15 model (dotted lines). Namely, Sc and Ti yields are greatly increased in the nucleosynthesis calculation of 2D jet-induced supernovae (Section 2.1). K, Sc, and V yields can also be affected by the neutrino process (Kobayashi et al. 2011a), whose effects are not included in any of models in this paper. Stellar rotation can enhance Cl, K, and Sc abundances, but not V (Prantzos et al. 2018). Cl, K, and Sc may also be enhanced by the O–C mergers during hydrostatic burning, which is seen in one of the 1D stellar evolution calculations ($15{M}_{\odot }$; Ritter et al. 2018a). In our models, [(Cl, K, Sc)/Fe] show a weak increase from [Fe/H]  ∼ −3 to ∼ −1, which is due to the metallicity dependence of SN II/HN yields. Note that the solar Cl abundance is increased by 0.37 dex, and the predicted [Cl/Fe] is negative overall, giving −0.8 at [Fe/H]= −2, which is as low as for K and Sc. Ti and V yields do not depend very much on the progenitor metallicity, and thus [(Ti, V)/Fe] show a plateau at [Fe/H] ≲ −1. At [Fe/H] ≳ −1, all of these elemental abundances show a weak decrease toward higher metallicities because of SNe Ia. K is also known for strong NLTE effects (Takeda et al. 2002; Reggiani et al. 2019; up to −0.7 dex), and the plotted data in Figure 21 are from NLTE analysis.

Iron-peak elements—Iron-peak elements are synthesized in thermonuclear explosions of supernovae, as well as in incomplete or complete Si-burning during explosive burning of core-collapse supernovae (K06), and therefore it is very important to obtain the exact abundances for constraining the explosion mechanism. Observationally, NLTE effects of iron-peak elements other than iron have not been well studied yet, except for a few cases (e.g., Bergemann & Gehren 2008; Bergemann et al. 2010; Bergemann & Cescutti 2010). However, Sneden et al. (2016) and Cowan et al. (2020) obtained consistent abundances between neutral and ionized lines using updated atomic data, except for Cr and Co, which implies that the NLTE effects may not be so large. Given these previous studies, it is a matter of urgency to check the NLTE effects for iron-peak elements with updated atomic data. In Figures 25–30, we compare our models to LTE observations. For Cr, as shown in Figures 20 and 21 of K06, the difference between Cr i and Cr ii abundances are significant, and we use only the Cr ii observations in this paper. The difference in the adopted solar abundances is up to 0.1 dex for iron-peak elements; there is a ∼0.1 dex decrease for Mn, Cu, and Zn, while there is a ∼0.1 dex increase for Co.

At −2.5 ≲ [Fe/H]  ≲ −1, [(Cr, Mn, Zn)/Fe] are consistent with the observed mean values (0.07, −0.56, 0.18 at [Fe/H] =−2, respectively). [Co i/Fe] is −0.20 at [Fe/H] = −2, which is ∼0.3 dex lower than the observations (Figure 28) and is slightly lower than in the K11 model (dashed line), but can be increased by the HN jet effects (dotted line). However, Cowan et al. (2020) showed a large difference between Co i and Co ii abundances, showing [Co ii/Fe] ∼0 at −3 ≲ [Fe/H]  ≲ −2.2, in contrast to the very high NLTE abundances in Bergemann et al. (2010). Our models are in good agreement with the Co ii observations, and it is necessary to increase the sample to discuss which of our models is the best. There is no such difference between Ni i and Ni ii abundances (Cowan et al. 2020). The predicted [Ni/Fe] is −0.19 at [Fe/H] = −2, which is ∼0.2 dex lower than the observations (Figure 29). Both Ni and Fe are produced in the complete Si-burning region (K06), and it is very difficult to change the ratio. A deeper mass cut could slightly increase ⁵⁸Ni yields but not ⁶⁰Ni yields, and the latter isotope forms the majority at low metallicities (Section 3.3).

At [Fe/H] ≳ −1, these abundance ratios stay roughly constant, except for [Mn/Fe] (see the next paragraph), because iron-peak elements are also produced by SNe Ia. There was an Ni overproduction problem by SNe Ia (dashed line in Figure 29), which is ameliorated in our models because of the new metallicity-dependent yields of SNe Ia (Section 2.1.3; Kobayashi et al. 2020). At [Fe/H] ≲ −2.5, observational data (namely, Reggiani et al. 2017) show an increase of [(Co i, Zn)/Fe] toward lower metallicities, which is not discussed here since inhomogeneous chemical enrichment is becoming increasingly important there.

Manganese—Mn is the most important element for constraining the physics of SNe Ia, since it is produced more by SNe Ia than SNe II/HNe relative to iron (Kobayashi & Nomoto 2009). Mn yields depend on the explosion model of SNe Ia, and thus indirectly depend on the progenitor model of SNe Ia. In this paper, we use the 2D delayed detonation model for the SN Ia yields given as a function of metallicity (Section 2.1).

At [Fe/H] ≲ −1, [Mn/Fe] shows a plateau with ∼ −0.56,−0.55, and −0.43 at [Fe/H] = −3, −2, and −1.1, which is determined by the IMF-weighted SN II/HN yields and is consistent with the plotted observations in Figure 26. The small difference in the adopted solar abundances is canceled out with a small reduction of Mn yields by failed supernovae. Recently, Eitner et al. (2020) suggested a large NLTE correction for Mn I lines (∼ −0.4 dex, up to 0.6 dex) and [Mn/Fe] is ∼0 at −4 ≲ [Fe/H]  ≲  0. However, using the same Mn atomic model, Amarsi et al. (2020) found a steeper NLTE [Mn/Fe] relation at [Fe/H] ≳ −2. Bergemann et al. (2019) also showed a positive or negative 3D effect depending on the spectral lines used and model parameters. Consequently, 3D-NLTE analysis for a large sample of stars is needed. Theoretically, Mn is synthesized in the incomplete Si-burning regions together with Cr, and thus higher [Mn/Fe] would result in higher [Cr/Fe], which is inconsistent with the Cr ii observations and even more inconsistent with Cr i observations. In Figure 27, we also show Mn ii observations. Since it is harder to obtain the Mn ii lines, there is not much literature on this. The [Mn/Fe] plateau value may be 0.1–0.2 dex higher than in the model but should not be as high as [Mn/Fe]  ∼ 0.

Above [Fe/H] ∼ −1, [Mn/Fe] shows an increase toward higher metallicities, which is caused by the delayed enrichment of SNe Ia and has been used to constrain the progenitors of SNe Ia (Seitenzahl et al. 2013; Cescutti & Kobayashi 2017; Eitner et al. 2020; Kobayashi et al. 2020). A similar evolutionary trend with a plateau and an increase was first found by Gratton (1989). ¹³ Feltzing et al. (2007) showed a steeper slope at [Fe/H] > 0 than in Reddy et al. (2003), which is better reproduced with the fiducial model than with the K11 model (dashed line).

Zinc—Zn is one of the most important elements for the physics of core-collapse supernovae as ${}_{30}^{64}$Zn is enhanced in the deepest region of HNe with higher explosion energy and entropy (K06), and thus [Zn/Fe] is increased with multidimensional effects (dotted line in Figure 30). Zn yields potentially depend on the neutrino processes and the resultant Y_e as well. However, it is worth noting that Zn can be enhanced at the stellar layer with ${Y}_{{\rm{e}}}\sim 0.5$, but Y_e too close to 0.5 gives too small ${}_{25}^{55}$Mn yields. From these dependencies, Y_e = 0.4997 is chosen for the incomplete Si-burning regions in K06. Neutron-rich isotopes of zinc (${}^{66-70}$Zn) can also be produced by neutron-capture processes, of which yields are larger for higher-metallicity massive SNe II (Section 3.3). The contribution from the s-process in AGB stars is very small, increasing [Zn/Fe] only by 0.004 dex at [Fe/H] = 0. As noted before, ECSNe can enhance Zn as well as Ni and Cu, but the contribution to GCE is small, up to 0.04 dex (magenta long-dashed lines in Figure 5).

The predicted [Zn/Fe] is about ∼0.2 for a wide range of metallicities if we apply such a large fraction of HNe (Section 2.1). This is ∼0.1 dex larger than in the K11 model, where the difference is mostly due to the adopted solar abundance. Our [Zn/Fe] ratios are in good agreement with the observational data, as well as the results from Roederer & Barklem (2018), who obtained Zn ii lines and found that the NLTE effect should be less than 0.1 dex, at [Fe/H] ≳ −2.5. At [Fe/H] ≲ −2.5, Sneden & Crocker (1988) first suggested an increase of [Zn/Fe]. More recent observations show a linear increase from [Fe/H] ∼ −1 to lower metallicities (Primas et al. 2000; Nissen et al. 2007; Saito et al. 2009), and the increase may be steeper with the NLTE correction (Takeda et al. 2005). This may imply that the HN fraction is larger in the earlier stages of galaxy formation (K06).

[Zn/Fe] observations seem to decrease from [Fe/H] ∼ −1 to $\sim -0.5$, and then slightly increase to [Fe/H]  ∼ 0 (Saito et al. 2009). The NLTE correction there is found to be mostly $\lesssim 0.1$ dex (Takeda et al. 2016). This up-turn trend is well reproduced with our fiducial model. With the metallicity-dependent HN fraction, the [Zn/Fe] ratio shows a continuous decrease from $[\mathrm{Fe}/{\rm{H}}]\sim -1$ to ∼0, which gives lower values than some of the observational data at high metallicities (Figure 6). A similar problem also arises for Co and Cu with the metallicity-dependent HN fraction. However, this problem is not seen in the chemodynamical simulations of Kobayashi & Nakasato (2011; Section 2.1).

Figure 31 shows the evolution of isotopic ratios against [Fe/H] for the solar neighborhood models. Comparing to the fiducial model including AGB and super-AGB stars (red solid lines), the models without super-AGB stars (red short-dashed lines) and with neither AGB nor super-AGB stars (red long-dashed lines) are also shown for C and N. The predicted ¹²C/¹³C ratio is 77.0 at [Fe/H] = 0, which is slightly lower than the solar ratio (89.4; AGSS09), but is 62.7 at 13.8 Gyr, which is in good agreement with the local ISM observations (68 ± 15 Milam et al. 2005; squares). Without AGB stars, the ¹²C/¹³C ratio would be too high. Super-AGB stars further decrease ¹²C/¹³C, which is consistent with the finding in Romano et al. (2019). At low metallicities, as discussed for N, the effect of inhomogeneous enrichment is important, which could explain the low ¹²C/¹³C ratios at low metallicities in stellar observations.

Zoom In Zoom Out Reset image size

The underproduction problem of ¹⁵N is known, and may require other sources such as novae (Romano & Matteucci 2003; Romano et al. 2019; see K11 for more details) and/or H-ingestions in massive stars (Pignatari et al. 2015). Super-AGB stars produce more ¹⁴N than ¹⁵N, which results in even larger ¹⁴N/¹⁵N ratios. N isotopic ratios are also observed in carbon stars, showing ¹⁴N/¹⁵N ratios of ∼1000 for N-type carbon stars (Hedrosa et al. 2013), which might require ¹⁵N production in the He-shell.

The predicted ¹⁶O/¹⁸O ratio is 484 at [Fe/H] = 0, which is in good agreement with the solar ratio (499; AGSS09), and is 389 at 13.8 Gyr, which is also in excellent agreement with the local ISM observations (385 ± 56; Polehampton et al. 2005). ¹⁷O is, however, overproduced by AGB and super-AGB stars, giving ¹⁶O/¹⁷O by a factor of ∼1.5 lower than the solar ratio. However, new sets of yields need to be calculated and tested using the new rate of the ¹⁷O(p, α)¹⁴N measured underground by the LUNA Collaboration (Bruno et al. 2016). The new rate is 2–2.5 times higher than the previous rate used in the models considered here and will result in a decrease of the contribution to ¹⁷O from AGB and super-AGB stars potentially of the same magnitude needed to reach the solar value.

The solar ${}^{\mathrm{21,22}}$Ne/²⁰Ne ratios are reasonably well reproduced with AGB stars. The predicted ²⁴Mg/${}^{\mathrm{25,26}}$Mg ratios are higher than the stellar observations, even with the inclusion of super-AGB stars. There is no significant difference among these models for the other ratios, and these are determined by the SN II/HN yields (K11); small mismatches for the solar ratios still remain. Because minor isotopes are enhanced by higher-metallicity SNe II/HNe, the major-to-minor isotopic ratios decrease as a function of metallicity. The model without failed SNe but with ${M}_{{\rm{u}}}=40{M}_{\odot }$ (green dotted lines in Figure 6) gives almost the same trends of isotopic ratios, while the model with the metal-dependent HN fraction (blue short-dashed lines in Figure 6) gives slightly shallower evolution of these isotopic ratios. The solar Zn isotopic ratios can be better explained with the metal-dependent HN fraction, although the Zn abundance is better explained without it.

Isotopic ratios are also available with the detailed analysis of molecular lines in radio observations, and an IMF variation is suggested from the low values at high redshifts (e.g., Zhang et al. 2018). This figure also shows some data for a high-redshift galaxy (small error bars), which are far from any of these theoretical predictions at $t=6.3\,\mathrm{Gyr}$ (corresponding to the observed redshift). Since these data are for a spiral galaxy, it is unlikely that the SFRs and/or the IMF are very different from our solar neighborhood models. These mismatches should be noted when the IMF is constrained from radio observations.

Figure 32 shows the evolution of neutron-capture elements in the solar neighborhood, for all neutron-capture elements that have stellar abundance estimates or upper limits in the literature; some of the elements including Au are measurable only in UV spectra from the Hubble Space Telescope, and thus the number of measurements is very limited. It is necessary to increase the sample to obtain a comprehensive picture on the origin and evolution of these heavy elements. The number of candidates is being increased by strategic surveys such as the R-process Alliance (Hansen et al. 2018). Two characteristic stars that show different r-process enrichment ¹⁴ are highlighted with the large gray filled and open squares, respectively. For the elements except for Sr–Zr and Ba–Gd, so-called r-II stars ¹⁵ are also plotted, but this may cause a selection bias; some elements are detected because of the large r-process enhancement. This figure shows our six GCE models switching the neutron-capture enrichment sources one by one. There are no differences in the evolution of the elements up to Ni when considering the different models shown in this figure. Ga and Ge are produced mainly from core-collapse supernovae, and the predicted trends are consistent with the observations (Sneden et al. 2003; Roederer et al. 2014c) although the sample is very limited.

Zoom In Zoom Out Reset image size

The s-process from AGB stars (blue long-dashed lines) can produce elements up to Pb, and the contribution appears from [Fe/H] ∼ −2 for light s-process elements (e.g., Sr, Y, Zr) and only from [Fe/H] ∼  −1.5 for heavy s-process elements (e.g., Pb). This is because the light s-process elements are produced also from intermediate-mass AGB stars, while the heavy s-process elements are mainly produced by low-mass AGB stars. At [Fe/H] = 0, ∼70% of the solar abundances of the elements belonging to the first s-process peak, Sr, Y, and Zr, are produced from AGB stars, while only ∼50% or less of the solar abundances of Mo, Ru, and Ag are of s-process origin, as is well known (Arlandini et al. 1999). The elements Ba, La, and Ce, belonging to the second s-process peak, are ∼50% overproduced, while the solar abundances of Pr and Nd are reproduced. The elements from Eu to Tm as well as Ir are typical r-process elements, with less than 30% of their solar abundances contributed by the s-process. The elements Yb and Hf have a significant contribution from the s-process. Finally, Pb belongs to the third s-process peak, and it is also overproduced by ∼30%.

The overproduction of the second (Ba) and third (Pb) s-process peak elements at [Fe/H] = 0 suggests that the contribution from the s-process with the adopted yield set is too large. This contribution can be reduced by decreasing the parameter ${M}_{\mathrm{mix}}$ in the AGB models, which controls the extent in mass of the region where the s-process elements are produced (Section 2.1). As noted in Section 2.1, ${M}_{\mathrm{mix}}$ values are already reduced from those in Karakas & Lugaro (2016; e.g., from 2 × 10⁻³ to 1 × 10⁻³ in the $2{M}_{\odot }$ models with Z = 0.0028 and 0.014). A decrease in ${M}_{\mathrm{mix}}$ brings the model predictions for the second peak s-process elements qualitatively closer to the direct observations of carbon-rich AGB stars at solar metallicity from Abia et al. (2002), although it is difficult to perform a detailed analysis given the large observational error bars (±0.4 dex). Smaller ${M}_{\mathrm{mix}}$ values than the standard considered here also provide a better coverage of the spread in isotopic ratios observed for Sr, Zr, and Ba in meteoritic stardust SiC grains (Lugaro et al. 2018).

With ECSNe (light-blue short-dashed lines), the enhancement is as small as ∼0.1 dex for [(Cu, Zn)/Fe] (Figure 5), but a larger enhancement is seen for As, Se, Rb, Sr, Y, Zr starting from [Fe/H] ∼ −3. Rb is produced more by ECSNe than by AGB stars. The ECSN contribution appears before AGB stars start to contribute because the progenitors of ECSNe are more massive than those of AGB stars at a given metallicity (Section 2.1). There is also a slightly earlier increase in Mo, while for heavier elements, the contribution from ECSNe is negligible. This result is different from Cescutti & Chiappini (2014) mainly because they assumed ECSNe from all $8\,-10{M}_{\odot }$ stars.

With ν-driven winds (green dotted–long-dashed lines), the elements from Sr to Ag are largely overproduced, which is a crucial problem. Therefore, the ν-driven winds are not included in the other models in this figure. Since ν-driven winds are associated with core-collapse supernovae, the contribution can appear at $[\mathrm{Fe}/{\rm{H}}]\ll -3$. This rapid contribution may explain some of the observations at $[\mathrm{Fe}/{\rm{H}}]\lesssim -3$.

With NS–NS mergers (olive dotted lines), the elements heavier than Zr show an increase starting from $[\mathrm{Fe}/{\rm{H}}]\sim -3$ due to the r-process. Although the progenitor masses of NSs are larger than those of ECSNe at a given metallicity, there is a time delay for the merging of two NSs (Section 2.1.2). The time delay is shorter for NS–BH mergers, and the first increase in the abundances is seen already at $[\mathrm{Fe}/{\rm{H}}]\sim -4$ if a similar r-process occurs in NS–BH mergers (orange short-dotted–dashed lines). However, this time delay is still not short enough to explain the observations at $[\mathrm{Fe}/{\rm{H}}]\lesssim -3$, although our one-zone models cannot give a strong conclusion at $[\mathrm{Fe}/{\rm{H}}]\lesssim -2.5$ where chemical enrichment takes place inhomogeneously. NS–NS/NS–BH mergers do not produce Pb, but Th and U.

Observations of metal-poor stars show that by [Fe/H] ≲ −3, there is already an enhancement of all neutron-capture elements, which requires production from a different site with a shorter time delay than NS–NS/NS–BH mergers (e.g., Wehmeyer et al. 2015; Haynes & Kobayashi 2019; Côté et al. 2019). MRSNe are very good candidates for the rapid enrichment as they are massive core-collapse supernovae with a very short time delay (10⁶ yr). In fact, the model with MRSNe (red solid lines) shows a plateau at $[\mathrm{Fe}/{\rm{H}}]\lesssim -3$, which is similar to ν-driven winds, but the elemental abundance ratios are much more consistent with observations than with ν-driven winds. From Sr to Ru, the plateau values are lower than with ν-driven winds, and are $\sim -1$ and ∼0 for [(Sr, Y)/Fe] and [(Zr, Mo, Ru)/Fe], respectively, which are in good agreement with the observations (see also Figures 33–35 for more detailed comparison). For Ag, both ν-driven winds and MRSNe seem to overproduce its abundance, although observational data have been provided by only a few studies (e.g., Hansen et al. 2012; Spite et al. 2018). Similar overproduction may also be seen for Pd and Cd. For Ba, the models with MRSNe or ν-driven winds can explain the low [Ba/Fe] of some metal-poor stars. However, AGB stars should contribute more at $-3\lesssim [\mathrm{Fe}/{\rm{H}}]\lesssim -2$ (Figure 36), which is possible with inhomogeneous chemical enrichment (see also Raiteri et al. 2015).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For the elements heavier than Sn, the plateau values are higher with MRSNe than with ν-driven winds, which is more consistent with the observations. The Te data are provided by only one study (Roederer et al. 2012), which seem to favor MRSNe over ν-driven winds. Although the model with MRSNe is in good agreement with the observations for Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, and Yb, the model prediction is lower than observed for La, Ce, and Pr at $[\mathrm{Fe}/{\rm{H}}]\lesssim -2$. Note that the atomic data for Pr, Dy, Tm, Yb, and Lu have been updated (Sneden et al. 2009), which may affect some of the observational data plotted here. Similar to Ba (Figure 36), these elements can be enhanced at $-3\lesssim$ $[\mathrm{Fe}/{\rm{H}}]\lesssim -2$ possibly by AGB stars with inhomogeneous chemical enrichment. In contrast, the MRSN model can reproduce the average trend of [Eu/Fe] very well (Figure 37). Note that the inhomogeneous enrichment could slightly increase the contribution from NSMs at $-3\,\lesssim$ $[\mathrm{Fe}/{\rm{H}}]\lesssim -2$. This model is also acceptable for Os, Ir, and Pt, while Au is underproduced in the model. Au measurements are available only for three well-known r-process enhanced stars: BD +17${}^{^\circ }3248$ (Cowan et al. 2002; filled upside-down triangle), CS 22892-052 (Sneden et al. 2003; open square), and CS 31082-001 (Barbuy et al. 2011; open circle). Finally, the MRSN model can also explain the observed Th abundances of metal-poor stars and the Sun. Note that our GCE model predictions are after the long-term decay at each time and are normalized by the proto-solar abundance (Section 2.2), and thus the lines are expected to go through [X/Fe] = $[\mathrm{Fe}/{\rm{H}}]=0$. The observational data, however, are normalized by the present-day solar abundance from AGSS09, assuming that the observed stars are as old as the Sun. Compared with Th, however, U may be overproduced in the nucleosynthesis yields; this is more serious for MRSNe, which give $M(\mathrm{Th})/M({\rm{U}})=0.18$, while NSMs give $M(\mathrm{Th})/M({\rm{U}})=0.58$. In our adopted solar abundances, ${M}_{\odot }(\mathrm{Th})/{M}_{\odot }({\rm{U}})=1.7$.

Zoom In Zoom Out Reset image size

Sr, Y, and Zr — Figures 33–37 compare more observational data to our elemental abundance tracks of the model with s-process only (dashed lines) and with s-process, ECSNe, NS–NS/NS–BH mergers, and MRSNe (the s + r model, solid lines). In Figure 33, the base level of [Sr/Fe] $\sim \,-0.8$ at $[\mathrm{Fe}/{\rm{H}}]\lesssim -3.5$ is caused by MRSNe. The average [Sr/Fe] increases from $[\mathrm{Fe}/{\rm{H}}]\sim -3.5$ due to ECSNe, from $[\mathrm{Fe}/{\rm{H}}]\sim -2.5$ due to AGB stars before decreasing at $[\mathrm{Fe}/{\rm{H}}]\gtrsim -1$ because of SNe Ia, and becomes [Sr/Fe]  = −0.064 at $[\mathrm{Fe}/{\rm{H}}]=0$ in the s + r model (solid line). The slope of the decrease becomes flat at $[\mathrm{Fe}/{\rm{H}}]\sim -0.3$ due to the increase of the AGB contribution (dashed line). This trend is in excellent agreement with the observational data, except for one star with low [Sr/Fe]. Differential analysis by Reggiani et al. (2017) gives slightly higher ratios than Zhao et al.'s (2016) NLTE abundances on the average. At $[\mathrm{Fe}/{\rm{H}}]\lesssim -2.5$, there is a large scatter, where LTE abundances by Roederer et al. (2014b) agree well with Andrievsky et al.'s (2011) and Zhao et al.'s (2016) NLTE abundances.

In Figure 34, starting from [Y/Fe] $\sim -0.4$ at $[\mathrm{Fe}/{\rm{H}}]\,\sim -3.5$, the average [Y/Fe] also increases gradually from $[\mathrm{Fe}/{\rm{H}}]\sim -3$ to ∼−1 before decreasing at $[\mathrm{Fe}/{\rm{H}}]\gtrsim -1$ due to SNe Ia, and becomes [Y/Fe] = −0.096 at $[\mathrm{Fe}/{\rm{H}}]=0$ in the s + r model (solid line). The predicted [Y/Fe] trend is also in excellent agreement with the observations, although the differential analysis by Reggiani et al. (2017) gives slightly higher ratios. Note that there are no NLTE abundances available at any metallicities for Y.

In Figure 35, the average [Zr/Fe] is rapidly enhanced to $\sim +0.4$ by MRSNe, stays roughly constant until $[\mathrm{Fe}/{\rm{H}}]\sim -1$, and then decreases at $[\mathrm{Fe}/{\rm{H}}]\gtrsim -1$ due to SNe Ia reaching [Zr/Fe]  = 0.098 at $[\mathrm{Fe}/{\rm{H}}]=0$ in the s + r model (solid line). Similar to Sr and Y, the predicted [Y/Fe] is slightly lower than the differential analysis by Reggiani et al. (2017) but is in excellent agreement with the other observations including the NLTE abundances from Zhao et al. (2016).

These three elements are similar in the sense that they are mainly produced by ECSNe and AGB stars, but the relative contributions are different. Nonetheless, our s + r model can reproduce the observations of the three elements consistently without introducing a free parameter, and without adding extra LEPP (Section 2.1). The large scatter at $[\mathrm{Fe}/{\rm{H}}]\lesssim -2.5$ may be caused by the rareness of ECSNe under the inhomogeneous enrichment; the stars with higher [(Sr, Y, Zn)/Fe] ratios may be locally enriched by ECSNe. It might be possible to constrain the mass ranges of ECSNe and the fate of super-AGB stars from the scatters in chemodynamical simulations.

Barium—Ba is the characteristic element for the s-process in AGB stars. In Figure 36, the average [Ba/Fe] is already enhanced up to the $\sim -1$ base level by the r-process, ¹⁶ which is consistent with the NLTE observations. Then, [Ba/Fe] shows an increase from $[\mathrm{Fe}/{\rm{H}}]\sim -2$ in the s + r model, but from $[\mathrm{Fe}/{\rm{H}}]\sim -3$ both in the NLTE and LTE observations. Finally, [Ba/Fe] shows a plateau at ∼0 from $[\mathrm{Fe}/{\rm{H}}]\sim -1$ in the s + r model, but from $[\mathrm{Fe}/{\rm{H}}]\sim -2$ in both NLTE and LTE observations. These mean that Ba enrichment is slower in the s + r model than in the observations; this mismatch should at least partially be caused by the lack of inhomogeneous enrichment in our GCE models, and should be tested together with its effect on the other s-process elements with chemodynamical simulations such as in Haynes & Kobayashi (2019). The dotted line shows a model with the s-process yield set recommended in Karakas & Lugaro (2016), where a larger ${M}_{\mathrm{mix}}$ at $Z\geqslant 0.0028$ was adopted. In this model, [Ba/Fe] is already overproduced at $[\mathrm{Fe}/{\rm{H}}]\sim -1$, and the model [Ba/Fe] is 0.37 at $[\mathrm{Fe}/{\rm{H}}]=0$, which is reduced to be 0.20 with the lower ${M}_{\mathrm{mix}}$ in this paper. The new yields with a smaller ${M}_{\mathrm{mix}}$ predict [(Zr,Ba)/Fe] ratios within 0.15 dex of the yields of Cristallo et al. (2015).

Europium—Different from Ba, Eu is mostly enhanced by the r-processes. In Figure 37, the average [Eu/Fe] ratio is already super-solar at $[\mathrm{Fe}/{\rm{H}}]\ll -3$ in the s + r model. This plateau value, $\sim +0.5$, depends on the MRSN rates, and 3% of HNe (Section 2.1.2) is chosen to match these observations. ¹⁷ From $[\mathrm{Fe}/{\rm{H}}]\sim -1$, the model [Eu/Fe] decreases by SNe Ia, reaching [Eu/Fe] = 0.038 at $[\mathrm{Fe}/{\rm{H}}]=0$, consistent with the solar ratio. This trend is in excellent agreement with the observations. At very high metallicities, although with a scatter, the NLTE abundances from Zhao et al. (2016) may be slightly lower than in the s + r model. The dotted–dashed line shows a model without MRSNe; clearly it is not possible to explain the observed [Eu/Fe] ratios with NSMs alone, and the contribution from MRSNe is necessary.

Lead—Pb is also a characteristic element for AGB stars and belongs to the third peak of the s-process, in contrast to Ba. The observational data are very limited, and can be 0.2–0.4 dex underestimated in the LTE analysis (Mashonkina et al. 2012). This is why we adopt the meteoritic solar abundance for Pb (Section 2.2). In Figure 38, the evolutionary trend is similar to that of Ba, and the model prediction is lower than the observations at $-2\lesssim$ $[\mathrm{Fe}/{\rm{H}}]\lesssim -1.5$, which could also be improved with inhomogeneous enrichment in chemodynamical simulations. At $[\mathrm{Fe}/{\rm{H}}]\lesssim -2$, a small amount of Pb is produced by MRSNe, but the predicted plateau value is lower than the four measurements at $[\mathrm{Fe}/{\rm{H}}]\sim -3$.

Zoom In Zoom Out Reset image size

Using our GCE model, which includes neutron-capture processes, we summarize the origin of elements in the form of a periodic table. In each box of Figure 39, the contribution from each chemical enrichment source is plotted as a function of time: Big Bang nucleosynthesis (black), AGB stars (green), core-collapse supernovae including SNe II, HNe, ECSNe, and MRSNe (blue), SNe Ia (red), and NSMs (magenta). It is important to note that the amounts returned via stellar mass loss are also included for AGB stars and core-collapse supernovae depending on the progenitor star mass. The x-axis of each box shows time from t = 0 (Big Bang) to 13.8 Gyr, while the y-axis shows the linear abundance relative to the Sun, X/X_⊙. The dotted lines indicate the observed solar values, i.e., $X/{X}_{\odot }=1$ and 4.6 Gyr for the age of the Sun. Since the Sun is slightly more metal-rich than the other stars in the solar neighborhood (Figure 1), the fiducial model goes through [O/Fe] = [Fe/H] = 0 slightly later compared with the Sun's age. Thus, a slightly faster star formation timescale (${\tau }_{{\rm{s}}}=4\,\mathrm{Gyr}$ instead of 4.7 Gyr) is adopted in this model. The evolutionary tracks of [X/Fe] are almost identical. The adopted star formation history is similar to the observed cosmic SFR history, and thus this figure can also be interpreted as the origin of elements in the universe. Note that Tc and Pm are radioactive. The origin of stable elements can be summarized as follows:

• 1.  
  H and most of He are produced in Big Bang nucleosynthesis. The small green and blue areas also include the amounts returned to the ISM via stellar mass loss and some He newly synthesized in stars. The Li model is very uncertain because the initial abundance and nucleosynthesis yields are uncertain (see also Grisoni et al. 2019). Be and B are supposed to be produced by cosmic rays (Prantzos et al. 1993), which are not included in our model either.
• 2.  
  49% of C, 51% of F, and 74% of N are produced by AGB stars (at t = 9.2 Gyr). Note that extra production from Wolf–Rayet stars is not included and may be important for F (Jönsson et al. 2014; Spitoni et al. 2018). For the elements from Ne to Ge, the newly synthesized amounts are very small for AGB stars, and the small green areas are mostly for mass loss.
• 3.  
  α elements (O, Ne, Mg, Si, S, Ar, and Ca) are mainly produced by core-collapse supernovae, but 22% of Si, 29% of S, 34% of Ar, and 39% of Ca come from SNe Ia. These fractions would become higher with sub-Ch-mass SNe Ia instead of Ch-mass SNe Ia in this model (Kobayashi et al. 2020). Therefore, in the [X/Fe]–[Fe/H] relations, the slopes of the decrease from $[\mathrm{Fe}/{\rm{H}}]\sim -1$ to ∼0 are shallower for these elements than those for O and Mg.
• 4.  
  A large fraction of Cr, Mn, Fe, and Ni are produced by SNe Ia. In classical works, most of Fe was thought to be produced by SNe Ia, but the fraction is only 60% in our model, and the rest is mainly produced by HNe. The inclusion of HNe is very important as it changes the cooling and star formation histories of the universe significantly. Co, Cu, Zn, Ga, and Ge are largely produced by HNe.
• 5.  
  Among neutron-capture elements, as predicted from nucleosynthesis yields, AGB stars are the main enrichment source for the s-process elements at the second (Ba) and third (Pb) peaks.
• 6.  
  32% of Sr, 22% of Y, and 44% of Zr can be produced from ECSNe even with our conservative mass ranges, which are included in the blue areas. Combined with the contributions from AGB stars, it is possible to perfectly reproduce the observed trends (Figures 33–35), and no extra LEPP is needed. The inclusion of ν-driven winds in GCE simulation results in a strong overproduction of the abundances of the elements from Sr to Sn with respect to the observations.
• 7.  
  For the heavier neutron-capture elements, contributions from both NS–NS/NS–BH mergers and MRSNe are necessary, and the latter is included in the blue areas.

Zoom In Zoom Out Reset image size

In this model, the O and Fe abundances go though the cross of the dotted lines, meaning [O/Fe] =  [Fe/H] = 0 at 4.6 Gyr ago. This is also the case for some important elements including N, Ca, Cr, Mn, Ni, Zn, Eu, and Th. Mg is slightly underproduced in the model. The underproduction of the elements around Ti is a long-standing problem. The s-process elements are slightly overproduced even with the updated s-process yields in this paper. Notably, Ag is overproduced by a factor of six, while Au is underproduced by a factor of five. U is also overproduced. These problems may require revising nuclear reaction rates.

As discussed in previous sections, these GCE predictions mainly depend on the input nucleosynthesis yields, and hence it is difficult to quantify the uncertainties, i.e., theoretical error bars. Very roughly speaking, α-element abundances can vary $\sim \pm 0.2$ dex depending on the detailed physics during hydrostatic stellar evolution, e.g., mass loss, convection, rotation, and magnetic fields. However, the ratios among α elements such as O/Mg and Si/S do not so much depend on these, but on the ¹²C(α, γ)¹⁶O reaction rate (K06). C, N, and odd-Z element abundances largely depend on rotation ($\sim +0.5$ dex), which can be found in Limongi & Chieffi (2018), as well as any mixing of hydrogen into the He-burning layer (K11).

Uncertainties in the treatment of mass loss and convection also affect AGB yields (e.g., Herwig 2005; Ventura & D'Antona 2005a, 2005b; Karakas & Lattanzio 2014; Karakas & Lugaro 2016 for C, N, F, and s-process elements). Observations of s-process elements in AGB stars can be used to constrain, for example, the efficiency of TDU mixing, but selection biases in observations mean that the whole stellar parameter space is not well sampled. Other observations including WDs in clusters (Marigo et al. 2020) or the number of carbon stars in a stellar population (Boyer et al. 2019) provide additional constraints but are also subject to their own biases. The uncertainties in AGB yields are difficult to quantify for the following reasons. If we vary one parameter, say we slow down the rate of mass loss in an intermediate-mass star with HBB, then the yields of C can change by a factor of ∼2, the yields of N by almost a factor of 10, while the yields of heavy elements can vary by almost a factor of 100 (using models from Karakas et al. 2012). Hence varying one input parameter can lead to different variations in yields for different elements, depending upon their production mechanism.

The uncertainties of the yields from core-collapse supernovae come from the lack of physical explosion models (Section 2.1) including neutrino heating and black hole formation. This could largely change the iron-peak element abundances relative to α elements, and thus we use the iron yields constrained from another independent observation—supernova light curves and spectra, which is not the case in the other supernova yields such as Woosley & Weaver (1995) and Limongi & Chieffi (2018). Once the iron yields are fixed, the ratios among iron-peak elements are not so flexible; in particular, the ratios among the elements formed in the same layer, i.e., Cr/Mn and Ni/Fe, are fixed (see more discussion on Mn and Ni in Figures 26 and 29). The effect of jet-like explosions can be quantified by the difference between the K11 and K15 models in Figures 22–24, but an additional difference can be caused by the neutrino process. These two can cause a nonlinear effect, and it is necessary to use 3D simulations to quantify the error bars. SN Ia yields also depend on an explosion mechanism, but the main uncertainties are caused by the modeling of progenitor systems including the WD masses (Section 2.1.3; see Kobayashi et al. 2020, for more discussion).

Among s-process elements, relative discrepancies between different abundances, such as the overproduction of Ba and Pb with respect to Sr, may be related to the physics of the nucleosynthetic models (Cristallo et al. 2015), for example, the extent of the PMZ ${M}_{\mathrm{mix}}$ in AGB stars discussed above and/or the occurrence of relatively slow, diffusive mixing within the pocket (Battino et al. 2019). The contribution of ECSNs and/or ν-driven winds to the abundances of Sr, Y, and Zr should also be considered, as well as the possible effects of the intermediate neutron-capture process (Hampel et al. 2016, 2019; the i process); see, e.g., Côté et al. (2018), whose resulting abundance pattern and stellar site are however currently unknown and debated (Banerjee et al. 2018; Clarkson et al. 2018; Denissenkov et al. 2019). Future analysis of the production of the isotopic rather than elemental abundances in the solar system will be a powerful tool to constrain the models in more detail.

The r-process nucleosynthesis is the cutting edge of nuclear astrophysics, and there are only a small number of yields available (Section 2.1.2). The dependence on initial conditions, e.g., the progenitor mass for ECSNe/MRSNe and the compact object masses for NSMs, have not been studied yet. The result also depends on the detailed modeling, e.g., numerical resolution, dimensionality, inclusion of general relativity, and modeling of neutrino physics, as well as nuclear reactions. It should also be kept in mind that major uncertainties are present in the nuclear physics inputs of the r-process model calculations, including the mostly theoretical predictions for nuclear masses and of fission fragments (Kajino et al. 2019). We note that all previous GCE models of the neutron-capture elements in the Milky Way (Travaglio et al. 2004; Prantzos et al. 2018) included the r-process yields as calculated using the r-residuals method, i.e., by subtracting the s-process contribution from the solar system abundances, and assumed the universal r-process to occur in SNe II. Here, instead we implemented a first-principle approach by including the r-process nucleosynthesis yields calculated for various r-process astrophysical sites, so that the mismatches between the model predictions and observations can be used to deepen our understanding of nuclear astrophysics.

Enrichment sources produce different elements on different timescales, and thus the time evolution of the elements varies as a function of location in a galaxy, depending on the star formation history. In Figure 40, we show the evolution of elemental abundance ratios [X/Fe] against [Fe/H] for the solar neighborhood (blue solid lines), halo (green short-dashed and cyan dotted lines), bulge (red long-dashed and olive dotted–short-dashed lines), and thick disk (magenta dotted–long-dashed lines), where the contributions from AGB stars, ECSNe, NSMs, and MRSNe are included (and those from the ν-driven winds are not).

Zoom In Zoom Out Reset image size

Bulge and thick disk—If the star formation timescale is shorter than in the solar neighborhood (blue solid lines), as in our bulge (red long-dashed and olive dotted–short-dashed lines) and thick disk (magenta dotted–long-dashed lines) models, the contribution from stars of a given lifetime appear at a higher metallicity than in the solar neighborhood. Intermediate-mass AGB stars, low-mass AGB stars, and SNe Ia start to contribute at $[\mathrm{Fe}/{\rm{H}}]\sim -2.5,-1.5$, and −1, respectively, in the solar neighborhood, but at a higher [Fe/H] in the bulge and thick disk models. The [(C, N, F)/Fe] ratios peak at higher metallicities (see K11 for more discussion). At $[\mathrm{Fe}/{\rm{H}}]\gtrsim -1$, [α/Fe] is higher and [Mn/Fe] is lower than in the solar neighborhood, which is consistent with observations (e.g., Bensby et al. 2004a; Johnson et al. 2014).

Similar behavior is also expected for neutron-capture elements. With our s + r models, [s/Fe] are lower in the bulge and thick disk than in the solar neighborhood, at a given metallicity with $[\mathrm{Fe}/{\rm{H}}]\gtrsim -1.5$. If the contribution from NSMs is larger, then [r/Fe] would also be lower. The difference between the two bulge models is seen only at the metal-rich end, where the outflow model shows slightly lower [α/Fe] and higher [r/Fe] than the wind model where star formation is totally quenched. These trends seem very different from observations. Johnson et al. (2012) showed that [(La, Nd, Eu)/Fe] ratios are positive at $[\mathrm{Fe}/{\rm{H}}]\lesssim -1$ and decrease from $[\mathrm{Fe}/{\rm{H}}]\sim -1$ to higher metallicities, which is similar to the [α/Fe]–[Fe/H] relations. This is not produced in our s + r models, and may suggest a different origin of neutron-capture elements. In Figure 41, the green dashed lines show the outflow bulge model with a different contribution of r-processes; the fraction of MRSNE to HNe decreases from 3% to 2%, so that the plateau value of [Eu/Fe] is consistent with observations. In the blue dotted lines, the contribution from AGB stars is halved as an experiment. [Zr/Fe] matches with the observations, but not [La/Fe] and [Nd/Fe]. However, it is still not possible to reproduce the abundances of Zr, La, and Nd, simultaneously. Similar observational results are also shown by Lucey et al. (2019).

Zoom In Zoom Out Reset image size

Halo—If the chemical enrichment efficiency is lower than in the solar neighborhood, as in our standard halo model (green short-dashed lines), the contribution from AGB stars and NSMs becomes larger compared with that from core-collapse supernovae, and thus [(s, r)/Fe] ratios are higher than in the solar neighborhood at $[\mathrm{Fe}/{\rm{H}}]\gtrsim -2$. If the chemical enrichment timescale is also shorter than in the solar neighborhood, as in our second halo model with stronger outflow (light-blue dotted lines), [(s, r)/Fe] ratios become even higher toward higher metallicities. The number of these relatively metal-rich halo stars should be very small, but by finding those stars with a large survey and measuring their various elemental abundances, it may be possible to reconstruct the star formation history of the Galactic halo.