Cosmic abundances and galactic chemical evolution

Measuring the composition of the universe

To know how much hydrogen, helium, carbon or iron there is in the universe we cannot collect samples from a star and bring them to the laboratory. What we can do is read the fingerprints that atoms leave in light: each element, heated in the atmosphere of a star, absorbs light at specific frequencies, forming an absorption spectrum with lines at positions exactly predictable from the quantum mechanics of atomic levels. By measuring the depth, shape and Doppler width of these lines, and comparing them with accurate radiative-transfer calculations in a model stellar atmosphere, we can deduce how much of that element is present in the photosphere and — under the standard mixing assumption — in the whole star below the outer convection zone.

The most detailed photograph we have of the “average” composition of the matter of the universe is that of the Sun — the nearest and best measured star — and that of the CI chondritic meteorites, the most primitive samples of the Solar System. Sun and CI chondrites are in excellent agreement for the non-volatile elements (within $\pm 10\%$ for most elements), a convergence that reflects the formation of the Solar System from gas and dust of homogeneous composition and that constitutes the reference point of all astrophysics. The four known CI chondrites — Ivuna, Alais, Orgueil, Tonk, fallen between 1806 and 1938 — have been analyzed with progressively better precision techniques (multicollector ICP-MS, neutron activation, secondary ion mass spectrometry, accelerator mass spectrometry) and constitute the “reference sample” of Solar System abundances for the heavy elements. The systematic differences between Sun and CI are well characterized: the Sun is slightly poorer in Li, Be and B because these elements are destroyed by proton burning in the layer below the convection zone during the main sequence, while the CI are poorer in volatiles (H, C, N, O, Ne, Ar) because these escaped during the formation and evolution of the asteroidal parent bodies.

Abundance scales

Cosmic abundances are expressed on different scales depending on the context. The spectroscopic logarithmic scale is the reference for comparisons among stars: $\log \epsilon(\mathrm{X}) \equiv \log_{10}(N_{\mathrm{X}}/N_{\mathrm{H}}) + 12$, normalized to $\log \epsilon(\mathrm{H}) = 12$ by definition. The mass fraction $X_i = M_i/M_{\mathrm{tot}}$, with $\sum_i X_i = 1$, is natural for stellar models: conventionally $X = X_\mathrm{H}$, $Y = X_\mathrm{He}$, and $Z = \sum_{i > 2} X_i$ is the so-called “metallicity”; for the modern Sun $X_\odot \approx 0{.}74$, $Y_\odot \approx 0{.}24$, $Z_\odot \approx 0{.}014$. The number fraction $Y_i = X_i/A_i$ is the natural one for writing nuclear reactions. For individual stars, the metallicity relative to the Sun is $[\mathrm{Fe/H}] = \log(N_\mathrm{Fe}/N_\mathrm{H})_\star - \log(N_\mathrm{Fe}/N_\mathrm{H})_\odot$, and analogous ratios $[\mathrm{X/Y}]$ measure relative abundances against the corresponding solar proportion. Iron anchors the metallicity scale not because it dominates the metal budget by mass — oxygen is far more abundant — but because it offers hundreds of measurable lines in optical spectra.

The standard compilations of reference solar abundances are three, produced by different groups with partially overlapping methodologies. The AGSS09 compilation (Asplund, Grevesse, Sauval, Scott 2009) was the first to systematically integrate 3D radiative-hydrodynamic simulations of the solar atmosphere and the NLTE (non-Local-Thermodynamic-Equilibrium) analysis of the main lines, and produced $Z_\odot = 0{.}0134$. The AAGS21 update [Asplund et al. 2021] further refines the atomic models for Fe-peak and heavy elements, bringing $Z_\odot = 0{.}0139$. The Lodders (2020) compilation [Lodders 2020] focuses on the meteoritic revisitation for the non-volatile elements and produces $Z_\odot = 0{.}0156$, slightly higher. The historical GS98 compilation (Grevesse-Sauval) and its updates [Grevesse et al. 2010] , based on pre-3D-NLTE 1D LTE atmospheres, gives $Z_\odot = 0{.}0170$ — about $0{.}13$-$0{.}15$ dex higher than AAGS21 for CNO. The differences among these compilations — apparently small numerically — have profound consequences for solar interior models, and are at the heart of the so-called Solar Modeling Problem discussed below.

The solar compilation

The solar photospheric abundances, measured in the outer layer of the star through the spectrum that reaches us on Earth, constitute the modern reference of astrophysics. Each new analysis method has progressively revised them downward by a fraction, and after the introduction of 3D simulations of solar convection and of the NLTE analysis of atomic and molecular lines (Asplund, Grevesse, Sauval and collaborators, in the 2000s), the most important abundances — C, N, O in particular — were reduced by a substantial fraction with respect to the values used for decades in stellar simulations. This revision matters because the photospheric abundances enter directly as input to solar interior models and to stellar models in general: changing $Z_\odot$ changes the opacity of the radiative zone, the thermal structure of the core, and the predicted evolution.

The values of the photospheric solar abundances in the three reference compilations for the main elements are summarized in the following table:

The overall metallicity $Z$ is $0{.}0134$ in AGSS09, $0{.}0139$ in AAGS21, $0{.}0170$ in GS98. The reduction of the CNO abundances is the main consequence of the 3D NLTE analysis: in the earlier 1D LTE models, the three-dimensional inhomogeneity of the convection and the deviation from the thermal distribution of atomic populations in the absorption lines were masked, systematically leading to overestimates of the photospheric abundances.

The measurement techniques vary significantly from element to element. Helium is not observable in the solar photosphere (its lines are too weak at $T_{\mathrm{eff}} = 5777$ K) and is measured indirectly through helioseismology — the inversion of the p-mode oscillation frequencies observed by networks such as BiSON and GONG — or through the spectroscopy of coronal prominences (at $T \gtrsim 10^{4}$ K). Carbon, nitrogen and oxygen are measured through a combination of atomic lines (permitted and forbidden C I, N I, O I) and molecular lines (CH, CN, CO, NH, OH); the analysis requires 3D NLTE simulations because the lines are saturated, sensitive to the thermal gradient of the photosphere, and sensitive to turbulent-mixing effects. Iron provides the basis of the metallicity scale thanks to hundreds of Fe I and Fe II lines available in the visible spectrum. The noble gases Ne and Ar are not observable in the photosphere and are measured indirectly from the solar wind or the coronal regions, with FIP (First Ionization Potential) coefficients applied to correct the abundance bias of the wind. For the low-abundance elements — Mn, V, Sc, and the s/r-process elements such as Eu and Ba — one uses a few single lines or data points with larger uncertainties ($\pm 0{.}05$-$0{.}15$ dex).

A technical controversy remains open on the solar oxygen abundance. AGSS09 and AAGS21 converge on $\log\epsilon(\mathrm{O}) = 8{.}69$, in tension with the requirements of the standard solar model (see the next section). Alternative analyses (Caffau et al. 2011 with CO5BOLD, Bergemann et al. 2021 with independent NLTE methods) suggest values $\sim 0{.}05$ dex higher that would reduce the tension. Community consensus is not yet consolidated, and this $\sim 0{.}05$ dex ambiguity on solar oxygen is — by an irony of fate — one of the most important numbers of quantitative stellar astrophysics.

The Solar Modeling Problem

There is a twenty-year tension that has animated the debate of the solar and stellar community: the standard solar model (SSM), built with the AGSS09/AAGS21 abundances, predicts an internal structure that is not in full agreement with the constraints derived from helioseismology, that is, from the frequencies of millions of acoustic oscillation modes observed with precision $< 10^{-4}$ on the surface of the Sun. The SSM that use the older GS98 abundances (with higher CNO) agree significantly better with all the helioseismic diagnostics. The open question is: is the 3D NLTE analysis wrong, or is the solar model incomplete?

The Solar Modeling Problem manifests itself concordantly in three independent diagnostics. The depth of the convection zone is $r_{\mathrm{cz}}/R_\odot = 0{.}7133 \pm 0{.}0005$ from helioseismology; AGSS09 SSM predict $0{.}722$, in tension at $\sim 1{.}5\sigma$ with the observed value, while GS98 SSM produce $0{.}7133$ in perfect agreement. The helium abundance in the convective surface $Y_s$ is $0{.}2485 \pm 0{.}0035$ from helioseismology (inversion of the small frequency separation); AGSS09 SSM predict $0{.}2426$, while GS98 SSM predict $0{.}2485$. The sound-speed profile $c(r)$ — derived from global inversion of $\sim 5000$ p-mode frequencies of Sun-as-a-star spectroscopy — shows differences up to $\sim 1\%$ between the AGSS09 model and the data, especially in the region just below the convection zone, while GS98 is consistent within the precision. AAGS21 SSM (with slightly higher metallicity than AGSS09) reduce the tension but do not resolve it. Updated SSM (Vinyoles et al. 2017, Magg et al. 2022) synthesize these seismic constraints.

The proposed solutions unfold along three complementary directions. The revision of opacities is the leading candidate: direct measurements of the Fe opacity at solar temperatures by Bailey et al. (2015) with the Sandia Z-machine showed values $\sim 25\%$ higher than the theoretical OP and OPAL calculations used as standard in SSM codes. A complete revision of the opacities in the radiative zone could absorb a large part of the tension, reducing the sensitivity of the SSM to the exact CNO abundances. Independent opacity measurements at the Z-machine and at other high-energy-density facilities are under way. The revision of the atomic diffusion and settling coefficients in stellar models modifies the predicted surface helium depletion and could contribute to the $Y_s$ gap. The addition of convective overshooting at the base of the convection zone marginally shifts the predicted $r_{\mathrm{cz}}$. And finally, further revisions of the photospheric abundances of C, N, O toward higher values — as suggested by the independent analyses of Caffau and Bergemann — could reduce the tension without requiring new physics.

The strongest independent nuclear diagnostic of the Solar Modeling Problem comes from the flux of solar CNO neutrinos measured by Borexino at Gran Sasso. The historic 2020 detection [Collaboration 2020] provided the first direct measurement of the neutrinos produced by the CNO cycle in the solar core, with integrated flux $(6{.}6^{+2{.}0}_{-0{.}9}) \times 10^{8}$ cm$^{-2}$s$^{-1}$ at 68% C.L. — compatible with GS98 SSM and in marginal tension (about $25\%$) with AGSS09/AAGS21 SSM. The updated 2022 Borexino measurement [Collaboration 2022] , with increased statistics and refined analysis, shifts the flux to $6{.}7^{+1{.}2}_{-0{.}8} \times 10^{8}$ cm$^{-2}$s$^{-1}$ and confirms the preference for high solar metallicity, without being decisive. The CNO flux is directly proportional to the C+N abundance in the solar core and is therefore an almost direct probe of $Z$ in the energy-production region. Future precision measurements from Hyper-Kamiokande and JUNO will further reduce the uncertainties and contribute to settling the question definitively. The original standard solar model framework that made these comparisons precise is represented by Bahcall, Serenelli and Basu [Bahcall et al. 2005] .

The final resolution of the Solar Modeling Problem plausibly requires a combination of abundances partially revised toward higher CNO values, new opacities incorporating updated laboratory measurements, and minor refinements of the mixing and diffusion modeling. The next five to ten years — with the Hyper-Kamiokande and JUNO results on the neutrino front, the laboratory high-EDX campaigns on the opacity front, and the new-generation 3D NLTE analyses on the spectroscopy front — promise to close the chapter opened by AGSS09 in 2009.

Abundances in stars and the ISM

Once the composition of the Sun is known, we can compare it with that of other stars of the Galaxy. The oldest, most metal-poor stars — which lived billions of years ago, in epochs when the Milky Way was younger and less enriched by the previous stellar generations — tell us how the Galactic chemical composition evolved in time. Stars in regions of the Galaxy with different star-formation histories (halo, thick disk, thin disk, bulge, globular clusters) tell us where and when the Galaxy accumulated its metals. The abundances of the present interstellar gas complete the picture: comparing them with the stellar ones, we can estimate how much “light” and “heavy” matter is still free in space compared with how much has been locked into now-inert stars or compact remnants.

The main chemical components of the Milky Way are characterized by their ranges of $[\mathrm{Fe/H}]$ and $[\alpha/\mathrm{Fe}]$, and by mean stellar age, according to the following schematic picture:

The evolution of the $[\alpha/\mathrm{Fe}]$ ratio with $[\mathrm{Fe/H}]$ — the classic “knee” of Galactic chemical chronography — is discussed in detail in the second part of this chapter, devoted to the chemical evolution of the Milky Way. Here we simply note that the chemical map of the Galaxy is today produced by large-scale spectroscopic surveys with unprecedented resolution and numbers of stars. APOGEE (Sloan Digital Sky Survey) has characterized $\sim 10^{5}$ stars in the H band at resolution $R \sim 22\,500$, covering the disk and the bulge. GALAH at the AAT has characterized $\sim 10^{5}$ stars in the optical at $R \sim 28\,000$, focused on the disk. The Gaia-ESO Survey has characterized $\sim 10^{4}$ stars in clusters at high resolution. The next-generation surveys — WEAVE at the WHT, 4MOST at VISTA, PFS at Subaru — will produce between $10^{6}$ and $10^{7}$ stars at $R = 5000$-$20000$ over the next five years. For the interstellar gas, the main probes are the absorption spectroscopy of DLAs (damped Lyman-$\alpha$ systems at $z = 2$-$4$ against the light of background QSOs), the emission spectroscopy of HII regions (in the Milky Way and in nearby galaxies), the diffuse interstellar bands for molecules, and the X-ray emission of the hot gas of the Galactic corona and halo.

Some specific data play a particular diagnostic role. The Spite lithium plateau in metal-poor halo stars — the observation that EMP stars with $T_{\mathrm{eff}} > 5800$ K all have $A(\mathrm{Li}) \approx 2{.}2$, independently of metallicity — is in tension with the predictions of Big Bang nucleosynthesis combined with the Planck cosmological parameters (which would predict $A(\mathrm{Li}) \approx 2{.}7$): it is the so-called cosmological lithium problem, discussed in chapters 2 and 8. The r-only stars, such as Sneden’s classic CS 22892-052, show a universal r-process pattern for $Z = 56$-$80$, scalable within $\pm 0{.}1$ dex to the solar pattern (see chapter 6), and are today catalogued in more than 200 specimens by the R-Process Alliance. The Carbon-Enhanced Metal-Poor (CEMP) stars represent about $25\%$ of the stars with $[\mathrm{Fe/H}] < -2$ and divide into subtypes CEMP-no (probably enriched by Pop III supernovae with partial fallback), CEMP-s (enriched by mass transfer from an AGB companion), CEMP-r (a single ancestral r-process event) and CEMP-rs (a complicated combination). The solar twins — stars similar to the Sun within $\pm 100$ K in $T_{\mathrm{eff}}$, $\pm 0{.}1$ dex in $\log g$ and $\pm 0{.}1$ dex in $[\mathrm{Fe/H}]$ — allow very-high-precision differential measurements ($< 0{.}01$ dex) and have revealed the so-called Sun’s chemical anomaly (Meléndez et al. 2009): the Sun turns out to be marginally poor in refractory elements (Fe, Si, Mg, Al, Ti) compared with solar twins of equal $[\mathrm{Fe/H}]$, by $\sim 0{.}04$ dex. The preferred interpretation is that the terrestrial planets sequestered refractory matter from the nebula before the complete formation of the Sun, leaving the residual gas that fed the photosphere slightly depleted — a small but measurable signature of planet formation.

Extremely metal-poor stars and Population III

The oldest and most metal-poor stars that we can observe today tell us a unique story: they were probably born from material enriched by a single primordial supernova (or by a small number of them), and they carry the direct chemical signature of those events in their spectra. By studying them, we reconstruct properties of the first stars of the universe — the Population III — that we cannot see directly because they lived and died billions of years ago, when the universe was a few hundred million years old and chemical enrichment had just begun.

A handful of Extremely Metal-Poor stars (EMP, $[\mathrm{Fe/H}] < -3$) and Ultra Metal-Poor stars (UMP, $[\mathrm{Fe/H}] < -4$) have been identified over the last twenty years, mainly by the Hamburg/ESO Survey, by SDSS/SEGUE, and by the Australian SkyMapper surveys. The most extreme are summarized in the following table:

The abundance patterns of these stars point to core-collapse SN progenitors of $\sim 25$-$60\,M_\odot$ with the mixing-fallback model (Umeda-Nomoto, Heger-Woosley): during the explosion, the deep Si-burning layers are partially mixed before falling into the residual black hole, and only the outermost $\alpha$-elements escape with very little Fe — consistent with $[\mathrm{Fe/H}] \ll -3$ but moderate or elevated $[\mathrm{X/Fe}]$ for C, N, O, Mg. The chemical signature typically includes $[\mathrm{C/Fe}] \gtrsim +1$ in CEMP-no (dominant among UMP), the absence of significant r-process or s-process enrichments, and a variable excess of N and Mg compatible with specific Pop III models. The best-fit yields for SMSS 0313 — the object with the most stringent metallicity limit known — suggest a Pop III progenitor of $\sim 50\,M_\odot$ in mixing-fallback regime with intermediate explosion energy.

The evidence for surviving pure Pop III stars (at strictly zero metallicity) remains uncertain. High-mass Pop III ($M > 10\,M_\odot$), born from the fragmentation of primordial HI-H$_2$ clouds without dust and without metals, died more than 13 Gyr ago and contribute only as progenitors of the observed EMP stars. Low-mass Pop III ($M < 0{.}8\,M_\odot$), if they formed, would still be alive today: their search through progressively deeper surveys has so far produced no candidates. SMSS 0313, with $[\mathrm{Fe/H}] < -7{.}3$, nevertheless has spectroscopically detected carbon and nitrogen, and therefore is not pure Pop III. The preferred interpretation is that low-mass Pop III never formed in significant numbers, because the fragmentation of primordial clouds at low masses requires an efficient cooling mechanism (dust, metals, heavy molecules) that was absent before the first enrichment — the so-called fragmentation threshold. JWST is searching for galaxies at $z = 10$-$15$ with spectral signatures compatible with dominant Pop III populations, and some candidates have been reported with preliminary interpretation; the definitive picture will require further high-resolution spectra in the coming years.

State of the art: observed abundances

The characterization of cosmic abundances is today a mature program, with reference compilations stable at the level of the statistical uncertainty and with residual systematics of order $0{.}05$-$0{.}1$ dex on many key elements. Three main quantitative questions remain open. The first is the definitive convergence of $Z_\odot$: the reconciliation among AAGS21, Lodders 2020, and the SSM requirements is played out over a dozen hundredths of a dex in the CNO abundances, and is the heart of the Solar Modeling Problem. The second is the completion of the EMP/UMP star census: identifying the few hundred most extreme objects in the Galactic halo, in the satellite dwarfs and in stellar streams is essential to reconstruct the initial mass function and the yields of the first stellar generation. The third is the complete chemical mapping of the Galaxy: the 4MOST and WEAVE surveys will produce within the next five years a catalog of $10^{7}$ stars with $> 20$ elements measured for each, which will allow the data-driven reconstruction of the chemical and dynamical history of the Milky Way.

The prospects over 5-10 years are concrete and convergent. On the solar front, Hyper-Kamiokande (data-taking from 2028) and JUNO (from 2025) will measure with precision the flux of solar $^{8}\mathrm{B}$ and CNO neutrinos, providing direct constraints on the $Z$ of the solar core and contributing to the resolution of the SMP. On the spectroscopic front, ELT, GMT and TMT will allow very-high-resolution spectroscopy of EMP stars at $V \sim 18$-$20$ — an order of magnitude fainter than those accessible today — and the identification of new UMP candidates. On the space front, Gaia DR4 (expected 2026-2027) will complete the 6D kinematics of $10^{9}$ stars, in combination with the metals from APOGEE/GALAH/4MOST/WEAVE, providing the basis for the self-consistent chemodynamics of the Galaxy. JWST and the future next-generation telescopes will spectrally characterize galaxies at $z = 8$-$15$ and potentially identify Pop III populations still alive in the most primitive accessible systems. The combination of these fronts promises to take observational cosmochemistry from a discipline of refinement to a discipline of complete mapping within the next decade.

The cosmic abundances measured in the Sun, in meteorites, in stars and in the interstellar gas are the final test bench of all nucleosynthesis: the mechanisms described in the previous chapters — quiescent burnings, s and r processes, p-process, explosive supernova nucleosynthesis, NSM kilonovae — must together quantitatively reproduce the curve of the solar abundances and its evolution over Galactic time. The second part of this chapter is devoted to how these sites are integrated into models of Galactic chemical evolution that make direct predictions for the observed $[\mathrm{X/H}]$ vs $[\mathrm{Fe/H}]$ ratios, and to how the combination of stellar yields, IMF, SFR and the delay time distribution of SNe Ia shapes the chemical signature of the Milky Way we see today.

A galaxy as a history of generations

A galaxy has not always had the composition it has today. When the Milky Way was young, its gas contained almost only hydrogen and helium, with minimal traces of lithium from primordial nucleosynthesis (chapter 2): all the rest of the periodic table was cooked and dispersed by stars over the following thirteen billion years. Each generation of stars that is born, lives and dies enriches the interstellar gas a little more with “metals” — in astrophysical jargon, any element heavier than helium. The stars of the following generation are born from richer gas and are therefore more metallic at birth. By studying the composition of stars of different ages and from different regions of the Galaxy we can read this history directly in the spectroscopic properties of their photospheres: we see the chemical evolution of the Milky Way engraved in the chemical abundances of its stars, from the present Solar System down to the oldest halo EMP stars with $[\mathrm{Fe/H}] < -4$.

The equations of GCE

Galactic chemical evolution (GCE) is described in quantitative form by a set of conservation equations over a Galactic region treated as a “box” or, in the more sophisticated models, as a collection of interconnected “zones”. In the single-zone model, the basic equations are:

$$\frac{d \sigma_{\mathrm{gas}}}{dt} = -\psi(t) + E(t) + I(t) - W(t)$$ $$\frac{d \sigma_\star}{dt} = \psi(t) - E(t)$$

where $\sigma_{\mathrm{gas}}$ is the surface density of gas, $\sigma_\star$ that of stars, $\psi(t)$ the star formation rate (SFR), $E(t)$ the rate of mass return from dead stars (SN ejecta, AGB winds), $I(t)$ the infall of gas (cosmological and satellite accretion), $W(t)$ the outflow (galactic winds from SN feedback). For each tracked element $i$, the evolution obeys an analogous equation weighted by the mass fraction:

$$\frac{d (\sigma_{\mathrm{gas}} X_i)}{dt} = -\psi(t) X_i(t) + E_i(t) + I_i(t) X_{i,\mathrm{infall}} - W_i(t) X_i(t),$$

where the chemical ejecta term is

$$E_i(t) = \int_{M_\ast}^{M_{\mathrm{up}}} (M - M_R)\,Y_i(M)\,\psi(t - \tau(M))\,\xi(M)\,dM,$$

with $Y_i(M)$ the yield of the star of mass $M$ for element $i$, $\xi(M)$ the initial mass function (IMF), $\tau(M)$ the stellar lifetime, and $M_R$ the mass of the compact remnant. The integral is the quantitative heart of the model: it transforms the star-formation history and individual stellar physics into a measurable prediction for the chemical evolution of the gas. The main ingredients are the IMF (Salpeter $\xi \propto M^{-2{.}35}$, Kroupa, Chabrier — the latter used as standard in modern models); the SFR-history law (for example Schmidt-Kennicutt $\psi \propto \sigma_{\mathrm{gas}}^{1{.}4}$ or more direct assumptions on the SFH); the stellar yields (Sukhbold-Woosley for core-collapse SNe, Nomoto-Kobayashi for SNe Ia, Karakas-Lugaro [Karakas & Lattanzio 2014] and Cristallo et al. [Cristallo et al. 2015] for AGB); the infall and outflow models; and the delay time distribution (DTD) of SNe Ia. For a formal and complete treatment we refer to the monographs of Pagel [Pagel 2009] and Matteucci [Matteucci 2021] .

The public reference GCE codes include NuPyCEE (Côté et al., Python single-zone with NSM, novae, super-AGB), OMEGA (Côté et al., multi-zone with SFH database), gce4py (Andrews et al., Python with customizable yields) and Chempy (Rybizki et al., a Bayesian framework for GCE parameters). They typically integrate $\sim 50$-$1000$ tracked chemical species over meshes of $10$-$100$ zones, with runtimes of a few hours per complete Milky Way model. The recent methodological review of the discipline is Matteucci (2021) [Matteucci 2021] , while the didactic reference text remains Pagel (2009) [Pagel 2009].

From closed boxes to real galaxies

The earliest analytic models used a closed box: no inflow, no outflow, instantaneous mixing, and sometimes the instantaneous recycling approximation. They remain useful because they show what chemical evolution would look like if enrichment were purely internal — and they fail in instructive ways. A closed box predicts too many long-lived low-metallicity stars compared with what is observed in the solar neighborhood: this is the classic G-dwarf problem. The solution is not a small correction to the yields; it requires gas infall, which dilutes the interstellar medium and sustains star formation over time.

Outflows are equally important in low-mass galaxies. Supernova feedback can expel enriched gas from shallow gravitational potentials: this is why dwarf galaxies have lower metallicities than large spirals at equal gas fraction, and why their $[\alpha/\mathrm{Fe}]$ knees fall at lower $[\mathrm{Fe/H}]$. A model that ignores winds may fit the Milky Way disk poorly and fail entirely for the dwarf spheroidals.

The $[\alpha/\mathrm{Fe}]$ knee

One of the most characteristic signatures of Galactic chemical evolution is the curve of the $\alpha$-elements/Fe ratio as a function of metallicity. For very old, metal-poor stars ($[\mathrm{Fe/H}] \ll -1$), the $[\alpha/\mathrm{Fe}]$ ratio is elevated — about $+0{.}3$-$+0{.}4$ — because only core-collapse supernovae had time to explode and enrich the gas with their $\alpha$-rich chemical signature (chapter 5). For progressively younger and more metallic stars, $[\alpha/\mathrm{Fe}]$ descends monotonically toward zero, because the SNe Ia — which produce much Fe but few $\alpha$-elements — enter the scene with a delay time of a few hundred million years and contribute progressively more to the Fe budget of the interstellar gas. The point where the curve bends — the knee — marks the epoch when the SNe Ia began contributing to the Fe pool, and its exact position is a direct chemical chronometer of the star-formation history.

Quantitatively, the $[\alpha/\mathrm{Fe}]$ ratio works as follows. At $[\mathrm{Fe/H}] \ll [\mathrm{Fe/H}]_{\mathrm{knee}}$ only core-collapse SNe have contributed and $[\alpha/\mathrm{Fe}] \approx +0{.}3$-$+0{.}4$ forms a plateau. At $[\mathrm{Fe/H}] \gg [\mathrm{Fe/H}]_{\mathrm{knee}}$ the SNe Ia dominate the Fe production and $[\alpha/\mathrm{Fe}] \to 0$, reaching the solar value. The knee position $[\mathrm{Fe/H}]_{\mathrm{knee}}$ depends on the historical SFR of the environment: a high SFR means that many core-collapse SNe enriched the gas in Fe quickly before the SNe Ia entered the scene, so the knee falls at higher $[\mathrm{Fe/H}]$; a low SFR, conversely, leaves time for the SNe Ia to begin contributing while the metallicity is still low, and the knee falls at low $[\mathrm{Fe/H}]$. The observed knee position in different Galactic environments is therefore a quantitative indicator of the mean SFR:

The delay time distribution (DTD) of the SNe Ia is the key physical parameter that determines the knee position for a given SFR. Observations in galaxy clusters (Maoz, Mannucci 2012) and the statistical analysis of SNe Ia in host galaxies with measured SFR support a power-law DTD $\Psi(\tau) \propto \tau^{-1}$ for $\tau > \tau_{\min} \approx 40$ Myr, physically motivated by the gravitational inspiral of WD binaries in the Double Degenerate scenario (chapter 5). The GCE models that quantitatively reproduce the Milky Way knee require this DTD. The SN Ia yields remain uncertain at $\pm 30\%$ for Fe and $\pm 50\%$ for Mn and Cr, and the different explosion scenarios (Chandrasekhar W7 deflagration, Chandra DDT, sub-Chandrasekhar double detonation) produce different yields. The Mn/Fe ratio is the most sensitive diagnostic to separate the channels: high for Chandra DDT (efficient $e^{-}$ capture at $\rho > 10^{9}$ g/cm$^{3}$), low for sub-Chandra at lower central density — and the Mn/Fe pattern observed as a function of $[\mathrm{Fe/H}]$ in disk and halo stars progressively constrains the partition among channels.

Nuclear yields as input

The GCE model is only as accurate as the yields of each type of dying star: core-collapse SNe, SNe Ia, AGB, NSM, novae. These yields are computed by the stellar nucleosynthesis models discussed in the previous chapters, and the yield uncertainties are — by far — the dominant uncertainties of modern GCE. The yield tables used in current GCE models cover the main contributors:

• Core-collapse SNe: Sukhbold-Woosley (2018) for $13$-$120\,M_\odot$ via the KEPLER code; Limongi-Chieffi (2018) FRANEC with differential rotation; Pignatari-Herwig (2016) NuGrid.
• SNe Ia: Iwamoto W7 (1999) for Chandrasekhar deflagration; Travaglio et al. (2004); Seitenzahl 3D-DDT (2013); Shen-Townsley for sub-Chandra.
• AGB: Karakas-Lugaro (2014) Monash [Karakas & Lattanzio 2014] ; Cristallo et al. (2015) FRUITY [Cristallo et al. 2015] ; Ventura-D’Antona (2018) ATON.
• NSM: Wanajo et al. (2014) for detailed kilonova; variants over the $Y_e$ distribution of the ejecta.
• Novae: José-Hernanz (2011); Denissenkov et al. (2014).

The IMF-integrated yields (Kroupa or Salpeter) per unit stellar generation — defined as $\eta_X = \int Y_X(M)\,\xi(M)\,dM / \int M\,\xi(M)\,dM$ — are tabulated in Kobayashi, Karakas and Lugaro (2020) [Kobayashi et al. 2020] and are direct input for modern GCE. The partition among dominant contributors for the main elements is summarized in the following table:

The sensitivity of GCE models to individual yields is significant and varies from element to element. The C, N, O yields are sensitive to the treatment of mass loss in massive stars and of Hot Bottom Burning (HBB) in intermediate-mass AGB, with $\pm 30\%$ variation among models. The Mn yield depends on the dominant SN Ia channel and produces an observable Mn/Fe ratio that distinguishes Pop II stars (low, sub-Chandra dominant) from Pop I (high, Chandra DDT contributing). The Eu yield depends on the NSM/collapsar mix and on the NSM DTD: the scatter of $[\mathrm{Eu/Fe}]$ vs $[\mathrm{Fe/H}]$ at low metallicities is a direct diagnostic of the rarity of individual r-process events. The Pb yield (strong s component) depends on the AGB mass loss at low metallicity and on the $^{13}\mathrm{C}$ pocket (chapter 4). The standard iterative calibration procedure is: yields as input → GCE model → comparison with the observed $[\mathrm{X/Fe}]$ vs $[\mathrm{Fe/H}]$ in large surveys → revision of ambiguous yields → iteration. The constraint is progressively refined by adding independent data such as those from presolar grains (chapter 4) and from the high-resolution spectroscopy of nearby stars.

Multi-zone models of the Milky Way

The first chemical-evolution models treated the Milky Way as a homogeneous box — a useful idealization for the pioneering calculations (Tinsley in the 1970s-80s), but manifestly inadequate for the real galaxy. We know today that our galaxy has a rich structure: a thin disk and a thick disk with different kinematics and chemistry, an inner and an outer halo, a bulge with its own evolutionary history, globular clusters with multiple-population compositions, and a series of satellite galaxies in various stages of accretion. Each of these environments has a different star-formation history and, consequently, a different chemical composition. Modern models divide the Galaxy into multiple zones, each with its own gas-star balance and with exchanges (gas infall, SN-driven outflow, radial mixing by stellar migration) among neighboring zones.

The standard multi-zone model of the Milky Way (Matteucci-Romano 2002 and successors) divides the Galaxy into halo + bulge + disk with distinct SFR and infall timescales, and the disk into $\sim 20$ concentric annuli between $R = 2$ kpc and $R = 22$ kpc, each with time-dependent SFR and infall. Radial mixing by stellar migration is implemented ad hoc, calibrated on the observed chemical gradients. The notable results of the model include the observed radial metallicity gradient $d[\mathrm{Fe/H}]/dR \approx -0{.}05$ dex/kpc in the outer disk — naturally explained by radius-dependent SFR and infall, with the inner disk faster in its formation and therefore chemically more evolved. The inside-out formation sees the bulge and the inner disk form first on a timescale of $\sim 1$ Gyr, the outer disk last on a timescale of $\sim 7$ Gyr, in agreement with the stellar ages mapped by Gaia DR3 and APOGEE. The two-infall model envisages a first rapid accretion phase (formation of halo + thick disk in $\tau \sim 1$ Gyr) followed by a second slow phase (formation of the thin disk in $\tau \sim 7$ Gyr), and quantitatively explains the observed “jump” of $[\alpha/\mathrm{Fe}]$ between thick and thin disk and the double sequence in $[\alpha/\mathrm{Fe}]$ vs $[\mathrm{Fe/H}]$ seen in APOGEE and GALAH.

On the cosmological front, the zoom-in hydrodynamic simulations of Milky Way-like galaxies — EAGLE, FIRE, IllustrisTNG, AURIGA, NIHAO — integrate N-body, SPH/AMR/MFM hydrodynamics, and sub-grid chemistry tracking $\sim 10$-$20$ chemical species with IMF-integrated yields, over cosmological volumes from the primordial collapse to today. They qualitatively reproduce the observed Galactic chemical evolution: the origin of the thick disk as a product of early internal star formation heated by gas-rich mergers or disk instabilities; the chemical inhomogeneity expected at low $[\mathrm{Fe/H}]$ with broad scatter consistent with the observations of EMP stars; the radial migration in which stars move in radius over time by scattering with spiral arms and the bar, mixing chemical populations of different radii and producing $[\mathrm{Fe/H}]$ dispersion at fixed age. The high-resolution simulations (FIRE-3, AURIGA Plus) reach $\sim 100$ pc resolution in the Galactic disk and trace the chemical structure of the disk in detail — a level of realism that allows quantitative comparison with the Gaia + APOGEE/GALAH maps of the present Milky Way.

Satellite galaxies and accretion flows

The Milky Way did not form in isolation: over its thirteen billion years of life it has swallowed dozens of smaller galaxies, accreting their stars and (in earlier epochs) their gas. By studying the chemical and kinematic signatures of halo stars — combining high-resolution spectroscopy with the proper motions and parallaxes measured by Gaia — astronomers can today distinguish the “indigenous” stars born in the original Milky Way from the “immigrant” ones accreted from satellite galaxies now dissolved or still undergoing dynamical mixing. This decomposition, unthinkable before Gaia DR2 (2018), is one of the most active frontiers of Galactic archaeology.

The identified and partially accreted satellite galaxies include several notable systems. The Sagittarius dwarf spheroidal (Sgr dSph) is in active accretion, with stellar streams clearly visible across many degrees of the sky. Gaia-Enceladus (or the Sausage) is a galaxy accreted about 10 Gyr ago, identified by Belokurov and Helmi in 2018 through the combination of $[\alpha/\mathrm{Fe}]$-$[\mathrm{Fe/H}]$ patterns and retrograde kinematics in the inner halo; its stars contribute significantly to the thick-disk and inner-halo populations, and the progenitor system probably had a mass of $\sim 10^{10}\,M_\odot$ at the epoch of accretion — comparable to the present LMC. Sequoia, Thamnos, the Helmi streams and other smaller streams have subsequently been identified with analogous techniques. The splashed stars of the thick disk — stars initially formed in the disk and then scattered vertically onto halo orbits by a massive accretion event — complete the picture of the inner-halo population.

The implications for GCE are profound. The chemical composition of the thick disk and of the inner halo is not that of a population formed in situ in a single evolution, but a mixture of heterogeneous populations with different star-formation histories. The integrated yields we observe therefore reflect the combination of the SFH of the original Milky Way and the SFHs of the accreted satellites. The Gaia-Enceladus stars, for example, show a mean $[\mathrm{Fe/H}] \approx -1{.}3$ (poorer than the in-situ thick disk), $[\alpha/\mathrm{Fe}] \approx +0{.}3$ (consistent with rapid formation before the SN Ia knee), and a specific $[\mathrm{Eu/Fe}]$ pattern distinguishable from that of the in-situ thick disk — a picture chemically consistent with a dwarf system of intermediate mass comparable to the surviving dSph (Sculptor, Fornax). The approach of chemical tagging — the identification of groups of co-formed stars through their multi-element chemical signature ($> 10$ elements) — is the promising frontier for disaggregating the accretion history of the Galaxy, and the APOGEE and GALAH surveys devote a significant fraction of their observing time precisely to this program.

Open problems

The general picture of Galactic chemical evolution is today consolidated in its main lines, but a series of open quantitative problems animates current research. Three families of questions are particularly alive.

The first is the primary nitrogen problem. Nitrogen is produced mainly as primary (that is, from carbon and oxygen synthesized in the same star) through Hot Bottom Burning in intermediate-mass AGB stars ($M = 4$-$7\,M_\odot$), which have timescales of $\sim 100$-$300$ Myr. At very low metallicity ($[\mathrm{Fe/H}] < -3$), these AGB have not yet had time to evolve and contribute to the interstellar gas when the first metal-poor stars form, so $[\mathrm{N/Fe}]$ should be very low or even $-\infty$. Observations instead show $[\mathrm{N/Fe}] \approx 0$ in many EMP stars, with significant scatter. The proposed solutions include fast rotation in massive Pop III stars (Hirschi 2007 and successors), which activates rotational mixing of N through the CNO cycle in the He shells during the main sequence, and the fragmentation of primordial clouds with a specific signature (Sharda 2021). The problem is not yet definitively solved.

The second is the problem of the $[\alpha/\mathrm{Fe}]$ plateaus. The $[\mathrm{O/Fe}]$, $[\mathrm{Mg/Fe}]$, $[\mathrm{Si/Fe}]$ ratios observed in metal-poor stars form remarkably flat plateaus around $+0{.}4$ for $-3 < [\mathrm{Fe/H}] < -1$, with observed scatter $< 0{.}1$ dex. This flatness requires either core-collapse SN yields that are very stable as a function of progenitor mass (in tension with what stellar models predict, which show $\sim 0{.}3$ dex variation), or efficient gas mixing on large scales that erases the event-by-event stochastic fluctuations. At $[\mathrm{Fe/H}] < -3$, the scatter grows significantly ($\pm 0{.}3$ dex), suggesting that the most extreme EMP stars have indeed seen few nucleosynthesis events — sometimes a single progenitor SN II — whose individual chemical signature is still readable in the spectrum.

The third is the problem of the missing Pop III stars. The theoretical models of first-star formation (Bromm, Yoshida, Hirano, Stacy) predict that Pop III had a top-heavy IMF with characteristic masses $M_\ast \sim 10$-$300\,M_\odot$ as a consequence of the absence of the metals and dust that trigger the fragmentation of primordial clouds. The highest-mass Pop III ($140 < M < 260\,M_\odot$) end their lives as pair-instability supernovae (PISN) with very distinctive yields: no Fe-peak element heavier than Ni, $^{56}\mathrm{Ni}$ produced up to $\sim 10\,M_\odot$ per event, anomalous $[\mathrm{Mg/Ca}]$ pattern and high $[\mathrm{Cr/Mn}]$, complete absence of r-process. No metal-poor star observed so far convincingly shows this chemical signature: the historical candidate was the UMP star SDSS J102915+172927 (the “Caffau star”), but later analyses showed that its pattern is more consistent with a standard core-collapse SN progenitor of $\sim 30\,M_\odot$ with fallback. Possible explanations include very rare PISN (perhaps $< 1\%$ of the EMP progenitors), or Pop III dominated by the intermediate masses ($10$-$40\,M_\odot$) that did most of the work of primordial enrichment, or the complete absence of low-mass Pop III from the sample of observed living stars (the minimum mass to survive 13 Gyr is $M < 0{.}8\,M_\odot$, and the simulations of primordial fragmentation do not routinely produce stars at these masses).

State of the art: chemical evolution

The picture of modern GCE is solid at first order: the nucleosynthesis mechanisms are identified and their yields calculated, the gas evolution is well described by the conservation equations, the chemical signatures observed in stars coherently reflect the star-formation history of the known Galactic environments. The methodological synthesis of reference is today Nomoto, Kobayashi and Tominaga (2013) [Nomoto et al. 2013] for the yield part, and Matteucci (2021) [Matteucci 2021] for the overall modeling part. Significant quantitative questions remain open in the three problems discussed above, and a series of refinements on stellar rotation, binarity, metallicity-dependent IMF, and 3D mixing in stellar atmospheres that marginally modify the predictions.

The prospects over 5-10 years are concrete and multidirectional. On the observational front, the 4MOST, WEAVE and PFS surveys will complete the spectroscopic mapping of the Milky Way with $10^{6}$-$10^{7}$ stars and $> 20$ elements per star, refining the GCE constraints on the scale of the whole Galaxy and identifying thousands of new EMP/UMP candidates for stellar archaeology. Gaia DR4 (expected 2026-2027) will provide precision 6D kinematics for the whole dataset, enabling the self-consistent chemodynamics of the Galactic population. JWST is searching for galaxies at $z = 10$-$15$ with spectral signatures compatible with dominant Pop III populations — candidates such as GN-z11 and the systems revealed in the JADES and CEERS surveys are undergoing detailed spectroscopic analysis. The next-generation 30-meter telescopes ELT, GMT and TMT will allow very-high-resolution spectroscopy of Pop II stars in satellite galaxies of the Milky Way and in globular clusters, reaching metallicities an order of magnitude lower than accessible today. On the theoretical front, the new-generation hydrodynamic models (FIRE-3, AURIGA Plus, IllustrisTNG-50) reach resolutions capable of chemically resolving the structure of the disk and the accretion history of the satellite galaxies, and the new-generation stellar models with self-consistent rotation and binarity produce more reliable yields. The combination of these fronts promises to take GCE from a discipline of qualitative synthesis to a discipline of quantitative archaeology within the next decade, able to reconstruct specific events and progenitor populations from the chemical signature of the observed stars.

Galactic chemical evolution is the statistical framework that connects all the previous nucleosynthesis sites to the stars and gas observed in the Milky Way today. It explains why the alpha elements trace massive stars, why iron records the delayed Type Ia enrichment, why europium marks rare r-process events, and why the oldest stars can preserve the signatures of individual early explosions. The next chapter closes the systematic part of the book by collecting the open frontiers of stellar nucleosynthesis, from uncertain nuclear data to the first stars and the most extreme explosive sites.