Stellar evolution and hydrostatic burning

A star’s life, in one page

A star is born from the gravitational collapse of a gas cloud, stabilizes when the nuclear reactions at its center supply enough thermal pressure to balance gravity, lives in hydrostatic equilibrium for most of its existence burning hydrogen into helium, and then enters a final phase that depends critically on the mass with which it was born. Low-mass stars like the Sun live ten billion years and die gently: they become red giants, shed their outer layers as a planetary nebula, and leave behind a dead degenerate core of carbon and oxygen — the CO white dwarf. Massive stars ($M > 8$-$10\,M_\odot$) live only a few million years and die violently: they explode as core-collapse supernovae, and the remnant is a neutron star or, for the most massive, a black hole. Nucleosynthesis occurs in all phases of stellar life, but in very different ways depending on the mass: for the same “physics”, the mass sets the temperatures, the pressures and the exposure times, and therefore which reactions are active and which yields are produced.

The structure equations

The internal structure of a star in hydrostatic equilibrium and spherical symmetry is described by four coupled ordinary differential equations [Kippenhahn et al. 2012]:

$$\frac{dP}{dr} = -\frac{Gm\rho}{r^2}, \quad \frac{dm}{dr} = 4\pi r^2 \rho, \quad \frac{dL}{dr} = 4\pi r^2 \rho\,\epsilon, \quad \frac{dT}{dr} = -\frac{3\kappa\rho L}{16\pi a c\,r^2 T^3}$$

where $P$ is the pressure, $m$ the mass within radius $r$, $L$ the luminosity, $T$ the temperature, $\rho$ the density, $\epsilon$ the rate of nuclear energy generation, $\kappa$ the Rosseland opacity. The last equation holds in the radiative case; in the convective case one uses the adiabatic gradient $\nabla_{\mathrm{ad}}$ or a mixing-length treatment. The four equations are closed by an equation of state $P(\rho, T, X_i)$, an opacity $\kappa(\rho, T, X_i)$ and a generation rate $\epsilon(\rho, T, X_i)$ that depend on the local composition $X_i$ — a system of hundreds of chemical species tracked for nucleosynthesis.

The classical evolutionary scheme follows the star along the HR diagram (luminosity vs effective temperature) through distinct phases: the Main Sequence (MS) with H burning in the core and duration $\tau_{\mathrm{MS}} \approx 10^{10}\,(M/M_\odot)^{-2{.}5}$ years; the subgiant and Red Giant Branch (RGB) phase with H burning in a shell and expansion of the envelope; He-burning with stable $3\alpha$ in the core; for stars of mass $M < 8\,M_\odot$ the AGB phase with double-shell burning, pulsing thermal instabilities, vigorous mass loss and the end as a CO or ONe white dwarf; for stars of mass $M > 8\,M_\odot$ the sequence of advanced burnings (C, Ne, O, Si in shells: final part of this chapter) followed by core collapse and SN II. The characteristic timescales that govern the evolution are the nuclear time $\tau_n \sim E_n/L$ for the MS, the Kelvin-Helmholtz time $\tau_{\mathrm{KH}} \sim GM^2/(RL)$ for the gravitational contractions, and the dynamical time $\tau_d \sim 1/\sqrt{G\rho}$ for the collapse. For the Sun: $\tau_n \approx 10^{10}$ years, $\tau_{\mathrm{KH}} \approx 3 \times 10^{7}$ years, $\tau_d \approx 30$ min. The short phases (pre-MS, giant phase) are traversed on $\tau_{\mathrm{KH}}$; the long ones (MS, HB) on $\tau_n$.

The stellar evolution codes in use today are MESA (Modules for Experiments in Stellar Astrophysics, open-source, the standard reference for low/intermediate-mass stars), GENEC (Geneva, focused on rotation), KEPLER (Stan Woosley, the historical reference for massive stars), FRANEC (Frascati-Rome, Limongi-Chieffi), STAREVOL (Lyon). The differences among codes are notable in the advanced phases (advanced burnings, TP-AGB) and depend on the treatment of convection, overshooting, semiconvection, rotation and mass loss — the main source of uncertainty on the final yields. The modern textbook treatment of stellar nuclear physics is Iliadis (2015) [Iliadis 2015].

Mass as the key parameter

Everything else in a star’s life — how long it lives, how it dies, which elements it produces and in what proportions — is largely determined by the mass with which it was born. From $0{.}08\,M_\odot$ to $\sim 100\,M_\odot$ there are three orders of magnitude in mass, five in evolutionary timescale and ten in luminosity: the same physical phenomenon changes its face as a function of the mass parameter, and the standard classification recognizes five mass bands with qualitatively different evolutionary fates.

Brown dwarfs ($M < 0{.}08\,M_\odot$) never reach the temperature for stable hydrogen ignition and cool by radiative emission over billions of years; they do not contribute to Galactic nucleosynthesis. Low-mass stars ($0{.}08 < M < 0{.}5\,M_\odot$) burn H very slowly for hundreds of billions of years; none of them has yet died in the history of the universe, and their contribution to nucleosynthesis is zero so far. Intermediate and Sun-like stars ($0{.}5 < M < 8\,M_\odot$) burn H and He, pass through the AGB phase with its thermal pulses, and end as CO or ONe white dwarfs; they produce carbon (in small part), nitrogen via Hot Bottom Burning in intermediate-mass AGB, and the s-process main component (Ba, La, Ce, Pb, chapter 4). Massive stars ($8 < M < 25\,M_\odot$) burn up to silicon through the sequence of advanced burnings, explode as core-collapse SNe and leave neutron stars; they produce the $\alpha$-elements (O, Ne, Mg, Si, Ca), the s-process weak component up to $A \sim 90$, and part of the heavy p-nuclei via the explosive gamma-process. Very massive stars ($M > 25\,M_\odot$) have intense radiative mass loss that can strip them of the H envelope before collapse (Wolf-Rayet stars), and their explosive fate may produce no luminous SN (direct black-hole formation for high compactness, chapter 5) or produce an exotic SN (collapsar, magnetar SN, hypernovae with associated long GRB).

The evolutionary properties as a function of zero-age mass are illustrated by the historical grids of Schaller, Schaerer, Meynet and Maeder (1992) and by their successors. The key transitions include: for $M \lesssim 1{.}3\,M_\odot$ hydrogen burning is dominated by the pp chain, above it the CNO cycle prevails — the consequence is that low-mass stars have radiative cores on the MS, massive stars convective cores; for $M < 2\,M_\odot$ helium burning starts in degenerate conditions through the helium flash, for $M > 2\,M_\odot$ it is quiescent, and the ignition mode determines the structure of the star in the HB phase; for $M < 8\,M_\odot$ carbon burning does not ignite and the star forms a CO or (in rare SAGB cases) ONe white dwarf; for $M > 8\,M_\odot$ the advanced burning proceeds in sequence with each phase progressively shorter because of neutrino cooling (final part of this chapter); for $M > 30$-$40\,M_\odot$ radiative mass loss dominates the evolution and the star can lose its entire H envelope before collapse. The initial mass function (IMF) — Salpeter’s $dN/dM \propto M^{-2{.}35}$ for $M > 0{.}5\,M_\odot$, Kroupa’s or Chabrier’s with a flat tail at low masses — is a fundamental input of GCE models (chapter 7).

Modern grids include effects that the historical Geneva 1992 grids neglected: rotation (with rotational mixing, anisotropic mass loss, brightening from internal merging), magnetism (Tayler-Spruit dynamo), more realistic treatments of convection and overshooting. The comparison among codes on the yields shows differences of $20$-$50\%$ in the mass of $\alpha$-elements for $M = 25\,M_\odot$ among MESA-rotating, GENEC-rotating and KEPLER-non-rotating; even larger differences on the heavy isotopes of the weak s-process (Sr, Y, Zr) between rotating and non-rotating models. The open effects that still limit predictivity include: the overshooting of convective cores, with extension $\alpha_{\mathrm{ov}}$ of the mixing beyond the Schwarzschild boundary, calibrated on open clusters at values $\alpha_{\mathrm{ov}} \approx 0{.}1$-$0{.}3\,H_P$ (with impact on MS lifetimes and on the yields of He, C, N, O); the treatment of rotation (Eddington-Sweet circulation vs shear instabilities), which varies the mixing at the core boundaries and hence the primary $^{14}\mathrm{N}$ yields in massive stars at low metallicity (impacting primary vs secondary nitrogen in GCE); mass loss (Vink’s laws for OB stars, de Jager for supergiants, Reimers/Bloecker for AGB, Nugis-Lamers for WR), with factor-2 differences on the final yields of WR stars and a correlated uncertainty on the SN progenitors that form.

Convection and mixing

Inside a star there are calm zones — convectively stable — where the matter does not remix and energy transport is radiative, and “boiling” zones — convectively unstable — where convective currents transport heat and mix the chemical composition like a pot of broth. The boundary between the two types of zone is not sharp: in the boundary layers occur the most subtle and least understood phenomena of all stellar physics, from overshooting to semiconvection to thermohaline mixing. Without convection and without the mixing mechanisms at its boundaries, the elements produced at depth would never rise to the surface, AGB stars could not carry the s-processed carbon toward the envelope (the third dredge-up), and stars could not enrich the galaxy with the products of their nucleosynthesis.

The Schwarzschild criterion establishes that a zone is convectively unstable when $\nabla_{\mathrm{rad}} > \nabla_{\mathrm{ad}}$, where $\nabla = d\ln T/d\ln P$. The Ledoux criterion generalizes it by including the chemical composition gradient: $\nabla_{\mathrm{rad}} > \nabla_{\mathrm{ad}} + (\varphi/\delta)\,\nabla_\mu$. The difference between the two criteria defines semiconvection: regions unstable by Schwarzschild but stable by Ledoux, where the mixing proceeds slowly with characteristic diffusive times and plays an important role in the He-burning phase of intermediate stars and in the C/O-burning shells of massive stars. The standard theory of convective transport used in 1D codes is the Mixing-Length Theory (MLT) of Böhm-Vitense, which parameterizes the heat transport through a mixing length $\ell_{\mathrm{MLT}} = \alpha_{\mathrm{MLT}}\,H_P$ with $\alpha_{\mathrm{MLT}} \sim 1{.}5$-$2{.}0$ calibrated on solar models. It is a local and phenomenological approximation, and becomes progressively less reliable at the convective boundaries where three-dimensional turbulent motions dominate.

The extra-convective mixing mechanisms that operate at the boundaries and that significantly modify the final yields include several families. Overshooting is the inertial penetration of convective eddies beyond the formal stability boundary; in 1D codes it is parameterized as a spatial extension (in units of $H_P$) or as a decreasing exponential profile (Herwig’s formulation [Herwig 2005] ). Thermohaline mixing, active when the mean molecular weight gradient $\mu$ inverts locally, is relevant at the RGB tip for the reduction of $^{3}\mathrm{He}$ and for the chemical signature observed in RGB-tip stars. Rotational mixing includes shear instabilities and the Eddington-Sweet meridional circulation (the formalism of Maeder and Zahn). The magnetic instabilities Tayler-Spruit dynamo and magneto-rotational instability (MRI) transport angular momentum and mix chemical composition in differentially rotating stars.

3D simulations of stellar convection (Meakin & Arnett 2007, Müller et al. 2016, Andrássy et al. 2020, Cristini et al.) have shown that the convective boundary is characterized by a slow but non-zero turbulent entrainment, with a rate calibrated as a function of the bulk Richardson number $Ri_B$. The 1D parameterizations derived from these calculations (Cristallo et al. 2009 for AGB, Pignatari et al. 2016 for massive stars) are significantly different from Herwig’s exponential overshooting and change both the s-process yields (larger $^{13}\mathrm{C}$ pocket, smaller pulse-to-pulse overlap factor) and the $^{12}\mathrm{C}/^{16}\mathrm{O}$ ratio in massive stars. Hot Bottom Burning (HBB) in intermediate-mass AGB ($4$-$8\,M_\odot$) requires that the base of the convective envelope penetrate into the underlying H-burning shell: it occurs for base temperatures $T_{\mathrm{base}} > 5 \times 10^{7}$ K, and radically modifies the surface composition (reduction of $^{12}\mathrm{C}$, increase of $^{14}\mathrm{N}$, $^{7}\mathrm{Li}$ via the Cameron-Fowler beryllium-transport mechanism). The quantitative details of HBB are sensitive to the treatment of convection and to the time step of the code, and lie at the heart of the origin of primary nitrogen in intermediate-mass AGB (see chapter 7 for the primary-nitrogen problem at low metallicities).

Mass loss

A star does not always keep the weight with which it is born: during its life it loses matter, especially in the advanced phases. Sun-like stars lose about half of their initial mass as a planetary nebula in the few $10^{4}$ years of the post-AGB phase; very massive stars ($M > 60\,M_\odot$) can lose more than $90\%$ of their initial mass before the explosive collapse, ending with residual masses of $\sim 10$-$20\,M_\odot$. This stellar “wind” is the main channel through which the elements cooked inside reach the interstellar medium and contribute to the Galactic budget — a channel alternative and complementary to the explosive SN ejecta, important in particular for the elements produced by AGB and WR stars.

The main mass-loss mechanisms are four. The dust-driven winds in AGB stars are the dominant mechanism of late mass loss: the condensation of dust (SiC, graphite, Mg/Fe-rich silicates) in the cool atmospheres at $T < 1500$ K couples the stellar radiation to the gas through absorption and scattering, accelerating the material to escape velocity, with mass-loss rates reaching $\dot{M} \sim 10^{-7}$-$10^{-4}\,M_\odot/\mathrm{yr}$ in the final phases (chapter 4). The line-driven radiative winds in OB and WR stars are driven by the radiative force on metal absorption lines in the UV; Vink’s empirical scaling law predicts $\dot{M} \propto L^{2{.}2}\,Z^{0{.}85}$, with a marked metallicity dependence that largely explains the evolutionary dichotomy between metal-rich massive stars (strong mass loss) and Pop III (almost no radiative mass loss). The LBV eruptions (Luminous Blue Variable) produce impulsive releases of $\sim 1\,M_\odot$ in episodes on timescales of $10^{3}$-$10^{4}$ years, and are dominant for stars $M > 50\,M_\odot$ in epochs of high instability close to the classical Eddington limit. Mass transfer in binary systems — via Roche-lobe overflow in semi-detached binaries or via common envelope in the late phases — is today recognized as the dominant channel for massive stars: Sana et al. (2012) showed that about $70\%$ of massive O stars belong to binary systems that interact during their lifetime, and mass transfer radically alters the SN progenitors (e.g., SNe Ib/c originate from stars that lost their envelope through RLOF, not through radiative winds).

The empirical mass-loss laws for AGB of Reimers (1975), Bloecker (1995), Vassiliadis & Wood (1993) differ significantly at high rates, with factor-2 consequences on the duration of the TP-AGB phase and on the final yields in C, N and s-process. For massive stars, the uncertainty on WR winds is the single largest contribution to the uncertainty on the final SN yields, and core-collapse SN models are directly sensitive to which WR mass-loss law is adopted (Nugis-Lamers, Hamann-Koesterke, Smith). The mass-loss rate on the main sequence is not zero for stars $M > 10\,M_\odot$: detections of N enhancement in O-star MS atmospheres suggest rotational mixing or binary transfer as relevant channels. For very massive stars at very low metallicity (Pop III at $Z = 0$, and extreme Pop II), the radiative mass loss is effectively absent (the $Z^{0{.}85}$ dependence): these stars remain massive until the explosion and can produce pair-instability supernovae ($140 < M < 260\,M_\odot$) with distinctive yields (high $[\alpha/\mathrm{Fe}]$, no r-process, specific $[\mathrm{Cr/Mn}]$, chapter 7), or direct collapse to a black hole for $M > 260\,M_\odot$.

Where everything happens: a map of the sites

To orient the reader in the following chapters of the book and in the synthesis processes treated in the first part, a map of which phase of stellar evolution produces which family of elements is useful. The Main Sequence of the Sun (and of all similar stars) burns H into He via the pp chains and CNO (next part of this chapter): it produces no new heavy elements, but provides measurable neutrinos that calibrate the stellar physics and the solar parameters. Helium burning in the red giant and Horizontal Branch phases transforms He into C and O via the $3\alpha$ reaction and via $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ (central part of this chapter): it is the main origin of C and O in the universe, and the final C/O ratio depends critically on the rate of the latter reaction, still the main source of uncertainty on massive-star yields. The thermally pulsing AGB phase operates the slow neutron capture (s-process main component, chapter 4), producing Ba, La, Ce, Nd, Pb, and modifies the surface composition via Hot Bottom Burning and third dredge-up, enriching the envelope in C, N, F and s-elements that are then expelled via the stellar wind (chapter 4).

The advanced burnings in massive stars (C, Ne, O, Si burning: final part of this chapter) produce the $\alpha$-elements and part of the iron-peak elements in the concentric shells of the pre-SN onion structure. The core-collapse SNe (chapter 5) activate shock-driven explosive nucleosynthesis in the pre-existing shells, with $\alpha$-rich freeze-out in the innermost region producing the $^{56}\mathrm{Ni}$ of the light curve, weak s-process fed by $^{22}\mathrm{Ne}(\alpha,n)$ in the C-burning shell, p-process (gamma-process) in the O/Ne shells, and possibly a limited r-process in the neutrino-driven winds. The SNe Ia (chapter 5) produce massive amounts of $^{56}\mathrm{Ni}$ (decaying to $^{56}\mathrm{Fe}$) through the detonation/deflagration of a CO white dwarf in a binary system, and are the main origin of iron in the universe. The classical novae (chapter 5) operate thermonuclear nucleosynthesis in the accreted layer on the surface of a CO or ONe white dwarf, with characteristic isotopes $^{15}\mathrm{N}$, $^{17}\mathrm{O}$, $^{22}\mathrm{Na}$. The X-ray bursts (chapter 5) activate the rp-process (rapid proton capture) on the atmosphere of accreting neutron stars, producing p-rich nuclei up to $A \sim 100$ confined to the NS surface. The neutron-star mergers (chapter 6) activate the r-process in the merger ejecta, producing gold, platinum, the actinides and the kilonova observed in GW170817.

The scheme just described is a first-approximation map: each site contributes to multiple elements with different weight, and the relative contributions to the Galactic budget are quantitatively discussed in the chapters on Galactic chemical evolution (chapter 7). The yield tables of Nomoto-Kobayashi, Limongi-Chieffi, Sukhbold-Woosley, Pignatari-Herwig for massive stars, and Karakas-Lugaro [Karakas & Lattanzio 2014] for AGB are the standard input for modern GCE models.

There are, in addition, boundary cases between mass bands that play a specific role in nucleosynthesis and do not fit cleanly into the five-band classification. The Super-AGB (SAGB) stars in the range $M \sim 7$-$10\,M_\odot$ ignite C-burning but not Ne-burning, and end as ONe white dwarfs or — in rare events — as electron-capture supernovae (ECSN) with an explosion mediated by $e^{-}$ capture on Mg and Ne; distinctive yields include high $^{60}\mathrm{Zn}$ and $^{50}\mathrm{Ti}$, and they could be the site of some rare nuclei of the Fe-Ni peak. Interacting binary stars with mass stripping, mass accretion, or mergers can radically change the explosion site (SNe Ib/c from stars that lost their envelope through RLOF, DD SNe Ia from the merger of two CO white dwarfs, blue stragglers in clusters originating from close-binary mergers). Massive Pop III stars ($M > 100\,M_\odot$) potentially end as pair-instability SNe or as direct collapse to a black hole, with a specific chemical signature observed controversially in some ultra-metal-poor stars.

State of the art and prospects

The evolutionary picture of stars is today consolidated in its main lines, and the new-generation stellar evolution codes (MESA, GENEC, FRANEC, KEPLER) quantitatively reproduce the observations at the various stages with residual uncertainties of $10$-$30\%$ on the yields. Significant quantitative questions remain open on three main fronts. The first is the treatment of convection and overshooting in 3D: the new-generation radiative-hydrodynamic MHD simulations (FleCSI, Cattaneo-Brun, Pratt et al.) produce progressively refined 1D parameterizations, which reduce the free-parameter space in the 1D evolution codes and improve the predictions of specific yields. The second is rotation and magnetism as channels of chemical mixing and angular-momentum transport: self-consistent MHD physics in differentially rotating stars is progressively integrated into 1D codes, and the asteroseismology results from Kepler/TESS/PLATO directly constrain the internal rotation profiles and hence the predicted mixing. The third is binarity as the dominant channel for massive stars: the new generation of binary evolution codes (BPASS by Eldridge-Stanway, COMPAS, MESA-binary) integrates single-star evolution with binary interactions (RLOF, common envelope, mergers), and produces complete grids of SN progenitors compatible with the observed statistics.

The prospects over 5-10 years are concrete and multidirectional. Asteroseismology with PLATO (launch 2026) and with the TESS extensions will produce internal rotation and mixing profiles for thousands of MS, RGB, HB and subgiant stars, directly constraining the parameters $\alpha_{\mathrm{ov}}$, $\alpha_{\mathrm{MLT}}$ and the rotational coefficients. The observing campaigns with JWST, ELT and the future above-atmosphere detectors will spectrally characterize AGB and RSG stars in satellite galaxies (LMC, SMC, local dwarfs) at different metallicities, constraining the yields as functions of $Z$. The 3D full-star simulations of stellar convection (Andrássy, Cristini, Müller et al.) will reach within the decade resolutions sufficient to quantitatively calibrate the 1D parameterizations. On the nuclear front, the measurements of LUNA-MV, JUNA, FRIB will reduce the uncertainties on the key reaction rates (in particular $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$) and on the beta rates for drip-line nuclei — the single largest contribution to the uncertainty on core-collapse SN yields. The combination of these converging fronts promises to take modern stellar evolution from a discipline of phenomenological calibration to a discipline of quantitative prediction within the next decade, with direct consequences for GCE and Galactic chemical archaeology.

Stellar evolution constitutes the physical framework into which all the nucleosynthesis mechanisms discussed in the previous chapters of the book fit: the initial mass determines temperatures, pressures and exposure times, and therefore which nuclear reactions are active in which phase, and which chemical signature the star imprints on the interstellar medium at the end of its life. Some specific complementary sites remain to be told in detail: the origin of the light nuclei Li, Be, B, which quiescent stellar nucleosynthesis does not produce and which come from cosmic spallation (chapter 2); the detailed physics of AGB stars and their thermal pulses as the laboratory of the s-process main component (chapter 4); the explosions of novae and X-ray bursts in accreting binary systems (chapter 5); and the astrophysics of presolar grains as the direct laboratory of individual stellar nucleosynthesis (chapter 4).

The Sun as a nuclear furnace

The Sun has been burning hydrogen for four and a half billion years, and has about five more to go before its core is exhausted and the expansion toward the red giant phase begins. The figures are everyday figures seen from very close up: every second, in the solar center, about $3{.}7 \times 10^{38}$ protons fuse into helium-4 nuclei, transforming $\sim 600$ million tonnes of hydrogen into $\sim 596$ million tonnes of helium; the difference, $\sim 4$ million tonnes per second, is converted into energy according to $E = mc^2$ and sustains the solar luminosity $L_\odot = 3{.}828 \times 10^{33}$ erg/s. That energy, once produced by the reactions at the center, undertakes a very long journey through the radiative zone: high-energy photons, kinetic energy of charged particles and thermalized radiation are absorbed, scattered and re-emitted billions of times, gradually losing energy, and take of order $10^{5}$ years before emerging as visible photons from the surface of the Sun. Today’s sunlight is the diary of nuclear reactions that happened during the last ice age.

There is, however, another form of “photograph” of the solar heart that is in real time: the neutrinos. The fusion of four protons into helium requires converting two of the four into neutrons, and each $p \to n$ conversion produces a positron and an electron neutrino. The neutrinos, having only weak interaction with matter, leave the Sun in a few seconds and reach the Earth in another eight minutes; they therefore carry with them a snapshot of the nuclear reactions happening now at the center. Measuring the flux of these neutrinos is the direct proof — and the only one — that the thermonuclear burning described by the models is actually at work in the depths of the Sun. The story of this measurement, from Davis’s pioneering experiments at Homestake to SNO and Borexino, is one of the most beautiful stories of physics of the second half of the twentieth century, and will return in the last section of this part of the chapter.

The total energy balance of the H → He burning is written as

$$4\,p \to {}^{4}\mathrm{He} + 2 e^{+} + 2 \nu_e, \qquad Q_{\mathrm{tot}} = 26{.}73 \, \mathrm{MeV}.$$

Of these 26.73 MeV, a fraction varying between $\sim 2\%$ and $\sim 27\%$ depending on the chain is carried away by the neutrinos and does not contribute to the observed luminosity; the rest is deposited as heat in the plasma and finally radiated. The figure of 26.73 MeV per helium produced is only $0{.}71\%$ of the rest-mass energy of the four starting protons — the same mass-defect fraction that Aston had measured in the 1920s, and that Eddington had indicated as the key to stellar energetics. The net reaction rate, and hence the luminosity of the star, is controlled by the first step of the chain, $p + p \to d + e^{+} + \nu_e$: a weak reaction with cross section $\sigma \sim 10^{-47}$ cm² at $E \sim 1$ MeV, so small that it has never been measured in the laboratory. It is precisely this slowness that gives the Sun a main-sequence life of order $10^{10}$ years: if the pp reaction were fast, the Sun would have burned its hydrogen in a few million years and the terrestrial biosphere would not have had time to exist.

Bethe in 1939 [Bethe 1939] identified two alternative and largely independent routes for the H → He burning: the pp chain (proton-proton), which starts directly from the fusion of two protons and proceeds via deuterium and helium-3 up to helium-4; and the CNO cycle, which uses carbon, nitrogen and oxygen nuclei as catalysts. The two options have wildly different temperature dependences — $\epsilon_{\mathrm{pp}} \propto T^{4}$ against $\epsilon_{\mathrm{CNO}} \propto T^{18}$ at the solar temperature — and the transition between the pp-dominated and the CNO-dominated regimes occurs around $T_c \approx 1{.}7 \times 10^{7}$ K, corresponding to a main-sequence stellar mass $M \approx 1{.}3\,M_\odot$. The Sun is just below the transition: the pp chain produces about 99% of the luminosity, the CNO cycle the remaining 1%. In more massive stars, like Sirius A ($M \approx 2\,M_\odot$), the ratio reverses and CNO dominates largely; in very massive main-sequence stars ($M \gtrsim 5\,M_\odot$) CNO contributes practically the entire luminosity.

The nuclear physics of the first step

The reaction $p + p \to d + e^{+} + \nu_e$ deserves a pause, because it is one of the most precise theoretical calculations of low-energy nuclear physics and has direct implications for all stellar energetics. It is a weak-process reaction on a two-nucleon system: the two protons must first overcome the Coulomb barrier by tunneling, then one of the two must transform into a neutron through the weak interaction, simultaneously with the formation of the deuterium bound state. The probability of the first process is the classic Gamow factor; the probability of the second, the weak matrix element, is calculated with ab initio methods starting from the measured nucleon-nucleon potentials (Argonne v18, Charge-Dependent Bonn) and electromagnetic and weak currents that include meson-exchange contributions. The result is the astrophysical factor

$$S_{11}(0) = (4{.}03 \pm 0{.}05) \times 10^{-25} \, \mathrm{MeV \cdot b}$$

with an uncertainty of about 1%, dominated by the ambiguities in the nuclear potentials and the exchange currents. It is a purely theoretical value — the reaction is too slow for any terrestrial accelerator to detect it directly — but it is considered reliable at the $1\%$ level, and this limit propagates onto the predicted flux of solar pp neutrinos, $\Phi_{\nu_{\mathrm{pp}}} \approx 6 \times 10^{10}$ cm⁻²s⁻¹ at Earth. Curiously, the residual uncertainties on the solar neutrino fluxes of all species are dominated not by nuclear physics but by stellar modeling: radiative opacities, initial composition of the Sun, gravitational diffusion of the heavy elements toward the center, and — above all — photospheric chemical abundances of C, N, O. On this last point a lively debate has opened over the last fifteen years, known as the Solar Modeling Problem, to which we shall return.

The pp chains

The pp chain is the H → He burning mechanism acting in low-mass stars, including solar-type main-sequence stars. It begins with the extremely slow fusion of two protons into a deuterium, continues with the capture of a proton on the deuterium to form helium-3, and then branches into three distinct terminations — called pp-I, pp-II and pp-III — distinguished by how the helium-3 is finally converted into helium-4. Each of the three terminations produces its own neutrino spectrum, with characteristic energies and fluxes, and since the three fluxes can be measured separately on Earth, each provides independent information on the conditions of the solar center.

The common initial sequence is

$$p + p \to d + e^{+} + \nu_e \quad (\langle E_\nu \rangle \approx 0{.}267 \, \mathrm{MeV}, \, E_{\nu,\mathrm{max}} = 0{.}42 \, \mathrm{MeV})$$ $$d + p \to {}^{3}\mathrm{He} + \gamma \quad (Q = 5{.}49 \, \mathrm{MeV}).$$

The deuterium produced in the first step is immediately burned by the second, with a mean lifetime in the solar plasma of a few seconds: deuterium is a transient species in the Sun, never accumulating to measurable concentrations. Helium-3, on the other hand, can accumulate for a while, because the next step is much slower, and reaches a stationary abundance that depends on the local temperature [Iliadis 2015].

At this point the story branches. In the pp-I termination, two $^{3}\mathrm{He}$ nuclei fuse directly:

$${}^{3}\mathrm{He} + {}^{3}\mathrm{He} \to {}^{4}\mathrm{He} + 2p, \qquad Q = 12{.}86 \, \mathrm{MeV}.$$

The total Q of the pp-I chain deposited in the plasma is $26{.}21$ MeV, with a neutrino loss of only $0{.}59$ MeV. It is the dominant termination in the Sun (86% of terminations), favored kinetically because it requires no further Coulomb-capture steps on nuclei with $Z > 2$.

In the pp-II termination, $^{3}\mathrm{He}$ captures a $^{4}\mathrm{He}$ nucleus (already abundant from BBN and pp-I), forms $^{7}\mathrm{Be}$, which in turn captures an electron to form $^{7}\mathrm{Li}$, which finally captures a proton to return two $^{4}\mathrm{He}$:

$${}^{3}\mathrm{He}(\alpha,\gamma)^{7}\mathrm{Be}(e^{-},\nu)^{7}\mathrm{Li}(p,\alpha)^{4}\mathrm{He}.$$

The neutrinos emitted by the electron capture of $^{7}\mathrm{Be}$ are monoenergetic (apart from a small thermal Doppler broadening), with two lines at 384 keV and 862 keV in 1:9 ratios depending on the final state of the $^{7}\mathrm{Li}$ (ground state or first excited). These are the celebrated Be neutrinos measured by Borexino with 3% precision in 2008. The fraction of pp-II terminations in the Sun is 14%.

The pp-III termination finally branches off from pp-II at the level of $^{7}\mathrm{Be}$: instead of capturing an electron, the $^{7}\mathrm{Be}$ captures a proton to form $^{8}\mathrm{B}$, which $\beta^{+}$-decays into $^{8}\mathrm{Be}$ (unstable), which immediately dissociates into two $\alpha$:

$${}^{7}\mathrm{Be}(p,\gamma)^{8}\mathrm{B}(\beta^{+}\nu)^{8}\mathrm{Be}^{*}(2\alpha)^{4}\mathrm{He}.$$

The neutrinos from the $\beta^{+}$ decay of $^{8}\mathrm{B}$ have a continuous spectrum with $\langle E_\nu \rangle \approx 7$ MeV and a cutoff at 14 MeV: they are by far the most energetic of the solar neutrinos, and the only ones accessible to detection through Cherenkov scattering in water or heavy water (SuperKamiokande, SNO). The fraction of pp-III terminations in the Sun is only $0{.}02\%$, but precisely because of the high energy its flux is observable, and it is the flux most sensitive to the central temperature of the Sun — $\Phi_{\nu B} \propto T_c^{\sim 25}$ — making it the most precise thermometer at our disposal for the solar heart.

Summarizing the main astrophysical factors of the pp-chain reactions, and their measurement status:

The cross section of the α capture on $^{3}\mathrm{He}$ was measured by LUNA down to the solar Gamow peak ($E_0 \approx 22$ keV) between 2007 and 2012, reaching a 5% uncertainty, almost at the level required for precision solar neutrinos ($\delta\Phi_{\nu_B}/\Phi_{\nu_B} \lesssim 5\%$). The cross section $^{7}\mathrm{Be}(p,\gamma)^{8}\mathrm{B}$ had instead a tormented history: measurements by Filippone (1983), Hammache (1998), Junghans (2003) gave values discordant by $\pm 20\%$ — the experimental difficulty is that the $^{7}\mathrm{Be}$ target is radioactive ($t_{1/2} = 53{.}3$ d) and must be produced fresh for each measurement. The compromise value adopted by Solar Fusion II [Adelberger et al. 2011] is $S_{17}(0) = 20{.}8 \pm 0{.}8$ eV·b, consistent with the modern indirect methods (Coulomb breakup of $^{8}\mathrm{B}$ on heavy targets, ANC analysis of the asymptotic tails of the wave function).

The CNO cycle

In stars more massive and hotter than the Sun, hydrogen does not fuse directly into helium: it uses carbon, nitrogen and oxygen as catalysts. These are nuclei already present in the star through the enrichment of previous generations — the Sun, for example, contains about $0{.}2\%$ by mass of CNO, inherited from the interstellar material that formed the solar system — and in a hot environment they successively capture one proton at a time, $\beta^{+}$-decay when necessary, and finally return the initial nucleus plus a newly formed helium-4. The net reaction is still $4p \to {}^{4}\mathrm{He}$, but the route is more convoluted and — above all — its speed depends much more steeply on temperature.

The main branch of the cycle, today denoted CNO-I (or the Bethe-Weizsäcker cycle, identified independently in 1938-39), is

$${}^{12}\mathrm{C}(p,\gamma)^{13}\mathrm{N}(\beta^{+}\nu)^{13}\mathrm{C}(p,\gamma)^{14}\mathrm{N}(p,\gamma)^{15}\mathrm{O}(\beta^{+}\nu)^{15}\mathrm{N}(p,\alpha)^{12}\mathrm{C}.$$

The total Q after neutrinos is $25{.}03$ MeV. The slow step that limits the speed of the cycle is $^{14}\mathrm{N}(p,\gamma)^{15}\mathrm{O}$: all the other reactions are at least an order of magnitude faster, with the result that in stars that have burned CNO for a significant time, nitrogen-14 accumulates at the expense of carbon and oxygen. The equilibrium ratio between $^{12}\mathrm{C}$ and $^{13}\mathrm{C}$ of the CNO cycle is $\approx 3{.}5$, far from the solar value $^{12}\mathrm{C}/^{13}\mathrm{C} \approx 89$: the observation of ratios $\sim 3{.}5$-$20$ in the atmospheres of red giants, AGB and evolved massive stars is a direct signature of completed CNO burning, and has proved one of the most powerful diagnostic tools for studying the mixing processes (first dredge-up, thermohaline mixing) that bring CNO-processed material back to the surface.

The temperature dependence of the main branch is

$$\epsilon_{\mathrm{CNO}} \propto X X_{\mathrm{CNO}} \rho T^{\nu}, \quad \nu = \frac{d \ln \epsilon}{d \ln T} \approx 18 \text{ at } T = 1{.}5 \times 10^{7} \, \mathrm{K},$$

against $\epsilon_{\mathrm{pp}} \propto X^{2} \rho T^{4}$ of the pp chain. The difference between exponent $18$ and exponent $4$ is enormous: a small variation in temperature changes the CNO rate by orders of magnitude, while the pp rate varies only modestly. An immediate structural consequence is that CNO-dominated stars have a convective core: the temperature gradient around the center is so steep that convection switches on and continuously mixes the core material, keeping it homogeneous in composition. The pp-dominated stars, conversely, have radiative cores: the burning leaves stratified composition profiles that are preserved up to the giant phase.

At higher temperatures the secondary CNO cycles activate. CNO-II: the $^{15}\mathrm{N}$ can capture a proton also through the $(p,\gamma)$ channel instead of only $(p,\alpha)$, forming $^{16}\mathrm{O}$, and from there the sequence $^{16}\mathrm{O}(p,\gamma)^{17}\mathrm{F}(\beta^{+}\nu)^{17}\mathrm{O}(p,\alpha)^{14}\mathrm{N}$ closes a cycle that returns to nitrogen. The $(p,\gamma)/(p,\alpha)$ branching on $^{15}\mathrm{N}$ is about $1:1000$ at solar temperatures, but grows with $T$. CNO-III, CNO-IV: the flow reaches $^{18}\mathrm{F}$ and $^{19}\mathrm{F}$ as internal branching points of the network. At still higher temperatures the NeNa and MgAl cycles open up, where protons are captured on isotopes of neon, sodium, magnesium and aluminum: these cycles are important in intermediate-mass stars in the AGB phase and in the thermonuclear flashes on white dwarfs (see chapters 4 and 5). They produce in particular $^{23}\mathrm{Na}$, $^{26}\mathrm{Al}$, $^{27}\mathrm{Al}$ — isotopes whose abundances in stellar material are diagnostic signatures of hydrogen-burning temperatures above $5 \times 10^{7}$ K.

The LUNA revolution on N(p,γ)O

The most recent history of the CNO cycle is dominated by one precise measurement: the cross section of $^{14}\mathrm{N}(p,\gamma)^{15}\mathrm{O}$, the slow step of the cycle, measured by LUNA at Gran Sasso in several campaigns from 2004 to 2015. The result — $S_{114}(0) = 1{.}57 \pm 0{.}13$ keV·b — is about half of the pre-LUNA standard value (Schröder et al. 1987, $S(0) \approx 3{.}2$ keV·b), with the systematic uncertainty reduced to about $8\%$. The direct consequence is a $\sim 50\%$ reduction of the predicted CNO cycle rate for the Sun, and hence a corresponding reduction of the flux of solar CNO neutrinos ($\nu_{\mathrm{CNO}} \equiv \nu_{^{13}N} + \nu_{^{15}O} + \nu_{^{17}F}$), which the post-LUNA SSM places at $\Phi_{\nu_{\mathrm{CNO}}} \approx 4{.}9 \times 10^{8}$ cm⁻²s⁻¹.

The observational confirmation came from Borexino: in 2020 the first direct measurement of the solar CNO neutrino flux [Collaboration 2020] , $\Phi_{\nu_{\mathrm{CNO}}} = 6{.}6^{+2{.}0}_{-0{.}9} \times 10^{8}$ cm⁻²s⁻¹, and in 2022 the updated measurement with accumulated statistics [Collaboration 2022] , $\Phi_{\nu_{\mathrm{CNO}}} = 6{.}7^{+1{.}2}_{-0{.}8} \times 10^{8}$ cm⁻²s⁻¹, are marginally in tension with the SSM-low-Z prediction (based on recent 3D NLTE photospheric abundances) and consistent with the SSM-high-Z prediction (Grevesse-Sauval 1998 composition). It is a tension at the $\sim 2\sigma$ level, not decisive but significant: the CNO neutrino measurement, directly sensitive to the C+N abundance in the solar core, is rapidly becoming a key piece for discriminating the two families of solar models within the Solar Modeling Problem (reviewed in detail in chapter 7).

The NeNa and MgAl cycles also produce an isotope particularly significant for γ-ray astronomy: $^{26}\mathrm{Al}$, with half-life $t_{1/2} = 7{.}17 \times 10^{5}$ years, decays emitting a γ photon at $1{.}809$ MeV. This photon is detected by the hard-γ astronomy missions (CGRO/COMPTEL in the 1990s, INTEGRAL/SPI since 2002), and the mapping of its emission across the Galactic plane allows an estimate of the total mass of live $^{26}\mathrm{Al}$ in the Galaxy: $\sim 2$-$3\,M_\odot$, with a regeneration timescale of $\sim 10^{6}$ years. The dominant sources are massive stars (in particular the winds of Wolf-Rayet stars and the ejecta of core-collapse supernovae), with additional contributions from novae and intermediate-mass AGB through the MgAl cycle in hot bottom burning. $^{26}\mathrm{Al}$ is one of the few medium-long-lived radioactive species observable live in our Galaxy, and constitutes a direct chronometer of ongoing nucleosynthesis.

Solar neutrinos as direct proof

Neutrinos are practically immaterial particles: their mean free path in matter of solar density is $\sim 10^{9}\,R_\odot$, so they leave the Sun practically undisturbed, carrying with them the information about what happened now at its center. The sunlight we see today, by contrast, is the product of reactions that occurred 100,000 years ago, “imprisoned” and scattered by the radiative layers: the two signals — light and neutrinos — are two views of the Sun offset by very different timescales. The solar neutrino experiments have transformed solar astrophysics from a modeling science to a directly observational one.

The story begins with Raymond Davis Jr. at the Homestake Mine in South Dakota, between 1968 and 1994: 615 tonnes of tetrachloroethylene ($\mathrm{C}_2\mathrm{Cl}_4$) confined in a deep mine, against which a solar electron neutrino can, with cross section $\sim 10^{-46}$ cm² at the energy peak, transform a $^{37}\mathrm{Cl}$ nucleus into $^{37}\mathrm{Ar}$ through inverse capture $\nu_e + {}^{37}\mathrm{Cl} \to {}^{37}\mathrm{Ar} + e^{-}$. The argon atoms produced — one every two-three days in a tank of hundreds of tonnes — were chemically extracted and counted by decay. The surprise was that the measured flux was about a third of that predicted by Bahcall’s solar models: the solar neutrino problem was born, and for almost thirty years the community was divided between those who attributed the deficit to errors in the stellar models and those who attributed it to new physics in neutrino propagation.

The resolution arrived in 2001-2002 from the Sudbury Neutrino Observatory (SNO), a heavy-water ($\mathrm{D}_2\mathrm{O}$) Cherenkov detector sensitive separately to $\nu_e$ through the charged-current reaction ($\nu_e + d \to p + p + e^{-}$, CC) and to all neutrino flavors through the neutral-current reaction ($\nu_x + d \to p + n + \nu_x$, NC). The measured CC/NC ratio — about one third — implied that only a minority of the solar neutrinos arrived at Earth as $\nu_e$, while the rest had become $\nu_\mu$ or $\nu_\tau$: direct proof of neutrino oscillation, and hence of non-zero neutrino mass [Ahmad et al. 2002] . The CC+NC sum agreed with the predicted SSM flux. The solar neutrino problem was a particle-physics problem, not an astrophysics one.

The oscillation mechanism is the Mikheyev-Smirnov-Wolfenstein (MSW) effect: in the passage of neutrinos through the dense solar plasma, the coherent interaction with the electrons modifies the propagation eigenstate and produces an adiabatic flavor conversion. The survival probability $P(\nu_e \to \nu_e)$ is therefore a function of energy: $\approx 0{.}55$ at low energies ($\nu_{\mathrm{pp}}$), just above $0{.}5$ at intermediate energies ($\nu_{\mathrm{Be}}$, $\nu_{\mathrm{pep}}$), and $\approx 0{.}33$ at high energies ($\nu_{\mathrm{B}}$). The oscillation parameters measured by combining SNO, KamLAND and Borexino indicate $\sin^{2}\theta_{12} \simeq 0{.}31$ and $\Delta m_{21}^{2} \simeq 7{.}4 \times 10^{-5}$ eV². Solar neutrino physics is today consolidated, and the spectral dependence of the survival probability is the direct test bench of the MSW mechanism.

Borexino, finally, an ultrapure liquid scintillator in the Gran Sasso Laboratory, in operation from 2007 to 2021, achieved the first spectroscopic measurement of solar neutrinos, separating the pp, Be, pep, B and finally CNO components. The overall picture of the SSM predictions (in two variants, based on the GS98 high-Z and AGSS09 low-Z solar compositions) and of the measurements is summarized in the following table (fluxes in cm⁻²s⁻¹):

The pp and pep fluxes, constrained by the conservation of the solar luminosity, are in full agreement with both predictions. The Be and B fluxes are more sensitive to the central temperature and to the opacity: the agreement is better with SSM-GS98. The CNO flux is the one that most sharply discriminates the two families, and is — at present — in $\sim 2\sigma$ tension with AGSS09. The reviews dedicated to the SSM and to the neutrino constraints are in [Serenelli 2016] and [Villante & Serenelli 2021] .

The Solar Modeling Problem in brief

The Solar Modeling Problem is the tension, which emerged around 2005, between the 3D NLTE determinations of the photospheric solar chemical abundances (Asplund and collaborators, AGSS09 and later) and the helioseismic constraints on the sound-speed profiles in the Sun. The new abundances, in particular for C, N, O, Ne, are lower by $\sim 30\%$ than the pre-2005 1D values (GS98), and this reduces the opacity in the radiative zone, modifies the temperature profile and produces a $\sim 1\%$ disagreement with the helioseismic data on the base of the convection zone, on the central density and on the initial helium fraction. The problem is discussed in depth in chapter 7, devoted to cosmic abundances; here it suffices to note that the measurement of the CNO neutrinos is the most direct observational probe of the C+N content of the solar core, and could in the coming years resolve or consolidate the currently ongoing tension. The most recent review of these abundances and of their comparisons is in [Asplund et al. 2021] .

With hydrogen burning completed, the core of the star remains composed almost entirely of helium-4. Main-sequence stars spend most of their life in this phase — the Sun is now about halfway through — but when the central hydrogen is exhausted, the helium core begins to contract gravitationally until it reaches temperatures sufficient for the next step of nucleosynthesis: the burning of helium into carbon and oxygen. It is the natural subject of the next part of this chapter.

The carbon gate

Without carbon there would be no organic molecules, no DNA, nor any form of life we could recognize as such. Carbon is the fourth most abundant element in the universe (after hydrogen, helium and oxygen), and yet its synthesis is one of the most subtle processes of all nucleosynthesis: it depends on the existence of an excited state of $^{12}\mathrm{C}$ with an energy very close to just the right threshold for the reaction that produces it to be astrophysically effective. This state — the Hoyle state, predicted theoretically by Fred Hoyle in 1953-1954 and confirmed experimentally by Cook, Fowler, Lauritsen and Lauritsen at Caltech in 1957 [Cook et al. 1957] — is one of the most important details of nuclear physics for the whole of cosmic chemistry. Hoyle famously made of it one of the most frequently cited proofs of the anthropic principle.

The problem is simple to state. To build a carbon nucleus starting from the only helium available after hydrogen burning, one would have to fuse three $^{4}\mathrm{He}$ nuclei. A simultaneous three-body collision, in a thermal environment, has negligible probability: the rate of a three-body reaction scales as $\rho^2$, and at stellar densities ($\rho \sim 10^4$-$10^6$ g/cm³) the frequency of triple collisions remains many trillion times lower than that of binary collisions. The two alternative binary steps are both blocked: $\alpha + \alpha \to {}^{8}\mathrm{Be}$ works, but $^{8}\mathrm{Be}$ is unstable and decays into two α with a characteristic time of order $10^{-16}$ s; and $\alpha + \alpha + p$, or $\alpha + p$, produce nothing bound. For the same reason, recalled in the previous chapter on BBN, cosmological synthesis fails to get past helium: the “gap” at $A = 5$ and at $A = 8$ blocks every standard chain of binary captures.

The universe manages to get around the gap thanks to a resonance. The $^{8}\mathrm{Be}$, though unstable, lives long enough (at helium-burning temperatures, $T \sim 10^{8}$ K) to establish a small dynamic-equilibrium population with the two α that form and dissociate it [Salpeter 1952] :

$$\alpha + \alpha \rightleftharpoons {}^{8}\mathrm{Be}, \qquad X({}^{8}\mathrm{Be})_{\mathrm{eq}} \sim 10^{-9}.$$

This minuscule but non-zero concentration of $^{8}\mathrm{Be}$ can capture a third α before dissociating. The cross section of the capture $^{8}\mathrm{Be}(\alpha,\gamma)^{12}\mathrm{C}$ is dominated by the resonance in the excited state $J^{\pi} = 0^{+}$ of $^{12}\mathrm{C}$ at $E_x = 7{.}654$ MeV — about 379 keV above the $3\alpha$ threshold, that is, about 288 keV above the $^{8}\mathrm{Be}+\alpha$ threshold — which is exactly the Hoyle state. Without this resonance, the cross section for the capture of the third α would be many orders of magnitude smaller, the rate of $^{12}\mathrm{C}$ production in stars would be astrophysically negligible, and the subsequent chemistry of the cosmos would be radically different.

Hoyle’s prediction

The argument with which Hoyle predicted the existence of the excited state is one of the few historical examples of a nuclear-physics prediction obtained on purely astrophysical grounds. Hoyle observed that carbon exists in significant abundance in the universe, and — in the absence of other known mechanisms to produce it — deduced that the $3\alpha$ had to be astrophysically active. Estimating the rate needed to produce the observed fraction of $^{12}\mathrm{C}$ in giant stars, and comparing with the expected cross section for the non-resonant $^{8}\mathrm{Be}(\alpha,\gamma)$ capture, Hoyle concluded that there had to exist a resonance at a very specific energy, close to the $3\alpha$ threshold of $^{12}\mathrm{C}$, with a $J^{\pi}$ compatible with the capture [Hoyle 1954] . He went to Caltech, convinced Fowler to look for the state, and in 1957 the group of Cook, Fowler, Lauritsen and Lauritsen found it where Hoyle had indicated [Cook et al. 1957] . It is one of the boldest and most thoroughly verified predictions of twentieth-century physics, and it opened the way to the whole program of stellar nucleosynthesis codified in the celebrated B²FH paper of 1957.

Kinetics of the 3α process

Combining the dynamic equilibrium of $^{8}\mathrm{Be}$ with the resonant capture of the third α, the thermal rate of the $3\alpha$ in the narrow-resonance approximation depends on the effective energy of the resonance with respect to the $3\alpha$ threshold, $E_r \simeq 379$ keV, not on the total Q-value of the reaction. The Q-value $Q_{3\alpha} = 7{.}275$ MeV is the energy released when three α become $^{12}\mathrm{C}$ in the ground state; the exponent that controls the thermal probability of the resonance is instead $\exp(-E_r/kT)$. At $T = 10^{8}$ K this factor is about $\exp(-44)$: an enormous suppression, but not the astronomically larger suppression one would get by erroneously using the Q-value in the Boltzmann term. In approximate form, the net rate of energy emission per unit mass is written

$$\epsilon_{3\alpha} \approx 5{.}1 \times 10^{8} \, X_\alpha^{3} \, \rho^{2} \, T_9^{-3} \, e^{-4{.}4/T_9} \, \mathrm{erg/g/s}$$

with $T_9 = T / 10^9 \,\mathrm{K}$. The same formula is often written with $T_8 = T/10^8\,\mathrm{K}$ as $\exp(-44/T_8)$; using $T_9$ therefore requires the exponent $-4{.}4/T_9$. The temperature dependence is extraordinarily steep: $\nu = d \ln \epsilon / d \ln T \approx 40$ at $T = 10^{8}$ K — an order of magnitude above the already steep CNO dependence, and almost ten times that of the pp chain. A direct consequence is that helium burning is extremely sensitive to the local temperature: small fluctuations translate into enormous rate variations, and in the degenerate regime (see the next section) this sensitivity produces a thermonuclear runaway, the celebrated helium flash.

The Hoyle state as a physical object

The structural properties of the Hoyle state make it one of the most studied and most subtle objects of low-energy nuclear physics. Characterized by spin-parity $J^{\pi} = 0^{+}$ and isospin $T = 0$, it presents a nucleon density significantly lower than that of the ground state of $^{12}\mathrm{C}$ (a larger rms radius than the ground state), and an internal structure well described in terms of a cluster of three weakly correlated α particles. The precise geometry is still debated: triangle models, α-condensate-like descriptions and ab initio approaches capture different aspects of the same state. Form-factor measurements from electron scattering, $3\alpha$ decay correlations and rotational-band candidates are today the main constraints on its structure [Freer & Fynbo 2014] .

Recent ab initio calculations have reached a predictive capability on the Hoyle state that was unthinkable until twenty years ago. The lattice Effective Field Theory of Epelbaum, Krebs, Lähde, Lee and Meißner [Epelbaum et al. 2011] reproduced the excitation energy within still significant uncertainties, using chiral effective interactions. It is an important confirmation that the Hoyle state can emerge from low-energy nuclear dynamics, and need not be treated as a phenomenological anomaly inserted by hand. Subsequent No-Core Shell Model calculations and algebraic cluster models have reached compatible results, each with a different theoretical language.

The decay of the Hoyle state is dominated by the $3\alpha$ channel: $\Gamma_{3\alpha} / \Gamma_\gamma \approx 2500$, with the radiative decay (via the $2^{+}$ state at 4.44 MeV) representing only $\sim 4 \times 10^{-4}$ of the total. Only this small fraction contributes to the net production of $^{12}\mathrm{C}$: for every $\sim 2500$ Hoyle states formed, only one decays radiatively to the ground state and contributes to cosmic carbon; the others break up again into three α. The shape of the $3\alpha$ decay has been the object of recent precision measurements (Smith et al. 2017, and later ones at JYFL and ATOMKI): the data confirm that $> 99{.}5\%$ of the decays proceed sequentially via $^{8}\mathrm{Be}$ + α (one α emitted, and then the remaining two from the breakup of the intermediate $^{8}\mathrm{Be}$), with less than $0{.}5\%$ of “democratic” decays in which the three α share energy simultaneously — a result that strongly constrains the internal structure of the state and its channel couplings.

The subsequent α capture and the birth of oxygen

Once $^{12}\mathrm{C}$ exists, the same stellar zone in which the $3\alpha$ is occurring can capture further α particles and produce heavier nuclei through the sequence

$${}^{4}\mathrm{He} \xrightarrow{3\alpha} {}^{12}\mathrm{C} \xrightarrow{(\alpha,\gamma)} {}^{16}\mathrm{O} \xrightarrow{(\alpha,\gamma)} {}^{20}\mathrm{Ne} \xrightarrow{(\alpha,\gamma)} {}^{24}\mathrm{Mg} \to \cdots$$

At He-burning temperatures ($T \sim 1{.}5$-$3 \times 10^{8}$ K) the sequence effectively stops at $^{16}\mathrm{O}$, because the rate of $^{16}\mathrm{O}(\alpha,\gamma)^{20}\mathrm{Ne}$ is suppressed by several orders of magnitude owing to the higher Coulomb barrier and the absence of advantageous resonances at the available energy (the first useful resonance in $^{20}\mathrm{Ne}$ is at $E_x = 4{.}97$ MeV, well above the α threshold). The further α captures toward Ne, Mg, Si will occur only in the more advanced phases (carbon, neon, oxygen burning), treated in the final part of this chapter.

The final product of He-burning is therefore a carbon-oxygen core — the celebrated “CO core” — whose exact composition depends on the competition between two reactions: the $3\alpha$, which produces $^{12}\mathrm{C}$, and $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$, which destroys it to form $^{16}\mathrm{O}$. The final C/O ratio is a very sensitive function of the burning temperature (higher favors $(\alpha,\gamma)$ at the expense of $3\alpha$, and hence shifts toward O), of the density (higher favors $3\alpha$, and is the regime of low-mass stars with degenerate cores), and — above all — of the value of the cross section $S(300\,\mathrm{keV})$ of $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$, which is the single largest nuclear uncertainty of all stellar nucleosynthesis. Typical C/O values at the end of He-burning, derived from stellar evolution models:

• Massive stars ($M \sim 15$-$25\,M_\odot$): $X(^{12}\mathrm{C}) \approx 0{.}20$-$0{.}25$, $X(^{16}\mathrm{O}) \approx 0{.}70$-$0{.}75$;
• Intermediate-mass stars (AGB, $M \sim 3$-$5\,M_\odot$): $X(^{12}\mathrm{C}) \approx 0{.}30$-$0{.}40$, $X(^{16}\mathrm{O}) \approx 0{.}50$-$0{.}60$;
• Low-mass stars ($M \sim 1$-$2\,M_\odot$): $X(^{12}\mathrm{C}) \approx 0{.}40$-$0{.}50$, $X(^{16}\mathrm{O}) \approx 0{.}40$-$0{.}50$.

The consequences of the C/O ratio are pervasive in stellar evolution and Galactic chemistry: in core-collapse supernovae it fixes the thickness of the Si, S and Ca mantles produced by the subsequent phases; in white dwarfs it is the fuel of Type Ia supernovae, and modulates the mass of $^{56}\mathrm{Ni}$ synthesized (more C means more availability of exoenergetic reactions, hence more $^{56}\mathrm{Ni}$ and more luminous supernovae); in AGB stars it determines whether the product of third dredge-up is an M-type star (O > C, silicate dust) or a C-type star (C > O, amorphous carbon and SiC dust), with implications for the mass loss, the dust formation and — ultimately — the composition of the enriched interstellar medium.

The “holy grail” of nuclear astrophysics

The reaction $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ has been called, in many reviews, the Holy Grail of stellar nucleosynthesis: no other single reaction produces a comparable uncertainty on the final product of nucleosynthesis. The problem is threefold. First: the cross section at the astrophysical Gamow peak ($E_0 \approx 300$ keV for $T = 2 \times 10^{8}$ K) is too small — about $10^{-17}$ barn — to be measured directly, even in the best current underground laboratories. Second: the extrapolation from laboratory energies (1-3 MeV) to the Gamow peak is dominated by the interference of three distinct contributions — E1 capture dominated by a subthreshold $1^{-}$ resonance at $-45$ keV below the α threshold, E2 capture dominated by a subthreshold $2^{+}$ resonance at $-245$ keV, and non-resonant direct capture — with a complex analytic structure that amplifies the uncertainties. Third: the existing data, accumulated over five decades by dozens of experimental groups with different techniques, are affected by systematics that are hard to combine.

The most complete and modern multi-channel R-matrix analysis is that of deBoer and collaborators in 2017 [deBoer et al. 2017] , which combines γ-capture data ($\alpha + {}^{12}\mathrm{C} \to {}^{16}\mathrm{O} + \gamma$), elastic scattering data $\alpha + {}^{12}\mathrm{C}$, $^{16}\mathrm{N}(\beta^{-}\alpha){}^{12}\mathrm{C}$ data (β-delayed α decay), nuclear transfer data ($^{12}\mathrm{C}({}^{6}\mathrm{Li},d){}^{16}\mathrm{O}$ to extract the ANC of the asymptotic tails) and inverse photodisintegration data $^{16}\mathrm{O}(\gamma,\alpha){}^{12}\mathrm{C}$. The result is $S_{E1+E2}(300\,\mathrm{keV}) = 140 \pm 21$ keV·b, with a $\sim 15\%$ uncertainty. The ongoing and planned experimental campaigns in underground laboratories, with inverse reactions and with transfer measurements, aim to reduce this uncertainty toward $5\%$. The effect on nucleosynthesis is significant: a $10$-$20\%$ variation in $S(300)$ shifts the final C/O ratio and produces cascading consequences on the abundances of the elements of the $A = 20$-$60$ group produced in the subsequent phases.

Secondary nucleosynthesis during He-burning

Besides the main production of C and O, during helium burning some secondary chains of great importance for the nucleosynthesis of the heavier elements activate. The most relevant is the sequence starting from the nitrogen-14 accumulated by the previous CNO-burning. The $^{14}\mathrm{N}$, surviving in significant concentration in the He core, is rapidly burned by α capture:

$${}^{14}\mathrm{N}(\alpha,\gamma)^{18}\mathrm{F}(\beta^{+}\nu)^{18}\mathrm{O}(\alpha,\gamma)^{22}\mathrm{Ne}.$$

At this point the $^{22}\mathrm{Ne}$, if the local temperature exceeds $\sim 2{.}5 \times 10^{8}$ K, can activate the reaction $^{22}\mathrm{Ne}(\alpha,n)^{25}\mathrm{Mg}$ — the main neutron source of the s-process weak component, active in the convective He-burning cores of massive stars and responsible for the production of the intermediate-mass s elements ($A \sim 60$-$90$). This is why the uncertainties on $^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$, through the modulation of the C/O ratio and hence of the availability of $^{22}\mathrm{Ne}$, propagate also onto the s-process predictions of massive stars: two distinct stages of nucleosynthesis, coupled by a chain of nuclear causality.

In intermediate-mass stars (AGB), the situation is different and even richer. During the thermally pulsing phase (TP-AGB, treated in detail in chapter 4), the convection between thermal pulses carries protons from the envelope into the He-rich intershell, where the capture $^{12}\mathrm{C}(p,\gamma){}^{13}\mathrm{N}(\beta^{+}\nu){}^{13}\mathrm{C}$ forms a $^{13}\mathrm{C}$ pocket. During the quiescent interpulse phase, the $^{13}\mathrm{C}$ burns through

$${}^{13}\mathrm{C}(\alpha,n)^{16}\mathrm{O}$$

at $T \approx 10^{8}$ K, providing the main neutron source of the s-process main component, responsible for the s elements of mass $A \sim 90$-$208$ (Sr, Y, Zr, Ba, La, Pb). The details are in chapter 4; here it suffices to note that helium burning is the primary source of the whole s-process, both weak and main, and its yields modulate the entire slow neutron nucleosynthesis of the Galaxy. The canonical review is Käppeler et al. 2011 [Käppeler et al. 2011] .

Where and how helium burning occurs

The way helium ignites depends critically on the stellar mass, because it determines the thermodynamic structure of the core at the moment of hydrogen exhaustion and hence the mode of transition to the new phase. For solar-type stars and for more massive ones the physics is qualitatively different.

In low-mass stars ($M \lesssim 2\,M_\odot$), the post-MS helium core contracts until it reaches densities $\sim 10^{6}$ g/cm³ in the electron-degenerate gas regime — degeneracy pressure dominates over thermal pressure, and the core temperature can rise independently of the pressure. When $T_c$ reaches $\sim 10^{8}$ K, the $3\alpha$ ignites: but in a degenerate gas the energy release does not cause expansion (the pressure is almost independent of $T$), and the temperature increases further, feeding a thermonuclear runaway. It is the celebrated helium flash: in a few seconds the burning rate grows by eight to ten orders of magnitude, reaching a peak luminosity $\sim 10^{11}\,L_\odot$ — comparable to the luminosity of an entire galaxy — confined to the core and thermalized by the convection that sets in immediately. The luminosity does not emerge at the surface because it is absorbed by the overlying envelope; but in a few seconds the core expands, removes the degeneracy, and settles into a regime of quiescent $3\alpha$ burning that will last $\sim 10^{8}$ years. The star then takes its place on the Horizontal Branch of the HR diagram (visible as the well-known horizontal sequence in the color-magnitude diagrams of globular clusters) and burns quietly until central He exhaustion.

In intermediate-mass stars ($2 \lesssim M / M_\odot \lesssim 8$), the helium core remains non-degenerate at ignition, because the greater compression and heating during the contraction keeps it thermal. He-burning ignites quiescently and the star settles into a blue loop or moves rapidly toward the asymptotic giant branch. The duration of the phase is of order $(10^{6}$-$10^{8})$ years, decreasing with mass.

In massive stars ($M \gtrsim 8\,M_\odot$), He-burning is even faster (for $M = 20\,M_\odot$, it lasts about $5 \times 10^{5}$ years) and follows hydrogen exhaustion after a short core-contraction gap. The He-burning of these stars occurs at higher temperatures, $T \sim 2{.}5$-$3 \times 10^{8}$ K, sufficient to activate $^{22}\mathrm{Ne}(\alpha,n)^{25}\mathrm{Mg}$ significantly. At the end, the carbon-oxygen core of the massive star continues to contract and carbon burning ignites — there is no pause between successive phases comparable to the long main-sequence life or the Horizontal Branch phase.

A short digression on anthropic fine-tuning

The fact that the Hoyle state sits close to the right energy for the $3\alpha$ to produce a significant abundance of $^{12}\mathrm{C}$ is one of the most frequently cited examples of anthropic “fine-tuning” in the cosmos. The natural quantitative question is: how fine is the tuning? The sensitivity calculations of Csótó, Oberhummer, Schlattl and Pichler and of [Schlattl et al. 2004] show that small effective variations of the nucleon-nucleon interaction can shift the energy of the Hoyle state by a few hundred keV, sensibly modifying the final C/O ratio. More precisely: an increase of the energy makes the capture of the third α less effective and reduces carbon production, while a decrease makes it more effective but also changes the thermal response of the star and the competition with $^{12}\mathrm{C}(\alpha,\gamma){}^{16}\mathrm{O}$. Stellar nucleosynthesis simulations with varied constants suggest that percent-level variations of the nuclear parameters would produce universes with C/O ratios very different from the observed one.

It should be said that the argument is less stringent than the popular literature suggests: more recent lattice EFT calculations [Epelbaum et al. 2011] and multi-parameter analyses show that there exist regions of “alternative” parameters in which the production of C and O is similar to the observed one, through different nuclear mechanisms (for example a $0^{+}$ resonance in $^{12}\mathrm{C}$ at a different position, or a combination of alternative reactions). The tuning is real but not as “miraculous” as sometimes presented, and it is an interesting example of the way the sensitivity of a system to small parameter variations can be real, yet also dependent on the particular parameterization of the problem.

With helium burning completed, the star possesses an inert carbon-oxygen core surrounded by active He-burning and H-burning shells. The subsequent fate depends once again on the mass: low-mass and intermediate-mass stars enter the AGB phase and then end as CO white dwarfs without further nuclear burnings (see chapter 4); massive stars continue the sequence of advanced burnings — carbon, neon, oxygen, silicon — up to the formation of an iron core and the gravitational collapse. These advanced phases, short and violent, are the subject of the final part of this chapter.

The onion structure

The last nuclear phases of a massive star — those separating helium exhaustion from the collapse of the iron core — constitute one of the most dramatic and rapid processes of all astrophysics. In less than a thousand years overall, and in a few hours in the final phases, the star burns in sequence four successive fuels — carbon, neon, oxygen, silicon — building an onion structure of concentric shells ever richer in heavy elements, separated by active reaction fronts. At the end, the center contains an inert core of iron and nickel, surrounded by still-active shells of Si-burning, O-burning, Ne-burning, C-burning, He-burning and H-burning. It is a thermally precarious configuration — the inert core is sustained only by the pressure of the degenerate electrons — and when its mass exceeds the Chandrasekhar limit the collapse is inevitable.

The temporal sequence is characteristically accelerating: each phase lasts much less than the previous one, and the last phases are completed on timescales ranging from a year to a day. For an archetypal star of $25\,M_\odot$ at solar metallicity (the precise numbers depend on the code and on the mixing parameters, but the order of magnitude is universal):

The deep cause of this acceleration is thermal neutrino cooling. At temperatures $T \gtrsim 5 \times 10^{8}$ K, the photons of the thermal bath have energies above the threshold for $e^{+}e^{-}$ pair production, and a small but growing fraction of the subsequent annihilations proceeds not through the channel $e^{+}e^{-} \to 2\gamma$ but through the weak channel $e^{+}e^{-} \to \nu \bar\nu$. Other thermal processes contribute — photoneutrinos ($\gamma + e \to e + \nu \bar\nu$), plasmon neutrinos ($\gamma_{\mathrm{pl}} \to \nu \bar\nu$), neutrino bremsstrahlung — and at Si-burning temperatures the total neutrino emission rate scales as $L_\nu \propto T^9$, with a neutrino luminosity of the core that exceeds $L_\gamma$ by several orders of magnitude. The neutrinos, being practically decoupled from the matter, leave the star in a few seconds and carry away energy directly from the core. The consequence is that the thermal balance of the core — which on the main sequence is dominated by radiative transport outward — is now dominated by neutrino transport, and to maintain hydrostatic equilibrium the star must burn fuel faster. The evolutionary timescale compresses by a factor $\sim T^{-9}$, and the entire Si-burning phase passes in a few hours.

A less obvious but equally important consequence is that the advanced burnings proceed in near absence of photonic feedback from the envelope. The neutrino luminosity of the core is not observable from the outside (except for dedicated detections such as the Kamiokande-IMB-Baksan coincidence with SN 1987A), and the envelope of the star does not “know” what is happening inside. A $25\,M_\odot$ star a few hours before collapse has practically the same surface luminosity and the same radius it had ten years earlier — the envelope is simply too large and too opaque to respond on such short timescales. Only gravitational waves and neutrinos, in principle, could give advance warning of the imminent collapse, and indeed a program of pre-supernova neutrino “alerts” (KamLAND, JUNO, SNO+) is the object of active research.

Core composition along the sequence

The composition of the burning material evolves continuously along the sequence. In broad strokes, for a $25\,M_\odot$ star at $Z = Z_\odot$ according to the calculations of Sukhbold et al. (2016) [Sukhbold et al. 2016] :

• start of C-burning: $^{12}\mathrm{C}/^{16}\mathrm{O} \approx 0{.}28/0{.}69$, traces of $^{20}\mathrm{Ne}$, $^{22}\mathrm{Ne}$ (residue of the α capture on $^{14}\mathrm{N}$ during He-burning);
• start of Ne-burning: $^{16}\mathrm{O}/^{20}\mathrm{Ne}/^{24}\mathrm{Mg} \approx 0{.}74/0{.}21/0{.}05$;
• start of O-burning: $^{16}\mathrm{O}/^{24}\mathrm{Mg}/^{28}\mathrm{Si} \approx 0{.}69/0{.}25/0{.}06$;
• start of Si-burning: $^{28}\mathrm{Si}/^{32}\mathrm{S}/$alphas $\approx 0{.}55/0{.}30/$ residues;
• at collapse: a core of Fe-peak nuclei with $\langle A \rangle \approx 53$, $\langle Z \rangle \approx 25$, $Y_e \approx 0{.}43$-$0{.}46$.

Each burning phase bequeaths to the next a material chemically very different from the one it started with, and at the same time modifies the thermal structure of the core (with increasing density, temperature and electron degeneracy).

Carbon burning

At temperatures of about $8 \times 10^{8}$ K — an order of magnitude higher than those of He-burning — two $^{12}\mathrm{C}$ nuclei possess enough kinetic energy to overcome their mutual Coulomb barrier ($E_C \approx 9$ MeV in center-of-mass coordinates) by tunneling. The initial reaction forms a compound nucleus $^{24}\mathrm{Mg}^{*}$ in an excited state at excitation energies of $\sim 14$-$16$ MeV, which decays rapidly along the open channels:

• $^{12}\mathrm{C} + ^{12}\mathrm{C} \to ^{20}\mathrm{Ne} + \alpha$, $Q = 4{.}62$ MeV, branching $\approx 0{.}56$;
• $^{12}\mathrm{C} + ^{12}\mathrm{C} \to ^{23}\mathrm{Na} + p$, $Q = 2{.}24$ MeV, branching $\approx 0{.}44$;
• $^{12}\mathrm{C} + ^{12}\mathrm{C} \to ^{23}\mathrm{Mg} + n$, $Q = -2{.}60$ MeV, branching $\lesssim 0{.}01$;
• $^{12}\mathrm{C} + ^{12}\mathrm{C} \to ^{24}\mathrm{Mg} + \gamma$, $Q = 13{.}93$ MeV, branching $\ll 1\%$.

The light particles produced (α, p, n) are immediately captured by other nuclei in secondary chains. The proton, for example, is captured preferentially by $^{12}\mathrm{C}$ (forming $^{13}\mathrm{N}$ which decays into $^{13}\mathrm{C}$) or by $^{23}\mathrm{Na}$ (regenerating $^{20}\mathrm{Ne}$ via $(p,\alpha)$); the α is captured by the products to build $^{24}\mathrm{Mg}$, $^{28}\mathrm{Si}$ and beyond; the small neutron flux feeds a weak s-process component that significantly produces the elements $A \sim 60$-$90$ (Cu, Zn, Ga, Ge, As, Se, Br, Kr) in the cores of massive stars. The reaction network of a detailed C-burning simulation typically counts $\sim 200$ nuclides and $\sim 1000$ reactions.

The mean yield of quiescent C-burning, for a $25\,M_\odot$ star, is characterized by post-phase mass fractions in the core: $X(^{20}\mathrm{Ne}) \approx 0{.}5$, $X(^{16}\mathrm{O}) \approx 0{.}3$, $X(^{24}\mathrm{Mg}) \approx 0{.}15$. The original carbon is completely consumed; the neon inherits a large part of the α flux; the magnesium is synthesized in significant quantity via the secondary $^{20}\mathrm{Ne}(\alpha,\gamma){}^{24}\mathrm{Mg}$. Sodium is one of the most diagnostic tracers of carbon burning in massive stars: its abundance in the ejecta of core-collapse supernovae is sensitive to $S_{12+12}$, to the p/α branching and to the mass cut of the SN.

The $^{12}\mathrm{C}+^{12}\mathrm{C}$ rate and the resonance debate

The cross section of $^{12}\mathrm{C} + {}^{12}\mathrm{C}$ at astrophysical energies ($E_0 \approx 1{.}5$ MeV in the center of mass, for $T = 8 \times 10^{8}$ K) is one of the most discussed open nuclear problems of the field. Direct laboratory measurements stop at $E_{\mathrm{cm}} \approx 2{.}1$ MeV, where the astrophysical factor $S^{*}(E)$ presents a complex resonant structure — a sequence of irregular peaks and troughs attributed to the formation of molecular states of $^{24}\mathrm{Mg}^{*}$ — and its extrapolation to the astrophysical Gamow peak is uncertain by up to a factor of 5, with a consequent 30% uncertainty on the $^{23}\mathrm{Na}$ yields and modifications of the evolutionary path.

The debate has concentrated in recent years on the possible existence of a subthreshold resonance near $E_{\mathrm{cm}} \approx 1{.}5$ MeV, which would amplify the rate by a factor $\sim 25$. The indirect measurement with the Trojan Horse Method by Tumino et al. (2018) [Tumino et al. 2018] reported evidence in favor of the resonance, while the direct measurements of the STELLA experiment at the Gran Sasso National Laboratories (Tan et al. 2020, [Tan et al. 2020] ) and the subsequent STELLA-2 measurements have excluded a strong resonance but left room for weaker contributions compatible with the TH data. The controversy is not yet completely resolved: even more sensitive measurements are under way with further improved setups at STELLA, and new experiments with isotopically enriched carbon targets are planned.

The nuclear uncertainty propagates non-trivially to the subsequent evolution. A higher C-burning rate accelerates the phase, reduces the mass of the carbon core at the moment of Ne-burning ignition, and produces a smaller compact core. This in turn modifies the mass cut of the final supernova and hence the mass of $^{56}\mathrm{Ni}$ expelled: two orders of magnitude in $S_{12+12}$ correspond to about 30% variation in the $^{56}\mathrm{Ni}$ mass of the ejecta for stars of $\sim 15$-$20\,M_\odot$. The $^{12}\mathrm{C}+^{12}\mathrm{C}$ rate is also one of the key parameters for determining the minimum mass of a star that explodes as a core-collapse supernova, versus slightly less massive stars that end as O-Ne-Mg white dwarfs after igniting only shell C-burning.

Neon burning

Neon is the first fuel of the sequence that does not burn by direct fusion. Its Coulomb barrier is too high — $^{20}\mathrm{Ne} + {}^{20}\mathrm{Ne}$ would require temperatures $> 3 \times 10^{9}$ K to have significant rates — and before reaching them the nuclear structure offers a cheaper route through photodisintegration. At $T \sim 1{.}5 \times 10^{9}$ K, the Wien tail of the Planck distribution of photons contains photons with energy $E_\gamma \gtrsim 5$ MeV in sufficient density to activate the reaction $^{20}\mathrm{Ne}(\gamma,\alpha)^{16}\mathrm{O}$ — the Q-value of the inverse direct capture is $Q_\alpha = 4{.}73$ MeV, well within thermal reach. The liberated α then combine rapidly with other $^{20}\mathrm{Ne}$ to form magnesium.

The reaction sequence of Ne-burning is written as

$$^{20}\mathrm{Ne}(\gamma,\alpha)^{16}\mathrm{O}, \quad ^{20}\mathrm{Ne}(\alpha,\gamma)^{24}\mathrm{Mg}, \quad ^{24}\mathrm{Mg}(\alpha,\gamma)^{28}\mathrm{Si}$$

with net balance $2 \,{}^{20}\mathrm{Ne} \to {}^{16}\mathrm{O} + {}^{24}\mathrm{Mg}$ and total Q of $4{.}58$ MeV. The photodisintegration rate is connected to the rate of the direct α captures via the principle of detailed balance, and the calculation requires accurate knowledge of the resonances of $^{24}\mathrm{Mg}$ at excitation energies $E_x \approx 13$-$15$ MeV. Some of these resonances are known from elastic scattering $(\alpha,\alpha')$ and from transfer reactions; others are less constrained, and the residual uncertainty on the Ne-burning rate is $\sim 20\%$. The balance of the reaction puts $^{20}\mathrm{Ne}$ in a quasi-equilibrium with $^{16}\mathrm{O}$ and $^{24}\mathrm{Mg}$, and the final composition is dominated by the competition between the $(\alpha,\gamma)$ branching toward Mg and the further $(\alpha,\gamma)$ branching toward Si. The typical post-Ne-burning yield is: $X(^{16}\mathrm{O}) \approx 0{.}7$, $X(^{24}\mathrm{Mg}) \approx 0{.}2$, $X(^{28}\mathrm{Si}) \approx 0{.}05$.

In parallel, the Ne-burning phase produces some important minor species through secondary chains fed by the α and the few residual protons: $^{27}\mathrm{Al}$ via $^{23}\mathrm{Na}(\alpha,p){}^{26}\mathrm{Mg}(\alpha,n){}^{29}\mathrm{Si}$ with subsequent recirculation of the p onto $^{26}\mathrm{Mg}$; $^{31}\mathrm{P}$ via $^{27}\mathrm{Al}(\alpha,p){}^{30}\mathrm{Si}$ and subsequent capture; a small residual s-process component fed by the neutron flux of the $^{22}\mathrm{Ne}(\alpha,n){}^{25}\mathrm{Mg}$ still active. The abundances of Al, P, K in SN II remnants carry the direct signature of this phase.

Oxygen burning

At temperatures of $\sim 2 \times 10^{9}$ K and densities of $\sim 10^{7}$ g/cm³, oxygen begins to burn by direct binary fusion. The initial reaction forms a compound nucleus $^{32}\mathrm{S}^{*}$ at excitation energies of $\sim 16$-$18$ MeV, which decays through a notable number of channels:

• $^{16}\mathrm{O} + ^{16}\mathrm{O} \to ^{28}\mathrm{Si} + \alpha$, $Q = 9{.}59$ MeV, branching $\approx 0{.}21$;
• $^{16}\mathrm{O} + ^{16}\mathrm{O} \to ^{31}\mathrm{P} + p$, $Q = 7{.}68$ MeV, branching $\approx 0{.}61$;
• $^{16}\mathrm{O} + ^{16}\mathrm{O} \to ^{31}\mathrm{S} + n$, $Q = 1{.}45$ MeV, branching $\approx 0{.}05$;
• $^{16}\mathrm{O} + ^{16}\mathrm{O} \to ^{30}\mathrm{P} + d$, $Q = -2{.}41$ MeV, branching $\approx 0{.}05$;
• $^{16}\mathrm{O} + ^{16}\mathrm{O} \to ^{32}\mathrm{S} + \gamma$, $Q = 16{.}54$ MeV, branching $\sim 0{.}01$.

The branchings have experimental uncertainties of $\sim 20\%$. The light particles (α, p, n, d) produced are immediately captured by the surrounding nuclei, feeding a very rich secondary nucleosynthesis. The dominant product of quiescent O-burning is $^{28}\mathrm{Si}$, with post-phase mass fractions in the core: $X(^{28}\mathrm{Si}) \approx 0{.}55$, $X(^{32}\mathrm{S}) \approx 0{.}25$, $X(^{36}\mathrm{Ar}) \approx 0{.}05$, plus minor contributions of Mg, Cl, K, Ca. The cross section of $^{16}\mathrm{O} + {}^{16}\mathrm{O}$ is measured in the laboratory down to $E \approx 6$ MeV (Hulke 1980, Spinka & Winkler 1974, recent measurements by the ELENA group at Caserta); the extrapolation to the astrophysical Gamow peak ($E_0 \approx 4$ MeV for $T = 2 \times 10^{9}$ K) introduces a $\sim 30\%$ uncertainty, comparable to that of $^{12}\mathrm{C} + {}^{12}\mathrm{C}$.

An important chemical characteristic of O-burning is the abundant production of the alpha-elements of the middle group: $^{24}\mathrm{Mg}$, $^{28}\mathrm{Si}$, $^{32}\mathrm{S}$, $^{36}\mathrm{Ar}$, $^{40}\mathrm{Ca}$. These are the “multiple-alpha” nuclei that dominate the composition of the ejecta of core-collapse supernovae, and their ratio with respect to iron in old-generation stars (the iconic $[\alpha/\mathrm{Fe}]$ of the Galactic chemical evolution diagrams) is the diagnostic signature for separating the contribution of SNe II from that of SNe Ia. In chapter 7 we shall see how this ratio is used to reconstruct the star-formation history of the Galaxy.

A secondary consequence of O-burning is the possible activation of the p-process (see chapter 5). During the shell O-burning phase or in the first moments of the supernova shock front, the photodisintegration of seed nuclei produced by the s-process in the previous phases — in particular nuclei around Ba, Ce, Sm — can generate the small population of “proton-rich” nuclei (p-nuclei) that characterizes the solar abundance of some rare isotopes. The complete network for O-burning coupled to the p-process typically counts $\sim 1000$ nuclides.

Silicon burning and nuclear statistical equilibrium

The last phase of quiescent thermonuclear burning is the fastest and the most conceptually subtle. At temperatures $T \gtrsim 3 \times 10^{9}$ K — corresponding to $kT \approx 250$ keV, and thermal photons of MeV — silicon does not burn by binary fusion $^{28}\mathrm{Si} + {}^{28}\mathrm{Si}$, because the Coulomb barrier ($E_C \approx 33$ MeV) is too high for tunneling to be significant on the required timescales. It burns instead through a mechanism of cascading photodisintegration followed by α recapture: the silicon is progressively dismantled by the thermal photons into lighter nuclei, the liberated α particles are recaptured by the heavier nuclei, and within a few hours the composition reaches a quasi-equilibrium regime dominated by the most tightly bound nuclei per nucleon — those of the iron peak ($A = 56$ or so).

The formal photodisintegration sequence is

$$^{28}\mathrm{Si}(\gamma,\alpha)^{24}\mathrm{Mg}(\gamma,\alpha)^{20}\mathrm{Ne}(\gamma,\alpha)^{16}\mathrm{O}(\gamma,\alpha)^{12}\mathrm{C}(\gamma,\alpha)2\alpha$$

but it is simultaneous with the α-capture chains in the opposite direction: the net flow goes toward the iron peak. At $T \gtrsim 5 \times 10^{9}$ K the regime of nuclear statistical equilibrium (NSE), already treated in chapter 2, is reached: all the fusion/photodisintegration and capture/emission reactions of light particles are in mutual equilibrium, and the composition of the plasma becomes a function only of the three thermodynamic parameters $(T, \rho, Y_e)$. The abundances are given by the Saha equation

$$Y(Z,N) = G(Z,N) \, A^{3/2} \, 2^{-A} \left( \frac{\rho N_A}{\theta} \right)^{A-1} e^{B(Z,N)/kT} \, Y_p^{Z} \, Y_n^{N}$$

with $\theta = (m_u kT / 2\pi \hbar^2)^{3/2}$, $G$ the partition function, $B$ the binding energy, and the free fractions $Y_p$, $Y_n$ constrained by the conservation laws $\sum_i A_i Y_i = 1$, $\sum_i Z_i Y_i = Y_e$. The exponential factor $e^{B/kT}$ favors the nuclide with the highest binding energy compatible with the given $Y_e$.

At neutron-proton symmetry ($Y_e = 0{.}5$), the winner of the NSE selection is $^{56}\mathrm{Ni}$ ($Z=N=28$, $B/A = 8{.}64$ MeV, near-doubly-magic). At $Y_e \lesssim 0{.}48$, the selection shifts toward $^{54}\mathrm{Fe}$ and $^{56}\mathrm{Fe}$ (stable iron); at $Y_e \lesssim 0{.}45$, $^{58}\mathrm{Ni}$, $^{60}\mathrm{Ni}$, $^{62}\mathrm{Ni}$ and more neutron-rich nuclei of the iron group appear in abundance. The value of $Y_e$ in the final phases is a critical parameter that evolves dynamically: in massive stars, $Y_e$ decreases during Si-burning through electron capture $e^{-} + p \to n + \nu_e$ at the high core densities, while in the thermonuclear Type Ia supernovae (explosions of CO white dwarfs) the $Y_e$ remains practically $0{.}5$ and the product is dominated by $^{56}\mathrm{Ni}$ — it is precisely the decay $^{56}\mathrm{Ni} \to {}^{56}\mathrm{Co} \to {}^{56}\mathrm{Fe}$ that sustains the light curve of these explosions.

Freeze-out and asymptotic regimes

The NSE is an instantaneous regime: as soon as the temperature drops below the critical threshold (typically $T \lesssim 4 \times 10^{9}$ K), the capture and photodisintegration rates are no longer sufficient to maintain the equilibrium and the composition freezes. The exact value of the composition at freeze-out depends not only on the temperature but also on the speed of the cooling — a parameter that enters through the expansion time $\tau_{\mathrm{exp}} = (\rho/\dot\rho)$ — and on the density at freeze-out. Two main regimes are distinguished:

• Normal freeze-out: high $\rho$ ($\gtrsim 10^7$ g/cm³), low entropy per nucleon. The triple-α reactions remain fast enough to convert the residual α into heavy nuclei, and the NSE persists until complete freeze-out. The product is almost purely $^{56}\mathrm{Ni}$ (for $Y_e = 0{.}5$). It is the regime of SNe Ia and of the deep core of core-collapse SNe.

• $\alpha$-rich freeze-out: low $\rho$ ($\lesssim 10^6$ g/cm³), high entropy per nucleon. The triple-α reactions fail to reconvert all the α into heavy nuclei before the cooling, and a non-negligible fraction of α (5-30%) remains in the final mixture together with the Fe-peak nuclei. The regime favors specific nuclei such as $^{44}\mathrm{Ti}$ ($\tau_{1/2} = 60$ yr, observable live), $^{48}\mathrm{Cr}$, $^{52}\mathrm{Fe}$. It is the typical regime of the outer layers of core-collapse SNe, where the fast expansion produces the high-entropy conditions.

The mass cut — the boundary between the material that remains trapped in the compact remnant (neutron star or black hole) and that which is expelled — controls how much of all this Fe-peak-rich material gets out in the ejecta and reaches the interstellar medium. The observed value of the $^{56}\mathrm{Ni}$ mass in the ejecta of core-collapse SNe is $\sim 0{.}07\,M_\odot$ for the archetypal case of SN 1987A, but varies by a factor $\sim 5$ among events and depends critically on the detailed explosion mechanism (see chapter 5). The detection of $^{44}\mathrm{Ti}$ in Cas A and SN 1987A by NuSTAR (in X-ray lines at 68 and 78 keV) is the direct observational confirmation of the α-rich freeze-out regime.

Electron capture and the role of weak rates

A crucial feature of Si-burning at very high densities ($\rho > 10^8$ g/cm³, typical of the final phases of a massive star’s core) is the dominance of electron capture on the Fe-peak nuclei. When the electron gas is degenerate, the Fermi energy of the electrons $E_F = (3\pi^2)^{1/3} \hbar c (n_e)^{1/3}$ can exceed the Q-thresholds of various reactions $e^{-} + (Z,A) \to (Z-1,A) + \nu_e$, and the matter progressively neutronizes: $Y_e$ falls from $\sim 0{.}5$ toward values around $0{.}43$. The electron-capture rates on Fe-peak nuclei are dominated by Gamow-Teller transitions, and their accurate determination — experimental ($(n,p)$ and $(d,{}^{2}\mathrm{He})$ in inverse kinematics) and theoretical (large-scale shell-model calculations) — is one of the most important efforts of nuclear astrophysics of the last twenty years. The standard compilation is that of Langanke & Martínez-Pinedo [Langanke & Martínez-Pinedo 2003] , which provides tabulated rates for hundreds of nuclei in the range $A = 45$-$65$ relevant to the pre-supernova core.

The role of these rates is twofold. On the one hand, they modulate the speed of neutronization of the core and hence the NSE composition; on the other, the associated neutrino losses (the electron-capture neutrinos leave the core without interacting) add a further cooling channel, accelerating the collapse. The transition from quiescent Si-burning to the pre-supernova collapse phase is not sharp: electron capture continues during the collapse itself, contributing to the final reduction of $Y_e$ to $\sim 0{.}3$ in the proto-neutron core.

From Si-burning to collapse

When the quiescent Si-burning in the core is exhausted, the star has at its center an inert nucleus of nickel and iron of mass $\sim 1{.}3$-$1{.}5\,M_\odot$, radius $\sim 3000$ km, density $\sim 10^{10}$ g/cm³, temperature $\sim 5 \times 10^{9}$ K. The pressure that sustains it against gravity is almost entirely the degeneracy pressure of the relativistic electrons — the thermal pressure is marginal. Around it, shells of Si-burning, O-burning, Ne-burning, C-burning, He-burning and H-burning continue to deposit processed material on the interface, progressively growing the mass of the inert core.

The limit to stability is the Chandrasekhar limit $M_{\mathrm{Ch}} = 5{.}83 \, Y_e^2 \, M_\odot$. For $Y_e \approx 0{.}43$, typical of the pre-supernova core, $M_{\mathrm{Ch}} \approx 1{.}08\,M_\odot$; as the core grows by accretion from the shells and $Y_e$ falls further by electron capture, the stability margin thins. Two mechanisms destabilize the core and trigger the collapse. The first is the photodisintegration of iron: at $T \gtrsim 7 \times 10^{9}$ K, thermal photons sufficiently energetic for the reaction $^{56}\mathrm{Fe} + \gamma \to 13\,\alpha + 4n$ (Q = $-124{.}4$ MeV) begin to subtract thermal energy from the gas and to reduce the pressure. It is a catastrophic endothermic process: the energy invested in the photodisintegration is subtracted from the pressure, accelerating the contraction and further increasing $T$ — a negative runaway. The second mechanism is the accelerated electron capture: as $\rho$ grows, the Fermi energy of the electrons exceeds the Q-thresholds of ever more nuclei, and the neutronization rate grows, reducing $Y_e$ and hence $M_{\mathrm{Ch}}$.

When these two mechanisms overcome the capacity of the degeneracy pressure to sustain the core, the collapse begins. In $\sim 100$ ms the core implodes from $R \sim 3000$ km to $R \sim 30$ km, reaching nuclear density ($\rho \sim 10^{14}$ g/cm³) at the center. The matter rebounds on the forming proto-neutron core, generating a hydrodynamic shock that propagates outward. The detail of how (or whether) this shock manages to explode the star — the so-called “supernova problem” that occupied theory for forty years — is the subject of chapter 5 on supernovae.

The final integrated yields of core-collapse supernovae, after the explosion and accounting for the mass cut, are the combined product of all the advanced burnings described in this chapter, modulated by the mass cut and by the α-rich freeze-out. For a $20\,M_\odot$ star at $Z = Z_\odot$, mass cut at $\sim 1{.}5\,M_\odot$ (indicative values, $\pm 30\%$ depending on the code):

The complete tables for stars of $11$-$120\,M_\odot$ at various metallicities are provided by several modeling groups: Limongi & Chieffi (FRANEC, [Limongi & Chieffi 2018] ), Sukhbold & Woosley (KEPLER, [Sukhbold et al. 2016] ), Pignatari & Herwig (NuGrid). The differences among the tables reflect different choices of uncertain nuclear reactions (in particular $^{12}\mathrm{C}(\alpha,\gamma){}^{16}\mathrm{O}$ and $^{12}\mathrm{C}+^{12}\mathrm{C}$), of mass-cut treatment, of convective mixing and of rotation. In chapter 7 we shall see how these yields, integrated over the initial mass function (IMF), provide the fundamental ingredient of Galactic chemical evolution models.

With the formation of the iron core the quiescent thermonuclear race is over. What follows is violent and different from everything that came before: the gravitational collapse, the formation of the proto-neutron core, and — perhaps — the explosion of the supernova, the subject of chapter 5. First, however, the story of nucleosynthesis passes through a quieter and equally fertile phase: the AGB stars, slow neutron capture and the meteoritic archive of presolar grains, the subject of the next chapter.