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The rest mass of a chemical atom in its ground state, a physical quantity expressed in SI mass units or the unified atomic mass unit. Here $m_i$ means the atomic mass of a neutral chemical atom of isotope $i$, not the mass of its bare atomic nucleus and not a population average.^[ref]

Atomic mass includes electrons and depends on nuclear binding energy; it is not just mass number times a universal nucleon mass. The atomic mass of neutral carbon-12 in its ground state is exactly $12\mathrm{u}$ by definition of the unified atomic mass unit. That exactness establishes the reference scale, rather than making the atomic masses of all isotopes exact integers in the same unit.

The central system of protons and neutrons in an atom. Its mass number $A$ is its baryon number in ordinary nuclear states; $Z$ is the proton number and $N$ the neutron number.

The nucleus contains almost all of an atom's mass and carries positive electric charge $Ze$, where $e$ is the elementary charge. The atomic number $Z$ identifies the chemical element; changing $N$ at fixed $Z$ gives another isotope. The mass number $A=Z+N$ counts nucleons, rather than specifying the exact nuclear mass: nuclear binding energy lowers the mass of a bound system relative to its separated constituents. The simplest nucleus is a single proton; other atomic nuclei can contain many protons and neutrons bound by the nuclear force.

A nuclear quantum state has properties such as spin and a spatial nuclear charge distribution. These are different questions from merely counting nucleons. A small-angle elastic scattering measurement can reveal a nuclear charge radius, while a wider range of momentum transfer tests the distribution's shape. The Hofstadter nuclear electron scattering experiment made this connection between scattering patterns and nuclear structure experimentally concrete.

Why can a star shine for so long, and why do hotter stellar cores favor different nuclear reactions? A useful theory must supply both an adequate energy reservoir and the right stellar energy-generation rate. Nuclear binding energy supplies the reservoir; the thermonuclear reaction rate, stellar composition and stellar structure determine how quickly it is spent. The proton–proton chain and CNO cycle are two routes for hydrogen burning with very different temperature sensitivities.

In his 1967 lecture, Hans Albrecht Bethe contrasted the short Kelvin-Helmholtz cooling time with long ages inferred from terrestrial and stellar evidence.^[ref]

For an original solar-scale estimate, choose mass $M=1.99\times 10^{30}\mathrm{kg}$, radius $R=6.96\times 10^8\mathrm{m}$ and luminosity $L=3.83\times 10^{26}\mathrm{W}$. The characteristic Kelvin-Helmholtz cooling time $G{M}^2/(RL)$ is about $31\mathrm{Myr}$; the precise gravitational reservoir has a structure-dependent coefficient. If a fraction $f=0.10$ of the mass can burn its initial hydrogen mass fraction $X=0.70$, with a chosen rest-energy conversion fraction $\eta =0.007$, the fuel estimate isThis assumes fixed luminosity and neglects neutrino losses. It demonstrates the reservoir advantage, rather than predicting the Sun's age or its full stellar evolution. Having enough fuel still does not identify a reaction: the fuel must release energy at a rate compatible with hydrostatic equilibrium and radiative transfer.

Hans Albrecht Bethe and Charles Louis Critchfield calculated deuterium formation from two protons in 1938, combining quantum tunnelling with a weak conversion of a proton into a neutron. Their calculated nuclear reaction cross-section was a theoretical input to hydrogen burning.^[ref] In 1939, Hans Albrecht Bethe analysed the catalytic carbon–nitrogen route and compared required stellar core temperatures with stellar structure estimates in Table VIII. His reported solar agreement used historical abundance and structure assumptions; it should not be read as a modern demonstration that the CNO cycle dominates the Sun.^[ref]

Hans Albrecht Bethe's 1967 lecture credits Carl Friedrich von Weizsäcker's independent 1938 carbon–nitrogen cycle discovery, distinguishing it from calculation of rates and temperature dependence.^[ref] The 1967 Nobel Prize in Physics recognized Hans Albrecht Bethe's contributions to nuclear reaction theory, especially stellar energy production.^[ref]

Laboratory nuclear reaction cross-sections were inputs, not measurements of an artificial star. The 1939 paper converted thick-target yields to nuclear reaction cross-sections using a range–energy relation and tabulated capture data.^[ref] A concrete apparatus example is the Cockcroft-Walton lithium disintegration experiment, with an accelerated proton beam, lithium target, zinc sulfide screen and beam-blocking control. Its nuclear reaction differs from both stellar burning routes. The 1967 lecture stresses extrapolation below laboratory access and tests for nearby nuclear levels.^[ref]

Binding energy per nucleon divides conventional positive nuclear binding energy by mass number. Typical well-bound nuclei have values of order $8\mathrm{MeV}$, and the broad maximum lies near iron-group nuclei. A mass correction added to free-nucleon masses instead has the opposite sign, $E_{\mathrm{corr}}/A=-B/A$.

Use natural units and a positive nuclear binding energy ${\epsilon }_D=m_n+m_p-m_D$. Chemical equilibrium of $n+p\leftrightarrow D+\gamma$ requires ${\mu }_D={\mu }_n+{\mu }_p$. The nonrelativistic Maxwell--Boltzmann number density givesHere $g_D=3$ and $g_n=g_p=2$ count the spin states. Taking $m_n\simeq m_p\simeq m_N$ and $m_D\simeq 2m_N$, then writing $X_i=n_i/n_B$, $n_B=\eta n_{\gamma }$ and using the photon number density yields the displayed abundance. The coefficient is approximately $8.14$. Omitting the spin-state ratio $3/4$ gives the rougher prefactor $16\zeta (3)/\sqrt{\pi }$. A convention $B=m_D-m_n-m_p<0$ instead writes the exponential as $e^{-B/T}$. The small baryon-to-photon ratio suppresses stable deuterium until the temperature is well below the binding scale, producing the deuterium bottleneck.

The chemical element with atomic number 26. Iron-group nuclear binding energy and silicon burning are central to the termination of ordinary energy-producing stellar fusion.

Isotopes of a chemical element have the same atomic number $Z$, hence the same number of protons, but different neutron numbers $N$ and therefore different mass numbers $A=Z+N$. For example, every neon atomic nucleus contains ten protons: neon-20, neon-21, and neon-22 contain ten, eleven, and twelve neutrons, respectively. They occupy one chemical element box in the periodic table, rather than three boxes.

Three labels answer different questions. The chemical element fixes $Z$; the isotope specifies which neutron number accompanies that $Z$; the ion label specifies the net electric charge. Neutral ${}_{10}^{22}\mathrm{Ne}$ and the ion ${}_{10}^{22}{\mathrm{Ne}}^{+}$ both contain ten protons and twelve neutrons; they contain ten and nine electrons, respectively. Ordinary ionization changes the electron count, leaving the isotope unchanged. Changing the neutron number can change nuclear binding energy and the possibility of radioactive decay. Isotopic abundances can help distinguish material sources, while radioactive decay can additionally supply chronological information.

A mass spectrum makes the distinction experimentally visible: a single chemical element can give separate ion peaks at different mass-to-charge ratios. Its bulk atomic weight is a weighted mean governed by isotopic abundances, rather than the location of one compulsory peak. The historical mass spectrograph experiment and the synthetic neon mass spectrum below illustrate these complementary observations.

Writing positive nuclear binding energy as $B(Z,N)$ and nuclear mass as $M(Z,N)$,The two-neutron nuclear separation energy is $S_{2n}(Z,N)=B(Z,N)-B(Z,N-2)$. In an independent-particle example, let the last occupied energy level have ${ϵ}_{-}=-8u$ and the next empty level ${ϵ}_{+}=-3u$, for a chosen positive energy unit $u$. With an unchanged core and no residual interactions, removal from the filled level costs $8u$, whereas removing the first particle added above it costs $3u$: the drop $5u$ equals the one-particle gap. These are invented model inputs, not measured nuclear values. Nuclear pairing, rearrangement of the nuclear mean field, and correlations complicate that identification in real masses; a finite difference of nuclear binding energies is not automatically a bare one-particle gap.

Nuclear Coulomb energy is the positive electrostatic mass-energy contribution from proton repulsion in a nucleus. A uniform-charge model gives scaling $Z^2/R$, and approximately constant nuclear density gives $R\propto A^{1/3}$. It reduces conventional positive nuclear binding energy.

The residual strong interaction between nucleons. Its long-distance part involves pion exchange and depends on spin and isospin. It is distinct from directly exchanging gluons between color-singlet nucleons.

Why do protons and neutrons bind into an atomic nucleus, even though protons repel electrically? An attractive nuclear force can lower the total energy below that of separated nucleons, producing nuclear binding energy. The deuteron, containing one proton and one neutron, is the simplest such bound state.

In Quantum chromodynamics, nucleons are composite colour singlets built from quarks and gluons. Their residual nuclear force has a longest-range strong contribution from one-pion exchange, with additional interactions at shorter distances.^[ref]

The mass deficit of a bound atomic nucleus relative to its separated protons and neutrons. If their rest masses are $m_p$ and $m_n$, the atomic nucleus has rest mass $M_{\mathrm{nuc}}$, and the positive nuclear binding energy is $B$, thenwith $c$ the speed of light. Releasing energy when the nucleons bind reduces the rest mass of the bound system without changing the exact count $A=Z+N$. A neutral atomic mass also includes $Z$ electron rest masses, reduced by the energy required to separate those bound electrons. Consequently $A-m_{\mathrm{a}}/\mathrm{u}$ is not the definition of the nuclear mass defect: the reference unit and constituent rest masses must also be accounted for.

The study of atomic nuclei, their structure, reactions and interactions. Nuclear forces between nucleons and nuclear binding energies connect it to the low-energy consequences of Quantum chromodynamics.

The energy needed to remove specified nucleons from a bound atomic nucleus in its ground state. It compares neighboring nuclear masses, and so can reveal a nuclear shell closure more clearly than a smoothly varying total nuclear binding energy.

Use $X_i=n_i/n_B$, so $X_D$ counts deuterium nuclei per baryon, not baryons bound into deuterium. For the radiative capture reaction, chemical equilibrium and the zero photon chemical potential give ${\mu }_D={\mu }_n+{\mu }_p$. Dividing the three nonrelativistic Maxwell--Boltzmann number densities removes all chemical potentials:Here $g_D=3$ and $g_n=g_p=2$. The binding convention in this PDF is a negative mass defect, $B_D=m_D-m_n-m_p<0$. The positive nuclear binding energy is ${\epsilon }_D=-B_D$, so the abundance is enhanced by $e^{-B_D/T}$, not by $e^{B_D/T}$.

Since $X_D/(X_n{X}_p)=n_B{n}_D/(n_n{n}_p)$, insert the baryon-to-photon ratio $n_B=\eta n_{\gamma }$ and the photon number density $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$. This givesAt equal nucleon mass $m_N$ in the prefactors, $m_D\simeq 2m_N$, the deuterium equilibrium abundance simplifies toThe dimensionless prefactor is about $8.14$. These expressions describe thermal equilibrium; residual primordial deuterium after nuclear reactions cease must instead be obtained from the reaction kinetics.

The elements have several complementary origins: primordial light nuclei, stellar and explosive fusion products, and nuclei built by neutron capture. Chemical enrichment also requires their ejection, rather than merely their production inside a star. Nucleosynthesis and transport into the interstellar medium must therefore be considered together.

Hydrogen and much of the helium are primordial. During Big Bang nucleosynthesis, a cooling, expanding plasma assembled surviving neutrons and protons into deuterium, helium-3 and especially helium-4, with small abundances of lithium-7 or its beryllium-7 progenitor. Unstable nuclei at mass numbers five and eight, together with the rapidly falling density and short available time, obstructed sustained production of heavier nuclei. The carbon, oxygen and heavier material in later generations consequently require other sites. Cosmic rays fragment heavier interstellar nuclei and contribute importantly to the light elements lithium, beryllium and boron, which ordinary stellar burning tends to destroy.

Burning stages depend on initial stellar mass. Stellar nuclear fusion can release energy when the products have greater nuclear binding energy than the reactants. The resulting stellar thermostat links nuclear burning to hydrostatic equilibrium: when fuel exhaustion lowers heating, contraction raises the central temperature until another fuel can ignite, if the core becomes hot enough. Higher nuclear charges bring larger Coulomb barriers, so later charged-particle burning generally requires progressively higher temperatures. Whether the core reaches them depends on its mass, electron degeneracy pressure, mixing and envelope loss.

An approximately solar-composition $1M_{\odot }$ star first spends most of its life on the main sequence, converting hydrogen into helium mainly through the proton–proton chain. The net conversion is four protons into one helium-4 nucleus, with positrons and neutrinos; the small CNO cycle contribution redistributes CNO catalyst abundances but is not a net source of carbon at this stage. Exhaustion of central hydrogen leaves a helium-rich core. A surrounding hydrogen-burning shell continues to add helium, while the core contracts and the envelope expands into a red giant.

The low-mass helium core becomes supported substantially by electron degeneracy pressure. At roughly $10^8\mathrm{K}$, the Triple-alpha process ignites, first making transient beryllium-8 and then carbon-12 through the Hoyle state. Because degeneracy initially weakens the usual expansion response, this ignition is a helium flash. It heats and lifts the degeneracy of the core, leading to a stable phase of core helium burning. The Triple-alpha process makes carbon; competing carbon-12 alpha capture makes oxygen:After central helium exhaustion, the star develops a carbon-oxygen core, a helium-burning shell and a more external hydrogen-burning shell. On the Asymptotic giant branch, recurrent thermal pulses of an asymptotic-giant-branch star arise from unstable helium-shell burning. Stellar dredge-up can bring processed material toward the surface, with its efficiency depending on mass and composition. The eventual stellar wind removes the envelope; the hot exposed core can illuminate a planetary nebula, and the remnant becomes a carbon-oxygen white dwarf. A usual isolated $1M_{\odot }$ star does not ignite carbon burning. In particular, an element synthesized in its core need not all escape: much of the carbon and oxygen remains locked in the white dwarf.

An approximately $5M_{\odot }$ star burns central hydrogen predominantly through the CNO cycle, usually in a convective core. The CNO nuclei act as catalysts; the slow nitrogen-14 proton-capture step tends to accumulate nitrogen-14 while converting hydrogen into helium. After central fuel exhaustion, a hydrogen-burning shell surrounds the contracting helium core. Unlike the $1M_{\odot }$ case, core helium burning normally begins before strong electron degeneracy, without a classical helium flash. The same Triple-alpha process and carbon-12 alpha capture create a carbon-oxygen core.

The $5M_{\odot }$ star then enters the Asymptotic giant branch, with hydrogen and helium burning in shells. Repeated thermal pulses and stellar dredge-up can enrich its envelope with freshly made carbon and s-process nuclei. Hot-bottom burning can occur when the base of the convective envelope is sufficiently hot: the CNO cycle processes dredged-up carbon into nitrogen, so a massive AGB star need not become carbon-rich despite making carbon below its envelope. Helium-shell reactions on neon-22 can also provide neutrons for the s-process. For an ordinary $5M_{\odot }$ model, strong envelope loss generally precedes carbon ignition, leaving a carbon-oxygen white dwarf. The exact threshold for carbon ignition and the size of these yields depend on metallicity, mass loss and mixing; they are not universal integers fixed solely by initial mass.

A $20M_{\odot }$ star reaches much more advanced stages. It starts with convective-core CNO cycle hydrogen burning, followed by core helium burning. After central helium exhaustion, contraction can ignite carbon burning at temperatures approaching $10^9\mathrm{K}$. Carbon-fusion channels make products including neon, sodium and magnesium. Neon burning then uses photodisintegration of neon-20 to release oxygen-16 and an alpha particle, followed by capture of that alpha particle on another neon-20 nucleus; the combined network releases energy and produces magnesium-24. Oxygen burning makes silicon-group nuclei through channels such asAt still higher temperature, silicon burning rearranges nuclei by photodisintegration and captures of protons, neutrons and alpha particles. It is not principally direct fusion of two silicon nuclei. Networks approach quasi-equilibrium and then nuclear statistical equilibrium, producing an iron-group core whose precise mixture depends on the electron fraction in stellar matter. Outer shells retain different stages of burning, giving an approximately layered structure. Increasing stellar neutrino energy loss makes advanced burning stages much shorter than hydrogen burning.

Near the iron group, nuclear binding energy per nucleon is close to its maximum. Further fusion cannot provide the sustained energy source required to support the core. As the core grows, electron capture reduces electron-pressure support and photodisintegration absorbs energy, assisting gravitational collapse. A successful core-collapse supernova ejects some of the envelope and processed shells, and leaves a neutron star or black hole. The shock also causes explosive burning, changing the pre-existing abundances. A $20M_{\odot }$ initial mass does not uniquely fix the explosion or remnant: envelope loss, rotation, composition and fallback all matter, and some collapses fail to eject much material.

Ejection determines the enrichment of the interstellar medium. Low- and intermediate-mass stars return helium, CNO-processed matter, dredged-up carbon and s-process products through stellar winds and envelope expulsion. Massive-star winds can expose hydrogen- or helium-burning products before collapse. Successful core-collapse supernovae eject substantial oxygen and other alpha-capture products, together with some iron-group matter formed in hydrostatic and explosive burning. Material that falls back or remains in a stellar remnant is withheld from the interstellar medium. Subsequent mixing, cooling and star formation distribute the escaped material into new stars. The distinction between production and ejection is why an internal burning inventory alone is not a stellar yield.

Binary stars provide an additional major iron-group source. In a Type Ia supernova, a carbon-oxygen white dwarf in a binary undergoes thermonuclear disruption after an appropriate accretion or merger history. Explosive carbon and oxygen burning makes intermediate-mass and iron-group nuclei. Much initially synthesized nickel-56 decays through cobalt-56 into iron-56; this radioactive decay also powers the supernova light curve. Such explosions have delay times and yields different from those of massive-star collapse, and help explain why iron enrichment need not track oxygen enrichment exactly.

Neutron capture builds much of the material beyond iron. Charged-particle reactions encounter increasing Coulomb barriers as nuclear charge grows, but a neutral neutron does not encounter that barrier. Neutron capture raises mass number, and subsequent beta-minus decay can turn a neutron into a proton and raise atomic number. The relative capture and decay times select different paths through the nuclear chart.

In the slow neutron-capture process, most unstable intermediates decay before the next capture, so the path stays near stable nuclei. In Asymptotic giant branch stars, important neutron sources areThe carbon-13 source operates in suitable partially mixed layers, while the hotter helium-shell pulses can activate the neon-22 source. Captures on pre-existing seed nuclei produce elements such as strontium and barium and, under suitable conditions, lead. Stellar dredge-up and stellar winds return these products to the interstellar medium. Massive stars also have a weak s-process, notably using neon-22 during helium- and carbon-burning phases; it need not produce the same abundance range as the main AGB contribution.

In the rapid neutron-capture process, very high neutron densities permit many captures before typical beta-minus decays. The path moves to neutron-rich nuclei, which later decay toward stability, producing heavy nuclei including lanthanides and actinides. Neutron-rich ejecta from neutron star mergers provide an established site: a merger can eject dynamical material and disk outflows, and radioactive decay of the newly made nuclei powers a kilonova. Special neutron-rich outflows associated with some massive-star collapses are additional candidate sites. It would be unjustified to assume that every ordinary core-collapse supernova produces the full heavy r-process pattern; the required neutron richness and expansion history are restrictive.

The two neutron-capture processes do not make every isotope beyond iron. Some proton-rich isotopes bypass their main paths. Gamma-process nucleosynthesis, for example, uses photodisintegration of pre-existing heavy seed nuclei in hot explosive material. Other proton-rich reaction paths can matter in suitable environments. Thus “heavier than iron” describes a family of production mechanisms rather than a single further ordinary fusion stage.

Binaries change both stellar evolution and the delivery of elements. Roche-lobe overflow, wind transfer and mergers can strip a donor, rejuvenate a mass-gaining star, alter core and envelope masses, and change the timing and character of an eventual explosion. Binary stripping can remove the hydrogen envelope of a massive core-collapse supernova progenitor. Accretion onto a white dwarf can cause a classical nova, ejecting hydrogen-burning products and particular isotopes while ordinarily leaving the white dwarf intact; this differs from the destructive Type Ia supernova. Compact binary stars can lose orbital energy through gravitational radiation and merge, providing the neutron star merger route to heavy r-process material. Consequently chemical enrichment cannot be inferred entirely from isolated-star tracks, even when a simplified population calculation treats binary components as evolving independently.

At neutrino decoupling, the interaction rate is comparable with the Hubble parameter. Equating the supplied rates givesHere the Planck mass is the unreduced $m_P=G^{-1/2}\simeq 1.22\times 10^{19}\mathrm{GeV}$ in natural units, $G_F\simeq 10^{-5}{\mathrm{GeV}}^{-2}$, and the effective number of relativistic energy degrees of freedom is $g_{\ast }=10.75$. The supplied rate gives an order-of-magnitude neutrino decoupling estimate; a precise neutron-proton conversion temperature uses its own weak-rate coefficients.

The chemical equilibrium condition for $n+{\nu }_e\leftrightarrow p+e^{-}$ is ${\mu }_n+{\mu }_{{\nu }_e}={\mu }_p+{\mu }_e$. With negligible lepton chemical potentials, ${\mu }_n={\mu }_p$. The nonrelativistic Maxwell--Boltzmann number density therefore givesbecause $g_n=g_p=2$ and the small mass difference can be neglected in the prefactor. This is the neutron-proton chemical equilibrium relation.

For the deuterium equilibrium abundance, ${\mu }_D={\mu }_n+{\mu }_p$ because photons have zero chemical potential. The sign convention in the paper is important: $B=m_D-m_n-m_p<0$, while the positive nuclear binding energy is ${\Delta }_D=-B$. Using spin multiplicities $g_D=3$, $g_n=g_p=2$ givesWith $n_B=\eta n_{\gamma }$, $m_n\simeq m_p\simeq m_N$, $m_D\simeq 2m_N$ and photon number density $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$, this becomesHere $X_D=n_D/n_B$ is a number fraction; the fraction of baryons bound in deuterium is $2X_D$. The formula estimates the onset of the deuterium bottleneck opening while depletion of free neutrons and protons remains small. Because $\eta \ll 1$, stable deuterium becomes abundant only far below the positive nuclear binding energy, not simply at $T={\Delta }_D$.

Let $T_f$ denote cosmological weak freeze-out of the neutron-to-proton ratio, $r_f=e^{-Q/T_f}$, and let $t_N$ mark efficient nuclear burning. Free-neutron decay after freeze-out givesAlmost every surviving neutron enters helium-4, whose atomic nucleus contains two neutrons and four baryons. Thus neutron-limited helium synthesis givesA longer neutron half-life leaves more neutrons available, increasing the primordial helium mass fraction. If the longer lifetime reflects weaker weak interactions, the accompanying earlier cosmological weak freeze-out can reinforce that effect.

During radiation domination, $H\propto \sqrt{G{g}_{\ast }}{T}^2$ whereas the weak conversion rate is proportional to $T^5$. Hence $T_f\propto (G{g}_{\ast }{)}^{1/6}$. Increasing the effective number of relativistic energy degrees of freedom raises $T_f$, increasing $r_f$, and shortens the interval for free-neutron decay after freeze-out. Both increase the primordial helium mass fraction while nuclear burning remains efficient. The same reasoning gives the primordial helium response to a varying gravitational constant: larger $G$ over this interval produces more helium, with fixed microscopic rates and the approximate Friedmann equation unchanged. Smaller $G$ has the opposite effect. A particular varying-$G$ model must specify its background expansion and time history; the simple trend must not be extrapolated to changes so large that nuclear burning itself fails.

A Type Ia supernova is a thermonuclear disruption associated with a carbon-oxygen white dwarf in a binary system. Electron degeneracy pressure supports the dense star, and initially depends much less on temperature than an ideal gas pressure does. Once carbon burning heats degenerate material, the immediate pressure response is insufficient to produce the usual expansion-and-cooling thermostat. The strongly temperature-sensitive nuclear reactions can therefore undergo a thermonuclear runaway.

One channel involves accretion toward the Chandrasekhar limit, about $1.4$ solar masses for a typical electron composition. Other proposed channels involve merging white dwarfs or an initial helium-layer detonation that triggers carbon burning below that mass. Thus the Chandrasekhar limit motivates an important model, but it is not an assumption that every Type Ia supernova has exactly the same progenitor mass or ignition history. Burning carbon and oxygen into more tightly bound nuclei releases enough nuclear binding energy to overcome the stellar gravitational binding energy. An ordinary successful event ejects the star rather than leaving the compact core characteristic of a core-collapse supernova.

The ejecta contain iron-group material, including radioactive nickel-56, and intermediate-mass elements from incomplete burning. The radioactively powered Type Ia supernova light curve is powered largely by the nickel-56 to cobalt-56 to iron-56 decay chain: its energetic products heat the ejecta, and radiative diffusion delays the emerging radiation. Near maximum light, ordinary Type Ia supernovae show prominent silicon absorption and lack the hydrogen lines defining Type II events. The spectral classification and the thermonuclear mechanism provide complementary descriptions.

The similar progenitors and radioactive power make Type Ia supernovae useful standard candles, but their peak luminosities are not identical. The relation between light curve width and peak brightness, together with color corrections, allows standardization for distance measurements. Extinction, composition and explosion diversity still introduce systematic uncertainties. A Type Ia event is a white-dwarf thermonuclear explosion with a radioactively powered light curve; its corrected luminosity makes it a valuable distance indicator.

Use natural units $\hslash =c=k_B=1$ and the unreduced Planck mass $M_P=G^{-1/2}$. Neutrino decoupling occurs when the weak interaction rate falls below the Hubble parameter. The weak-decoupling temperature estimate equates the given rate with the radiation-era expansion rate:Here $g_{\ast }$ counts the effective number of relativistic energy degrees of freedom, and the Fermi constant is $G_F\simeq 10^{-5}{\mathrm{GeV}}^{-2}$. The omitted reaction-rate coefficients make this an order-of-magnitude estimate: decoupling occurs at a temperature of order one MeV.

The charge-conserving channel is $n+{\nu }_e\leftrightarrow p+e^{-}$, together with $p+{\bar{\nu }}_e\leftrightarrow n+e^{+}$. The positron sign in the printed neutron-capture channel violates electric charge conservation. Chemical equilibrium requires ${\mu }_n+{\mu }_{{\nu }_e}={\mu }_p+{\mu }_e$. Negligible lepton chemical potentials therefore give ${\mu }_n\simeq {\mu }_p$. Since neutrons and protons are nonrelativistic particles, their Maxwell-Boltzmann distribution givesBoth nucleons have two spin states, and their masses agree closely in the prefactor. Thus neutron-proton chemical equilibrium impliesThe heavier neutron is suppressed by a Boltzmann factor.

For the deuterium equilibrium abundance, retain the paper's convention $B=m_D-m_n-m_p<0$; the positive nuclear binding energy is ${\epsilon }_D=-B\simeq 2.22\mathrm{MeV}$. Chemical equilibrium of $n+p\leftrightarrow D+\gamma$ gives ${\mu }_D={\mu }_n+{\mu }_p$ because the equilibrium photon chemical potential vanishes. Using the nonrelativistic Maxwell--Boltzmann number density for each species and dividing givesHere $g_D=3$ and $g_n=g_p=2$. With the baryon-to-photon ratio $\eta =n_B/n_{\gamma }$, the photon number density $n_{\gamma }=2\zeta (3)T^3/{\pi }^2$, and $m_D\simeq 2m_N$, $m_n\simeq m_p\simeq m_N$, this becomesIn this convention $X_D=n_D/n_B$ counts nuclei, so its contribution to the fraction of baryons in deuterium is $2X_D$.

The deuterium bottleneck explains why helium-4 formation waits until $T$ is far below ${\epsilon }_D$. At higher temperature, there are vastly more photons than baryons; even the dilute energetic tail can destroy newly formed deuterium. A crude estimate balances ${\eta }^{-1}{e}^{-{\epsilon }_D/T}$ against one, giving $T\sim {\epsilon }_D/\log ({\eta }^{-1})\sim 0.1\mathrm{MeV}$ for $\eta$ of order $10^{-9}$. The power-law factors in the deuterium equilibrium abundance and the nuclear reaction rates refine this estimate. Once stable deuterium accumulates, further reactions can efficiently place nearly all surviving neutrons into helium-4.

Write $r_N=n_n/n_p$ at the onset of nuclear burning. Each helium-4 nucleus requires two neutrons and contains four baryons. Neutron-limited helium synthesis therefore gives the primordial helium mass fractionFor a realistic rough cosmological weak freeze-out estimate, $r_f$ is about $1/6$ and free-neutron decay after freeze-out reduces it to about $1/7$, giving roughly one quarter of the baryonic mass in helium-4. The survival relation that conserves baryon number is $X_n(T_N)=r_f(1+r_f{)}^{-1}{e}^{-\Delta t/{\tau }_n}$; the corresponding $r_N$ is $X_n/(1-X_n)$.

The rough neutrino decoupling rate by itself does not determine that one-quarter value. Substituting $T_d=1.8\mathrm{MeV}$ literally into $e^{-Q/T_d}$ would give $r\simeq 0.49$ and $Y_4\simeq 0.65$ before neutron decay. Neutron-proton chemical equilibrium persists to a lower temperature, approximately $0.7$--$0.8\mathrm{MeV}$ once the actual conversion rates are included. The rate scaling fixes the MeV scale; accurate abundances need the conversion coefficients and the delay through the deuterium bottleneck. This sequence is described in David Tong's discussion of nucleosynthesis.

Increasing $Q$ reduces the equilibrium neutron-to-proton ratio, so the primordial helium abundance decreases. Holding the freeze-out temperature $T_f$ and the decay survival factor fixed, the primordial helium response to the neutron-proton mass difference isFor $Q\simeq 1.29\mathrm{MeV}$ and $T_f=0.7$--$0.8\mathrm{MeV}$ this is about $0.85$--$0.87$, a reduction of roughly fifteen percent. A larger $Q$ also changes neutron decay and detailed weak rates; their quantitative effects require more microscopic information. The sign from the Boltzmann factor is already clear.

Write $n_{56}$ and $n_{54}$ for the number densities of nickel-56 and iron-54. Both nuclei contain 28 neutrons, while their proton numbers differ by two. Dividing the positive-exponential nuclear statistical equilibrium formulas printed in the PDF therefore gives the nickel–iron nuclear equilibrium ratioThe neutron-density factors cancel. $K$ contains the isotope-dependent statistical and translational mass factors suppressed in the supplied proportionality; it is constant in this simplified calculation. The TeX conversion incorrectly reverses the exponential sign, so the PDF sign is essential here.

In the specified restricted composition, equality of total proton and neutron numbers meansConsequentlyDifferentiation, with the suppressed prefactors held constant, givesUsing the paper's supplied signed energy difference producesWith that negative value, $F$ tends to zero at both temperature endpoints, increases below this temperature and decreases above it. At the stationary point $d^2\log F/d{T}^2=-3/T^2<0$, so it is indeed a maximum for the printed toy data.

There is a genuine physical-data qualification: the stated negative energy difference is inconsistent with interpreting $Q$ as the conventional positive total nuclear binding energy of these actual isotopes. Using atomic masses, whose electron masses cancel in this reaction,The masses used are $1.00782503223\mathrm{u}$, $53.93960899\mathrm{u}$ and $55.94212855\mathrm{u}$, from NIST hydrogen masses, NIST iron masses and NIST nickel masses. Small electronic binding corrections do not change the sign. With a positive $\Delta B$, the simplified $F(T)$ above is strictly decreasing for $T>0$ and has no finite maximum. Thus the numerical maximum follows from the supplied negative parameter, not from consistent physical binding energies; changing the TeX sign to recover a maximum would not repair the PDF's data.

Nuclear statistical equilibrium is appropriate when the relevant forward and inverse strong or electromagnetic reactions are fast compared with changes of temperature, density and composition. It is useful in sufficiently hot late silicon-burning material and in the hottest explosive burning regions of massive stars. Some burning zones reach only quasi-equilibrium, and cooling or expansion eventually freezes the abundances out; an equilibrium expression cannot by itself predict the full ashes when these timescale conditions fail. Temperature, density, conserved baryon number and the electron fraction in stellar matter must all be specified. Negligible free-particle contributions in bulk bookkeeping are not the same as exactly zero free densities in the equilibrium formula.

Strong and electromagnetic reactions preserve total charge and baryon number, so they conserve the proton-to-neutron ratio. Weak reactions need not: electron capture and beta-plus decay convert protons into neutrons and lower it. Such conversions occur in hydrogen-burning networks and in the nitrogen-to-neon chain during helium burning. At high density in advanced stellar cores and core collapse, electron capture becomes particularly important; neutrino emission also affects energy and lepton loss. Beta-minus decay changes the ratio in the opposite direction and can compete during later relaxation. Nuclear equilibrium for strong reactions does not automatically imply weak equilibrium or a fixed proton-to-neutron ratio throughout all stellar evolution.

Stellar nucleosynthesis and the subsequent return of matter explain most elements heavier than the primordial hydrogen and helium. Big Bang nucleosynthesis supplied predominantly those two elements, with small light-isotope abundances; stars do not make every element from an initially metal-free mixture. Different stellar masses reach different central temperatures and leave different remnants. The mass comparisons below describe representative isolated, near-solar-composition stars; mass loss and binary interaction can alter the final core. A nucleus contributes to later generations only if it is eventually expelled into the interstellar medium, rather than remaining locked in a remnant.

A $1M_{\odot }$ star spends its main sequence life converting hydrogen to helium, predominantly through the proton–proton chain. After core hydrogen burning ends, a hydrogen-burning shell supplies an expanding red giant envelope and builds an electron-degenerate helium core. First dredge-up exposes material whose CNO cycle has redistributed the original carbon and nitrogen abundances. The helium flash starts core helium burning: the Triple-alpha process makes carbon, and carbon-12 alpha capture makes oxygen. Once core helium is exhausted, the Asymptotic giant branch has a carbon-oxygen core with helium-burning shell and hydrogen-burning shell. Shell instability causes AGB thermal pulses; subsequent third dredge-up, where efficient enough, can bring carbon and neutron-capture products into the envelope. Its efficiency at such low mass depends on composition and mixing. The star does not ordinarily ignite core carbon burning. Stellar winds remove the envelope, a hot remnant may illuminate a planetary nebula, and a carbon-oxygen white dwarf retains much of the core's synthesized matter.

A $5M_{\odot }$ star generally uses the more temperature-sensitive CNO cycle for main-sequence hydrogen burning, with a substantial convective core. Its hotter helium core ignites core helium burning without the low-mass degeneracy-driven helium flash, producing carbon and oxygen. After core helium exhaustion it develops double-shell Asymptotic giant branch burning and a degenerate carbon-oxygen core. Second dredge-up and repeated third dredge-up alter the envelope composition. In sufficiently hot models, hot-bottom burning at the base of the convective envelope converts dredged-up carbon into nitrogen through the CNO cycle; consequently a carbon-producing shell need not produce a carbon-rich final envelope. Hot helium-shell pulses can activate the ${}^{22}\mathrm{Ne}(\alpha ,n){}^{25}\mathrm{Mg}$ neutron source. Strong stellar winds expel these processed layers. A conventional isolated five-solar-mass model ends as a carbon-oxygen white dwarf, below the mass range that normally develops sustained core carbon burning; mass loss and mixing govern the exact boundaries between outcomes.

A $32M_{\odot }$ star has core CNO cycle hydrogen burning, followed by core helium burning. The core then contracts through carbon burning, neon burning, oxygen burning and silicon burning, with active shells outside the successive exhausted cores. Carbon burning produces chiefly nuclei such as neon and magnesium. Neon burning combines photodisintegration and alpha capture; oxygen burning makes silicon-group nuclei. Silicon burning is a network of captures and photodisintegration approaching nuclear statistical equilibrium, rather than simply fusion of pairs of silicon nuclei. It produces an iron-group mixture governed by temperature, density and weak reactions. The maximum in nuclear binding energy per nucleon near the iron group prevents continued exothermic fusion from supplying ordinary hydrostatic support. Stellar neutrino energy loss accelerates the advanced stages; electron capture and photodisintegration help remove support during iron-core collapse.

The massive star can return products both before and during collapse. Radiation-driven stellar winds may expose and expel layers enriched in helium, nitrogen, carbon and oxygen. A successful core-collapse supernova ejects additional hydrostatic products and creates further nuclei by shock-driven explosive burning, including alpha-rich and iron-group products. The nickel-56 decay chain converts radioactive nickel into iron through cobalt. Initial mass alone does not fix the explosion or remnant: substantial fallback or a failed supernova can leave a black hole and retain much of the core material. Thus the possible core-collapse supernova contribution is not a guarantee that a thirty-two-solar-mass star ejects every element it made.

For nuclei well beyond the iron group, neutron capture is a principal assembly mechanism, avoiding the large Coulomb barrier for charged-particle fusion. Beta-minus decay converts captured neutrons into protons and raises the atomic number. In the s-process, captures are slower than relevant beta decays, so the sequence usually follows nuclei near stability. The main site is the Asymptotic giant branch: ${}^{13}\mathrm{C}(\alpha ,n){}^{16}\mathrm{O}$ is an important neutron source in low-mass intershell material, while ${}^{22}\mathrm{Ne}(\alpha ,n){}^{25}\mathrm{Mg}$ becomes important in hotter pulses and in massive-star helium/carbon burning. Third dredge-up and stellar winds make these products available to the interstellar medium. Barium and lead illustrate heavy products; the detailed yield depends on seed abundance, neutron exposure and branching at unstable nuclei.

In the r-process, neutron captures outrun beta decays, building very neutron-rich nuclei before they decay toward stable heavy isotopes. This needs much larger neutron densities than the s-process and can produce very heavy nuclei, including uranium. Neutron-rich ejecta from neutron star mergers provide an established site: observed kilonova emission and spectroscopy support heavy-element synthesis in such ejecta. Rare magnetically driven stellar-collapse outflows are additional candidate sites. An ordinary core-collapse supernova cannot automatically be assumed to supply a complete heavy r-process pattern.

Not every heavy isotope belongs to the two capture sequences. Gamma-process nucleosynthesis in hot supernova material uses photodisintegration on pre-existing heavy seed nuclei through neutron, proton and alpha-emission channels, helping produce proton-rich isotopes bypassed by the main capture paths. This is distinct from simply using charged-particle fusion to build every element beyond the iron group.

Finally, binary stars change both production and delivery. Roche-lobe overflow and wind mass transfer in a binary star strip envelopes, change subsequent core growth, and expose processed layers. Mass transfer from an Asymptotic giant branch donor can enrich a surviving companion in carbon and s-process products, as in a barium star; this redistributes existing yields rather than creating them in the recipient. Accretion onto a white dwarf can trigger a classical nova with explosive hydrogen-burning products. Thermonuclear disruption of a carbon-oxygen white dwarf in a Type Ia supernova, through suitable accretion or the Double-degenerate Type Ia supernova scenario, supplies a major delayed iron-group contribution, including iron; it is not the same event as collapse of a massive iron core. Compact binaries containing two neutron stars can later merge and eject r-process material. Thus binaries affect stellar mass histories, enrich companions, open thermonuclear explosion channels, and create heavy-element-producing compact-remnant mergers.

The key distinction is between making nuclei and ejecting them: ordinary low/intermediate-mass stars mainly return processed envelopes, massive stars add advanced and explosive burning products when those layers escape, and neutron-capture environments supply much of the material beyond iron.