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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.

For a slowly evolving spherical star, use a Lagrangian enclosed mass coordinate $m$. Let $u$ be specific thermal internal energy and $D/Dt$ follow a mass element. The stellar energy balance equation isThe outward luminosity is energy transported through the mass shell, ${ϵ}_{\mathrm{nuc}}$ is nuclear rest-mass energy released per unit mass and time, and ${ϵ}_{\nu }$ is escaping neutrino power per unit mass. Define the gravothermal stellar energy generation by ${ϵ}_{\mathrm{grav}}=-Du/Dt-PD(1/\rho )/Dt$. At fixed composition it is $-TDS/Dt$ by the first law of thermodynamics. During composition changes the full internal-energy derivative must use the appropriate equation of state, with nuclear rest-mass changes counted only once. If a tabulated nuclear rate already subtracts reaction-neutrino energy, that loss must not be subtracted a second time in ${ϵ}_{\nu }$.

In a nearly steady main sequence star, stellar nuclear fusion supplies most of the luminosity. Before stable hydrogen ignition, Kelvin-Helmholtz contraction releases gravitational energy. In a gas-supported hydrostatic star the stellar virial theorem gives $U\simeq -\Omega /2$, so about half the change in gravitational binding energy heats the star and half is available for radiation. The associated timescale is $G{M}^2/(RL)$. After nuclear burning ends, a white dwarf can shine by losing stored thermal energy; crystallization releases latent heat, and composition separation can add gravitational energy. These are heat and gravitational reservoirs rather than sustained hydrogen fusion.

Stellar nuclear fusion is possible because lighter nuclei can combine into products with larger nuclear binding energy. The energy per reaction is $Q=(m_{\mathrm{initial}}-m_{\mathrm{final}})c^2$, including the relevant particles consistently. Although thermal energies are below the Coulomb barrier, quantum tunnelling permits fusion. The thermonuclear reaction rate averages a cross-section over the thermal relative-speed distribution:For nonresonant charged-particle reactions the energy weighting contains $\exp [-E/(kT)-\sqrt{E_G/E}]$; the compromise between the thermal tail and penetration produces the Gamow peak. A resonance can greatly enhance the rate. Density, abundances and temperature therefore all affect the stellar energy-generation rate, and power laws such as $T^4$ or $T^{16}$ are local approximations.

For the proton–proton chain, first produce deuterium and helium-3:The initial reaction is slow because it converts a proton to a neutron through the weak interaction. It is the bottleneck that allows long hydrogen-burning lifetimes. The alternative pep reaction $p+e^{-}+p\to {}^2\mathrm{H}+{\nu }_e$ also feeds the same chain. There are three principal proton-proton chain branches. In pp I, two helium-3 nuclei terminate the chain:In pp II, helium-3 first captures an existing helium-4 nucleus, then electron capture and proton capture finish the branch:In pp III, the beryllium-7 instead captures a proton:Relative branch weights change with temperature and composition. All branches convert four protons into a net helium-4 nucleus, with two weak conversions and two neutrinos, allowing for the electron consumed in pp II. After including positron annihilation, the atomic-mass energy budget is approximately $26.7\mathrm{MeV}$ per net helium-4 nucleus, but the deposited heat is smaller by the branch-dependent escaping-neutrino energy. Near ordinary low-mass main-sequence conditions, ${ϵ}_{pp}\propto \rho X_H^2{T}^4$ is a useful local approximation.

The CNO cycle provides an alternative catalyzed hydrogen burning route in hotter cores. The CNO-I cycle, also called the CN cycle, consists ofThe carbon seed is regenerated: the net reaction again consumes four protons and produces one helium-4 nucleus, two positrons and two neutrinos. The slow ${}^{14}\mathrm{N}(p,\gamma ){}^{15}\mathrm{O}$ reaction governs the ordinary cycle and causes nitrogen-14 to accumulate. The CNO-II cycle branches through ${}^{15}\mathrm{N}(p,\gamma ){}^{16}\mathrm{O}$, ${}^{16}\mathrm{O}(p,\gamma ){}^{17}\mathrm{F}$, ${}^{17}\mathrm{F}\to {}^{17}\mathrm{O}+e^{+}+{\nu }_e$, and ${}^{17}\mathrm{O}(p,\alpha ){}^{14}\mathrm{N}$, after which the main cycle continues. Thus CNO nuclei are catalytic in the closed cycles, although their relative abundances change. A local rate is ${ϵ}_{\mathrm{CNO}}\propto \rho X_H{X}_{\mathrm{CNO}}{T}^{\eta }$ with a much steeper exponent than the PP chain; the preceding homology problem specifies $\eta =16$. The concentration of this heating near the centre favors a convective core in hotter, more massive hydrogen-burning stars.

Once hydrogen is exhausted in a core, contraction can raise its temperature enough for core helium burning. The Triple-alpha process overcomes the absence of stable mass-5 and mass-8 nuclei by maintaining a tiny transient beryllium-8 population:The Hoyle state, an excited carbon-12 resonance near the three-alpha threshold, makes the second capture efficient enough despite beryllium-8's very short lifetime. The net conversion is $3{}^4\mathrm{He}\to {}^{12}\mathrm{C}$, releasing about $7.27\mathrm{MeV}$. Because three alpha particles are required, the specific rate scales as ${\rho }^2{Y}^3$; a schematic resonant rate isIts local temperature exponent is $-3+44/T_8$, of order forty near $T_8=1$. A degenerate core cannot expand promptly to regulate this increase, giving a helium flash; in a nondegenerate core the stellar thermostat permits stable helium burning. The competing ${}^{12}\mathrm{C}(\alpha ,\gamma ){}^{16}\mathrm{O}$ reaction determines much of the eventual carbon-oxygen mixture.

If the core becomes hot enough, carbon burning follows, for example through ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}\to {}^{20}\mathrm{Ne}+\alpha$ and ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}\to {}^{23}\mathrm{Na}+p$. Neon burning begins with photodisintegration, ${}^{20}\mathrm{Ne}+\gamma \to {}^{16}\mathrm{O}+\alpha$, followed by alpha capture such as ${}^{20}\mathrm{Ne}+\alpha \to {}^{24}\mathrm{Mg}+\gamma$. The first step consumes heat, but the combined rearrangement can release net energy. Oxygen burning includes ${}^{16}\mathrm{O}+{}^{16}\mathrm{O}\to {}^{28}\mathrm{Si}+\alpha$ and channels producing phosphorus and other nearby nuclei. At still higher temperatures, silicon burning proceeds through photodisintegration and particle captures in a reaction network approaching quasi-statistical equilibrium, producing iron-group nuclei. It is not simply direct silicon-plus-silicon fusion; the products depend on the electron fraction and weak-interaction timescale.

The increasing nuclear binding energy per nucleon supplies progressively less energy as products approach the iron group. Further fusion past that region cannot provide sustained net heat to support a core. Massive stars can therefore end with an unstable iron-group core; lower-mass stars do not attain all these burning stages and instead leave remnants such as carbon-oxygen white dwarfs. Heavy-element neutron captures can synthesize nuclei beyond the iron group without being the principal hydrostatic power source.

Finally, stellar neutrino energy loss competes with every heat source. Reaction neutrinos accompany the proton–proton and CNO chains. Hot or dense material also emits neutrino pairs through $e^{-}+e^{+}\to \nu +\bar{\nu }$, plasmon decay, $e^{-}+\gamma \to e^{-}+\nu +\bar{\nu }$, and electron-ion bremsstrahlung. These neutrinos usually escape, unlike the photons whose transport is diffusive; neutrino trapping requires much more extreme collapse conditions. Late burning has smaller fuel-energy reservoirs and strong neutrino cooling, so its duration is much shorter than core hydrogen burning. The surface luminosity is the escaping photon power, $L=4\pi R^2\sigma T_{\mathrm{eff}}^4$, not the sum of photons and unobserved neutrino power. Radiative diffusion in a star, convection and electron conduction redistribute energy, while the energy-balance equation distinguishes true sources, losses and storage. Accretion or mass loss adds boundary energy and mechanical work when present, rather than changing the nuclear reaction budget.

Letbe the scalar moment of inertia and total kinetic energy. Sincesubstitution of the Euler momentum equation reduces the stress contribution, by the divergence theorem and isotropic pressure $\mathbf{P}=P1$, toFor self-gravity, $\int \rho \mathbf{r}\cdot \mathbf{F}dV=\Omega$, where $\Omega <0$ is the gravitational potential energy. Thus the stellar virial theorem is

Apply hydrostatic equilibrium to the isothermal core, taking its boundary pressure to be $P_c$. Its ideal-gas pressure integral scales as $\int P,dV\propto M_c{T}_c$, its volume as $R_c^3$, and its gravitational energy as $-G{M}_c^2/R_c$. The virial theorem therefore givesor, after absorbing fixed dimensional and structural factors into positive constants,Hydrogen-burning reactions are extremely temperature-sensitive, so expansion cools and suppresses burning while contraction heats and enhances it. This stellar thermostat keeps the shell and adjoining isothermal core near an approximately fixed $T_c$.

At fixed $M_c,T_c$, differentiating $P_c(R_c)$ gives

For a homologous envelope, hydrostatic balance gives $P_c\propto G{M}^2/R^4$ and the ideal-gas temperature scale gives $T_c\propto GM/R$. Eliminating $R$ yieldsup to composition and gravitational constants common to the sequence. A matching core exists only if this required pressure does not exceed $P_{c,\max }\propto T_c^4/M_c^2$. ThereforeThis maximum fractional isothermal-core mass is the mechanism behind the Schönberg-Chandrasekhar limit.