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Nucleosynthesis is the production of atomic nuclei through nuclear reactions. It includes Big Bang nucleosynthesis, stellar nucleosynthesis and production in explosive or compact-remnant environments. Making nuclei and delivering them to the interstellar medium are separate steps in chemical enrichment.

The energy-conserving self-similar blast wave requires a stage in which the original stellar radius, ejecta mass, ambient pressure and cooling have ceased to supply important scales. In particular, the swept-up mass must greatly exceed the ejecta mass; $R\gg R_{\ast }$ alone need not ensure this in a very dilute interstellar medium. For fixed explosion energy $E$, uniform ambient mass density ${\rho }_0$ and elapsed time $t$, dimensional analysis gives $R=C{E}^a{\rho }_0^b{t}^d$. Matching mass, length and time gives $a=1/5$, $b=-1/5$, $d=2/5$. Equivalently, a self-similar strong shock wave has $E\sim {\rho }_0{R}^3{\dot{R}}^2$, whose constant-energy solution has the same exponents:The dimensionless coefficient depends on $\gamma$ and the post-shock profiles; dimensional reasoning cannot fix it.

For completeness, obtain the Strong-shock Rankine-Hugoniot conditions in the shock frame. Let $v_1=V-u$ denote downstream speed relative to the outward-moving front. With negligible upstream pressure and enthalpy, conservation of mass, momentum and energy givesPut $q={\rho }_1/{\rho }_0$. Substituting $v_1=V/q$ into the last two equations and selecting the compressive, nontrivial branch yields $q=(\gamma +1)/(\gamma -1)$. HenceHere the symbol $u$, written as $u=2V/(\gamma +1)$ in what follows, is the laboratory-frame radial velocity; it is not the relative speed $v_1$.

Use the thin-shell approximation for a spherical blast wave. Set $W=4\pi R^3/3$ and $M_s={\rho }_{0}W$. Equating the swept-up mass to $4\pi R^{2}d{\rho }_1$ gives $d/R=(\gamma -1)/[3(\gamma +1)]$. The growing shell's momentum is $M_{s}u$. Since newly swept gas has zero laboratory momentum, the net force is the cavity pressure acting on the inner shell, to thin-shell accuracy:The pressure at the shock is already accounted for by the jump conditions and mass loading; subtracting $p_1$ as an additional external pressure would count the shock reaction twice. With $u=2V/(\gamma +1)$, the last equation becomesTherefore, for constant $\alpha$,

The shell volume is $M_s/{\rho }_1=W/q$. Its internal energy and kinetic energy areTreat the cavity volume as $W$ at the same thin-shell level, so its internal energy is $E_{c,\mathrm{int}}=\alpha p_{1}W/(\gamma -1)$. Their sum is proportional to $R^3{V}^2$ for constant $\alpha$ and $\gamma$. Constant total energy thus requires $3+6(\alpha -1)=0$, giving $\alpha =1/2$. ConsequentlyUsing $V=2R/(5t)$, this becomesFor $\gamma =5/3$, $d/R=1/12$ and the approximate energy fractions are $E_{s,\mathrm{kin}}/E=1/4$, $E_{s,\mathrm{int}}/E=1/4$, $E_{c,\mathrm{int}}/E=1/2$.

This is an approximate normalization, not an exact treatment of finite shell thickness. If instead one subtracts the shell volume and uses $W_c=W(1-1/q)$ for the cavity, while retaining the other shell assumptions, the energy coefficient becomes $2(2\gamma -1)/[(\gamma -1)(\gamma +1{)}^2]$ and ${\xi }_0^5=75(\gamma -1)(\gamma +1{)}^2/[32\pi (2\gamma -1)]$. The difference is an order-$d/R$ geometric correction of this crude shell model; a consistent exact coefficient requires the full self-similar profiles.

In the idealized mass budget all swept gas is assigned to the shell, so the cavity inertia is neglected. The hot cavity's residual mass density is not specified, so its kinetic energy cannot be assigned a unique numerical fraction from the stated data. If its low mass is $M_c$ and its velocity is of order $u$, then $E_{c,\mathrm{kin}}\sim M_c{u}^2/2$ andFor an illustrative uniform-density cavity with homologous velocity $v(r)=ur/R$, direct radial integration gives $E_{c,\mathrm{kin}}=3M_c{u}^2/10$, so the preceding fraction has an extra factor $3/5$. The cavity can carry substantial internal energy despite negligible kinetic energy because it is hot. Neglect of its inertia is justified in the swept-mass-dominated stage, not merely by its small initial radius.

The calculation describes the adiabatic expansion stage of a supernova remnant, giving an estimate of its age or explosion energy from its radius, expansion speed and ambient mass density. For example $t\simeq 2R/(5V)$, and $E\propto {\rho }_0{R}^3{V}^2$. Such blasts inject energy and momentum into the interstellar medium and redistribute stellar ejecta. The early ejecta-dominated stage, radiatively cooled late shell, nonuniform ambient medium, finite ambient sound speed, magnetic fields, nonspherical geometry and energy transfer to cosmic rays violate different assumptions. Even in the adiabatic spherical stage, the real profiles are not a uniform thin shell and a uniform-pressure cavity, so ${\xi }_0$ here is an estimate.

Use the instantaneous recycling approximation and treat the interstellar medium as well mixed. If star formation initially consumes mass $dM$, returning a fraction $\beta$ leaves locked stellar mass $ds=(1-\beta )dM$. The original metal mass removed from the interstellar medium is $Z_{i}dM$. Of this original material, $\beta f{Z}_{i}dM$ returns and remains in the galaxy. In addition, the stellar yield produces fresh metal mass $y_{i}ds$, of which the retained part is $f{y}_{i}ds$. Therefore metal accounting givesThis is the retained-ejecta chemical evolution equation. It assumes that the same retention fraction applies to original and newly synthesized metals. Pristine galactic gas inflow can change $g$ without contributing to $d(g{Z}_i)$; enriched galactic gas inflow would require an additional metal source. Here $0\le \beta <1$, and $0\le f\le 1$.

A star inherits the gas-phase metallicity at its birth. Suppose enrichment is monotone and a fixed initial mass function, with an appropriate surviving tracer selection, produces $\kappa$ observable stars per unit locked mass. For two birth abundances $Z_a<Z_b$, the relative count isThus $s(Z_i)$ is proportional to the cumulative metallicity distribution function; its derivative gives the differential metallicity distribution function. The normalized distribution after final locked mass $s_{\infty }$ is $s_{\infty }^{-1}ds/d{Z}_i$. For logarithmic abundances, the logarithmic metallicity distribution instead has $dN/d{\log }_{10}{Z}_i=(\ln 10)Z_{i}dN/d{Z}_i$.

The fixed initial mass function and survival selection matter: a mass distribution is not automatically the number distribution of every presently observable star. If enrichment is nonmonotone, add the contributions $\kappa |ds/d{Z}_i|$ from all birth-time branches attaining that abundance, rather than using one inverse.

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.

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.

Moving radially outward from the quasar, the standard active-galactic-nucleus wind bubble contains: the freely expanding fast wind; a reverse shock that thermalizes it; a hot shocked-wind bubble; a contact discontinuity; a dense swept-up interstellar shell behind a forward shock; and finally the undisturbed interstellar medium.

Let $M_s(R)$ be the swept-up shell mass. If inverse-Compton and atomic radiative cooling remove the shocked wind's thermal energy faster than the bubble expands, the reverse shock is momentum driven andIf cooling is slow, the shocked wind remains hot and the bubble is energy driven:supplemented by the bubble energy equationThe hot bubble stores wind energy and performs work for much longer than the direct photon momentum-crossing time. Equating wind power with shell kinetic power gives the characteristic momentum boostwhen the shell is much slower than the nuclear wind.

Let $S=s/s_{\infty }$, let the interstellar medium have mass $g=s_{\infty }S(1-S)$, and retain fraction $f=S$ of returned material. For $0\le \beta <1$ put $q=(2-\beta )/(1-\beta )$. The integrating factor for the metal mass is $S^{1/(1-\beta )}/(1-S)$. A finite initial gas-phase metallicity removes the homogeneous solution and gives the displayed expression, equivalently $Z=y{\sum }_{n\ge 1}{S}^n/(q+n)$ for $0\le S<1$. The increasing initial gas mass requires pristine galactic gas inflow: gas accounting gives $d{M}_{\mathrm{in}}/ds=q(1-S)$. A fixed initial mass function makes the birth metallicity distribution function proportional to $ds/dZ$ when enrichment is monotone.

Under the instantaneous recycling approximation, forming mass $dM$ locks $ds=(1-\beta )dM$ into stars. If a fraction $f$ of returned material remains in the interstellar medium, the retained original metal mass is $\beta fZdM$, and retained newly made metal mass is $fyds$, where $y$ is the stellar yield per locked mass. Subtracting the metal mass $ZdM$ initially consumed gives the displayed equation. Pristine galactic gas inflow adds gas but no metals; enriched inflow adds its metal mass separately. The retained fraction applies to both old and newly made metals in this idealization.