Stellar Nucleosynthesis

Stars are self-regulating fusion reactors held together by gravity. A main-sequence star sits in hydrostatic equilibrium: the outward pressure of a hot plasma balances its own weight, and the temperature at the center is whatever fusion rate replaces the energy radiated from the surface. Because the thermonuclear rate depends so steeply on temperature, this balance is stable — a small contraction raises $T$ , raises the rate, and restores the pressure. The sequence of nuclei a star builds is set by which Coulomb barriers its central temperature can overcome, and the reactions run in a definite order from hydrogen up to the iron peak, past which fusion no longer releases energy.¹

Hydrogen burning: the proton-proton chain

The net result of hydrogen burning is $4 p \to^{4} He + 2 e^{+} + 2 ν_{e}$ , releasing $Q = 26.73 MeV$ once the positrons annihilate. The barrier problem is severe: the first step must fuse two protons, and there is no bound diproton, so the reaction proceeds only if one proton converts to a neutron by the weak interaction during the brief collision,

p + p \to^{2} H + e^{+} + ν_{e}, Q = 0.42 MeV .

This is a weak process happening inside a Coulomb-suppressed collision, so its rate is minute — the mean time for a given proton in the solar core to undergo it is billions of years. That slowness is why the Sun burns for ten billion years rather than exploding. The deuteron then captures a proton and two helium-3 nuclei combine, in the branch called pp-I:

^{2} H + p \to^{3} He + γ (5.49 MeV),^{3} He +^{3} He \to^{4} He + 2 p (12.86 MeV) .

The two neutrinos of pp-I carry away about $0.6 MeV$ , leaving roughly $26.2 MeV$ to heat the star. When helium has accumulated, the $^{3} He$ can instead capture a $^{4} He$ , opening the pp-II and pp-III branches through $^{7} Be$ , which either captures an electron to $^{7} Li$ or a proton to $^{8} B$ . The $^{8} B$ positron decay produces the highest-energy solar neutrinos, up to $15 MeV$ , and although this branch is rare it dominates the neutrino signal that terrestrial detectors can see.¹

The proton-proton chain fuses four protons into helium-4; after the slow weak-interaction first step it splits into three branches, pp-I completing through two helium-3 nuclei and pp-II/pp-III proceeding through beryllium-7.

The CNO cycle

In stars more massive than about $1.3$ solar masses the central temperature exceeds $1.7 \times 1 0^{7} K$ , and hydrogen burns faster through a catalytic cycle that uses pre-existing carbon, nitrogen, and oxygen. The CNO cycle captures four protons onto a carbon-12 seed and returns the carbon at the end, ejecting one helium-4:

^{12} C (p, γ)^{13} N (β^{+} ν)^{13} C (p, γ)^{14} N (p, γ)^{15} O (β^{+} ν)^{15} N (p, α)^{12} C .

The net reaction and energy release are identical to the pp chain, $4p \to {}^{4}\mathrm{He}

2e^+ + 2\nu_e $w i t h$ Q = 26.73\ \mathrm{MeV} $, b u tt h e t e m p er a t u r e d e p e n d e n ce i s f a r s t ee p er b ec a u se t h er a t e - l imi t in g p r o t o n c a pt u r es m u s tp e n e t r a t e t h e$ Z = 6 $-$ 8 $C o u l o mbba r r i er s . T h es l o w es t s t e p i s$ {}^{14}\mathrm{N}(p,\gamma){}^{15}\mathrm{O} $, so t h ecy c l e d r i v es m os t o f t h ec a t a l y s t in t o$ {}^{14}\mathrm{N} $, a s i g na t u r eo f C N O p r ocess in g . T h e pp c hain sc a l esr o ug h l y a s$ T^4 $n e a r so l a r co n d i t i o n s an d t h e C N O cy c l e a s$ T^{16} $-$ T^{20} $, so t h e tw o m ec hani s m scr oss a t ab o u t$ 1.7\times10^7\ \mathrm{K} $: t h e S u n r u n so n pp (C N O co n t r ib u t es ab o u t$ 1%$), while more massive, hotter stars run on CNO.

The CNO cycle is catalytic, threading four proton captures and two positron decays around a loop that consumes four protons, emits one helium-4, and regenerates the carbon-12 seed; the nitrogen-14 proton capture is the bottleneck.

Helium burning and the triple-alpha bottleneck

When core hydrogen is exhausted the star contracts and heats until helium can fuse near $1 0^{8} K$ . Building carbon from helium is blocked by two gaps: there is no stable nucleus at mass $5$ or at mass $8$ . Two alphas make $^{8} Be$ , which is unbound by $92 keV$ and flies apart in $1 0^{- 16} s$ . The path to carbon exists only because of two resonances stacked in coincidence. A tiny equilibrium population of $^{8} Be$ survives long enough for a third alpha to be captured, and that capture is resonant with an excited $0^{+}$ state of $^{12} C$ at $7.65 MeV$ — the Hoyle state, predicted from the requirement that carbon be produced at all before it was found experimentally:

α + α ⇌^{8} Be,^{8} Be + α \to^{12} C^{*} (7.65) \to^{12} C + γ .

The overall triple-alpha reaction $3 α \to^{12} C$ releases $Q = 7.275 MeV$ . Because it needs three bodies, the rate scales as the cube of the alpha density and, through the two resonances, as roughly $T^{40}$ near $1 0^{8} K$ — the most temperature-sensitive reaction in a star. Once carbon is present it captures another alpha, $^{12} C (α, γ)^{16} O$ , and some oxygen captures a further alpha to $^{20} Ne$ , setting the carbon-to-oxygen ratio that later burning stages inherit.¹

The triple-alpha process crosses the mass-8 gap through the short-lived beryllium-8 resonance and the Hoyle resonance of carbon-12; the equilibrium of the first step and the resonant capture of the second make carbon production possible.

Advanced burning to the iron peak

A massive star burns through successive fuels, each igniting when the ash of the previous stage contracts and heats enough to overcome the next barrier. The stages run faster as they proceed, because each releases less energy per nucleon while radiating from an ever hotter, denser core.¹

Carbon burning ( $\sim 5 \times 1 0^{8} K$ ): $^{12} C +^{12} C \to^{20} Ne + α$ , $^{23} Na + p$ , $^{24} Mg + γ$ .
Neon burning ( $\sim 1.2 \times 1 0^{9} K$ ): photodisintegration $^{20} Ne (γ, α)^{16} O$ frees alphas that recombine onto other neon, building $^{24} Mg$ .
Oxygen burning ( $\sim 2 \times 1 0^{9} K$ ): $^{16} O +^{16} O \to^{28} Si + α$ , ${}^{31}\mathrm{P}
- p $,$ {}^{31}\mathrm{S} + n$.
Silicon burning ( $\sim 3 \times 1 0^{9} K$ ): photodisintegration and re-capture reach a quasi-equilibrium that funnels nuclei toward the most bound species near $^{56} Ni$ , which decays to $^{56} Fe$ .

Each stage lasts far less time than the last — hydrogen burning of a $25$ -solar-mass star takes millions of years, silicon burning about a day — and the star develops an onion-shell structure with the heaviest ash at the center. Fusion halts at the iron peak because $B / A$ is maximal there: fusing iron would consume energy rather than release it, so no further exothermic fusion can support the core. When the inert iron core exceeds the Chandrasekhar mass it collapses, and the star's fusion history ends.

A massive star near the end of its life has concentric burning shells, each fusing the ash of the layer outside it, with an inert iron core at the center where exothermic fusion stops.

Building the heavy elements by neutron capture

Charged-particle fusion cannot climb past iron: the Coulomb barrier grows with $Z$ while the energy return turns negative. Elements heavier than the iron peak are assembled instead by neutron capture, which has no barrier. A nucleus captures a neutron, $(n, γ)$ , moving one step to the right in $A$ ; if the product is unstable it $β^{-}$ -decays, raising $Z$ and moving up the chart. Two regimes are distinguished by whether capture or decay is faster.

The s-process (slow): the neutron flux is low, so between captures an unstable nucleus has time to $β^{-}$ -decay. The path hugs the valley of stability, stepping up along it one mass unit at a time. It runs in thermally pulsing AGB stars, with neutrons supplied by $^{13} C (α, n)^{16} O$ and $^{22} Ne (α, n)^{25} Mg$ , and terminates in a cycle at $^{208} Pb$ – $^{209} Bi$ . Abundance peaks appear at the magic neutron numbers $N = 50, 82, 126$ ( $A \approx 88, 138, 208$ ), where the small capture cross section makes nuclei pile up.
The r-process (rapid): the neutron flux is enormous, so many captures occur before any $β^{-}$ -decay. The path runs far out on the neutron-rich side, along a contour of nearly constant neutron separation energy, pausing at waiting points where a magic neutron number lowers the capture rate until a $β^{-}$ -decay lets it advance. It requires an explosive, neutron-rich site — neutron-star mergers, confirmed by the kilonova of a merger event, and possibly core-collapse supernovae. After the flux ceases, the neutron-rich isotopes decay back to stability, producing abundance peaks at $A \approx 80, 130, 195$ , displaced below the s-process peaks because the freeze-out happens at fixed neutron number.

On the chart of nuclides the s-process climbs in a staircase along the valley of stability, while the r-process runs horizontally into neutron-rich territory and then decays diagonally back to stability after the neutron flux stops.

The solar neutrino confirmation

The reactions of the pp chain and CNO cycle can be checked directly because each weak step emits a neutrino that leaves the star immediately, carrying a fingerprint of the reaction that made it. The Sun's photons take $1 0^{5}$ years to random-walk out of the core; its neutrinos arrive in $8$ minutes. Ray Davis's chlorine experiment in the 1960s measured a flux of $^{8} B$ neutrinos about a third of the standard solar model prediction, the solar neutrino problem. The deficit was not a failure of the fusion model but of particle physics: electron neutrinos oscillate into muon and tau flavors on the way out, which the chlorine detector could not see. The Sudbury Neutrino Observatory settled it by measuring both the electron-flavor flux and the total flux over all flavors; the total matched the solar model while the electron flux was suppressed, confirming both the pp-chain energy generation and neutrino flavor mixing.² The same weak process that starts hydrogen burning also set the neutron-to-proton ratio of the early universe, the subject of the next lesson.

Krane, Introductory Nuclear Physics, §14.3 (Fusion in Stars): the proton-proton chain with its three branches, the CNO catalytic cycle, the triple-alpha process through the $^{8} Be$ and Hoyle resonances ( $Q = 7.275 MeV$ ), the advanced burning stages to the iron peak, and the s- and r-process neutron-capture paths. The synthesis framework is that of Burbidge, Burbidge, Fowler, and Hoyle. Reaction $Q$ -values and cross sections are compiled by the NNDC, https://www.nndc.bnl.gov/. ↩ ↩² ↩³ ↩⁴
Krane, §14.3, and Tipler & Llewellyn, Modern Physics, Ch. 13 (solar fusion). The $^{8} B$ solar neutrinos, the Davis chlorine deficit, and its resolution by neutrino oscillation as established by the Sudbury Neutrino Observatory. Neutrino-oscillation parameters are reviewed by the Particle Data Group, https://pdg.lbl.gov/. ↩