Hydrogen Burning: pp Chains and the CNO Cycle

Hydrogen burning converts four protons into one helium-4 nucleus. The net reaction,

releases $Q=26.73\text{MeV}$ once the two positrons annihilate with ambient electrons. No single collision accomplishes this; four protons meeting at one point is vanishingly unlikely, and two of them must convert to neutrons by the weak interaction. The transformation proceeds through a sequence of two-body reactions, organized into two networks that dominate in different temperature regimes: the proton–proton (pp) chain and the carbon–nitrogen–oxygen (CNO) cycle.

The mass defect

The energy released is the binding energy locked up when four nucleons assemble into a tightly bound helium nucleus. Using atomic masses,

so the mass defect is $\Delta m=0.028698\text{u}$, which the mass–energy relation converts to

The fraction of the rest mass released is $\Delta m/(4m_{{}^1\text{H}})=0.71\%$. This efficiency, an order of magnitude larger than any chemical process and smaller than the several-percent yields of later burning stages and of accretion onto compact objects, sets the nuclear timescale of the Sun: at $L_{\odot }=3.83\times 10^{26}\text{W}$, converting $0.71\%$ of the Sun's hydrogen supplies power for roughly $10^{10}\text{yr}$.

The proton–proton chain

The chain opens with the slowest reaction in all of stellar physics,

Two protons must not only tunnel through the Coulomb barrier but simultaneously undergo a ${\beta }^{+}$ conversion of one proton to a neutron, a weak-interaction process, during the fleeting moment of contact. The resulting deuteron is the only bound two-nucleon state; the diproton is unbound, so without the weak conversion the collision produces nothing. The combined smallness of the barrier penetration and the weak matrix element makes the mean lifetime of a proton against this reaction about $10^{10}\text{yr}$ at the solar center. Every later step is faster by many orders of magnitude, so this first reaction throttles the entire chain and fixes the Sun's luminosity and lifetime.¹

The deuteron captures a proton almost immediately,

with a mean lifetime of order seconds, so deuterium never accumulates. The chain then completes by one of three branches, distinguished by the fate of ${}^3\text{He}$.

pp-I closes when two helium-3 nuclei react:

Reaching one ${}^4\text{He}$ this way requires the first two reactions to run twice. At the solar center pp-I produces about $69\%$ of the helium.

pp-II and pp-III begin instead with a capture on a pre-existing ${}^4\text{He}$,

Beryllium-7 then either captures an electron (pp-II) or a proton (pp-III):

At the solar center pp-II accounts for about $31\%$ of the helium and pp-III for only about $0.3\%$. Small as it is, pp-III matters out of proportion to its rate: the ${\beta }^{+}$ decay of ${}^8\text{B}$ emits neutrinos with energies up to $\sim 15\text{MeV}$, the only solar neutrinos energetic enough for the early chlorine and water-Cherenkov detectors, so this rare branch dominated the historical solar-neutrino measurements.

Neutrino losses per branch

Each branch emits its neutrinos at different energies, and the neutrinos escape the star immediately, carrying their energy away without contributing to the pressure or luminosity. The effective heat deposited per helium nucleus is therefore branch-dependent, always less than the full $26.73\text{MeV}$:

The pp-I branch loses only about $2\%$ of $Q$ to neutrinos, while pp-III loses nearly a third because the ${}^8\text{B}$ neutrino is so energetic. The neutrino spectrum is a direct probe of which branches operate, and measuring it tests the standard solar model at the level of individual reactions.

The CNO cycle

Where carbon, nitrogen, and oxygen are already present, a second network fuses hydrogen using those nuclei as catalysts that are consumed and regenerated. The main branch, CNO-I, is a closed loop of six reactions:

Summing the loop, four protons enter and one ${}^4\text{He}$ leaves, with the same net $Q=26.73\text{MeV}$ minus the energy of two neutrinos from the ${}^{13}\text{N}$ and ${}^{15}\text{O}$ decays; the ${}^{12}\text{C}$ reappears unchanged. The rate-limiting reaction is the proton capture on nitrogen-14, ${}^{14}\text{N}(p,\gamma ){}^{15}\text{O}$, which has the highest Coulomb barrier of the loop and a small cross section. Because it is the slowest link, the catalytic material piles up as ${}^{14}\text{N}$: a star running the CNO cycle converts most of its initial carbon and oxygen into nitrogen, the nucleosynthetic origin of much of the galaxy's ${}^{14}\text{N}$.

About one proton capture on ${}^{15}\text{N}$ in a thousand takes the alternative channel ${}^{15}\text{N}(p,\gamma ){}^{16}\text{O}$ rather than releasing helium. This opens the CNO-II branch, which threads through ${}^{16}\text{O}$, ${}^{17}\text{F}$, and ${}^{17}\text{O}$ before returning to ${}^{14}\text{N}$, extending the catalytic network to oxygen isotopes without changing the net hydrogen-to-helium conversion.

The crossover and the main-sequence division

The two networks differ sharply in temperature sensitivity, for the reason derived in the Gamow-peak analysis: the CNO reactions run against a charge product $Z_1{Z}_2=6$–$7$ rather than the $Z_1{Z}_2=1$ of the first pp step, so their Gamow peaks sit far higher on the thermal tail. Near $1.5\times 10^7\text{K}$ the local power laws are

with $X$ the hydrogen mass fraction and $X_{\mathrm{CNO}}$ the catalyst abundance. The gentle $T^4$ of the pp chain dominates at low temperature; the steep $T^{\sim 20}$ of the CNO cycle overtakes it above a crossover near $T\approx 1.8\times 10^7\text{K}$ for solar composition. The Sun's central temperature, $1.57\times 10^7\text{K}$, sits just below the crossover, so the Sun generates about $99\%$ of its power through the pp chain and only $\sim 1\%$ through CNO.

This crossover organizes the main sequence. Stars below about $1.2M_{\odot }$ have central temperatures under the crossover and burn hydrogen by the pp chain with a radiative core; stars above it run the CNO cycle, whose extreme temperature sensitivity concentrates the energy generation in a small central region and drives a convective core. The structural difference between the lower and upper main sequence, developed in the main sequence and its structure, follows directly from which of these two networks supplies the star's luminosity. The next lesson turns to the fuel that follows hydrogen, helium and the triple-alpha process, whose ignition requires the far higher temperatures the Gamow scaling demands for charge-2 nuclei.