Proton–proton chain

The proton–proton chain converts hydrogen to helium and dominates hydrogen burning in stars near and below the Sun's mass. Near $2\times 10^7\mathrm{K}$, its specific energy generation is roughly proportional to $\rho T^4$.

The proton–proton chain first produces deuterium by a weak reaction and then helium-3 by proton capture. pp I ends with ${}^3\mathrm{He}+{}^3\mathrm{He}\to {}^4\mathrm{He}+2p$. pp II passes through beryllium-7 electron capture to lithium-7 and then proton capture. pp III passes through boron-8 beta-plus decay and beryllium-8 breakup. Each branch produces one net helium-4 nucleus from four protons, but the neutrino spectrum and deposited heat differ. In pp II the net weak reactions involve one emitted positron and one captured electron.

Present solar hydrogen is most depleted in the central burning region and rises toward the envelope. helium-3 is efficiently destroyed in the hot centre, increases outward as its equilibrium abundance rises, and develops an off-centre maximum where equilibration becomes slow. Still farther out its production becomes weak, leaving a smaller envelope abundance. Convective mixing tends to flatten the outer profiles. This qualitative pattern is not itself a quantitative solar model.

An effective Proton–proton chain network can track protons, deuterium, helium-3, helium-4 and lithium-7 after eliminating a rapidly adjusting beryllium intermediate. Identical-reactant event rates include a factor one half. Stoichiometric changes must count both helium-4 products of lithium-7 proton capture. The weighted baryon mass density $n_1+2n_2+3n_3+4n_4+7n_7$ is conserved by the nuclear reactions at fixed volume.

With $A={\lambda }_{33}$, $B={\lambda }_{34}{n}_4$ and $C={\lambda }_{11}{n}_1^2/2$, helium-3 evolves as ${\dot{n}}_3=C-A{n}_3^2-B{n}_3$ after deuterium elimination. The positive root is its unique attracting equilibrium for a fixed background. In the pp-I limit it reduces to $n_1\sqrt{{\lambda }_{11}/(2{\lambda }_{33})}$, making equilibrium abundance strongly temperature dependent.

A small abundance displacement from the helium-3 equilibrium abundance obeys $\dot{x}=-x/\tau$ to linear order. Rapid relaxation relative to background evolution allows the abundance to track its equilibrium. If the equilibrium drifts, the displacement equation also contains $-{\dot{n}}_{3e}$. In the pp-I limit the rate is $n_1\sqrt{2{\lambda }_{11}{\lambda }_{33}}$, so both production and destruction temperature dependences matter.

Equilibrium fails where the helium-3 relaxation time becomes comparable to stellar age. In a fixed-density pp-I scaling with reaction exponents four and sixteen, the equilibrium abundance scales as $T^{-6}$ and the relaxation time as $T^{-10}$. Central normalization then places the transition near six million kelvin; using the full supplied exponential factors gives a somewhat higher estimate. A precise radius or temperature requires a stellar mass density/composition profile and the second branch contribution.

In the pp-I limit, helium-3 relaxation time is ${\tau }^{-1}=n_1\sqrt{2{\lambda }_{11}{\lambda }_{33}}$. At fixed proton number density, each thermonuclear reaction rate coefficient obeys $\lambda (T)/\lambda (T_0)=(T/T_0{)}^{-2/3}\exp [-\eta (T_0)((T_0/T{)}^{1/3}-1)]$. Taking their geometric mean gives the displayed relaxation ratio with $\bar{\eta }=[{\eta }_{11}(T_0)+{\eta }_{33}(T_0)]/2$. A local power-law approximation instead gives $\tau \propto T^{-(\alpha +\beta )/2}$. These are estimates rather than a complete radial model: the proton number density, composition and pp-II branch change along a real stellar composition profile.

deuterium proton capture is much faster than the weak reaction that produces deuterium. Its abundance therefore adjusts so that ${\lambda }_{21}{n}_2{n}_1\simeq {\lambda }_{11}{n}_1^2/2$. Substituting this balance eliminates the fast intermediate from the slower proton and helium-3 evolution equations. This is a quasi-steady approximation, not an assumption that hydrogen itself is in nuclear equilibrium.