Helium-3 equilibrium abundance ($n_{3e}=(-B+\sqrt{B^2+4AC})/(2A)$)

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.