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

With

A = λ_{33}

B = λ_{34} n_{4}

and

C = λ_{11} n_{1}^{2} /2

, helium-3 evolves as

\overset{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} λ_{11} / (2 λ_{33})

, making equilibrium abundance strongly temperature dependent.

Table of contents
- Helium-3 relaxation time Helium-3 equilibrium abundance
  - Temperature for helium-3 freeze-out Helium-3 relaxation time
    - Helium-3 freeze-out estimate with Gamow factors Temperature for helium-3 freeze-out

Helium-3 relaxation time ( $τ^{- 1} = 2 A n_{3 e} + B = B^{2} + 4 A C$ )

 0  0

A small abundance displacement from the helium-3 equilibrium abundance obeys

\overset{x}{˙} = - x / τ

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

- \overset{n}{˙}_{3 e}

. In the pp-I limit the rate is

n_{1} 2 λ_{11} λ_{33}

, so both production and destruction temperature dependences matter.

Temperature for helium-3 freeze-out ( $τ (T) \sim t_{⊙}$ )

 0  0

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.

Helium-3 freeze-out estimate with Gamow factors ( $τ (T) / τ (T_{0}) = (T / T_{0})^{2/3} exp {\overset{η}{ˉ} [(T_{0} / T)^{1/3} - 1]}$ )

 0  0

In the pp-I limit, helium-3 relaxation time is

τ^{- 1} = n_{1} 2 λ_{11} λ_{33}

. At fixed proton number density, each thermonuclear reaction rate coefficient obeys

λ (T) / λ (T_{0}) = (T / T_{0})^{- 2/3} exp [- η (T_{0}) ((T_{0} / T)^{1/3} - 1)]

. Taking their geometric mean gives the displayed relaxation ratio with

\overset{η}{ˉ} = [η_{11} (T_{0}) + η_{33} (T_{0})] /2

. A local power-law approximation instead gives

τ \propto T^{- (α + β) /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.