Solution

In the nonrelativistic regime,

E (p) = m c^{2} + \frac{p ^{2}}{2 m} + O (\frac{p ^{4}}{m ^{3} c ^{2}}) .

The assumption

E - μ ≫ k_{B} T

n ≃ \frac{4 π g _{s}}{h ^{3}} e^{(μ - m c^{2}) / (k_{B} T)} \int_{0}^{\infty} p^{2} e^{- p^{2} / (2 m k_{B} T)} d p = \frac{4 π g _{s}}{h ^{3}} e^{(μ - m c^{2}) / (k_{B} T)} \frac{π}{4} (2 m k_{B} T)^{3/2} .

n = g_{s} (\frac{2 πm k _{B} T}{h ^{2}})^{3/2} exp (\frac{μ - m c ^{2}}{k _{B} T}) .

For

p + e^{-} \leftrightarrow H + γ

μ_{p} + μ_{e} = μ_{H} .

Applying the number-density formula to each massive species and eliminating the chemical potentials gives

\frac{n _{e} n _{p}}{n _{H}} = \frac{g _{e} g _{p}}{g _{H}} (\frac{2 π k _{B} T}{h ^{2}})^{3/2} (\frac{m _{e} m _{p}}{m _{H}})^{3/2} exp (- \frac{( m _{p} + m _{e} - m _{H} ) c ^{2}}{k _{B} T}) .

Use charge neutrality,

n_{p} ≃ n_{e}

, the mass approximation

m_{H} ≃ m_{p}

, and the convention in the question that suppresses the order-one internal-degeneracy ratio

g_{e} g_{p} / g_{H}

E_{bind} = (m_{p} + m_{e} - m_{H}) c^{2},

\frac{n _{e}^{2}}{n _{H}} ≃ (\frac{2 π m _{e} k _{B} T}{h ^{2}})^{3/2} exp (- \frac{E _{bind}}{k _{B} T}) .

Solved by gpt-5.6-sol high.

 Ancestors (10)