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Home/Astronomy & The Sky/Hydrogen Saha Equilibrium (recombination)

Hydrogen Saha Equilibrium (recombination)

n_p n_e / n_H vs T and ionized fraction X; steep Boltzmann knee ~3000 K scale (density dependent).

Gas (pure H)

4

n_b = n_H + n_p. Lower n_b moves the X(T) transition toward ~3000 K (toy dilute limit); cosmic densities need much higher T for the same X in Saha equilibrium.

Temperature sweep

1200
14000
3000

Shortcuts

  • •Drag sliders — T range, n_b, marker T
  • •R — reset to defaults

Measured values

n_b1.000e+4m⁻³
Saha n_p n_e / n_H @ 3000 K1.136e+4m⁻³
(n_p n_e / n_H) / n_b @ marker T1.136e+0
X at marker T0.6397

About this model

For pure hydrogen in thermal equilibrium, H ⇌ p + e⁻, the Saha relation gives n_p n_e / n_H = (g_p g_e / g_H) (2π m_e k_B T / h²)^{3/2} exp(−χ_H / (k_B T)) in SI number-density units for an ideal non-degenerate plasma (ground-state g factors). With total baryon density n_b = n_H + n_p and charge neutrality n_e = n_p, the ionized fraction X = n_p / n_b satisfies X²/(1−X) = (n_p n_e / n_H) / n_b. The Boltzmann exponential makes the Saha product rise steeply over thousands of kelvin; the knee in X(T) moves to higher T when n_b is large because recombination requires (n_p n_e / n_H) ~ n_b. Real cosmic recombination is kinetic (Peebles) and not this equilibrium curve alone.

Who it's for: Introductory astrophysics after ideal gas and atomic structure; complements FLRW and CMB overview pages.

Key terms

  • Saha equation
  • hydrogen recombination
  • ionization fraction
  • Boltzmann factor
  • baryon density

How it works

For pure hydrogen in thermal equilibrium, H ⇌ p + e⁻, the Saha relation fixes n_p n_e / n_H as a function of T (ideal, non-degenerate gas). With g factors for ground hydrogen, n_p n_e / n_H = (2π m_e k_B T / h²)^{3/2} exp(−χ_H / (k_B T)) in SI (same dimensions as a number density). Fixing total baryon density n_b = n_H + n_p gives the ionized fraction X = n_p / n_b from X²/(1−X) = (n_p n_e / n_H) / n_b. The Boltzmann factor makes n_p n_e / n_H jump steeply near k_B T ~ χ_H/10–15 (thousands of kelvin); with cosmic-scale n_b the Saha X curve shifts to higher T than the naive “3000 K” cartoon because (n_p n_e / n_H)/n_b must be order unity — real recombination also needs Peebles kinetics, not plotted here.

Frequently asked questions

Why is the “3000 K” knee not universal?
The Saha product grows with T, but X is fixed by comparing that product to n_b. Higher n_b pushes the transition to higher T in equilibrium; cosmology also needs radiative transfer and rate equations beyond Saha.
Why g_p g_e / g_H = 2 here?
For ground hydrogen, g_H = 2 (spin), g_p = g_e = 2 for free proton and electron spins, giving a ratio of 2 in the textbook convention used on this page.