Saha-Boltzmann ionization of hydrogen

What you are seeing: the ionization fraction $x = n_+/n_\text{tot}$ of a pure-hydrogen plasma as a function of temperature, at the user-set total number density. The Saha equation $x^2/(1-x) = \mathrm{Saha}(T) / n_\text{tot}$ with $\mathrm{Saha}(T) = (2\pi m_e k_B T / h^2)^{3/2} \exp(-\chi / k_B T)$ is solved as a closed-form quadratic in $x$.

For solar-photosphere densities ($n \sim 10^{23}$ m$^{-3}$, $T = 5800$ K), hydrogen is only about $10^{-4}$ ionized; for the chromosphere ($T \sim 10^4$ K) it crosses 50 percent. The classic ratio $\chi / k_B \approx 158000$ K is deceptive because the prefactor $\propto T^{3/2}/n$ pushes the half-ionization temperature down by an order of magnitude.