# The Saha equation

Degree of ionization of atomic hydrogen in thermal equilibrium can be described by the Saha equation $\frac{1-X}{X^2}=n\lambda_{Te}^3e^{\frac{I}{kT}},$ where $X=n_e/n$ is the degree of ionization, $n_e$ and $n$ are concentrations of electrons and atoms (both neutral and ionized) respectively, $\lambda_{Te}^2=\frac{2\pi\hbar^2}{m_e kT}$ is the electron's thermal de Broglie wave length and $I=13.6eV$ is the ionization energy for hydrogen. It is often used in astrophysics for description of stellar dynamics.

### Problem 1

Using the Saha equation, determine the hydrogen ionization degree

a) 100 seconds after the Big Bang;

b) at the epoch of recombination;

c) at present time.

Assume for simplicity $\Omega=1$.

### Problem 2

Assuming that the ionization degree at the last scattering was equal to $10\%$, determine the decoupling temperature using the Saha equation.

### Problem 3

How many iterations in the Saha equation are needed in order to obtain the decoupling temperature with accuracy $1K$? Write down analytically the approximate result.

### Problem 4

Estimate the duration of the epoch of recombination: how long did it take for hydrogen ionization degree to change from $90\%$ to $10\%$ according to the Saha equation?

### Problem 5

Using the Saha equation, determine the hydrogen ionization degree in the center of the Sun ($\rho=100g/cm^3$, $T=1.5\cdot10^7K$).