(updated for 2013)
How do we deal with the distribution over different states of ionization r? In thermodynamic equilibrium, the Saha equation gives:
where and are the partition functions as discussed in the previous section. The partition function for electrons is given by
For a derivation of this, see pages 103-104 of this handout from Bowers and Deeming.
If and are approximated by the first terms in their sums (i.e. if the ground state dominates their level populations), then
where is the energy required to ionize from the ground (j = 1) level. Ultimately, this is just a function of and . This assumes that the only relevant ionization process is via thermal collision (i.e. shocks, strong ionizing sources, etc. are ignored).