2. Using the procedure outlined below, for a stellar atmosphere that is composed of pure helium, find the temperature at which half of the He I atoms have been ionized. The ionization energies of neutral helium and singly ionized helium are 24.6 eV and 54.4 eV, respectively. The partition functions are $Z_I = 1$, $Z_{II} = 2$, $Z_{III} = 1$. Electron pressure is 20 N m$^{-2}$.
(a) 10 points How do $N_{II}/N_I$ and $N_{III}/N_{II}$ compare to each other at temperatures of 5000 K, 15,000 K, and 25,000 K? Show the ratios in numbers and compare them. What does this mean?
(b) 10 points Show that $N_{II}/N_{total} = N_{II}/(N_I + N_{II} + N_{III})$ can be expressed in terms of ratios $N_{II}/N_I$ and $N_{III}/N_{II}$.
(c) 10 points Using a programming language of your choice make a graph that shows $N_{II}/N_{total}$ as a function of temperature that varies from 5000 K to 25,000 K. What is the temperature at which half of the He I atoms have been ionized (these atmospheres are found on white dwarfs of certain spectral type)?
