1. Partition Function

Write a little program (e.g., in Python) to evaluate the partition function for atomichydrogen, as a function of gas temperature Z(T) =kupperk=1 g(k)exp[(k)/kBT] where all symbols have their usual meaning (see the lecture notes).

a. Consider a stellar atmosphere (photosphere) with atomichydrogen number densityof nH= 10^16cm^3. For this density, where should the sum in the partition function betruncated? Or put differently: What is the appropriate k upper here?

b. Next, numerically evaluate the partition function for a range of temperatures,3,000 KT30,000 K.

c. Plot your Z(T) vs.T. On the same plot, insert a (say, dashed) line marking thesimple short-cut that Z(T)g1(i.e., that most atoms are in the ground state). This plot should not just be drawn by hand, but instead find a way to have this generated (and printed out) by a computer.

d. Very briefly: Comparing the exact numerical result with the simple most-atoms-in-the-ground-state approximation, what do you conclude? How good/bad is this approximation?