The surface temperature of the Sun is 5900 K, but sunspots are only about
4300 K.
Above is a photograph of the Sun followed by a close-up of a sunspot.

Consider two hydrogen atoms, one located on the sunspot, the other on the hotter surface
of the Sun. The energy levels of the hydrogen atom are given by the equation

E_{n} = -13.6 eV /n^{2} ( n = 1, 2, 3,...).

Define P_{2}(surface, 5900 K) to be the probability of the hydrogen
atom being in the first excited state (n=2) on the Sun's surface.
Define P_{2}(sunspot, 4300 K) to be the probability of the hydrogen
atom being in the first excited state (n=2) on a sunspot.

Calculate the ratio P_{2}(surface, 5900 K) / P_{2}(sunspot, 4300 K).
Assume that you can ignore energy states higher than the first excited state, n=2 (why?).
Also, ignore any degeneracy: assume that the hydrogen atom has one and only one state for each energy level.

Note: In such problems it is helpful to use the Boltzmann constant
in electron-volt units, k = 8.617×10^{-5} eV/K.