This homework must be submitted on paper; you may write out your work on paper, or type it into some word processor and then print it. Either way, staple together all pages and write your name legibly at the top of each page.
This assignment asks "Just how hard is it to measure properties of the atmosphere of a planet circling another star?" The answer is (obviously) "pretty hard!", but going through this exercise may help you to see some of the reasons. Please look at the material in this lecture from another course:
You don't need to read it all in detail! But you ought to find some clues for answering the questions below.
We'll go over the answers in class, so don't worry too much if you aren't sure what to do.
Astronomers find an Earth-like planet circling a star very much like the Sun, AND the system is very close to us. This is an ideal case for us to look for an atmosphere. We'll assume that the planet has an atmosphere which is uniform in density (which is unrealistic, but makes the calculations easy).
stellar radius Rs = 7 x 108 m
planet radius Rp = 6.37 x 106 m
planet's atmosphere height h = 50 km = 5 x 104 m
See the section using an absorption line and the figures therein.
We'll observe this star through two narrow filters, each of width w = 1 nm. We'll use two filters each of width 1 nm, "A" at a wavelength of 545 nm, and "B" at a wavelength of 550 nm. The atmosphere doesn't absorb any of the light through filter "A", but there's a strong absorption line due to a molecule in the atmosphere at a wavelength of 550 nm, so the atmosphere absorbs all the light in filter "B".
The goal is to measure the light from the star as the planet passes in front of it. We'll measure exactly how many photons we receive through filter "A", and how many through filter "B". The difference will give us evidence for the atmosphere.
One last piece of information: if we use a big 10-meter telescope to observe the star by itself, when the planet is not passing in front it, we collect this many photons through the two filters:
filter "A": 9.000 x 109 photons per sec
filter "B": 9.000 x 109 photons per sec
As you can see, it's the same brightness through both filters. Very simple.
Now, what happens if the planet passes in front of the star? The planet blocks some of the star's light, so we will collect fewer photons. The atmosphere will also block some light through filter "B", so we should collect a bit less light through that filter.