Question #1 (20%) - non-math
The Hubble law predicts that the Andromeda galaxy should be moving away from the Milky Way with the recession velocity of about 50 km/s. Observations show that the Andromeda galaxy is actually moving towards us with the velocity of about 275 km/s, in a drastic disagreement with the Hubble law. Can this observation be used to rule out the Big Bang theory? Explain.

The Hubble law only applies to the freely moving objects. In other words, only objects which have no forces acting upon them participate in the expansion of the universe. The mutual gravity of the Milky Way and Andromeda is pulling them toward each other (from our reference point it looks like Andromeda is coming toward us), causing a deviation from the Hubble law. Thus, the answer is no: Andromeda and the Milky Way do not have to follow the Hubble law.

Question #2 (20%) - non-math
Two astronomers, Jack and Jill, decided to measure the mass of the Andromeda galaxy. Jack simply estimated the total mass in stars that he could see with his telescope. Jill however used the appropriately modified version of the third law of Kepler to measure the mass of Andromeda based on orbits of Andromeda satellites (dwarf galaxies that orbit Andromeda in much the same way as Earth orbits the Sun). To their surprise, they find that their results disagree strongly. Who found the larger value for the mass of Andromeda and why? Whose number is the correct one?

Jill will find a larger and more accurate value, because she measures the total gravity of Andromeda. She account for everything that has gravity: stars, the dark matter (the main thing here), gas, planets, cosmic dust, the trash from alien spaceships, the alien spaceships themselves, etc. Jack only measures the stars, so he misses everything that has mass but gives no light.

Question #3 (20%) - non-math
We know observationally that there are galaxies and quasars at very large cosmological redshifts, the record holder is a quasar at a redshift of about 6. However, not only expansion of the universe can cause a redshift -- a strong gravitational field in the vicinity of a black hole also causes a redshift. For example, the last stable orbit around the black hole has a redshift of about 0.3. Give an argument why astronomers nevertheless think that quasars are at cosmological distances -- why can not a quasar be just a small but very bright blob of hot gas sitting very close to (but still outside of) the horizon of a black hole? Remember that we do not know the size of a quasar, i.e. you cannot use the size as an argument.

If a quasar with the redshift of 6 is a blob of gas orbiting a black hole, it has to be closer to the horizon than the last stable orbit at the redshift of 0.3 - the redshift is larger where gravity is stronger. But then it cannot orbit a black hole - it is closer than the last stable orbit, so it will be sucked into the black hole. Therefore any object that has a redshift of more than 0.3 has to be at a cosmological distance (unless it is an alien spaceship hovering just above the horizon of the black hole by the power of its engine).

There is a different correct answer to this question: since quasars actually reside at the centers of galaxies, if you see a galaxy around the quasar, you will know that the quasar is far away. This is a very difficult observation: only recently astronomers managed to see galaxies around the quasars (because quasars are so bright, they outshine everything around them, just like the Sun outshines stars during the day so that we do not see them -- but they are still there!), and only around a few nearest ones. But it is still a right answer to the question (it however relies on the knowledge of astronomy which you are not required to have in this class).

Question #4 (20%) - non-math
Astronomer Z is very unhappy with the Big Bang theory -- he really does not like a universe which has a beginning. But we know observationally that the universe expands. So, Z invented a model in which the universe exists forever, but still expands. Can we use the Olber paradox (see H&H, pp. 319-320) to rule out his model? Please explain.

The Olber paradox is the contradiction between the observed darkness of the night sky and the prediction of Newtonian cosmology that in the static infinite in space and time universe the sky should be ablaze day and night. In Z's model the universe expands, and the light from distant stars will be redshifted into the infrared band of the electromagnetic spectrum and becomes invisible. Thus the night sky will be dark in Z's model, and we cannot use the Olber paradox to rule this model out.

Question #5 (20%) - non-math
Astronomer X developed another model of the universe in which the universe was so hot and dense during some early epoch that nuclear reactions during that epoch created enough oxygen atoms to fit the existing observations of the oxygen abundance in the universe. Give a strong argument against such a cosmological model.

If all the observed oxygen was produced in the early universe, then the stars which we observe today cannot produce any oxygen. But of course we cannot tell the stars what to do, they operate according to the laws of physics, and they produce oxygen no matter what happened in the early universe. Thus in order to avoid overproducing oxygen in X's model, that model should contain no stars -- not a very plausible model given what we can see in the window in a sunny day.

Question #6 (20%) - math
Imagine that you built a time machine and went into the past. But something broke down, and you actually went much much further than you had planned -- billions of years back in time, before the Earth or the Sun even existed. You found yourself hanging in the empty space and felt pretty uncomfortable. Thus you decided to have a glass of cold water. To cool the water quickly, you poured it into a closed container and put it outside of your time capsule, exposed to the empty space. To your great amazement, the water started to boil. Estimate the minimum possible value of the cosmological redshift z at which you ended up (water boils at 373 degrees Kelvin). [Hint: see H&H, p. 355 for variation of CMB]

The water will boil because there is something to heat it up in the "empty space", and this something is radiation, the Cosmic Microwave Background radiation (CMB). Of course, today the CMB is very cold, less than 3 degrees Kelvin. At this temperature even the air freezes out into a solid. But early on the CMB was hotter, so if we go far enough in the past, we can reach the time when the CMB was hot enough to boil the water. Since the temperature of the CMB is T_CMB=2.73*(1+z) degrees Kelvin, this temperature is equal to the temperature of boiling water (373 degrees Kelvin) when 1+z = 373/2.73=136.6 and thus z = 135.6.