1) Two face-on binary systems, Alpha and Beta, have companion stars with identical orbital periods. The two systems also have the same angular size as seen from Earth. Parallax measurements show that system Alpha is further away. Based on this information, which system has the more massive central star, Alpha or Beta? Justify your answer fully.

2) About 90% of the visible matter in our galay is Hydrogen gas. Most of this is at very low temperatures. We can detect it by looking for a spectral emission line with a wavelength of 21-cm. This same line is not visible in stars, even though stars are composed largely of Hydrogen, because it is a "forbidden line", not seen in high density gas like that found in the atmosphere of a star.

a) (7 pts) Explain why Hydrogen gas in our galaxy would not be visible if not for the existence of the 21-cm emission line.

b) (7 pts) Explain why forbidden lines like the 21-cm transition are only visible in very low density gas.

3) Suppose two stars, Gamma and Delta, are observed to have the same apparent luminosity and same red color. Observations of the spectral line widths of the two stars show that they have the same size (radius). We also know that there is a substantial amount of interstellar gas and dust (ISM) along the line of sight to star Delta but very little along the line of sight to Gamma.

a) (6 pts) We use spectral line information to find out about the amount of ISM along the line of sight to a star. How and why do ISM spectral lines differ from the spectral lines in stellar atmospheres?

b) (8 pts) Assuming that the ISM has done little to change the apparent luminosity of either star, which one of these two stars is likely further away, based on your knowledge of the effects of reddening? Justify your answer.

4) Over the course of the main sequence lifetime of the Sun, its absolute luminosity has gradually grown due to changes within the core.

a) (8 pts) Explain what has happened within the Sun to make it brighter.

b) (6 pts) Explain the "Faint Sun Paradox" here on Earth and how it has been resolved by a study of Earthıs past properties.

5) Cepheid variables undergo periods of expansion and contraction. These changes can be understood through the mechanism of hydrostatic equilibrium (HSE).

a) (6 pts) During the expansion phase of a Cepheid's pulsation, state and explain what is happening to the temperature and outward-pushing pressure of the star.

b) (8 pts) Suppose Cepheids are observed in two clusters, Kappa and Theta. The Cepheid in cluster Kappa shows a period twice as long as the other Cepheid, and both stars have the same apparent luminosity. Explain why the period and absolute luminosity of Cepheids are related, then explain which cluster is further away from us (and how we know it).

6) Black holes in binary systems are often thought to exist based on the effects they have on their companions or their surroundings.

a) (7 pts) Explain how observations of a black holeıs companion star can "prove" the existence of a black hole.

b) (7 pts) Suppose the black hole is found to be 10 solar masses in part (a). Now we replace the black hole with a 10 solar-mass star. Assume that the 10 solar-mass star and its companion are separated by a large enough distance that they donıt touch. The distance between the centers of the two objects is identical in both cases. Would the orbital velocity of the companion star change? Why or why not?

7) Suppose our galaxy had no dark matter. What would the rotation curve for the disk of the galaxy look like? Draw in what you think it would look like on the graph below. The Keplerian curve (which we see in our solar system) is included for reference. If the galaxyıs rotation curve would exactly match this, just write "no change" on the graph below. After drawing in your curve, explain why you drew it the way you did (or wrote "no change").

8) One possible component of the dark matter in our galaxy are planet-sized masses known as MACHO's. Until recently, observation of these objects has been impossible.

a) (7 pts) Explain how we detect MACHO's.

b) (7 pts) Some distant stars vary in brightness for reasons unrelated to MACHO's. Give an example of one such variation, and explain how and why it is different from the variation related to MACHO's.