1) Parallax is one of the most important distance determination techniques in Astronomy, since it is the only one based on pure mathematics and measurements, rather than relying on any assumptions about stellar properties.

a) (7 pts) Explain how and why the parallax angle of a star is related to its distance from the Earth.

b) (7 pts) If we were to compile a catalog of 1000 stars with the largest parallax angles, would this catalog constitute a representative sample? Why or why not?

2) Star A and B have the same apparent luminosity as seen from Earth. ISM measurements tell us that there is no gas or dust along the line of sight to either star. Both stars have the same spectral line widths as well. Star A shows higher ionization species in its spectrum.

a) (8 pts) Which star is larger? Explain your reasoning.

b) (6 pts) Which star is closer? Explain your reasoning?

3) On the H-R Diagram below, do the following. No explanations needed for this problem.

a) (7 pts) Draw in the approximate location of a star that is the same temperature as the Sun but is much larger than the Sun. Mark this location with an "A".

b) (7 pts) Draw in the approximate location of a star with 4 times the size as the Sun but only half the Sun's temperature. Mark this location with a "B".

---

4) Two face-on binary systems, System A and System B, are both measured to have identical orbital distances from the central star to the companion star. System A's companion, however, is found to have an orbital period twice as long as System B.

a) (4 pts) Explain why the orbital velocity for this system is not directly observable.

b) (10 pts) Which system's central star is more massive? Justify your answer either mathematically or in a couple of sentences.

---

5) The interstellar medium (ISM) affects our estimates of stellar absolute luminosities and thus, our estimates of the distances to many stars.

a) (8 pts) Explain what reddening is and why it makes us think stars are closer than they really are.

b) (6 pts) Forbidden lines are only seen in the very low density environment of the ISM. Explain why low density is required in order for an atom to emit a forbidden line.

---

6) The densest ISM clouds often will collapse to form stars, but not all collapsing clouds form stars.

a) (6 pts) Explain why only clouds with a mass above about 8% of the sunıs mass ever form stars.

b) (8 pts) In dense ISM clouds, molecules will often form. In fact, in star forming regions, complex organic molecules sometimes form. Explain how the light from forming stars helps create these exotic molecules.

---

7) During the main sequence, the Sun will gradually grow in size. This may result in changes in the Earthıs climate.

a) (7 pts) Explain why the Sun has increased in luminosity during the first five billion years of its lifetime.

b) (7 pts) What is the Faint Sun Paradox? How has the Earth "resolved" this paradox?

---

8) Giant stars often fuse Helium in their cores. Sometimes, their outer layers grow unstable, and the resulting variations in their luminosities allow us to determine distance to these stars with amazing accuracy.

a) (7 pts) Explain why Helium fusion requires a higher density and a higher temperature than Hydrogen fusion.

b) (7 pts) Explain how we use the Cepheid Period - Absolute Luminosity relation to determine the distances to Cepheids in other galaxies.