## Answers to Example Questions for the Second Exam

1. Use Kepler's third law : (M1 + M2) = a3/P2; so (M1 + M2) = 503/1002 = 12.5 solar masses (notice we have 50 for a since it is the semi-major axis, ie half the orbit major axis).

2. Use the mass ratio equation : M1/M2 = V2/V1. The mass ratio is 15/30 = 0.5. Hence the individual masses are 12.5 x 1/3 and 12.5 x 2/3 = 4.17 and 8.34 solar masses.

3. 11 solar luminosities. Use the mass luminosity relation L proportional to M3.5, so that a 2 solar mass star has a luminosity 23.5 = 11 (approx) times greater than a 1 solar mass star (which has a luminosity of 1 solar luminosity, obviously).

4. a. The eclipsing binary allows you to measure the velocities of the stars with no ambiguities about the angle of the velocity (we know the plane of the orbit is edge on). Also, by timing the length of the eclipses, one can estimate the relative sizes of the stars. Less of this is possible with the other kinds of binary star.

5. Hydrostatic Equilibrium refers to the balance between the inward force of gravity and the outward force of pressure. In an object like the Sun, this occurs throughout the interior. It results in the stability of the sun, ie the fact that the sun is neither contracting nor expanding.

6. b

7. (a) Cold, dense molecular clouds (seen with molecular rotation/vibration radio waves, eg for CO). (b) Warm neutral atomic hydrogen (HI) clouds (seen with the 21cm radio transition from electron spin flip). (c) hot ionized hydrogen (HII) clouds (seen as pink light from the Balmer series). (d) very hot thin gas at about 1 million degrees (seen using X-rays).

8. b. Dust preferentially scatters blue light and so an object seen through dust appears REDDER than without the dust (eg the setting sun). The other statements are all true.

9. b. Protostars enter the HR diagram from the right hand side - ie they are cool and red (they emerge from cold dense clouds). The other statements are all true.

10. True. Remember, it is the O and B stars which are sufficiently hot to produce the necessary UV radiation to ionize the hydrogen and create an HII region. Most other stars aren't hot enough to do this.

11. False. A star which is gradually using its hydrogen is on the main sequence, where it remains pretty much fixed during most of its life (there is a slight increase in luminosity, but not much). The main sequence does NOT represence an evolutionary track of a single star, but the locus of stars of different masses when they are in the core hydrogen burning phase of their lives.

12. d. The star is becoming a red giant, and hence it is clearly increasing in size and also getting cooler (redder).

13. The full sequence for the sun is : Protostar, main sequence, red giant, horizontal branch, asymptotic giant branch, planetary nebula, white dwarf. The corresponding energy generation is : graviational contraction, hydrogen core burning, hydrogen shell burning, helium core and hydrogen shell burning, helium shell and hydrogen shell burning, no energy production (in planetary nebula and white dwarf stages).

14. Nope, the sun becomes a white dwarf (which is the hot compact carbon/oxygen core) by ejecting the envelope to reveal the core inside. Hence, becoming a white dwarf inherently involves the loss of the stars atmosphere.

15. b. Note that c is wrong because there can be no white dwarf stars with more than 1.4 solar masses (the Chandrasekhar mass), since electron degeneracy pressure fails above this limit. Note that d is wrong because the white dwarfs have NO nuclear reactions occurring, they are simply glowing from their stored heat, slowly cooling down.

16. a. For a cluster of stars, as time passes stars evolve off the main sequence from the top left (O stars) gradually down to the lower right (ie through A,B,F, etc). Hence, turn off at A2 is BEFORE the turnoff of F2, hence the first cluster (A2) is the younger.

17. d. Remember, as the high mass star evolves, it generates an onion shell structure with different elements burning in each shell. Therefore it is NOT very uniform in composition in its center.

18. True. An iron atom's nucleus is the most tightly bound of all the nuclei (it lies at the bottom of the binding energy curve), and hence changing it into either a lighter nucleus (by fission) or a heavier one (by fusion) REQUIRES energy.

19. False. The collapse of the iron core from earth size to Charlottesville size takes less than 0.1 seconds. It is the blast wave that takes an hour or two to reach the surface of the star, when we see it explode.

20. False. Recall that 99% of the gravitational energy released during core collapse emerges as neutrinos; 1% emerges as kinetic energy (the blast wave moving up through the star and ejecting the envelope); and only 0.01% in the form of light -- despite the fact that it is the light that we see and which draws our attention to the supernova explosion.

21. False. For most matter, more stuff fills a larger volume. But for objects supported by degeneracy pressure (either White dwarfs, or neutron stars) more mass crushes the star down more, and it is SMALLER.

22. False. The Crab pulsar is a neutron star, and so it CANNOT have a mass larger than the Oppenheimer-Volkoff limit which is about 2-3 solar masses. Hence 10 solar masses is too high and would be a black hole.

23. d. The first three are all true. Although some (very important) pulsars are in binary systems, most are not -- they were formed from single massive star supernovae.

24. a. Recall the example of the Crab pulsar, for which the luminosity of the Crab nebula is equal to the loss of rotational energy of its central pulsar, which is slowing down at just the rate needed to power the nebula.

25. b. Recall, hydrogen detonation on the white dwarf surface produces a nova explosion, while helium detonation on the surface of a neutron star produces an X-ray burster. Both aquire the matter onto their surface from a companion in a binary star.

26. True. Recall that the Schwarzschild radius is PROPORTIONAL to the mass of the black hole (ie double the mass, double the Schwarzschild radius). Recall also that the constant of proportionality is 3km per 1 solar mass; ie a 1 solar mass black hole has Rs=3km, a 10 solar mass black hole has Rs=30 km etc.

27. b. Recall that close to a black hole time slows down, so frequencies of emitted light run slower, and hence there is a "gravitational redshift".

28. d. Contrary to simple expectation, it is not easy to detect a black hole directly, you need to look at its effects on nearby material. If there is gas nearby, it will heat as it falls into the hole and emit X-rays. Since the region is very small and chaotic, the emission will vary rapidly. This is the most effective way to find black holes.