40. Estimate what neutrino rest mass (in eV) would provide

1. (I) Using the definitions of the parsec and the light-year, show that 1 pc = 3.26 ly.

2. (I) A star exhibits a parallax of 0.38 seconds of arc. How far away is it?

3. (I) The parallax angle of a star is 0.00019. How far away is the star?

4. (1) A star is 36 pc away. What is its parallax angle? State (a) in seconds of arc, and (b) in degrees.

5. (I) What is the parallax angle for a star that is 55 ly away? How many parsecs is this?

6. (I) If one star is twice as far away from us as a second star, will the parallax angle of the farther star be greater or less than that of the nearer star? By what factor?

7. (II) A star is 35 pc away. How long does it take for its light to reach us?

8. (II) We saw earlier (Chapter 14) that the rate energy-reaches the Earth from the Sun (the solar constant) is about 1.3 X 103W/m2. What is (a) the apparent brightness / of the Sun. and (b) the absolute luminosity L of the Sun?

9. (II) What is the relative brightness of the Sun as seen from Jupiter as compared to its brightness from Earth? (Jupiter is 5.2 times farther from the Sun than the Earth.)

10. (II) Estimate the angular width that our Galaxy would subtend if observed from the nearest galaxy to us (Table 33-1). Compare to the angular width of the Moon from Earth.

11. (II) When our Sun becomes a red giant, what will be its average density if it expands out to the orbit of Earth (1.5 X 10" m from the Sun)?

12. (II) When our Sun becomes a white dwarf, it is expected to be about the size of the Moon. What angular width will it subtend from the present distance to Earth?

13. (II) Calculate the density of a white dwarf whose mass is equal to the Suns and whose radius is equal to the Earths. How many times larger than Earth's density is this?

14. (II) A neutron star whose mass is 1.5 solar masses has a radius of about 11 km. Calculate its average density and compare to that for a white dwarf (Problem 13) and to that of nuclear matter.

15. (II) Calculate the (2-values for the He burning reactions of F.q. 33-2. (The mass of the very unstable $Bc is 8.005305 u.)

16. (II) Suppose two stars of the same apparent brightness / are also believed to be the same size. The spectrum of one star peaks at 800 nm whereas that of the other peaks at 400 nm. Use Wiens law (Section 27-2) and the Stefan-Boltzmann equation (Eq. 14-5) to estimate their relative distances from us. [Hint: see Examples 33-4 and 33-5.]

17. (Ill) Stars located in a certain cluster are assumed to be about the same distance from us. Two such stars have spectra that peak at A| = 500 nm and A2 = 700 nm. and the ratio of their apparent brightness is l\/l2 = 0.091. Estimate their relative sizes (give ratio of their diameters). [Him: use the Stefan-Boltzmann equation. Eq. 14-5.]

18. (I) Show that the Schwarzschild radius for a star with mass equal to that (a) of our Sun is 2.95 km. and (b) of Earth is 8.9 mm.

19. (II) What is the Schwarzschild radius for a typical galaxy (like ours)?

20. (II) Describe a triangle, drawn on the surface of a sphere, for which the sum of the angles is (a) 359. and (b) 180.

21. (II) What is the maximum sum-of-the-angles for a triangle on a sphere?

22. (I) If a galaxy is traveling away from us at 1.0% of the speed of light, roughly how far away is it?

23. (I) The redshift of a galaxy indicates a velocity of 3500 km/s. How far away is it?

24. (I) Estimate the speed of a galaxy (relative to us) that is near the observable "edge" of the universe, say 12 billion light-years away.

25. (II) Estimate the observed wavelength for the 656-nm line in the Balmer series of hydrogen in the spectrum of a galaxy whose distance from us is (a) 1.0 X 106ly. (b) 1.0 X 108 ly. (c) 1.0 X 10loly.

26. (II) Estimate the speed of a galaxy, and its distance from us. if the wavelength for the hydrogen line at 434 nm is measured on Earth as being 610 nm.

27. (II) What is the speed of a galaxy with z = 0.60?

28. (II) What would be the redshift parameter z for a galaxy-traveling away from us at v = 0.50c?

29. (II) Starting from Eq. 33-3, show that the Doppler shift in wavelength is AA/A<> % v/c (Eq. 33-5b) for v c. [Him: use the binomial expansion.]

30. (I) Calculate the wavelength at the peak of the blackbody radiation distribution at 2.7 K using Wiens law.

31. (II) The critical density for closure of the universe is pc % 10-26 kg/m3. State pc in terms of the average number of nucleons per cubic meter.

32. (II) The scale of the universe (the average distance between galaxies) at any one moment is believed to have been inversely proportional to the absolute temperature. Estimate the size of the universe, compared to today, at (<?) 1 = 106yr, (/>) t = 1 s, (c) t = 106s. and (d)l = 10~35 s.

33. (II) At approximately what time had the universe cooled below the threshold temperature for producing (a) kaons (M ^ 500 Me V/c2), (b) Y (A/* 9500 Me V/c2). and (c) muons (A/ = 100MeV/c2)?

34. Suppose that three main-sequence stars could undergo the three changes represented by the three arrows. A. B. and C, in the H-R diagram of Fig. 33-28. For each case, describe the changes in temperature, luminosity, and size.

35. Assume that the nearest stars to us have an absolute luminosity about the same as the Sun's. Their apparent brightness, however, is about 1011 times fainter than the Sun. From this, estimate the distance to the nearest stars. (Newton did this calculation, although he made a numerical error of a factor of 100.)

36. Use conservation of angular momentum to estimate the angular velocity of a neutron star which has collapsed to a diameter of 20 km. from a star whose radius was equal to that of our Sun (7 x 108m), of mass 1.5 times that of the Sun. and which rotated (like our Sun) about once a month.

37. By what factor does the rotational kinetic energy change when the star in Problem 36 collapses to a neutron star?

38. A certain pulsar, believed to be a neutron star of mass 1.5 times that of the Sun. with diameter 20 km. is observed to have a rotation speed of 1.0 rev/s. If it loses rotational kinetic energy at the rate of 1 part in 109 per day, which is all transformed into radiation, what is the power output of the star ?

39. The nearest large galaxy to our Galaxy is about 2 X 106 ly away. If both galaxies have a mass of 3 x 1041 kg, with what gravitational force does each galaxy attract the other?

40. Estimate what neutrino rest mass (in eV) would provide the critical density to close the universe. Assume the neutrino density is, like photons, about 109 times that of nucleons, and that nucleons make up only (a) 2% of the mass needed, or (/>) 5% of the mass needed.

41. Two stars, whose spectra peak at 600 nm and 400 nm. respectively, both lie on the main sequence. Use Wiens law. the Stefan-Boltzmann equation, and the H-R diagram (Fig. 33-6) to estimate the ratio of their diameters. [Him: sec Examples 33-4 and 33-5.]

42. Suppose we can measure distances with parallax at 100 parsecs. What is our minimum angular resolution (in degrees), based on this information?

43. Through some coincidence, the Balmer lines from singly ionized helium in a distant star happen to overlap with the Balmer lines from hydrogen (Fig. 27-22) in the Sun. How fast is the star receding from us?

44. What is the temperature that corresponds to 1.8-TeV collisions at the Fermilab collider? To what era in cosmological history does this correspond? [Hint, see Fig. 33-25.]

45. Astronomers have recently measured the rotation of gas around what might be a supermassive black hole of about 2 billion solar masses at the center of a galaxy. If the radius from the galactic center to the gas clouds is 60 light-years, what Doppler shift AA/A,, do you estimate they saw?

46. A galaxy is moving away from Earth. The blue hydrogen line at 434 nm emitted from the galaxy is measured on Earth to be 650 nm. (a) How fast is the galaxy moving? (b) How far is it from Earth?

47. In the later stages of stellar evolution, a star (if massive enough) will begin fusing carbon nuclei to form, for example, magnesium: sC + 'lc - ftMg + y. (a) How much energy is released in this reaction (see Appendix B). (b) How much kinetic energy must each carbon nucleus have (assume equal) in a head-on collision if they are just to touch (use Eq. 301) so that the strong force can come into play? (c) What temperature does this kinetic energy correspond to?

48. Consider the reaction gO + *gO -> gSi + }He. and answer the same questions as in Problem 47.

49. How large would the Sun be if its density equaled the critical density of the universe. pc 10- 26kg/m3? Express your answer in light-years and compare with the Earth-Sun distance and the size of our Galaxy.