Questions

Review Questions

1. Explain the difference between a star’s apparent brightness and its luminosity.

2. Describe how the parallax method of finding a star’s distance is similar to binocular (two-eye) vision in humans.

3. Why does it take at least six months to make a measurement of a star’s parallax?

4. Why are measurements of stellar parallax difficult to make? What are the advantages of making these measurements from orbit?

5. What is the inverse-square law? Use it to explain why an ordinary lightbulb can appear brighter than a star, even though the lightbulb emits far less light energy per second.

6. Briefly describe how you would determine the luminosity of a nearby star. Of what value is knowing the luminosity of various stars?

7. Which are more common, stars more luminous than the Sun or stars less luminous than the Sun?

8. Why is the magnitude scale called a “backward” scale? What is the difference between apparent magnitude and absolute magnitude?

9. The star Zubenelgenubi (from the Arabic for “scorpion’s southern claw”) has apparent magnitude +2.75, while the star Sulafat (Arabic for “tortoise”) has apparent magnitude +3.25. Which star appears brighter? From this information alone, what can you conclude about the luminosities of these stars? Explain your answer.

10. Explain why the color ratios of a star are related to the star’s surface temperature.

11. Would it be possible for a star to appear bright when viewed through a U filter or a V filter, but dim when viewed through a B filter? Explain your answer.

12. Which gives a more accurate measure of a star’s surface temperature, its color ratios or its spectral lines? Explain.

13. Menkalinan (Arabic for “shoulder of the rein-holder”) is an A2 star in the constellation Auriga (the Charioteer). What is its spectral class? What is its spectral type? Which gives a more precise description of the spectrum of Menkalinan?

14. What are the most prominent absorption lines you would expect to find in the spectrum of a star with a surface temperature of (a) 35,000 K, (b) 2800 K, and (c) 5800 K (like the Sun)? Briefly describe why these stars have such different spectra even though they have essentially the same chemical composition.

15. A fellow student expresses the opinion that since the Sun’s spectrum has only weak absorption lines of hydrogen, this element cannot be a major constituent of the Sun. How would you enlighten this person?

16. If a red star and a blue star both have the same radius and both are the same distance from Earth, which one looks brighter in the night sky? Explain why.

17. If a red star and a blue star both appear equally bright and both are the same distance from Earth, which one has the larger radius? Explain why.

18. If a red star and a blue star both have the same radius and both appear equally bright, which one is farther from Earth? Explain why.

19. Sketch a Hertzsprung-Russell diagram. Indicate the regions on your diagram occupied by (a) main-sequence stars, (b) red giants, (c) supergiants, (d) white dwarfs, and (e) the Sun.

20. Most of the bright stars in the night sky (see Appendix 5) are giants and supergiants. How can this be, if giants and supergiants make up only 1% of the population of stars?

21. Explain why the dashed lines in Figure 17-15b slope down and to the right.

22. Some giant and supergiant stars are of the same spectral type (G2) as the Sun. What aspects of the spectrum of a G2 star would you concentrate on to determine the star’s luminosity class? Explain what you would look for.

23. Briefly describe how you would determine the distance to a star whose parallax is too small to measure.

24. What information about stars do astronomers learn from binary systems that cannot be learned in any other way? What measurements do they make of binary systems to garner this information?

25. Suppose that you want to determine the temperature, diameter, and luminosity of an isolated star (not a member of a binary system). Which of these physical quantities require you to know the distance to the star? Explain your answer.

26. What is the mass-luminosity relation? Does it apply to stars of all kinds?

27. Use Figure 17-22 to (a) estimate the mass of a main-sequence star that is 1000 times as luminous as the Sun, and (b) estimate the luminosity of a main-sequence star whose mass is one-fifth that of the Sun. Explain your answers.

28. Which is more massive, a red main-sequence star or a blue main-sequence star? Which has the greater radius? Explain your answers.

29. How do white dwarfs differ from brown dwarfs? Which are more massive? Which are larger in radius? Which are denser?

30. Sketch the radial velocity curves of a binary consisting of two identical stars moving in circular orbits that are (a) perpendicular to our line of sight and (b) parallel to our line of sight.

31. Give two reasons why a visual binary star is unlikely to also be a spectroscopic binary star.

32. Sketch the light curve of an eclipsing binary consisting of two identical stars in highly elongated orbits oriented so that (a) their major axes are pointed toward Earth and (b) their major axes are perpendicular to our line of sight.

Advanced Questions

Questions preceded by an asterisk (*) involve the topic discussed in the Boxes.

33. Find the average distance from the Sun to Neptune in parsecs. Compared to Neptune, how many times farther away from the Sun is Proxima Centauri?

34. Suppose that a dim star were located 2 million AU from the Sun. Find (a) the distance to the star in parsecs and (b) the parallax angle of the star. Would this angle be measurable with present-day techniques?

35. The star GJ 1156 has a parallax angle of 0.153 arcsec. How far away is the star?

36. *Kapteyn’s star (named after the Dutch astronomer who found it) has a parallax of 0.255 arcsec, a proper motion of 8.67 arcsec per year, and a radial velocity of +246 km/s. (a) What is the star’s tangential velocity? (b) What is the star’s actual speed relative to the Sun? (c) Is Kapteyn’s star moving toward the Sun or away from the Sun? Explain.

37. *How far away is a star that has a proper motion of 0.08 arcseconds per year and a tangential velocity of 40 km/s? For a star at this distance, what would its tangential velocity have to be in order for it to exhibit the same proper motion as Barnard’s star (see Box 17-1)?

38. *The space velocity of a certain star is 120 km/s and its radial velocity is 72 km/s. Find the star’s tangential velocity.

39. *In the spectrum of a particular star, the Balmer line H_α has a wavelength of 656.15 nm. The laboratory value for the wavelength of H_α is 656.28 nm. (a) Find the star’s radial velocity. (b) Is this star approaching us or moving away? Explain your answer. (c) Find the wavelength at which you would expect to find H_α in the spectrum of this star, given that the laboratory wavelength of H_α is 486.13 nm. (d) Do your answers depend on the distance from the Sun to this star? Why or why not?

40. *Derive the equation given in Box 17-1 relating proper motion and tangential velocity. (Hint: See Box 1-1.)

41. How much dimmer does the Sun appear from Neptune than from Earth? (Hint: The average distance between a planet and the Sun equals the semimajor axis of the planet’s orbit.)

42. Stars A and B are both equally bright as seen from Earth, but A is 120 pc away while B is 24 pc away. Which star has the greater luminosity? How many times greater is it?

43. Stars C and D both have the same luminosity, but C is 32 pc from Earth while D is 128 pc from Earth. Which star appears brighter as seen from Earth? How many times brighter is it?

44. Suppose two stars have the same apparent brightness, but one star is 8 times farther away than the other. What is the ratio of their luminosities? Which one is more luminous, the closer star or the farther star?

45. The solar constant, equal to 1370 W/m², is the amount of light energy from the Sun that falls on 1 square meter of Earth’s surface in 1 second (see Section 17-2). What would the distance between Earth and the Sun have to be in order for the solar constant to be 1 watt per square meter (1 W/m²)?

46. The star Procyon in Canis Minor (the Small Dog) is a prominent star in the winter sky, with an apparent brightness 1.3 × 10⁻¹¹ that of the Sun. It is also one of the nearest stars, being only 3.50 pc from Earth. What is the luminosity of Procyon? Express your answer as a multiple of the Sun’s luminosity.

47. The star HIP 92403 (also called Ross 154) is only 2.97 pc from Earth but can be seen only with a telescope, because it is 60 times dimmer than the dimmest star visible to the unaided eye. How close to us would this star have to be in order for it to be visible without a telescope? Give your answer in parsecs and in AU. Compare with the semimajor axis of Pluto’s orbit around the Sun.

48. *The star HIP 72509 has an apparent magnitude of +12.1 and a parallax angle of 0.222 arcsecond. (a) Determine its absolute magnitude. (b) Find the approximate ratio of the luminosity of HIP 72509 to the Sun’s luminosity.

49. *Suppose you can just barely see a twelfth-magnitude star through an amateur’s 6-inch telescope. What is the magnitude of the dimmest star you could see through a 60-inch telescope?

50. *A certain type of variable star is known to have an average absolute magnitude of 0.0. Such stars are observed in a particular star cluster to have an average apparent magnitude of +14.0. What is the distance to that star cluster?

51. *(a) Find the absolute magnitudes of the brightest and dimmest of the labeled stars in Figure 17-6b. Assume that all of these stars are 110 pc from Earth. (b) If a star in the Pleiades cluster is just bright enough to be seen from Earth with the naked eye, what is its absolute magnitude? Is such a star more or less luminous than the Sun? Explain.

52. (a) On a copy of Figure 17-8, sketch the intensity curve for a blackbody at a temperature of 3000 K. Note that this figure shows a smaller wavelength range than Figure 17-7a. (b) Repeat part (a) for a blackbody at 12,000 K (see Figure 17-7c). (c) Use your sketches from parts (a) and (b) to explain why the color ratios b_V/b_B and b_B/b_U are less than 1 for very hot stars but greater than 1 for very cool stars.

53. *Astronomers usually express a star’s color using apparent magnitudes. The star’s apparent magnitude as viewed through a B filter is called m_B, and its apparent magnitude as viewed through a V filter is m_V. The difference m_B − m_V is called the B–V color index (“B minus V”). Is the B–V color index positive or negative for very hot stars? What about very cool stars? Explain your answers.

54. *(See Question 53.) The B–V color index is related to the color ratio b_V/b_B by the equation
    (a) Explain why this equation is correct. (b) Use the data in Table 17-1 to calculate the B–V color indices for Bellatrix, the Sun, and Betelgeuse. From your results, describe a simple rule that relates the value of the B–V color index to a star’s color.

55. The bright star Rigel in the constellation Orion has a surface temperature about 1.6 times that of the Sun. Its luminosity is about 64,000 L_⊙. What is Rigel’s radius compared to the radius of the Sun?

56. (See Figure 17-12.) What temperature and spectral classification would you give to a star with equal line strengths of hydrogen (H) and neutral helium (He I)? Explain your answer.

57. The Sun’s surface temperature is 5800 K. Using Figure 17-12, arrange the following absorption lines in the Sun’s spectrum from the strongest to the weakest, and explain your reasoning: (i) neutral calcium; (ii) singly ionized calcium; (iii) neutral iron; (iv) singly ionized iron.

58. Star P has one-half the radius of star Q. Stars P and Q have surface temperatures 4000 K and 8000 K, respectively. Which star has the greater luminosity? How many times greater is it?

59. Star X has 12 times the luminosity of star Y. Stars X and Y have surface temperatures 3500 K and 7800 K, respectively. Which star has the larger radius? How many times larger is it?

60. Suppose a star experiences an outburst in which its surface temperature doubles but its average density (its mass divided by its volume) decreases by a factor of 8. The mass of the star stays the same. By what factors do the star’s radius and luminosity change?

61. The Sun experiences solar flares (see Section 16-10). The amount of energy radiated by even the strongest solar flare is not enough to have an appreciable effect on the Sun’s luminosity. But when a flare of the same size occurs on a main-sequence star of spectral class M, the star’s brightness can increase by as much as a factor of 2. Why should there be an appreciable increase in brightness for a main-sequence M star but not for the Sun?

62. The bright star Zubeneschmali (β Librae) is of spectral type B8 and has a luminosity of 130 L_⊙. What is the star’s approximate surface temperature? How does its radius compare to that of the Sun?

63. Castor (α Geminorum) is an A1 V star with an apparent brightness of 4.4 × 10⁻¹² that of the Sun. Determine the approximate distance from Earth to Castor (in parsecs).

64. A brown dwarf called CoD–33°7795 B has a luminosity of 0.0025L_⊙. It has a relatively high surface temperature of 2550 K, which suggests that it is very young and has not yet had time to cool down by emitting radiation. (a) What is this brown dwarf’s spectral class? (b) Find the radius of CoD–33°7795 B. Express your answer in terms of the Sun’s radius and in kilometers. How does this compare to the radius of Jupiter? Is the name “dwarf” justified?

65. The star HD 3651 shown in Figure 17-13 has a mass of 0.79 M_⊙. Its brown dwarf companion, HD 3651B, has about 40 times the mass of Jupiter. The average distance between the two stars is about 480 AU. How long does it take the two stars to complete one orbit around each other?

66. The visual binary 70 Ophiuchi (see the accompanying figure) has a period of 87.7 years. The parallax of 70 Ophiuchi is 0.2 arcsec, and the apparent length of the semimajor axis as seen through a telescope is 4.5 arcsec. (a) What is the distance to 70 Ophiuchi in parsecs? (b) What is the actual length of the semimajor axis in AU? (c) What is the sum of the masses of the two stars? Give your answer in solar masses.

    

67. An astronomer observing a binary star finds that one of the stars orbits the other once every 5 years at a distance of 10 AU. (a) Find the sum of the masses of the two stars. (b) If the mass ratio of the system is M₁/M₂ = 0.25, find the individual masses of the stars. Give your answers in terms of the mass of the Sun.

Discussion Questions

68. From its orbit around Earth, the Hipparcos satellite could measure stellar parallax angles with acceptable accuracy only if the angles were larger than about 0.002 arcsec. Discuss the advantages or disadvantages of making parallax measurements from a satellite in a large solar orbit, say at the distance of Jupiter from the Sun. If this satellite can also measure parallax angles of 0.002 arcsec, what is the distance of the most remote stars that can be accurately determined? How much bigger a volume of space would be covered compared to Earth-based observations? How many more stars would you expect to be contained in that volume?

69. *As seen from the starship Enterprise in the Star Trek television series and movies, stars appear to move across the sky due to the starship’s motion. How fast would the Enterprise have to move in order for a star 1 pc away to appear to move 1° per second? (Hint: The speed of the star as seen from the Enterprise is the same as the speed of the Enterprise relative to the star.) How does this compare with the speed of light? Do you think the stars appear to move as seen from an orbiting space shuttle, which moves at about 8 km/s?

70. It is desirable to be able to measure the radial velocity of stars (using the Doppler effect) to an accuracy of 1 km/s or better. One complication is that radial velocities refer to the motion of the star relative to the Sun, while the observations are made using a telescope on Earth. Is it important to take into account the motion of Earth around the Sun? Is it important to take into account Earth’s rotational motion? To answer this question, you will have to calculate Earth’s orbital speed and the speed of a point on Earth’s equator (the part of Earth’s surface that moves at the greatest speed because of the planet’s rotation). If one or both of these effects are of importance, how do you suppose astronomers compensate for them?

Web/eBook Questions

71. Search the World Wide Web for information about Gaia, a European Space Agency (ESA) spacecraft planned to extend the work carried out by Hipparcos. When is the spacecraft planned to be launched? How will Gaia compare to Hipparcos? For how many more stars will it be able to measure parallaxes? What other types of research will it carry out?

72. Search the World Wide Web for recent discoveries about brown dwarfs. Are all brown dwarfs found orbiting normal stars, or are they also found orbiting other brown dwarfs? Are any found in isolation (that is, not part of a binary system)? The Sun experiences flares (see Section 16-10), as do other normal stars; is there any evidence that brown dwarfs also experience flares? If so, is there anything unusual about these flares? What is the lowest temperature for a brown dwarf found so far?