Questions

Review Questions

1. How did the ancient Greeks explain why the Sun and the Moon slowly change their positions relative to the background stars?

2. In what direction does a planet move relative to the stars when it is in direct motion? When it is in retrograde motion? How do these compare with the direction in which we see the Sun move relative to the stars?

3. (a) In what direction does a planet move relative to the horizon over the course of one night? (b) The answer to (a) is the same whether the planet is in direct motion or retrograde motion. What does this tell you about the speed at which planets move on the celestial sphere?

4. What is an epicycle? How is it important in Ptolemy’s explanation of the retrograde motions of the planets?

5. What is the significance of Occam’s razor as a tool for analyzing theories?

6. How did the models of Aristarchus and Copernicus explain the retrograde motion of the planets?

7. How did Copernicus determine that the orbits of Mercury and Venus must be smaller than Earth’s orbit? How did he determine that the orbits of Mars, Jupiter, and Saturn must be larger than Earth’s orbit?

8. At what configuration (for example, superior conjunction, greatest eastern elongation, and so on) would it be best to observe Mercury or Venus with an Earth-based telescope? At what configuration would it be best to observe Mars, Jupiter, or Saturn? Explain your answers.

9. Is it ever possible to see Mercury at midnight? Explain your answer.

10. Which planets can never be seen at opposition? Which planets can never be seen at inferior conjunction? Explain your answers.

11. What is the difference between the synodic period and the sidereal period of a planet?

12. What is parallax? What did Tycho Brahe conclude from his attempt to measure the parallax of a supernova and a comet?

13. What observations did Tycho Brahe make in an attempt to test the heliocentric model? What were his results? Explain why modern astronomers get different results.

14. What are the foci of an ellipse? If the Sun is at one focus of a planet’s orbit, what is at the other focus?

15. What are Kepler’s three laws? Why are they important?

16. At what point in a planet’s elliptical orbit does it move fastest? At what point does it move slowest? At what point does it sweep out an area at the fastest rate?

17. A line joining the Sun and an asteroid is found to sweep out an area of 6.3 AU² during 2010. How much area is swept out during 2011? Over a period of five years?

18. The orbit of a spacecraft about the Sun has a perihelion distance of 0.1 AU and an aphelion distance of 0.4 AU. What is the semimajor axis of the spacecraft’s orbit? What is its orbital period?

19. A comet with a period of 125 years moves in a highly elongated orbit about the Sun. At perihelion, the comet comes very close to the Sun’s surface. What is the comet’s average distance from the Sun? What is the farthest it can get from the Sun?

20. What observations did Galileo make that reinforced the heliocentric model? Why did these observations contradict the older model of Ptolemy? Why could these observations not have been made before Galileo’s time?

21. Why does Venus have its largest angular diameter when it is new and its smallest angular diameter when it is full?

22. What are Newton’s three laws? Give an everyday example of each law.

23. How much force do you have to exert on a 3-kg brick to give it an acceleration of 2 m/s²? If you double this force, what is the brick’s acceleration? Explain your answer.

24. What is the difference between weight and mass?

25. What is your weight in pounds and in newtons? What is your mass in kilograms?

26. Suppose that Earth were moved to a distance of 3.0 AU from the Sun. How much stronger or weaker would the Sun’s gravitational pull be on Earth? Explain your answer.

27. How far would you have to go from Earth to be completely beyond the pull of its gravity? Explain your answer.

28. A cannonball is shot horizontally from a barrel off a building. Name two forms of energy the cannonball has as it exits the barrel. How do these two forms of energy increase or decrease during the cannonball’s flight? Before the cannonball was fired, where was the energy stored?

29. A satellite is in circular orbit. What two forms of energy are part of the satellite’s orbital energy? Would its orbital energy need to increase or decrease in order to orbit at a larger distance from Earth?

30. If an object loses orbital energy through air drag, is energy still conserved? If so, where does the energy go?

31. Calculate the escape speed for Earth and show that it is 11.2 km/s. (Consult Appendix 2 for planetary data.)

32. Including the effects of Earth’s atmosphere, would you expect the real escape speed for a hypothetical cannonball to be greater or less than 11.2 km/s? Why?

33. What are conic sections? In what way are they related to the orbits of planets in the solar system?

34. Why was the discovery of Neptune an important confirmation of Newton’s law of universal gravitation?

35. What is a tidal force? How do tidal forces produce tides in Earth’s oceans?

36. What is the difference between spring tides and neap tides?

Advanced Questions

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

37. Figure 4-2 shows the retrograde motion of Mars as seen from Earth. Sketch a similar figure that shows how Earth would appear to move against the background of stars during this same time period as seen by an observer on Mars.

38. *The synodic period of Mercury (an inferior planet) is 115.88 days. Calculate its sidereal period in days.

39. *Table 4-1 shows that the synodic period is greater than the sidereal period for Mercury, but the synodic period is less than the sidereal period for Jupiter. Draw diagrams like the one in Box 4-1 to explain why this is so.

40. *A general rule for superior planets is that the greater the average distance from the planet to the Sun, the more frequently that planet will be at opposition. Explain how this rule comes about.

41. In 2006, Mercury was at greatest western elongation on April 8, August 7, and November 25. It was at greatest eastern elongation on February 24, June 20, and October 17. Does Mercury take longer to go from eastern to western elongation, or vice versa? Explain why, using Figure 4-6.

42. Explain why the semimajor axis of a planet’s orbit is equal to the average of the distance from the Sun to the planet at perihelion (the perihelion distance) and the distance from the Sun to the planet at aphelion (the aphelion distance).

43. A certain comet is 2 AU from the Sun at perihelion and 16 AU from the Sun at aphelion. (a) Find the semimajor axis of the comet’s orbit. (b) Find the sidereal period of the orbit.

44. A comet orbits the Sun with a sidereal period of 64.0 years. (a) Find the semimajor axis of the orbit. (b) At aphelion, the comet is 31.5 AU from the Sun. How far is it from the Sun at perihelion?

45. One trajectory that can be used to send spacecraft from Earth to Mars is an elliptical orbit that has the Sun at one focus, its perihelion at Earth, and its aphelion at Mars. The spacecraft is launched from Earth and coasts along this ellipse until it reaches Mars, when a rocket is fired to either put the spacecraft into orbit around Mars or cause it to land on Mars. (a) Find the semimajor axis of the ellipse. (Hint: Draw a picture showing the Sun and the orbits of Earth, Mars, and the spacecraft. Treat the orbits of Earth and Mars as circles.) (b) Calculate how long (in days) such a one-way trip to Mars would take.

46. The mass of the Moon is 7.35 × 10²² kg, while that of Earth is 5.98 × 10²⁴ kg. The average distance from the center of the Moon to the center of Earth is 384,400 km. What is the size of the gravitational force that Earth exerts on the Moon? What is the size of the gravitational force that the Moon exerts on Earth? How do your answers compare with the force between the Sun and Earth calculated in the text?

47. The mass of Saturn is approximately 100 times that of Earth, and the semimajor axis of Saturn’s orbit is approximately 10 AU. To this approximation, how does the gravitational force that the Sun exerts on Saturn compare to the gravitational force that the Sun exerts on Earth? How do the accelerations of Saturn and Earth compare?

48. Suppose that you traveled to a planet with 4 times the mass and 4 times the diameter of Earth. Would you weigh more or less on that planet than on Earth? By what factor?

49. On Earth, a 50-kg astronaut weighs 490 newtons. What would she weigh if she landed on Jupiter’s moon Callisto? What fraction is this of her weight on Earth? See Appendix 3 for relevant data about Callisto.

50. Except for some rare and exotic microbes, the Sun provides energy for life on Earth. Can you describe how energy in sunlight can end up in a kangaroo jumping through the air? In your description, can you name three forms of energy involved, other than sunlight?

51. You’re talking with other students about how the energy of a satellite in circular orbit changes with altitude. One student says that lower orbits have more energy because Kepler’s second law says that closer planets have higher speeds. Another student argues that orbits at larger distances must have more energy because rockets have to expend more energy to launch satellites farther into to space. Which student reaches the right conclusion?

52. Some argue that life might have started on either Mars or Earth and spread to the other planet by microbe-carrying debris ejected during very large asteroid impacts. Consult Appendix 2 for planetary data and calculate the escape speed for both Earth and Mars. Assuming all else being equal, for which planet is it easier for rocks to escape?

53. Imagine a planet like Earth orbiting a star with 4 times the mass of the Sun. If the semimajor axis of the planet’s orbit (a) is 1 AU, what would be the planet’s sidereal period? (Hint: Use Newton’s form of Kepler’s third law. Compared with the case of Earth orbiting the Sun, by what factor has the quantity m₁ + m₂ changed? Has a changed? By what factor must P² change?)

54. A satellite is said to be in a “geosynchronous” orbit if it appears always to remain over the exact same spot on rotating Earth. (a) What is the period of this orbit? (b) At what distance from the center of Earth must such a satellite be placed into orbit? (Hint: Use Newton’s form of Kepler’s third law.) (c) Explain why the orbit must be in the plane of Earth’s equator.

55. Figure 4-23 shows the lunar module Eagle in orbit around the Moon after completing the first successful lunar landing in July 1969. (The photograph was taken from the command module Columbia, in which the astronauts returned to Earth.) The spacecraft orbited 111 km above the surface of the Moon. Calculate the period of the spacecraft’s orbit. See Appendix 3 for relevant data about the Moon.

56. *In Box 4-4 we analyze the orbit of Jupiter’s moon Io. Look up information about the orbits of Jupiter’s three other large moons (Europa, Ganymede, and Callisto) in Appendix 3. Demonstrate that these data are in agreement with Newton’s form of Kepler’s third law.

57. *Suppose a newly discovered asteroid is in a circular orbit with synodic period 1.25 years. The asteroid lies between the orbits of Mars and Jupiter. (a) Find the sidereal period of the orbit. (b) Find the distance from the asteroid to the Sun.

58. The average distance from the Moon to the center of Earth is 384,400 km, and the diameter of Earth is 12,756 km. Calculate the gravitational force that the Moon exerts (a) on a 1-kg rock at the point on Earth’s surface closest to the Moon, and (b) on a 1-kg rock at the point on Earth’s surface farthest from the Moon. (c) Find the difference between the two forces you calculated in parts (a) and (b). This difference is the tidal force pulling these two rocks away from each other, like the 1-ball and 3-ball in Figure 4-26. Explain why tidal forces cause only a very small deformation of Earth.

Discussion Questions

59. Which planet would you expect to exhibit the greatest variation in apparent brightness as seen from Earth? Which planet would you expect to exhibit the greatest variation in angular diameter? Explain your answers.

60. Use two thumbtacks, a loop of string, and a pencil to draw several ellipses. Describe how the shape of an ellipse varies as the distance between the thumbtacks changes.

Web/eBook Questions

61. (a) Search the World Wide Web for information about Kepler. Before he realized that the planets move on elliptical paths, what other models of planetary motion did he consider? What was Kepler’s idea of “the music of the spheres”? (b) Search the World Wide Web for information about Galileo. What were his contributions to physics? Which of Galileo’s new ideas were later used by Newton to construct his laws of motion? (c) Search the World Wide Web for information about Newton. What were some of the contributions that he made to physics other than developing his laws of motion? What contributions did he make to mathematics?

62. Monitoring the Retrograde Motion of Mars. Watching Mars night after night reveals that it changes its position with respect to the background stars. To track its motion, access and view the animation “The Path of Mars in 2016” in Chapter 4 of the Universe Web site or eBook. (a) Through which constellations does Mars move? (b) On approximately what date does Mars stop its direct (west-to-east) motion and begin its retrograde motion? (Hint: Use the “Stop” function on your animation controls.) (c) Over how many days does Mars move retrograde?