SUMMARY

Geocentric models of the universe were based on the assumption that the Sun, the Moon, and the planets all orbit Earth. The most successful and long-lived of these was the Ptolemaic model. Unlike the Sun and the Moon, planets sometimes appear to temporarily reverse their direction of motion (from night to night) relative to the stars and then resume their normal "forward" course. This phenomenon is called retrograde motion.To account for retrograde motion within the geocentric picture, it was necessary to suppose that planets moved on small circles called epicycles, whose centers orbited Earth on larger circles called deferents. The heliocentric view of the solar system holds that Earth, like all the planets, orbits the Sun. This model accounts for retrograde motion and the observed size and brightness variations of the planets in a much more straightforward way than the geocentric model. The widespread realization during the Renaissance that the solar system is Sun centered, and not Earth centered, is known as the Copernican revolution, in honor of Nicholas Copernicus, who laid the foundations of the modern heliocentric model.

Galileo Galilei is often regarded as the father of experimental science. His telescopic observations of the Moon, the Sun, Venus, and Jupiter played a crucial role in supporting and strengthening the Copernican picture of the solar system. Johannes Kepler improved on Copernicus’s model with his three laws of planetary motion: (1) Planetary orbits are ellipses, with the Sun at one focus. (2) A planet moves faster as its orbit takes it closer to the Sun. (3) The semi-major axis of the orbit is related in a simple way to the planet’s orbit period. Most planets move on orbits whose eccentricities are quite small, so their paths differ only slightly from perfect circles. The distance from Earth to the Sun is called the astronomical unit. Nowadays, the astronomical unit is determined by bouncing radar signals off the planet Venus and measuring the time the signal takes to return.

Isaac Newton succeeded in explaining Kepler’s laws in terms of a few general physical principles, now known as Newtonian mechanics. The tendency of a body to keep moving at constant velocity is called inertia. The greater the body’s mass, the greater its inertia. To change the velocity, a force must be applied. The rate of change of velocity, called acceleration, is equal to the applied force divided by the body’s mass. To explain planetary orbits, Newton postulated that gravity attracts the planets to the Sun. Every object with any mass is surrounded by a gravitational field, whose strength decreases with distance according to an inverse-square law. This field determines the gravitational force exerted by the object on any other body in the universe. Newton’s laws imply that a planet does not orbit the precise center of the Sun, but instead that both the planet and the Sun orbit the common center of mass of the two bodies. For an object to escape from the gravitational pull of another, its speed must exceed the escape speed of the second body. In this case, the motion is said to be unbound, and the orbital path is no longer an ellipse, although it is still described by Newton’s laws.



SELF-TEST: TRUE OR FALSE?

1. Aristotle was first to propose that all planets revolve around the Sun. HINT

2. The teachings of Aristotle remained unchallenged until the eighteenth century A.D. HINT

3. Ptolemy was responsible for a geocentric model that was successful at predicting the positions of the planets, Moon, and the Sun. HINT

4. The heliocentric model of the universe holds that Earth is at the center, and everything else moves around it. HINT

5. Kepler’s discoveries regarding the orbital motion of the planets were based mainly on observations made by Copernicus. HINT

6. The Sun’s location in a planet’s orbit is at the center. HINT

7. The semi-major axis of an orbit is half the major axis. HINT

8. A circle has an eccentricity of zero. HINT

9. The astronomical unit is a distance equal to the semi-major axis of Earth’s orbit around the Sun. HINT

10. The speed of a planet orbiting the Sun is independent of the planet’s position in its orbit. HINT

11. Kepler’s laws work for only the six planets known in his time. HINT

12. Kepler never knew the true distances between the planets and the Sun, only their relative distances. HINT

13. Galileo’s observations of the sky were made with the naked eye. HINT

14. Using his laws of motion and gravity, Newton was able to prove Kepler’s laws. HINT

15. You throw a baseball to someone; before the ball is caught, it is temporarily in orbit around Earth’s center. HINT



SELF-TEST: FILL IN THE BLANK

1. Stonehenge was used as a _____ by people in the Stone Age. HINT

2. Accurate records of comets and "guest" stars were kept over many centuries by _____ astrologers. HINT

3. The astronomical knowledge of ancient Greece was kept alive and augmented by _____ astronomers. HINT

4. The apparent "backward" (westward motion) of the planets Mars, Jupiter, or Saturn in the sky relative to the stars is known as _____ motion. HINT

5. Observation, theory, and testing are the cornerstones of the _____.HINT

6. The heliocentric model was reinvented by _____.HINT

7. Central to the heliocentric model is the assertion that the observed motions of the planets and the Sun are the result of _____ motion around the Sun. HINT

8. Kepler’s laws were based on observational data obtained by _____.HINT

9. Kepler discovered that the shape of an orbit is an _____ , not a _____, as had previously been believed. HINT

10. Kepler’s third law relates the _____ of the orbital period to the _____ of a planet’s semi-major axis. HINT

11. Galileo discovered _____ orbiting Jupiter, the _____ of Venus, and the Sun’s rotation from observations of _____. HINT

12. The modern method of measuring the astronomical unit uses _____ measurements of a planet or asteroid. HINT

13. Newton’s first law states that a moving object will continue to move in a straight line with constant speed unless acted upon by a _____.HINT

14. Newton’s law of gravity states that the gravitational force between two objects depends on the _____ of their masses and inversely on the _____ of their separation. HINT

15. Newton discovered that, in Kepler’s third law, the orbital period depends on the semi-major axis and on the sum of the _____ of the two objects involved. HINT



REVIEW AND DISCUSSION

1. What contributions to modern astronomy were made by Chinese and Islamic astronomers during the Dark Ages of medieval Europe? HINT

2. Briefly describe the geocentric model of the universe. HINT

3. The benefit of our current knowledge lets us see flaws in the Ptolemaic model of the universe. What is its basic flaw? HINT

4. What was the great contribution of Copernicus to our knowledge of the solar system? What was still a flaw in the Copernican model? HINT

5. What is a theory? Can a theory ever be proved to be true? HINT

6. When were Copernicus’s ideas finally accepted? HINT

7. What is the Copernican principle? HINT

8. What discoveries of Galileo helped confirm the views of Copernicus, and how? HINT

9. Briefly describe Kepler’s three laws of planetary motion. HINT

10. How did Tycho Brahe contribute to Kepler’s laws? HINT

11. If radio waves cannot be reflected from the Sun, how can radar be used to find the distance from Earth to the Sun? HINT

12. How did astronomers determine the scale of the solar system prior to the invention of radar? HINT

13. What does it mean to say that Kepler’s laws are empirical? HINT

14. What are Newton’s laws of motion and gravity? HINT

15. List the two modifications made by Newton to Kepler’s laws. HINT

16. Why do we say that a baseball falls toward Earth, and not Earth toward the baseball? HINT

17. Why would a baseball go higher if it were thrown up from the surface of the Moon than if it were thrown with the same velocity from the surface of Earth? HINT

18. In what sense is the Moon falling toward Earth? HINT

19. What is the meaning of the term escape speed? HINT

20. What would happen to Earth if the Sun’s gravity were suddenly "turned off"? HINT



PROBLEMS Algorithmic versions of these questions are available in the Practice Problems Module of the Companion Website.

The number of squares preceding each problem indicate's the approximate level of difficulty.

1. Tycho Brahe’s observations of the stars and planets were accurate to about one arc minute (1’). To what distance does this angle correspond at the distance of (a) the Moon; (b) the Sun; and (c) Saturn (at closest approach)? 000 km from Earth at closest approach; or (e) the nearest star, about three lightyears from Earth. Which is correct? HINT

2. To an observer on Earth, through what angle will Mars appear to move relative to the stars over the course of 24 hours, when the two planets are at closest approach? Assume for simplicity that Earth and Mars move on circular orbits of radii 1.0 A.U. and 1.5 A.U., respectively, in exactly the same plane. Will the apparent motion be prograde or retrograde? HINT

3. Using the data in Table 2.1, show that Pluto is closer to the Sun at perihelion (the point of closest approach to the Sun in its orbit) than Neptune is at any point in its orbit. HINT

4. An asteroid has a perihelion distance of 2.0 A.U. and an aphelion distance of 4.0 A.U. Calculate its orbital semi-major axis, eccentricity, and period. HINT

5. A spacecraft has an orbit that just grazes Earth’s orbit at aphelion and just grazes Venus’s orbit at perihelion. Assuming that Earth and Venus are in the right places at the right times, how long will the spacecraft take to travel from Earth to Venus? HINT

6. Halley’s comet has a perihelion distance of 0.6 A.U. and an orbital period of 76 years. What is its aphelion distance from the Sun? HINT

7. What is the maximum possible parallax of Mercury during a solar transit, as seen from either end of a 3000-km baseline on Earth? HINT

8. How long would a radar signal take to complete a round trip between Earth and Mars when the two planets are 0.7 A.U. apart? HINT

9. Jupiter’s moon Callisto orbits the planet at a distance of 1.88 million km. Callisto’s orbital period about Jupiter is 16.7 days. What is the mass of Jupiter? [Assume that Callisto’s mass is negligible compared with that of Jupiter, and use the modified version of Kepler’s third law (Section 2.7).] HINT

10. The Sun moves in a roughly circular orbit around the center of the Milky Way Galaxy, at a distance of 26,000 light- years. The orbit speed is approximately 220 km/s. Calculate the Sun’s orbital period and centripetal acceleration, and use these numbers to estimate the mass of our galaxy. HINT

11. At what distance from the Sun would a planet’s orbital period be one million years? What would be the orbital period at a distance of one light-year? HINT

12. The acceleration due to gravity at Earth’s surface is 9.80 m/s2. What is the gravitational acceleration at altitudes of (a) 100 km; (b) 1000 km; (c) 10,000 km? Take Earth’s radius to be 6400 km. HINT

13. What would be the speed of a spacecraft moving in a circular orbit at each of the three altitudes listed in the previous problem? In each case, how does the centripetal acceleration (More Precisely 2-2) compare with the gravitational acceleration? HINT

14. Use Newton’s law of gravity to calculate the force of gravity between you and Earth. Convert your answer, which will be in newtons, to pounds using the conversion 4.45 N equals one pound. What do you normally call this force? HINT

15. The Moon’s mass is 7.4 1022 kg and its radius is 1700 km. What is the speed of a spacecraft moving in a circular orbit just above the lunar surface? What is the escape speed from the Moon? HINT



COLLABORATIVE EXERCISES

1. Tourist Attraction or Sacred Ground. Your group has been asked to arbitrate a dispute between a tour bus company and a nearby Native American tribe. The dispute surrounds an ancient medicine wheel recently discovered by a team of university archeologists, Using sketches as necessary, compose a legal brief that describes what a medicine wheel is designed to do astronomically and summarize the opposing positions of the two groups.

2. Galileo's Observations. Your group should select what it believes to be Galileo's single most important astronomical observation, why it was most important, and explain what he observed using sketches.

3. Binary Star Observing Proposal. Write a one-page proposal for "telescope-time" to determine the total mass of stars in a binary star system. Explain what a binary star system is and describe exactly which measurement each member in your group will need to make in order to completely determine the star system's total mass.

4. Galileo's Dialogue. Galileo's Dialogue Concerning the Two World Chief Systems described fictional conversations between three people. Create a short play using this style describing Kepler's Laws of Planetary Motion using each person in your group.



RESEARCHING THE WEB To complete the following exercises, go to the online Destinations Module for Chapter 2 on the Companion Website for Astronomy Today 4/e.

1. Access the "Kepler’s Laws Java Orbit Simulator" and, while holding a piece of paper over the screen, trace the orbits and show that this particular simulator is or is not a valid depiction of Kepler’s second law by determining the area swept out at perigee and apogee.

2. Access the "Keplerian Elements Tutorial" and describe the difference between a satellite’s orbital inclination and its argument of perigee.

3. Access the "Laws of Science Abused by Science Fiction" page and list the five most commonly misused physics laws in science fiction.



PROJECTS

1. Look in an almanac for the date of opposition of one or all of these bright planets: Mars, Jupiter, and Saturn. At opposition, these planets are at their closest points to Earth and are at their largest and brightest in the night sky. Observe these planets. How long before opposition does each planet’s retrograde motion begin? How long afterward does it end?

2. Draw an ellipse. (See Figure 2.13) You’ll need two pins, a piece of string, and a pencil. Tie the string in a loop and place it around the pins. Place the pencil inside the loop and run it around the inside of the string, holding the loop taut. The two pins will be at the foci of the ellipse. What is the eccentricity of the ellipse you have drawn?

3. Use a small telescope to replicate Galileo’s observations of Jupiter’s four largest moons. Note the moons’ brightnesses and their locations with respect to Jupiter. If you watch over a period of several nights, draw what you see; you’ll notice that these moons change their positions as they orbit the giant planet. Check the charts given monthly in Astronomy or Sky & Telescope magazines to identify each moon you see.



SKYCHART III PROJECTS The SkyChart III Student Version planetarium program on which these exercises are based is included as a separately executable program on the CD in the back of this text.

1. Observe the apparent motion of Jupiter against the background stars by centering the display on Jupiter with VIEW/Center Planet/Jupiter. Deselect DRAW/Horizon Mask, Constellations, and Grid Lines. With a 180° Field of View set ANIMATION/Trail For/Jupiter. Set DRAW/Sky Background/Black and set the time to approximately midnight. Animate the image with time steps of one week beginning April 1, 2001, until approximately September 1, 2004. If the animation runs too fast on your computer, use time steps of one day. During this time, count how many times the motion of Jupiter stops and reverses itself. You will observe that after moving in retrograde motion for a while, Jupiter will again stop and reverse its motion, illustrating the epicycle the ancients imagined.

2. The Mesopotamians were well aware of the cycles of the heavens. Venus (the Star of Ishtar) was symbolized by an eight-pointed star. It turns out there is a reason for the number eight. Venus experiences a cycle of approximately eight years. Venus is located at the same location with respect to the "fixed" stars every eight years.In this time, it has switched back and forth between morning and evening star five times. Using an appropriate Time Step and using the Trails feature in the ANIMATION menu, show that this period is correct and that Venus switches back and forth five times. Center on the vernal equinox (R.A.: 00 00 and Dec. 00 00, Equatorial Coordinates).

In addition to the Practice Problems and Destinations modules, the Companion Website at http://www.prenhall.com/chaisson provides for each chapter an additional true-false, multiple choice, and labeling quiz, as well as additional annotated images, animations, and links to related Websites.