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 Copernicuss 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 planets 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 Keplers 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 bodys 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 bodys 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. Newtons 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 Newtons laws.
|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 Brahes 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 Earths orbit at aphelion and just grazes Venuss 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. Halleys 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. Jupiters moon Callisto orbits the planet at a distance of 1.88 million km. Callistos orbital period about Jupiter is 16.7 days. What is the mass of Jupiter? [Assume that Callistos mass is negligible compared with that of Jupiter, and use the modified version of Keplers 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 Suns 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 planets 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 Earths 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 Earths 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 Newtons 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 Moons 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
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 "Keplers 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 Keplers second law by determining the area swept out at perigee and apogee.
2. Access the "Keplerian Elements Tutorial" and describe the difference between a satellites 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.
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 planets retrograde motion begin? How long afterward does it end?
2. Draw an ellipse. (See Figure 2.13) Youll 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 Galileos observations of Jupiters 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; youll 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.