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# Roger A. Freedman • William J. Kaufmann III

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Roger A. Freedman • William J. Kaufmann III
Universe Eighth Edition CHAPTER 4 Gravitation and the Waltz of the Planets

Reading Assignment – Chapter 5 (The Nature of Light)
Homework – Chapter 3 Quiz due Wednesday 9/15 by 9 PM An astronaut orbits the Earth attached to the International Space Station’s manipulator arm. (NASA)

An astronaut orbits the Earth attached to the International Space
Station’s manipulator arm. (NASA)

Figure 4-1 A Merry-Go-Round Analogy
(a) Two children walk at different speeds around a rotating merrygo- round with its wooden horses. (b) In an analogous way, the ancient Greeks imagined that the Sun and Moon move around the rotating celestial sphere with its fixed stars. Thus, the Sun and Moon move from east to west across the sky every day and also move slowly eastward from one night to the next relative to the background of stars.

Figure 4-2 The Path of Mars in 2011–2012
From October 2011 through August 2012, Mars will move across the zodiacal constellations Cancer, Leo, and Virgo. Mars’s motion will be direct (from west to east, or from right to left in this figure) most of the time but will be retrograde (from east to west, or from left to right in this figure) during February and March Notice that the speed of Mars relative to the stars is not constant: The planet travels farther across the sky from October 1 to December 1 than it does from December 1 to February 1.

Figure 4-3 A Geocentric Explanation of Retrograde Motion (a) The ancient Greeks imagined that each planet moves along an epicycle, which in turn moves along a deferent centered approximately on the Earth. The planet moves along the epicycle more rapidly than the epicycle moves along the deferent. (b) At most times the eastward motion of the planet on the epicycle adds to the eastward motion of the epicycle on the deferent. Then the planet moves eastward in direct motion as seen from Earth. (c) When the planet is on the inside of the deferent, its motion along the epicycle is westward. Because this motion is faster than the eastward motion of the epicycle on the deferent, the planet appears from Earth to be moving westward in retrograde motion.

Figure 4-3 A Geocentric Explanation of Retrograde Motion (a) The ancient Greeks imagined that each planet moves along an epicycle, which in turn moves along a deferent centered approximately on the Earth. The planet moves along the epicycle more rapidly than the epicycle moves along the deferent. (b) At most times the eastward motion of the planet on the epicycle adds to the eastward motion of the epicycle on the deferent. Then the planet moves eastward in direct motion as seen from Earth. (c) When the planet is on the inside of the deferent, its motion along the epicycle is westward. Because this motion is faster than the eastward motion of the epicycle on the deferent, the planet appears from Earth to be moving westward in retrograde motion.

Epicycles are used in a geocentric model to explain the
eastward motion of Mars. westward motion of Mars. phases of Venus. appearance of Venus in the early morning or late evening sky. phases of the moon. Q4.3

Epicycles are used in a geocentric model to explain the
eastward motion of Mars. westward motion of Mars. phases of Venus. appearance of Venus in the early morning or late evening sky. phases of the moon. A4.3

Figure 4-4 Nicolaus Copernicus (1473–1543)
Copernicus was the first person to work out the details of a heliocentric system in which the planets, including the Earth, orbit the Sun. (E. Lessing/Magnum)

Figure 4-5 A Heliocentric Explanation of Retrograde Motion
In the heliocentric model of Aristarchus, the Earth and the other planets orbit the Sun. The Earth travels around the Sun more rapidly than Mars. Consequently, as the Earth overtakes and passes this slower-moving planet, Mars appears for a few months (from points 4 through 6) to fall behind and move backward with respect to the background of stars.

Figure 4-5 A Heliocentric Explanation of Retrograde Motion
In the heliocentric model of Aristarchus, the Earth and the other planets orbit the Sun. The Earth travels around the Sun more rapidly than Mars. Consequently, as the Earth overtakes and passes this slower-moving planet, Mars appears for a few months (from points 4 through 6) to fall behind and move backward with respect to the background of stars.

Figure 4-6 Planetary Orbits and Configurations
When and where in the sky a planet can be seen from Earth depends on the size of its orbit and its location on that orbit. Inferior planets have orbits smaller than the Earth’s, while superior planets have orbits larger than the Earth’s. (Note that in this figure you are looking down onto the solar system from a point far above the Earth’s northern hemisphere.)

Figure 4-6 Planetary Orbits and Configurations
When and where in the sky a planet can be seen from Earth depends on the size of its orbit and its location on that orbit. Inferior planets have orbits smaller than the Earth’s, while superior planets have orbits larger than the Earth’s. (Note that in this figure you are looking down onto the solar system from a point far above the Earth’s northern hemisphere.)

Determining the period of a planet takes some care, because
the Earth, from which we must make the observations, is also moving. Realizing this, Copernicus was careful to distinguish between two different periods of each planet. The synodic period is the time that elapses between two successive identical configurations as seen from Earth—from one opposition to the next, for example, or from one conjunction to the next. The sidereal period is the true orbital period of a planet, the time it takes the planet to complete one full orbit of the Sun relative to the stars. The synodic period of a planet can be determined by observing the sky, but the sidereal period has to be found by calculation. Copernicus figured out how to do this (Box 4-1). Table 4-1 shows the results for all of the planets.

Let P be the planet’s sidereal
period, S the planet’s synodic period, and E the Earth’s sidereal period or sidereal year (see Section 2-8), which Copernicus knew to be nearly 3651⁄4 days. The rate at which the Earth moves around its orbit is the number of degrees around the orbit divided by the time to complete the orbit, or 360°/E (equal to a little less than 1° per day). Similarly, the rate at which the inferior planet moves along its orbit is 360°/P. During a given time interval, the angular distance that the Earth moves around its orbit is its rate, (360°/E), multiplied by the length of the time interval. Thus, during a time S, or one synodic period of the inferior planet, the Earth covers an angular distance of (360°/E)S around its orbit. In that same time, the inferior planet covers an angular distance of (360°/P)S. Note, however, that the inferior planet has gained one full lap on the Earth, and hence has covered 360° more than the Earth has (see the figure). Thus, (360°/P)S = (360°/E)S °.

When Jupiter is at opposition (see diagram), it is overhead (i. e
When Jupiter is at opposition (see diagram), it is overhead (i.e., on the meridian) at 6:00 A.M. 12:00 noon. 6:00 P.M. midnight. never at night. Q4.1

When Jupiter is at opposition (see diagram), it is overhead (i. e
When Jupiter is at opposition (see diagram), it is overhead (i.e., on the meridian) at 6:00 A.M. 12:00 noon. 6:00 P.M. midnight. never at night. A4.1

Figure 4-7 A Nearby Object Shows a Parallax Shift
Tycho Brahe argued that if an object is near the Earth, an observer would have to look in different directions to see that object over the course of a night and its position relative to the background stars would change. Tycho failed to measure such changes for a supernova in 1572 and a comet in He therefore concluded that these objects were far from the Earth.

Figure Ellipses (a) To draw an ellipse, use two thumbtacks to secure the ends of a piece of string, then use a pencil to pull the string taut. If you move the pencil while keeping the string taut, the pencil traces out an ellipse. The thumbtacks are located at the two foci of the ellipse. The major axis is the greatest distance across the ellipse; the semimajor axis is half of this distance.

Figure 4-10 Ellipses (b) A series of ellipses with the same major axis but different eccentricities. An ellipse can have any eccentricity from e 0 (a circle) to just under e 1 (virtually a straight line).

Figure 4-11 Kepler’s First and Second Laws
According to Kepler’s first law, a planet travels around the Sun along an elliptical orbit with the Sun at one focus. According to his second law, a planet moves fastest when closest to the Sun (at perihelion) and slowest when farthest from the Sun (at aphelion). As the planet moves, an imaginary line joining the planet and the Sun sweeps out equal areas in equal intervals of time (from A to B or from C to D). By using these laws in his calculations, Kepler found a perfect fit to the apparent motions of the planets.

A planet in an elliptical orbit around the Sun moves most rapidly when it is
at aphelion. at perihelion. halfway between perihelion and aphelion. closer to aphelion than perihelion. None of the above; it has a constant speed. Q4.6

A planet in an elliptical orbit around the Sun moves most rapidly when it is
at aphelion. at perihelion. halfway between perihelion and aphelion. closer to aphelion than perihelion. None of the above; it has a constant speed. A4.6

Note: A graph of P2 vs. a3 will yield a straight line.

Kepler’s three laws were initially derived from fundamental physics laws. were initially found from observation and are empirical rules that describe how the planets move. were handed down from the ancient Greeks. were discovered by Kepler in ancient Babylonian texts. are now known to be grossly incorrect. Q4.5

Kepler’s three laws were initially derived from fundamental physics laws. were initially found from observation and are empirical rules that describe how the planets move. were handed down from the ancient Greeks. were discovered by Kepler in ancient Babylonian texts. are now known to be grossly incorrect. A4.5

Two planets are in circular orbits around similar stars
Two planets are in circular orbits around similar stars. Each is the same distance from its parent star. One planet is Jupiter’s size, whereas the other is Earth’s size. Which of the following is true? The Jupiter-size planet has a much longer period than the Earth-size planet. The Earth-size planet has a much longer period than the Jupiter-size planet. Both planets have the same period. The Earth-size planet has a period exactly twice that of the Jupiter-size planet. Not enough information is given to determine the correct answer. Q4.7

Two planets are in circular orbits around similar stars
Two planets are in circular orbits around similar stars. Each is the same distance from its parent star. One planet is Jupiter’s size, whereas the other is Earth’s size. Which of the following is true? The Jupiter-size planet has a much longer period than the Earth-size planet. The Earth-size planet has a much longer period than the Jupiter-size planet. Both planets have the same period. The Earth-size planet has a period exactly twice that of the Jupiter-size planet. Not enough information is given to determine the correct answer. A4.7

Figure 4-12 Galileo Galilei (1564–1642)
Galileo was one of the first people to use a telescope to observe the heavens. He discovered craters on the Moon, sunspots on the Sun, the phases of Venus, and four moons orbiting Jupiter. His observations strongly suggested that the Earth orbits the Sun, not vice versa. (Eric Lessing/Art Resource)

Figure 4-12 Galileo Galilei (1564–1642)
Galileo was one of the first people to use a telescope to observe the heavens. He discovered craters on the Moon, sunspots on the Sun, the phases of Venus, and four moons orbiting Jupiter. His observations strongly suggested that the Earth orbits the Sun, not vice versa. (Eric Lessing/Art Resource)

Figure 4-13 The Phases of Venus
This series of photographs shows how the appearance of Venus changes as it moves along its orbit. The number below each view is the angular diameter of the planet in arcseconds. Venus has the largest angular diameter when it is a crescent, and the smallest angular diameter when it is gibbous (nearly full). (New Mexico State University Observatory)

Figure 4-14 The Changing Appearance of Venus Explained in a Heliocentric Model
A heliocentric model, in which the Earth and Venus both orbit the Sun, provides a natural explanation for the changing appearance of Venus shown in Figure 4-13.

Figure 4-15 The Appearance of Venus in the Ptolemaic Model
In the geocentric Ptolemaic model the deferents of Venus and the Sun rotate together, with the epicycle of Venus centered on a line (shown dashed) that connects the Sun and the Earth. In this model an Earth observer would never see Venus as more than half illuminated. (At positions a and c, Venus appears in a “new” phase; at positions b and d, it appears as a crescent. Compare with Figure 3-2, which shows the phases of the Moon.) Because Galileo saw Venus in nearly fully illuminated phases, he concluded that the Ptolemaic model must be incorrect.

Figure 4-16 Jupiter and Its Largest Moons
This photograph, taken by an amateur astronomer with a small telescope, shows the four Galilean satellites alongside an overexposed image of Jupiter. Each satellite is bright enough to be seen with the unaided eye, were it not overwhelmed by the glare of Jupiter. (Courtesy of C. Holmes)

Figure 4-17 Early Observations of Jupiter’s Moons
In 1610 Galileo discovered four “stars” that move back and forth across Jupiter from one night to the next. He concluded that these are four moons that orbit Jupiter, much as our Moon orbits the Earth. This drawing shows notations made by Jesuit observers on successive nights in The circle represents Jupiter and the stars its moons. Compare the drawing numbered 13 with the photograph in Figure (Yerkes Observatory)

Figure 4-20 An Explanation of Orbits
If a ball is dropped from a great height above the Earth’s surface, it falls straight down (A). If the ball is thrown with some horizontal speed, it follows a curved path before hitting the ground (B, C). If thrown with just the right speed (E), the ball goes into circular orbit; the ball’s path curves but it never gets any closer to the Earth’s surface. If the ball is thrown with a speed that is slightly less (D) or slightly more (F) than the speed for a circular orbit, the ball’s orbit is an ellipse.

Which of the following objects has zero acceleration?
A car just starting out from a stop light A car traveling at 70 miles per hour on a long straight interstate highway A car exiting an interstate on a curved off-ramp A car slowing down for a red light A car skidding to a stop to avoid an accident Q4.9

Which of the following objects has zero acceleration?
A car just starting out from a stop light A car traveling at 70 miles per hour on a long straight interstate highway A car exiting an interstate on a curved off-ramp A car slowing down for a red light A car skidding to a stop to avoid an accident Q4.9

Because it has been so successful in explaining and predicting
many important phenomena, Newtonian mechanics has become the cornerstone of modern physical science. Even today, as we send astronauts into Earth orbit and spacecraft to the outer planets, Newton’s equations are used to calculate the orbits and trajectories of these spacecraft. The Cosmic Connections figure summarizes some of the ways that gravity plays an important role on scales from apples to galaxies.

Figure 4-24 The Origin of Tidal Forces
(a) Imagine three identical billiard balls placed some distance from a planet and released. (b) The closer a ball is to the planet, the more gravitational force the planet exerts on it. Thus, a short time after the balls are released, the blue 2-ball has moved farther toward the planet than the yellow 1-ball, and the red 3-ball has moved farther still. (c) From the perspective of the 2-ball in the center, it appears that forces have pushed the 1-ball away from the planet and pulled the 3-ball toward the planet. These forces are called tidal forces.

Figure 4-25 Tidal Forces on the Earth
(a) The Moon exerts different gravitational pulls at different locations on the Earth. (b) At any location, the tidal force equals the Moon’s gravitational pull at that location minus the gravitational pull of the Moon at the center of the Earth. These tidal forces tend to deform the Earth into a nonspherical shape.

Figure 4-26 High and Low Tides
(a) The gravitational forces of the Moon and Sun deform the Earth’s oceans, giving rise to low and high tides. (b), (c) The strength of the tides depends on the relative positions of the Sun, Moon, and Earth.

Figure 4-27 Tidal Forces on a Galaxy
For millions of years the galaxies NGC 2207 and IC 2163 have been moving ponderously past each other. The larger galaxy’s tremendous tidal forces have drawn a streamer of material a hundred thousand light-years long out of IC If you lived on a planet orbiting a star within this streamer, you would have a magnificent view of both galaxies. NGC 2207 and IC 2163 are respectively 143,000 light-years and 101,000 light-years in diameter. Both galaxies are 114 million light-years away in the constellation Canis Major. (NASA and the Hubble Heritage Team, AURA/STScI)

The tidal force of the Moon on the Earth occurs because
the Moon pulls more strongly on the near side of the Earth than the far side of the Earth. the Moon pulls more strongly on the far side of the Earth than the near side of the Earth. the Moon pulls equally strongly on the near side and the far side of the Earth. the Moon keeps its same face toward the Earth. the Moon is in a synchronous orbit. Q4.14

The tidal force of the Moon on the Earth occurs because
the Moon pulls more strongly on the near side of the Earth than the far side of the Earth. the Moon pulls more strongly on the far side of the Earth than the near side of the Earth. the Moon pulls equally strongly on the near side and the far side of the Earth. the Moon keeps its same face toward the Earth. the Moon is in a synchronous orbit. A4.14

Key Ideas Copernicus’s Heliocentric Model: Copernicus’s heliocentric (Sun-centered) theory simplified the general explanation of planetary motions. In a heliocentric system, the Earth is one of the planets orbiting the Sun. A planet undergoes retrograde motion as seen from Earth when the Earth and the planet pass each other. The sidereal period of a planet, its true orbital period, is measured with respect to the stars. Its synodic period is measured with respect to the Earth and the Sun (for example, from one opposition to the next).

Key Ideas Apparent Motions of the Planets: Like the Sun and Moon, the planets move on the celestial sphere with respect to the background of stars. Most of the time a planet moves eastward in direct motion, in the same direction as the Sun and the Moon, but from time to time it moves westward in retrograde motion. The Ancient Geocentric Model: Ancient astronomers believed the Earth to be at the center of the universe. They invented a complex system of epicycles and deferents to explain the direct and retrograde motions of the planets on the celestial sphere.

Key Ideas Kepler’s Improved Heliocentric Model and Elliptical Orbits: Copernicus thought that the orbits of the planets were combinations of circles. Using data collected by Tycho Brahe, Kepler deduced three laws of planetary motion. (1) the orbits are in fact ellipses (2) a planet’s speed varies as it moves around its elliptical orbit (3) the orbital period of a planet is related to the size of its orbit.

Key Ideas Evidence for the Heliocentric Model: The invention of the telescope led Galileo to new discoveries that supported a heliocentric model. These included his observations of the phases of Venus and of the motions of four moons around Jupiter. Newton’s Laws of Motion: Isaac Newton developed three principles, called the laws of motion, that apply to the motions of objects on Earth as well as in space.

Key Ideas (1) the tendency of an object to maintain a constant velocity, (2) the relationship between the net outside force on an object and the object’s acceleration, and (3) the principle of action and reaction. These laws and Newton’s law of universal gravitation can be used to deduce Kepler’s laws. They lead to extremely accurate descriptions of planetary motions. The mass of an object is a measure of the amount of matter in the object. Its weight is a measure of the force with which the gravity of some other object pulls on it.

Key Ideas In general, the path of one object about another, such as that of a planet or comet about the Sun, is one of the curves called conic sections: circle, ellipse, parabola, or hyperbola. Tidal Forces: Tidal forces are caused by differences in the gravitational pull that one object exerts on different parts of a second object. The tidal forces of the Moon and Sun produce tides in the Earth’s oceans. The tidal forces of the Earth have locked the Moon into synchronous rotation.

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