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CHAPTER 3 Gravity and the Rise of Modern Astronomy CHAPTER OUTLINE 31 Galileo Galilei and the Telescope 1. Galileo was born in 1564 and was a contemporary of Kepler. He built his first telescope in 1609, shortly after hearing about telescopes being constructed in the Netherlands. 2. Galileo was the first person to use a telescope to study the sky. He made 5 important observations that affected the comparison between the geocentric and heliocentric theories. (a) Mountains and valleys on the Moon (b) Sunspots (c) More stars than can be observed with the naked eye (d) Four moons of Jupiter (e) Complete cycle of phases of Venus Observing the Moon, the Sun, and the Stars 1. Though Galileo’s first three observations do not disprove the geocentric model, they cast doubt on its basic assumption of perfection in the heavens. 2. The existence of stars too dim to be seen with the naked eye also cast doubt on the literal interpretation of some Biblical passages. Jupiter’s Moons 1. In 1610 Galileo discovered that Jupiter had four satellites of its own, now known as the Galilean moons of Jupiter. 2. The motion of Jupiter and its orbiting moons contradicted the Ptolemaic notions that the Earth is the center of all things and that if the Earth moved through space it would leave behind the Moon. The Phases of Venus 1. Galileo observed that Venus goes through a full set of phases: full, gibbous, quarter, crescent. 2. Venus’s full set of phases cannot be explained by the Ptolemaic model but can be explained by the heliocentric model. 3. The Ptolemaic model predicts that Venus will always appear in a crescent phase, which is not borne out by the observations. 4. Also, the heliocentric model explains the correlation between Venus’ phases and its corresponding observed sizes. 5. Galileo is credited with setting the standard for studying nature through reliance on observation and experimentation to test hypotheses. 6. The year Galileo died—1642—is the year Isaac Newton was born. Newton took the work of Galileo and Kepler and created a new theory of motion. 32 Isaac Newton’s Grand Synthesis Newton’s First Two Laws of Motion 1. Newton’s First Law (Law of Inertia): Unless a net, outside force, acts upon an object, the object will maintain a constant speed in a straight line (if initially moving), or remain at rest (if initially at rest). 2. Inertia is the tendency of an object to resist a change in its motion. 3. The first law indicates that a net force is necessary for an object to change its speed and/or its direction of motion (i.e., to accelerate). 4. Newton’s second law quantifies and extends the first law. It tells us how much force is necessary to produce a certain acceleration of an object. An Important Digression — Mass and Weight 1. Mass is the quantifiable property of an object that is a measure of its inertia. It is an intrinsic property of an object and independent of location. 2. Mass is NOT volume or weight. (The weight of an object on Earth is simply the downward force experienced by the object due to its gravitational interaction with the Earth.) 3. The international (SI) unit of mass is the kilogram. A kilogram weighs about 2.2 pounds on Earth. Back to Newton’s Second Law 1. Newton’s Second Law: A net external force applied to an object causes it to accelerate at a rate that is inversely proportional to its mass: Acceleration = net force / mass, or F = m a. 2. When the net force is zero, there is no acceleration. Newton’s Third Law 1. Newton’s Third Law: When object X exerts a force on object Y, object Y exerts an equal and opposite force back on X. 2. The Third Law is sometimes stated as “For every action there is an equal and opposite reaction,” but the first statement is more precise in terms of physical forces. 33 Motion in a Circle 1. Motion of an object in a circle at constant speed (uniform circular motion) is an example of acceleration causing a change in direction. 2. Centripetal (“centerseeking”) force is the force directed toward the center of the curve along which the object is moving. Centripetal force is simply a label we apply to a net force that causes an object to move in a curve. 34 The Law of Universal Gravitation 1. Aristotle believed that earthly laws do not hold in the heavens, but Newton sought to apply his laws of motion there, too. Newton hypothesized that the centripetal force that moves the Moon in a circle about the Earth was the same force that cause an object here on Earth to fall to the ground. 2. The law of universal gravitation states that between every two objects there is an attractive force, the magnitude of which is directly proportional to the mass of each object and inversely proportional to the square of the distance between the centers of the objects. 3. In equation form: F = Gm_{1}m_{2} / d^{2}, where G is a constant, m_{1}_{ }and_{ }m_{2 }are the masses, and d is the distance between their centers. 4. Weight is the gravitational force between an object and the planetary/stellar body where the object is located. 5. According to Newton, gravity not only makes objects fall to Earth but keeps the Moon in orbit around the Earth and keeps the planets in orbit around the Sun. His laws could explain the planets’ motions and why Kepler’s laws worked. Arriving at the Law of Universal Gravitation 1. Whether or not force is proportional to mass can be tested by showing that weight is proportional to mass here on Earth. 2. To test the dependence of force on distance, Newton compared accelerations of objects near the Earth’s surface to the Moon’s acceleration in orbit around the Earth. 3. Because the distance from the center of the Earth to the Moon is about 60 times the distance from the center of the Earth to its surface, the centripetal acceleration of the Moon should be (1/60)^{2} or 1/3600 of the acceleration of gravity on Earth. Newton’s calculations showed this to be the case and confirmed the validity of his theory of gravitation. 35 Newton’s Laws and Kepler’s Laws 1. Kepler arrived at the laws of planetary motion because they worked. Based on his laws of motion and his law of gravity, Newton was able not only to prove Kepler’s laws but also expand beyond them. 2. Kepler’s first law (that the planets move in elliptical orbits) can be derived from Newton’s laws but requires calculus. 3. Kepler’s second law (that an imaginary line connecting a planet and the Sun sweeps out equal areas in equal times) can also be derived from Newton’s laws. As planets orbit the Sun they show a change in both speed and direction. 4. Newton showed mathematically that Kepler’s third law—the perioddistance relationship—derives from the inverse square law for gravitation. 5. Newton’s modified version of Kepler’s third law, a_{(AU)}^{3}/P_{(yrs)}^{2} = (m_{1} + m_{2})/m_{Sun}, is valid for any two objects orbiting each other as a result of their mutual gravitational attraction (a binary system). For the case where the two objects are the Sun and a planet, Newton’s modified version reduces to Kepler’s third law because a planet’s mass is much smaller than the Sun’s. 36 Orbits and the Center of Mass 1. Center of mass is the average location of the various masses in a system, weighted according to how far each is from that point. The center of mass of a system is not always the same as the center of gravity. 2. For the Sun–Earth system, the Sun is 330,000 times more massive than the Earth and, thus, the center of mass of this system is essentially at the center of the Sun. 3. For the EarthMoon system, the Earth’s mass is only 81 times that of the Moon, and the center of mass for this system is inside the Earth, about 4,800 km from its center. 3. The relationship between the distances of two objects from their center of mass be used to determine their relative masses. 4. When we consider the motion of the Earth around the Sun, it is the center of mass of the EarthMoon system that follows an elliptical path. 37 Beyond Newton 1. Newtonian ideas succeed in describing a wide range of observations based on a few basic ideas: for example, Kepler’s laws can be derived from them, they explain tides and precession, and their use predicted the existence of the planet Neptune and the return of comet Halley. 2. Newton’s laws provide a way to measure things quantitatively and predict the motion of things. They were the first “laws” ever that could be shown to hold for both the heavens and the Earth. They offer a unifying view of the universe. 3. Newton’s work confirmed the belief of the ancient Greeks that nature is explainable. 38 General Theory of Relativity 1. We defined mass as the measure of the inertia of an object. However, in Newton’s law of gravity, mass determines the strength of gravitational attraction. The same quantity measures two seemingly different physical properties. 2. Experiments show that the measures of inertia and gravitational attraction are identical to one part in a trillion. This is not a coincidence. 3. Einstein, in his General Theory of Relativity, showed mathematically that the two types of masses are indeed equivalent. 4. According to the principle of equivalence, effects of acceleration are indistinguishable from gravitational effects. 5. Space warp, or the curvature of space, is used by Einstein to explain the similarity of gravity and acceleration as well as the bending of light near a massive object. 6. For Einstein, as objects move, they follow the curvature of space created by the presence of mass. (“Matter tells space how to curve, and space curvature tells matter how to move.”) This view differs from Newton’s attractive gravitational force between objects. 39 Gravitation and Einstein 1. The many tests of Einstein’s general theory of relativity show that it provides a better description of gravity than Newton’s theory. Test 1: The Gravitational Bending of Light 1. The general theory predicts that light will curve in the presence of a massive object. This prediction, made in 1915, was first confirmed during a solar eclipse in 1919. 2. The bending of starlight by a massive object (the Sun, for example) is fully explained by the general theory of relativity but is not by Newtonian mechanics. Test 2: The Orbit of Mercury 1. 19th century measurements showed that Mercury’s elliptical orbit slowly precesses about the Sun at the rate of 574 arcseconds per century. (Precession of an orbit corresponds to the change in orientation of the major axis of the elliptical path of an object.) 2. Calculations based on Newtonian mechanics can account for 531 arcseconds of this precession of Mercury’s orbit. However, 43 arcseconds are unaccounted for by classical theory. 3. Einstein’s general theory of relativity was able to explain the “mysterious” 43 arcseconds of precession by considering the properties of curved space. The Sun’s mass curves space in its vicinity and Mercury’s orbit is affected by this space curving. Additional Tests 1. According to the general theory of relativity, massive objects warp not only space around them but also time: time slows down in the presence of gravity. 2. This effect was tested in 1960 and the results (discussed in Chapter 15) were in complete agreement with the theory. 3. Also, the theory predicts the existence of gravitational waves, ripples in the curvature of space produced by changes in the distribution of matter. 4. Strong indirect evidence for gravitational waves comes from observations of a binary system of stars (discussed in Chapter 15). 5. Newton’s theory of gravity does not include any effect that gravity might have on time and does not predict the existence of gravitational waves. 6. All tests of Einstein’s relativity theory have confirmed it, and the theory forms the basis of modern astronomy. We still talk about Newtonian ideas because they work very well in the realm of our everyday experiences, but Einstein’s theory is essential in some cases such as when discussing black holes. The Correspondence Principle 1. The correspondence principle states that the predictions of a new theory must agree with the theory it replaces in cases where the previous theory has been found to be correct. 2. The general theory of relativity is in accord with the correspondence principle. It agrees with Newtonian mechanics where the older theory provided correct results. Advancing the Model: Travel to the Moon
Advancing the Model: The Special Theory of Relativity 1. The special theory of relativity is based on two principles. (a) All laws of physics are the same for all nonaccelerating observers, no matter what the speed of those observers. (b) The speed of light is the same for all nonaccelerating observers, no matter what their motion relative to the source of the light.. 2. The special theory of relativity predicts that: (a) the observed length along the line of motion of a moving object becomes less than its length when measured at rest; (b) the observed passage of time becomes slower for the moving object; (c) the observed inertia of an object becomes greater than its inertia when at rest; and (d) mass can be transformed to energy and vice versa. 3. Experimental results confirm the predictions made by the special theory of relativity. 