The Historical Quest to Model the Solar System

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The progress of science is generally regarded as a kind of clean, rational advance along a straight line; in fact, it has followed a zig-zag course, at times almost more bewildering than the evolution of political thought. The history of cosmic theories, in particular, may without exaggeration be called a history of collective obsessions and controlled schizophrenias; and the manner in which some of the most important individual discoveries were arrived at reminds one more of a sleepwalker’s performance than an electronic brain. Arthur Koestler, The Sleepwalkers, 1959
The Historical Quest to Model the
Solar System
Part 1 • Discovering the Science of Astronomy5-2
In this chapter we follow the way our knowledge of the layout of the Solar System developed from the time of the Greek Earth-centered view to the Sun-center view during the renaissance in Europe. We will be interested in how the ideas that developed in
one era were a natural outgrowth of the state of astronomical observations and degree of sophistication of the times. In previous chapters we briefly discussed some historical aspects of astronomy; for example, we considered the idea of a spherical Earth, and how this concept slowly came to be accepted by 100 b.c.e. by Greek thinkers through the gradual accumulation of evidence from observations of both the Earth and the heavens. Yet this concept was established relatively rapidly when compared to the slow acceptance of the idea of a Sun-centered Solar System.
5.1 Greek Astronomy Although we saw in the last chapter that a variety of cultures were intimately involved with phenomena of the sky, it was the Greek culture that made progress toward the models we have today.
What the Greeks Inherited The Babylonians and Egyptians left the Greeks an extensive body of astronomical knowledge. However, the astronomy of both ancient Egypt and Babylonia was the province of a priestly aristocracy; as a consequence, practical and political considerations often took precedence over theoretical inquiries. Egyptian astronomers, for example, had discovered that when the bright star Sirius could just be seen rising in the east before the Sun, the flooding of the Nile was imminent. Such knowledge gave tremendous power to the priesthood and inevitably involved its members closely with the state. Similarly, in Babylonia, astronomer-priests had acquired a considerable amount of information concerning the motions of the Moon, Sun, and planets and had found certain regular cycles that enabled them to predict some eclipses—a power that was frequently used for political purposes.
The ancient Egyptians and Babylonians knew the length of the year and the different types of calendars, both solar and lunar. The Egyptians had learned the rudiments of simple mathematics, algebra, and geometry. Sundials had been invented, and systems of timekeeping were in existence. The Babylonians had made systematic observations of the positions of heavenly objects and had practical methods for predicting the positions of the Moon, Sun, and planets. Figure 5-1, for example, shows a table of data for Jupiter. The bottom part of the figure describes the method of calculation. The Babylonian value for the length of the synodic
period of the Moon was not surpassed in accuracy until the end of the nineteenth century. Both cultures had attempted to construct a cosmology that placed Earth and humanity in their proper position in relation to the universe and the gods, but they never attempted to construct a truly consistent theoretical framework for their cosmology in the way the Greeks did.
Aristotle The Greeks enjoyed philosophy, by which they meant a broad attempt to understand all things. Aristotle (384–322 B.C.E.) viewed the universe in the abstract and argued that simply thinking logically about the universe was sufficient to provide an understanding of it.
At the time of Aristotle, the thinking was that there were four elements making up the universe — earth, air, fire and water. The tendency of objects to fall to the ground was explained as follows: objects made of “earthy material” fell to the ground because they moved toward similar
Chapter opening photo: Galileo showing the heavens to the public in Venice with one of his telescopes.
FIGURE 5-1. A clay tablet from ancient Mesopotamia containing astronomical observations of Jupiter in the top part, and a description of the method of calculation in the bottom part.
5-3Chapter 5 • The Historical Quest to Model the Solar System
these two time intervals were in fact nearly equal, implying that rays of light from the Sun are arriving in nearly parallel lines for all parts of the Moon’s orbit. From this one concludes that the Sun must be many times farther away than the Moon.
Inquiry 5-1 What did Aristarchus assume about the Moon’s orbital shape and motion?
Aristarchus was also able to estimate the relative sizes of the Earth and Moon by timing the duration of lunar eclipses. Figure 5-3 shows the principle of this determination. The length of time it takes the entire Moon to enter the Earth’s shadow depends directly on (that is, is proportional to) the diameter of the Moon. The length of time it takes for one part of the Moon — say, its leading edge — to completely cross the Earth’s shadow is proportional to the diameter of the Earth. By comparing these two times, we can estimate the ratio between the diameters of the Earth and Moon, as Aristarchus did.1
Inquiry 5-2 Suppose it takes an hour for the entire Moon to enter the Earth’s shadow and four hours for the edge of the Moon to cross the shadow. How many times larger than the Moon’s diameter is the Earth’s diameter?
1 Actually, the diameter of the Earth’s shadow at the distance of the Moon is slightly less than
the diameter of the Earth; however, this leads to only a small error.
materials. Thus, an object’s “natural place” was close to the Earth, to which it moved. Similarly, fire rises towards its “natural place,” which is away from the “earthy materials,” which is up.
In addition, he taught that continuous movement required continuous contact between the mover and the moving object to perpetuate the motion. And, as discussed later in this chapter, he argued that planets would move in circular orbits. Such ideas came from philosophy and did not need to be tested because logic said it was right. These ideas were the mainstay of scientific knowledge until the seventeenth century.
Aristarchus of Samos When Greek philosophical thinking was coupled with their highly developed mathematical skills, the Greeks advanced so rapidly in astronomy from 600 B.C.E. to 100 B.C.E. that their era was without question one of the shining examples of scientific discovery prior to the year 1500 C.E. Progress in understanding the structure of our solar system came especially quickly with the emergence of the Alexandrian school of Greek astronomers, particularly the work of Aristarchus of Samos (c. 300 B.C.E.), who combined careful observations and sharp reasoning to draw inferences about the relative sizes and distances of the Earth, Moon, and Sun.
Aristarchus demonstrated that the Sun was many times farther from the Earth than the Moon by using a reasoning process illustrated in Figure 5-2. You can see that if the Sun were quite close to the Earth–Moon system, then the time interval from first quarter Moon to third quarter Moon would be longer than the time interval from third quarter to first quarter. Aristarchus observed that
Sun
M1
Moon at first quarter
Moon at third quarter
FIGURE 5-2. Aristarchus’s method to show that the Sun is farther from the Earth than the Moon. If the Sun were very near the Earth-Moon system, the times between quarter phases would be a lot different. However, if the Sun were much more distant than the Moon, the lines S-M1 and S-M3 would be nearly parallel, and the time interval between first and third quarters would be nearly equal to the time interval between third and first quarters.
Earth's shadow
ABC
FIGURE 5-3. The principle of Aristarchus’s measurement of the Moon’s diameter compared to that of Earth. The time it takes the Moon to enter the Earth’s shadow (between points A and B) is proportional to the Moon’s diameter. The time the Moon takes to cross the Earth’s shadow, between A and C, is proportional to the Earth’s shadow diameter, which is nearly that of the Earth.
Finally, Aristarchus knew that during a total eclipse of the Sun, the Moon was just able to cover the Sun, and therefore, the angular sizes of the Moon and Sun in the sky were about the same. He could then reason that the actual sizes of the Moon and Sun were proportional to their distances, and in this way he could estimate the diameter of the Sun (Figure 5-4). Unfortunately, although his estimate of the distance to the Moon was quite good, his estimate of the distance to the Sun was about 10 times too small. For this reason, his estimate of the Sun’s diameter was also 10 times too small. Nevertheless, he was able to show that the distance to the Sun was considerably greater than that to the Moon, and that the Sun’s diameter was much greater than the Earth’s.
Inquiry 5-3 Aristarchus was the first to propose that the Earth goes around the Sun, rather than vice versa. Suggest one factor that may have led him to this conclusion.
Eratosthenes Another classic experiment of antiquity was the determination of the Earth’s circumference by Eratosthenes (c. 200 B.C.E.). The conclusions of Aristarchus concerning the relative sizes and distances of the Earth, Moon, and Sun were all in terms of the then-unknown size of the Earth. Their actual sizes could not be known until the Earth’s size was known.
Eratosthenes had heard that at Syene, near the modern Aswan in Egypt, there was a deep well, and that on a certain day of the year the Sun stood directly overhead so that its reflection could be seen in the water in the bottom of the well. Eratosthenes was also able to observe that on that same day of the year in Alexandria, where he lived, the Sun was not directly overhead but was 7° south of the zenith. He determined this angle with

= 360
angle at C .
Inquiry 5-4 Assuming that the angle at C is 7° and that the distance of Alexandria from Syene is 5000 stadia (the units used by the ancient Greeks; stadia is the plural of stadium), what is the diameter of the Earth in stadia? Although the exact length of a stadium is unknown, it is thought to be about 0.16 of a kilometer (one-tenth of a mile). Using this value, compute the radius of the Earth.
Inquiry 5-5 What does Eratosthenes’ experiment assume about the shape of the Earth?
Inquiry 5-6 If the Earth were flat, what would be the value of the angle at C?
Small, nearby object Large, distant object
FIGURE 5-4. Two objects with the same angular sizes have diameters that are proportional to their distances.
FIGURE 5-5. Eratosthenes’s measurement of the diameter of the Earth. The difference in altitude of the Sun is proportional to the distance between the two points, which is known. This allows the length of one degree on the Earth’s surface to be determined.
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Hipparchus Perhaps the greatest of all ancient astronomers was Hipparchus (c. 150 B.C.E.). Many of the conclusions he drew were so sophisticated that it takes some knowledge of astronomy to appreciate how great his contributions were. He built an observatory, constructed the best astronomical instruments up to that time, and established a program of careful and systematic observations that resulted in the compilation of a great star catalog with 850 entries. His catalog used a celestial coordinate system similar to our modern one for cataloging the sky. It was Hipparchus who originated a system, which is still in use today in modified form, for estimating the brightness of stars. In addition, he used the older Babylonian observations and deduced Earth’s precession (Section 4.6), which is so slow that it takes almost 26,000 years for it to complete one cycle. Finally, he greatly developed trigonometry, which was, and still is, a useful tool for astronomy.
Other Deductions of the Greek Astronomers: The Distances of the Planets The Greeks estimated the relative distances of the planets from Earth using principles still in use today for determining distances to astronomical objects. They reasoned that the more distant a planet was, the more slowly it would appear to move across the sky. The effect is similar to what happens when we compare the apparent motion of a high-flying airplane with that of one that is flying low. The distant airplane appears to move slowly across the sky, whereas the low-flying one is seen for only a short time and then is gone. In the same way, the Greeks could put most of the naked-eye planets in order of their distance from Earth by assuming that increasing distances corresponded to slower apparent motions. The argument fails with Mercury and Venus, however, because it places Mercury closer to Earth.
Inquiry 5-7 What assumption results in Mercury being placed closer to Earth than Venus?
We have another, independent determination of relative planetary distances from their brightnesses.
We use an analogy: when you are driving at night on a two-lane road and wish to pass the car in front of you, you pass only if the headlights of an oncoming car are faint. When you do this, you are making an implicit assumption: all car headlights have about the same intrinsic brightness, with their apparent brightness depending on the distance. Similarly, if we assume that all planets have the same intrinsic brightness, then their apparent brightness as seen from Earth would depend on their distances from us. But, all the planets do not have the same intrinsic brightness, because their differences in size, distance from the Sun, surfaces, and atmospheric properties affect the amount of light they reflect in our direction. However, even allowing for these uncertainties, it is still possible to use this principle to rank the planets approximately in order of distance from the Earth.
The Apparent Motions of the Planets in Relation to the Stars Three additional observations of planetary motion were important in determining the details of the models the Greeks developed for the Solar System. We now summarize these observations, which were discussed in detail in the previous chapter:
Because the planets are considerably closer to us than the fixed stars, they appear to move against the starry background. Observations of Mars, Jupiter, and Saturn showed them to move generally eastward on the celestial sphere. In other words, stars move with the celestial sphere whereas the Sun, Moon, and planets move on the celestial sphere.
Occasionally, however, as discussed in Chapter 4, a planet’s motion becomes retrograde in which it changes from eastward to westward for up to several months before it slows down and again reverses its direction, resuming its normal easterly motion (see Figure 4-28).
Venus and Mercury are never more than 48° and 28°, respectively, from the Sun.
The Geocentric Model of the Solar System Where should the center of the system be? There were really only two obvious candidates — the Earth and the Sun. This question was considered carefully by Greek philosophers, and the fact that ultimately they reached an incorrect conclusion provides an interesting example of why the scientific method is not the simple turn-the-crank-and-the-answers- fall-out process that some sources describe it to be.
If the Sun is at the center of the Solar System, then Earth moves around it in space. Such a hypothesis provides a testable prediction. As shown in Figure 5-6, some of the stars ought to shift their apparent positions in the sky as the Earth moves from one side of its orbit to the other. Such parallax effects, as they are called, were looked for by many Greek observers, including Hipparchus, but were never found. The Greeks therefore concluded that the Earth was stationary in space.
Aristarchus, however, apparently supported the hypothesis that the Earth orbits the Sun, if surviving works of Archimedes and Plutarch are correct. Unfortunately, the work in which he put forth his hypothesis is lost, and apparently no other Greek astronomers held to this opinion. The model of a geocentric (Earth-centered) system easily won out over the heliocentric (Sun-centered) one.
Inquiry 5-8 What hypothesis might explain why stellar parallaxes were not observed, even though the Earth does in fact orbit the Sun?
The Hypothesis of Circular Motion A second feature of the Greek models of the Solar System provides another excellent example of the way in which scientific models can go astray — inflexible assumptions. Ever since the time of Pythagoras (c. 570–500 b.c.e.) and Aristotle, the
circle and the sphere had been considered to be the most perfect geometrical figures. Such perfection was expected in the natural world. (Even today, symmetry and simplicity in scientific thought are important concepts.) Thus, the great astronomer Hipparchus assumed the widely accepted idea that celestial objects, being perfect, could move only in circular orbits. In addition, this circular motion had to be uniform: an object moved the same angular distance in its orbit each day. However, the observations of retrograde motion made it impossible to model the Solar System with a simple geocentric system. Hipparchus cleverly got around this problem by extending the idea of uniform motion to include motion that was a combination of circular motions. In this way, as we will now see, it was eventually possible to model the retrograde motion of the planets using uniform circular motion.
The idea of combining circles geometrically is due to Apollonius of Perga (c. 265–190 b.c.e.), but Hipparchus was the first to apply the idea to actual celestial objects when he proposed a hypothesis to explain the motion of the Sun and Moon. However, he did not have enough data to apply it to the planets; this final step was carried out by the astronomer Ptolemy (whom we discuss shortly).
The explanation of retrograde motion by means of combinations of circular motion is illustrated in Figure 5-7a. The planet moves around a small circle called an epicycle, and the center of the epicycle moves around on a larger circle called the deferent. Because one can adjust the relative sizes of the epicycle and deferent, and the speeds with which motion takes place on each, it is not difficult for the model to produce complicated apparent motions. In particular, if the planet moves around the epicycle faster than the epicycle moves around the deferent, retrograde motion will be observed at certain points along the orbit, as shown in Figure 5-7b.
An explanation for the observation that Mercury and Venus were never far from the Sun (28° for Mercury and 48° for Venus) became possible by adding one more feature to the epicycle-deferent model. If the center of the epicycle of Venus, for example, is attached firmly to the line joining the Earth and Sun, then Venus will always remain near the Sun, as shown in Figure 5-8. In a similar fashion, Mercury’s epicycle is attached to the Earth- Sun line. Now, as the Sun moves along its deferent at a rate of about 1° per day, it carries Mercury and
Orbit o
s
FIGURE 5-6. The prediction of stellar parallax in a heliocentric universe. The apparent position of a star with respect to the background stars would appear to change if the Earth went around the Sun. (Not to scale.)
Venus along with it and the model explains the available observations.
Inquiry 5-9 What phases of Venus are possible for the model shown in Figure 5-8? (Hint: Make an enlarged drawing and shade the parts of Venus that are not illuminated by the Sun.)
Ptolemy The last of the great ancient astronomers was Claudius Ptolemy (c. 150 C.E.). In fact, the geocentric system that was previously discussed was passed down bearing his name — the Ptolemaic system. Ptolemy completed the explanation of planetary motion in terms of combinations of circles and added a number of complex refinements to Hipparchus’ system to improve the model’s agreement with observations.
Ptolemy’s aim was to produce a model that correctly predicted observations, which remains a key goal of science today. In that sense, he succeeded admirably. The Greeks believed that a true knowledge of reality was confined to the gods alone, and the best that humans could do was to produce descriptions of the observed world that would correctly predict the results of observation and experiment. Leaving aside the gods, these ideas are close to the spirit of modern science, in which even everyday concepts such as force and mass, which modern physicists think they understand quite well, have meaning only in reference to models and the measuring processes that are appropriate to them.
Models: Revisited Models are powerful tools not only to interpret observations, but to suggest possible new observations. In fact, an incorrect model can often lead one to ask the wrong questions, and thus can delay understanding. For example, Kapteyn’s model for the universe (described in Section 3.5) was wrong and delayed our understanding of Earth’s position in the Galaxy. In astronomy, models necessarily become more and more uncertain as the objects they represent become more distant, because we have less and less data on which to base the model. Despite this, as we will see throughout this book, models play a central role in our quest to understand the universe.
Inquiry 5-10 What are some models, in the sense just described, that are used in everyday life? (You may want to think about such fields as politics, economics, education, philosophy, psychology, etc.)
5.2 Astronomy during the Middle Ages No significant astronomical advances were made for the next 13 centuries. Greek civilization declined and the Greeks became subservient to the Romans. (This had already taken place by the time of Ptolemy.) If the Greeks were scientists, the
FIGURE 5-7. A geocentric explanation of retrograde motion. (a) The planet moves on an epicycle, which moves about the Earth on a deferent. (b) The motion resulting from a planet moving about an epicycle as it moves around the deferent.
FIGURE 5-8. An explanation of the motion of Venus, using epicycles. According to this model, Venus is always closer to Earth than the Sun is, and the angular separation of Venus and the Sun is never large.
Deferent Epicycle
Earth Planet
Deferent Epicycle
Romans were engineers, doing little basic science. Progress in science came to a virtual standstill in the West.
A different story emerged in the Middle East. The astronomical knowledge that originated in Greece reached the Arab world, and scientists and scholars there continued the pursuit of knowledge. There was a flowering of literature, art, and science throughout the Middle East. One of the most important events was the preservation of much of the scientific knowledge of the ancient world, at least in translation. Ptolemy’s great work, for example, survived as a book called the Almagest, Arabic for “the greatest.” In this form the Arabs eventually transmitted Ptolemy’s writings back to the West.
A considerable amount of original work was done in the Islamic world. Much of it was directed toward practical matters, such as navigation. Beautiful and intricate instruments such as the astrolabe (Figure 5-9) were perfected. Much theoretical work was accomplished, particularly in mathematics, aided by the Arabic system of numbers (that had come from India), as well as by perfection of the methods of algebra (the word algebra is Arabic). Imagine doing your income taxes using Roman numerals!
At the same time in China, an indigenous tradition of astronomy made important progress during long periods of political stability. Astronomy
was always closely connected with the state, and changes in dynasty were frequently accompanied by calendar reforms, providing secure employment for court astronomers. Early on, the Chinese developed the view that space was infinite in extent and that the stars floated independently in it. They were tireless and systematic observers of heavenly events; often our only records of novae and comets are Chinese. They invented advanced instrumentation for observing the sky and measuring time, including elaborate water-driven clocks. In many ways their science was more advanced than what the Greeks had accomplished. The influence of Chinese science on the West has been shown to have been much greater than has been generally recognized.
In Europe, by contrast, intellectual decadence was widespread. The principles of Greek science, received from the Arabs, were adopted as dogma by the increasingly powerful Roman Catholic Church. It was argued that there was only a finite amount of knowledge, and that the Greeks had discovered it all! Discovery was no longer necessary; humans had only to memorize what the Greeks had found to be true. The Ptolemaic picture became frozen into a rigid image of the universe with no room for independent thought and inquiry that would challenge this view of the universe.
5.3 The Heliocentric Hypothesis Once the renaissance in Europe began in the 14th century, changes began to occur in most fields of human endeavor: architecture, art, literature, philosophy, religion, and science. We will now look at the contribution of three scientists whose work challenged the status quo and helped move Western thought to a Sun-centered view.
Nicolaus Copernicus During the Middle Ages, tinkering with the Ptolemaic model had continued, but in an uncreative way — for example, by adding more epicycles. In fact, this model had become so complicated that King Alfonso of Castile is said to have remarked, when having the contemporary version of the Ptolemaic hypothesis explained to him, that if he had been around when the world was created he could have taught the Creator a thing or two.
FIGURE 5-9. A Persian astrolabe. The small leaves mark the positions of bright stars. By hanging the astrolabe by the ring and sighting along the movable sight, we can measure the altitude of a star.
The human mind cannot be contained forever. By the early sixteenth century the Renaissance was shaking the antique castles of Western intellectualism. Nicolaus Copernicus (1473– 1543), a Polish prelate with a strong mathematical bent as well as a new vision of the heavens (Figure 5-10), found himself in this changing atmosphere. As a student in Italy, Copernicus read of Aristarchus’s heliocentric hypothesis. Wherever he got the idea of placing the Sun at the center of the universe (Figure 5-11), there is no doubt that he felt this was the proper place, for as he himself says, “For who would place this lamp of a very beautiful temple in another or better place than this wherefrom it can illuminate everything at the same time?” An important benefit of the heliocentric model is that it allowed the sidereal orbital periods and relative distances of the planets to be determined from the observed synodic periods of the planets (as discussed at the end of Section 4.8).
Moreover, in a heliocentric model, the apparent retrograde motion of a planet like Mars (or any planet more distant from the Sun than the Earth) is a natural consequence of the relative motions of the Earth and Mars. Figure 5-12 shows how this comes about. At some time Mars will appear in direction 1. As each planet orbits the Sun, Mars’s line-of-sight position in relation to the background
stars will move toward the east, as indicated by lines 1 through 4. However, between times 4 and 6 Mars will appear to move backward, in a retrograde direction, toward the west. At point 6 Mars will have resumed its normal easterly track. As you can see, Mars shows retrograde motion whenever the Earth laps it in its orbit. The greater orbital speed of the Earth results in the apparent backward motion of Mars. One sees a similar effect when a fast automobile passes a slower one on the highway. Viewed from the faster car, the slower one appears to move backward when seen against the distant landscape.
FIGURE 5-10. Nicolaus Copernicus (1473–1543) proposed the daring idea that the Sun, not the Earth, was at the center of the universe.
FIGURE 5-11. The Copernican model of the Solar System.
FIGURE 5-12. Retrograde motion is a natural consequence of the heliocentric hypothesis and occurs as Earth approaches and then passes the planet. The loop appears because the orbital planes of the Earth and the planet are not identical.
Earth
Mars
Jupiter
Saturn
Mercury
Venus
Sun
Inquiry 5-11 What would be the relative positions of the Earth, Sun, and Mars for the retrograde motion to occur?
Despite its natural and simple explanation of retrograde motion and elegant positioning of the Sun in the center of the Solar System, many objections were raised against the Copernican hypothesis. For example, some people contended that if the Earth moved it would leave the Moon behind, an argument that was hard to answer because the concept of gravity had not yet been developed. More seriously, Copernicus’s hypothesis in this simple form could not explain the observations as accurately as Ptolemy’s hypothesis. For this reason it was necessary to introduce a large number of small epicycles into Copernicus’s heliocentric model to explain numerous small deviations of the observations from the idealized circular motion. However, Copernicus was opposed to these complicated refinements, feeling that they detracted from the perfection of uniform circular motion and were unworthy of celestial objects. The end result was that he actually needed more epicycles than Ptolemy to get equivalent accuracy, and this marred the aesthetic simplicity of his basic heliocentric hypothesis, which used circular orbits.
Inquiry 5-12 Employing the principle of Occam’s razor (the simplest model is preferred), from the point of view of astronomers of the time, which model of the Solar System would have been preferable, Ptolemy’s or Copernicus’s?
Unquestionably, Copernicus was aware of the difficulties with his model. For example, the model predicted that parallax should be observed for nearby stars, which it was not. However, he realized that the lack of parallax meant the stars were far away and the parallax was too small to be observed with the instruments then available. In addition, he was a rather timid man, and it is possibly for these reasons that publication of his hypotheses was delayed until he was literally on his deathbed. His work appeared in the year 1543 titled De Revolutionibus Orbium Cœlestium, which means On the Revolutions of the Heavenly Spheres. The book contained an unauthorized preface by the
Lutheran cleric Andreas Osiander stating that the book expounded a mathematical model and was not to be construed as a representation of reality. It is doubtful that Copernicus would have sanctioned its inclusion had he known of it.
Copernicus hypothesized a heliocentric universe; he did not prove it. Validation of his ideas was to require input from additional scientists over the following decades and centuries.
Tycho Brahe In 1546, a Danish nobleman was born who was to become the first great observational astronomer of the modern era — Tycho Brahe (pronounced Tee-ko Bra-he) (Figure 5-13). Tycho became interested in astronomy as a teenager when he observed a predicted eclipse of the Sun. Later, in 1572, he observed a supernova, an exploding star, that was so bright it could be observed in daytime. He favored the heliocentric hypothesis, but the lack of supporting observations caused him to rethink his position. Through careful observations, Tycho showed that the daily rotation of the Earth caused no parallactic shifts in the supernova’s position — in fact, the star did not move at all during the many months it was visible. He therefore concluded that it must be well beyond the Moon, and probably as distant as the stars. This caused instant problems for the older understanding of the universe, which had assumed that the heavens were unchangeable. Five years later, Tycho proved that the comet of 1577 was also more distant than the Moon. Because comets had been believed to be “exhalations of the Earth” this posed additional problems for the older ideas.
The fortunate combination of Tycho’s noble birth and his astronomical talents soon earned him the finest observatory in the Western world up to that time. It was financed from the coffers of King Frederick II of Denmark and was located on an island off the Danish coast. Since optical instruments had not yet been invented, the instruments were sighting devices similar to the quadrant, protractor and cross-staff, but larger in size and constructed with the greatest possible precision. Tycho’s principal instrument was the great mural quadrant (Figure 5-14). By sighting along the movable pointers in a manner not unlike sighting a gun, he could aim the pointers at two objects and read the angle between them with great accuracy for the time (about 0.5 minute of arc).
Tycho distinguished himself by the great care he brought to his observations. This included making multiple observations and averaging them together to minimize random uncertainties. So painstaking was his work that his observations had unprecedented accuracy and precision, not to be improved upon until the invention of the telescope. Another valuable aspect of his observations was that he observed continuously and systematically over many years, as Hipparchus had done. As a result, he amassed a large body of data of consistently high accuracy that was suitable for detailed analysis. Why an aristocrat such as Tycho could become as obsessed with astronomy as he was is something of a mystery, and it is miraculous that his obsession led him to produce such excellent astronomical data. Another fortunate historical accident is that this body of work eventually fell into the hands of Johannes Kepler (1571–1630), who, as it turned out, was uniquely equipped to put it to good use.
Tycho developed an alternative Solar System model with the Earth at the center, the Sun orbiting the Earth, and the other planets orbiting the Sun (Figure 5-15). This Tychonic system had the advantages of reproducing the observed motions of the planets while predicting no stellar parallax. It
became the preferred model in the late 16th and 17th centuries and was, in fact, accepted by the Roman Catholic Church until some critical observations, discussed later, were made.
Johannes Kepler Although Tycho was a great observer, he realized that he was not a strong mathematician, and he longed for a collaborator who could properly analyze and interpret his high quality observations. His accumulated observations on the positions of Mars posed the greatest difficulties for theoretical interpretation because Mars deviated from its predicted position more than any other planet. Having moved to Prague toward the end of his life, Tycho acquired a young assistant who, by his brilliance in mathematics, transcended his lower- class origins. Johannes Kepler (Figure 5-16) had sufficient genius to overthrow the sterile hypotheses of the geocentric universe and perfect circular motion and to substitute a new and truer description of the Solar System.
In his book The Sleepwalkers, Arthur Koestler describes the contrast between Tycho and Kepler. Tycho was an aristocrat by birth and arrogant by
FIGURE 5-13. Tycho Brahe (1546–1601) was the greatest naked- eye observer in the history of astronomy.
FIGURE 5-14. Tycho’s great mural quadrant, his principal instrument for measuring star positions.
nature, accustomed to power and privilege. He ran his observatory like a court, arriving for the night in full formal dress and arrogantly ordering his assistants about. Always sure of himself, in his college days he had lost his nose in a duel over a point of mathematics, and ever afterward he wore a false nose made of silver or bronze. He had a dwarf servant named Jepp, who followed his master about like a pet and received scraps of food at the supper table. Kepler, by contrast, had no advantages of birth and reached a position of eminence by virtue of brilliance and sheer tenacity. He had little in common with Tycho, other than irascibility and an interest in astronomy. During much of his life, he had to earn his living by casting horoscopes, and in fact he cast his own horoscope every day. He was a compulsive individual who kept meticulous notes on everything he thought and did. A hypochondriac as well, Kepler recorded an hour-by-hour chronicle of his physical maladies. But because his diary also included his scientific efforts, we have been left a detailed account of the paths through which he wandered in making his momentous discoveries.
Kepler took Tycho’s observations of Mars (literally took them, because Tycho’s heirs had other plans for the data!) and set out to find a geometric curve that would represent its motion accurately. (Recent evidence indicates that Tycho may have been poisoned. Was this done by Kepler to obtain the data?) The calculations were extremely tedious, and he made many mistakes. Without calculators, every calculation had to be done by long multiplication or division. Yet, something in Kepler’s character kept him working persistently at the problem. Time after time he rejected solutions that had taken him months, even years, to work out, because they failed to agree with the observations as accurately as he knew they should. At last, after a total of eight years, he boldly rejected the hallowed ancient idea that planetary motion must take place
on circular paths, thus ending two millennia of tradition. He describes in his diaries the fear and trembling he suffered in his mind when he took this step, which you will learn about next.
5.4 Kepler’s Laws of Planetary Motion Kepler’s laws of planetary motion are not laws in the sense that we use the word today, because there was at that time no conception of the physical forces that caused these motions. His laws would be better described as empirical descriptions of planetary motions, meaning they were derived solely on the basis of observation with no theoretical underpinning. However, for the first time they gave a description that was as accurate as the best available data allowed. Although the word “law” is a misnomer, it has been in use for many years and we are stuck with it.
Kepler found that Mars moved around the Sun on a mathematical curve called an ellipse. Figure 5-17 shows a simple way to draw this curve. The two points located by pins are called the foci of the ellipse (foci is the plural of focus). Ellipses
FIGURE 5-16. Johannes Kepler (1571–1630) refined Copernicus’s heliocentric hypothesis with the laws of elliptical motion.
Figure 5-15. The Tychonic model of the solar system, in which the Earth is at the center, the Sun orbits it, and the other planets orbit the Sun.
are characterized by the length of their longest dimension (called the major axis — B in the figure) and their degree of non-circularity. If the two foci coincide, we have a circle. The further apart the foci, the greater the flattening, or eccentricity, of the ellipse.
Kepler’s first law can be stated in this way: The orbits of the planets are ellipses, with the Sun at one focus.
Figure 5-18 illustrates Kepler’s first law. The flattening of the ellipse is much exaggerated — no planet has an orbit this elliptical. In fact, if one were to observe these orbits from outside the Solar System, it would be difficult using the naked eye to distinguish most of them from circles. This is the principal reason why it took so long to discover the true shapes of the planetary orbits. As shown, the second focus is empty. Eccentric ellipses are also found; comets are examples of objects that move in highly eccentric ellipses.
The distance of a planet from the Sun varies as the planet moves. For example, Earth is almost 5 million kilometers closer to the Sun in January than in July. Kepler found that as Mars’s distance from the Sun varied, its orbital speed also varied, being greatest when the planet was closest to the Sun and least when it was farthest away. He found even more: there is a definite relationship between a planet’s distance from the Sun and its speed in its orbit.
Figure 5-19 indicates four positions on an elliptical orbit. During the interval of time between points A and B, a line from the planet to the Sun sweeps out the long, skinny area shaded on the left. Between the equal time interval from C to D, the line from the planet to the Sun sweeps out the fat area on the right. Kepler found that this area is equal in size to the area swept out between times A and B. In other words, because the time intervals from A to B and C to D are equal, and because the distance from C to D is greater than that from A to B, the planet must be moving faster from C to D than from A to B. For example, the motion of the Earth is more rapid in January than in July.
Kepler’s second law of planetary motion can be formulated as follows:
The line joining a planet and the Sun sweeps out equal areas in equal amounts of time.
Kepler’s third law was discovered much later than the first two. It appears, almost as an afterthought, in a rambling, mystical work of Kepler’s titled Mysterium Cosmographicum, or Cosmic Mystery. The emphasis of this work lay in
FIGURE 5-17. Drawing an ellipse with two pins and a loop of string. The pins mark the position of the two foci of the ellipse. The eccentricity of an ellipse is defined to be e = F/B, where F is the distance between the two foci and B is the longest axis (the major axis) of the ellipse. It characterizes how far the ellipse deviates from being a perfect circle.
FIGURE 5-19. Kepler’s second law: If the planet takes the same amount of time to go from A to B as it does to go from C to D, then the two shaded areas are equal.
FIGURE 5-18. Kepler’s first law: Planets move in ellipses with the Sun at one focus.
Drawn ellipse
mystical speculation on the cosmos rather than on the mathematical relationship expressed in the third law.
Kepler’s third law is a relationship between the average distance of a planet from the Sun (which equals half the length of the major axis, or semi- major axis) and the planet’s sidereal period, the length of time it takes for the planet to orbit once around the Sun. The third law is most easily expressed mathematically. It states that if P is the orbital period of a planet measured in years and A its average distance from the Sun measured in astronomical units, then
The period squared equals the average distance cubed.
Expressed symbolically:
P 2 = A 3.
This equation clearly works for planet Earth. Earth’s orbital period is 1 year and its average distance from the Sun is 1 AU, so P = 1 and A = 1, and 12 = 13.

A = P 23 .

P = 9 9 = 9 × 3= 27 years.

A = P 23 = 823 = 643 = 4 AU
because 4 × 4 × 4 = 64. Problems using Kepler’s third law are more easily solved with an electronic calculator.
Inquiry 5-13 The average distance of Mars from the Sun is 1.52 AU. To the nearest tenth of a year, what is its orbital period?
Inquiry 5-14 Uranus’s orbital period is 84 years. To the nearest tenth of an AU, what is its average distance from the Sun? (Hint: 20 × 20 × 20 = 8000.)
The sidereal period is not a directly observable quantity because we are on a moving Earth. What is observable, however, is the planet’s synodic period, which is the time interval between successive occurrences of a given planetary configuration (described in Section 4.8). For example, the synodic period is the time interval between, say, two successive oppositions or conjunctions.
To clarify the difference between synodic and sidereal periods, we draw an analogy using two runners on a track. The sidereal period is the time from the starting line back to the starting line. The synodic period is the time for one runner to lap the other one — that is, to move ahead of and then catch-up to the other runner again. Because both runners are moving, one runner cannot directly measure the sidereal period of the other. However, each runner could easily determine the time it takes to lap the other runner — the synodic period. If you then know your own sidereal period, you can compute the other runner’s sidereal period. (The mathematics and some examples are given in Appendix A6.)
In summary, we observe an object’s synodic period, compute the sidereal period, plug it into Kepler’s third law, and determine the planet’s semi- major axis. Figure 5-20 illustrates the procedure and the logic involved.
Kepler’s three laws, as described here, are exact only in the idealized case of two isolated objects. In the real Solar System, the large number of objects that interact with one another causes deviations from the ideal. Nevertheless, the laws are accurate enough to describe the motions of the planets and agree with the best naked-eye observations, which happened to be Tycho’s. Had Tycho’s observations been somewhat less accurate, or had Kepler been less exacting in his criteria for agreement between hypothesis and observation, Kepler’s three laws might not have been discovered until much later in history.
Kepler’s three laws overthrew the earlier Greek ideas; no longer did we believe that the planets moved with uniform speeds in circular orbits. These laws are general in that they apply to any two objects circling each other under the influence of
gravity. They may be applied to natural or artificial satellites orbiting planets, stars orbiting stars, stars orbiting galaxies, or even galaxies orbiting galaxies. Thus, we will return to Kepler’s laws at many places throughout our study of astronomy.
5.5 The Search for Underlying Laws Kepler’s works are purely empirical, meaning they were derived solely on the basis of observation with no theoretical underpinning. Nevertheless, Galileo’s experimental work on the motion of objects and Newton’s beautiful and general theoretical development of his laws of motion show the validity of Kepler’s work.
Galileo Galilei Galileo Galilei (1564–1642), shown in the chapter opening illustration, was a contemporary of Kepler’s, was an early convert to the heliocentric hypothesis, and argued vigorously for it. In his later years this brought him into conflict with the Inquisition, and he was forced to recant his beliefs and live under a form of house arrest — a pitiful, partially blind old man, fearful of the instruments of torture that apparently had been shown to him.
Kepler’s laws, successful as they were in predicting the positions of the planets in the sky,
were no proof that the Earth moves. Skeptics could reasonably claim that the three laws were merely mathematical tricks that happened to give accurate predictions and not in any way the truth about the physical nature of the universe.
Galileo clung to the hypothesis of circular motion for heavenly objects while conducting experiments in dynamics that eventually overthrew it. Remember, the ideas of Aristotle had been the accepted ones for centuries. And, they were what the Catholic Church, as the arbiter of authority, said was truth.
Galileo, in a revolutionary move, decided to let observation, not human reason, be the ultimate arbiter of physical reality. He performed experiments to decide whether a particular hypothesis was true. Galileo’s insistence on making and using observations to find scientific “truth” led the way to the modern approach to science with the use of inductive logic. For example, he investigated the motions of objects under various conditions by rolling balls down inclined planes, thus showing that a state of motion was as natural as a state of rest, and that to change either state required an outside influence. In another famous case, it had been claimed by believers in Aristotelian physics that an object that was twice as heavy as a second one would fall twice as fast. Perhaps by actual demonstration, or maybe only with thought experiments, Galileo showed that such objects accelerated at nearly the same rate and hypothesized that only the resistance of the air kept the rates from being identical. The truth of Galileo’s position was demonstrated dramatically when the Apollo 15 astronauts dropped a feather and a hammer together in the vacuum of the Moon. The two objects reached the lunar surface simultaneously. You, too, can perform Galileo’s experiment by simultaneously dropping, say, a paper clip and a heavy shoe.
In performing his experiments, Galileo formulated the concept of inertia, which is an object’s tendency to resist a change in its motion. The amount of inertia is measured by what we call mass, which is, loosely, a measure of the amount of matter in an object. The greater the mass, the more inertia, and the harder it is to set an object into motion. Furthermore, the greater the mass, the more inertia, and the harder it is to change an object’s motion.
Galileo’s greatest fame comes from his discoveries in astronomy, even though his
Observations
Which, along with
Results in the
Figure 5-20. The procedure for finding a planet’s semi-major axis given an observation of its synodic period.
contributions to physics were probably equally important. Upon learning of the invention of the telescope, he constructed one of his own and turned it to the heavens. His most important astronomical discoveries can be divided into three groups: those that argued most strongly for the heliocentric model (numbers 1 and 2 below), those that showed the universe is not perfect and unchangeable (numbers 3 and 4), and other observations (numbers 5 and 6).
1. He found that the planets show phases, as the Moon does. This was particularly important in the case of Venus, which exhibits all phases. In the Ptolemaic model, Venus could only show new and crescent phases, because it would always be closer to Earth than is the Sun and show only a small fraction of the illuminated side of the planet (Figure 5-21a). Only if Venus were sometimes farther from the Earth than the Sun could a larger fraction of the illuminated side be seen from Earth and thus exhibit a gibbous phase (Figure 5-21b). This discovery showed that the Ptolemaic picture of the universe must be incorrect. An observation such as this is an example of a critical observation, which is one that, by itself, is sufficient to favor one model over another. In this case, however, preconceived ideas won out over observation, and the true meaning of this observation was discounted.
2. He discovered the four brightest moons of Jupiter by following the variations in their positions from night to night and showing that they always remained close to the moving planet.
3. He examined the Moon in detail, and discussed and named many of its surface features — its mountains, the dark areas called maria (“seas”), craters, and the like.
4. He studied sunspots by observing the Sun at sunrise or sunset when the Sun is not too bright and not so harmful to the eyes. By following sunspots across the visible disk of the Sun, he showed that the Sun spins. Together with his observations of the Moon, this discovery struck a blow to the older assumptions of astronomy that had maintained that the heavenly objects were perfect, without blemishes. The existence of sunspots, mountains, and craters showed that these bodies were neither perfect nor without blemishes.
5. He found that the diffuse band of light across the sky, known as the Via Lactea or the Milky Way, actually consists of myriad stars too faint to be seen by the naked eye. This discovery was the first step toward a modern view of the nature of our galaxy.
6. While observing with a telescope, he discovered that the planets had a visible disk, whereas the stars remained infinitesimal points even at the highest magnification.
Inquiry 5-15 What might be the reason for Galileo’s sixth observation?
Inquiry 5-16 One of the arguments against Copernicus’s hypothesis was that the Earth could not be moving because it would leave the Moon behind. How does Galileo’s second observation help disprove that contention?
Isaac Newton As important as the contributions of Galileo and Kepler were in altering long-held ideas about astronomy and physics, they did not create a single comprehensive system that unified the physics of the heavens with the physics of the Earth. Such
(b)
(a)
FIGURE 5-21. (a) Venus’s phases under the Ptolemaic model, as viewed from Earth. The path shown is Venus’s epicycle; see Figure 5-8 for another view. Note that only crescent phases are seen (b) Venus’s phases in a heliocentric model, as viewed from Earth. Because Venus can go behind the Sun, a full set of phases including gibbous phases is possible.
unification awaited the genius of Isaac Newton (1642–1727), who was born in the year of Galileo’s death (Figure 5-22). Newton provided us with a system of physical laws — a theory of motion — that is so fundamental that even today they are the basis for much scientific activity. It would be difficult here to give an adequate idea of the power of Newton’s intellect and the importance of his accomplishments; we can only scratch the surface. As a young man, Newton invented the tools of the branch of mathematics known as calculus, did fundamental work on optics and color vision, and developed the reflecting telescope, as well as proposing his famous laws of motion and gravity. He made such an impression on his contemporaries that the poet Alexander Pope was moved to write: “Nature and Nature’s laws lay hid in night:/God said Let Newton be! and all was light.”
To begin to understand Newton’s contributions to the study of moving objects we first define some terms. Velocity describes the change in position of an object divided by the time interval over which the change occurs. Velocity describes not only the speed of an object but its direction. For example, while a car traveling 100 km per hour north moves at the same speed as a car traveling 100 km per hour east, their velocities are different because their directions of motion are different.2 Motion is rarely constant in either amount or direction. Any change in an object’s velocity, in either its speed or direction, is called acceleration. Acceleration is the term used even if you are slowing down. Even when traveling at a constant speed on a Ferris wheel, you are accelerating because the direction is continuously changing.
Inquiry 5-17 Use the definition of acceleration to list the accelerators in a standard passenger car.
Newton is most renowned for his three laws of motion and his law of gravity. His three laws of motion, first published in his Philosophiae Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy), are generalized statements — a theory — concerning the motions and interactions of objects. The first law concerns inertia and is often referred to as the law of inertia. 2 Velocity is an example of a vector, a quantity having size and direction. speed, on the other hand, is called a scalar because it has only a size associated with it.
In the absence of an outside influence, an object at rest will stay at rest, while an object in motion will continue to move at a constant speed in a straight line.
As an example consider a box that is not tied down in the back of a pick-up truck. If the box is located toward the front of the truck bed and the truck accelerates quickly forward, the box will move toward the rear because it “wants” to stay at rest in relation to the street as the truck moves forward. On the other hand, if the box is toward the rear of the truck bed and the brakes are applied rapidly, the box will move toward the front, since it continues moving in the original direction. The more mass the box has, the more inertia it has, and the more the box will resist motion and tend to stay in its original location.
Newton’s first law, the law of inertia, was revolutionary. Since the time of Aristotle, it had been assumed that an object required some continual action on it to remain in motion, unless that motion was a part of the natural motion of heavy or light substances. The philosophers of the Middle Ages had refined Aristotle’s view by asserting that an object was imparted a certain
FIGURE 5-22. Sir Isaac Newton (1642–1727) laid a firm theoretical basis in mathematics and physics for planetary motions, which would eventually reign unchallenged until Einstein’s theories in the early 1900’s.
quantity of “impetus” by the hand or bow string that launched it, and that it would continue to move until it had used up its impetus. Impetus, however, was an example of what Newton referred to as an occult quality, for it could not be perceived in any fashion. It was merely supposed to exist in order to preserve Aristotle’s ideas. Thus, all motion seemed to require the invocation of an unknown substance to explain it. Newton’s simple statement swept away all that chaos and mental clutter, replacing it with brilliant simplicity.
Inquiry 5-18 According to Aristotle, the planets continue to move because that is their nature. What reason would Newton have given?
What happens if there is an outside influence? In answering this question, Newton called any outside influence acting on an object and causing it to accelerate a force. Thus the second law of motion:
The total force on an object is equal to the product of the object’s mass times its acceleration.
This expresses the famous formula F = Ma, which is the basis for studying the motions of particles of all sizes, from submicroscopic atoms to stars in their orbits in the Galaxy. This concept turned out to be enormously powerful in stating that forces produce accelerations, and the size of the acceleration depends on the mass. Furthermore, an object can accelerate only if there is a force acting on it; remove the force and the object no longer accelerates, but continues moving at the same speed and in the same direction. Finally, because acceleration has size and direction, so does force.
If an object accelerates, either its speed or its direction of motion changes, or both change. If an object accelerates in the direction of motion, speed increases. Acceleration in the direction opposite to the direction of motion decreases the speed. A sideways acceleration changes its direction of motion. Any acceleration (forward, backward, or sideways) requires an external force that points in the direction of the acceleration. Further, the greater an object’s mass, the larger the force required to produce a given acceleration.
Inquiry 5-19 As an example of Newton’s second law, consider a game of kick-the-can played with two cans — one empty and one filled with concrete. Which can has the greater mass? If someone came along and kicked the two cans with exactly the same force, which can would have the greater acceleration? Explain this in terms of what you have learned.
In the third law, Newton shows that forces always come in pairs.
For every force that an object exerts on a second object, there is an equal and opposite force exerted by the second object on the first.
As an example of pairs of forces, when someone touches you, you also touch that person; one touch cannot occur without the other. If you and a friend are in rowboats on a lake and one of you pushes against the other with an oar, both boats will move, but in opposite directions. The simple act of sitting in a chair illustrates this law: you exert a downward force on the chair, and the chair exerts an equal and opposite force on you. While a falling brick exerts a force on the air molecules it encounters, pushing them ahead of it, the air molecules exert an equal and opposite force on the brick, thus slightly slowing its motion. A final example is a rocket. If one force is the rocket engine pushing gases backward, the opposite force is that of the gases pushing on the rocket engine in the forward direction. That is why rockets work in the vacuum of space; rockets do not operate because of the exhaust gases pushing back on the Earth.
Inquiry 5-20 In the kick-the-can game in Inquiry 5-19, what does this third law predict about the effect on the foot that kicks the two cans?
Newton’s Law of Gravity Newton’s three laws, especially the second, enable us to calculate the acceleration of an object and
hence its motion, but we must calculate the forces first. Heavenly objects such as the Sun, Moon, and planets cannot exert forces on each other directly, since they do not touch each other. Newton proposed that they exert an attractive force on each other at a distance, that is, across empty space. He called this force gravitation. His universal law of gravitation expresses the gravitational force of attraction between any two objects. What he discovered is that gravity is a characteristic of any object that has mass. This means that you and the person next to you have a gravitational attraction between one another.

d2
where F is the force exerted by an object of mass M1 on an object of mass M2, d is the distance between the centers of the two objects, and G is a numerical constant called the universal gravitational constant. The value of the universal constant depends on the units used to express the other quantities in the equation. From this formula, it is possible to calculate how the force between the objects varies with the distance between them, and therefore what the acceleration is at each moment.
Inquiry 5-21 If the distance between mass M
1 and mass M 2
is made four times greater, by what factor is the gravitational force between them changed? Is it increased or decreased?
FIGURE 5-23. The motion of the Moon around the Earth according to Newton. The Moon “falls” toward the Earth just enough to keep it on a curved elliptical path around the planet.
Inquiry 5-22 Which force is greater: Earth’s gravitational pull on the Moon or the Moon’s gravitational pull on the Earth?
Although the idea that objects of different masses fall at the same rate was experimentally shown by Galileo to be true, it is not intuitive. However, it is consistent with Newton’s universal law of gravitation and his second law, as shown mathematically in Appendix A7.
A measure of Newton’s genius is that he proposed that his laws of motion and gravity applied not only to objects on Earth but also to all objects in the universe. His insight occurred when he compared the force of gravity exerted by the Earth on an object, allegedly a falling apple, with the gravitational force required to keep the Moon in its orbit around the Earth. As shown in Figure 5-23, if there were no gravitational force exerted by the Earth on the Moon, it would travel in a straight line past the Earth in accordance with Newton’s first law. To make the Moon travel around the Earth requires a force toward the Earth. The Earth’s gravitational force on the Moon causes it to deviate from a straight line and follow a curved path around the Earth. For each kilometer the Moon moves in its orbit, the Moon must fall 0.0014 m (0.14 cm) toward the Earth in order to stay in its elliptical orbit. In other words, the Earth’s force of gravity causes the Moon to accelerate just enough to maintain its distance from the Earth. For this reason, the Moon’s orbital motion can be described as resulting from the Moon’s falling toward the Earth’s center! From such considerations Newton
Earth
Force of gravity
Part 1 • Discovering the Science of Astronomy20
found that the actual force required to hold the Moon in its orbit around the Earth agreed well with the value he computed theoretically. Thus, observations showed his description to be a valid scientific theory. It is one that can be applied throughout the universe anywhere and at any time.
Weight Weight is the force one object exerts on another object. In particular, your weight is the gravitational force exerted by the Earth’s mass on your body. Your weight depends on the mass of your body, the mass of the Earth, and your distance from the Earth’s center, which is the Earth’s radius. For this reason, your weight would be different should you travel to a different planet having a different mass and size than the Earth; your mass, however, would be the same. For example, a person weighing 120 lbs on Earth would weigh 20 lbs on the Moon but 305 lbs on Jupiter. Interested readers can find weight expressed mathematically in Appendix A8.
To understand why an astronaut floats in the International Space Station even
though there is still gravity in space, you should do Discovery 5-1 at this time.
Momentum You probably have some intuitive feel for the word momentum. A train moving at 30 km per hour has more momentum than a bicycle moving at the same speed; it would have more impact and effect if it ran into something! For an object moving in a straight line, linear momentum is defined as the object’s mass times its velocity.
Newton’s first law can also be expressed as the principle of the conservation of linear momentum. This says that for an isolated system of objects without any outside force acting on them, the sum of the linear momenta of all objects in a system is always the same. For example, when a billiard ball slows down after hitting another ball, the struck ball must speed up by the same amount (since the masses are equal).
Most objects move in curved paths, and the concept of angular momentum comes into play. The amount of angular momentum possessed by a planet in a circular orbit around the Sun is given by:
Angular momentum = Mass × Speed × Distance from planet to Sun.
Expressed symbolically, if M is the mass of an object and v its speed when at a distance d, the angular momentum is given by
Angular momentum = Mvd.
Like linear momentum, angular momentum is conserved for an isolated system. If the mass of an orbiting planet in an elliptical orbit remains constant, the only way for angular momentum to remain constant is for the speed of the planet to increase as the distance decreases, and vice versa. This is exactly what Kepler’s second law says. The concept of angular momentum conservation is important in understanding not only the motions of planets but also such diverse subjects as the formation of the Solar System, the motions of stars, and the shapes of galaxies. As applied to the orbits of the planets, Kepler’s second law states that the angular momentum of an orbiting planet remains constant.
A spinning object also has angular momentum. In this case, d represents the distance from the rotation axis to the surface — in other words, the radius. As a spinning gas cloud collapses to form a star, conservation of angular momentum requires that it increase its rotation speed, just as an ice skater increases rotation speed as the arms are brought in (Figure 5-24).
For simplicity we have defined angular momentum in terms of circular orbits, but the
concept can be extended to elliptical orbits.
Inquiry 5-23 Suppose a planet’s farthest distance from the Sun is twice as great as its nearest distance. How many times greater is its orbital speed when the planet is nearest the Sun than when it is farthest away?
Newton’s Generalization of Orbital Motion While Kepler showed that planets moved in elliptical rather than circular orbits, Newton was able to show that moving objects could be in other types of orbits, too. The other possible orbital shapes
21Chapter 5 • The Historical Quest to Model the Solar System
are the parabola and the hyperbola. (Objects in such orbits are said to be in parabolic and hyperbolic orbits.) The various orbital shapes are best visualized geometrically and then understood by looking at the energy of the orbiting object.
Circles, ellipses, parabolas, and hyperbolas are all shapes known as conic sections because they are most readily visualized as formed from a cone (Figure 5-25). Conic sections result where a plane intersects the cone. For example, the curve formed by the intersection between a plane perpendicular to the cone’s axis and the cone will be a circle, as shown in purple. If the plane is not perpendicular to the axis the intersection will be an ellipse (red). If the plane is parallel to the cone’s axis, the intersecting shape is a hyperbola (yellow). Finally, if the plane is parallel to the cone’s sides, the shape will be a parabola (orange).
Thus, objects moving in elliptical and circular orbits are in closed, repeating orbits; planets in elliptical orbits around the Sun are gravitationally bound to the Sun, which governs its motion. Objects in parabolic or hyperbolic orbits, however, are not bound but are in open orbits; the motions are not repeated and the orbiting object will continue moving away from the Sun forever. Examples of such objects are the Voyager spacecrafts that are escaping from the Solar System and the non- periodic comets (Chapter 7).
The type of orbit a body has is determined by the amount of energy the orbiting body has. Orbital motion includes two types of energy.
The first is potential energy. A rock sitting on a cliff has gravitational potential energy because it has the potential for motion if it were to fall off. Orbiting bodies have potential energy because of the gravitational attraction of the attracting body.
The second type of energy is kinetic energy, which is energy of motion. All moving bodies have kinetic energy. The more massive the body the greater the kinetic energy, and the faster it is moving the greater the kinetic energy. Thus, even a small asteroid moving at, say, 20 km/s contains significant kinetic energy such that if it collided with Earth it would make a big dent in the surface.
The total energy is the sum of the kinetic and potential energies. The total energy is constant, which is the concept called the conservation of energy. Thus, since the potential energy decreases as an orbiting body approaches the Sun, its kinetic energy (that is, its speed) must increase.
Objects in elliptical orbits have more potential energy than kinetic energy and will therefore stay bound. If there is more kinetic energy than
FIGURE 5-24. An ice skater’s spin rate increases due to conservation of angular momentum when the arms are pulled in.
Figure 5-25. Defining the ellipse, parabola, and hyperbola from in terms of sections of a cone
(a) (b)
potential energy, the orbit is open and the orbiting body will escape.
We can summarize this discussion by considering the motion of a rocket. A rocket with the right amount of fuel would go into a bound orbit around the Earth (a circular or elliptical orbit). With just the right amount of additional energy, the rocket would travel in an open, parabolic orbit with exactly the right speed to escape from Earth. Additional energy beyond that amount would cause the rocket to be in an open hyperbolic orbit and to escape Earth’s gravity entirely.
The ideas presented here will be revisited in numerous contexts, from the orbits of comets, to molecules in planetary atmospheres, to the escape of stars, and to galaxies in collisions.
Newton’s Form of Kepler’s Third Law Newton was able to show, using the calculus that he invented to solve the problem of orbital motion, that Kepler’s empirical laws of planetary motion were direct consequences of his own more fundamental theory of motion. For example, he was able to show mathematically that a planet governed by a gravitational force that varies inversely with the square of the distance from the Sun would follow an elliptical orbit. This was a major accomplishment! In addition, using his laws he found Kepler’s third law was even more general than originally proposed. In Newton’s formulation, it becomes
(M1 + M2)P 2 = A 3
where again P is measured in years and A in astronomical units, and the additional factor (M1 + M2) is the sum of the masses of the two objects in orbit about each other, expressed in terms of the mass of the Sun.
When applied to the planets, (M1 + M2) is almost precisely equal to 1, because the mass of the Sun is so much greater than that of any of the planets. (This is why Kepler did not discover the mass factor.) The form in which Newton wrote Kepler’s third law turns out to be much more useful than Kepler’s form. For example, if you observe the orbital period and separation of a planet’s moon, you can compute the mass of the planet; or, by observing the size of a double star’s orbit and the
period of the orbiting stars, you can calculate the total mass of the stars in the binary system. The revised ideas even apply to galaxies in orbit around each other. In fact, much of the observational data we have on the masses of objects in the universe come from applying Newton’s form of Kepler’s third law.3
As an example of its use, if two stars identical to the Sun (so the mass of each in solar units is 1) orbit about each other at a distance of 0.1 AU, we can use Newton’s form of Kepler’s third law as follows:
(1 + 1)P 2 = (0.1) 3 P 2 = 1/2 × 10 –3
P = 2.2 × 10 –2 years = 8.2 days. However, if the binary system had a star identical
to the Sun and one six times as massive, the period would be 4.4 days.
In this example, the masses of the two objects are the same or not much different. If one object is many times more massive than the other (e.g., a star orbiting a galaxy), the sum of the two masses is
basically the same as the more massive object.
Inquiry 5-24 Suppose a double star is observed to have a period of 10 years and an average distance between the two stars of 8 AU. What is the sum of the masses of the two stars in the system?
Inquiry 5-25 Consider a 10 solar mass star orbiting a galaxy whose mass is 10 11 solar masses. What is the value of the sum of the masses? How long should it take such a star to orbit the galaxy if it is located 10,000 parsecs from the galaxy’s center?

G A3
where G is the Universal gravitational constant from Newton’s gravitation law and has a value dependent on the units chosen for the masses, P and A. If the value in the back of the book is used, then the masses are in kilograms, P is in seconds and A is in meters. This form must be used if the units of the variables are not in terms of the Sun’s mass, years, and AU.
as universal as Newton had claimed. Newton himself had difficulty explaining the motion of the Moon with accuracy comparable to observations because the gravitational force of the Sun, as well as that of the Earth, was important and altered the orbit away from purely elliptical motion. Eventually, however, the mathematical problems of computing the Moon’s motion were solved satisfactorily. In a similar way, as telescopes and observational techniques improved, the gravitational effects of one planet on another explained observed deviations from purely elliptical motion.
An early triumph of Newton’s theory of motion and gravity was the discovery by Edmund Halley, a contemporary of Newton, that several comets observed in the past were in fact one and the same object. Halley predicted that it would reappear in 1758. Although Halley died in 1742, its reappearance right on time electrified Europe and led to a great flood of comet hunting. The comet, of course, was named for Halley. It returned again, right on schedule during late 1985; its next appointment near the Sun will be in the year 2061. Due to gravitational perturbations by the planets, the exact date cannot yet be determined.
In 1781, another discovery was made that at first led to one of Newtonian theory’s greatest challenges and later to one of its greatest triumphs. Sir William Herschel, during a routine survey of the sky, discovered a new object that, unlike the stars, showed a clearly visible disk in his telescope. It was soon apparent that the object was moving with respect to the stars, and when the theory of gravity was used to establish its orbit, it was found to be a new planet farther from the Sun than any previously known. Eventually the new planet was named Uranus, after the Greek god of the heavens.
The discovery of a new planet at first seemed to be a confirmation of Newtonian physics. But after many years of observations, it became evident that something was wrong. No matter what was done, the orbit of Uranus could not be brought into agreement with the predictions of the law of gravity. In late 1845 and early 1846, two young mathematicians simultaneously proposed that there was another, as yet undiscovered planet that was affecting Uranus’s orbit and causing it to deviate from the predictions of Newton’s laws. The English astronomer John Couch Adams sent
his calculations to the Astronomer Royal, who failed to appreciate the importance of the young man’s work. The Frenchman U. J. J. Leverrier had better luck. His predictions were sent to the Berlin Observatory, where the new planet was found almost immediately, close to the predicted position. Both Adams and Leverrier now share the honor of Neptune’s discovery.4
In conclusion, the point is that Newton’s theory makes wide-ranging testable predictions that observations over hundreds of years verify.
5.6 Evidence in Favor of the Heliocentric Hypothesis You have now learned about the history of the ideas leading up to the development of the heliocentric model of our Solar System. In addition, you have learned the concepts that govern all motions in the universe. The question now is, what observational evidence shows the heliocentric model is valid? All that is required to validate it is clear evidence that the Earth moves. Astronomers now sought such observational evidence.
It took so long to demonstrate the Earth’s motion because it is not so easy to do. Although one might argue that the observation of different constellations at different times of the year unambiguously demonstrates the Earth’s motion, one could argue that the presence of different constellations results from a motion of the celestial sphere rather than the Earth. Or one might argue that the presence of seasons proves the Earth’s motion. However, hypothesizing a north-south motion of the Sun on the celestial sphere could explain what produces the seasons. What we need are authoritative arguments that cannot be explained away like these two can.
It was not until the year 1729 that definitive evidence of Earth’s motion was first observed. In that year James Bradley, the British Astronomer Royal, was trying to observe stellar parallax (see Figure 5-6). However, rather than observing parallax, he discovered instead a phenomenon called the aberration of light. (The name does not imply that there is anything wrong with light.) The aberration of light is an apparent shift in a star’s position as a result of the motion of the Earth and the finite velocity of light. It is readily understood 4 Galileo recorded an object in his notes in the year 1613 that turns out to have been Neptune, according to Charles Kowal of Palomar Observatory. Of course, Galileo did not recognize that he had seen a planet, so the honor of Neptune’s discovery rightfully belongs to Adams and Leverrier.
by means of an analogy. Imagine that you are in a rainstorm with the raindrops coming straight down. To catch the rain in a tall bucket, you hold it pointing upward, as shown in Figure 5-26a, and the raindrops will fall straight down to the bottom of the bucket without hitting the sides. On the other hand, if you run with the bucket, you will have to tilt it in the direction of your motion to keep the drops from hitting the bucket’s side (Figure 5-26b).
In the same way, a telescope on the moving Earth must be tilted very slightly in the direction of its motion if light entering the top of the telescope tube is to pass all the way through and arrive at the eyepiece centered at the bottom. The effect is not large: during the course of the year, a star’s observed position deviates a maximum of 20 seconds of arc on either side of its true position. However, eighteenth-century telescopes were sufficiently accurate to enable this effect to be detected. Because the deviation depends directly on the speed of the Earth in relation to the speed of light, a measurement of the deviation gives a numerical value for the speed of the Earth as it moves through space in its orbit around the Sun. The value is about one ten-thousandth of the speed of light, or 30 km/s (19 miles/s). Thus, the first actual verification of the moving Earth came in 1729 with the discovery of the aberration of starlight.
The discovery of parallax, predicted by the Greeks and searched for in vain by Tycho Brahe, was delayed for yet another century. In the year 1838 the great astronomer Friedrich Wilhelm Bessel finally succeeded in measuring the parallax of the nearby faint star 61 Cygni. Why did parallax take so long to find? The answer is that the stars are so far away that the amount of their parallactic shift is extremely small. In the case of 61 Cygni, for example, the total shift is only about 0.6 " (0.00017 degrees), less than a thirtieth of the effect of aberration. This is approximately the angular size of a U.S. penny viewed from six kilometers away — a very small angle!
Figure 5-27a illustrates the effect of parallax for a star that is located in the plane of the Earth’s orbit (the easiest case to visualize). When the Earth is at point A, the nearby star will be seen at position A ' with respect to the more distant background stars. Six months later, when the Earth is at point B, the nearby star appears to be located at point B '. The parallactic shift in the position of the star is the angle A-star-B. Clearly, the more distant the star, the smaller the parallactic shift will be.
If we were to take photographs of the star field when the Earth is at points A and B in its orbit, they might look like Figure 5-27b. The nearby star would appear to shift its position in relation to the more distant stars, as shown. Assuming that the background stars are much more distant than the nearby stars, the angle A-star-B of Figure 5-27a will be practically equal to the angular shift in position shown in Figure 5-27b.
Using parallax to find the distance to stars will be discussed in Chapter 15.
Inquiry 5-26 If the parallax to the nearby star in Figure 5-27 was observed from a planet farther from the Sun than the Earth, would the parallactic shift be larger or smaller than when observed from the Earth? If observed from Mars, which has a distance of 1.5 AU, what would be the parallactic shift to the same nearby star?
(a)
(b)
FIGURE 5-26. (a) A stationary bucket in the rain. Raindrops fall straight to the bottom. (b) Moving a bucket in the rain. It must be tilted to keep the drops from hitting the side.
5.7 Observational Evidence of the Earth’s Rotation Although every schoolchild can tell you that the Earth spins, few college graduates can explain how we know this simplest of all astronomical facts. As evidence, most people would cite the observation of the Sun and stars rising in the east and setting in the west. One could argue, as the Greeks did, that what is observed is simply the motion of a star-studded crystalline sphere around a stationary Earth once every 24 hours, with the Sun moving on its own sphere at its own slightly slower rate! Thus, just as in the case of the Earth’s motion, unique and authoritative observations are required.
Foucault Pendulum A famous demonstration of the rotation of the Earth is the pendulum experiment first performed by the nineteenth-century French physicist Jean- Bernard-Léon Foucault. In this experiment, a massive pendulum is hung from a long wire and set in motion. Pegs placed in a circle around the pendulum are successively knocked down over time due to the apparently changing plane in which the pendulum swings (Figure 5-28). To construct the simplest possible explanation of what is observed, imagine that we suspend a pendulum on a long cable from a support that has been placed above the North Pole (Figure 5-29). We make the coupling between the cable and the support as
friction-free as possible. If we set the pendulum to swinging back and forth, it will swing along a line that we can mark by driving pegs into the ice. Because of Newton’s first law of motion, the plane defined by the swinging cable will be fixed in space in relation to the distant stars. If the Earth were not rotating, the line along which the pendulum would swing would not change. However, because the Earth turns beneath the pendulum, the line of the pendulum swing will appear to change (as viewed on the turning Earth) in a direction that is opposite to the Earth’s rotation. Thus, the observed change in the direction of the pendulum swing over time is actually caused by Earth’s rotation.
At other latitudes, such as the latitude of Paris where the experiment was originally performed, the rotation will be slower but still evident. Only at the equator does the pendulum swing not change.
Although the motion of Foucault’s pendulum is a demonstration of Earth’s rotation on its axis, the concepts involved in the explanation depend on theories and models from other areas, in particular Newton’s first law. Without it, the Foucault pendulum experiment would have made no sense.
Inquiry 5-27 Why doesn’t the plane of oscillation of the pendulum change for an observer at the equator? (Hint: Consider what the pendulum’s apparent direction of swing would be in the Southern Hemisphere.)
FIGURE 5-27. (a) The effect of parallax on a nearby star (not to scale). The apparent direction toward the star

The Historical Quest to Model the Solar System

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