(Background, above) Most of our understanding of the universe has come within the last century. In an early twentieth-century classroom at Harvard College, as at other schools, astronomy came into its own as a viable and important subject of study.
(Inset A) Harlow Shapley (18851972), seen here at Harvard sitting at his huge circular desk that he could spin to work on different topics of astronomical interest. He discovered our place in the "suburbs" of the Milky Way, dispelling the notion that the Sun resides at the center of the universe.
(Inset B) Annie Cannon (18631941), one of the greatest astronomical cataloguers of all time, carefully analyzed photographic plates to classify nearly a million stars over the course of fifty years of work at Harvard Observatory.
(Inset C) Maria Mitchell (18181889) in her observatory on Nantucket Island. The first woman professional astronomer in the United States, she taught at Vassar College and made important contributions to several areas of astronomy, including the photographic study of the Sun's surface.
(Inset D) Edwin Hubble (18891953), here posing in front of one of the telescopes on Mount Wilson, in California, is often credited with having discovered the expansion of the universe.
Studying this chapter will enable you to:
Relate how some ancient civilizations attempted to explain the heavens in terms of Earth-centered models of the universe.
Summarize the role of Renaissance science in the history of astronomy.
Explain how the observed motions of the planets led to our modern view of a Sun-centered solar system.
Sketch the major contributions of Galileo and Kepler to the development of our understanding of the solar system.
State Kepler's laws of planetary motion.
Explain how Kepler's laws enable us to construct a scale model of the solar system, and the technique used to determine the actual size of the planetary orbits.
State Newton's laws of motion and universal gravitation, and explain how they account for Kepler's laws.
Explain how the law of gravitation enables us to measure the masses of astronomical bodies. Living in the Space Age, we have become accustomed to the modern view of our place in the universe. Images of our planet taken from space leave little doubt that Earth is round, and no one seriously questions the idea that we orbit the Sun. Yet there was a time, not so long ago, when our ancestors maintained that Earth was flat and lay at the center of all things. Our view of the universe--and of ourselves--has undergone a radical transformation since those early days. Earth has become a planet like many others, and humankind has been torn from its throne at the center of the cosmos and relegated to a rather unremarkable position on the periphery of the Milky Way Galaxy. But we have been amply compensated for our loss of prominence--we have gained a wealth of scientific knowledge in the process. The story of how all this came about is the story of the rise of the scientific method and the genesis of modern astronomy.
Back Many ancient cultures took a keen interest in the changing nighttime sky. The records and artifacts that have survived until the present make that abundantly clear. But unlike today, the major driving force behind the development of astronomy in those early societies was probably neither scientific nor religious in nature. Instead, it was decidedly practical and very down to earth. Seafarers needed to navigate their vessels, and farmers had to know when to plant their crops. In a very real sense, then, human survival depended on knowledge of the heavens. As a result, the ability to predict the arrival of the seasons, as well as other astronomical events, was undoubtedly a highly prized, and perhaps also jealously guarded, skill.
In Chapter 1 we saw that the human brain's ability to perceive patterns in the stars led to the "invention" of constellations as a convenient means of labeling regions of the celestial sphere. The realization that these patterns returned to the night sky at the same time each year met the need for a practical means of tracking the seasons. Many separate cultures, all over the world, built large and elaborate structures to serve, at least in part, as primitive calendars. In some cases, the keepers of the secrets of the sky eventually enshrined their knowledge in myth and ritual, so that these astronomical sites were often also used for religious rites.
Perhaps the best known such site is Stonehenge, located on Salisbury Plain, in England, and shown in Figure 2.1. This ancient stone circle, which today is one of the most popular tourist attractions in Britain, dates back to the Stone Age. Researchers believe it was an early astronomical observatory of sorts--not in the modern sense of the term (a place for making new observations and discoveries), but rather a kind of three-dimensional calendar or almanac, enabling its builders and their descendents to identify important dates by means of specific celestial events. Its construction apparently spanned a period of some 17 centuries, beginning around 2800 B.C. Additions and modifications continued up to about 1100 B.C., indicating its ongoing importance to the Stone Age and later Bronze Age people who built, maintained, and used Stonehenge. The largest stones shown in Figure 2.1 weigh up to 50 tons and were transported from quarries many miles away.
Figure 2.1 Stonehenge was probably constructed as a primitive calendar and almanac. The fact that the largest stones were carried to the site from many miles away attests to the importance of this structure to its Stone Age builders. The inset shows sunrise at Stonehenge on the summer solstice. As seen from the center of the stone circle, the Sun rose directly over the "heel stone" on the longest day of the year.
Many of the stones are aligned so that they point toward important astronomical events. For example, the line joining the center of the inner circle to the so-called heel stone, set off at some distance from the rest of the structure, points in the direction of the rising Sun on the summer solstice. Other alignments are related to the rising and setting of the Sun and the Moon at various other times of the year. Although the accurate alignments (within a degree or so) of the stones of Stonehenge were first noted in the eighteenth century, it is only relatively recently--in the second half of the twentieth century, in fact--that the scientific community has credited Stone Age technology with the ability to carry out such a precise feat of engineering. While some of Stonehenge's purposes remain uncertain and controversial, the site's function as an astronomical almanac seems well established. Although Stonehenge is the most impressive and the best preserved, other stone circles, found all over Europe, are believed to have performed similar functions.
Many ancient cultures are now known to have been capable of similarly precise accomplishments. The Big Horn Medicine Wheel in Wyoming (Figure 2.2a) is similar to Stonehenge in design--and, presumably, intent--although it is somewhat simpler in execution. The Medicine Wheel's alignments with the rising and setting Sun and with some bright stars indicate that its builders--the Plains Indians--had much more than a passing familiarity with the changing nighttime sky. Figure 2.2b shows the Caracol temple, built by the Mayans around 1000 A.D. in Mexico's Yucatán peninsula. This temple is much more sophisticated than Stonehenge, but it probably played a similar role as an astronomical observatory. Its many windows are accurately aligned with astronomical events, such as sunrise and sunset at the solstices and equinoxes and the risings and settings of the planet Venus. Astronomy was of more than mere academic interest to the Mayans, however. Caracol was also the site of countless human sacrifices, carried out when Venus appeared in the morning or evening sky.
Figure 2.2 (a) The Big Horn Medicine Wheel, in Wyoming, was built by the Plains Indians. Its spokes and other features are aligned with risings and settings of the Sun and other stars. (b) Caracol temple in Mexico. The many windows of this Mayan construct are aligned with astronomical events, indicating that at least part of Caracol's function was to keep track of the seasons and the heavens.
The ancient Chinese too observed the heavens. Their astrology attached great importance to "omens" such as comets and "guest stars"--stars that appeared suddenly in the sky and then slowly faded away--and they kept careful and extensive records of such events. Twentieth-century astronomers still turn to the Chinese records to obtain observational data recorded during the Dark Ages (roughly from the fifth to the tenth century A.D.), when turmoil in Europe largely halted the progress of Western science. Perhaps the best-known guest star was one that appeared in 1054 A.D. and was visible in the daytime sky for many months. We now know that the event was actually a supernova: the explosion of a giant star, which scattered most of its mass into space. It left behind a remnant that is still detectable today, nine centuries later. The Chinese data are a prime source of historical information for supernova research. A vital link between the astronomy of ancient Greece and that of medieval Europe was provided by Arab astronomers (Figure 2.3). For six centuries, from the depths of the Dark Ages to the beginning of the Renaissance, Islamic astronomy flourished and grew, preserving and augmenting the knowledge of the Greeks. The Arab influence on modern astronomy is subtle but quite pervasive. Many of the mathematical techniques involved in trigonometry were developed by Muslim astronomers in response to very practical problems, such as determining the precise dates of holy days or the direction of Mecca from any given location on Earth. Astronomical terms like "zenith" and "azimuth" and the names of many stars, such as Rigel, Betelgeuse, and Vega, all bear witness to this extended period of Muslim scholarship.
Figure 2.3 Arab astronomers at work, as depicted in a medieval manuscript.
Astronomy, we see, is not the property of any one culture, civilization, or era. The same ideas, the same tools, and even the same misconceptions have been invented and reinvented by human societies all over the world, in response to the same basic driving forces. Astronomy came into being because people believed that there was a practical benefit in being able to predict the positions of the stars, but its roots go much deeper than that. The need to understand where we came from, and how we fit into the cosmos, is an integral part of human nature.
The Greeks of antiquity, and undoubtedly civilizations before them, built models of the universe. The study of the workings of the universe on the very largest scales is called cosmology. Today, cosmology entails looking at the universe on scales so large that even entire galaxies can be regarded as mere points of light scattered throughout space. To the Greeks, however, the universe was basically the solar system--namely, the Sun, Earth, Moon, and the planets known at that time. The stars beyond were surely part of the universe, but they were considered to be fixed, unchanging beacons on a mammoth celestial dome. The Greeks did not consider the Sun, the Moon, and the planets to be part of the celestial sphere, however. Those objects had patterns of behavior that set them apart.
Over the course of a night, the stars slide smoothly across the sky. Over the course of a month, the Moon moves smoothly and steadily along its path on the sky relative to the stars, passing through its familiar cycle of phases. Over the course of a year, the Sun progresses along the ecliptic at an almost constant rate, varying little in brightness from day to day. In short, the behavior of both Sun and Moon seemed fairly simple and orderly. But the Greeks were also aware of five other bodies in the sky--the planets Mercury, Venus, Mars, Jupiter, and Saturn--whose behavior was not so easy to grasp. Their motions ultimately led to the downfall of an entire theory of the solar system and to a fundamental change in humankind's view of the universe.
Planets do not behave in as regular and predictable a fashion as the Sun, Moon, and stars. They vary in brightness, and they don't maintain a fixed position in the sky. Unlike the Sun and the Moon, with their regular paths, the planets seem to wander around the celestial sphere--indeed, the word planet derives from the Greek word planetes, meaning wanderer. Planets never stray far from the ecliptic and generally traverse the celestial sphere from west to east, as the Sun does. The planets, however, seem to speed up and slow down during their journeys. They even appear to loop back and forth relative to the stars, as shown in Figure 2.4. In other words, there are periods when a planet's eastward motion (relative to the stars) stops, and the planet appears to move westward in the sky for a month or two before reversing direction again and continuing on its eastward journey.
Figure 2.4 Most of the time, planets move from west to east relative to the background stars. Occasionally, however, they change direction and temporarily undergo retrograde motion before looping back. The image at the top shows an actual retrograde loop in the motion of the planet Mars. The lower image depicts the movements of several planets over the course of several years, as reproduced on the inside dome of a planetarium. The motion of the planets relative to the stars (represented as unmoving points) produces continuous streaks on the planetarium "sky." Motion in the eastward sense is usually referred to as direct, or prograde, motion; the backward (westward) loops are known as retrograde motion. Mars, Jupiter, and Saturn are always brightest--which means closest to Earth--during the retrograde portions of their orbits. Obviously this planetary behavior requires an explanation more complex than the relatively simple motions of the Moon and the Sun. The occasional retrograde loops of some planets, and the brightness variations of all of them, necessitated major modifications to the Greek's basic cosmological model describing the Sun, the Moon, and the stars. The earliest models of the solar system followed the teachings of the Greek philosopher Aristotle, and were geocentric in nature. These geocentric models held that Earth lay at the center of the universe and that all other bodies moved around it. (Recall Figures 1.7 and 1.10a, which illustrate the basic geocentric view.) Aristotle's views were immensely influential--so much so, in fact, that his teachings carried great weight even centuries after his death. The geocentric model went largely unchallenged until the sixteenth century A.D.
Actually, history records that some ancient Greek astronomers reasoned differently about the motions of heavenly bodies. Foremost among them was Aristarchus of Samos (310230 B.C.), who proposed that all the planets, including the Earth, revolve around the Sun and, furthermore, that Earth rotates on its axis once each day. This, he argued, would create an apparent motion of the sky--a simple idea that is familiar to anyone who has ridden on a merry-go-round and watched the landscape appear to move past them as they go. However, Aristarchus's description of the heavens, though essentially correct, did not gain widespread acceptance during his lifetime. Aristotle's influence was too strong, his followers too numerous, his writings too comprehensive. The Aristotelian school presented some simple and (at the time) compelling arguments in favor of their views. First, of course, Earth doesn't feel as if it's moving. And if it were, wouldn't there be a strong wind as we moved at high speed around the Sun? Then again, considering that the vantage point from which we view the stars changes over the course of a year, why don't we see stellar parallax? Nowadays we might be inclined to dismiss the first two points as merely naive, but the third is a valid argument and the reasoning is essentially sound. We now know that there is stellar parallax as Earth orbits the Sun. However, because the stars are so distant, it amounts to less than 1´´, even for the closest stars. Early astronomers simply would not have noticed it. We will encounter many other instances in astronomy where correct reasoning has led to the wrong conclusions because it was based on inadequate data.
The earliest geocentric models of the universe employed what Aristotle, and Plato before him, had taught was the perfect form: the circle. The simplest possible description--uniform motion around a circle having Earth at its center--provided a fairly good approximation of the orbits of the Sun and the Moon, but it could not possibly account for observed variations in planetary brightness or retrograde motion. A more complex model was needed to describe the planets.
The first step toward this new model modified the idea that the planets moved on circles centered on Earth. Instead, each planet was taken to move uniformly around a small circle, called an epicycle, whose center moved uniformly around Earth on a second and larger circle, known as the deferent (see Figure 2.5). The motion was now composed of two separate circular orbits, which created the possibility that, at some times, the planet's apparent motion in the sky could be retrograde. Also, the distance from the planet to Earth would vary, accounting for changes in brightness. By tinkering with the relative sizes of the epicycle and the deferent, with the planet's speed on the epicycle, and with the epicycle's speed along the deferent, this "epicyclic" motion could be brought into fairly good agreement with the observed paths of the planets in the sky. Moreover, this model had good predictive power, at least to the accuracy of observations at the time.
Figure 2.5 In the geocentric model of the solar system, the observed motions of the planets made it impossible to assume that they moved on simple circular paths around Earth. Instead, each planet was thought to follow a small circular orbit (the epicycle) about an imaginary point that itself traveled in a large, circular orbit (the deferent) about Earth.
However, as time went by and the number and the quality of observations increased, it became clear that the simple epicyclic model was not perfect. Small corrections had to be introduced to bring it into line with new observations. The center of the deferents had to be shifted slightly from the center of Earth, and the motion of the epicycles had to be imagined uniform with respect not to Earth but to yet another point in space. Around A.D. 140, a Greek astronomer named Ptolemy constructed perhaps the best geocentric model of all time. Illustrated in simplified form in Figure 2.6, it explained remarkably well the observed paths of the five planets then known, as well as the paths of the Sun and the Moon. However, to achieve its explanatory and predictive power, the full Ptolemaic model required a series of no fewer than 80 distinct circles. To account for the paths of the Sun, the Moon, and all the nine planets (and their moons) that we know today would require a vastly more complex set. Nevertheless, Ptolemy's text on the topic, Syntaxis (better known today by its Arabic name Almagest--"the greatest") provided the intellectual framework for all discussion of the universe for well over a thousand years.
Figure 2.6 The basic features, drawn roughly to scale, of the geocentric model of the inner solar system that enjoyed widespread popularity prior to the Renaissance. To avoid confusion, we have drawn partial paths (dashed) of only two planets, Venus and Jupiter. Today, our scientific training leads us to seek simplicity, because simplicity in the physical sciences has so often proved to be an indicator of truth. We would regard the intricacy of a model as complicated as the Ptolemaic system as a clear sign of a fundamentally flawed theory. With the benefit of hindsight, we now recognize that the major error lay in the assumption of a geocentric universe. This was compounded by the insistence on uniform circular motion, whose basis was largely philosophical, rather than scientific, in nature.
Back The Ptolemaic picture of the universe survived, more or less intact, for almost 13 centuries. Given the scope of the cultural, scientific, and technological changes that have occurred in even the last 50 years, it may be difficult for us to grasp how any theory, and especially one so erroneous, could have persisted for such a long time. Whatever the reasons, the Ptolemaic model of the solar system became deeply embedded in European culture at all levels and lasted until the fifteenth century. Then came the Renaissance, a rebirth of artistic, philosophical, and scientific inquiry. Western thought moved away from the passive acceptance of ancient dogma and static beliefs toward critical thinking and observational testing. In astronomy, thinking (theory) and looking (observation) merged to produce a model of the solar system that was simpler than the geocentric one embraced by the ancients. A sixteenth-century Polish cleric, Nicholas Copernicus (see Figure 2.7), rediscovered Aristarchus's heliocentric model--one centered on the Sun--and showed how, in its harmony and organization, it better fit the observed facts than did the tangled geocentric cosmology.
Figure 2.7 Nicholas Copernicus (14731543).
Copernicus asserted that Earth spins on its axis every day and, like the other planets, orbits the Sun. Only the Moon, he said, orbits Earth. The observed daily and seasonal changes in the heavens can be understood in terms of these simple motions. The seven crucial statements that form the basis for what is now known as the Copernican revolution are summarized in Interlude 2-1. The Copernican view stands in stark contrast to the conventional beliefs of the preceding two millennia and presents a more ordered and natural explanation of the observed facts than any geocentric model could provide. We have already seen in Chapter 1 how this picture accounts for the motions of the Sun and the stars. Figure 2.8 shows how it explains both the varying brightness of the planets and their observed looping motions. If we suppose that Earth moves faster than the planet Mars, then every so often we will "overtake" Mars. Mars will appear to move backwards in the sky, in much the same way as a car we overtake on the highway seems to slip backwards relative to us. Note that the looping motions are now only apparent, not real, as they were in the Ptolemaic view.
Figure 2.8 The Copernican model of the solar system explains the varying brightnesses of the planets, something the Ptolemaic system largely ignored. Here, for example, when Earth and Mars are relatively close to one another in their respective orbits (as at position 6), Mars seems brighter; when farther away (as at position 1), Mars seems dimmer. Also, because the line of sight from Earth to Mars changes as the two planets smoothly orbit the Sun, Mars would appear to loop back and forth, undergoing retrograde motion. The line of sight changes because Earth, on the inside track, moves faster in its orbit than Mars moves along its path.
Copernicus's major motivation for introducing the heliocentric model was simplicity. Even so, he was still influenced by Greek thinking and clung to the idea of circles to model the planets' motions. To bring his theory into agreement with observations, he was forced to retain the idea of epicyclic motion, though with the deferent centered on the Sun rather than on Earth, and with the epicycles being smaller than in the Ptolemaic picture. Thus, he retained unnecessary complexity and actually gained little in accuracy over the geocentric model. The heliocentric model did indeed rectify some small discrepancies and inconsistencies in the Ptolemaic system, but for Copernicus, the primary attraction of heliocentricity was its simplicity, its being "more pleasing to the mind." His theory was more something he felt than he could prove. To the present day, scientists still are guided by simplicity, symmetry, and beauty in modeling all aspects of the universe.
Despite the support of some observational data, neither his fellow scholars nor the general public easily accepted Copernicus's model. For the learned, heliocentricity went against the grain of much previous thinking and violated many of the religious teachings of the time, largely because it relegated Earth to a noncentral and undistinguished place within the solar system and the universe. In the heliocentric model, Earth became just one of several planets. And Copernicus's work had little impact on the general populace of his time, partly because it was published in Latin, which most people could not read. Only after Copernicus's death, when others--notably Galileo Galilei--popularized his ideas, did the Roman Catholic church take them seriously enough to bother banning them. Copernicus's writings on the heliocentric universe were placed on the Index of Prohibited Books in 1616, 73 years after they were first published. They remained there until the end of the eighteenth century.
Back In the century following the death of Copernicus and the publication of his theory of the solar system, two scientists--Galileo Galilei and Johannes Kepler--made indelible imprints on the study of astronomy. Contemporaries, they were aware of each other's work, and corresponded from time to time about their theories. Each achieved fame for his discoveries and made great strides in popularizing the Copernican viewpoint. Yet, in their approach to astronomy--and in their personalities--they were as different as night and day.
Back Galileo Galilei (Figure 2.9) was an Italian mathematician and philosopher. By his willingness to perform experiments to test his ideas--a rather radical approach in those days (see Interlude 2-2)--and by embracing the brand-new technology of the telescope, he revolutionized the way science was done, so much so that he is now widely regarded as the father of experimental science. The telescope was invented in Holland in the early seventeenth century. Hearing of the invention--but without seeing one--Galileo built a telescope for himself in 1609 and aimed it at the sky. What he saw conflicted greatly with the philosophy of Aristotle, and provided much new data to support the ideas of Copernicus.*
*In fact, Galileo had already abandoned Aristotle in favor of Copernicus, although he had not published these beliefs at the time he began his telescopic observations.
Figure 2.9 Galileo Galilei (15641642).
Using his telescope, Galileo discovered that the Moon had mountains, valleys, and craters. Its surface was reminiscent more of Earth's than that of a perfect, unblemished celestial orb, which the Moon was conventionally held to be. Looking at the Sun--something that should never be done directly and that eventually blinded Galileo--he saw dark blemishes. These blemishes, which we now call sunspots and which we will study in Chapter 16, directly contradicted the ancient Greek notion that the Sun was a perfect, jewellike body. Furthermore, he could see that the blemishes moved across the face of the Sun. From this he inferred that the Sun rotates on an axis roughly perpendicular to the ecliptic about once a month. In studying the planet Jupiter, Galileo saw four small points of light, invisible to the naked eye, orbiting it. He recognized that Jupiter and its natural satellites (now called the Galilean moons) were a system similar to the Sun and its family of planets, but on a smaller scale. That another planet had moons--and thus that some body other than Earth could be the center of motion--conflicted directly with Aristotelianism. To Galileo, the moons of Jupiter provided the strongest support for the Copernican model. Clearly, the Earth was not the center of all things.
Galileo also discovered that Venus showed a complete cycle of phases, like those of our Moon (see Figure 2.10). The insets show how the full and crescent views of Venus must be caused by the planet's motion around the Sun. The observations of the phases of Venus were further strong evidence that the Earth is not the center of things, and that at least one planet orbited the Sun.
Figure 2.10 The phases of Venus, rendered at different points in the planet's orbit. If Venus orbits the Sun and is closer to the Sun than is Earth, as Copernicus maintained, then Venus should display phases, much as our Moon does. As shown here, when directly between Earth and the Sun, Venus's unlit side faces us, and the planet is invisible to us. As Venus moves in its orbit (at a faster speed than Earth moves in its orbit), progressively more of its illuminated face is visible from Earth. Note also the connection between orbital phase and the apparent size of the planet. Venus seems much larger in its crescent phase than when it is full because it is much closer to us during its crescent phase. (The insets are actual photographs of Venus at two of its crescent phases.)
Galileo published his findings, and his controversial conclusions supporting the Copernican theory, in 1610, in a book called Sidereus Nuncius (The Starry Messenger). In reporting these and other wondrous observations made with his new telescope, Galileo was challenging the scientific establishment and religious dogma of the time and aggressively urging people to change their basic view of the universe. He was (literally) playing with fire. He certainly was aware that only a few years earlier, in 1600, the astronomer Giordano Bruno had been burned at the stake in Rome for his heretical teaching that the Earth orbited the Sun. By all accounts, however, Galileo delighted in publicly ridiculing and irritating his Aristotelian colleagues. In 1616, his ideas were judged heretical, and Copernicus's works were banned by the Roman Church. Galileo was instructed to abandon his cosmological pursuits.
But Galileo would not desist. In 1632, he raised the stakes by publishing Dialogue Concerning the Two Chief World Systems, which compared the Ptolemaic and Copernican models. The book presented a discussion among three people: one of them a dull-witted Aristotelian, whose views time and again were roundly defeated by the arguments of one of his companions, an articulate proponent of the heliocentric system. To make the book accessible to a wide popular audience, Galileo wrote it in Italian rather than Latin. These actions brought Galileo into direct conflict with the Church. Eventually, the Inquisition forced him, under threat of torture, to retract his claim that the Earth orbits the Sun, and he was placed under house arrest in 1633; he remained imprisoned for the rest of his life. Not until 1992 were Galileo's "crimes" publicly forgiven by the Church. But the damage to the orthodox view of the universe was done, and the Copernican genie was out of the bottle once and for all.
The Copernican episode is a good example of how the scientific method, though affected at any given time by the subjective whims, human biases, and sheer luck of researchers, does ultimately lead to a definite degree of objectivity. Over time, many groups of scientists checking, confirming, and refining experimental tests will neutralize the subjective attitudes of individuals. Usually one generation of scientists can bring sufficient objectivity to bear on a problem, though some especially revolutionary concepts are so swamped by tradition, religion, and politics that more time is necessary. In the case of heliocentricity, objective confirmation was not obtained until about three centuries after Copernicus published his work and more than 2000 years after Aristarchus had proposed the concept. Nonetheless, that objectivity did in fact eventually prevail.
The development and eventual acceptance of the heliocentric model were milestones in our thinking. Understanding our planetary system freed us from an Earth-centered view of the universe and eventually enabled us to realize that Earth orbits only one of myriad similar stars in the Milky Way Galaxy, which is itself one of myriad galaxies. This removal of the Earth from any position of great cosmic significance is generally known, even today, by the term Copernican principle.
At about the same time that Galileo was becoming famous for his telescopic observations, Johannes Kepler (see Figure 2.11), a German mathematician and astronomer, announced his discovery of a set of simple empirical "laws" that accurately described the motions of the planets. While Galileo was the first "modern" observer, Kepler was a pure theorist; he based his work almost entirely on the observations of another (in part because of his own poor eyesight). Those observations, which predated the telescope by several decades, had been made by Kepler's employer, Tycho Brahe (15461601), arguably one of the greatest observational astronomers who ever lived.
Figure 2.11 Johannes Kepler (15711630).
Tycho, as he is often called, was both an eccentric aristocrat and a talented observer. He was born in Denmark and educated at some of the best universities in Europe, where he studied astrology, alchemy, and medicine. By all accounts, he was impossibly rude and insulting to virtually everyone he met, an attitude that cost him dearly and ultimately resulted in his leaving Denmark in 1597. He moved to Prague, which happens to be fairly close to Graz, in Austria, where Kepler lived. Kepler joined Tycho in Prague in 1600. There he was put to work trying to find a theory that could explain Brahe's planetary data. When Tycho died a year later, Kepler inherited not only Brahe's position as Imperial Mathematician of the Holy Roman Empire (then actually located in Eastern Europe), but also his most priceless possession: the accumulated observations of the planets, spanning several decades. Tycho's observations, though made with the naked eye, were nevertheless of very high quality. In most cases, his measured positions of stars and planets were accurate to within about 1´. Kepler set to work seeking a unifying principle to explain in detail the motions of the planets, without the need for epicycles. The effort was to occupy much of the remaining 29 years of his life.
Kepler and Tycho held different theories of the solar system. Brahe never fully accepted the Copernican view. Instead, he had his own "hybrid" cosmology, in which the Sun orbited the Earth, but the other planets orbited the Sun--a picture that, from a purely observational viewpoint, was just as consistent with observations as Copernicus's and philosophically more satisfying to Tycho.
In contrast, Kepler accepted the heliocentric picture of the solar system, but was careful not to say so in such a way as to antagonize his contemporaries. He was concerned about the relationship of his work to established Church doctrine, not so much from fear of retribution, but simply because he was a religious man. Kepler's goal was to find a simple, elegant description of the solar system that fit Tycho's complex mass of detailed observations. In the end, he found that it was necessary to abandon Copernicus's original simple idea of circular planetary orbits. As a result, an even greater simplicity emerged. After long years studying Brahe's planetary data, and after many false starts and blind alleys, Kepler developed the laws of planetary motion that now bear his name.
Kepler determined the shape of each planet's orbit by triangulation (see Section 1.9)--not from different points on Earth, but from different points on Earth's orbit. For the data, he used Brahe's detailed observations that had been made at many different times of the year. By using a portion of Earth's orbit as a baseline, Kepler was able to measure the relative sizes of the other planetary orbits. Noting where the planets were on successive nights, he also found the speeds at which the planets moved. We do not know how many geometrical shapes Kepler tried for the orbits before he hit upon the correct one. His difficult task was made even more complex because he had to determine Earth's own orbit, too. Nevertheless, he eventually succeeded in summarizing the motions of all the known planets, including Earth, in just three laws, the laws of planetary motion.
Figure 2.12 illustrates a means of constructing an ellipse, which is simply an elongated circle. Take a length of string and attach it to a piece of paper using two thumbtacks. Then, keeping the string taut at all times, use a pencil to trace out the curve shown in the diagram. Almost any ellipse can be drawn in this way by varying the length of the string or the distance between the tacks. The two points where the string is pinned are each called a focus (plural: foci) of the ellipse. The long axis of the ellipse, containing the two foci, is known as the major axis. We conventionally refer to half the length of this long axis--the semi-major axis--as a measure of the ellipse's "size." The eccentricity of the ellipse is the ratio of the distance between the foci to the length of the major axis. Notice that a circle is a special kind of ellipse in which the two foci coincide, so that the eccentricity is zero. The semi-major axis of a circle is simply its radius.
Figure 2.12 Any ellipse can be drawn with the aid of a string, a pencil, and two thumbtacks. The wider the separation of the foci, the more elongated, or eccentric, is the ellipse. In the special case where the two foci are at the same place, the drawn curve is a circle.
These two numbers--the semi-major axis and the eccentricity--are all that we need to describe the size and shape of the orbital path. From them, we can derive other useful quantities. For example, if a planet's orbit has semi-major axis a and eccentricity e, we can compute that its perihelion (closest approach to the Sun) is at a distance a (l e) and that its aphelion (greatest distance from the Sun) is a (l + e). Thus, for example, a (hypothetical) planet with a semi-major axis of 400 million km and an eccentricity of 0.5 would range between 400 × (1 0.5) = 200 million km and 400 x (1 + 0.5) = 600 million km from the Sun over the course of one complete orbit.
None of the planets' elliptical orbits is nearly as elongated as the one shown in Figure 2.12. With two exceptions, the paths of Mercury and Pluto, the planetary orbits have such small eccentricities that our eyes would have trouble distinguishing them from true circles. Only because the orbits are so nearly circular were the Ptolemaic and Copernican models able to come as close as they did to describing reality.
Kepler's substitution of elliptical for circular orbits was no small advance. It amounted to abandoning an aesthetic bias--the belief in the perfection of the circle--that had governed astronomy since Greek antiquity. And it was another heavy blow to Aristotelian philosophy. Even Galileo, not known for his conservatism in these scholarly matters, clung to the idea of circular motion and never accepted that the planets move on elliptical paths.
Kepler's second law, illustrated in Figure 2.13, addresses the speed at which a planet traverses different parts of its orbit:
An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal intervals of time.
Figure 2.13 A diagram illustrating Kepler's second law: Equal areas are swept out in equal intervals of time. The three shaded areas (A, B, and C) are equal. Note that an object would travel the length of each of the three arrows in the same amount of time. Therefore, planets move faster when closer to the Sun. While orbiting the Sun, a planet traces the arcs labeled A, B, and C in Figure 2.13 in equal times. But notice that the distance traveled by the planet along arc C is greater than the distance traveled along arc A or arc B. Because the time is the same and the distance is different, the speed must vary. When a planet is close to the Sun, as in sector C, it moves much faster than when farther away, as in sector A. This law is not restricted to planets. It applies to any orbiting object. Spy satellites, for example, move very rapidly as they swoop close to Earth's surface--not because they are propelled with powerful on-board rockets, but because any object in a highly elliptical orbit moves much faster as it approaches the focus where the object around which it orbits (in this case, the Earth) is located.
By taking into account the relative speeds and positions of the planets in their elliptical orbits about the Sun, Kepler's first two laws explained the variations in planetary brightness and some observed peculiar nonuniform motions that could not be accommodated within the assumption of circular motion, even with the inclusion of epicycles. Gone at last were the circles within circles that rolled across the sky. Kepler's modification of Copernicus's theory to allow the possibility of elliptical orbits both greatly simplified the model of the solar system and at the same time provided much greater predictive accuracy than had previously been possible.
Kepler published these two laws in 1609, stating that he had proved them only for the orbit of Mars. Ten years later, he extended the first and second laws to all the known planets and added a third law relating the size of a planet's orbit to its sidereal orbital period--the time needed for the planet to complete one circuit around the Sun. Kepler's third law states that The square of a planet's orbital period is proportional to the cube of its semi-major axis.
In other words, the planet's "year"--or, more technically, its (sidereal) orbital period P--increases more rapidly than does the size of its semi-major axis a, according to the rule P2 a3 (the symbol means "is proportional to"). For example, Earth has, by definition, an orbital semi-major axis of 1 astronomical unit (1 A.U.--see the More Precisely feature on p. 13) and an orbital period of 1 Earth year. The planet Venus, orbiting at a distance of roughly 0.7 A.U., takes only 0.6 Earth years--about 225 days--to complete one circuit. By contrast, Saturn, almost 10 A.U. out, takes considerably more than 10 Earth years--in fact, nearly 30 years--to orbit the Sun just once.
Kepler's third law becomes particularly simple when we choose the astronomical unit as our unit of length and the (Earth) year as our unit of time. If we do this, the constant of proportionality in the above relation becomes equal to one, and we can replace proportionality (" ") by equality ("="). In other words, for any planet, we can write
P2 (in Earth years) = a3 (in astronomical units).
Table 2-1 presents basic data describing the orbits of the nine planets now known. Renaissance astronomers knew these properties for the innermost six planets and used them to construct the currently accepted heliocentric model of the solar system. The second column presents each planet's orbital semi-major axis, measured in astronomical units; the third column gives the orbital period, in Earth years. The fourth column lists the planets' orbital eccentricities. For purposes of verifying Kepler's third law, the rightmost column lists the ratio P2/a3. As we have just seen, in the units used in the table, the third law implies that this number should equal 1 in all cases.
The main points to be grasped from Table 2-1 are these: (1) with the exception of Mercury and Pluto, the planets' orbits are very nearly circular (that is, their eccentricities are close to zero), and (2) the farther a planet is from the Sun, the greater is its orbital period, in precise agreement with Kepler's third law, to within the four-digit accuracy of the numbers in the table. For example, in the case of Pluto, verify for yourself that 39.533 = 248.62 (at least, to three significant figures). Most important, note that Kepler's laws are exactly obeyed by all the known planets, not just by the six on which he based his conclusions.
Back Kepler's laws allow us to construct a scale model of the solar system, with the correct shapes and relative sizes of all the planetary orbits, but they do not tell us the actual size of any orbit. We can express the distance to each planet only in terms of the distance from Earth to the Sun. Why is this? Because Kepler's triangulation measurements all used a portion of Earth's orbit as a baseline, distances could be expressed only relative to the size of that orbit, which was not itself determined. Thus our model of the solar system would be analogous to a road map of the United States showing the relative positions of cities and towns, but lacking the all-important scale marker indicating distances in kilometers or miles. For example, we would know that Kansas City is about three times more distant from New York than it is from Chicago, but we would not know the actual mileage between any two points on the map.
If we could somehow determine the value of the astronomical unit--in kilometers, say--we would be able to add the vital scale marker to our map of the solar system and compute the exact distances between the Sun and each of the planets. We might propose using triangulation to measure the distance from Earth to the Sun directly. However, we would find it impossible to measure the Sun's parallax using Earth's diameter as a baseline. The Sun is too bright, too big, and too fuzzy for us to distinguish any apparent displacement relative to the field of distant stars. To measure the Sun's distance from Earth, we must resort to some other method.
Before the middle of the twentieth century, the most accurate measurements of the astronomical unit were made using triangulation on the planets Mercury and Venus during their rare transits of the Sun--that is, during the brief periods when those planets passed directly between the Sun and the Earth (as shown for the case of Mercury in Figure 2.14). Because the time at which a transit occurs can be determined with great precision, astronomers can use this information to make very accurate measurements of a planet's position in the sky. They can then use simple geometry to compute the distance to the planet by combining observations made from different locations on Earth, just as discussed earlier in Chapter 1. For example, the parallax of Venus at closest approach to Earth, as seen from two diametrically opposite points on Earth (separated by about 13,000 km), is about 1 arc minute--at the limit of naked-eye capabilities, but easily measurable telescopically. This parallax represents a distance of 45 million km.
Figure 2.14 A solar transit of Mercury. Such transits happen only about once per decade, because Mercury's orbit does not quite coincide with the plane of the ecliptic. Transits of Venus are even rarer, occurring only about twice per century. Knowing the distance to Venus, we can immediately compute the magnitude of the astronomical unit. Figure 2.15 is an idealized diagram of the SunEarthVenus orbital geometry. The planetary orbits are drawn as circles here, but in reality they are slight ellipses. This is a subtle difference, and we can correct for it using detailed knowledge of orbital motions. Assuming for the sake of simplicity that the orbits are perfect circles, we see from the figure that the distance from Earth to Venus at closest approach is approximately 0.3 A.U. Knowing that 0.3 A.U. is 45,000,000 km makes determining 1 A.U. straightforward--the answer is 45,000,000/0.3, or 150,000,000 km.
Figure 2.15 Simplified geometry of the orbits of Earth and Venus as they move around the Sun. The wavy lines represent the paths along which radar signals might be transmitted toward Venus and received back at Earth at the moment when Venus is at its minimum distance from Earth. Because the radius of Earth's orbit is 1 A.U. and that of Venus is about 0.7 A.U., we know that this distance is 0.3 A.U. Thus, radar measurements allow us to determine the astronomical unit in kilometers. The modern method for deriving the absolute scale of the solar system uses radar rather than triangulation. The word radar is an acronym for radio detection and ranging. In this technique, radio waves are transmitted toward an astronomical body, such as a planet. Their returning echo indicates the body's direction and range, or distance, in absolute terms (that is, in kilometers rather than in astronomical units). The calculations involved in using radar to measure astronomical distances resemble those used to derive the distance between two cities if our car's speed and travel time are known. If we multiply the round-trip travel time of the radar signal (the time elapsed between transmission of the signal and reception of the echo) by the speed of light (300,000 km/s, which is also the speed of radio waves), we obtain twice the distance to the target planet (back and forth). The round-trip travel time can be measured with high precision--in fact, well enough to determine the dimensions of the orbit of Venus to an accuracy of 1 km. Through precise radar ranging, the astronomical unit is now known to be 149,597,870 km. In this text, we will round this number off to a value of 1.5 × 108 km.
Having determined the value of the astronomical unit, we can reexpress the sizes of the other planetary orbits in terms of more familiar units, such as miles or kilometers. The entire scale of the solar system can then be calibrated to high precision.
What prevents the planets from flying off into space or from falling into the Sun? What causes the planets to revolve about the Sun, apparently endlessly? To be sure, the motions of the planets obey Kepler's three laws, but only by considering something more fundamental than those laws can we understand these motions. The heliocentric system was secured when, in the seventeenth century, the British mathematician Isaac Newton (Figure 2.16) developed a deeper understanding of the way all objects move and interact with one another as they do.
Figure 2.16 Isaac Newton (16421727).
Isaac Newton was born in Lincolnshire, England, on Christmas Day in 1642, the year that Galileo died. Newton studied at Trinity College of Cambridge University, but when the bubonic plague reached Cambridge in 1665, he returned to the relative safety of his home for two years. During that time he made probably the most famous of his discoveries, the law of gravity (although it is but one of the many major scientific advances Newton is responsible for). However, either because he regarded the theory as incomplete or possibly because he was afraid that he would be attacked or plagiarized by his colleagues, he did not tell anyone of his monumental achievement for almost 20 years. It was not until 1684, when Newton was discussing with Edmund Halley (of Halley's comet fame) the leading astronomical problem of the day--Why do the planets move according to Kepler's laws?--that he astounded his companion by remarking casually that he had solved the problem in its entirety nearly two decades before!
Prompted by Halley, Newton published his theories in perhaps the most influential physics book ever written: Philosophiae Naturalis Principia Mathematica (or The Mathematical Principles of Natural Philosophy--what we would today call "science"), usually known simply as Newton's Principia. The ideas expressed in that work form the basis for what today is known as Newtonian mechanics. Three basic laws of motion, the law of gravity, and the calculus (which Newton also invented) are sufficient to explain and quantify virtually all of the complex dynamic behavior we see on Earth and throughout the universe. Newton's laws are listed in the More Precisely feature below.
Figure 2.17 illustrates Newton's first law of motion. The first law simply states that a moving object will, in principle, move forever in a straight line unless some external force changes its direction of motion. For example, the object might glance off a brick wall or be hit with a baseball bat; in either case, a force changes the original motion of the object. The tendency of an object to keep moving at the same speed and in the same direction unless acted upon by a force is known as inertia. A familiar measure of an object's inertia is its mass--loosely speaking, the total amount of matter it contains. The greater an object's mass, the more inertia it has, and the greater is the force needed to change its state of motion.
Figure 2.17 An object at rest will remain at rest (a) until some force acts on it (b). It will then remain in that state of uniform motion until another force acts on it. The arrow in (c) shows a second force acting at a direction different from the first, which causes the object to change direction.
Newton's first law contrasts sharply with the view of Aristotle, who incorrectly maintained that the natural state of an object was to be at rest--most probably an opinion based on Aristotle's observations of the effect of friction. In our discussion, we will neglect the familiar concept of friction--the force that slows balls rolling along the ground, blocks sliding across tabletops, and baseballs moving through the air. In any case, it is not an issue for the planets because there is no appreciable friction in outer space. The fallacy in Aristotle's argument was first realized and exposed by Galileo, who conceived of the notion of inertia long before Newton formalized it into a law.
The rate of change of the velocity of an object--speeding up, slowing down, or simply changing direction--is called its acceleration. Newton's second law states that the acceleration of an object is directly proportional to the applied force and inversely proportional to its mass--that is, the greater the force acting on the object, or the smaller the mass of the object, the greater its acceleration. Thus, if two objects are pulled with the same force, the more massive one will accelerate less; if two identical objects are pulled with different forces, the one experiencing the greater force will accelerate more. Finally, Newton's third law simply tells us that forces cannot occur in isolation--if body A exerts a force on body B, then body B necessarily exerts a force on body A that is equal in magnitude, but oppositely directed.
Only in extreme circumstances do Newton's laws break down, and this fact was not realized until the twentieth century, when Albert Einstein's theories of relativity once again revolutionized our view of the universe (see Chapter 22). Most of the time, however, Newtonian mechanics provides an excellent description of the motion of planets, stars, and galaxies through the cosmos.
Forces can act instantaneously or continuously. To a good approximation, the force from a baseball bat that hits a home run can be thought of as being instantaneous, or momentary, in nature. A good example of a continuous force is the one that prevents the baseball from zooming off into space--gravity, the phenomenon that started Newton on the path to the discovery of his laws. Newton hypothesized that any object having mass always exerts an attractive gravitational force on all other massive objects. The more massive an object, the stronger its gravitational pull. The continuous pull of Earth's gravity can be visualized if we consider a baseball thrown upward. Figure 2.18 illustrates how the baseball's path changes continuously. The baseball, having some mass of its own, also exerts a gravitational pull on Earth. By Newton's third law, this force is equal and opposite to the weight of the ball (the force with which Earth attracts it). But, by Newton's second law, the Earth has a much greater effect on the light baseball than the baseball has on the much more massive Earth. The ball and Earth each feel the same gravitational force, but the Earth's acceleration is much smaller.
Figure 2.18 A ball thrown up from the surface of a massive object such as a planet is pulled continuously by the gravity of that planet (and, conversely, the gravity of the ball continuously pulls the planet). Now consider the trajectory of a baseball batted from the surface of the Moon, which has much less mass than Earth has. Because the pull of gravity is about one-sixth as great on the Moon as on Earth, a baseball's path changes more slowly near the Moon. A typical home run in a ballpark on Earth would travel nearly half a mile on the Moon. The Moon, less massive than Earth, has less gravitational influence on the baseball. The magnitude of the gravitational force, then, depends on the masses of the attracting bodies. Theoretical insight, as well as detailed experiments, tells us that the force is in fact directly proportional to the product of the two masses.
Studying the motions of the planets around the Sun reveals a second aspect of the gravitational force. At all locations equidistant from the Sun's center, the gravitational force has the same strength, and it is always directed toward the Sun. Furthermore, in much the same way that temperature decreases with distance from a fire, gravity weakens with distance from any object that has mass.
Forces that decrease with distance from their source are encountered throughout all of science. Many of them, including gravity, decrease in proportion to the square of the distance. They are said to obey an inverse-square law. As shown in Figure 2.19, inverse-square forces decrease rapidly with distance from their source. For example, tripling the distance makes the force 32 = 9 times weaker, while multiplying the distance by 5 results in a force that is 52 = 25 times weaker. Despite this rapid decrease, the force never quite reaches zero. The gravitational pull of an object having some mass can never be completely extinguished.
Figure 2.19 Inverse-square forces rapidly weaken with distance from their source. The strength of the gravitational force decreases with the square of the distance from the Sun. The force never quite diminishes to zero, however, no matter how far away from the Sun we go. We can combine the preceding statements about mass and distance to form a law of gravity that dictates the way in which all material objects attract one another. As a proportionality, Newton's law of gravity is
gravitational force . This relationship is a compact way of stating that the gravitational pull between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance separating them. (See the More Precisely feature above for a fuller statement of this law.)
To Newton, gravity was a force that acted at a distance, with no obvious way in which it was actually transmitted from place to place. Newton was not satisfied with this explanation, but he had none better. To appreciate the modern view of gravity, consider any piece of matter having some mass--it could be smaller than an atom or larger than a galaxy. Extending outward from this object in all directions is a gravitational field produced by the matter. We now regard such a field as a property of space itself--a property that determines the influence of one massive object on another. All other matter "feels" the field as a gravitational force.
Figure 2.20 The Sun's inward pull of gravity on a planet competes with the planet's tendency to continue moving in a straight line. These two effects combine, causing the planet to move smoothly along an intermediate path, which continually "falls around" the Sun. This unending tug-of-war between the Sun's gravity and the planet's inertia results in a stable orbit.
The Sunplanet interaction sketched here is analogous to what occurs when you whirl a rock at the end of a string above your head. The Sun's gravitational field is your hand and the string, and the planet is the rock at the end of that string. The tension in the string provides the force necessary for the rock to move in a circular path. If you were suddenly to release the string--which would be like eliminating the Sun's gravity--the rock would fly away along a tangent to the circle, in accordance with Newton's first law. In the solar system, at this very moment, Earth is moving under the combined influence of these two effects: the competition between gravity and inertia. The net result is a stable orbit, despite our continuous rapid motion through space. In fact, Earth orbits the Sun at a speed of about 30 km/s, or some 70,000 miles per hour. (Verify this for yourself by calculating how fast Earth must move to complete a circle of radius 1 A.U.--and hence of circumference 2¼ A.U., or 940 million km--in 1 year, or 3.2 × 107 seconds. The answer is 9.4 × 108 km/3.1 × 107s, or 30.3 km/s.)
Before we leave this topic, let's make one final, very important point. We can use Newtonian mechanics to calculate the relationship between the distance (r) and the speed (v) of a planet moving in a circular orbit around the Sun (of mass m). By calculating the force required to keep the planet moving in a circle, and comparing it with the gravitational force due to the Sun, we can show that the circular speed is
v = ,
where the gravitational constant G is defined in the More Precisely feature above. Because we have measured G in the laboratory on Earth and because we know the length of a year and the size of the astronomical unit, we can use Newtonian mechanics to weigh the Sun. Inserting the known values of v = 30 km/s, r = 1 A.U. = 1.5 × 1011 m, and G = 6.7 × 10-11 Nm2/kg2, we can calculate the mass of the Sun to be 2 × 1030 kg--an enormous mass by terrestrial standards. Similarly, knowing the distance to the Moon and the length of the (sidereal) month, we can measure the mass of Earth to be 6 × 1024 kg.
In fact, this is how basically all masses are measured in astronomy. Because we can't just go out and weigh an astronomical object when we need to know its mass, we must look for its gravitational influence on something else. This principle applies to planets, stars, galaxies, and even clusters of galaxies--very different objects, but all subject to the same physical laws.
The orbit of a planet around the Sun is an ellipse, with the common center of mass of the planet and the Sun at one focus.
As shown in Figure 2.21, however, the center of mass for two objects of comparable mass does not lie within either object. For identical masses (Figure 2.21a), the orbits are identical ellipses, with a common focus located midway between the two objects. For unequal masses (as in Figure 2.21b), the elliptical orbits still share a focus and both have the same eccentricity, but the more massive object moves more slowly, and on a tighter orbit. (Note that Kepler's second law, as stated earlier, continues to apply without modification to each orbit separately, but the rates at which the two orbits sweep out area are different.) In the case of a planet orbiting the much more massive Sun (see Figure 2.21c), the path traced out by the Sun's center is smaller than the Sun itself.
Figure 2.21 (a) The orbits of two bodies (stars, for example) with equal masses, under the influence of their mutual gravity, are identical ellipses with a common focus. That focus is not at the center of either star but instead is located at the center of mass of the pair, midway between them. The positions of the two bodies at three different times are indicated by the pairs of numbers. (Notice that a line joining the bodies always passes through the common focus.) (b) The orbits of two bodies, one of which is twice as massive as the other. Again, the elliptical orbits have a common focus, and the two ellipses have the same eccentricity. However, in accordance with Newton's laws of motion, the more massive body moves more slowly, and in a smaller orbit, staying closer to the center of mass (at the common focus). In this particular case, the larger ellipse is twice the size of the smaller one. (c) In this extreme case of a hypothetical planet orbiting the Sun, the common focus of the two orbits lies inside the Sun.
The change to Kepler's third law is also small in the case of a planet orbiting the Sun but very important in other circumstances, such as when we consider the orbital motion of two stars that are bound to one another by gravity. Following through the mathematics of Newton's theory, we find that the true relationship between the semi-major axis a of the planet's orbit relative to the Sun and its orbital period P is
where Mtotal is the combined mass of the two objects. Notice that Newton's restatement of Kepler's third law preserves the proportionality between P2 and a3, but now the proportionality includes Mtotal, so it is not quite the same for all the planets. The Sun's mass is so great, however, that the differences in Mtotal among the various combinations of the Sun and the other planets are almost unnoticeable, so Kepler's third law, as originally stated, is a very good approximation. Just as before, we can once again choose units to simplify this proportionality. Expressing distance in astronomical units, time in years, and mass in units of the mass of the Sun, we can now say:
P2 (in Earth years) = .
This modified form of Kepler's third law is true in all circumstances, inside or outside the solar system.
Figure 2.22 The effect of launch speed on the trajectory of a satellite. With too low a speed at point A the satellite will simply fall back to Earth. Given enough speed, however, the satellite will go into orbit--it "falls around the Earth." As the initial speed at point A is increased, the orbit will become more and more elongated. When the initial speed exceeds the escape velocity, the satellite will become unbound from Earth and will escape along a hyperbolic trajectory. Some space vehicles, such as the robot probes that visit the other planets, attain enough speed to escape our planet's gravitational field and move away from Earth forever. This speed, known as the escape velocity, is about 41 percent greater (actually, = 1.41421... times greater) than the speed of a circular orbit at any given radius.* At less than escape velocity, the old adage "what goes up must come down" (or at least stay in orbit) still applies. At more than escape velocity, our spacecraft will leave Earth for good (neglecting the effect of air resistance on our way through Earth's atmosphere and assuming that we don't turn the craft around using an on-board rocket motor). Planets, stars, galaxies--all gravitating bodies--have escape velocities. No matter how massive the body, gravity decreases with distance. As a result, the escape velocity diminishes with increasing separation. The farther we go from Earth (or any gravitating body), the easier it becomes to escape.
*In terms of our earlier formula, the escape velocity is just vescape = .
The speed of a satellite in a circular orbit just above Earth's atmosphere is 7.9 km/s (roughly 18,000 mph). The satellite would have to travel at 11.2 km/s (about 25,000 mph) to escape from Earth altogether. In the case of an object exceeding the escape velocity, the motion is said to be unbound, and the orbit is no longer an ellipse. In fact, the path of the spacecraft relative to Earth is a related geometrical figure called a hyperbola. In the intermediate case, where the spacecraft has exactly the escape velocity and so has just enough energy to get away, the orbital trajectory is an intermediate geometrical shape--a parabola. If we simply change the word ellipse to hyperbola or parabola, the modified version of Kepler's first law still applies, as does Kepler's second law. (Kepler's third law does not extend to unbound orbits because it doesn't make sense to talk about a period in those cases.)
Newton's laws explain the paths of objects moving at any point in space near any gravitating body. These laws provide a firm physical and mathematical foundation for Copernicus's heliocentric model of the solar system and for Kepler's laws of planetary motions. But they do much more than that. Newtonian gravitation governs not only the planets, moons, and satellites in their elliptical orbits, but also the stars and galaxies in their motion throughout our universe--as well as apples falling to the ground.