Chapter 0
Figure 0.16 Solar Eclipse Types INTERACTIVE (a) The Moon's shadow consists of two parts: the umbra, where no sunlight is seen, and the penumbra, where a portion of the Sun is visible. (b) Situated in the umbra, we see a total eclipse; in the penumbra, we see a partial eclipse. (c) If the Moon is too far from Earth at the moment of the eclipse, the umbra does not reach Earth and there is no region of totality; instead, an annular eclipse is seen. (Note that these figures are not drawn to scale.) (Insets: NOAA; G. Schneider)
Because the Moon's orbit is slightly inclined to the plane of the ecliptic (Figure 0.13b), we don't see a lunar eclipse at every full Moon. The Moon is usually above or below the ecliptic when a full Moon occurs and, hence, is untouched by Earth's shadow. Similarly, most new Moons do not result in solar eclipses. The chance that a new (or full) Moon will occur just as the Moon happens to cross the ecliptic plane (so Earth, Moon, and Sun are perfectly aligned, as illustrated in Figure 0.17) is quite low. As a result, eclipses are relatively infrequent events. On average, there
Figure 0.4 The Celestial Sphere INTERACTIVE Planet Earth sits fixed at the hub of the celestial sphere. This is one of the simplest possible models of the universe, but it doesn't agree with the facts that astronomers now know about the universe. description of the heavens, astronomers still use the idea as a convenient fiction that helps us visualize the positions of stars in the sky. The point where Earth's rotation axis (the line through the center around which the planet rotates) intersects the celestial sphere in the Northern Hemisphere is known as the north celestial pole; it is directly above Earth's North Pole. The star Polaris happens to lie close to the north celestial pole, which is why its direction indicates due north. In the Southern Hemisphere, the extension of Earth's axis in the opposite direction defines the south celestial pole. (There are no bright stars conveniently located near the south celestial pole and hence no "southern Pole Star.") Midway between the north and south celestial poles lies the celestial equator, representing the intersection of Earth's equatorial plane (the plane through Earth's center, perpendicular to the rotation axis) with the celestial sphere.
Celestial Coordinates The simplest method of locating stars in the sky is to specify their constellation and then rank the stars in that constellation in order of brightness. The brightest star is denoted by the Greek letter α (alpha), the second brightest by β (beta), and so on. For example, Betelgeuse and Rigel, the two brightest stars in the constellation Orion, are also known as α Orionis and β Orionis, respectively (see Figures 0.2 and 0.3). (Precise observations show that Rigel is actually brighter than Betelgeuse, but the names are now permanent.) Because there are many more stars in any given constellation than there are letters in the Greek alphabet, this method is of limited use. However, for naked-eye astronomy, where only bright stars are involved, it is quite satisfactory. For more precise measurements, astronomers find it helpful to use a system of celestial coordinates on the sky. If we think of the stars as being attached to the celestial sphere centered on Earth, then the familiar system of angular measurement on Earth's surface—latitude and longitude (Figure 0.6a)—extends quite naturally to the sky. The celestial analogs of latitude and longitude are called declination and right ascension, respectively (Figure 0.6b). Just as latitude and longitude are tied to Earth, right ascension and declination are fixed on the celestial sphere. Although the stars appear to move across the sky because of Earth's rotation, their celestial coordinates remain constant over the course of a night.
Figure 0.22 Scientific Method Scientific theories evolve through a combination of observation, theory, and prediction, which in turn suggests new observations. The process can begin at any point in the cycle, and it continues forever—or until the theory fails to explain an observation or makes a demonstrably false prediction. Scientific theories share several important defining characteristics: They must be testable; that is, they must admit the possibility that both their underlying assumptions and their predictions can be exposed to experimental verification. This feature separates science from, for example, religion, since ultimately divine revelations or scriptures cannot be challenged within a religious framework: We can't design an experiment to "understand the mind of God." Testability also distinguishes science from a pseudoscience such as astrology, whose underlying assumptions and predictions have been repeatedly tested and never verified, with no apparent impact on the views of those who continue to believe in it! They must continually be tested, and their consequences tested, too. This is the basic circle of scientific progress depicted in Figure 0.22. They should be simple. This is a practical outcome of centuries of scientific experience—the most successful theories tend to be the simplest ones that fit the facts. This view is often encapsulated in a principle known as Occam's razor (after the 14th-century English philosopher William of Ockham): 0.1 Discovery Sizing Up Planet Earth In about 200 b.c. a Greek philosopher named Eratosthenes (276-194 b.c.) used simple geometric reasoning to calculate the size of our planet. He knew that at noon on the first day of summer, observers in the city of Syene (now called Aswan), in Egypt, saw the Sun pass directly overhead. This was evident from the fact that vertical objects cast no shadows and sunlight reached to the very bottoms of deep wells, as shown in the insets in the accompanying figure. However, at noon of the same day in Alexandria, a city 5000 stadia to the north, the Sun was seen to be displaced slightly from the vertical. The stadium was a Greek unit of length, roughly equal to 0.16 km—the modern town of Aswan lies about 780 (5000 * 0.16) km south of Alexandria. Using the simple technique of measuring the length of the shadow of a vertical stick and applying elementary geometry, Eratosthenes determined the angular displacement of the Sun from the vertical at Alexandria to be 7.2°. What could have caused this discrepancy between the two measurements? As illustrated in the figure, the explanation is simply that Earth's surface is not flat, but curved. Our planet is a sphere. Eratosthenes was not the first to realize that Earth is spherical—the philosopher Aristotle had done that over 100 years earlier (Section 0.5)—but he was apparently the first to build on this knowledge, combining geometry with direct measurement to infer our planet's size. Here's how he did it. Rays of light reaching Earth from a very distant object, such as the Sun, travel almost parallel to one another. Consequently, as shown in the figure, the angle measured at Alexandria between the Sun's rays and the vertical (that is, the line joining Alexandria to the center of Earth) is equal to the angle between Syene and Alexandria, as seen from Earth's center. (For the sake of clarity, this angle has been exaggerated in the drawing.) The size of this angle in turn is proportional to the fraction of Earth's circumference that lies between Syene and Alexandria:
Earth's circumference is therefore 50 × 5000 or 250,000 stadia, or about 40,000 km. Earth's radius is therefore 250,000/2π stadia, or 6366 km. The correct values for Earth's circumference and radius, now measured accurately by orbiting spacecraft, are 40,070 km and 6378 km, respectively. Eratosthenes' reasoning was a remarkable accomplishment. More than 20 centuries ago he estimated the circumference of Earth to within 1 percent accuracy, using only simple geometry. A person making measurements on only a small portion of Earth's surface was able to compute the size of the entire planet on the basis of observation and pure logic—an early triumph of scientific reasoning.
0.1 More Precisely Angular Measure The size and scale of astronomical objects are often specified by measuring lengths and angles. The concept of length measurement is fairly intuitive. The concept of angular measurement may be less familiar, but it too can become second nature if you remember a few simple facts: A full circle contains 360 arc degrees (or 360°). Therefore, the half circle that stretches from horizon to horizon, passing directly overhead and spanning the portion of the sky visible to one person at any one time, contains 180°. Each 1° increment can be further subdivided into fractions of an arc degree, called arc minutes; there are 60 arc minutes (60′) in 1 arc degree. Both the Sun and the Moon project an angular size of 30 arc minutes on the sky. Your little finger, held at arm's length, does about the same, covering about a 40-arc minute slice of the 180° horizon-to-horizon arc. An arc minute can be divided into 60 arc seconds (60″). Put another way, an arc minute is 1/60 of an arc degree, and an arc second is 1/60 × 1/60 = 1/3600 of an arc degree. An arc second is an extremely small unit of angular measure—it is the angular size of a centimeter-size object (a dime, say) at a distance of about 2 kilometers (a little over a mile). The above figure illustrates this subdivision of the circle into progressively smaller units.
Earth's orbit face on, instead of almost edge-on, as in Figure 0.10. Notice that the orbit is almost perfectly circular, so the distance from Earth to the Sun varies very little (in fact, by only about 3 percent) over the course of a year—not nearly enough to explain the seasonal changes in temperature. What's more, Earth is actually closest to the Sun in early January, the dead of winter in the Northern Hemisphere, so distance from the Sun cannot be the main factor controlling our climate. The two points where the ecliptic intersects the celestial equator (Figure 0.9)—that is, where Earth's rotation axis is perpendicular to the line joining Earth to the Sun (Figure 0.10)—are known as equinoxes. On those dates, day and night are of equal duration. (The word equinox derives from the Latin for "equal night.") In the fall (in Earth's Northern Hemisphere), as the Sun crosses from the northern into the southern celestial hemisphere, we have the autumnal equinox (on September 21). The vernal equinox occurs in spring, on or near March 21, as the Sun crosses the celestial equator moving north. The vernal equinox plays an important role in human timekeeping. The interval of time from one vernal equinox to the next—365.242 solar days—is known as one tropical year.
0.3 The Motion of the Moon Early astronomers had very practical reasons for studying the sky. Some stars (such as Polaris) served as navigational guides, while others served as primitive calendars to predict planting and harvesting seasons. By observing repeating patterns in the sky and associating them with events on Earth, astronomers began to establish concrete connections between celestial events and everyday life and took the first steps toward true scientific understanding of the heavens. In a real sense, then, human survival depended on astronomical knowledge. The ability to predict and even explain astronomical events was undoubtedly a highly prized, and perhaps jealously guarded, skill. The Moon also played an important role in ancient astronomy. Calendars and religious observances were often tied to its phases and cycles, and even today the calendars of most of the world's major religions are still based wholly or partly on the lunar orbit. The Moon's regularly changing appearance (as well as its less regular, but much more spectacular, eclipses) was an integral part of the framework within which ancient astronomers sought to understand the universe. We will study the Moon's physical properties in more detail in Chapter 5. Here we continue our inventory of the sky with a brief description of the motion of our nearest neighbor in space. Lunar Phases Apart from the Sun, the Moon is by far the brightest object in the sky. Like the Sun, the Moon appears to move relative to the background stars. Unlike the Sun, however, the explanation for this motion is the obvious one—the Moon really does revolve around Earth. Data Points Lunar Phases More than half of all students tested using Mastering had difficulty ordering the Moon's phases as the Moon orbits Earth. Some points to remember: From night to night the Moon moves from west to east across the sky relative to the stars—right to left, as viewed from the Northern Hemisphere. The sunlit part of the moon grows (waxes) from west to east between the new and full phases and shrinks (wanes) from west to east from full to new. The Moon's appearance undergoes a regular cycle of changes, or phases, taking a little more than 29 days to complete. (The word month is derived from the word Moon.) Figure 0.13(a) illustrates the appearance of the Moon at different times in this monthly cycle. Starting from the new Moon, which is all but invisible in the sky, the Moon appears to wax (grow) a little each night and is visible as a growing crescent (frame 1 of Figure 0.13a). One week after new Moon, half of the lunar disk (the circular face we would see if the Moon were completely illuminated) can be seen (frame 2). This phase is known as a quarter Moon. During the next week, the Moon continues to wax, passing through the gibbous phase (more than half of the lunar disk visible, frame 3), until 2 weeks after new Moon the full Moon (frame 4) is visible. During the next 2 weeks, the Moon wanes (shrinks), passing in turn through the gibbous, quarter, and crescent phases (frames 5-7), eventually becoming new again. The location of the Moon in the sky, as seen from Earth, depends on its phase. For example, the full Moon rises in the east as the Sun sets in the west, while the first quarter Moon actually rises at noon, but often only becomes visible late in the day as the Sun's light fades. By this time the Moon is already high in the sky. These connections between lunar phase and rising/setting times are indicated on Figure 0.13(a). Unlike the Sun and the other stars, the Moon emits no light of its own. Instead, it shines by reflected sunlight, giving rise to the phases we see. As indicated in Figure 0.13(a), half of the Moon's surface is illuminated by the Sun at any moment, but not all of the Moon's sunlit face can be seen because of the Moon's position with respect to Earth and the Sun. When the Moon is full, we see the entire "day lit" face because the Sun and the Moon are in opposite directions from Earth in the sky. The Sun's light is not blocked by Earth at the full phase because, as shown in Figure 0.13(b), the Moon's orbit is inclined at a small angle (5.2°) to the plane of the ecliptic, so the alignment of the three bodies is not perfect. (The sizes of Earth and the Moon are greatly exaggerated in these figures.) In the case of a new Moon, the Moon and the Sun are in nearly the same part of the sky, and the sunlit side of the Moon is oriented away from us—at new Moon the Sun is almost behind the Moon, from our perspective. Self-Guided Tutorial Phases of the Moon Notice, by the way, that the Moon always keeps the same face toward Earth—as indicated on the figure, it rotates on its axis in exactly the same time it takes to orbit Earth. This is called synchronous rotation.We will discuss the reason for it in Chapter 5. As it revolves around Earth, the Moon's position in the sky changes with respect to the stars. In one sidereal month (27.3 days), the Moon completes one revolution and returns to its starting point on the celestial sphere, having traced out a great circle in the sky. The time required for the Moon to complete a full cycle of phases, one synodic month, is a little longer—about 29.5 days. The synodic month is a little longer than the sidereal month for the same basic reason that a solar day is slightly longer than a sidereal day (Figure 0.7): Because of Earth's motion around the Sun, the Moon must complete slightly more than one full revolution to return to the same phase in its orbit. Eclipses From time to time—but only at new or full Moon—the Sun, Earth, and the Moon line up precisely, and we observe the spectacular phenomenon known as an eclipse. When the Sun and the Moon are in exactly opposite directions as seen from Earth, Earth's shadow sweeps across the Moon, temporarily blocking the Sun's light and darkening the Moon in a lunar eclipse, as illustrated in
Figure 0.13 Lunar Phases (a) Because the Moon orbits Earth, this top-down view shows how the visible fraction of the lunar sunlit face varies from night to night, although the Moon always keeps the same face toward our planet. (Note the location of the small, straight arrows that mark the same point on the lunar surface at each phase shown.) The complete cycle of lunar phases takes about 29 days to complete. Rising and setting times for some phases are also indicated. (b) A side view shows how the Moon's orbit is inclined at about 5° to the ecliptic, so not all orbital configurations produce an eclipse.
0.4 The Measurement of Distance So far we have considered only the directions to the Sun, Moon, and stars, as seen from Earth. But knowing the direction to an object is only part of the information needed to locate it in space. Before we can make a systematic study of the heavens, we must find a way of measuring distances, too. One distance-measurement method, called triangulation, is based on the principles of Euclidean geometry and is widely used today in both terrestrial and astronomical settings. Modern surveyors use this age-old geometrical idea to measure the distances to faraway objects. In astronomy, it forms the foundation of the family of distance-measurement techniques that together make up the cosmic distance scale. Imagine trying to measure the distance to a tree on the other side of a river. The most direct method would be to lay a tape across the river, but that's not always practical. A smart surveyor would make the measurement by visualizing an imaginary triangle (hence, the term triangulation), sighting the tree (measuring its direction) on the far side of the river from two positions on the near side, as illustrated in Figure 0.19. The simplest triangle is a right triangle, in which one of the angles is exactly 90°, so it is often convenient to set up one observation
Figure 0.19 Triangulation Surveyors often use simple geometry to estimate the distance to a faraway object by triangulation. By measuring the angles at A and B and the length of the baseline, the distance can be calculated without the need for direct measurement (or getting wet!). position directly opposite the object, as at point A, although this isn't necessary for the method to work. The surveyor then moves to another observation position at point B, noting the distance covered between A and B. This distance is called the baseline of the imaginary triangle. Finally, the surveyor, standing at point B, sights toward the tree and notes the angle formed at point B by the intersection of this sight line and the baseline. No further observations are required. Knowing the length of one side (AB) and two angles (the right angle at A and the angle at B) of the triangle, and using elementary trigonometry, the surveyor can construct the remaining sides and angles and so establish the distance from A to the tree. Obviously, for a fixed baseline, the triangle becomes longer and narrower as the tree's distance from A increases. Narrow triangles cause problems because it is hard to measure the angles at A and B with sufficient accuracy. A surveyor on Earth can "fatten" the triangle by lengthening the baseline, but in astronomy there are limits on how long a baseline we can choose. For example, consider an imaginary triangle extending from Earth to a nearby object in space, perhaps a neighboring planet. The triangle is now extremely long and narrow, even for a relatively nearby object (by cosmic standards). Figure 0.20(a) illustrates a case in which the longest baseline possible on Earth—Earth's diameter, measured from point A to point B—is used. In principle, we could sight the planet from opposite sides of Earth, measuring the angles at A and B. However, in practice it is easier to measure the third angle of the imaginary triangle. Here's how.
Figure 0.1 Size and Scale in the Universe The bottom right of this figure shows humans on Earth, a view that widens progressively in the other four scenes from bottom to top—Earth, the solar system, a galaxy, and truly deep space. The numbers within the dashed zooms indicate approximately the increase in scale between successive images: Earth is 10 million times larger than humans, our solar system in turn is some million times larger than Earth, and so on. (See the Preface for an explanation of the icon at the bottom, which here indicates that these images were made in visible light.)
Figure 0.2 Constellation Orion INTERACTIVE (a) A photograph of the group of bright stars that make up the constellation Orion. (b) The stars connected to show the pattern visualized by the Greeks: the outline of a hunter. The Greek letters serve to identify some of the brighter stars in the constellation (see Figure 0.3). You can easily find Orion in the northern winter sky by identifying the line of three bright stars in the hunter's "belt." (P. Sanz/Alamy) Pleiades, the seven daughters of the giant Atlas. According to Greek mythology, the gods placed the Pleiades among the stars to protect them from Orion, who still stalks them nightly across the sky. Many other constellations have similarly fabulous connections with ancient cultures. The stars making up a particular constellation are generally not close together in space. They merely are bright enough to observe with the naked eye and happen to lie in the same direction in the sky as seen from Earth. Figure 0.3 illustrates this point for Orion, showing the true relationships between that constellation's brightest stars. Although constellation patterns have no real significance, the terminology is still used today. Constellations provide a convenient means for astronomers to specify large areas of the sky, much as geologists use continents or politicians use voting precincts to identify certain localities on Earth. In all, there are 88 constellations, most of them visible from North America at some time during the year.
Figure 0.20 Parallax INTERACTIVE (a) A triangle can be imagined to extend from Earth to a nearby object in space. The group of stars at the top represents a background field of very distant stars. (b) Hypothetical photographs of the same star field showing the nearby object's apparent shift, relative to the distant unshifted stars. The observers each sight toward the planet, taking note of its position relative to some distant stars seen on the plane of the sky. The observer at A sees the planet at apparent location A′ (pronounced "A prime") relative to those stars, as indicated in Figure 0.20(a). The observer at B sees the planet at location B′. If each observer takes a photograph of the same region of the sky, the planet will appear at slightly different places in the two images, as shown in Figure 0.20(b). (The positions of the background stars appear unchanged because of their much greater distance from the observer.) This apparent shift of a foreground object relative to the background as the observer's location changes is known as parallax. The size of the shift in Figure 0.20(b), measured as an angle on the celestial sphere, is equal to the third angle of the imaginary triangle in Figure 0.20(a). The closer an object is to the observer, the larger the parallax. To see this for yourself, hold a pencil vertically just in front of your nose (see Figure 0.21). Look at some far-off object—a distant wall, say. Close one eye, then open it while closing the other. You should see a large shift in the apparent position of the pencil relative to the wall—a large parallax. In this example, one eye corresponds to point A in Figure 0.20, the other eye to point B. The distance between your eyeballs is the baseline, the pencil represents the planet, and the distant wall the remote field of stars. Now hold the pencil at arm's length, corresponding to a more distant object (but still not as far away as the distant stars). The apparent shift of the pencil will be smaller. By moving the pencil farther away, you are narrowing the triangle and decreasing the parallax. If you were to paste the pencil to the wall, corresponding to the case where the object of interest is as far away as the background star field, blinking would produce no apparent shift of the pencil at all. The amount of parallax is inversely proportional to an object's distance. Small parallax implies large distance. Conversely, large parallax implies small distance. Knowing the amount of parallax (as an angle) and the length of the baseline, we can easily derive the distance through triangulation. Surveyors of the land routinely use these simple geometric techniques to map out planet Earth (Discovery 0-1 presents an early example). As surveyors of the sky, astronomers use the same basic principles to chart the universe.
Figure 0.21 Parallax Geometry Parallax is inversely proportional to an object's distance. An object near your nose has a much larger parallax than an object held at arm's length.
0.5 Science and the Scientific Method Science is a step-by-step process for investigating the physical world, based on natural laws and observed phenomena. However, the scientific facts just presented did not come easily or quickly. Progress in science is often slow and intermittent and may require a great deal of patience before significant progress is made. The earliest known descriptions of the universe were based largely on imagination and mythology and made little attempt to explain the workings of the heavens in terms of testable earthly experience. However, history shows that some early scientists did come to realize the importance of careful observation and testing to the formulation of their ideas. The success of their approach changed, slowly but surely, the way science was done and opened the door to a fuller understanding of nature. Experimentation and observation became central parts of the process of inquiry. Theories and Models To be effective, a theory—the framework of ideas and assumptions used to explain some set of observations and make predictions about the real world—must be continually tested. Scientists accomplish this by using a theory to construct a theoretical model of a physical object (such as a planet or a star) or phenomenon (such as gravity or light), accounting for its known properties. The model then makes further predictions about the object's properties or perhaps how it might behave or change under new circumstances. If experiments and observations favor those predictions, the theory can be further developed and refined. If they do not, the theory must be reformulated or rejected, no matter how appealing it originally seemed. The process is illustrated schematically in Figure 0.22. This approach to investigation, combining thinking and doing—that is, theory and experiment—is known as the scientific method. It lies at the heart of modern science, separating science from pseudoscience, fact from fiction. Notice that there is no end point to the process depicted in Figure 0.22. A theory can be invalidated by a single wrong prediction, but no amount of observation or experimentation can ever prove it "correct." Theories simply become more and more widely accepted as their predictions are repeatedly confirmed. The process can fail at any point in the cycle. If a theory cannot explain an experimental result or observation, or if its predictions are demonstrated to be untrue, it must be discarded or amended. And if it makes no predictions at all, then it has no scientific value.
Figure 0.22 Scientific Method Scientific theories evolve through a combination of observation, theory, and prediction, which in turn suggests new observations. The process can begin at any point in the cycle, and it continues forever—or until the theory fails to explain an observation or makes a demonstrably false prediction. Scientific theories share several important defining characteristics: They must be testable; that is, they must admit the possibility that both their underlying assumptions and their predictions can be exposed to experimental verification. This feature separates science from, for example, religion, since ultimately divine revelations or scriptures cannot be challenged within a religious framework: We can't design an experiment to "understand the mind of God." Testability also distinguishes science from a pseudoscience such as astrology, whose underlying assumptions and predictions have been repeatedly tested and never verified, with no apparent impact on the views of those who continue to believe in it! They must continually be tested, and their consequences tested, too. This is the basic circle of scientific progress depicted in Figure 0.22. They should be simple. This is a practical outcome of centuries of scientific experience—the most successful theories tend to be the simplest ones that fit the facts. This view is often encapsulated in a principle known as Occam's razor (after the 14th-century English philosopher William of Ockham): 0.1 Discovery Sizing Up Planet Earth In about 200 b.c. a Greek philosopher named Eratosthenes (276-194 b.c.) used simple geometric reasoning to calculate the size of our planet. He knew that at noon on the first day of summer, observers in the city of Syene (now called Aswan), in Egypt, saw the Sun pass directly overhead. This was evident from the fact that vertical objects cast no shadows and sunlight reached to the very bottoms of deep wells, as shown in the insets in the accompanying figure. However, at noon of the same day in Alexandria, a city 5000 stadia to the north, the Sun was seen to be displaced slightly from the vertical. The stadium was a Greek unit of length, roughly equal to 0.16 km—the modern town of Aswan lies about 780 (5000 * 0.16) km south of Alexandria. Using the simple technique of measuring the length of the shadow of a vertical stick and applying elementary geometry, Eratosthenes determined the angular displacement of the Sun from the vertical at Alexandria to be 7.2°. What could have caused this discrepancy between the two measurements? As illustrated in the figure, the explanation is simply that Earth's surface is not flat, but curved. Our planet is a sphere. Eratosthenes was not the first to realize that Earth is spherical—the philosopher Aristotle had done that over 100 years earlier (Section 0.5)—but he was apparently the first to build on this knowledge, combining geometry with direct measurement to infer our planet's size. Here's how he did it. Rays of light reaching Earth from a very distant object, such as the Sun, travel almost parallel to one another. Consequently, as shown in the figure, the angle measured at Alexandria between the Sun's rays and the vertical (that is, the line joining Alexandria to the center of Earth) is equal to the angle between Syene and Alexandria, as seen from Earth's center. (For the sake of clarity, this angle has been exaggerated in the drawing.) The size of this angle in turn is proportional to the fraction of Earth's circumference that lies between Syene and Alexandria:
Earth's circumference is therefore 50 × 5000 or 250,000 stadia, or about 40,000 km. Earth's radius is therefore 250,000/2π stadia, or 6366 km. The correct values for Earth's circumference and radius, now measured accurately by orbiting spacecraft, are 40,070 km and 6378 km, respectively. Eratosthenes' reasoning was a remarkable accomplishment. More than 20 centuries ago he estimated the circumference of Earth to within 1 percent accuracy, using only simple geometry. A person making measurements on only a small portion of Earth's surface was able to compute the size of the entire planet on the basis of observation and pure logic—an early triumph of scientific reasoning.
Figure 0.23 A Lunar Eclipse These photographs show Earth's shadow sweeping across the Moon during an eclipse. Aristotle reasoned that Earth was the cause of the shadow and concluded that Earth must be round. (G. Schneider) The Universe Today Our conception of the cosmos has changed a lot since ancient times. The modern universe is much larger, far more complex, and infinitely stranger than anything early astronomers ever imagined. In sharp contrast to the predictable and orderly heavens of the ancients, the universe we inhabit today is dynamic, expanding, evolving, and yet (we must admit) apparently dominated by fundamental forces that still lie beyond our understanding. Despite this, scientific inquiry today is guided by the same fundamental principles that led our ancestors to uncover the basic workings of the universe—gravity, light, relativity, quantum physics, and the Big Bang that brought our cosmos into being. Experiment and observation are integral parts of the process of modern science. Untestable theories, or theories unsupported by experimental evidence, rarely gain any measure of acceptance in scientific circles. Observation, theory, and testing are cornerstones of the scientific method, a technique whose power will be demonstrated again and again throughout our text. Concept Check Can a theory ever become a "fact," scientifically speaking? The Big Question Take another look at the spectacular photo at the start of this chapter (p. 4). Think about all those stars—about 100 billion in our Galaxy alone. We cannot help but wonder: Are there planets around some of those stars and perhaps intelligent beings on some of those planets? This grandest of all unsolved questions about the universe now lies at the heart of modern astronomy.
The Celestial Sphere Over the course of a night, the constellations appear to move across the sky from east to west. However, ancient sky-watchers noted that the relative positions of stars (to each other) remained unchanged as this nightly march took place. It was natural for early astronomers to conclude that the stars were attached to a celestial sphere surrounding Earth—a canopy of stars like an astronomical painting on a vast heavenly ceiling. Figure 0.4 shows how early astronomers pictured the stars as moving with this celestial sphere as it turned around a fixed Earth. Figure 0.5 shows how stars appear to move in circles around a point in the sky very close to the star Polaris (better known as the Pole Star or the North Star). To early astronomers, this point represented the axis around which the celestial sphere turned. From our modern standpoint, the apparent motion of the stars is the result of the spin, or rotation, not of the celestial sphere, but of Earth. Even though we now know that a revolving celestial sphere is an incorrect
Figure 0.3 Orion in 3D The true three-dimensional relationships among the most prominent stars in Orion. The distances in light-years were measured by the European Hipparcos satellite in the 1990s. (See Section 10.1.)
Figure 0.7 Solar and Sidereal Days A sidereal day is Earth's true rotation period—the time taken for our planet to return to the same orientation in space relative to the distant stars. A solar day is the time from one noon to the next. The difference in duration between the two is easily explained because Earth revolves around the Sun at the same time it rotates on its axis. Frames (a) and (b) are one sidereal day apart, when Earth rotates exactly once on its axis and also moves a little in its solar orbit—approximately 1°. Consequently, between noon at point A on one day and noon at the same point the next day, Earth actually rotates through about 361° (c), and the solar day exceeds the sidereal day by about 4 minutes. Note that the diagrams are not drawn to scale; the 1° angle is much smaller than shown here.
Figure 0.8 The Zodiac interactive The night side of Earth faces a different set of constellations at different times of the year. The 12 constellations named here make up the astrological zodiac. The arrows indicate the most prominent zodiacal constellations in the night sky at various times of year. For example, in June, when the Sun is "in" Gemini, Sagittarius and Capricornus are visible at night.
0.2 Earth's Orbital Motion Day-to-Day Changes We measure time by the Sun. The rhythm of day and night is central to our lives, so it is not surprising that the period of time from one sunrise (or noon, or sunset) to the next, the 24-hour solar day, is our basic social time unit. As we have just seen, this apparent daily progress of the Sun and other stars across the sky, known as diurnal motion is a consequence of Earth's rotation. But the stars' positions in the sky do not repeat themselves exactly from one night to the next. Each night, the whole celestial sphere appears shifted a little compared with the night before—you can confirm this for yourself by noting over the course of a week or two which stars are visible near the horizon just after sunset or just before dawn. Because of this shift, a day measured by the stars—called a sidereal day after the Latin word sidus, meaning "star"—differs in length from a solar day. The reason for the difference in length between a solar day and a sidereal day is sketched in Figure 0.7. Earth moves in two ways simultaneously: it rotates on its central axis while at the same time revolving around the Sun. Each time Earth rotates once on its axis, it also moves a small distance along its orbit. Therefore, each day Earth has to rotate through slightly more than 360° in order for the Sun to return to the same apparent location in the sky. As a result, the interval of time between noon one day and noon the next (a solar day) is slightly greater than the true rotation period (one sidereal day). Our planet takes 365 days to orbit the Sun, so the additional angle is 360°/365= 0.986°. Because Earth takes about 3.9 minutes to rotate through this angle, the solar day is 3.9 minutes longer than the sidereal day.
Seasonal Changes Because Earth revolves around the Sun, our planet's darkened hemisphere faces in a slightly different direction each night. The change is only about 1° per night (Figure 0.7)—too small to be easily discerned with the naked eye from one evening to the next. However, the change is clearly noticeable over the course of weeks and months, as illustrated in Figure 0.8. In 6 months, Earth moves to the opposite side of its orbit, and we face an entirely different group of stars and constellations at night. Because of this motion, the Sun appears, to an observer on Earth, to move slowly (at a rate of 1° per day) relative to the background stars over the course of a year. This apparent motion of the Sun on the sky traces out a path on the celestial sphere known as the ecliptic. The 12 constellations through which the Sun passes during the year as it moves along the ecliptic—that is, the constellations we would see looking in the direction of the Sun if they weren't overwhelmed by the Sun's light—had special significance for astrologers of old. They are collectively known as the zodiac. As illustrated in Figure 0.9, the ecliptic forms a great circle on the celestial sphere, inclined at an angle of about 23.5° to the celestial equator. In reality, as shown in Figure 0.10, the plane defined by the ecliptic is the plane of Earth's orbit around the Sun. Its tilt is a consequence of the inclination of our planet's rotation axis to its orbital plane. The point on the ecliptic where the Sun is at its northernmost point above the celestial equator (see Figure 0.9) is known as the summer solstice (from the Latin words sol, meaning "sun," and stare, "to stand"). As indicated in Figure 0.10, it represents the point on Earth's orbit where our planet's North Pole is oriented closest to the Sun. This occurs on or near June 21—the exact date varies slightly from year to year because the actual length of a year is not a whole number of days. As Earth rotates on that date, points north of the equator spend the greatest fraction of their time in sunlight, so the summer solstice corresponds to the longest day of the year (that is, the greatest number of daylight hours—Earth's rotation period doesn't change!) in Earth's Northern Hemisphere and the shortest day in Earth's Southern Hemisphere.
Figure 0.9 Ecliptic The seasons result from the changing height of the Sun above the celestial equator. At the summer solstice, the Sun is at its northernmost point on its path around the ecliptic. It is therefore highest in the sky, as seen from Earth's Northern Hemisphere, and the days are longest. The reverse is true at the winter solstice. At the vernal and autumnal equinoxes, when the Sun crosses the celestial equator, day and night are of equal length. Six months later, the Sun is at its southernmost point below the celestial equator (Figure 0.9)—or, equivalently, Earth's North Pole is oriented farthest from the Sun (Figure 0.10). We have reached the winter solstice (December 21), the shortest day in Earth's Northern Hemisphere and the longest in the Southern Hemisphere.
The tilt of Earth's rotation axis relative to the ecliptic is responsible for the seasons we experience—the marked difference in temperature between the hot summer and cold winter months. As illustrated in Figure 0.10, two factors combine to cause this variation. First, there are more hours of daylight during the summer than in winter. To see why this is, look at the yellow lines on the surfaces of the Earths in the figure. (For definiteness, they correspond to a latitude of 45° north—roughly that of the Great Lakes or the south of France.) A much larger fraction of the line is sunlit in the summertime, and more daylight means more solar heating. Second, as illustrated in the insets in Figure 0.10, when the Sun is high in the sky in summer, rays of sunlight striking Earth's surface are more concentrated—spread out over a smaller area—than in winter. As a result, the Sun feels hotter. Therefore summer, when the Sun is highest above the horizon and the days are longest, is generally much warmer than winter, when the Sun is low and the days are short.
