(Background Above) One of the most easily recognizable star fields in the winter nighttime sky, the familiar constellation Orion. This field of view spans roughly 100 light-years, or 1015 kilometers. (See also Figure 1.6.)

(Inset A) If we magnify the wide view of the Orion constellation, shown above, by a million times, we enter into the realm of the largest stars, with sizes of about a billion kilometers. Such a star is seen in this false-color image of the red-giant star Betelguese (which is actually the bright star at the upper left of the Orion constellation).

(Inset B) Another magnification of a million brings us to the scale of typical moons--roughly 1,000 kilometers--represented here by Ariel, one of the many moons of Uranus.

(Inset C) With yet another million-times magnification, we reach scales of meters, represented here by an astronomer at the controls of her telescope.

(Inset D) At a final magnification of an additional million, we reach the scale of molecules (about 10-6 meter), represented by this coiled DNA molecule of a rat's liver.

LEARNING GOALS

Studying this chapter will enable you to:

Explain the concept of the celestial sphere and the conventions of angular measurement that enable us to locate objects in the sky.

Describe how the Sun, the Moon, and the stars appear to change their positions from night to night and from month to month.

Account for these apparent motions in terms of the actual motions of the Earth and the Moon.

Show how the relative motions of the Earth, the Sun, and the Moon lead to eclipses.

Explain the simple geometric reasoning that allows astronomers to measure the distances of faraway objects.

Nature offers no greater splendor than the starry sky on a clear, dark night. Silent and jeweled with the constellations of ancient myth and legend, the night sky has inspired wonder throughout the ages--a wonder that leads our imaginations far from the confines of Earth and the pace of the present day, and out into the distant reaches of space and cosmic time itself. Astronomy, born in response to that wonder, is built on two of the most basic traits of human nature: the need to explore and the need to understand. Through the interplay of curiosity, discovery, and analysis--the keys to exploration and understanding--people have sought answers to questions about the universe since the earliest times. Astronomy is the oldest of all the sciences, yet never has it been more exciting than it is today.

In all of human history, there have been only two periods in which our understanding of the universe has been revolutionized within a single lifetime. The first spanned the years from the middle of the sixteenth century to the early part of the seventeenth, when the work of Copernicus, Kepler, and Galileo established beyond reasonable doubt the fact that our Earth is not the unmoving center of the entire universe, but in fact revolves about the Sun. The second is now underway. In the late twentieth century we have begun to break away from planet Earth, and in doing so we have achieved a whole new perspective on the universe in which we live.

Of all the scientific insights attained to date, one stands out boldly: Earth is neither central nor special. We inhabit no unique place in the universe. Astronomical research, especially within the past few decades, strongly suggests that we live on what seems to be an ordinary rocky planet called Earth, which is one of 9 known planets orbiting an average star called the Sun, which is one star near the edge of a huge collection of stars called the Milky Way galaxy, which is one galaxy among countless billions of others spread throughout the observable universe.

We are connected to these distant realms of space and time not only by our imaginations, but also through a common cosmic heritage: Most of the chemical elements in our bodies were created billions of years ago in the hot centers of long-vanished stars. Their fuel supply spent, these giant stars died in huge explosions, scattering afar the elements created deep within their cores. Eventually, this matter collected into clouds of gas that slowly collapsed to give birth to a new generation of stars. In this way, the Sun and its family of planets were formed nearly 5 billion years ago. Everything on Earth embodies atoms from other parts of the universe and from a past far more remote than the beginning of human evolution.

Although ours is the only planetary system we know of, others may orbit many of the billions upon billions of stars in the universe. Elsewhere, other beings, perhaps with an intelligence much greater than our own, might at this very moment be gazing in wonder at their own nighttime sky. Our own Sun might be nothing more than an insignificant point of light to them, if it is visible at all. If such beings exist, they too must share our cosmic origin.

Before going any further, let us clarify just what we mean by "the universe." We might define it poetically as the vast tracts of space and enormous stretches of time populated sparsely by stars and galaxies glowing in the dark. More scientifically, the universe is the totality of all space, time, matter, and energy. Consult Figures 1.1 through 1.4, and put some of these objects in perspective by studying Figure 1.5.

Take another look at the galaxy in Figure 1.3. This galaxy, whose catalog name is M83, is a swarm of about a hundred billion stars--more stars than people who have ever lived on Earth. The entire assemblage is spread across some 100,000 light years. A light year is the distance traveled by light, at a velocity of about 300,000 kilometers per second, in a year. It equals about 10 trillion kilometers (or around 6 trillion miles). Typical galactic systems are truly "astronomical" in size.

Figure 1.1 The Earth is a planet, a mostly solid object, though it has some liquid in its oceans and its core, and gas in its atmosphere. (In this view, you can clearly see the North and South American continents.)

Figure 1.2 The Sun is a star, a very hot ball of gas. Much bigger than the Earth, the Sun is held together by its own gravity.

Figure 1.3 A typical galaxy is a collection of a hundred billion stars, each separated by vast regions of nearly empty space. This galaxy is the eighty-third entry in the catalog compiled by the eighteenth-century French astronomer Charles Messier--M83 for short. Our Sun is a rather undistinguished star near the edge of another such galaxy, called the Milky Way.

Figure 1.4 This photograph shows a portion of the Coma cluster of galaxies, some 300 million light years from Earth. Each galaxy contains hundreds of billions of stars, probably planets, and possibly living creatures.

Figure 1.5 This artist's conception puts each of the previous figures in perspective. The bottom of this figure shows spacecraft (and astronauts) in Earth orbit, a view that widens progressively in each of the next five cubes drawn from bottom to top--the Earth, the planetary system, the local neighborhood of stars, the Milky Way Galaxy, and the closest cluster of galaxies. The numbers indicate approximately the increase in scale between each successive image.

A thousand (1000), a million (1,000,000), a billion (1,000,000,000), and even a trillion (1,000,000,000,000)--these words occur regularly in everyday speech. But let's take a moment to understand the magnitude of these numbers and to appreciate the differences among them. One thousand is easy enough to understand; at the rate of one number per second, you could count to a thousand in about 16 minutes. However, if you wanted to count to a million, you would need more than 2 weeks of counting at the rate of one number per second, 16 hours per day (allowing 8 hours per day for sleep). To count from 1 to a billion at the same rate of one number per second and 16 hours per day would take nearly 50 years--the better part of an entire human lifetime.

In this text we will consider spatial domains spanning not just billions of kilometers, but billions of light years. We will discuss objects containing not just trillions of atoms, but trillions of stars. We will contemplate time intervals of not just billions of seconds or hours, but billions of years. You will need to become familiar with--and comfortable with--such enormous numbers. A good way to begin is to try and recognize just how much larger than a thousand is a million, and how much larger still is a billion. (The More Precisely feature below explains the convenient method used by scientists for writing and manipulating very large and very small numbers.)

How have we come to know the universe around us? How do we know the proper perspective sketched in Figure 1.5? Our study of the universe, the subject of astronomy, begins by examining the sky.

CONSTELLATIONS IN THE SKY

Between sunset and sunrise on a clear night, we can see some 3000 points of light. Include the view from the opposite side of Earth, and nearly 6000 stars are visible to the unaided eye. A natural human tendency is to see patterns, and so people connected the brightest stars into configurations called constellations, which ancient astronomers named after mythological beings, heroes, and animals--whatever was important to them. Figure 1.6 shows a constellation especially prominent in the nighttime sky from October through March: the "hunter" named Orion. Orion was a mythical Greek hero famed for his great beauty and stature, his hunting prowess, and his amorous pursuit of the Pleiades, the seven daughters of the giant Atlas. According to Greek mythology, in order to protect the Pleiades from Orion, the gods placed them among the stars, where Orion nightly stalks them across the sky but never catches them. Many constellations have similarly fabulous connections with ancient cultures.

Figure 1.6 (a) A photograph of the group of bright stars that make up the constellation Orion. (b) The stars are connected to show the pattern visualized by the Greeks: the outline of a hunter. You can easily find this constellation in the winter sky by identifying the line of three bright stars in the hunter's "belt." (c) The true relationships between the stars, in three dimensions.

Generally speaking, the stars that make up any particular constellation are not actually close to one another, even by astronomical standards. These stars merely are bright enough to observe with the naked eye and happen to lie in roughly the same direction in the sky as seen from the Earth. That these stars have become associated with one another over the centuries is a tribute to the power of the human brain, which is extremely good at recognizing patterns and relationships between objects even when no true connection exists.

Perhaps not surprisingly, the patterns seen have a strong cultural bias--the astronomers of ancient China saw mythical figures different from those seen by the ancient Greeks, the Babylonians, and the people of other cultures, even though they were all looking at the same stars in the nighttime sky. Interestingly, different cultures often made the same basic groupings of stars, despite widely varying interpretations of what they saw. For example, the group of seven stars usually known in North America as "the Big Dipper" is known as "the Wagon" or "the Plough" in Western Europe. The ancient Greeks regarded these same stars as the tail of "the Great Bear," the Egyptians saw them as the leg of an ox, the Siberians as a stag. Some Native Americans saw two mythical brothers, others an ermine, others still a funeral procession. The Chinese saw the pattern as a minor government official, dealing with the day-to-day concerns of the emperor.

The origins of most constellations, and of their names, date back to the dawn of recorded history. Some constellations served as navigational guides. For example, the star Polaris, part of the Little Dipper, signals north, and the near-constancy of its location in the sky, from hour to hour and night to night, has aided travelers for centuries. Other constellations served as primitive calendars to predict planting and harvesting seasons. For example, many cultures knew well that the appearance of certain stars on the horizon just before daybreak signaled the beginning of spring and the end of winter. Such knowledge provided the foundations for the science of astronomy.

In many societies, people came to believe that there were other benefits in being able to trace the regularly changing positions of heavenly bodies. The relative positions of stars and planets at a person's birth were carefully studied by astrologers, who used the data to make predictions about that person's destiny. Thus, in a sense, astronomy and astrology arose from the same basic impulse--the desire to "see" into the future--and indeed, for a long time they were indistinguishable from one another. Today most people recognize that astrology is nothing more than an amusing diversion (although millions still study their horoscopes in the newspaper every morning!). Nevertheless, the ancient astrological terminology--the names of the constellations and some terms used to describe the locations and motions of the planets, for example--is still used throughout the astronomical world. The constellations still help astronomers to specify large areas of the sky, much as geologists use continents or politicians use voting precincts to identify certain localities on planet Earth. In all, there are 88 constellations, most of them visible from North America at some time during the year.

THE CELESTIAL SPHERE

Following the constellations nightly, ancient sky-watchers noted that the star patterns seemed unchanging. It was natural for them to conclude that the stars must be firmly attached to a celestial sphere surrounding Earth--a canopy of stars resembling an astronomical painting on a heavenly ceiling. Figure 1.7 shows how early astronomers pictured the stars as moving with this celestial sphere as it turned around a fixed, unmoving Earth.

Figure 1.7 Planet Earth sits fixed at the hub of the celestial sphere, which contains all the stars. This is one of the simplest possible models of the universe, but it doesn't agree with all the facts that astronomers know about the universe.

For the most part, stars rise in the east, move across the sky, and set in the west each night. Some sweep out a large arc high above the horizon, but others appear to move very little. In fact, closer scrutiny (or time-lapse photography--see Figure 1.8) shows that all stars 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 the ancients, this point represented the axis around which the entire celestial sphere turned.

Figure 1.8 Time-lapse photographs of the northern sky. Each trail is the path of a single star across the nighttime sky. The duration of the exposure in (a) is about 1 hour, that in (b) is about 5 hours. How can you tell that this is so? The center of the concentric circles is near the north star, Polaris, whose short, bright arc is prominently visible in (b).

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. Polaris indicates the direction--due north--in which Earth's rotation axis points. Even though we now know that the celestial sphere is an incorrect description of the heavens, we still use the idea as a convenient fiction that helps us visualize the positions of stars in the sky. The point where Earth's axis intersects the celestial sphere in the Northern Hemisphere is known as the north celestial pole, and it is directly above Earth's North Pole. In the Southern Hemisphere, the extension of Earth's axis in the opposite direction defines the south celestial pole, directly above Earth's South Pole. Midway between the north and south celestial poles lies the celestial equator, representing the intersection of Earth's equatorial plane with the celestial sphere. These parts of the celestial sphere are marked on Figure 1.7.

DAY-TO-DAY CHANGES

Back We measure time by the Sun. Because the rhythm of day and night is central to our lives, 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. The daily progress of the Sun and the other stars across the sky is known as diurnal motion. As we have just seen, it is a consequence of the rotation of Earth. But the star's positions in the sky do not repeat themselves exactly from one night to the next. In fact, each night, the whole celestial sphere appears to be shifted a little relative to the horizon, compared with the night before. The easiest way to confirm this difference is by noticing the stars that are visible just after sunset or just before dawn. You will find that they are in slightly different locations from the previous night. Because of this shift, a day measured by the stars--called a sidereal day after the Latin word sidus, meaning star--differs from a solar day. Evidently, there is more to the apparent motion of the heavens than just simple rotation. In fact, the motion of the Earth relative to the Sun--Earth's revolution--is also of great importance.

The reason for the difference between a solar day and a sidereal day is sketched in Figure 1.9. Each time Earth rotates once on its axis, it also moves a small distance along its orbit about the Sun. Earth therefore has to rotate through slightly more than 360° for the Sun to return to the same apparent location in the sky. Thus, the interval of time between noon one day and noon the next (a solar day) is slightly greater than one 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 (that is, one sidereal day is roughly 23h56m long.) From the point of view of the ancients, the "fact" that the Sun moved relative to the stars made it necessary to envisage not just one celestial sphere, but two: one for the stars and one for the Sun, each spinning about the Earth but at slightly different rates and with different orientations.

Figure 1.9 The difference between a solar and a sidereal day can be easily explained once we understand that Earth revolves around the Sun at the same time as it rotates on its axis. A solar day is the time from one noon to the next. In that time, Earth also moves a little in its solar orbit. Because Earth completes one circuit (360°) around the Sun in 1 year (365 days), it moves through nearly 1° in 1 day. Thus, between noon at point A on one day and noon at the same point the next day, Earth actually rotates through about 361°. Consequently, the solar day exceeds the sidereal day (360° rotation) by about 4 minutes. Note that the diagram is not drawn to scale, so that the true 1° angle is in reality much smaller than shown here.

SEASONAL CHANGES

Seen from Earth, the Sun appears to move relative to the other stars. That is because Earth orbits the Sun, completing one orbit in 365.242 solar days--a period of time known as one tropical year. By the end of that time, the Sun has "traversed the sky" and returned to its starting position on the celestial sphere, and the cycle begins anew. The apparent motion of the Sun in the sky, expressed relative to the stars, follows a path known as the ecliptic. As illustrated in Figure 1.10(a), the ecliptic forms a great circle on the celestial sphere, inclined at an angle of about 23.5° to the celestial equator. This tilt is just a consequence of the inclination of Earth's rotation axis to the plane of its orbit, as shown in Figure 1.10(b).

The point on the ecliptic where the Sun is at its northernmost point above the celestial equator is known as the summer solstice (from the Latin word sol, meaning sun). As shown in Figure 1.10, it represents the point in Earth's orbit where our planet's North Pole points 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, points north of the equator spend the greatest fraction of their time in sunlight on that date, so the summer solstice corresponds to the longest day of the year in the Northern Hemisphere and the shortest day in the Southern Hemisphere. Six months later, the Sun is at its southernmost point, and we have reached the winter solstice (December 21)--the shortest day in the Northern Hemisphere and the longest in the Southern Hemisphere. These two effects--the height of the Sun above the horizon and the length of the day-- combine to account for the seasons we experience. In summer in the Northern Hemisphere, the Sun is high in the sky and the days are long, so temperatures are generally much higher than in winter, when the Sun is low and the days are short.

Figure 1.10 (a) The apparent path of the Sun on the celestial sphere and (b) its actual relation to Earth's rotation and revolution. The seasons result from the changing height of the Sun above the horizon. At the summer solstice (the points marked 1), the Sun is highest in the sky, as seen from the northern hemisphere, and the days are longest. In the "celestial sphere" picture (a), the Sun is at its northernmost point on its path around the ecliptic; in reality (b), the summer solstice corresponds to the point on Earth's orbit where our planet's north pole points most nearly toward the Sun. The reverse is true at the winter solstice (point 3). At the vernal and autumnal equinoxes, day and night are of equal length. These are the times when, as seen from Earth (a), the Sun crosses the celestial equator. They correspond to the points in Earth's orbit when our planet's axis is perpendicular to the line joining the Earth and Sun (b).

The two points where the ecliptic intersects the celestial equator 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 the Northern Hemisphere), as the Sun crosses from the Northern into the Southern Hemisphere, we have the autumnal equinox (on September 21). The vernalequinox occurs in Northern spring, on or near March 21, as the Sun crosses the celestial equator moving north. Because of its association with the end of winter and the start of a new growing season,the vernal equinox was particularly important to early astronomers and astrologers.

SUMMER AND WINTER CONSTELLATIONS

Figure 1.11 (a) illustrates the major stars visible from most locations in the United States on clear summer evenings. The brightest stars--Vega, Deneb, and Altair--form a conspicuous triangle high above the constellations Sagittarius and Capricornus, which are low on the southern horizon. In the winter sky, however, these stars have been replaced, as shown in Figure 1.11(b), by several well-known constellations that include Orion, Leo, and Gemini. In the constellation Canis Major lies Sirius (the Dog Star), the brightest star in the sky. Year after year, the same stars and constellations return, each in its proper season. Every winter evening, Orion is high overhead; every summer, it is gone. For more detailed maps of the sky at different seasons, consult the star charts at the end of the book.

Figure 1.11 (a) A typical summer sky above the United States. Some prominent stars (labeled in larger print) and constellations (labeled in small capital letters) are shown. (b) A typical winter sky above the United States.

The reason for these regular seasonal changes is, once again, the revolution of the Earth around the Sun. Earth's darkened hemisphere faces in a slightly different direction in space each evening. The change in direction is only about 1° per night (see Figure 1.9)--too small to be easily noticed with the naked eye from one evening to the next, but clearly noticeable over the course of weeks and months, as illustrated in Figure 1.12. After 6 months, the Earth has reached the opposite side of its orbit, and at night we face an entirely different group of stars and constellations. Ancient astronomers would have said that the Sun has moved to the opposite side of the celestial sphere, so that a different set of stars is visible at night.

Figure 1.12 The view of the night sky changes as the Earth moves in its orbit about the Sun. As drawn here, the night side of Earth faces a different set of constellations at different times of the year.

The 12 constellations through which the Sun passes 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. These constellations are collectively known as the zodiac. The time required for the constellations to complete one cycle around the sky and to return to their starting points as seen from a given point on the Earth is one sidereal year. Earth completes exactly one orbit around the Sun in this time. One sidereal year is 365.256 solar days long, about 20 minutes longer than a tropical year. We will return to the reason for this slight discrepancy in a moment.

Back The simplest method of locating stars in the sky is to specify their constellation and then rank the stars in it in order of brightness. The brightest star is denoted by the Greek letter (alpha), the second brightest by (beta), and so on. Thus, the two brightest stars in the constellation Orion--Betelgeuse and Rigel--are also known as Orionis and Orionis, respectively. (Precise recent 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 utility. However, for naked-eye astronomy, where only bright stars are involved, it is quite satisfactory.

For more precise measurements, astronomers find it helpful to lay down 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 latitude and longitude on Earth's surface extends quite naturally to cover the sky. The celestial analogs of latitude and longitude on the Earth's surface are known as declination and right ascension, respectively. Figure 1.13 illustrates the meanings of right ascension and declination on the celestial sphere and compares them with longitude and latitude on Earth.

Figure 1.13 The right ascension and declination of a star on the celestial sphere are defined similarly to longitude and latitude on the surface of the Earth. Just as longitude and latitude allow us to locate a point on the surface of Earth, right ascension and declination specify locations on the sky. For example, to find Philadelphia on Earth, look 75° west of the Greenwich Meridian and 40° north of the Equator. Similarly, to locate the star Betelgeuse on the celestial sphere, look 5h54m east of the vernal equinox (the line on the sky with a right ascension of zero) and 7°24´ north of the celestial equator.

Declination (dec) is measured in degrees (°) north or south of the celestial equator, just as latitude is measured in degrees north or south of Earth's equator. (See the More Precisely feature above for a discussion of angular measure.) Thus, the celestial equator is at a declination of 0°, the north celestial pole is at +90°, and the south celestial pole is at -90° (the minus sign here just means "south of the celestial equator").

Right ascension (RA) is measured in units called hours, minutes, and seconds, and it increases in the eastward direction. The angular units used to measure right ascension are constructed to parallel the units of time; the two sets of units are connected by the rotation of the Earth (or of the celestial sphere). In 24 hours, Earth rotates once on its axis, or through 360°. Thus, in a time period of 1 hour, Earth rotates through 360°/24 = 15°, or 1h. In 1 minute of time, Earth rotates through 1m; in 1 second, Earth rotates through 1s. The choice of zero right ascension is quite arbitrary--it is conventionally taken to be the position of the Sun in the sky at the instant of the vernal equinox, when the Sun happens to lie between the constellations Pisces and Aquarius.

Although the units of right ascension were originally defined in this way to assist astronomical observation, their names are rather unfortunate, as these are angular measures, not units of time. Moreover, the angular units used for right ascension are not the same units as defined in the More Precisely feature, which discusses angular measure. In fact, right ascension 1m = 15°/60 = 0.25°, or 15 arc minutes (15´). Similarly, right ascension 1s = 15 arc seconds (15´´). If you remember that the units of right ascension are used only for that one purpose, and that all angular measurements except right ascension use arc minutes (´) and arc seconds (´´), you should avoid undue confusion.

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 the Earth's rotation, their celestial coordinates remain constant over the course of a night. Thus, we have a quantitative alternative to the use of constellations in specifying the positions of stars in the sky. For example, the stars Rigel and Betelgeuse mentioned earlier can be precisely located by looking in the directions 5h13m36s (RA), -8°13´ (dec) and 5h54m0s (RA), 07°24´ (dec), respectively. The coordinates of Betelgeuse on the celestial sphere are marked on Figure 1.13.

Back The Moon is our nearest neighbor in space. Apart from the Sun, it is by far the brightest object in the sky. Unlike the Sun and the other stars, however, it emits no light of its own. Instead, it shines by reflecting sunlight. Another difference is that the Moon's appearance changes from night to night--in fact, on some nights it cannot be seen at all. Also, the Moon's daily rising and setting and its nightly motion through the sky differ from the motion of the celestial sphere. Like the Sun, the Moon appears to move relative to the stars--it crosses the sky at a rate of about 12° per day, which means it moves an angular distance equal to its own diameter--30´--in about an hour. Today, we explain these observations in terms of the Moon's revolution around the Earth. For ancient astronomers, however, the Moon's motion meant having to introduce yet another sphere in the sky, with a motion separate from either that of the stars or that carrying the Sun.

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 actually derived from the word Moon.) Figure 1.14 illustrates the appearance of the Moon at different times in this monthly cycle. Starting from the so-called new Moon, which is all but invisible in the sky, the Moon appears to wax (or grow) a little each night and is visible as a growing crescent (panel 1 of Figure 1.14). One week after new Moon, half of the lunar disk can be seen (panel 2). This phase is known as a quarter Moon. During the next week, the Moon continues to wax, passing through the gibbous phase (panel 3) until, two weeks after new Moon, the full Moon (panel 4) is visible. During the next 2 weeks, the Moon wanes (or shrinks), passing in turn through the gibbous, quarter, and crescent phases (panels 5­7), eventually becoming new again. The waxing and waning phases are not merely time reversals of each other, however. The waxing Moon grows from the western edge of the disk, while the waning Moon shrinks toward the eastern edge.

Figure 1.14 Because the Moon orbits the Earth, the visible fraction of the sunlit face differs from night to night. The complete cycle of lunar phases takes 29 days to complete.

The Moon doesn't actually change its size and shape on a monthly basis, of course; the full circular disk of the Moon is present at all times. Why then don't we always see a full Moon? As illustrated in Figure 1.14, half of the Moon's surface is illuminated by the Sun at any instant. However, 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 "daylit" face because the Sun and the Moon are in opposite directions from the Earth in the sky. In the case of a new Moon, the Moon and the Sun are in almost the same part of the sky, and the sunlit side of the Moon is oriented away from us. During the new Moon, the Sun must be almost behind the Moon, from our perspective.

As the Moon revolves around Earth, its position in the sky changes with respect to the stars. In one sidereal month (27.3 days), the Moon returns to the starting point on its 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 reason that a solar day is slightly longer than a sidereal day: Because of the motion of the Earth around the Sun, the Moon must complete slightly more than one full revolution to return to the same phase in its orbit (see Figure 1.15).

Figure 1.15 The difference between a synodic and a sidereal month stems from the motion of the Earth relative to the Sun. Because the Earth orbits the Sun in 365 days, in the 29.5 days from one new Moon to the next (one synodic month), Earth moves through an angle of approximately 29°. Thus the Moon must revolve more than 360° between new Moons. The sidereal month, which is the time taken for the Moon to revolve through exactly 360°, relative to the stars, is about 2 days shorter.

Back From time to time--but only at new or full Moon--the Sun and the Moon line up precisely as seen from Earth, 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 1.16. From Earth, we see the curved edge of Earth's shadow begin to cut across the face of the full Moon and slowly eat its way into the lunar disk. Usually, the alignment of the Sun, Earth, and Moon is imperfect, and the shadow never completely covers the Moon. Such an occurrence is known as a partial eclipse. Occasionally, however, the entire lunar surface is obscured in a total eclipse, as shown in Figure 1.16. Total lunar eclipses last only as long as is needed for the Moon to pass through Earth's shadow--no more than about 100 minutes. During that time, the Moon often acquires an eerie, deep red coloration--the result of a small amount of sunlight being refracted (bent) by Earth's atmosphere onto the lunar surface, preventing the shadow from being completely black. It is perhaps understandable that many primitive cultures interpreted lunar eclipses as harbingers of disaster.

Figure 1.16 A lunar eclipse occurs when the Moon passes through Earth's shadow. At these times we see a darkened, copper-colored Moon, as shown in the inset photograph. The coloration is caused by sunlight deflected by the Earth's atmosphere onto the Moon's surface. (Note that this figure is not drawn to scale.)

When the Moon and the Sun are in exactly the same direction, an even more awe-inspiring event occurs. The Moon passes directly in front of the Sun, briefly turning day into night in a solar eclipse. In a total solar eclipse, when the alignment is perfect, planets and some stars become visible in the daytime as the Sun's light is reduced to nearly nothing. We can also see the Sun's ghostly outer atmosphere, or corona (Figure 1.17a).* In a partial solar eclipse, the Moon's path is slightly "off center," and only a portion of the Sun's face is covered. In either case, the sight of the Sun apparently being swallowed up by the black disk of the Moon is disconcerting even today. It must surely have inspired fear in early observers. Small wonder, then, that the ability to predict such events was a highly prized skill.

*Actually, although a total solar eclipse is undeniably a spectacular occurrence, the visibility of the corona is probably the most important astronomical aspect of such an event today. It enables us to study this otherwise hard-to-see part of our Sun.

Figure 1.17 (a) During a total solar eclipse, the Sun's corona becomes visible as an irregularly shaped halo surrounding the blotted-out disk of the Sun. This was the July 1991 eclipse, as seen from the Baja Peninsula. (b) During an annular eclipse, the Moon fails to completely hide the Sun, so a thin ring of light remains. No corona is seen in this case because even the small amount of the Sun still visible completely overwhelms the corona's faint glow. This was the December 1973 eclipse, as seen from Algiers. (The gray fuzzy areas at top left and right are clouds in Earth's atmosphere.)

Unlike a lunar eclipse, which is simultaneously visible from all locations on the nighttime side of Earth, a total solar eclipse can be seen from only a small portion of the daytime side. The Moon's shadow on Earth's surface is about 7000 kilometers wide--roughly twice the diameter of the Moon. Outside of that shadow, no eclipse is seen. However, only within the central region of the shadow, the umbra, is the eclipse total. Within the shadow but outside the umbra, in the penumbra, the eclipse is partial, with less and less of the Sun being obscured the farther one travels from the shadow's center. The connections between the umbra, the penumbra, and the relative locations of Earth, Sun, and Moon are illustrated in Figure 1.18. One of the reasons that total solar eclipses are rare is that, although the penumbra is some 7000 kilometers across, the umbra is always very small--even under the most favorable circumstances, its diameter never exceeds 270 kilometers. Moreover, because the shadow sweeps across the Earth's surface at over 1700 kilometers per hour, the duration of a total eclipse at any given point can never exceed 7.5 minutes.

Figure 1.18 (a) The Moon's shadow on Earth during a solar eclipse consists of the umbra, where the eclipse is total, and the penumbra, where the Sun is only partially obscured. If the Moon is too far from Earth at the moment of the eclipse, there is no region of totality; instead, an annular eclipse is seen. (b) Actual photographs taken by an Earth-orbiting weather satellite of the Moon's shadow projected onto the Earth's surface (near the Baja Peninsula) during the total solar eclipse of July 11, 1991.

The Moon's orbit around Earth is not exactly circular. Thus, the Moon may be far enough from Earth at the moment of an eclipse that its disk fails to cover the disk of the Sun completely, even though their centers coincide. In that case, there is no region of totality--the umbra never reaches Earth at all, and a thin ring of sunlight can still be seen surrounding the Moon. Such an occurrence, called an annular eclipse, is depicted in Figures 1.17(b) and 1.18. Roughly half of all solar eclipses are annular in nature.

If the Moon orbited Earth in exactly the same plane as Earth orbits the Sun, there would be precisely one solar and one lunar eclipse every synodic month, with the two alternating at half-month intervals. However, the Moon's orbit is actually slightly inclined to the ecliptic, at an angle of 5.2°, so that the chance of an Earth­Moon­Sun alignment occurring just as the Moon crosses the ecliptic plane is greatly reduced. Figure 1.19 illustrates some possible configurations of the three bodies. If the Moon happens to lie above or below the plane of the ecliptic when new (or full), a solar (or lunar) eclipse cannot occur. Such a configuration is termed unfavorable for producing an eclipse. In a favorable configuration, on the other hand, the Moon is new or full just as it crosses the ecliptic plane, and eclipses are seen. Unfavorable configurations are much more common than favorable ones.

Figure 1.19 (a) An eclipse occurs when Earth, Moon, and Sun are precisely aligned. If the Moon's orbital plane lay in exactly the plane of the ecliptic, this alignment would occur once a month. However, the Moon's orbit is inclined at about 5° to the ecliptic, so not all configurations are actually favorable for producing an eclipse. (b) For an eclipse to occur, the line of intersection of the two planes must lie along Earth­Sun line. Thus, eclipses can occur only at specific times of the year.

Eclipses are relatively rare events. Moreover, they can occur only at certain times of the year. As indicated on Figure 1.19(b), the two points on the Moon's orbit where it crosses the ecliptic plane are known as the nodes of the orbit. The line joining them, which is also the line of intersection of Earth's and the Moon's orbital planes, is known as the line of nodes. Times when the line of nodes is not directed toward the Sun are unfavorable for eclipses. However, when the line of nodes briefly lies along the Earth­Sun line, eclipses are possible. These two periods, known as eclipse seasons, are the only times at which an eclipse can occur. Notice that there is no guarantee that an eclipse will occur. For a solar eclipse, we must have a new Moon during an eclipse season. Similarly, a lunar eclipse can occur only at full Moon during an eclipse season.

In fact, the gravitational tug of the Sun causes the Moon's orbital orientation, and hence the line of nodes, to change slowly with time. The result is that the eclipse seasons gradually progress backward through the calendar, occurring about 20 days earlier each year and taking 18.6 years to make one complete circuit. This phenomenon is known as the regression of the line of nodes. In 1991, the eclipse seasons were in January and July; on July 11, a total eclipse actually occurred, visible in Hawaii, Mexico, and parts of Central and South America. Three years later, in 1994, the eclipse seasons were in May and October; on May 10, an annular eclipse was visible across much of the continental United States. Because we know the orbits of Earth and the Moon to great accuracy, we can predict eclipses far into the future. Figure 1.20 shows the location and duration of all total and annular eclipses of the Sun from 1995 to 2005.

Figure 1.20 Regions of Earth that will see total or annular solar eclipses between the years 1995 and 2005. Each track represents the path of the Moon's umbra across Earth's surface during an eclipse.

The solar eclipses that we do see highlight a remarkable cosmic coincidence. Although the Sun is many times farther away from Earth than is the Moon, it is also much larger. In fact, the ratio of distances is almost exactly the same as the ratio of sizes, so the Sun and the Moon both have roughly the same angular diameter--about half a degree seen from Earth. Thus, the Moon covers the face of the Sun almost exactly. If the Moon were larger, we would never see annular eclipses, and total eclipses would be much more common. If the Moon were a little smaller, we would see only annular eclipses.

Back Earth has many motions--it spins on its axis, it travels around the Sun, and it moves with the Sun through the Galaxy. We have just seen how some of these motions can account for the changing nighttime sky and the changing seasons. In fact, the situation is even more complicated. Like a spinning top that rotates rapidly on its own axis while that axis slowly revolves about the vertical, Earth's axis changes its direction over the course of time (although the angle between the axis and a line perpendicular to the plane of the ecliptic remains close to 23.5°). This kind of change is called precession. Figure 1.21 illustrates Earth's precession, which is caused mostly by the gravitational pulls of the Moon and the Sun. During a complete cycle of precession, taking about 26,000 years, Earth's axis traces out a cone.

Because of Earth's precession, the length of time from one vernal equinox to the next--one tropical year--is not quite the same as the time required for Earth to complete one orbit--one sidereal year. Recall that the vernal equinox occurs when Earth's rotation axis is perpendicular to the line joining Earth and Sun, and the Sun is crossing the celestial equator moving from south to north. In the absence of precession, this would occur exactly once per orbit, and the tropical and sidereal years would be identical. However, because of the slow precessional shift in the orientation of Earth's rotation axis, the instant when the axis is next perpendicular to the line to the Sun occurs slightly sooner than we would otherwise expect. Consequently, the vernal equinox drifts slowly around the zodiac over the course of the precession cycle. This is the cause of the 20-minute discrepancy between the two "years" mentioned earlier.

The tropical year is the year that our calendars measure. If our timekeeping were tied to the sidereal year, the seasons would slowly march around the calendar as the Earth precessed--13,000 years from now, summer in the Northern Hemisphere would be at its height in late February! By using the tropical year instead, we ensure that July and August will always be summer months. However, in 13,000 years' time, Orion will be a summer constellation.

There are many more complexities to Earth's motion. For example, the cone traced out by Earth's precession is not as clean as that drawn in Figure 1.21. Because of the combined gravitational influence of the Moon and the planet Jupiter, the tilt of Earth's axis with respect to the ecliptic varies back and forth between 22° and 24°. This variation is very slow, with one cycle completed only every 41,000 years. In addition, the regression of the line of nodes of the Moon's orbit in turn causes our planet's rotation axis to wobble ever so slightly, changing the angle between Earth's axis and the ecliptic by plus or minus 9 arc seconds every 18.6 years. This additional motion, superimposed on Earth's precession, is known as nutation. Finally, the Sun itself travels through space. Currently, the Sun is moving at a speed of about 20 kilometers per second (relative to our neighboring stars) toward Vega, a bright star almost directly overhead in the early evening autumn sky. Along with the other planets that make up the solar system, the Earth is just along for the ride, tracing out a corkscrew path as it travels through space.

Figure 1.21 Earth's axis currently points nearly toward the star Polaris. Some 12,000 years from now--nearly half-way through one cycle of precession--Earth's axis will point toward a star called Vega, which will then be the "North Star". Five thousand years ago, the North Star was a star named Thuban in the constellation Draco.

Back We have seen a little of how astronomers--ancient and modern-- track and record the positions of the stars on the sky. But knowing the directions in which objects lie is only part of the information needed to locate them in space. Before we can make a systematic study of the heavens, we must find a way of measuring distances, too. One such method is called triangulation. It is based on the principles of Euclidean geometry and finds widespread application today in both terrestrial and astronomical settings. Today's engineers, especially surveyors, use these age-old geometrical ideas to measure indirectly the distance to faraway objects. In astronomical contexts, triangulation 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 is to lay a tape across the river, but that's not the simplest way. A smart surveyor would make the measurement by visualizing an imaginary triangle, sighting the tree on the far side of the river from two positions on the near side, as illustrated in Figure 1.22. The simplest possible triangle is a right triangle, in which one of the angles is exactly 90°, so it is usually convenient to set up one observation position directly opposite the object, as at point A. The surveyor then moves to another observation position at point B, noting the distance covered between points A and B. This distance is the baseline of the imaginary triangle. Finally, the surveyor sights toward the tree whose distance is to be measured and notes the angle at point B. No further observations are required. The rest of the problem is a matter of calculation. Knowing the value of one side (AB) and two angles (the right angle itself, at point A, and the angle at point B) of the right triangle, the surveyor can geometrically construct the remaining sides and angles and so establish the distance to the tree.

Figure 1.22 Surveyors often use simple geometry and trigonometry to estimate the distance to a faraway object.

To use triangulation to measure distances, a surveyor must be familiar with trigonometry, the mathematics of geometrical angles. However, even if we knew no trigonometry at all, we could still solve the problem by graphical means, as shown in Figure 1.23. Suppose that we pace off the baseline AB, measuring it to be 450 m, and measure the angle between the baseline and the line from B to the tree to be 52°, as illustrated in the figure. We can transfer the problem to paper by letting one box on our graph represent 25 m on the ground. Drawing the line AB on paper, completing the other two sides of the triangle, at angles of 90° (at A) and 52° (at B), we measure the distance on paper from A to the tree to be 23 boxes--that is, 575 m. We have solved the real problem by modeling it on paper. The point to remember here is this: Nothing more complex than basic geometry is needed to infer the distance, the size, and even the shape of an object too far away or too inaccessible for direct measurement.

Figure 1.23 We don't even need trigonometry to estimate distances indirectly. Scaled estimates, like this one on a piece of paper, often suffice.

Triangles with larger baselines are needed if we are to measure greater distances. Figure 1.24 shows a triangle having a fixed baseline between two observation positions at points A and B. Note how the triangle becomes narrower as an object's distance becomes progressively greater. Narrow triangles cause problems because the angles at points A and B are hard to measure accurately. The measurements can be made easier by "fattening" the triangle--in other words, by lengthening the baseline.

Figure 1.24 A triangle of fixed baseline (distance between points A and B) is narrower the farther away the object. As shown here, the imaginary triangle is much thinner when estimating the distance to a remote hill than it is when estimating the distance to a nearby flower.

Now consider an imaginary triangle extending from Earth to a nearby object in space, perhaps the Moon or a neighboring planet. The imaginary triangle is extremely long and narrow, even for the nearest cosmic objects. Figure 1.25(a) illustrates the case in which Earth's diameter, measured from point A to point B, is the baseline. In principle, two observers could sight the object from opposite sides of the Earth and thus measure the triangle's angles at points A and B. In practice, though, these angles cannot be accurately measured. It is actually easier to measure the third angle of the imaginary triangle, namely the very small one near the object. Here's how.

Figure 1.25 (a) This imaginary triangle extends from Earth to a nearby object in space (such as a planet). 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 displacement, or shift, relative to the distant, undisplaced stars.

The observers on either side of Earth sight toward the object, taking note of its position relative to some distant stars seen on the plane of the sky. The observer at point A sees the object projected against a field of very distant stars. Call its apparent location A´, as indicated in Figure 1.25(a). Similarly, the object appears projected at point B´ to the observer at point B. If each observer takes a photograph of the appropriate region of the sky, the object will appear at slightly different places in the two images. In other words, the object's photographic image is slightly displaced, or shifted, relative to the field of distant background stars, as shown in Figure 1.25(b). The background stars themselves appear undisplaced because of their much greater distance from the observer. This apparent displacement 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 1.25(b), measured as an angle on the celestial sphere, is equal to the very small angle shown in Figure 1.25(a). For historical reasons, one-half of this angle is called the parallactic angle.

The closer an object is to the observer, the larger is its parallax. To understand this concept, hold a pencil vertically in front of your nose, as sketched in Figure 1.26. Concentrate on some far-off object--say, a distant wall. Close one eye, then open it while closing the other. By blinking in this way, you should be able to see a large shift of the apparent position of the pencil projected onto the distant wall. In this example, one eye corresponds to point A, the other eye to point B, the distance between your eyeballs to the baseline, the pencil to the nearby object, and the distant wall to a remote field of stars. If you now hold the pencil at arm's length, corresponding to a more distant object but one still not as far away as the distant stars, the apparent shift of the pencil will be less. By moving the pencil farther away, we are narrowing the triangle and decreasing the parallax (and, in the process, making its accurate measurement more difficult). 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.

Figure 1.26 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.

The amount of parallax is thus inversely proportional to an object's distance. Small parallax implies large distance. Conversely, large parallax implies small distance. Knowing the amount of parallax and the length of the baseline, we can easily derive that distance through triangulation.

As surveyors of the sky, we use the same basic information as the surveyor of the land. The calculation is basically the same. Only the means used to obtain the angles are different. The following More Precisely feature illustrates application of simple geometrical logic to solve a distance-measurement problem--in this case, the determination of the radius of the Earth. We will see many instances of similar reasoning throughout this text.