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16 Properties of Stars

LEARNING GOALS

16.1 Snapshot of the Heavens 16.5 The Hertzsprung–Russell Diagram • How can we learn about the lives of stars, which last • What is the Hertzsprung–Russell (H–R) diagram? millions to billions of years? • What are the major features of the H–R diagram? • What are the two main elements in all stars? • How do stars differ along the main sequence? • What two basic physical properties do astronomers • What determines the length of time a star spends on use to classify stars? the main sequence? • What are Cepheid variable stars, and why are they 16.2 Stellar Luminosity important to astronomers? • What is luminosity, and how do we determine it? • How do we measure the distance to nearby stars? 16.6 Star Clusters • How does the magnitude of a star relate to its • What are the two major types of star cluster? apparent brightness? • Why are star clusters useful for studying stellar evolution? 16.3 Stellar Surface Temperature • How do we measure the age of a star cluster? • How are stars classified into spectral types? • What determines a star’s spectral type? 16.4 Stellar Masses • What is the most important property of a star? • What are the three major classes of binary star systems? • How do we measure stellar masses?

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“All men have the stars,” he answered, mysterious points of light in the sky. We now know that all “but they are not the same things for stars form in great clouds of gas and dust. Each star begins different people. For some, who are its life with roughly the same chemical composition: About travelers, the stars are guides. For others three-quarters of the star’s mass at birth is hydrogen, and they are no more than little lights in the about one-quarter is helium, with no more than about 2% sky. For others, who are scholars, they consisting of elements heavier than helium. During most are problems. For my businessman they of any star’s life, the rate at which it generates energy de- were wealth. But all these stars are pends on the same type of balance between the inward pull silent. You—you alone—will have the of gravity and the outward push of internal pressure that stars as no one else has them.” governs the rate of fusion in our Sun. Despite these similarities, stars appear different from Antoine de Saint-Exupéry, from The Little Prince one another for two primary reasons: They differ in mass, and we see different stars at different stages of their lives. The key that finally unlocked these secrets of stars was n a clear, dark night, a few thousand stars an appropriate classification system. Before the twentieth are visible to the naked eye. Many more century, humans classified stars primarily by their bright- ness and location in our sky. The names of the brightest become visible through binoculars, and O stars within each constellation still bear Greek letters desig- with a powerful telescope we can see so many stars nating their order of brightness. For example, the brightest that we could never hope to count them. Like indi- star in the constellation Centaurus is Alpha Centauri, the second brightest is Beta Centauri, the third brightest is vidual people, each individual star is unique. Like the Gamma Centauri, and so on. However, a star’s brightness human family, all stars share much in common. and membership in a constellation tell us little about its true Today, we know that stars are born from clouds nature. A star that appears bright could be either extremely luminous or unusually nearby, and two stars that appear of interstellar gas, shine brilliantly by nuclear fusion right next to each other in our sky might not be true neigh- for millions or billions of years, and then die, some- bors if they lie at significantly different distances from Earth. times in dramatic ways. This chapter outlines how Today, astronomers classify a star primarily according to its luminosity and surface temperature. Our task in this we study and categorize stars and how we have come chapter is to learn how this extraordinarily effective classi- to realize that stars, like people, change over their fication system reveals the true natures of stars and their lifetime. life cycles. We begin by investigating how to determine a star’s luminosity, surface temperature, and mass.

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a Measuring Cosmic Distances Tutorial, Lesson 2 16.1 Snapshot of the Heavens Imagine that an alien spaceship flies by Earth on a simple 16.2 Stellar Luminosity but short mission: The visitors have just 1 minute to learn A star’s luminosity is the total amount of power it radiates everything they can about the human race. In 60 seconds, into space, which can be stated in watts. For example, the they will see next to nothing of each individual person’s Sun’s luminosity is 3.8 1026 watts [Section 15.2].We can- life. Instead, they will obtain a collective “snapshot” of hu- not measure a star’s luminosity directly, because its bright- manity that shows people from all stages of life engaged in ness in our sky depends on its distance as well as its true their daily activities. From this snapshot alone, they must luminosity. For example, our Sun and Alpha Centauri A piece together their entire understanding of human beings (the brightest of three stars in the Alpha Centauri system) and their lives, from birth to death. are similar in luminosity, but Alpha Centauri A is a feeble We face a similar problem when we look at the stars. point of light in the night sky, while our Sun provides Compared with stellar lifetimes of millions or billions enough light and heat to sustain life on Earth. The differ- of years, the few hundred years humans have spent study- ence in brightness arises because Alpha Centauri A is ing stars with telescopes is rather like the aliens’ 1-minute about 270,000 times farther from Earth than is the Sun. glimpse of humanity. We see only a brief moment in any More precisely, we define the apparent brightness of star’s life, and our collective snapshot of the heavens con- any star in our sky as the amount of light reaching us per sists of such frozen moments for billions of stars. From this unit area (Figure 16.1). (A more technical term for appar- snapshot, we try to reconstruct the life cycles of stars while ent brightness is flux.) The apparent brightness of any light also analyzing what makes one star different from another. source obeys an inverse square law with distance, similar Thanks to the efforts of hundreds of astronomers to the inverse square law that describes the force of grav- studying this snapshot of the heavens, stars are no longer ity [Section 5.3].Ifwe viewed the Sun from twice Earth’s

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COMMON MISCONCEPTIONS Luminosity is the total amount of power (energy per second) Photos of Stars the star radiates into space.

Photographs of stars, star clusters, and galaxies convey a great deal of information, but they also contain a few arti- facts that are not real. For example, different stars seem to have different sizes in photographs, but stars are so far away that they should all appear as mere points of light. Stellar sizes in photographs are an artifact of how our instruments record light. Because of the problem of overexposure, brighter stars tend to appear larger than dimmer stars. Overexposure can be a particular problem for photo- graphs of globular clusters of stars and photographs of galaxies. These objects are so much brighter near their centers than in their outskirts that the centers are al- most always overexposed in photographs that show the Apparent brightness is the amount of starlight outskirts. That is why globular clusters and galaxies often Not to scale! reaching Earth (energy look in photographs as if their central regions contain a per second per square single bright blob, when in fact the centers contain many meter). individual stars separated by vast amounts of space. Figure 16.1 Luminosity is a measure of power, and apparent Spikes around bright stars in photographs, often brightness is a measure of power per unit area. making the pattern of a cross with a star at the center, are another such artifact. These spikes are not real but rather are created by the interaction of starlight with the supports holding the secondary mirror in the telescope [Section 7.2]. The spikes generally occur only with point sources of light like stars, and not with larger objects like galaxies. When you look at a photograph showing many galaxies (for example, Figure 20.1), you can tell which objects are stars by looking for the spikes.

2 distance, it would appear dimmer by a factor of 2 4. If 1 AU we viewed it from 10 times Earth’s distance, it would ap- 2 AU pear 102 100 times dimmer. From 270,000 times Earth’s distance, it would look like Alpha Centauri A—dimmer 3 AU by a factor of 270,0002,or about 70 billion. Figure 16.2 shows why apparent brightness follows an inverse square law. The same total amount of light must pass through each imaginary sphere surrounding the star. Figure 16.2 The inverse square law for light. At greater distances If we focus our attention on the light passing through a from a star, the same amount of light passes through an area that small square on the sphere located at 1 AU, we see that the gets larger with the square of the distance. The amount of light same amount of light must pass through four squares of per unit area therefore declines with the square of the distance. the same size on the sphere located at 2 AU. Thus, each 1 1 square on the sphere at 2 AU receives only 22 4 as much light as the square on the sphere at 1 AU. Similarly, the same nosity, and distance of any light source. We will call it the amount of light passes through nine squares of the same luminosity–distance formula: size on the sphere located at 3 AU. Thus, each of these luminosity 1 1 squares receives only 2 as much light as the square apparent brightness 3 9 4p (distance)2 on the sphere at 1 AU. Generalizing, we see that the amount of light received per unit area decreases with increasing dis- Because the standard units of luminosity are watts, the tance by the square of the distance—an inverse square law. units of apparent brightness are watts per square meter. This inverse square law leads to a very simple and Because we can always measure the apparent brightness important formula relating the apparent brightness, lumi- of a star, this formula provides a way to calculate a star’s

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luminosity if we can first measure its distance or to calcu- When we measure the apparent brightness in visible late a star’s distance if we somehow know its luminosity. light, we can calculate only the star’s visible-light luminosity. (The luminosity–distance formula is strictly correct only Similarly, when we observe a star with a spaceborne X-ray if interstellar dust does not absorb or scatter the starlight telescope, we measure only the apparent brightness in X rays along its path to Earth.) and can calculate only the star’s X-ray luminosity. We will Although watts are the standard units for luminosity, use the terms total luminosity and total apparent bright- it’s often more meaningful to describe stellar luminosities ness to describe the luminosity and apparent brightness we in comparison to the Sun by using units of solar luminos- would measure if we could detect photons across the entire 26 ity: LSun 3.8 10 watts. For example, Proxima Cen- electromagnetic spectrum. (Astronomers refer to the total tauri, the nearest of the three stars in the Alpha Centauri luminosity as the bolometric luminosity.) system and hence the nearest star besides our Sun, is only about 0.0006 times as luminous as the Sun, or 0.0006L . Sun Measuring Distance Through Stellar Parallax Betelgeuse, the bright left-shoulder star of Orion, has a luminosity of 38,000LSun, meaning that it is 38,000 times Once we have measured a star’s apparent brightness, the more luminous than the Sun. next step in determining its luminosity is to measure its distance. The most direct way to measure the distances Measuring Apparent Brightness to stars is with stellar parallax, the small annual shifts in a star’s apparent position caused by Earth’s motion around We can measure a star’s apparent brightness by using a the Sun [Section 2.6]. detector, such as a CCD, that records how much energy Recall that you can observe parallax of your finger by strikes its light-sensitive surface each second. For example, holding it at arm’s length and looking at it alternately with such a detector would record an apparent brightness of first one eye closed and then the other. Astronomers mea- 2.7 108 watt per square meter from Alpha Centauri A. sure stellar parallax by comparing observations of a nearby The only difficulties involved in measuring apparent bright- star made 6 months apart (Figure 16.3). The nearby star ness are making sure the detector is properly calibrated appears to shift against the background of more distant and, for ground-based telescopes, taking into account the stars because we are observing it from two opposite points absorption of light by Earth’s atmosphere. of Earth’s orbit. The star’s parallax angle is defined as half No detector can record light of all wavelengths, so we the star’s annual back-and-forth shift. necessarily measure apparent brightness in only some small Measuring stellar parallax is difficult because stars are range of the complete spectrum. For example, the human so far away, making their parallax angles very small. Even eye is sensitive to visible light but does not respond to the nearest star, Proxima Centauri, has a parallax angle of ultraviolet or infrared photons. Thus, when we perceive only 0.77 arcsecond. For increasingly distant stars, the paral- a star’s brightness, our eyes are measuring the apparent lax angles quickly become too small to measure even with brightness only in the visible region of the spectrum. our highest-resolution telescopes. Current technology

Mathematical Insight 16.1 The Luminosity–Distance Formula

We can derive the luminosity–distance formula by extending the Example: What is the Sun’s apparent brightness as seen from idea illustrated in Figure 16.2. Suppose we are located a distance d Earth? from a star with luminosity L. The apparent brightness of the star 26 Solution: The Sun’s luminosity is LSun 3.8 10 watts, and is the power per unit area that we receive at our distance d. We Earth’s distance from the Sun is d 1.5 1011 meters. Thus, the can find this apparent brightness by imagining that we are part Sun’s apparent brightness is: of a giant sphere with radius d, similar to any one of the three L 3.8 1026 watts spheres in Figure 16.2. The surface area of this giant sphere is 4p d2, and the star’s entire luminosity L must pass through 4p d2 4p (1.5 1011 m)2 this surface area. (The surface area of any sphere is 4 radius2.) p 1.3 103 watts/m2 Thus, the apparent brightness at distance d is the power per unit area passing through the sphere: The Sun’s apparent brightness is about 1,300 watts per square meter at Earth’s distance. It is the maximum power per unit area star’s luminosity apparent brightness that could be collected by a detector on Earth that directly faces surface area of imaginary sphere the Sun, such as a solar power (or photovoltaic) cell. In reality, solar L collectors usually collect less power because Earth’s atmosphere absorbs some sunlight, particularly when it is cloudy. 4p d2 This is our luminosity–distance formula.

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distant stars By definition, the distance to an object with a parallax angle of 1 arcsecond is 1 parsec,abbreviated pc.(The word parsec comes from the words parallax and arcsecond.) With Every January, Every July, a little geometry and Figure 16.3 (see Mathematical Insight we see this: we see this: 16.2), it is possible to show that: nearby star 1 pc 3.26 light-years 3.09 1013 km

p If we use units of arcseconds for the parallax angle, a simple formula allows us to calculate distances in parsecs: d 1 d (in parsecs) p (in arcseconds) For example, the distance to a star with a parallax 1 angle of 2 arcsecond is 2 parsecs, the distance to a star with 1 AU 1 Not to scale a parallax angle of 10 arcsecond is 10 parsecs, and the dis- 1 tance to a star with a parallax angle of 100 arcsecond is July January 100 parsecs. Astronomers often express distances in par- Figure 16.3 Parallax makes the apparent position of a nearby secs or light-years interchangeably. You can convert quickly star shift back and forth with respect to distant stars over the between them by remembering that 1 pc 3.26 light-years. course of each year. If we measure the parallax angle p in arc- 1 Thus, 10 parsecs is about 32.6 light-years; 1,000 parsecs, seconds, the distance d to the star in parsecs is p. The angle in this figure is greatly exaggerated: All stars have parallax angles or 1 kiloparsec (1 kpc), is about 3,260 light-years; and 1 mil- of less than 1 arcsecond. lion parsecs, or 1 megaparsec (1 Mpc), is about 3.26 mil- lion light-years. Enough stars have measurable parallax to give us a allows us to measure parallax only for stars within a few fairly good sample of the many different types of stars. hundred light-years—not much farther than what we call For example, we know of more than 300 stars within about our local solar neighborhood in the vast, 100,000-light-year- 33 light-years (10 parsecs) of the Sun. About half are diameter Milky Way Galaxy. binary star systems consisting of two orbiting stars or

Mathematical Insight 16.2 The Parallax Formula

Here is one of several ways to derive the formula relating a star’s We need one more fact from geometry to derive the parallax distance and parallax angle. Figure 16.3 shows that the parallax formula given in the text. As long as the parallax angle, p, is small, angle p is part of a right triangle, the side opposite p is the Earth– sin p is proportional to p. For example, sin 2 is twice as large as 1 Sun distance of 1 AU, and the hypotenuse is the distance d to the sin 1 , and sin 2 is half as large as sin 1 .(You can verify these 1 object. You may recall that the sine of an angle in a right triangle examples with your calculator.) Thus, if we use 2 instead of 1 for is the length of its opposite side divided by the length of the the parallax angle in the formula above, we get a distance of 2 pc 1 hypotenuse. In this case, we find: instead of 1 pc. Similarly, if we use a parallax angle of 10 ,we get a distance of 10 pc. Generalizing, we get the simple parallax for- length of opposite side 1AU sin p mula given in the text: length of hypotenuse d 1 If we solve for d, the formula becomes: d (in parsecs) p (in arcseconds) 1 AU d Example: Sirius, the brightest star in our night sky, has a mea- sin p sured parallax angle of 0.379.How far away is Sirius in parsecs? By definition, 1 parsec is the distance to an object with In light-years? a parallax angle of 1 arcsecond (1 ), or 1/3,600 degree (be- Solution: From the formula, the distance to Sirius in parsecs is: cause that 1° 60 and 160). Substituting these numbers into the parallax formula and using a calculator to find that 1 d (in pc) 2.64 pc sin 14.84814 10 6,we get: 0.379 1 AU 1 AU Because 1 pc 3.26 light-years, this distance is equivalent to: 1 pc 206,265 AU 6 sin 1 4.84814 10 light-years 2.64 pc 3.26 8.60 light-years That is, 1 parsec 206,265 AU, which is equivalent to 3.09 pc 1013 km or 3.26 light-years. (Recall that 1 AU 149.6 million km.)

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multiple star systems containing three or more stars. Most magnitudes, and a few bright stars have apparent magni- are tiny, dim red stars such as Proxima Centauri—so dim tudes less than 1—which means brighter than magnitude 1. that we cannot see them with the naked eye, despite the For example, the brightest star in the night sky, Sirius, has fact that they are relatively close. A few nearby stars, such an apparent magnitude of 1.46. Appendix F gives the ap- as Sirius (2.6 parsecs), Vega (8 parsecs), Altair (5 parsecs), parent magnitudes and solar luminosities for nearby stars and Fomalhaut (7 parsecs), are white in color and bright and the brightest stars. in our sky, but most of the brightest stars in the sky lie far- The modern magnitude system also defines absolute ther away. Because so many nearby stars appear dim while magnitudes as a way of describing stellar luminosities. A many more distant stars appear bright, their luminosities star’s absolute magnitude is the apparent magnitude it must span a wide range. would have if it were at a distance of 10 parsecs from Earth. For example, the Sun’s absolute magnitude is about 4.8, The Magnitude System meaning that the Sun would have an apparent magnitude of 4.8 if it were 10 parsecs away from us—bright enough Many amateur and professional astronomers describe stellar to be visible, but not conspicuous, on a dark night. brightness using the ancient magnitude system devised by Understanding the magnitude system is worthwhile the Greek astronomer Hipparchus (c. 190–120 B.C.). The because it is still commonly used. However, for the cal- magnitude system originally classified stars according to how culations in this book, it’s much easier to work with the bright they look to our eyes—the only instruments avail- luminosity–distance formula, so we will avoid using able in ancient times. The brightest stars received the desig- magnitude formulas in this book. nation “first magnitude,”the next brightest “second magni- ypla om ce n . o c r o

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a The Hertzsprung–Russell Diagram Tutorial, Lessons 1–3 We call these descriptions apparent magnitudes because they compare how bright different stars appear in the sky. Star charts (such as those in Appendix J) often use dots of 16.3 Stellar Surface Temperature different sizes to represent the apparent magnitudes of stars. The second basic property of stars (besides luminosity) In modern times, the magnitude system has been ex- needed for modern stellar classification is surface tempera- tended and more precisely defined (see Mathematical In- ture. Measuring a star’s surface temperature is somewhat sight 16.3). As a result, stars can have fractional apparent easier than measuring its luminosity because the measure-

Mathematical Insight 16.3 The Modern Magnitude Scale

The modern magnitude system is defined so that each difference Solution: We imagine that our eye sees “Star 1” with magnitude 5 of 5 magnitudes corresponds to a factor of exactly 100 in bright- and the telescope detects “Star 2” with magnitude 30. Then we ness. For example, a magnitude 1 star is 100 times brighter than compare: a magnitude 6 star, and a magnitude 3 star is 100 times brighter apparent brightness of Star 1 than a magnitude 8 star. Because 5 magnitudes corresponds to a (1001/5)30 5 (1001/5)25 factor of 100 in brightness, a single magnitude corresponds to a apparent brightness of Star 2 factor of (100)1/5 2.512. 1005 1010 The following formula summarizes the relationship between 10 stars of different magnitudes: The magnitude 5 star is 10 ,or 10 billion, times brighter than the magnitude 30 star, so the telescope is 10 billion times more sensi- apparent brightness of Star 1 m m tive than the human eye. (1001/5) 2 1 apparent brightness of Star 2 Example 2: The Sun has an absolute magnitude of about 4.8. Polaris, the North Star, has an absolute magnitude of 3.6. where m1 and m2 are the apparent magnitudes of Stars 1 and 2, respectively. If we replace the apparent magnitudes with absolute How much more luminous is Polaris than the Sun? magnitudes (designated M instead of m), the same formula ap- Solution: We use Polaris as Star 1 and the Sun as Star 2: plies to stellar luminosities: luminosity of Polaris (1001/5)4.8 ( 3.6) (1001/5)8.4 luminosity of Star 1 (1001/5)M 2 M1 luminosity of Sun luminosity of Star 2 1001.7 2,500 Example 1: On a clear night, stars dimmer than magnitude 5 are Polaris is about 2,500 times more luminous than the Sun. quite difficult to see. Today, sensitive instruments on large tele- scopes can detect objects as faint as magnitude 30. How much more sensitive are such telescopes than the human eye?

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Figure 16.4 This Hubble Space Telescope view through the heart of our Milky Way Galaxy reveals that stars emit light of many different colors. VIS

ment is not affected by the star’s distance. Instead, we de- Spectral Type termine surface temperature directly from the star’s color or spectrum. One note of caution: We can measure only a The emission and absorption lines in a star’s spectrum star’s surface temperature, not its interior temperature. (In- provide an independent and more accurate way to measure terior temperatures are calculated with theoretical models its surface temperature. Stars displaying spectral lines of highly ionized elements must be fairly hot, while stars dis- [Section 15.3].) When astronomers speak of the “tempera- ture” of a star, they usually mean the surface temperature playing spectral lines of molecules must be relatively cool unless they say otherwise. [Section 6.4].Astronomers classify stars according to surface A star’s surface temperature determines the color of temperature by assigning a spectral type determined from the spectral lines present in a star’s spectrum. light it emits [Section 6.4].A red star is cooler than a yellow star, which in turn is cooler than a blue star. The naked The hottest stars, with the bluest colors, are called spec- eye can distinguish colors only for the brightest stars, but tral type O, followed in order of declining surface tempera- colors become more evident when we view stars through ture by spectral types B, A, F, G, K, and M. The time-honored binoculars or a telescope (Figure 16.4). mnemonic for remembering this sequence, OBAFGKM, Astronomers can determine the “color” of a star more is “Oh Be A Fine Girl/Guy, Kiss Me!” Table 16.1 summarizes precisely by comparing its apparent brightness as viewed the characteristics of each spectral type. Each spectral type is subdivided into numbered sub- through two different filters [Section 7.3].For example, a cool star such as Betelgeuse, with a surface temperature of about categories (e.g., B0, B1,...,B9). The larger the number, the 3,400 K, emits more red light than blue light and therefore cooler the star. For example, the Sun is designated spectral looks much brighter when viewed through a red filter than type G2, which means it is slightly hotter than a G3 star but when viewed through a blue filter. In contrast, a hotter star cooler than a G1 star. such as Sirius, with a surface temperature of about 9,400 K, emits more blue light than red light and looks brighter through a blue filter than through a red filter.

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THINK ABOUT IT Table 16.1 The Spectral Sequence Invent your own mnemonic for the OBAFGKM sequence. To help get you thinking, here are two examples: (1) Only Bungling Spectral Temperature Astronomers Forget Generally Known Mnemonics; and (2) Only Type Example(s) Range Business Acts For Good, Karl Marx. O Stars of >30,000 K Orion’s Belt History of the Spectral Sequence You may wonder why the spectral types follow the peculiar BRigel 30,000 K–10,000 K order of OBAFGKM. The answer lies in the history of stel- lar spectroscopy. Astronomical research never paid well, and many astron- ASirius 10,000 K–7,500 K omers of the 1800s were able to do research only because of family wealth. One such astronomer was Henry Draper (1837–1882), an early pioneer of stellar spectroscopy. After FPolaris 7,500 K–6,000 K Draper died in 1882, his widow made a series of large do- nations to Harvard College Observatory for the purpose of building upon his work. The observatory director, Edward GSun, Alpha 6,000 K–5,000 K Pickering (1846–1919), used the gifts to improve the facili- Centauri A ties and to hire numerous assistants, whom he called “com- puters.”Pickering added money of his own, as did other KArcturus 5,000 K–3,500 K wealthy donors. Most of Pickering’s hired computers were women who had studied physics or astronomy at women’s colleges such as Wellesley and Radcliffe. Women had few opportunities to MBetelgeuse, <3,500 K Proxima advance in science at the time. Harvard, for example, did not Centauri allow women as either students or faculty. Pickering’s project of studying and classifying stellar spectra provided plenty of work and opportunity for his computers, and many of the Harvard Observatory women ended up among the most prominent astronomers of the late 1800s and early 1900s. One of the first computers was Williamina Fleming glance. During her lifetime, she personally classified over (1857–1911). Following Pickering’s suggestion, Fleming clas- 400,000 stars. She became the first woman ever awarded an sified stellar spectra according to the strength of their hy- honorary degree by Oxford University, and in 1929 the drogen lines: type A for the strongest hydrogen lines, type B League of Women Voters named her one of the 12 greatest for slightly weaker hydrogen lines, and so on to type O, for living American women. stars with the weakest hydrogen lines. Pickering published Fleming’s classifications of more than 10,000 stars in 1890. As more stellar spectra were obtained and the spec- tra were studied in greater detail, it became clear that the classification scheme based solely on hydrogen lines was inadequate. Ultimately, the task of finding a better classifi- cation scheme fell to Annie Jump Cannon (1863–1941), who joined Pickering’s team in 1896 (Figure 16.5). Building on the work of Fleming and another of Pickering’s computers, Antonia Maury (1866–1952), Cannon soon realized that the spectral classes fell into a natural order—but not the alphabetical order determined by hydrogen lines alone. Moreover, she found that some of the original classes over- lapped others and could be eliminated. Cannon discovered that the natural sequence consisted of just a few of Picker- ing’s original classes in the order OBAFGKM and also added the subdivisions by number. Figure 16.5 Women astronomers pose with Edward Pickering Cannon became so adept that she could properly clas- at Harvard College Observatory in 1913. Annie Jump Cannon is sify a stellar spectrum with little more than a momentary fifth from the left in the back row.

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Brightest Key Absorption Wavelength Line Features (color) Typical Spectrum

Lines of ionized <97 nm hydrogen helium, weak (ultraviolet)* hydrogen lines O Lines of neutral 97–290 nm helium, moderate (ultraviolet)* B hydrogen lines Very strong 290–390 nm hydrogen lines (violet)* A

Moderate hydrogen 390–480 nm lines, moderate lines (blue)* F of ionized calcium Weak hydrogen 480–580 nm lines, strong lines (yellow) G of ionized calcium Lines of neutral and 580–830 nm K singly ionized metals, (red) some molecules

Molecular lines >830 nm M strong (infrared) ionized titanium sodium titanium calcium oxide oxide

*All stars above 6,000 K look more or less white to the human eye because they emit plenty of radiation at all visible wavelengths.

The astronomical community adopted Cannon’s levels, ionized hydrogen can neither emit nor absorb its system of stellar classification in 1910. However, no one usual specific wavelengths of light. At the other end of the at that time knew why spectra followed the OBAFGKM spectral sequence, M stars are cool enough for some par- sequence. Many astronomers guessed, incorrectly, that ticularly stable molecules to form, explaining their strong the different sets of spectral lines reflected different com- molecular absorption lines. Payne-Gaposchkin described positions for the stars. The correct answer—that all stars her work and her conclusions in a dissertation published in are made primarily of hydrogen and helium and that a star’s 1925. A later review of twentieth-century astronomy called surface temperature determines the strength of its spec- her work “undoubtedly the most brilliant Ph.D. thesis ever tral lines—was discovered by Cecilia Payne-Gaposchkin written in astronomy.” (1900–1979), another woman working at Harvard ypla om ce n . o c r o

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a The Hertzsprung–Russell Diagram Tutorial, Lessons 1–3 Relying on insights from what was then the newly develop- ing science of quantum me- 16.4 Stellar Masses chanics, Payne-Gaposchkin The most important property of a star is its mass, but stellar showed that the differences in masses are harder to measure than luminosities or surface spectral lines from star to star temperatures. The most dependable method for “weigh- merely reflected changes in the ing” a star relies on Newton’s version of Kepler’s third law ionization level of the emitting [Section 5.3].This law can be applied only when we can atoms. For example, O stars have measure both the orbital period and the average distance weak hydrogen lines because, between the stars (semimajor axis) of the orbiting star at their high surface tempera- Cecilia Payne-Gaposchkin system. Thus, in most cases we can measure stellar masses tures, nearly all their hydrogen only in binary star systems in which we have determined is ionized. Without an electron to “jump” between energy the orbital properties of the two stars.

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Mizar is a visual binary. Alcor Mizar B

Mizar Spectroscopy shows that each of the visual “stars” is itself a binary.

Mizar A

Figure 16.6 Mizar looks like one star to the naked eye but is actually a system of four stars. Through a telescope Mizar appears to be a visual binary made up of two stars, Mizar A and Mizar B, that gradually change positions, indicating that they orbit every few thousand years. However, each of these two “stars” is actually a spectroscopic binary, making a total of four stars. (The star Alcor appears very close to Mizar to the naked eye but does not orbit it.)

A

B

1900 1910 1920 1930 1940 1950 1960 1970 Figure 16.7 Each frame represents the relative positions of Sirius A and Sirius B at 10-year intervals from 1900 to 1970. The back-and-forth “wobble” of Sirius A allowed astronomers to infer the existence of Sirius B even before the two stars could be resolved in telescopic photos.

Types of Binary Star Systems sky as if it were a member of a visual binary, but its companion is too dim to be seen. For example, slow About half of all stars orbit a companion star of some kind. shifts in the position of Sirius, the brightest star in the These star systems fall into three classes: sky, revealed it to be a binary star long before its com- panion was discovered (Figure 16.7). ● A visual binary is a pair of stars that we can see dis- tinctly (with a telescope) as the stars orbit each other. ● An eclipsing binary is a pair of stars that orbit in the Mizar, the second star in the handle of the Big Dipper, plane of our line of sight (Figure 16.8). When neither is one example of a visual binary (Figure 16.6). Some- star is eclipsed, we see the combined light of both stars. times we observe a star slowly shifting position in the When one star eclipses the other, the apparent bright-

We see light We see light We see light We see light from both from all of B, from both only from A. A and B. some of A. A and B.

B B

B

A AA A

Figure 16.8 The apparent brightness of brightness apparent an eclipsing binary system drops when either star eclipses the other. time

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ness of the system drops because some of the light is Star B spectrum at time 1: 1 blocked from our view. A light curve, or graph of ap- approaching, therefore blueshifted approaching us parent brightness against time, reveals the pattern of B the eclipses. The most famous example of an eclipsing binary is Algol, the “demon star” in the constellation Perseus (algol is Arabic for “the ghoul”). Algol becomes to Earth A three times dimmer for a few hours about every 3 days as the brighter of its two stars is eclipsed by its dimmer companion. ● If a binary system is neither visual nor eclipsing, we B may be able to detect its binary nature by observing 2 Doppler shifts in its spectral lines [Section 6.5].Such Star B spectrum at time 2: receding from us systems are called spectroscopic binary systems. If one receding, therefore redshifted star is orbiting another, it periodically moves toward Figure 16.9 The spectral lines of a star in a binary system are us and away from us in its orbit. Its spectral lines show alternately blueshifted as it comes toward us in its orbit and red- shifted as it moves away from us. blueshifts and redshifts as a result of this motion (Fig- ure 16.9). Sometimes we see two sets of lines shifting back and forth—one set from each of the two stars Measuring Masses in Binary Systems in the system (a double-lined spectroscopic binary). Other times we see a set of shifting lines from only one Even for a binary system, we can apply Newton’s version of star because its companion is too dim to be detected Kepler’s third law only if we can measure both the orbital (a single-lined spectroscopic binary). Each of the two period and the separation of the two stars. Measuring orbital stars in the visual binary Mizar is itself a spectroscopic period is fairly easy. In a visual binary, we simply observe binary (see Figure 16.6). how long each orbit takes (or extrapolate from part of an

Orbital Separation and Newton’s Version Mathematical Insight 16.4 of Kepler’s Third Law

Measurements of stellar masses rely on Newton’s version of Kep- Star 2. What are the masses of the two stars? Assume that each of ler’s third law [Section 5.3],for which we need to know the orbital the two stars traces a circular orbit around their center of mass. period p and semimajor axis a. As described in the text, it’s gen- Solution: We will find the masses by using Newton’s version of erally easy to measure p for binary star systems. We can rarely Kepler’s third law, solved for the masses: measure a directly, but we can calculate it in cases in which we 4p2 4p2 a3 can measure the orbital velocity of one star relative to the other. p2 a3 ⇒ (M M ) 1 2 2 If we assume that the first star traces a circle of radius a G(M1 M2) G p around its companion, the circumference of its orbit is 2pa. Be- We are given the orbital period p 6.2 107 s, and we find the cause the star makes one circuit of this circumference in one orbital semimajor axis a of the system from the given orbital velocity v: period p, its velocity relative to its companion is: 7 pv (6.2 10 s) (100,000 m/s) distance traveled in one orbit 2pa a v 2p 2p period of one orbit p 9.9 1011 m Solving for a, we find: Now we calculate the sum of the stellar masses by substituting the pv values of p, a, and the gravitational constant G [Section 5.3] a into 2p the mass equation above: 4 2 (9.9 1011 m)3 Once we know both p and a, we can use Newton’s version p (M1 M2) 3 7 2 of Kepler’s third law to calculate the sum of the masses of the two m (6.2 10 s) 6.67 10 11 stars (M1 M2). We can then calculate the individual masses kg s2 of the two stars by taking advantage of the fact that the relative 32 velocities of the two stars around their common center of mass 1.5 10 kg are inversely proportional to their relative masses. Because the lines of Star 1 shift twice as far as those of Star 2, Example: The spectral lines of two stars in a particular eclipsing we know that Star 1 moves twice as fast as Star 2, and hence that binary system shift back and forth with a period of 2 years (p Star 1 is half as massive as Star 2. In other words, Star 2 is twice 6.2 107 seconds). The lines of one star (Star 1) shift twice as far as massive as Star 1. Using this fact and their combined mass of as the lines of the other (Star 2). The amount of Doppler shift 1.5 1032 kg, we conclude that the mass of Star 2 is 1.0 1032 kg indicates an orbital speed of v 100,000 m/s for Star 1 relative to and the mass of Star 1 is 0.5 1032 kg.

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orbit). In an eclipsing binary, we measure the time between Hertzsprung and Russell based their diagrams on the eclipses. In a spectroscopic binary, we measure the time it spectral sequence OBAFGKM. takes the spectral lines to shift back and forth. ● The vertical axis represents stellar luminosity, in units Determining the average separation of the stars in a of the Sun’s luminosity (L ). Stellar luminosities binary system is usually much more difficult. Except in rare Sun span a wide range, so we keep the graph compact by cases in which we can measure the separation directly, we can making each tick mark represent a luminosity 10 times calculate the separation only if we know the actual orbital larger than the prior tick mark. speeds of the stars from their Doppler shifts. Unfortunately, a Doppler shift tells us only the portion of a star’s velocity Each location on the diagram represents a unique that is directly toward us or away from us [Section 6.5].Be- combination of spectral type and luminosity. For example, cause orbiting stars generally do not move directly along the dot representing the Sun in Figure 16.10 corresponds our line of sight, their actual velocities can be significantly to the Sun’s spectral type, G2, and its luminosity, 1LSun. greater than those we measure through the Doppler effect. Because luminosity increases upward on the diagram and The exceptions are eclipsing binary stars. Because these surface temperature increases leftward, stars near the upper stars orbit in the plane of our line of sight, their Doppler left are hot and luminous. Similarly, stars near the upper shifts can tell us their true orbital velocities.* Eclipsing right are cool and luminous, stars near the lower right are binaries are therefore particularly important to the study cool and dim, and stars near the lower left are hot and dim. of stellar masses. As an added bonus, eclipsing binaries allow us to measure stellar radii directly. Because we know THINK ABOUT IT how fast the stars are moving across our line of sight as one eclipses the other, we can determine their radii by timing Explain how the colors of the stars in Figure 16.10 help indi- how long each eclipse lasts. cate stellar surface temperature. Do these colors tell us anything about interior temperatures? Why or why not?

THINK ABOUT IT The H–R diagram also provides direct information Suppose two orbiting stars are moving in a plane perpendicular about stellar radii, because a star’s luminosity depends on to our line of sight. Would the spectral features of these stars both its surface temperature and its surface area or radius. appear shifted in any way? Explain. Recall that surface temperature determines the amount ypla om ce n . o c r o

t m

s of power emitted by the star per unit area: Higher tempera-

a The Hertzsprung–Russell Diagram Tutorial, Lessons 1–3 ture means greater power output per unit area [Section 6.4]. Thus, if two stars have the same surface temperature, one 16.5 The Hertzsprung–Russell can be more luminous than the other only if it is larger in Diagram size. Stellar radii therefore must increase as we go from the high-temperature, low-luminosity corner on the lower left During the first decade of the twentieth century, a similar of the H–R diagram to the low-temperature, high-luminosity thought occurred independently to astronomers Ejnar Hertz- corner on the upper right. sprung, working in Denmark, and Henry Norris Russell, working in the United States at Princeton University: Each decided to make a graph plotting stellar luminosities on one Patterns in the H–R Diagram axis and spectral types on the other. Such graphs are now Figure 16.10 also shows that stars do not fall randomly called Hertzsprung–Russell (H–R) diagrams.Soon after throughout the H–R diagram but instead fall into several they began making their graphs, Hertzsprung and Russell distinct groups: uncovered some previously unsuspected patterns in the prop- erties of stars. As we will see shortly, understanding these ● Most stars fall somewhere along the main sequence, patterns and the H–R diagram is central to the study of stars. the prominent streak running from the upper left to the lower right on the H–R diagram. Our Sun is a main- A Basic H–R Diagram sequence star. ● The stars along the top are called supergiants because Figure 16.10 displays an example of an H–R diagram. they are very large in addition to being very bright. ● The horizontal axis represents stellar surface tempera- ● Just below the supergiants are the giants,which are ture, which, as we’ve discussed, corresponds to spectral somewhat smaller in radius and lower in luminosity type. Temperature increases from right to left because (but still much larger and brighter than main- sequence stars of the same spectral type). *In other binaries, we can calculate an actual orbital velocity from the ● velocity obtained by the Doppler effect if we also know the system’s orbital The stars near the lower left are small in radius and inclination. Astronomers have developed techniques for determining appear white in color because of their high tempera- orbital inclination in a relatively small number of cases. ture. We call these stars white dwarfs.

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When classifying a star, astronomers generally report sequence stars, and luminosity classes II and IV are inter- both the star’s spectral type and a luminosity class that mediate to the others. For example, the complete spectral describes the region of the H–R diagram in which the star classification of our Sun is G2 V. The G2 spectral type means falls. Table 16.2 summarizes the luminosity classes: Lumi- it is yellow in color, and the luminosity class V means it is nosity class I represents supergiants, luminosity class III a main-sequence star. Betelgeuse is M2 I, making it a red represents giants, luminosity class V represents main- supergiant. Proxima Centauri is M5 V—similar in color and

10 2 10 3 Solar Radii Solar Radii 10 6 60MSun 30M 10 Solar Radii Sun Deneb

10 5 Centauri Rigel SUPERGIANTS Spica Canopus Betelgeuse Lifetime 10M 4 Sun 10 10 7 yrs Bellatrix Antares 1 Solar Radius Polaris 3 10 MAIN 6MSun Achernar GIANTS

Lifetime SEQUENCE Aldebaran 8 3M Arcturus 10 2 10 yrs Sun Vega Pollux 0.1 Solar Radius Sirius Procyon 10 1.5M Altair Sun Lifetime Centauri A 10 9 yrs 1 Sun 1MSun Centauri B Eridani Sirius Ceti luminosity (solar units) 10 2 Solar Radius 61 Cygni A 0.1 61 Cygni B Lifetime Lacaille 9352 10 10 yrs 0.3MSun Gliese 725 A Sirius B 10 2 WHITE Gliese 725 B 0.1M DWARFS Lifetime Barnard’s Star Sun 10 3 11 Ross 128 Solar Radius 10 yrs 10 3 Wolf 359 Proxima Centauri Procyon B DX Cancri

10 4

10 5

30,000 10,000 6,000 3,000 increasing temperature surface temperature (Kelvin) decreasing temperature

Figure 16.10 An H–R diagram, one of astronomy’s most important tools, shows how the surface tem- peratures of stars (plotted along the horizontal axis) relate to their luminosities (plotted along the ver- tical axis). Several of the brightest stars in the sky are plotted here, along with a few of those closest to Earth. They are not drawn to scale—the diagonal lines, labeled in solar radii, indicate how large they are compared to the Sun. The lifetime and mass labels apply only to main-sequence stars (see Figure 16.11). (Star positions on this diagram are based on data from the Hipparcos satellite.)

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surface temperature to Betelgeuse, but far dimmer because Table 16.2 Stellar Luminosity Classes of its much smaller size. White dwarfs are usually designated with the letters wd rather than with a Roman numeral. Class Description

ISupergiants The Main Sequence II Bright giants The common trait of main-sequence stars is that, like our III Giants Sun, they are fusing hydrogen into helium in their cores. Because stars spend the majority of their lives fusing hydro- IV Subgiants gen, most stars fall somewhere along the main sequence VMain sequence of the H–R diagram. Why do main-sequence stars span such a wide range of luminosities and surface temperatures? By measuring the masses of stars in binary systems, astronomers have dis- The nuclear burning rate, and hence the luminosity, is very covered that stellar masses decrease downward along the sensitive to mass. For example, a 10MSun star on the main main sequence (Figure 16.11). At the upper end of the main sequence is about 10,000 times more luminous than the Sun. sequence, the hot, luminous O stars can have masses as The relationship between mass and surface tempera- high as 100 times that of the Sun (100MSun). On the lower ture is a little subtler. In general, a very luminous star must end, cool, dim M stars may have as little as 0.08 times the either be very large or have a very high surface tempera- mass of the Sun (0.08MSun). Many more stars fall on the ture, or some combination of both. Stars on the upper end lower end of the main sequence than on the upper end, of the main sequence are thousands of times more lumi- which tells us that low-mass stars are much more common nous than the Sun but only about 10 times larger than the than high-mass stars. Sun in radius. Thus, their surfaces must be significantly The orderly arrangement of stellar masses along the hotter than the Sun’s surface to account for their high lu- main sequence tells us that mass is the most important minosities. Main-sequence stars more massive than the attribute of a hydrogen-burning star. Luminosity depends Sun therefore have higher surface temperatures than the directly on mass because the weight of a star’s outer layers Sun, and those less massive than the Sun have lower sur- determines the nuclear burning rate in its core. More weight face temperatures. That is why the main sequence slices means the star must sustain a higher nuclear burning rate diagonally from the upper left to the lower right on the in order to maintain gravitational equilibrium [Section 15.3]. H–R diagram.

Mathematical Insight 16.5 Calculating Stellar Radii

26 Almost all stars are too distant for us to measure their radii directly. LBet 38,000 LSun 38,000 3.8 10 watts However, we can calculate a star’s radius from its luminosity with 1.4 1031 watts the aid of the thermal radiation laws. As given in Mathematical Insight 6.2, the amount of thermal radiation emitted by a star of Now we can use the formula derived above to calculate the radius surface temperature T is: of Betelgeuse: 4 emitted power per unit area sT L r where the constant s 5.7 10 8 watt/(m2 Kelvin4). 4psT 4 The luminosity L of a star is its power per unit area multi- plied by its total surface area. If the star has radius r, its surface 1.4 1031 watts area is given by the formula 4 r2.Thus: p watt 8 4 2 4 4p 5.7 10 (3,400 K) L 4pr sT m2 K4 With a bit of algebra, we can solve this formula for the star’s 1.4 1031 watts radius r: 11 watts 3.8 10 m 7 L 9.6 10 2 r m 4psT 4 The radius of Betelgeuse is about 380 billion meters or, equiva- Example: Betelgeuse has a luminosity of 38,000LSun and a surface lently, 380 million kilometers. Note that this is more than twice temperature of about 3,400 K. What is its radius? the Earth–Sun distance of 150 million kilometers. Solution: First, we must make our units consistent by convert- ing the luminosity of Betelgeuse into watts. Remembering that 26 LSun 3.8 10 watts, we find:

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Main-Sequence Lifetimes The fact that massive stars exist at all at the present time tells us that stars must form continuously in our A star has a limited supply of core hydrogen and therefore galaxy. The massive, bright O stars in our galaxy today can remain as a hydrogen-fusing main-sequence star for formed only recently and will die long before they have only a limited time—the star’s main-sequence lifetime a chance to complete even one orbit around the center of (or hydrogen-burning lifetime). Because stars spend the the galaxy. vast majority of their lives fusing hydrogen into helium, we sometimes refer to the main-sequence lifetime as sim- ply the “lifetime.”Like masses, stellar lifetimes vary in an THINK ABOUT IT orderly way as we move up the main sequence: Massive Would you expect to find life on planets orbiting massive stars near the upper end of the main sequence have shorter O stars? Why or why not? (Hint: Compare the lifetime of an lives than less massive stars near the lower end (see Fig- O star to the amount of time that passed from the forma- ure 16.11). tion of our solar system to the origin of life on Earth.) Why do more massive stars live shorter lives? A star’s lifetime depends on both its mass and its luminosity. Its On the other end of the scale, a 0.3-solar-mass star mass determines how much hydrogen fuel the star initially emits a luminosity just 0.01 times that of the Sun and con- contains in its core. Its luminosity determines how rap- sequently lives roughly 0.3/0.01 30 times longer than idly the star uses up its fuel. Massive stars live shorter lives the Sun. In a universe that is now about 14 billion years because, even though they start their lives with a larger old, even the most ancient of these small, dim M stars still supply of hydrogen, they consume their hydrogen at a pro- survive and will continue to shine faintly for hundreds of digious rate. billions of years to come. The main-sequence lifetime of our Sun is about 10 bil- lion years [Section 15.1].A 30-solar-mass star has 30 times more hydrogen than the Sun but burns it with a luminosity Giants, Supergiants, and White Dwarfs some 300,000 times greater. Consequently, its lifetime is Giants and supergiants are stars nearing the ends of their roughly 30/300,000 1/10,000 as long as the Sun’s—cor- lives because they have already exhausted their core hydro- responding to a lifetime of only a few million years. Cos- gen. Surprisingly, stars grow more luminous when they mically speaking, a few million years is a remarkably short begin to run out of fuel. As we will discuss in the next chap- time, which is one reason why massive stars are so rare: ter, a star generates energy furiously during the last stages Most of the massive stars that have ever been born are long of its life as it tries to stave off the inevitable crushing force since dead. (A second reason is that lower-mass stars form of gravity. As ever-greater amounts of power well up from in larger numbers than higher-mass stars [Section 17.2].) the core, the outer layers of the star expand, making it a giant or supergiant. The largest of these celestial behemoths have radii more than 1,000 times the radius of the Sun. If

6 60 our Sun were this big, it would engulf the planets out to 10 MSun 30MSun Jupiter. 10 5 Spica Because they are so bright, we can see giants and super- 10M giants even if they are not especially close to us. Many of 10 4 Lifetime Sun 107 yrs 6 the brightest stars visible to the naked eye are giants or 3 MSun 10 MAIN Achernar supergiants. They are often identifiable by their reddish SEQUENCE 2 Lifetime 10 3MSun color. Nevertheless, giants and supergiants are rarer than 108 yrs Sirius main-sequence stars. In our snapshot of the heavens, we 1.5 10 MSun catch most stars in the act of hydrogen burning and rela- Lifetime 9 1 10 yrs Sun 1MSun tively few in a later stage of life. Giants and supergiants eventually run out of fuel en- 0.1 Lifetime 0.3MSun tirely. A giant with a mass similar to that of our Sun ulti- 1010 yrs

luminosity (solar units) 2 10 0.1 mately ejects its outer layers, leaving behind a “dead” core Lifetime MSun 11 in which all nuclear fusion has ceased. White dwarfs are 10 3 10 yrs Proxima Centauri these remaining embers of former giants. They are hot 10 4 because they are essentially exposed stellar cores, but they 10 5 are dim because they lack an energy source and radiate only their leftover heat into space. A typical white dwarf 30,000 10,000 6,000 3,000 is no larger in size than Earth, although it may have a mass surface temperature (Kelvin) as great as that of our Sun. (Giants and supergiants with Figure 16.11 Along the main sequence, more massive stars are masses much larger than that of the Sun ultimately explode, brighter and hotter but have shorter lifetimes. (Stellar masses are leaving behind neutron stars or black holes as corpses 30 given in units of solar masses: 1MSun 2 10 kg.) [Section 17.4].)

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period 6 10 2 10 Solar Radii Cepheids with periods of days 5 10 Solar Radii 10 10 3 Solar R 10 4 adii Polaris 3 1 Solar Radius 10 instability strip variable stars with 10 2 periods of hours (called RR Lyrae apparent brightness apparent 10 0.1 Solar Radius variables)

1 Sun 0507525 100 125 150 175 200 10 2 0.1 Solar Radius time (days)

luminosity (solar units) 2 Figure 16.12 A typical light curve for a Cepheid variable star. 10 3 Cepheids are giant, whitish stars whose luminosities regularly pulsate 10 10 3 over periods of a few days to about a hundred days. The pulsation Solar Radius period of this Cepheid is about 50 days. 10 4

10 5

Pulsating Variable Stars 30,000 10,000 6,000 3,000 surface temperature (Kelvin) Not all stars shine steadily like our Sun. Any star that sig- Figure 16.13 An H–R diagram with the instability strip highlighted. nificantly varies in brightness with time is called a variable star. A particularly important type of variable star has a pe- culiar problem with achieving the proper balance between the power welling up from its core and the power being the luminosity–distance formula. In fact, as we’ll discuss in radiated from its surface. Sometimes the upper layers of Chapter 20, Cepheids provide our primary means of mea- such a star are too opaque, so energy and pressure build up suring distances to other galaxies and thus teach us the true beneath the photosphere and the star expands in size. How- scale of the cosmos. The next time you look at the North ever, this expansion puffs the upper layers outward, making Star, Polaris, gaze upon it with renewed appreciation. Not them too transparent. So much energy then escapes that only has it guided generations of navigators in the North- the underlying pressure drops, and the star contracts again. ern Hemisphere, but it is also one of these special Cepheid In a futile quest for a steady equilibrium, the atmo- variable stars.

sphere of such a pulsating variable star alternately expands ypla om ce n . o c r o

t m

and contracts, causing the star to rise and fall in luminos- s

a Stellar Evolution Tutorial, Lessons 1, 4 ity. Figure 16.12 shows a typical light curve for a pulsating variable star, with the star’s brightness graphed against time. Any pulsating variable star has its own particular period 16.6 Star Clusters between peaks in luminosity, which we can discover easily All stars are born from giant clouds of gas. Because a single from its light curve. These periods can range from as short interstellar cloud can contain enough material to form many as several hours to as long as several years. stars, stars almost inevitably form in groups. In our snap- Most pulsating variable stars inhabit a strip (called the shot of the heavens, many stars still congregate in the groups instability strip) on the H–R diagram that lies between the in which they formed. These groups are of two basic types: main sequence and the red giants (Figure 16.13). A special modest-size open clusters and densely packed globular category of very luminous pulsating variables lies in the clusters. upper portion of this strip: the Cepheid variables,or Open clusters of stars are always found in the disk of Cepheids (so named because the first identified star of the galaxy (see Figure 1.18). They can contain up to sev- this type was the star Delta Cephei). eral thousand stars and typically span about 30 light-years Cepheids fluctuate in luminosity with periods of a few (10 parsecs). The most famous open cluster is the Pleiades, a days to a few months. In 1912, another woman astronomer prominent clump of stars in the constellation Taurus (Fig- at Harvard, Henrietta Leavitt, discovered that the periods of ure 16.14). The Pleiades are often called the Seven Sisters, these stars are very closely related to their luminosities: The although only six of the cluster’s several thousand stars are longer the period, the more luminous the star. This period– easily visible to the naked eye. Other cultures have other luminosity relation holds because larger (and hence more names for this beautiful group of stars. In Japanese it is luminous) Cepheids take longer to pulsate in and out in size. called Subaru, which is why the logo for Subaru automo- Once we have measured the period of a Cepheid vari- biles is a diagram of the Pleiades. able, we can use the period–luminosity relation to deter- Globular clusters are found primarily in the halo of mine its luminosity. We can then calculate its distance with our galaxy, although some are in the disk. A globular clus-

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Figure 16.14 A photo of the Pleiades, a nearby open cluster of stars. The most prominent stars in this open cluster are of spectral type B, indicating that the Pleiades are no more than 100 million years old, relatively young for a star cluster. The region shown here is about 11 light-years across. VIS

ter can contain more than a million stars concentrated in a ball typically from 60 to 150 light-years across (20 to 50 par- secs). Its innermost part can have 10,000 stars packed within a region just a few light-years across (Figure 16.15). The view from a planet in a globular cluster would be mar- velous, with thousands of stars lying closer than Alpha Centauri is to the Sun. Because a globular cluster’s stars nestle so closely, they engage in an intricate and complex dance choreographed by gravity. Some stars zoom from the cluster’s core to its outskirts and back again at speeds approaching the escape velocity from the cluster, while others orbit the dense core more closely. When two stars pass especially close to each other, the gravitational pull between them deflects their trajectories, altering their speeds and sending them careen- ing off in new directions. Occasionally, a close encounter boosts one star’s velocity enough to eject it from the clus- ter. Through such ejections, globular clusters gradually lose stars and grow more compact. Star clusters are extremely useful to astronomers for two key reasons: 1. All the stars in a cluster lie at about the same distance from Earth. 2. Cosmically speaking, all the stars in a cluster formed at about the same time (i.e., within a few million years Figure 16.15 This globular VIS of one another). cluster, known as M 80, is over 12 billion years old. The prominent Astronomers can therefore use star clusters as laboratories reddish stars in this Hubble Space for comparing the properties of stars, as yardsticks for Telescope photo are red giant stars nearing the ends of their lives. The region pictured here is about 15 light-years across. measuring distances in the universe [Section 20.3], and as timepieces for measuring the age of our galaxy. We can use clusters as timepieces because we can de- termine their ages from H–R diagrams of cluster stars.

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6 10 2 10 3 6 10 2 10 3 10 Solar R Solar Radii 10 Solar Radii Solar Radii adii 10 Solar Radii 10 5 10 Solar Radii 10 5

4 4 Lifetime 10 10 107 yrs 1 Solar Radius 10 3 1 Solar Radius 10 3

10 2 10 2 Lifetime 108 yrs 0.1 Solar Radius 10 0.1 Solar Radius 10 Lifetime 1 1 109 yrs Lifetime 0.1 0.1 1010 yrs luminosity (solar units) 10 2 luminosity (solar units) 10 2 Lifetime h + Persei Ð 14 million years 1011 yrs 3 3 10 10 Pleiades Ð 100 million years Hyades Ð 650 million years 10 4 10 4 NGC 188 Ð 7 billion years 10 5 10 5

30,000 10,000 6,000 3,000 30,000 10,000 6,000 3,000 surface temperature (Kelvin) surface temperature (Kelvin) Figure 16.16 An H–R diagram for the stars of the Pleiades. Figure 16.17 This H–R diagram shows stars from four clusters Triangles represent individual stars. The Pleiades cluster is missing with very different ages. Each star cluster has a different main- its upper main-sequence stars, indicating that these stars have sequence turnoff point. The youngest cluster, h Persei, still already ended their hydrogen-burning lives. The main-sequence contains main-sequence O and B stars (only the most massive turnoff point at about spectral type B6 tells us that the Pleiades stars are shown), indicating that it is only about 14 million years are about 100 million years old. old. Stars at the main-sequence turnoff point for the oldest cluster, NGC 188, are only slightly more luminous and massive than the Sun, indicating an age of 7 billion years.

To understand how the process works, consider Figure 16.16, which shows an H–R diagram for the Pleiades. Most of case, we determine the cluster’s age from the lifetimes of the stars in the Pleiades fall along the standard main se- the stars at its main-sequence turnoff point: quence, with one important exception: The Pleiades’ stars trail away to the right of the main sequence at the upper age of the cluster lifetime of stars at main- end. That is, the hot, short-lived O stars are missing from sequence turnoff point the main sequence. Apparently, the Pleiades are old enough Stars in a particular cluster that once resided above the for its O stars to have already ended their hydrogen-burning turnoff point on the main sequence have already exhausted lives. At the same time, they are young enough for some their core supply of hydrogen, while stars below the turn- B stars to still survive on the main sequence. off point remain on the main sequence. The precise point on the H–R diagram at which the Pleiades’ main sequence diverges from the standard main sequence is called the main-sequence turnoff point. In THINK ABOUT IT this cluster, it occurs around spectral type B6. The main- Suppose a star cluster is precisely 10 billion years old. On sequence lifetime of a B6 star is roughly 100 million years, an H–R diagram, where would you expect to find its main- so this must be the age of the Pleiades. Any star in the sequence turnoff point? Would you expect this cluster to have Pleiades that was born with a main-sequence spectral type any main-sequence stars of spectral type A? Would you expect hotter than B6 had a lifetime shorter than 100 million years it to have main-sequence stars of spectral type K? Explain. and hence is no longer found on the main sequence. Stars (Hint: What is the lifetime of our Sun?) with lifetimes longer than 100 million years are still fusing hydrogen and hence remain as main-sequence stars. Over The technique of identifying main-sequence turnoff the next few billion years, the B stars in the Pleiades will points is our most powerful tool for evaluating the ages of die out, followed by the A stars and the F stars. Thus, if we star clusters. We’ve learned, for example, that most open could make an H–R diagram for the Pleiades every few clusters are relatively young and that very few are older than million years, we would find that the main sequence grad- about 5 billion years. In contrast, the stars at the main- ually grows shorter. sequence turnoff points in globular clusters are usually Comparing the H–R diagrams of other open clusters less massive than our Sun (Figure 16.18). Because stars like makes this effect more apparent (Figure 16.17). In each our Sun have a lifetime of about 10 billion years and these

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16 billion years, making them the oldest known objects in 6 10 2 10 3 10 Solar Radii Solar Radii the galaxy. In fact, globular clusters place a constraint on 10 5 10 Solar Radii the possible age of the universe: If stars in globular clusters are 12 billion years old, then the universe must be at least 4 10 this old. 10 3 1 Solar Radius

10 2 THE BIG PICTURE 10 0.1 Solar Radius Sun Putting Chapter 16 into Context 1 We have classified the diverse families of stars visible in 0.1 10 2 Lifetime Solar Radius 1010 yrs the night sky. Much of what we know about stars, gal- luminosity (solar units) 10 2 axies, and the universe itself is based on the fundamental

3 properties of stars introduced in this chapter. Make sure 10 you understand the following “big picture” ideas: 4 10 ● All stars are made primarily of hydrogen and helium, 10 5 at least at the time they form. The differences between stars come about primarily because of differences in 30,000 10,000 6,000 3,000 mass and age. surface temperature (Kelvin) ● Much of what we know about stars comes from study- Figure 16.18 This H–R diagram shows stars from the globular ing the patterns that appear when we plot stellar sur- cluster Palomar 3. The main-sequence turnoff point is in the vicin- ity of stars like our Sun, indicating an age for this cluster of around face temperatures and luminosities in an H–R diagram. 10 billion years. A more technical analysis of this cluster places its Thus, the H–R diagram is one of the most important age at around 12–14 billion years. (Stars in globular clusters tend tools of astronomers. to contain virtually no elements other than hydrogen and helium. Because of their different composition, these stars are somewhat ● Stars spend most of their lives as main-sequence stars, bluer and more luminous than stars of the same mass and compo- fusing hydrogen into helium in their cores. The most sition as our Sun.) massive stars live only a few million years, while the least massive stars will survive until the universe is many times its present age. stars have already died in globular clusters, we conclude that globular-cluster stars are older than 10 billion years. ● Much of what we know about the universe comes from More precise studies of the turnoff points in globular studies of star clusters. Here again, H–R diagrams clusters, coupled with theoretical calculations of stellar play a vital role. For example, H–R diagrams of star lifetimes, place the ages of these clusters at between 12 and clusters allow us to determine their ages.

SUMMARY OF KEY CONCEPTS

16.1 Snapshot of the Heavens 16.2 Stellar Luminosity • How can we learn about the lives of stars, which last • What is luminosity, and how do we determine it? A millions to billions of years? By taking observations star’s luminosity is the total power (energy per unit of many stars, we can study stars in many phases of time) that it radiates into space. It can be calculated their life, just as we might study how humans age from a star’s measured apparent brightness and dis- by observing all the humans living in a particular tance, using the luminosity–distance formula: village at one time. luminosity apparent brightness • What are the two main elements in all stars? All stars 4p (distance)2 are made primarily of hydrogen and helium at birth. • How do we measure the distance to nearby stars? The • What two basic physical properties do astronomers distance to nearby stars can be measured by paral- use to classify stars? Stars are classified by their lumi- lax, the shift in the apparent position of a star with nosity and surface temperature, which depend pri- respect to more distant stars as Earth moves around marily on a star’s mass and its stage of life. the Sun. continued

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• How does the magnitude of a star relate to its appar- diagonally from lower right to upper left. The giants ent brightness? The magnitude scale runs backward, and supergiants inhabit the upper-right region of so a star of magnitude 5 is brighter than a star of mag- the diagram, above the main sequence. The white nitude 18. dwarfs are found near the lower left, below the main sequence. 16.3 Stellar Surface Temperature • How do stars differ along the main sequence? All • How are stars classified into spectral types? From main-sequence stars are fusing hydrogen to helium hottest to coolest, the major spectral types are O, B, in their cores. Stars near the lower right of the main A, F, G, K, and M. These types are futher subdivided sequence are lower in mass and have longer lifetimes into numbered categories. For example, the hottest than stars further up the main sequence. Lower-mass A stars are type A0 and the coolest A stars are type main-sequence stars are much more common than A9, which is slightly hotter than F0. higher-mass stars. • What determines a star’s spectral type? The main • What determines the length of time a star spends on factor in determining a star’s spectral type is its sur- the main sequence? A star’s mass determines how face temperature. Spectral type does not depend much hydrogen fuel it has and how fast it fuses that much on composition, because the compositions of hydrogen into helium. The most massive stars have stars—primarily hydrogen and helium—are nearly the shortest lifetimes because they fuse their hydro- the same. gen at a much faster rate than do lower-mass stars. • What are Cepheid variable stars, and why are they 16.4 Stellar Masses important to astronomers? Cepheid variables are • What is the most important property of a star? A star’s very luminous pulsating variable stars that follow most important property is its mass, which deter- a period–luminosity relation, which means we can mines its luminosity and spectral type at each stage calculate luminosity by measuring pulsation period. of its life. Once we know a Cepheid’s luminosity, we can cal- culate its distance with the luminosity–distance • What are the three major classes of binary star sys- formula. This technique enables us to measure dis- tems? A visual binary is a pair of orbiting stars that tances to many other galaxies in which we have we can see distinctly through a telescope. An eclips- observed these variable stars. ing binary reveals its binary nature because of peri- odic dimming that occurs when one star eclipses the 16.6 Star Clusters other as viewed from Earth. A spectroscopic binary reveals its binary nature when we see the spectral • What are the two major types of star cluster? Open lines of one or both stars shifting back and forth as clusters contain up to several thousand stars and the stars orbit each other. are found in the disk of the galaxy. Globular clusters are much denser, containing hundreds of thousands • How do we measure stellar masses? We can directly of stars, and are found mainly in the halo of the gal- measure stellar mass only in binary systems for which axy. Globular-cluster stars are among the oldest stars we are able to determine the period and separation known, with estimated ages of up to 12–14 billion of the two orbiting stars. We can then calculate the years. Open clusters are generally much younger system’s mass using Newton’s version of Kepler’s than globular clusters. third law. • Why are star clusters useful for studying stellar evolu- 16.5 The Hertzsprung–Russell Diagram tion? The stars in star clusters are all at roughly the same distance and, because they were born at about • What is the Hertzsprung–Russell (H–R) diagram? the same time, are all about the same age. The H–R diagram is the most important classifica- tion tool in stellar astronomy. Stars are located on • How do we measure the age of a star cluster? The age the H–R diagram by their surface temperature (or of a cluster is equal to the main-sequence lifetime spectral type) along the horizontal axis and their of the hottest, most luminous main-sequence stars luminosity along the vertical axis. Surface tempera- remaining in the cluster. On an H–R diagram of ture decreases from left to right on the H–R diagram. the cluster, these stars sit farthest to the upper left and define the main-sequence turnoff point of the • What are the major features of the H–R diagram? cluster. Most stars occupy the main sequence, which extends

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True Statements? 18. Luminosity Classes. What do we mean by a star’s luminosity class? On your sketch of the H–R diagram from problem Decide whether each of the following statements is true or false 17, identify the regions for luminosity classes I, III, and V. and clearly explain how you know. 19. Pulsating Variables. What are pulsating variable stars? Why 1. Two stars that look very different must be made of different do they vary periodically in brightness? kinds of elements. 20. H–R Diagrams of Star Clusters. Explain why H–R diagrams 2. Sirius is the brightest star in the night sky, but if we moved look different for star clusters of different ages. How does it 10 times farther away it would look only one-tenth as the location of the main-sequence turnoff point tell us the bright. age of the star cluster? 3. Sirius looks brighter than Alpha Centauri, but we know 21. Stellar Data. Consider the following data table for several that Alpha Centauri is closer because its apparent position bright stars. Mv is absolute magnitude, and mv is apparent in the sky shifts by a larger amount as Earth orbits the Sun. magnitude. 4. Stars that look red-hot have hotter surfaces than stars that Spectral Luminosity look blue. Star Mv mv Type Class 5. Some of the stars on the main sequence of the H–R dia- gram are not converting hydrogen into helium. Aldebaran 0.2 0.9 K5 III 6. The smallest, hottest stars are plotted in the lower left-hand Alpha Centauri A 4.4 0.0 G2 V portion of the H–R diagram. Antares 4.5 0.9 M1 I 7. Stars that begin their lives with the most mass live longer Canopus 3.1 0.7 F0 II than less massive stars because it takes them a lot longer to use up their hydrogen fuel. Fomalhaut 2.0 1.2 A3 V 8. Star clusters with lots of bright, blue stars are generally Regulus 0.6 1.4 B7 V younger than clusters that don’t have any such stars. Sirius 1.4 1.4 A1 V 9. All giants, supergiants, and white dwarfs were once main- Spica 3.6 0.9 B1 V sequence stars. 10. Most of the stars in the sky are more massive than the Sun. Answer each of the following questions, including a brief explanation with each answer. Problems a. Which star appears brightest in our sky?

11. Similarities and Differences. What basic composition are all b. Which star appears faintest in our sky? stars born with? Why do stars differ from one another? c. Which star has the greatest luminosity? 12. Across the Spectrum. Explain why we sometimes talk about d. Which star has the least luminosity? wavelength-specific (e.g., visible-light or X-ray) luminosity or apparent brightness, rather than total luminosity and e. Which star has the highest surface temperature? total apparent brightness. f. Which star has the lowest surface temperature? 13. Determining Parallax. Briefly explain how we calculate a g. Which star is most similar to the Sun? star’s distance in parsecs by measuring its parallax angle in arcseconds. h. Which star is a red supergiant? 14. Magnitudes. What is the magnitude system? Briefly explain i. Which star has the largest radius? what we mean by the apparent magnitude and absolute j. Which stars have finished burning hydrogen in their magnitude of a star. cores? 15. Deciphering Stellar Spectra. Briefly summarize the roles of k. Among the main-sequence stars listed, which one is the Annie Jump Cannon and Cecilia Payne-Gaposchkin in dis- most massive? covering the spectral sequence and its meaning. l. Among the main-sequence stars listed, which one has 16. Eclipsing Binaries. Describe why eclipsing binaries are so the longest lifetime? important for measuring masses of stars. 22. Data Tables. Study the spectral types listed in Appendix F 17. Basic H–R Diagram. Draw a sketch of a basic Hertzsprung– for the 20 brightest stars and for the stars within 12 light- Russell (H–R) diagram. Label the main sequence, giants, years. Why do you think the two lists are so different? Explain. supergiants, and white dwarfs. Where on this diagram do we find stars that are cool and dim? Cool and luminous? *23. The Inverse Square Law for Light. Earth is about 150 million Hot and dim? Hot and bright? km from the Sun, and the apparent brightness of the Sun

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in our sky is about 1,300 watts/m2.Using these two facts *26. Parallax and Distance. Use the parallax formula to calculate and the inverse square law for light, determine the apparent the distance to each of the following stars. Give your an- brightness we would measure for the Sun if we were located swers in both parsecs and light-years. at the following positions. a. Alpha Centauri: parallax angle of 0.742. a. Half Earth’s distance from the Sun. b. Procyon: parallax angle of 0.286. b. Twice Earth’s distance from the Sun. *27. The Magnitude System. Use the definitions of the magni- c. Five times Earth’s distance from the Sun. tude system to answer each of the following questions. *24. The Luminosity of Alpha Centauri A. Alpha Centauri A lies a. Which is brighter in our sky, a star with apparent mag- at a distance of 4.4 light-years and has an apparent bright- nitude 2 or a star with apparent magnitude 7? By how ness in our night sky of 2.7 108 watt/m2.Recall that much? 1 light-year 9.5 1012 km 9.5 1015 m. b. Which has a greater luminosity, a star with absolute a. Use the luminosity–distance formula to calculate the magnitude 4 or a star with absolute magnitude 6? luminosity of Alpha Centauri A. By how much? b. Suppose you have a light bulb that emits 100 watts of *28. Measuring Stellar Mass. The spectral lines of two stars in a visible light. (Note: This is not the case for a standard particular eclipsing binary system shift back and forth with 100-watt light bulb, in which most of the 100 watts goes a period of 6 months. The lines of both stars shift by equal to heat and only about 10–15 watts is emitted as visible amounts, and the amount of the Doppler shift indicates light.) How far away would you have to put the light that each star has an orbital speed of 80,000 m/s. What are bulb for it to have the same apparent brightness as Alpha the masses of the two stars? Assume that each of the two Centauri A in our sky? (Hint: Use 100 watts as L in the stars traces a circular orbit around their center of mass. luminosity–distance formula, and use the apparent (Hint: See Mathematical Insight 16.4.) brightness given above for Alpha Centauri A. Then solve *29. Calculating Stellar Radii. Sirius A has a luminosity of for the distance.) 26LSun and a surface temperature of about 9,400 K. What *25. More Practice with the Luminosity–Distance Formula. Use is its radius? (Hint: See Mathematical Insight 16.5.) the luminosity–distance formula to answer each of the following questions. Discussion Question a. Suppose a star has the same luminosity as our Sun 26 (3.8 10 watts) but is located at a distance of 10 light- 30. Classification. Edward Pickering’s team of female “comput- years. What is its apparent brightness? ers” at Harvard University made many important contribu- b. Suppose a star has the same apparent brightness as tions to astronomy, particularly in the area of systematic Alpha Centauri A (2.7 108 watt/m2) but is located stellar classification. Why do you think rapid advances in at a distance of 200 light-years. What is its luminosity? our understanding of stars followed so quickly on the heels of their efforts? Can you think of other areas in science 26 c. Suppose a star has a luminosity of 8 10 watts and where huge advances in understanding followed directly 12 2 an apparent brightness of 3.5 10 watt/m .How from improved systems of classification? far away is it? Give your answer in both kilometers and light-years. d. Suppose a star has a luminosity of 5 1029 watts and an apparent brightness of 9 1015 watt/m2.How far away is it? Give your answer in both kilometers and light-years.

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MEDIA EXPLORATIONS For a complete list of media resources available, go to www.astronomyplace.com, and choose Chapter 16 from the pull-down menu.

ypla om ce n . o c r o

t m s 4. Why is there a relationship between luminosity and mass a Astronomy Place Web Tutorials for main-sequence stars on the H–R diagram? Tutorial Review of Key Concepts Measuring Cosmic Distances Tutorial, Lesson 2 1. Explain how we measure distances with stellar parallax. Use the interactive Tutorials at www.astronomyplace.com to Give an example. review key concepts from this chapter. 2. Explain why we cannot use parallax to measure the distance Hertzsprung–Russell Diagram Tutorial to all stars. Lesson 1 The Hertzsprung–Russell (H–R) Diagram Stellar Evolution Tutorial, Lesson 4 Lesson 2 Determining Stellar Radii 1. In the animation at the beginning of Lesson 4, list the order Lesson 3 The Main Sequence in which you saw the three differently colored stars in the cluster disappear, and explain why they disappeared in this order. 2. In the second animation in Lesson 4, in what order did you see stars on the main sequence disappear? Explain the reason for this. 3. How does the age of a dim star cluster of mostly small stars compare to a bright cluster with some giants in it? Explain your answer. Measuring Cosmic Distances Tutorial r: SkyG ge a a z y e

o r Lesson 2 Stellar Parallax V Exploring the Sky and Solar System

Of the many activities available on the Voyager: SkyGazer CD- ROM accompanying your book, use the following files to observe key phenomena covered in this chapter. Go to the File: Basics folder for the following demonstrations: 1. Large Stars 2. More Stars 3. Star Color and Size Stellar Evolution Tutorial Go to the File: Demo folder for the following demonstrations: Lesson 1 Main-Sequence Lifetimes 1. Circling the Hyades Lesson 4 Cluster Dating 2. Flying Around Pleiades 3. The Tail of Scorpius

Web Projects

Take advantage of the useful web links on www.astronomyplace. com to assist you with the following projects. 1. Women in Astronomy. Until fairly recently, men greatly Supplementary Tutorial Exercises outnumbered women in professional astronomy. Neverthe- less, many women made crucial discoveries in astronomy Use the interactive Tutorial Lessons to explore the following throughout history. Do some research about the life and questions. discoveries of a woman astronomer from any time period, Hertzsprung–Russell Diagram Tutorial, Lessons 1–3 and write a two- to three-page scientific biography. 1. If one star appears brighter than another, can you be sure 2. The Hipparcos Mission. The European Space Agency’s Hip- that it is more luminous? Why or why not? parcos mission, which operated from 1989 to 1993, made 2. Answer each part of this question with either high, low, precise parallax measurements for more than 40,000 stars. left, or right. On an H–R diagram, where will a star be if Learn about how Hipparcos allowed astronomers to mea- it is hot? Cool? Bright? Dim? sure smaller parallax angles than they could from the ground and how Hipparcos discoveries have affected our knowl- 3. Why is there a relationship between stellar radii and loca- edge of the universe. Write a one- to two-page report on your tions on the H–R diagram? findings.

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