Measuring the Stars - Understanding Our Universe PDF

10 Measuring the Stars The stars in the night sky have different brightnesses for reasons that you will learn about here in Chapter 10. Study the star chart for the current season, found in Appendix 4. About how many more faint stars (magnitude 4) than bright stars (magnitude 1) are there in this c...

10 Measuring the Stars The stars in the night sky have different brightnesses for reasons that you will learn about here in Chapter 10. Study the star chart for the current season, found in Appendix 4. About how many more faint stars (magnitude 4) than bright stars (magnitude 1) are there in this chart? Notice the scale bar that shows how the size of the dot relates to the magnitude of the star. Next, count the number of stars of each magnitude and make a table of your results. Does this more careful analysis agree with your estimate? Take this star chart outside on a clear night, and find all the stars with magnitude 1 that are above the horizon. This will orient you to the sky. Now find some stars with magnitudes 2, 3, 4, and so on. If you can find no stars fainter than magnitude 3, then the “limiting magnitude” of your observing site is 3. What is the limiting magnitude of your observing site on this date? What sources of light are making it difficult to see stars fainter than this? EXPERIMENT SETUP PREDICTION SKETCH OF RESULTS LEARNING GOALS 1 Use the brightnesses of nearby stars and their distances from Earth to discover how luminous they are. 2 Infer the temperatures of stars from their color, and combine luminosity and temperature to infer the radius. 3 Determine the chemical compositions and masses of stars. 4 Classify stars and organize this information on a Hertzsprung-Russell diagram. 5 Find the luminosity, temperature, and size of a main-sequence star from its position on the Hertzsprung- Russell diagram. Humans, by nature, are curious. How far away is that star? How big is it? How luminous is it? Asking questions is an important aspect of how science works. Unlike our exploration of the Solar System, though, we cannot answer these questions by sending space probes to a star to take detailed pictures. Even to the most powerful telescopes, nearly every star is just a point of light in the night sky. But, by applying the science of light, matter, and motion to observations of those point sources, scientists find patterns and build a remarkably detailed picture of the physical properties of stars. Many of these patterns are evident in the Hertzsprung-Russell diagram (H-R diagram), an important organizing diagram in astronomy that we will begin to explore in this chapter. 10.1 The Luminosity of a Star Can Be Found from Its Brightness and Distance Brightness refers to how bright an individual star appears in our sky, while luminosity refers to how much light the star actually emits. Determining the brightness of a star is a conceptually straightforward task that usually involves comparing it to other nearby stars whose brightnesses are known. The distance to a star must be known before it is possible to determine the luminosity—does the star appear faint because it emits very little light or because it is very far away? Finding the distance is somewhat involved, depending on the method used, which in turn depends on whether the star is relatively near or relatively far. In this section, you will learn how astronomers find the distances to nearby stars and how to combine that information with the star’s brightness to find the luminosity. The Brightness of a Star Two thousand years ago, the Greek astronomer Hipparchus classified the VOCABULARY ALERT brightest stars he could see as being “of the first magnitude” and the faintest as being “of the sixth magnitude.” We know now that the brightest stars he saw are about 100 times brighter than the dimmest stars. Magnitude has come to mean a measure of a star’s brightness in the sky, but note that the magnitude of the object decreases as its brightness increases. It is also an example of logarithmic behavior, so that an object of magnitude 2 is much more than twice as bright as an object of magnitude 4. Each decrease of 2.5 magnitudes corresponds to an increase in brightness by a factor of 10; an object with a magnitude of M = 1 is 100 times brighter than an object with a magnitude of M = 6. Hipparchus himself must have had typical eyesight, because an average person under dark skies can see stars only as faint as 6th magnitude. Modern telescopes can see much “deeper” than this. The Hubble Space Telescope can detect stars as faint as 30th magnitude—4 billion times fainter than what the naked eye can see. Objects that are brighter than 1st magnitude in this system have smaller magnitudes; that is, magnitudes less than 1 or even negative. For example, Sirius, the brightest star in the sky in visible wavelengths, has a magnitude of −1.46. Venus, at magnitude −4.4, is bright enough to cast shadows. The magnitude of the full Moon is −12.7, and that of the Sun is −26.7. Thus the Sun is 14.0 magnitudes (about 400,000 times) brighter than the full Moon. Again, take note: The magnitude scale is backward, so that larger numbers mean fainter stars. A star is often observed through an optical filter that lets through only a small range of wavelengths. The brightness of the star generally varies depending on the filter used, so astronomers use special symbols to represent magnitudes through these filters. Two of the most common filters are a blue filter (B) and a “visual” or yellow-green filter (V). The term visual is used because yellow-green light roughly corresponds to the range of wavelengths to which our eyes are most sensitive. The magnitude of a star, as we have discussed it, is called the star’s apparent magnitude, because it is the brightness of the star as it appears to us in our sky. Stars are found at different distances from us, so a star’s apparent magnitude does not tell us how much light it actually emits. To find out how much light it emits—its luminosity— we must know the distance from us to the star. Then we can put it on a scale with all other stars of known distances, and we can calculate how bright they would be if they were all located at the same distance from us. This is called the absolute magnitude: the brightness that would be measured for each star if every star were the same distance from Earth. That reference distance is 32.6 light-years, or 10 parsecs, as will be explained shortly. Astronomers Use Parallax to Measure Distances to Nearby Stars Astronomy in Action: Parallax Hold up your finger in front of you, quite close to your nose. View it with your right eye only and then with your left eye only. Each eye sends a slightly different image to your brain, and so your finger appears to move back and forth relative to the background behind it. Now hold up your finger at arm’s length, and blink your right eye, then your left eye. Your finger appears to move much less. The way your brain combines the information from your two eyes to determine the distance to an object is called stereoscopic vision. Figure 10.1a shows an overhead view of the experiment you just performed with your finger. The left eye sees the blue pencil nearly directly between the objects on the bookshelf, but the right eye sees the blue pencil to the left of both objects. The apparent position of the pink pencil also varies. Because the pink pencil is closer to the observer, its apparent position changes more than the apparent position of the blue pencil—it moves from the right of the blue pencil to the left of the blue pencil, so it must have shifted farther. (a) (b)  Figure 10.1 (a) Stereoscopic vision allows you to judge the distance to an object by comparing the view from each eye. (b) Similarly, comparing views from different places in Earth’s orbit allows us to determine the distance to stars. As Earth moves around the Sun, the apparent positions of nearby stars change more than the apparent positions of more distant stars. (The diagram is not to scale.) This is the starting point for measuring the distances to nearby stars. Stereoscopic vision allows you to judge the distances of objects as far away as 10 meters, but beyond that it is of little use. Each of your two eyes has an identical view of a mountain several kilometers away—all you can determine is that the mountain is too far away for you to judge its distance stereoscopically. The distance over which our stereoscopic vision works is limited by the separation between our two eyes—about 6 centimeters (cm). If you could separate your eyes by several meters, you could judge the distances to objects that were about half a kilometer away. Although we cannot literally take our eyes out of our heads and hold them apart at arm’s length, we can “take a step to the side” to see that the apparent movement of an object is larger when measured across a longer baseline. If you took a photograph from each location, you could compare the photographs to determine which objects were closer and which were farther away. Similarly, if an astronomer takes a picture of the sky tonight and then waits 6 months to take another picture, the distance between the two locations is the diameter of Earth’s orbit (2 astronomical units [AU]) (Figure 10.1b). This long baseline yields very powerful stereoscopic vision. This illustration mimics the illustration in Figure 10.1a, viewing the observer from “above.” The change in position of Earth over 6 months is like the distance between the right eye and the left eye in Figure 10.1a. The nearby (pink and blue) stars are like the pink and blue pencils, while the distant yellow stars are like the objects on the bookcase. Because of the shift in perspective as Earth orbits the Sun, nearby stars appear to shift their positions. The pink star, which is closer, appears to move farther than the more distant blue star. Over the course of a full year, the nearby star appears to move and then move back again with respect to distant background stars, returning to its original position 1 year later. We can determine the distance to the star using the amount of this shift, the distance from Earth to the Sun, and geometry. what if... What if you measure the distances to all the visible stars in the constellation Sagittarius: Would you expect the distances to be similar or very different for these stars, and what would that imply? Figure 10.2a shows the same configuration as Figure 10.1b. Look first at star A, the closer star. When Earth is at the top of the figure, it forms a right triangle with the Sun and star A at the other corners, as shown in Figure 10.2b. (Remember that a right triangle is one with a 90° angle in it.) The short leg of the triangle is the distance from Earth to the Sun, which is 1 AU. The long leg of the triangle is the distance from the Sun to star A. The small angle at the star A corner of the triangle measures the change in the apparent position of the star. This change in position, measured as an angle, is known as the parallax of the star. The apparent motion of a star across the sky, described earlier for Figure 10.1b, is equal to twice the parallax.  (a) (b) Figure 10.2 (a) The parallax (p) of a star is inversely proportional to its distance. More distant stars have smaller parallax angles. (The diagram is not to scale.) (b) Earth, the Sun, and the target star form a right triangle when Earth is in the best location to measure the parallax angle of the target star. The parallax of a real star is tiny, much smaller than any of the angles shown in Figure 10.2. Parallax angles are so small that we need special units for them. Just as an hour on the clock is divided into minutes and seconds, a degree that measures an angle can be divided into arcminutes and arcseconds. An arcminute (arcmin) is 1/60 of a degree, and an arcsecond (arcsec) is 1/60 of an arcminute, and therefore 1/3,600 of a degree. An arcsecond is an important astronomical unit, and it can be difficult to imagine how small it is. Imagine a person standing directly in front of you, uncomfortably close. To look from one of their eyes to the other, you need to turn your own eyes through a (small) angle. An arcsecond is the size of the angle that your eyes would have to move if that person were 10.8 kilometers (km; equivalent to 6.71 miles) away. An arcsecond is really small. The distances to real stars are large, and in this book we normally use units of light-years to describe them. As we saw in Chapter 4, 1 light-year is the distance that light travels in 1 year—about 9.5 trillion km. We use this unit because it is the unit you are most likely to see in a newspaper article or a popular book about astronomy. When astronomers discuss distances to stars and galaxies, however, they often use the parsec (pc), which is short for parallax arcsecond: 1 parsec is the distance to a star that has a parallax of 1 arcsecond. This unit was invented because it makes the mathematics easy when calculating parallax distances. One parsec is equal to 3.26 light-years. More distant stars make longer and skinnier triangles and have smaller parallax angles. Returning to Figure 10.2, note that star B is twice as far away as star A, so its right triangle is longer and skinnier than the triangle for star A. Because the triangle is longer and skinnier, star B’s parallax is half the parallax of star A. Stars that are farther away have smaller parallax angles than stars that are closer to us. In other words, the parallax of a star is inversely proportional to its distance: When one goes up, the other goes down. When the parallax is expressed in arcseconds, then the distance in parsecs can be found by dividing 1 by the parallax (see Working It Out 10.1). working it out 10.1 Parallax and Distance Recall from earlier chapters that “inversely proportional” means that on one side of the equation a variable is in the numerator, while on the other side a different variable is in the denominator. The relationship between distance (d) and parallax (p) is an inverse proportion: The parsec has been adopted by astronomers because it makes the relationship between distance and parallax easier than using light-years: or Notice that the proportionality sign has turned into an equals sign: You don’t have to remember any constants if the distance is in parsecs and the parallax is in arcseconds. Suppose that the parallax of a star is measured to be 0.5 arcsec. To find the distance to the star, we substitute that into the parallax equation for p: Suppose that the parallax of a star is measured to be 0.01 arcsec. What is its distance in light-years? First, we find its distance in parsecs: Then, we convert to light-years by remembering that a parsec is 3.26 light-years: The star closest to us after the Sun is Proxima Centauri. Located at a distance of 4.22 light-years, Proxima Centauri is a faint member of a system of three stars called Alpha Centauri. What is this star’s parallax? First, we must convert from light-years back to parsecs: Then we find the parallax from the distance: Solve for p to get: Then insert our value for the distance in parsecs: This star has a parallax of only 0.77 arcsec. Because Proxima Centauri is the closest star to us, every other star we observe will have a smaller parallax than this. When astronomers began to apply parallax to stars in the sky, they discovered that stars are very distant objects (see the example in Working It Out 10.1). The first successful parallax measurement was made by F. W. Bessel (1784–1846), who in 1838 reported a parallax of 0.314 arcsec for the star 61 Cygni. This finding implied that 61 Cygni was 3.2 pc away, or 660,000 times as far away as the Sun. With this one measurement, Bessel increased the known size of the universe by a factor of 10,000. Today, only about 60 stars are known within about 5 parsecs of the Sun. In the neighborhood of the Sun, each star (or star system) has about 7 cubic parsecs of space all to itself. Space is mostly empty. Astronomers worked hard to find the parallax of known stars for more VOCABULARY ALERT than 150 years. But most stars were so far away that this motion relative to background stars was too small to measure with ground-based telescopes. Knowledge of our stellar neighborhood took a tremendous step forward during the 1990s, when the Hipparcos satellite measured the positions and parallax angles of 120,000 stars. The accuracy of each Hipparcos parallax measurement is about ± 0.001 arcsec. Because of this observational uncertainty, measurements of the distances to stars are likewise uncertain. For example, a star with a Hipparcos parallax of 0.004 ± 0.001 arcsec really has a parallax between 0.003 arcsec and 0.005 arcsec. This gives a corresponding distance range of 200–330 pc from Earth. Since 2014, the European Space Agency’s Gaia satellite has been using parallax to measure the positions and distances to many more stars (more than 1 billion astronomical objects all together) to construct the largest and most precise three-dimensional map of the Milky Way Galaxy and its neighborhood ever made. Methods other than parallax—to be discussed later—are used for more distant stars. Finding Luminosity Although the brightness of a star is directly measurable, it does not immediately tell us much about the star itself. As illustrated in Figure 10.3, an apparently bright star in the night sky may in fact be intrinsically dim but close by. Conversely, a faint star may actually be very luminous, but because it is very far away, it appears faint to us. We can specifically say that its apparent brightness, which measures the starlight that reaches us, is inversely proportional to the square of our distance from the star. If we know this distance, we can then use our measurement of the star’s apparent brightness to find its luminosity. Two everyday concepts—stereoscopic vision and the fact that objects appear brighter when closer—have given us the tools we need to measure the distances and therefore the luminosities of the closest stars. Once the vast distances are known, it is clear that stars are not merely faint points of light in the night sky; they are extraordinarily powerful beacons located at great distances. Figure 10.3 The brightness of a visible star in our sky depends on both its luminosity—how much light it emits—and its distance from us. When brightness and distance are measured, luminosity can be calculated.  Figure 10.4 The distribution of the luminosities of stars is plotted here logarithmically, so that increments are in powers of 10. For example, for every million stars with a luminosity equal to the Sun, there are only about 100 with a luminosity 10,000 times greater. The range of possible luminosities for stars is very large. The Sun provides a convenient basis for measuring the properties of stars, including their luminosities. (We compare stars to the Sun so often that properties of the Sun have a special subscript: for example, LSun is the luminosity of the Sun.) The most luminous stars are more than a million times the luminosity of the Sun. The least luminous stars have luminosities less than 1/10,000 LSun. The most luminous stars are therefore more than 10 billion times more luminous than the least luminous stars. Very few stars are near the upper end of this range of luminosities, and the vast majority of stars are far less luminous than our Sun. Figure 10.4 shows the relative number of stars compared to their luminosity in solar units. CHECK YOUR UNDERSTANDING 10.1 It is necessary to know ______ in order to find the luminosity of a star. 1. both the distance and the brightness 2. either the distance or the brightness Answers to Check Your Understanding questions are in the back of the book. Glossary brightness The apparent intensity of light from a luminous object, which depends on both the luminosity of a source and its distance. Units at the detector: watts per square meter (W/m2). filter An instrument component that transmits a limited wavelength range of electromagnetic radiation. For the optical range, such components are typically made of different kinds of glass and take on the hue of the light they transmit. apparent magnitude A measure of the brightness of a celestial object, generally a star, as it appears to us on our sky. Compare absolute magnitude. absolute magnitude A measure of the intrinsic brightness of a celestial object, generally a star. Specifically, the apparent brightness of an object, such as a star, if it were located at a standard distance of 10 parsecs (pc). Compare apparent magnitude. stereoscopic vision The way the brains of humans and some animals combine the different information from two eyes to perceive the distances to surrounding objects. parallax 1. The apparent shift in the position of one object relative to another object, caused by the changing perspective of the observer. 2. In astronomy, the displacement in the apparent position of a nearby star caused by the changing location of Earth in its orbit. parsec (pc) The distance to a star with a parallax of 1 arcsecond, using a base of 1 astronomical unit (AU). One parsec is approximately 3.26 light-years. deep In astronomy, far from Earth or distant. uncertainty A description of the accuracy of a measurement, sometimes expressed as a percentage, more often as an interval. The uncertainty gives the range over which one might expect to obtain measurements if the experiment were repeated multiple times. VOCABULARY ALERT deep In everyday language, deep has many meanings, such as referring to depth in the ocean or to a profound idea. Astronomers use deep to refer to an object’s distance—how deep it is in space. Because distant objects are typically fainter, deep and faint are often related and used somewhat interchangeably. VOCABULARY ALERT uncertainty An uncertain distance does not mean that the distance is unknown. Uncertainty is a way of expressing how precisely the distance (or any other measurement) is known. For scientists, the uncertainty is sometimes the most important number, and they often get very excited when a new experiment reduces the uncertainty, even when it does not change the value. When astronomers discovered that the age of the universe was 13.8 billion years, with an uncertainty of 0.1 billion years, many astronomers were more excited about the 0.1 than the 13.8. This is because the measurement was so much more precise than any that came before it. To better understand uncertainty, consider your speed while driving down the road. If your digital speedometer says 10 kilometers per hour (km/h), you might actually be traveling 10.4 km/h or 9.6 km/h. The precision of your speedometer is limited to the nearest 1 km/h, but that doesn’t mean you don’t have any idea about your speed. You are certainly not traveling 100 km/h, for example. arcminute (arcmin) A minute of arc (′), a unit used for measuring angles. An arcminute is 1⁄60 of a degree of arc. arcsecond (arcsec)  A second of arc (″), a unit used for measuring very small angles. An arcsecond is 1⁄60 of an arcminute or 1/3,600 of a degree of arc. magnitude A system used by astronomers to describe the brightness or luminosity of stars. The brighter the star, the smaller its magnitude. 10.2 Radiation Tells Us the Temperature, Size, and Composition of Stars Studying the details of the light from stars yields an enormous amount of information. In this section, we focus on how astronomers determine temperature, radius, and composition of stars from the colors in their spectra. Wien’s Law Revisited: The Color and Surface Temperature of Stars Look back at Working It Out 5.1 to refresh your understanding of the Stefan-Boltzmann law (which states that, among same-sized objects, the hotter objects are more luminous) and Wien’s law (which states that hotter objects are bluer). In this section, we will use these two laws to measure the temperatures and sizes of stars. We will also develop a more detailed understanding of emission lines mentioned in Chapters 4 and 5, and we will use that to obtain information about the composition of stars. Wien’s law shows that hotter objects emit bluer light. Stars with VOCABULARY ALERT especially hot surfaces are blue; stars with especially cool surfaces are red; and stars like our Sun, which emit almost equal amounts of red, yellow, and blue, are white. (The Sun appears yellow in Earth’s sky because the red and blue light are bent and scattered by Earth’s atmosphere.) If you measure the wavelength at which a star’s spectrum is brightest (or “peaks”), then Wien’s law will tell you the temperature of the star’s surface. The color of a star tells us only about the temperature at the surface, because this layer is giving off most of the radiation that we see. Stellar interiors are far hotter than stellar surfaces. AstroTour: Stellar Spectrum In practice, it is usually unnecessary to obtain a complete spectrum of a star to determine its temperature. Instead, astronomers often measure the colors of stars by comparing the brightness at two different, specific wavelengths. From a pair of pictures of a group of stars, each taken through a different filter (B and V filters, for example, as described in the previous section), astronomers can find an approximate value of the surface temperature of every star in the picture—perhaps hundreds or even thousands—all at once. When we do, we find there are many more cool stars than hot stars. We also discover that most stars are cooler than the Sun. The Stefan-Boltzmann Law and Finding the Sizes of Stars Stars are so far away that the vast majority of them cannot be imaged as more than point sources. The size of a star must be inferred from other measurements, such as the temperature and the luminosity. The temperature of a star can be found directly, either from Wien’s law, as illustrated in Figure 10.5a, or from another method. The luminosity of a star can be found from its brightness and its distance, as we discussed in the previous section. (a) (b)  (c) Figure 10.5 (a) The temperature of a star can be found from its color by using Wien’s law. (b) The luminosity depends on both the temperature and the size of the star. (c) Once the temperature and the luminosity are known, the size of the star can be calculated from the Stefan-Boltzmann law. The temperature of a star is one factor that influences its luminosity. The size of the star is also important. If a large star and a small star are the same temperature, they will emit the same energy from every equal-sized patch of surface, but the large star has more patches, so it is more luminous altogether. Conversely, if two stars are the same size, the hotter one will be more luminous than the cooler one. This is an application of the Stefan-Boltzmann law, shown in Figure 10.5b. Because the luminosity, the temperature, and the radius of the star are all related, combining the luminosity with the temperature allows us to determine the radius of the star, as shown in Figure 10.5c. Recall that we carried out a calculation relating these three quantities in Working It Out 5.1. In that case, Earth was the example, but the laws of physics remain the same for planets and for stars. The relationship between luminosity, temperature, and radius has been used to estimate the radii of thousands of stars. The radius of the Sun, written as RSun, is about 700,000 km. The smallest stars, called white dwarfs, have radii that are only about 1 percent of the radius of the Sun (R = 0.01 RSun). The largest stars, called red supergiants, can have radii more than 1,000 times that of the Sun. There are many more small stars than large stars. Most stars are smaller than the Sun. Atomic Energy Levels So far, we have concentrated on what we can learn about stars by applying our understanding of blackbody radiation. However, the spectra of stars are not smooth, continuous blackbody spectra. Instead, when we pass the light of stars through a prism, we see dark and bright lines at specific wavelengths in their spectra. These lines have been used to determine much of what we know about the universe. To understand these lines, we must first know how light interacts with matter. You learned a little about light and matter in Chapter 4, but here in Chapter 10, we will explore more about how they interact. Electrons in an atom can have only particular energies. These energies are known as “energy states” of the atom. We can imagine these energy states as a set of shelves in a bookcase, as shown in Figure 10.6a. The energy of an atom might correspond to the energy of one shelf or to the energy of the next shelf, but the energy of the atom will never be found between the two states, just as a book will never be found floating between two shelves. Astronomers keep track of the allowed states of an atom using energy level diagrams, as shown in Figure 10.6b, where each energy level is like a shelf on the bookcase. Both of these metaphors (the bookcase and the energy level diagram) are simplifications of a three-dimensional system: Atoms are three-dimensional objects formed of a very small, dense nucleus surrounded by a cloud of electrons. Changing the energy state of an atom changes the shape and size of the electron cloud that surrounds the nucleus, loosely sketched in Figure 10.6c. (a)  (b) (c) Figure 10.6 (a) Energy states of an atom are analogous to shelves in a bookcase. You can move a book from one shelf to another, but books can never be placed between shelves. (b) Energy level diagrams are used to represent the different amounts of energy that an atom can have. (c) Atoms exist in one allowed energy state or another but never in between. There is no level below the ground state.  The lowest possible energy state for a system (or part of a system) such as an atom is called the ground state. When the atom is in the ground state, the electron has its minimum energy. It can’t give up any more energy to move to a lower state, because there isn’t a lower state. An atom will remain in its ground state forever unless it absorbs energy from outside the system. Energy levels above the ground state are called excited states. An atom in an excited state can transition to the ground state by getting rid of the “extra” energy all at once. It does this when the electron emits a photon. The atom goes from one energy state to another, but it never has an amount of energy in between. Money is similarly quantized, as shown in Figure 10.7a. If you have a penny, a nickel, and a dime, you have 16 cents. If you give away the nickel, you are left with 11 cents. You never had an intermediate amount of money, such as 12 cents or 13.6 cents. You had 16 cents, and then you had 11 cents. Atoms do not accept and give away money to change energy states, but they do accept and give away photons. Atoms falling from a higher-energy state with energy E2 to a lower-energy state, E1, lose an amount of energy exactly equal to the difference in energy levels, E2 – E1. Therefore, the energy of the photon emitted must be Ephoton = E2 – E1. This change is illustrated in Figure 10.7b, where the green downward arrow indicates that the atom went from the higher state to the lower state. In order for the electron to move back into energy state E2, the atom would need to absorb exactly the same amount of energy, Ephoton = E2 – E1. Electron transitions between these states lead to two different types of spectra: emission spectra, in which atoms are falling to lower-energy states, and absorption spectra, in which atoms are jumping to higher-energy states. (a)  (b) Figure 10.7 (a) The quantized energy associated with transitions between energy states is analogous to individual coins in a handful. (b) Similarly, an atom can give up photons with only specific energies. A photon with energy E2− E1 is emitted when an atom in the higher-energy state decays to the lower-energy state. AstroTour: Atomic Energy Levels and the Bohr Model Astronomy in Action: Emission and Absorption Recall from Chapter 4 that the energy E, wavelength λ, and frequency f of photons are all related (E = hf and λ = c/f, where h and c are constants). When an excited atom goes from energy state E2 to state E1, the photon emitted by the atom has energy Ephoton = E2 – E1, a specific wavelength λ2→1, and a specific frequency f2→1. Therefore, these emitted photons have a specific color, and every photon emitted in any transition from E2 to E1 will have this same color. Because of this connection between energy and wavelength, the energy level structure of an atom determines the wavelengths of the photons it emits, and hence the color of the light that the atom gives off. An atom can emit photons with energies corresponding only to the difference between two of its allowed energy states. AstroTour: Light as a Wave, Light as a Photon An atom sitting in its ground state will remain there forever unless it absorbs just the right amount of energy to kick it up to an excited state. In general, an atom absorbs energy from a photon, or it collides with another atom (or perhaps a free electron) and absorbs some of the other atom’s energy. Atoms moving from a lower-energy state E1 to a higher-energy state E2 can absorb only the amount of energy E2 – E1, whether it comes in the form of photons or collisions. Imagine a cloud of hot gas consisting of atoms with only two energy states, E2 and E1, as shown in Figure 10.8. Because the gas is hot, the atoms are continually zooming around and colliding, thus getting kicked up from the ground state E1 into the higher-energy state E2. Any atom in the higher-energy state quickly drops to the lower energy state and emits a photon in a random direction. This emitted light contains only photons with the specific energy Ephoton = E2 – E1. In other words, all of the light coming from the cloud is the same color. If passed through a slit and a prism, it forms a single bright line of one color, called an emission line. This is how some neon signs work: Each color in a “neon” sign comes from a different gas (not necessarily neon) trapped inside the glass tubes. A spectrum like the one shown to the right in Figure 10.8 is an emission spectrum, identifiable because the continuum spectrum is missing; it is dominated by a narrow emission line of a particular color. Real objects have atoms with more than two energy states; therefore, emission spectra of real objects have more than one emission line. Figure 10.8 A hot cloud of gas containing atoms with two energy states, E1 and E2 (left), emits photons with energy E2 − E1. When these photons pass through an astronomical instrument (middle), they appear in the spectrum (right) as a single emission line. When an object shines because it is hot, it emits all the colors of light, because the particles are traveling at a wide range of speeds, colliding from all directions and losing energy in random amounts. This produces a blackbody spectrum. In Figure 10.9a, we imagine viewing a hot filament through a spectrometer. Because all the colors are present, the spectrometer creates a complete rainbow (a complete spectrum) on a detector. Notice that the green light shows up in the same place on the detector here as it did in Figure 10.8. All light of the same color has the same wavelength, so a system like this will always place the green light at the same location on the detector.  (a) (b) Figure 10.9 (a) When passed through a prism, white light produces a spectrum containing all colors. (b) If light of all colors passed through a cloud of atoms with only two possible states, photons with energy E2 − E1 would be absorbed, leading to the dark absorption line in the spectrum. When the white light passes through a cool cloud of gas, however, some photons will be absorbed. Imagine that, as in the hot cloud of gas shown in Figure 10.8, the atoms in this cool cloud have only two energy states. Almost all of the photons will pass through the cloud of gas unaffected because they do not have the right amount of energy (E2 – E1) to be absorbed by atoms of the gas, but photons with just the right amount of energy will be absorbed. As a result, they will be missing from the spectrum. We will see a sharp, dark line at the wavelength corresponding to this energy, as shown in Figure 10.9b. This process is called absorption, and the dark line is called an absorption line. Figure 10.10 shows such absorption lines in the spectrum of a star. The spectrum is shown in two different ways here: as a rainbow with light missing, and then again as a graph of the brightness at every wavelength. Comparing the top and bottom versions of the spectrum, you can see that the brightness drops abruptly at a particular wavelength where there are dark lines. Places between the lines are brighter and therefore higher on the graph than the absorption lines. (a)  (b) Figure 10.10 Absorption lines in the spectrum of a star may be viewed two ways: (a) The camera attached to the telescope captures an image of the “rainbow” with dark lines where absorption has occurred. (b) Astronomers measure the brightness at each wavelength and make a graph that shows the shape of the absorption line in more detail. AstroTour: Atomic Energy Levels and Light Emission and Absorption For any element, the absorption lines occur at exactly the same wavelength as the emission lines. The energy difference between the two levels is the same whether the electron in the atom is emitting a photon or absorbing one, so the energy of the photon involved will be the same in either case. The spectrum shown in Figure 10.10 is typical of an absorption spectrum: The blackbody spectrum of the object is bright, with dark lines superimposed on it where light has been absorbed by atoms. When an atom absorbs a photon, it may quickly return to its previous energy state, emitting a photon with the same energy as the photon it just absorbed. If the atom emits a photon just like the one it absorbed, you might reasonably ask why the absorption matters at all, because the photon taken out of the spectrum was replaced by an identical one. The photon was replaced, it’s true, but all of the absorbed photons were originally traveling in the same direction, whereas the emitted photons are now traveling in random directions. In other words, most of the photons have been diverted from their original paths. If you look at a white light through the cloud, you will observe an absorption line at a wavelength of λ1→2, as shown in Figure 10.9b. But if you look at the cloud from another direction, you will observe an emission line at the same wavelength, λ2→1, as shown in Figure 10.8. Emission and Absorption Lines Are the Spectral Fingerprints of Atoms Real atoms can occupy many more than just two possible energy states, so an atom of a given element is capable of emitting and absorbing photons at many different wavelengths. In an atom with three energy states, for example, the electron might jump from state 3 to state 2, or from state 3 to state 1, or from state 2 to state 1. Its spectrum will have three distinct emission lines. Each element’s atoms will produce a unique set of lines. Every atom of hydrogen, for example, has the same energy states available to it, so all hydrogen atoms have the same emission and absorption lines. Figure 10.11a shows the energy level diagram of hydrogen. Figure 10.11b shows the spectrum of emission lines for hydrogen in the visible part of the spectrum. Figure 10.11c displays this same information as a graph. Each different element has a unique set of available energy states and therefore a unique set of wavelengths at which it can emit or absorb radiation. These unique sets of wavelengths serve as unmistakable spectral “fingerprints” for each element. Figure 10.11d shows the sets of emission lines from several different kinds of atoms. Figure 10.11 (a) Electrons make transitions between the energy states of the hydrogen atom. Transitions from higher levels to level E2 emit photons in the visible part of the spectrum. (b) The light from a hydrogen lamp produces an emission spectrum. This is the image a camera would produce if you took a picture of the spectrum. (c) The brightness at every wavelength can be measured to produce a graph of the brightness of spectral lines versus their wavelength. (d) Emission spectra from several other types of gases. If the spectral lines of hydrogen, helium, carbon, oxygen, or any other VOCABULARY ALERT element are present in the light from a distant object, then that element is present in that object. The strengths of various absorption lines tell us not only what kinds of atoms are present in the gas but also the abundance of each. However, we must take great care in interpreting spectra to account properly for the temperature and density of the gas in the atmosphere of a star. This type of analysis is how Cecilia Payne- Gaposchkin, in 1925, figured out that stars are made mostly of hydrogen and helium. Typically, more than 90 percent of the atoms in the atmosphere of a star are identified as hydrogen, while helium accounts for most of what remains. All other elements are present only in very small amounts. The spectral lines also tell us about other physical properties of stars, such as pressure and magnetic-field strength. In addition, by making use of the Doppler shift (see Chapter 5), we can measure rotation rates, motions within the atmosphere, expansion and contraction of the star, “winds” driven away from stars, and other dynamic properties of stars. Classification of Stars what if... What if you observe two stars that have very similar peak wavelengths for their spectra but very different strengths of spectral lines: What can you conclude from your observations? Although the hot “surface” of a star emits radiation with a spectrum very close to a smooth blackbody curve, this light must then escape through the outer layers of the star’s atmosphere. The atoms and molecules in the cooler layers of the star’s atmosphere leave their absorption line fingerprints in this light, as shown in Figure 10.12. These atoms and molecules, along with any heated gas that might be found in the vicinity of the star, can also produce emission lines in stellar spectra. Although absorption and emission lines complicate how we use the laws of blackbody radiation to interpret light from stars, spectral lines more than make up for this trouble by providing a wealth of information about the state of the gas in a star’s atmosphere.  Figure 10.12 Absorption and emission lines both appear in the spectra of stars. The blackbody spectrum is the light emitted from a hot object, just because it is hot (recall Chapter 5). As that light passes through a gas, some of it is absorbed, producing an absorption spectrum. Hot gas also emits light and produces emission lines in the spectra of some stars. The spectra of stars were first classified during the late 1800s, long before stars, atoms, or radiation were well understood. Stars with the strongest hydrogen lines were labeled “A stars,” stars with somewhat weaker hydrogen lines were labeled “B stars,” and so on. The classification we use today is based on the prominence of particular absorption lines seen in the spectra.  Figure 10.13 Spectra of stars with different spectral types are shown, ranging from hot blue O stars to cool red M stars. Hotter stars are more luminous at shorter wavelengths. The dark lines are absorption lines. Our Sun is a G2 star. Annie Jump Cannon (1863–1941) led an effort at the Harvard College Observatory to examine and classify systematically the spectra of hundreds of thousands of stars. In the end, the team grouped the stars into seven types that were subsequently reordered on the basis of surface temperatures. The hottest stars, with surface temperatures above 30,000 kelvins (K), are classified as O stars. O stars have weak absorption lines, even from hydrogen and helium. The coolest stars—M stars—have temperatures as low as 2800 K. M stars show myriad lines from many different types of atoms and molecules. The complete sequence of spectral types of stars, from hottest to coolest, is O, B, A, F, G, K, M. Spectra of stars of different types are shown in Figure 10.13. The boundaries between spectral types are imprecise. A hotter-than-average G star is very similar to a cooler-than-average F star. Brown dwarfs, described as “failed stars” in Chapter 5, are even cooler than M stars and have classifications L and T. Astronomers divide the main spectral types into subclasses by adding numbers to the letter designations, so that each star is classified by a letter followed by a number. For example, the hottest B stars are called B0 stars, slightly cooler B stars are called B1 stars, and so on. The coolest B stars are B9 stars, which are only slightly hotter than A0 stars. The Sun is a G2 star. Figure 10.13 shows that the brightest part of the spectrum shifts from the left (blue) end to the right (red) end, as the temperature decreases. For example, compare a B5 star with a surface temperature of 15,400 K with a K5 star at 4350 K. This is a visual statement of Wien’s Law: that hotter objects are bluer. The absorption lines in stellar spectra change with temperature as well. The temperature of the gas in the atmosphere of a star affects the state of the atoms in that gas, which in turn affects the energy level transitions available to absorb radiation. In O stars, the temperature is so high that most atoms have had one or more electrons stripped from them by energetic collisions within the gas. Few transitions are available in the visible part of the electromagnetic spectrum, so the visible spectrum of an O star is relatively featureless. At lower temperatures, there are more atoms that can absorb light in the visible part of the spectrum, so the visible spectra of cooler stars are more complex than the spectra of O stars, as shown in Figure 10.13. Most absorption lines have a temperature at which they are strongest. For example, absorption lines from hydrogen are most prominent at temperatures of about 10,000 K, which is the surface temperature of an A star. (Recall that spectral-type A stars were classified as “A” because they are the stars with the strongest lines of hydrogen in their spectra.) At the very lowest stellar temperatures, atoms in the atmosphere of the star form molecules. Molecules such as titanium oxide (TiO) are responsible for much of the absorption in the atmospheres of cool M stars. Because different spectral lines are formed at different temperatures, these absorption lines can be used to measure a star’s temperature directly. The temperatures of stars measured in this way agree extremely well with the temperatures of stars measured by use of Wien’s law, again confirming that the physical laws that apply on Earth apply to stars also. CHECK YOUR UNDERSTANDING 10.2 If a star has very weak hydrogen lines and is blue, what does that most likely mean? 1. The star is too hot for hydrogen lines to form. 2. The star has very little hydrogen. 3. The star is too cold for hydrogen lines to form. 4. The star is moving too fast to measure the lines. Answers to Check Your Understanding questions are in the back of the book. Glossary ground state The lowest possible energy state for a system or part of a system, such as an atom, molecule, or particle. Compare excited state. spectral type A classification system for stars that is based on the presence and relative strength of absorption lines in their spectra. Spectral type is related to the surface temperature of a star. emission spectrum The spectrum of the light emitted from an object; it may contain emission lines. Compare absorption spectrum. absorption spectrum A spectrum showing absorption lines. Compare emission spectrum. surface The outermost layer of something. In astronomy, this is often defined to be the visible surface. strength 1. Magnitude of a force. 2. In spectroscopy, brightness of an emission line or depth of an absorption line. VOCABULARY ALERT surface In everyday language, we don’t use surface to refer to a layer within a gaseous body. But here, astronomers mean the part of the star that gives off most of the radiation that we see. A star’s surface is not solid like the surface of a terrestrial planet, and stars usually have more layers outside of this “surface.” The Sun’s corona, for example, lies above the photosphere, which is the surface that we see. VOCABULARY ALERT strength In this context, strength means how bright the emission line is or how faint the absorption line is. More atoms interact with more photons, so they either add more light (in the case of emission) or remove more light (in the case of absorption), making a stronger line. A strong emission line may rise high above the rest of the spectrum, add light to a wide region of the spectrum, or both. excited state An energy level of a particular atom, molecule, or particle that is higher than its ground state. Compare ground state. 10.3 The Mass of a Star Can Be Determined in Some Binary Systems The amount of light from a star and the star’s size are not good measures of its mass. A star’s radius changes throughout its lifetime, as does its temperature, while its mass remains nearly constant. The mass, therefore, does not determine the radius or the temperature of the star. However, the mass does determine how the star interacts with objects around it. If we replaced the Sun with a more massive star, that star would exert a larger gravitational force on Earth, no matter how large its radius or how low its temperature. When astronomers are trying to determine the masses of astronomical objects, they almost always wind up looking for the effects of gravity. In Chapter 3, you learned that Kepler’s laws of planetary motion are the result of gravity, and you learned that the orbit of a planet can be used to measure the mass of the Sun. Similarly, Newton’s version of Kepler’s laws can be used to find the mass of a star when two stars orbit each other. About half of the higher-mass stars in the sky are actually systems consisting of several stars orbiting under the influence of their mutual gravity. Most of these are binary stars in which two stars orbit each other. Most stars are single, however, so their masses cannot be found this way. Binary Stars Orbit a Common Center of Mass The center of mass is the balance point of a system. If two objects were sitting on a seesaw in a gravitational field, the support of the seesaw would have to be directly under the center of mass for the objects to balance, as shown in Figure 10.14. In a binary system, the two stars orbit the center of mass, a point in space that is seldom located inside either star but usually somewhere in between. Figure 10.14 The center of mass of two objects is the “balance” point on a line joining the centers of the two masses. When Newton applied his laws of motion to the problem of orbits, he found that two objects must move in elliptical orbits around each other, and that their common center of mass lies at one focus shared by both of the ellipses, as shown in Figure 10.15. The center of mass, which lies along the line between the two objects, remains stationary. The two objects will always be found on exactly opposite sides of the center of mass.  Figure 10.15 In a binary star system, the two stars orbit on elliptical paths about their common center of mass. In this case, star 2 has twice the mass of star 1. The eccentricity of the orbits is 0.5. There are equal time steps between the frames. Because the orbit of the less massive star is larger than the orbit of the more massive star, the less massive star has farther to go than the more massive star. But it must cover that distance in the same amount of time, so the less massive star must be moving faster than the more massive star. The velocity of a star in a binary system is inversely proportional to its mass. Imagine that you are watching a binary star as shown in Figure 10.16a. As seen from above, two stars orbit the common center of mass. The less massive star (star 1) must complete its orbit in the same time that the more massive star does. Because the less massive star has farther to go around the center of mass, it must move more quickly. In this view from above, no determination of the Doppler shift (recall Chapter 5) can be made because all the motion is in the plane of the sky, and none is toward or away from the observer. Figure 10.16 (a) The view from “above” the binary system shows that both stars orbit a common center of mass. (b) The spectrum of the combined system (seen edge-on) shows that the spectral lines of each star shift back and forth. (c) Graphing the Doppler shifts of stars 1 and 2 versus time reveals that star 2 has half the maximum Doppler shift, so star 2 is twice as massive as star 1. P is the period of the orbit. When the system is edge-on to the observer, however, the observer can take advantage of the Doppler shift to find out about the motion. Observations of the spectrum of the combined system (Figure 10.16b) show that the spectral lines of the stars shift back and forth as the stars move toward and away from the observer. When star 2 approaches, star 1 recedes. The light coming from star 2 will be shifted to shorter wavelengths by the Doppler effect as it approaches, so the light will be blueshifted, and the light coming from star 1 will be shifted to longer wavelengths as it recedes, so the light will be redshifted. Halfway around the orbit, the situation will be reversed: lines from star 2 will be redshifted, and lines from star 1 will be blueshifted. The less massive star has a larger orbit—and consequently moves more quickly—than the more massive star. Comparing the maximum Doppler shift for star 2 with the maximum Doppler shift for star 1 (Figure 10.16c) gives the ratio of the masses of the two stars. That is, we can find in Figure 10.16 that star 2 is 2 times as massive as star 1, but we can’t determine the actual mass of either star from these observations alone. Kepler’s Third Law and Total Mass of a Binary System In Chapter 3, we ignored the complexity of the motion of two objects around their common center of mass; we assumed that one object was so much more massive than the other that it remained nearly stationary. Now, however, this very complexity enables us to measure the masses of the two stars in a binary system. Recall that the “period” is the time it takes for one complete orbit. If we can measure the period of the binary system and the average separation between the two stars, then Kepler’s third law gives us the total mass in the system: the sum of the two masses. The analysis in the previous subsection gives the ratio of the two masses, so we now have two different relationships between two different unknowns. This is all we need to determine the mass of each star separately. In other words, if we know that star 2 is 2 times as massive as star 1, and we know that star 1 and star 2 together are 3 times as massive as the Sun, then we can calculate separate values for the masses of star 1 and star 2. Figure 10.17 ★ WHAT AN ASTRONOMER SEES The two stars of this visual binary are resolved. These stars are two components of Alpha Centauri, the nearest star system to the Sun. When looking at images of stars, astronomers know that the brighter stars appear larger on the image. So in this image, α Cen A is significantly brighter than α Cen B. Knowing that these two stars are part of a system of stars, an astronomer will further conclude that they are the same distance away. She will then know that α Cen A is not only brighter but also more luminous than α Cen B. Depending on the type of system, there are two ways to measure the average distance between the stars and the period. In a visual binary system, shown in Figure 10.17, the system is close enough to Earth, and the stars are far enough from each other, that we can take pictures that show the two stars separately. We can directly measure the shapes and period of the orbits of the two stars, just by watching them as they orbit each other. In many binary systems, however, the two stars are so close together and so far away from us that we cannot actually see the stars separately. We know these stars belong to binary systems only because we see the Doppler shift in the spectral lines of the two stars; these are called spectroscopic binary stars. If a binary system is viewed nearly edge-on, so that one star passes in front of the other, it is called an eclipsing binary. An observer will see a dip in brightness as one star passes in front of (or eclipses) the other (Figure 10.18). During the orbit, each star has a moment when it is moving directly toward or directly away from Earth. The total speed of each star can be found from the Doppler shift, and these speeds can be used to find the mass, as described.  Figure 10.18 In an eclipsing binary system, the system is viewed nearly edge-on, so that the stars repeatedly pass behind one another, blocking some of the light. Even though the blue star here is smaller, it is significantly more luminous because its temperature is higher. When the blue star passes in front of the larger, cooler star, less light is blocked than when it passes behind the red star. The shape of the dips in the light curve of an eclipsing binary can reveal information about the relative size and surface brightness of the two stars. what if... What if three stars were orbiting about their common center of mass, all in the same plane: Could you still use measured velocities to determine the orbital radii, and would Kepler’s third law give you the mass of the system? The range of stellar masses found in this way is not nearly as great as the range of stellar luminosities. The least massive stars have masses of about 0.08 MSun; the most massive stars appear to have masses greater than 200 MSun. Why does the mass of a star have any limits? These limits are determined solely by the physical processes that go on deep in the interior of the star. You will learn in the chapters ahead that a minimum stellar mass is necessary to ignite the nuclear furnace that keeps a star shining, but the furnace can run out of control if the stellar mass is too great. Thus, although the most luminous stars are 1010 (10 billion) times more luminous than the least luminous ones, the most massive stars are only about 103 (a thousand) times more massive than the least massive stars. CHECK YOUR UNDERSTANDING 10.3 To find the masses of both stars in a binary system, you must find the ______ of each star, the _______ of the orbit, and the average distance between the stars. 1. temperature; period 2. speed; magnitude 3. speed; period 4. temperature; size Answers to Check Your Understanding questions are in the back of the book. Glossary binary star A system in which two stars are in gravitationally bound orbits around their common center of mass. center of mass 1. The location associated with an object system at which we may regard the entire mass of the system as being concentrated. 2. The point in any isolated system that moves according to Newton’s first law of motion. visual binary A pair of stars in which both stars can be seen individually from Earth. Compare eclipsing binary and spectroscopic binary. spectroscopic binary A pair of stars whose existence and properties are revealed only by the Doppler shifts of their spectral lines. Most spectroscopic binaries are close pairs. Compare eclipsing binary and visual binary. eclipsing binary A pair of stars in which the orbital plane is oriented such that each star appears to pass in front of the other as seen from Earth. Compare spectroscopic binary and visual binary. 10.4 The Hertzsprung-Russell Diagram Is the Key to Understanding Stars We have come a long way in our effort to measure the physical properties of stars. However, just knowing some of the basic properties of stars does not mean that we understand stars. The next step involves looking for patterns in the properties we have determined. The first astronomers to take this step were Ejnar Hertzsprung (1873–1967) and Henry Norris Russell (1877–1957). In the early part of the 20th century, Hertzsprung and Russell studied the properties of stars independently; each plotted the luminosities of stars versus their surface temperatures. The resulting plot is referred to as the Hertzsprung-Russell diagram, or simply the H-R diagram. Astronomers use the H- R diagram to track the evolution of stars, from birth to death. In this section, we take a first look at this important diagram and the way stars are organized within it. AstroTour: H-R Diagram Reading the Hertzsprung-Russell Diagram what if... What if Hertzsprung and Russell had made their diagram a plot of stellar radius versus temperature: How would the main sequence appear in such a diagram? We begin with the layout of the H-R diagram, shown in Figure 10.19. The surface temperature is plotted on the horizontal axis (the x-axis), but it is plotted backward: Temperature is high on the left and low on the right. As a result, hot blue stars are on the left side of the H-R diagram, whereas cool red stars are on the right. Temperature is plotted logarithmically, which means that the size of an interval along the axis from a point representing a star with a surface temperature of 40,000 K to one with a surface temperature of 20,000 K—a temperature change by a factor of 2—is the same as the size of an interval between points representing a star with a temperature of 10,000 K and a star with a temperature of 5000 K, which is also a temperature change by a factor of 2. The temperature axis is sometimes labeled with another characteristic that corresponds to temperature, such as the color or the spectral type, as shown at the bottom of Figure 10.19. Interactive Simulation: H-R Diagram  Figure 10.19 The Hertzsprung-Russell (H-R) diagram is used to plot the properties of stars. More luminous stars are at the top of the diagram. Hotter stars are on the left. Stars of the same radius (R) lie along the dotted lines moving from upper left to lower right. Along the vertical axis (the y-axis), we plot the luminosity of stars—the total amount of energy that a star radiates each second. More luminous stars are toward the top of the diagram, and less luminous stars are toward the bottom. Like temperature, luminosities are plotted logarithmically, in this case with each step along the left y-axis corresponding to a multiplicative factor of 10. To understand why the plotting is done this way, recall that the most luminous stars are 10 billion times more luminous than the least luminous stars, yet all of these stars must fit on the same plot. Sometimes the luminosity axis is labeled with the absolute visual magnitude instead of luminosity, as shown on the right y-axis. Recall from Section 10.2 that the temperature, the luminosity, and the radius are all related. Because each point on the H-R diagram is specified by a surface temperature and a luminosity, we can find the radius of a star at that point as well. A star in the upper right corner of the H-R diagram is very cool, so each square meter of its surface radiates a small amount of energy. This star is also extremely luminous, however, so it must be huge to account for the feeble amount of radiation coming from each square meter of its surface. Conversely, a star in the lower left corner of the H-R diagram is very hot, which means that a large amount of energy is coming from each square meter of its surface. This star has a very low luminosity, however, so it must be very small. Starting in the lower left corner of the H-R diagram and then moving up and to the right takes you to larger and larger stars. Moving down and to the left takes you to smaller and smaller stars. All stars with the same radius lie along slanted lines across the H-R diagram, as shown by dotted lines in Figure 10.19. The Main Sequence VOCABULARY ALERT  Figure 10.20 An H-R diagram for more than 4 million stars plotted from data obtained by the Gaia satellite clearly shows the main sequence. Most of the stars lie along this band running from the upper left of the diagram toward the lower right. Figure 10.20 shows more than 4 million nearby stars plotted on an H-R diagram. The data are based on observations obtained by the Gaia satellite. There are so many stars here that plotting them all as individual dots leads to confusion. The colors in the middle of the diagram show the density of stars in that area; yellow means higher density, and red means lower density. A quick look at this diagram immediately reveals a remarkable fact, one that was first discovered in the original diagrams of Hertzsprung and Russell. About 90 percent of the stars in the sky lie in a well-defined region running across the H-R diagram from lower right to upper left, known as the main sequence. On the left end of the main sequence are the O stars: hotter, larger, and more luminous than the Sun. On the right end of the main sequence are the M stars: cooler, smaller, and fainter than the Sun. If you know that a star lies on the main sequence and has a particular temperature (both of which can be determined from the star’s spectrum), then you also know its luminosity and radius. Combining the luminosity with the brightness yields the distance to the main-sequence star. This method of determining distances to main-sequence stars from their spectra, luminosity, and brightness is called spectroscopic parallax. Despite the similarity between the names, this method is very different from the parallax method that uses geometry, discussed earlier in this chapter. Spectroscopic parallax is useful at much larger distances than the geometric method of parallax, although it is less precise.  Figure 10.21 The main sequence of the H-R diagram is a sequence of masses. From a combination of observations of binary star masses, parallax, luminosity measurements, and mathematical models of the interiors of stars, astronomers have determined that stars of different masses lie on different parts of the main sequence. If a main-sequence star is less massive than the Sun, it will be smaller, cooler, redder, and less luminous than the Sun; it will be located to the lower right of the Sun on the main sequence. Conversely, if a main- sequence star is more massive than the Sun, it will be larger, hotter, bluer, and more luminous than the Sun; it will be located to the upper left of the Sun on the main sequence. The mass of a main-sequence star determines where on the main sequence the star will lie, as shown in Figure 10.21. For main-sequence stars with similar chemical compositions, the mass alone determines all of the star’s other characteristics. The mass of a main-sequence star determines the star’s radius, its surface temperature, luminosity, internal structure, lifetime, evolutionary path, and final fate. Therefore, a star’s mass is its most important characteristic. Stars Not on the Main Sequence Although 90 percent of stars are main-sequence stars, some stars are found in the upper right portion of the H-R diagram, well above the main sequence. These stars must be large, cool giants with radii hundreds or thousands of times the radius of the Sun. At the other extreme are stars found in the lower left corner of the H-R diagram. These stars are tiny, comparable to the size of Earth. Their small surface areas explain why they have such low luminosities despite having such high temperatures. Stars that lie off the main sequence on the H-R diagram can be identified by their luminosities (determined by their distances) or by slight differences in their spectral lines. The density and temperature of gas in a star’s atmosphere affect the width of a star’s spectral lines. In general, hotter stars have broader lines. Puffed-up red giant stars above the main sequence have lower densities and lower temperatures compared to main-sequence stars. When using the H-R diagram to estimate the distance to a star by the spectroscopic parallax method, astronomers must know whether the star is on, above, or below the main sequence in order to find the star’s luminosity. Stars both on and off the main sequence have a property called luminosity class, which tells us the size of the star. Supergiant stars, which are the largest stars that we see, are luminosity class I, bright giants are class II, giants are class III, subgiants are class IV, main-sequence stars are class V, and white dwarfs are class WD. The Sun is a G2V star. Luminosity classes I through IV lie above the main sequence, whereas class WD falls below and to the left of the main sequence. The labels in Figure 10.19 show the approximate locations of some of these classes of stars on the H-R diagram. Thus, the complete spectral classification of a star includes both its spectral type, which tells us temperature and color, and its luminosity class, which indicates size. The existence of the main sequence, together with the fact that the mass of a main-sequence star determines where on the main sequence it will lie, is a grand pattern that points to the possibility of a deep understanding of what stars are and how they shine. The existence of stars that do not follow this pattern raises yet more questions. In the coming chapters, you will learn that the main sequence tells us what stars are and how they work, and that stars off the main sequence tell us how stars form, how they evolve, and how they die. CHECK YOUR UNDERSTANDING 10.4 Suppose you are studying a star with a luminosity of 100 LSun and a surface temperature of 4000 K. According to the H-R diagram, this star is a 1. main-sequence star. 2. giant red star. 3. white dwarf. 4. giant blue star. Answers to Check Your Understanding questions are in the back of the book. Glossary H-R diagram The Hertzsprung-Russell diagram, which is a plot of the luminosities versus the surface temperatures of stars. The evolving properties of stars are plotted as tracks across the H-R diagram. main sequence The strip on the H-R diagram where most stars are found. Main-sequence stars are fusing hydrogen to helium in their cores. spectroscopic parallax Use of the spectroscopically determined luminosity and the observed brightness of a star to determine the star’s distance. luminosity class A spectral classification based on stellar size, from the largest supergiants to the smallest white dwarfs. confusion A condition where data points are so numerous or densely packed that they are difficult to distinguish. VOCABULARY ALERT confusion In common language, confusion means the state of being bewildered, or unclear. In science, the word “confusion” is closely related to this everyday term. “Confusion” means that the data points or the stars or the leaves on the tree overlap so that you can’t tell them apart. Your colleagues will then be “confused” about which object you might be talking about. reading Astronomy News Mystery of nearby SS Cygni star system finally resolved by John P. Millis, for redOrbit.com – Your Universe Online Astronomers are still finding out about stars. Sometimes we are lucky enough to get new data that help settle a long-standing issue. In 1990, the Hubble Space Telescope measured the distance to a nearby star system known as SS Cygni, composed of a low-mass main-sequence star and a compact object known as a white dwarf—a stellar remnant about the mass of our Sun, but compressed to the size of Earth. The distance measured by Hubble puzzled scientists, as the measured brightness of the system was considerably higher than expected. If correct, it would call into question the mechanisms by which a white dwarf interacts with a nearby companion. “If SS Cygni was actually as far away as Hubble measured, then it was far too bright to be what we thought it was, and we would have had to rethink the physics of how systems like this worked,” noted James Miller-Jones from the Curtin University campus of the International Centre for Radio Astronomy Research (ICRAR). Miller-Jones and other astronomers have used two of the most powerful radio telescope networks in the world— the VLBA [Very Long Baseline Array] and EVLA [Expanded Very Large Array]—to measure the distance to SS Cygni, attempting to resolve the dilemma created by the Hubble result. The team used a method known as parallax, whereby the system is observed at various points during Earth’s orbit around the Sun, and then the position of the system is measured against the fixed, distant background. “If you hold your finger out at arm’s length and move your head from side to side, you should see your finger appear to wobble against the background. If you move your finger closer to your head, you’ll see it starts to wobble more. We did the exact same thing with SS Cygni—we measured how far it moved against some very distant galaxies as Earth moved around the Sun,” noted Miller-Jones. “The wobble we were detecting is the equivalent of trying to see someone stand up in New York from as far as away as Sydney.” The team found that SS Cygni is about 372 light-years from Earth, considerably closer than previous measurements made using the Hubble Space Telescope. “The pull of gas off a nearby star onto the white dwarf in SS Cygni is the same process that happens when neutron stars and black holes are orbiting with a nearby companion, so a lot of effort has gone in to understanding how this works,” explained Miller-Jones. “Our new distance measurement has solved the puzzle of SS Cygni’s brightness, it fits our theories after all.” EVALUATING THE NEWS 1. In the second paragraph, the article states that “the distance measured” was a puzzle, because “the measured brightness of the system was considerably higher than expected.” Did the author really mean “brightness” or “luminosity” here? How can you tell? 2. Evaluate the author’s description of parallax. Is his explanation understandable to an average reader and also correct? 3. Given the distance of 372 light-years, calculate the measured parallax. (This is the opposite calculation of the one the astronomers did.) Is the analogy of the person in New York as viewed from Sydney approximately correct? 4. Sketch an H-R diagram. Label the locations of the two stars of SS Cygni.  5. How do the studies described in this article reflect what you learned about the scientific method in Chapter 1? Source: John P. Millis, “Mystery of SS Cygni Star System Finally Resolved,” from RedOrbit.com, May 23, 2013. Used by permission of Dr. John P. Millis for redOrbit.com – Your Universe Online. SUMMARY Finding the distances to stars is a difficult but important task for astronomers. Parallax and spectroscopic parallax are two of the methods that astronomers use to determine distances to stars. Combining the brightness with the distance yields the luminosity. Combining the luminosity and the temperature yields the radius. Careful study of the light from a star, including its spectral lines, gives the temperature, size, and composition. Study of binary systems gives the mass of stars of various spectral types, which we can extend to all stars of the same spectral type. The H-R diagram shows the relationship among the various physical properties of stars. The major determining factor in all the properties of a star is its mass. 1 The luminosity of a star is found by combining stellar distance from Earth with the brightness of the star in the sky. The luminosity is the total energy emitted from the star each second. 2 The temperature of a star is determined by its color, with blue stars being hotter and red stars being cooler. Combining this temperature information with the luminosity of the star gives the radius. 3 Spectral lines carry a great deal of information about a star: temperature, composition, and mass, the motions of individual stars, and indirectly the stellar radius. 4 The H-R diagram is a key to understanding stellar properties. Temperature increases to the left, so that hotter stars lie on the left side of the diagram, while cooler stars lie on the right. Luminosity increases vertically, so that the most lumious stars lie near the top of the diagram. Ninety percent of stars lie along the main sequence. 5 The mass of a main-sequence star is the fundamental determining factor of all of the star’s other characteristics: luminosity, temperature, and size. The main sequence on the H-R diagram is a sequence of masses.