Chapter 20, Chapter Review

Stars spend most of their lives on the main sequence, in the core-hydrogen burning phase of stellar evolution, stably fusing hydrogen into helium at their centers. Stars leave the main sequence when the hydrogen in their cores is exhausted. For the Sun, which is about halfway through its main-sequence lifetime, this stage will occur about 5 billion years from now. Low-mass stars evolve much more slowly than the Sun, and high-mass stars evolve much faster.

When the central nuclear fires in the interior of a solar-mass star cease, the helium in the star’s core is still too cool to fuse into anything heavier. With no internal energy source, the helium core is unable to support itself against its own gravity and begins to shrink. The star at this stage is in the hydrogen-shell burning phase, in which the nonburning helium at the center is surrounded by a layer of burning hydrogen. The energy released by the contracting helium core heats the hydrogen-burning shell, greatly increasing the nuclear reaction rates there. As a result, the star becomes much brighter while the envelope expands and cools. A low-mass star like the Sun moves off the main sequence on the H–R diagram first along the subgiant branch, then almost vertically up the red-giant branch.

As the helium core contracts, it heats up. Eventually, the core of a star more than 0.25 times the mass of the Sun reaches the point at which helium begins to fuse into carbon. The net effect of the fusion reactions is that three helium nuclei (or alpha particles) combine to form a nucleus of carbon in the triple-alpha process. In a star like the Sun, conditions at the onset of helium burning are such that the electrons in the core have become degenerate—they can be thought of as tiny, hard spheres that once brought into contact, present stiff resistance to being compressed any further. This electron degeneracy pressure makes the core unable to “react” to the new energy source, and helium burning begins violently in the helium flash. The flash expands the core and reduces the star’s luminosity, sending it onto the horizontal branch of the H–R diagram. The star now has a core of burning helium surrounded by a shell of burning hydrogen.

As helium burns in the core, it forms an inner core of nonburning carbon. The carbon core shrinks and heats the overlying burning layers, and the star once again becomes a red giant. It reenters the red-giant region of the H–R diagram along the asymptotic-giant branch, becoming an extremely luminous red-supergiant star. The core of a low-mass star never becomes hot enough to fuse carbon. Such a star continues to ascend the asymptotic-giant branch until its envelope is ejected into space as a planetary nebula. At that point the core becomes visible as a hot, faint, and extremely dense white dwarf. The planetary nebula diffuses into space, carrying helium and some carbon into the interstellar medium. The white dwarf cools and fades, eventually becoming a cold black dwarf. Most white dwarfs are composed of carbon and oxygen, although stars in binary systems may give rise to helium white dwarfs, while more massive stars may become neon-oxygen white dwarfs.

Evolutionary changes happen more rapidly for high-mass stars than for low-mass stars because larger mass results in higher central temperatures. High-mass stars do not experience a helium flash and attain central temperatures high enough to fuse carbon. They form heavier and heavier elements in their cores, at a more and more rapid pace, and eventually die explosively.

The theory of stellar evolution can be tested by observing star clusters, all of whose stars formed at the same time. As time goes by, the most massive stars evolve off the main sequence first, then the intermediate-mass stars, and so on. At any instant, no stars with masses above the cluster’s main-sequence turnoff mass remain on the main sequence. Stars below this mass have not yet evolved into giants and so still lie on the main sequence. By comparing a particular cluster’s main-sequence turnoff mass with theoretical predictions, astronomers can measure the age of the cluster.

Stars in binary systems can evolve quite differently from isolated stars because of interactions with their companions. Each star is surrounded by a teardrop-shaped Roche lobe, which defines the region of space within which matter “belongs” to the star. As a star in a binary evolves into the giant phase it may overflow its Roche lobe, forming a mass-transfer binary as gas flows from the giant onto its companion. If both stars overflow their Roche lobes, a contact binary results. Stellar evolution in binaries can produce states not achievable in single stars. In a sufficiently wide binary, both stars evolve as though they were isolated.

SELF-TEST: TRUE OR FALSE?

1. “High-mass” stars are conventionally taken to have masses greater than that of the Sun. HINT

2. All the red-dwarf stars that ever formed are still on the main sequence today. HINT

3. Once on the main sequence, gravity is no longer important in determining a star’s internal structure. HINT

7. As a red giant, the Sun will have a core that is smaller than it was when the Sun was on the main sequence. HINT

8. When helium starts to fuse inside a solar-mass red giant, it does so slowly at first, and the rate of fusion increases gradually over many years. HINT

10. A planetary nebula is the disk of matter around a star that will eventually form a planetary system. HINT

13. A star cluster with an age of 500 million years will contain many O-type stars. HINT

14. In a binary-star system, it is never possible for the lower-mass star to be more evolved than the higher-mass companion. HINT

15. In a mass-transfer binary, one of the stars has filled its Roche lobe. HINT

SELF-TEST: FILL IN THE BLANK

1. A main-sequence star doesn’t collapse because of the outward _____ produced by hot gases in the stellar interior. HINT

3. While a star is on the main sequence _____ is slowly depleted and _____ builds up in the core. HINT

5. At the end of its main-sequence lifetime, a star’s core starts to _____. HINT

7. Just before helium fusion begins in the Sun, the core’s outward pressure will be provided mainly by electron _____. HINT

8. As a star ascends the asymptotic-giant branch, its _____ core is shrinking in size and growing in mass. HINT

9. As a red supergiant the Sun will eventually become about ____ times its present size. HINT

10. The various stages of stellar evolution predicted by theory can be tested using observations of stars in _____. HINT

11. By the time the envelope of a red supergiant is ejected, the core has shrunk to a diameter of about _____. HINT

12. A typical white dwarf has the following properties: about half a solar mass, fairly _____ surface temperature, small size, and _____ luminosity. HINT

13. As time goes by, the temperature and the luminosity of a white dwarf both _____. HINT

14. As a star cluster ages, the luminosity of the main-sequence turnoff _____. HINT

15. Whether being a member of a binary-star system will affect the evolution of a star depends largely on the _____ of the two stars in the binary. HINT

REVIEW AND DISCUSSION

4. Why is the depletion of hydrogen in the core of a star such an important event? HINT

6. Roughly how big (in A.U.) will the Sun become when it enters the red-giant phase? HINT

7. How long does it take for a star like the Sun to evolve from the main sequence to the top of the red-giant branch? HINT

10. Describe an important way in which winds from red-giant stars are linked to the interstellar medium. HINT

11. How do the late evolutionary stages of high-mass stars differ from those of low-mass stars? HINT

13. What is a planetary nebula? Why do many planetary nebulae appear as rings? HINT

15. Can you think of a way in which a helium white dwarf might exist today? HINT

20. Why is it odd that the binary system Algol consists of a low-mass red giant orbiting a high-mass main-sequence star? How did Algol come to be in this configuration? HINT

PROBLEMS

Algorithmic versions of these questions are available in the Practice Problems Module of the Companion Website.

The number of squares preceding each problem indicates the approximate level of difficulty.

The Sun will evolve off the main sequence when roughly 10 percent of its hydrogen has been fused into helium. Using the data given in Section 16.5 and Table 16.2, calculate the total amount of mass destroyed (that is, converted into energy) and the total energy released by the fusion of this amount of matter. HINT

Use the radius–luminosity–temperature relation to calculate the radius of a red supergiant with temperature 3000 K (half the solar value) and luminosity 10,000 solar luminosities. (Sec. 17.3) How many planets of our solar system would this star engulf? HINT

What would be the luminosity of the Sun if its surface temperature was 3000 K and its radius was (a) 1 A.U., (b) 5 A.U.? HINT

Use the radius–luminosity–temperature relation to calculate the radius of a 12,000 K (twice the temperature of the Sun), 0.0004-solar-luminosity white dwarf. HINT

Use the graph in More Precisely 20-1 to estimate the factor by which CNO energy production outstrips proton–proton energy production in a 10-solar-mass star with a central temperature of 25 million K. What do you think the factor would be if the abundances of C, N, and O were just one-tenth the solar value? HINT

A main-sequence star at a distance of 20 pc is barely visible through a certain telescope. The star subsequently ascends the giant branch, during which time its temperature drops by a factor of three and its radius increases 100-fold. What is the new maximum distance at which the star would still be visible using the same telescope? HINT

A Sunlike star goes through a rapid luminosity change between stages 8 and 9, when the luminosity increases by about a factor of 100 in 10⁵ years. On average, how rapidly does the star’s absolute magnitude change, in magnitudes per year? Do you think this change would be noticeable in a distant star within a human lifetime? HINT

Calculate the average density of a red-giant core of 0.25 solar-mass and radius 15,000 km. Compare this with the average density of the giant’s envelope, if it has a 0.5 solar-mass and its radius is 0.5 A.U. Compare each with the central density of the Sun. (Sec. 16.2) HINT

How long will it take the Sun’s planetary nebula, expanding at a speed of 50 km/s, to reach the orbit of Neptune? How long to reach the nearest star? HINT

10.

What are the escape speed (in km/s) and surface gravity (relative to Earth’s gravity) of Sirius B? (See Table 20.2.) HINT

11.

A 15-solar-mass blue supergiant with a surface temperature of 20,000 K becomes a red supergiant with the same total luminosity and a temperature of 4000 K. By what factor does its radius change? HINT

12.

The radius of Betelgeuse varies by about 60 percent within a period of three years. If the star’s surface temperature remains constant, by how much does its absolute magnitude change during this time? HINT

13.

The Sun will reside on the main sequence for 10¹⁰ years. If the luminosity of a main-sequence star is proportional to the fourth power of the star’s mass, what mass star is just now leaving the main sequence in a cluster that formed (a) 400 million years ago, (b) 2 billion years ago? HINT

14.

In roughly 5 billion years, the Sun will eject its envelope as a planetary nebula. Before then, suppose it loses 20 percent of its mass on the giant branch. (a) If Jupiter stays in a circular orbit while this mass is being lost and its angular momentum stays constant, what will be the planet’s orbital radius and period when the Sun’s mass has fallen to 0.8 solar masses? (b) When the planetary nebula is formed, the Sun loses a further 0.3 solar masses rapidly enough that we can regard the loss as immediate. Jupiter’s instantaneous velocity is unchanged, but the orbit is no longer circular, due to the Sun’s smaller mass. The planet’s location at that moment becomes the perihelion of the new orbit, and the orbital semimajor axis increases (in this case) by a factor of 2.5. What are the eccentricity and period of the new orbit? HINT

15.

From the discussions presented in More Precisely 2-3 and More Precisely 15-1, it may be shown that the angular momentum of a circular binary system, of separation r and component masses m₁ and m₂, is proportional to m₁

m₂

(if the total mass is constant). Such a binary has component masses one and two times the mass of the Sun, respectively, and an orbital period of two years. (a) What is its orbital separation r? (b) Mass transfer moves 0.2 solar masses of material from the more massive to the less massive star, keeping the total mass of the system fixed and conserving angular momentum. If the binary remains circular, calculate its new separation and orbital period. HINT

COLLABORATIVE EXERCISES

1. Evolutionary Sequences. As a plot of luminosity vs. temperature, the H–R diagram is useful for describing how stars evolve over time even though “time” is not the label on either axis. As a group, create an imaginary graph of “dollars of financial income” (vertical axis) vs. “weight” (horizontal axis) and use it to describe the past and future life cycle of one of your group members. Clearly label your diagram and provide a figure caption clearly explaining each life phase.

RESEARCHING ON THE WEB

To complete the following exercises, go to the online Destinations Module for Chapter 20 on the Companion Website for Astronomy Today 4/e.

1. Access the "Story of a Rarely Seen Stellar Explosion" page and describe what a Sakurai’s object is, where it is located in the sky, and how it relates to the evolutionary future of our Sun.

2. Access the "Hertzsprung–Russell Diagram" page and define the four luminosity classes shown on the H–R Diagram.

1. Can you find the Hyades cluster? It lies about 46 pc away in the constellation Taurus, making up the “face” of the bull. It appears to surround the very bright star Aldebaran, the Bull’s eye, which makes it easy to locate in the sky. Aldebaran is a red giant, probably on the asymptotic-giant branch of its evolution. Despite appearances, it is not part of the Hyades cluster. In fact it lies only about half as far away—some 20 pc from Earth.

2. Now look for the Double Cluster in Perseus, h and chi Persei. These two young clusters probably formed together, and now move together through space. They lie about 2500 pc away, and are barely visible to the naked eye just east of the “W” of Cassiopeia.

3. Find a library that has the Astrophysical Journal. Find an article from the late 1950s and 1960s that gives the photometry of a star cluster like the Pleiades or Hyades. Plot a color–magnitude diagram (V vs. B–V; see Section 17.6). Determine the V magnitude of the main-sequence turnoff, and hence estimate the age of the cluster. Compare your age with that given in the article.

SKYCHART III PROJECTS

The SkyChart III Student Version planetarium program on which these exercises are based is included as a separately executable program on the CD in the back of this text.

An H–R diagram in Figure 17.14 relates the surface temperature of a star to its luminosity. In this and the following exercise you will generate an H–R diagram using spreadsheet software. SkyChart does not give us the temperature, so use the color index (see exercise 2 in Chapter 3) as its proxy. You will also need to use the absolute magnitude. SkyChart gives the apparent magnitude of stars and it can be converted to an absolute magnitude if you know the distance. Pick a region in the sky with plenty of stars (a field of about 10° should be sufficient). Select 50 stars and record their magnitude, color index (B–V), and distance in pc. Some information may be missing on a particular star, so just skip it and use another. The first manipulation will be constructing a graph of distance versus apparent magnitude. Looking at your graph, can you identify a general trend? Is there more than one trend at work? Common sense tells us that stars farther away will appear less bright. In fact, the luminosity will decrease as 1/D2, where D is the distance to the star, so a star twice as far away will appear one-fourth as bright. Does your graph support this? If it does not, can you give reasons why? What other factors besides distance play a role in a star’s apparent magnitude?

Now generate the H–R diagram. To do this, you will have to convert the apparent magnitude, m, to an absolute magnitude, M, using the following equation: M = 5-5log(D) + m, where D is the distance to the star measured in parsecs, pc. We can easily handle this equation using spreadsheet software (such as Excel). Enter in column A the name of the star, in column B the color index, in column C the distance, and in column D the magnitude. In the cell E2, enter the following expression: = 5-5*LOG10(C2) + D2. Select E2 again and select Copy from the Edit menu. Now highlight E3 through to the end of your data and select Paste from the Edit menu. You have just figured out the absolute magnitudes for all your data points. The last calculation that needs to be done is to determine the luminosities of the stars from their absolute magnitudes according to: L = 2.512^{(Mo 2 M)}, where L is the luminosity in solar units, M is the absolute magnitude and M_o is the absolute magnitude of the Sun, which is 4.78. In cell F2, enter the equation: = 2.512^( 4.78 2 E2 ). Now Copy the formula from G2 and Paste it to cells G3 to the end of the data. Now generate a graph of luminosity versus color index, with the vertical axis on a logarithmic scale. What is the coolest star you recorded? How luminous is it? What is the least luminous star you recorded? Estimate how hot it is. What is the most luminous star you recorded? Estimate how hot it is. How does your H–R diagram compare to the various H–R diagrams in this chapter?

In addition to the Practice Problems and Destinations modules, the Companion Website at http://www.prenhall.com/chaisson provides for each chapter an additional true-false, multiple choice, and labeling quiz, as well as additional annotated images, animations, and links to related Websites.

SUMMARY

SELF-TEST: TRUE OR FALSE?

SELF-TEST: FILL IN THE BLANK

REVIEW AND DISCUSSION

PROBLEMS Algorithmic versions of these questions are available in the Practice Problems Module of the Companion Website.

COLLABORATIVE EXERCISES

SKYCHART III PROJECTS The SkyChart III Student Version planetarium program on which these exercises are based is included as a separately executable program on the CD in the back of this text.