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, it 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 explosively, 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 carbon into the interstellar medium. The white dwarf cools and fades, eventually becoming a cold black dwarf.
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 do 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.
1. "Low-mass" stars are conventionally taken to have masses less than about 8 solar masses. (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)
4. The Sun will get brighter as it begins to run out of fuel in its core. (Hint)
5. As a star evolves away from the main sequence it gets larger. (Hint)
6. As a star evolves away from the main sequence it gets hotter. (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; the rate of fusion increases gradually over many years. (Hint)
9. With the onset of helium fusion a red giant gets brighter. (Hint)
10. A planetary nebula is the disk of matter around a star that will eventually form a planetary system. (Hint)
11. For a high-mass star there is no helium flash. (Hint)
12. High-mass stars can fuse carbon and oxygen in their cores. (Hint)
13. A star cluster with an age of 100 million years will still 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)
1. A main-sequence star doesn't collapse because of the outward _____ produced by hot gases in the stellar interior. (Hint)
2. The Sun will leave the main sequence in about _____ years. (Hint)
3. While a star is on the main sequence _____ is slowly depleted and _____ builds up in the core. (Hint)
4. A temperature of at least _____ is needed to fuse helium. (Hint)
5. At the end of its main-sequence lifetime, a star's core starts to _____. (Hint)
6. When helium fuses, it produces _____ and releases _____.
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. (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 down 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)
1. Why don't stars live forever? Which types of stars live the longest? (Hint)
2. What is main-sequence equilibrium? (Hint)
3. How long can a star like the Sun keep burning hydrogen in its core? (Hint)
4. Why is the depletion of hydrogen in the core of a star such an important event? (Hint)
5. What makes an ordinary star become a red giant? (Hint)
6. 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)
8. What is the helium flash? (Hint)
9. Describe an important way in which winds from red giant stars are linked to the interstellar medium. (Hint)
10. How do the late evolutionary stages of high-mass stars differ from those of low-mass stars? (Hint)
11. What is a planetary nebula? Why do many planetary nebulae appear as rings? (Hint)
12. What are white dwarfs? What is their ultimate fate? (Hint)
13. Do many black dwarfs exist in the Galaxy? (Hint)
14. What are the Roche lobes of a binary system? (Hint)
15. 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)
1. 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, calculate the total amount of mass destroyed (that is, converted into energy) and the total energy released. (Hint)
2. Use the radius—luminosity—temperature relation (L 
R2T4; see Section 17.3) to calculate the radius of a red supergiant with temperature 3000 K (half the solar value) and luminosity 10,000 solar luminosities. How many planets of our solar system would this star engulf? (b) Repeat your calculation for a 12,000 K (twice the temperature of the Sun), 0.0004 solar luminosity white dwarf.
3. 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 3 and its radius increases 100-fold. What is the new maximum distance at which the star will still be visible using the same telescope? (Hint)
4. The Sun will reside on the main sequence for 1010 years. If the luminosity of a main-sequence star is proportional to the cube of the star's mass, what mass star is just now leaving the main sequence in a cluster that formed 400 million years ago? (Hint)
5. A Sun-like star goes through its most 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?
6. Calculate the average density of a red-giant core of mass 0.25 solar mass and radius 15,000 km. Compare this with the average density of the giant's envelope, if its mass is 0.5 solar mass and its radius is 0.5 A.U. Compare each with the average core density of the Sun.
7. How many years are spent by a Sun-like star during its post—main-sequence evolution? (Use Table 20.1 and consider stages 8 through 12.) What is the probability that oberservers will happen to catch a star during one of its rapid periods of evolution—say, between stages 8 and 9—if they examine a large number of randomly chosen post—main-sequence stars? How would the odds change if all stars (including those on the main sequence) were included in the search?
8. 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?
9. What are the escape speed and surface gravity of Sirius B? (See Table 20.2.) (Hint)
10. 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, of separation R and component masses m1 and m2, is proportional to m1 
m2 
(if the total mass is held fixed). A circular binary has component masses one and two times the mass of the Sun, respectively, and an orbital period of 2 years. (a) What is its orbital separation (semi-major axis)? (b) Mass transfer moves 0.2 solar mass 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.
1. You can tour the Galaxy without ever leaving Earth, just by looking up. In the winter sky, you'll find the red supergiant Betelgeuse in the constellation Orion. It's easy to see because it's one of the brightest stars visible in our night sky. Betelgeuse is a variable star, with a period of about 6.5 years. Its brightness changes as the star expands and contracts. At maximum size, Betelgeuse fills a volume of space that would extend from the Sun beyond the orbit of Jupiter. Betelgeuse is thought to be about 10 to 15 times more massive than our Sun. It is probably between 4 and 10 million years old—and in the final stages of its evolution.
A similar star can be found shining prominently in midsummer. This is the red supergiant Antares in the constellation Scorpius. Depending on the time of year, can you find one of these stars? Why are these stars red?
2. 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.
