REQUIRED READINGS: Text, Ch. 20, and all that follows here. You may find it helpful to review Lessons 3 (the Sun) and Lesson 4 (observations of stars) before reading this section.

 In Lesson 4 we described how astronomers classify stars according to intrinsic properties [mass M, surface temperature T, luminosity L, radius R] and plot these properties on diagrams relating pairs of these properties [L&T; M&L; M&R]. On such plots, more than 90% of the stars cluster in a thin band called the main sequence. If one of these properties -- say, M -- is specified, all other properties -- L, R, T, are also determined. Clearly, all these main sequence (M-S) stars have many things in common; the only reason they differ from each other is that they have different masses. Why is this true? It turns out that we can understand these relationships as the result of three basic physical principles.

In his famous 1920 article, The Internal Constitution of the Stars, Arthur S. Eddington described the three basic principles that form our modern understanding of the Sun and other main sequence stars. They are: 

1. Hydrostatic Equilibrium: this principle says that the outward force due to gas pressure must balance the inward force due to gravity.
2. Energy transport: this principle describes how the heat energy inside the star leaks out through the gas inside the star, eventually emerging as the radiation we see.
3. Energy generation: this principle describes how nuclear fusion reactions in the central part of the star replenish the heat energy that leaks out.

 Hydrostatic Equilibrium: This principle says that since M-S stars are neither expanding nor contracting, there must be a nearly exact balance between the outward pressure force of the hot gas within the star and the inward tug on the gas due to gravity. Since scientists knew (from laboratory experiments) how pressure depends on gas density and temperature, and how gravity depends on mass and radius, they could write an equation setting these forces equal. From this equation, they could infer the central temperature, T_c, of a star from its mass M and radius R. The result is

where M and R are measured in units of the Sun's mass and radius, respectively. Note that T_c is much hotter than the surface temperature of the star. For example the Sun (with T_c = 1.6 x 10⁷ K) has a surface temperature of about 6,000 K.

With this equation and the observed mass-radius relation, we can deduce a very important clue to the structure of main sequence stars. The observations (see text, Figure 17.18a and Lesson 4) show that main sequence stars have radius that is almost proportional to mass: R = M^0.8. Plugging this observed fact into the above equation for central temperature gives the equation T_c = (1.6 x 10⁷ K) M^0.2, which implies that the central temperature of a main sequence star is extremely insensitive to the star's mass. For example, a blue giant star with mass M = 100 Solar masses will have T_c = 4 x 10⁷ K, while a red dwarf with mass M = 0.1 Solar masses will have T_c = 1.0 x 10⁷ K. While M ranges over a factor of 1000, Tc ranges over a factor < 4!

Energy Transport: this principle describes how the tremendous heat energy in the interior of a star leaks out as the radiation we observe. Again, the appropriate equation follows from laboratory experiments. At the enormous temperatures inside stars, most of the radiation is X-rays. But even by 1920, people had been measuring how X-rays pass through matter -- specifically, how far X-rays could travel before they are absorbed or deflected by atoms. Eddington refers to these experiments in The Internal Constitution of the Stars.

In class we had a little demonstration in which we threw dice to illustrate how radiation moves through matter by diffusion, or random walk. The point is that if a photon is scattered by atoms in random directions many times, it will take much longer to escape from the center of the star than it would if it could travel out directly. For example, a photon would require only two seconds to travel from the center of the Sun to the surface if it could travel freely. But actually, it takes several thousand years for a photon to leak out of the Sun by diffusion.

The experiments showing how X-rays pass through matter and a theory for diffusion are sufficient to determine the luminosity of a main sequence star from its mass, radius, and central temperature, i.e., to determine the mathematical function L(M,R,T_c). Moreover, since the principle of hydrostatic equilibrium gives T_c(M,R), and we can determine T_c from the observed mass-radius relation, the mass alone is sufficient to determine the luminosity, i.e., these relations give the mass-luminosity relation L(M). In fact, this theory gives the formula L = M^3.5, which is just the relationship between luminosity and mass that main sequence stars are seen to obey (see Lesson 4).

 Energy generation: we have already discussed the idea that the Sun derives its energy from nuclear fusion reactions in its core. Eddington understood that the energy that would be released by the fusion of four hydrogen atoms to form helium was hundreds of times greater than the Sun's internal heat energy. He realized that such fusion reactions must be going on inside the Sun; otherwise it would collapse in about 20 million years, as Kelvin and Helmholtz said.

But Eddington did not know how such reactions could take place. Indeed, he recognized there was an unsolved mystery. To quote from The Internal Constitution of the Stars: "But is it possible to admit that such a transmutation is occurring? It is difficult to assert, but perhaps more difficult to deny, that this is going on."

It would be another 30 years until this problem was solved. The solution was found in the 1950's by Professor Hans Bethe of Cornell University. Like many physicists of that time, Bethe had worked on the development of the atomic bomb and the hydrogen bomb, and so he knew a lot more about nuclear reactions than Eddington knew in the 1920s. In fact, the key insight about how nuclear fusion reactions could occur was provided by George Gamow, who was a Professor here at the University of Colorado. You will hear a lot more about Gamow later in this course. Bethe understood Gamow's theory of nuclear reactions and he also knew a lot of detailed experimental information about fusion reactions.

The reason that fusion reactions require such high temperatures to occur is that atomic nuclei repel each other very strongly by electrical forces. The nuclei cannot react with each other unless they come very close. This is unlikely enough for two nuclei; it would be virtually impossible for four hydrogen atoms to get close enough to stick together all at the same time. Therefore, Bethe understood that the fusion of hydrogen into helium must take place by a sequence of reactions in which other elements are formed as intermediate steps. He discovered two such sequences of reactions, both of which play a role in converting hydrogen into helium in main sequence stars. The first, which dominates in stars with masses comparable to the Sun and less, is called the proton-proton chain, which is illustrated below.

Click here for a movie showing these reactions in sequence.

The second, which dominates in more massive main sequence stars, is called the CNO cycle. This reaction sequence is described on p. 422 of your text and here. Click here for another movie showing the CNO cycle.

Stability of Main Sequence Stars: This temperature sensitivity of thermonuclear reactions creates an elegant kind of thermostat that regulates the central temperature of main sequence stars, T_c, at a constant equilibrium value. If T_c rises above this value, the nuclear reactions pump heat energy into the stellar interior at a greater rate than the energy can escape by radiative diffusion. This extra heat energy causes the interior of the star to expand a bit, and this expansion actually causes the interior temperature to decrease a bit. But that reduces the rate of the thermonuclear reactions, which causes the temperature to drop back to the equilibrium value for which the thermonuclear energy production exactly balances the heat loss by radiative diffusion.

Armed with an understanding of how main sequence stars burn by fusing hydrogen into helium, astrophysicists were ready to return to the old problem that Eddington addressed so eloquently in The Internal Constitution of the Stars. How long can stars run?

Using the mass-luminosity relationship, L(M) = M^3.5 (see Lesson 4), it's easy to generalize this argument into an equation that gives the main-sequence lifetime, t_MS, of any star in terms of its mass:

 t_MS = t_Sun M/L(M) = 10¹⁰ years M/M^3.5 = 10¹⁰ years M^-2.5 ,

We know now that the Sun has a main sequence lifetime of 10 billion years and that it has already lived for about 4.5 billion years. So it has already converted about half the hydrogen in its core into helium (see text, Fig. 20.2 and here;). In another 5.5 billion years, the core of the Sun will be pure helium. What happens then?

When the core shrinks, the compression due to gravity will actually cause the gas to become hotter, not cooler. Moreover, the shrinking core is surrounded by hydrogen gas that was too cool to burn when the star was a main-sequence star. But now, as the core shrinks and becomes hotter, the temperature of the hydrogen surrounding this core will rise above the kindling temperature for hydrogen fusion reactions, and the hydrogen will begin to burn in a thin shell surrounding the helium core, which is not burning, as illustrated in Figure 20.3 of your text or here. This shell burns fiercely, producing more luminosity than the core of the star did while it still had hydrogen to burn. In fact, the luminosity produced by the shell is so great that it cannot leak out of the envelope as fast as it is produced. Instead, it causes the outer envelope of the star to expand.

We call the resulting star a red giant. The radius of its photosphere swells up by a factor of 100 or more compared to the radius it had as a main sequence star, and its luminosity increases by a factor of a few hundred. Its interior structure is totally different from that of a main-sequence star (e.g., the Sun, Fig. 16.4). The distended outer part of the star has very low density. But the inner core of the star, which contains about half the mass, is very small (about the size of Jupiter) and very dense.

Helium burning: the Sun will be a red giant for about 100 million years or so, as its helium core slowly continues to contract and get hotter. But then something remarkable happens. At temperatures above about 10⁸ K (100 million), helium can fuse into carbon in a process called the triple-alpha reaction. It is described in your text, p. 424. See also this diagram and this movie. Since one carbon nucleus has less mass than the sum of three helium nuclei, this reaction releases energy (in the form of gamma ray photons). At sufficiently high temperatures some of the carbon will also fuse with another helium nucleus to form oxygen. But the energy yield per unit mass in these reactions is less (by a factor of about 1/7) than that from the fusion of hydrogen into helium.

When the temperature of the helium in the core of the red giant star rises above 10⁸ K, the triple-alpha reaction turns on suddenly, in a phenomenon called the helium flash. The core suddenly expands due to the increased energy production. This expansion causes the hydrogen burning shell to diminish in luminosity, and this causes the outer envelope of the star to shrink back again, from a few hundred solar radii to about 10 solar radii. The resulting star is called a horizontal branch star, and its interior structure is illustrated schematically here.

Horizontal branch stars are called so because of their locations on the Hertzsprung-Russell diagram, as illustrated in the figure below, which is a H-R diagram for a dense cluster of stars called M3 (which we shall discuss in more detail below. [Click here to see an image of M3.]

But, after about another 50 million years or so, the helium in the core of the HB star all turns to carbon and oxygen, and these elements can't burn at the temperatures (100 - 200 million degrees) in the core of the HB star. Therefore, the carbon-oxygen core begins to shrink again. As it becomes hotter and denser, the surrounding helium begins to burn furiously in a shell. Once again, the shell burning produces more luminosity than can escape through the envelope of the star, so the star swells up again to a red giant star. But this star has a different internal structure than it did the last time it was a red giant (before it became a horizontal branch star). It contains a very dense carbon/oxygen core that is not burning, surrounded by a burning shell of helium, surrounded by a burning shell of hydrogen, surrounded by a hydrogen envelope. The internal structure is illustrated in Fig. 20.7 and here.

These kinds of red giant stars, called asymptotic giant branch stars, or AGB stars, are indicated on the H-R diagram above (see also text, Figure 20.13). From the outside, they are indistinguishable from red giant stars. So we can't be sure, for example, whether Betelgeuse has a core composed of helium or of carbon/oxygen.

To review the evolution of stars through the red giant and AGB stages, you should play with this wonderful stellar evolution simulator, by Terry Herter of Cornell University.

The most luminous main sequence stars known have luminosities equal to about 10 million Suns. These blue supergiants are very rare. There is a cluster of them in a nebula called 30 Doradus in a nearby galaxy, the Large Magellanic Cloud, that you can see here and read about here. Just a few months ago, astronomers from UCLA, using a new infrared camera on the Hubble Space Telescope, discovered the current record holder, which you can read about here.

Some of the most luminous red giant stars are seen to pulsate, becoming brighter and dimmer over periods of a year or so. These are called "long period variables." AGB stars (described above) are even more unstable than red giants. Toward the end of their lifetimes, they are likely to expel almost all their hydrogen and helium envelope, leaving the carbon/oxygen core exposed. The expelled gas escapes into interstellar space and is seen as a glowing shell, called a planetary nebula. The nebula glows in emission lines because it is heated and ionized by ultraviolet radiation from the carbon/oxygen core. (The name planetary nebula is a misnomer. Planetary nebulae have nothing to do with planetary systems. But the astronomers who first observed them thought they might be the sites of planet formation, and the name stuck.)

Planetary nebulae are among the most beautiful objects in the sky. Some are fairly smooth and symmetrical, such as the famous Ring Nebula in Lyra (a.k.a. M57), which you can see easily through the telescopes at the Sommers Bausch Observatory. Another one is the Helix Nebula (a.k.a. NGC7293). A Close-up of the Helix Nebula taken with the Hubble Space Telescope shows that the gas in this nebula is still being blown away by the tremendous radiation and stellar wind from its central star. Other planetary nebulae have strange and wonderful shapes, indicating that the mass ejection process is very complicated. You can find several images of planetary nebulae from the Hubble Space Telescope here. Be sure to read the captions.

As we have discussed, when the core of a star burns up all its fuel, it must begin to contract as it radiates away its heat energy. It contracts because the lost heat implies a loss of pressure support. Up to now, the only pressure that we have been considering is heat pressure. But at very high gas density, such as that encountered in the carbon/oxygen core of an AGB star, a new kind of pressure, called degeneracy pressure, becomes comparable to or greater than the heat pressure. As we shall discuss below, this degeneracy pressure stops the contraction of the carbon/oxygen core.

Scientists gained a better understanding of this degeneracy pressure during the 1930s when they developed theory of quantum mechanics, which explains the structure of atoms, molecules, and liquid and solid matter. With this theory, we can understand how matter at high density resists compression.

If the carbon/oxygen core that remains after a star has shed its outer envelope has mass less than 1.4 solar masses, it has come to the end of its life story. It has no nuclear reactions and it will have no further major structural changes. It will take many millions of years, however, before it cools off enough that it becomes invisible. We call such a star a white dwarf star.

In fact, astronomers have observed many white dwarf stars. Indeed, the brightest star in the sky, Sirius, is actually a binary star system consisting of Sirius A, a fairly luminous main sequence star (mass about 6 Suns), and Sirius B, a white dwarf. Click here for an image of Sirius A and B. The radius of Sirius B, as inferred from Stefan's Law (see Lesson 4), is actually less than the radius of Earth. We know now that there are billions of white dwarf stars in the Milky Way galaxy, but we haven't observed very many because they are so faint and hard to find.

The theory the structure of white dwarf stars was first worked out in 1930 by S. Chandrasekhar while he was a graduate student at Cambridge University. Chandrasekhar's theory showed that a white dwarf star with mass equal to that of the Sun would have a radius less than that of the Earth. That implies that the average density of such a star would be greater than a million times the average density of the Earth -- a one ounce shot glass of white dwarf matter would weigh more than 100 tons!

Chandrasekhar's theory also showed that the greater the mass of such a star, the smaller its radius would be. In fact, he found that if the star's mass exceeded 1.4 times the Sun's mass, the degeneracy pressure would not be strong enough to resist gravity and the star would collapse. We call this value of mass -- 1.4 Suns -- the Chandrasekhar limit*. As you will see, this limit plays a fundamental role in determining the final fates of stars.

*Arthur S. Eddington, also at Cambridge University, didn't believe this result, and in fact ridiculed the idea. But he was wrong.

 6. EVOLUTION OF STAR CLUSTERS

 As we discussed in Lesson 5, stars are not born in isolation; they are born in clusters containing many thousands of stars. Moreover, the timescale for a cluster of stars to be born from a dense interstellar gas cloud is a few million years -- far less than the typical lifetimes of most stars, which range from ten million years for the most massive stars to billions of years for stars like the Sun.

Before reading this section, it might be a good idea to review the observations of star clusters, which are discussed in your text, Section 17.10 and in Lesson 4. Now we are ready to address the last question in the summary of Lesson 4: Why are the types of stars found in globular clusters so different from those in open clusters? We now know the answer: globular clusters are very old, while open clusters are relatively young. As you will see, our understanding of this fact gives us a powerful tool to determine the ages of astronomical systems, even the universe itself.

The study of star clusters is an example of population dynamics, a technique that has wide-ranging application in all sorts of natural and social sciences. As the name suggests, the same technique can be applied to the study of human populations. For example, suppose we take a census of a large city and count people by height. We might find that about 80% -- 4/5 -- of all the inhabitants were between 4'8" and 6' 2" tall, and that about 20% of people were shorter. If we had data on ages as well, we would see that most of the people shorter than 4'8" were less than 16 years old. We might infer that 20% of the people were short because the average lifetime of people is about 80 years, so that people spend only 1/5 of their lifetimes under 16 years old, i.e., under 4'8". The logic is sound if the population of the city is stationary, so that roughly the same numbers of people are born and die each year.

Since we now know something about the life cycles of stars, we can regard clusters of stars as analogues of towns and apply the same kind of reasoning to analyze their populations. But there are some differences. The most important one is that stars, unlike human beings, do not all have similar life spans. The most massive stars remain on the main sequence (i.e., burn hydrogen in their cores) for only about 10⁷ (10 million) years, while stars like the Sun remain on the main sequence for about 10¹⁰ (10 billion) years. That means that in a newborn star cluster, of age less than 10⁷ years, even the most massive blue giant stars will still be on the main sequence, as illustrated here. But after 10⁷ years, the most massive blue giants will have consumed all the hydrogen in their cores and will have become red giants (while the less massive stars will still be on the main sequence), as illustrated by the schematic color-magnitude diagram here. Compare this diagram with that of the Pleiades, shown below.

As time goes on, less massive stars also become red giants, and some of the stars that were previously red giants have evolved into white dwarf stars. This sequence of events is described hereBut the diagrams are incomplete because they don't show the horizontal branch stars that are also found in these clusters. For example, compare the schematic color-magnitude diagram of a 10 billion year old globular cluster here with the real color-magnitude diagram of the globular cluster M3, which shows many horizontal branch stars as well as red giant stars. The observations used to construct this diagram were not sensitive enough to see the many white dwarf stars that must exist in M3, but the Hubble Space Telescope found many white dwarfs in the globular cluster M4. Below is a color-magnitude diagram of M4, showing the faint main sequence and white dwarf stars detected there (this diagram does not show the brighter red giants and horizontal branch stars in M4).

We define the main sequence turnoff of a star cluster as the position on the H-R diagram of the most luminous star that remains on the main sequence. For example, the main sequence turnoff is located right next to the label "blue dwarf stars" in the diagram above.) As the diagrams above show, this point moves to lower luminosity and temperature as the star cluster ages. Since we have a theory that tells us how long stars of a given mass (or luminosity) can remain on the main sequence, we can infer the age of a star cluster from the location of the main sequence turnoff of its H-R diagram.

The globular clusters are the oldest star clusters known. By comparing the main sequence turnoff points of their H-R diagrams with the theory of stellar evolution, astronomers find that the oldest globular clusters are 11.5 billion years old with an uncertainty of +/- 1.3 billion years (data are shown here). This tells us something important about the universe itself. Of course, it must be at least as old as any star cluster that it contains -- i.e., at least 10.2 billion years old.

Mass Transfer in Binary Systems: In all orbiting systems, there is a location called the Inner Lagrange Point (also called the L1 point), which is the place where the gravitational pull of the two objects is exactly equal, as illustrated in Figure 20.21 and 20.22 of your text. For the Earth-Sun system, for example, the L1 point is located at a point about 20 Earth diameters away from Earth toward the Sun. The SOHO satellites are located there. As Figure 20.21 shows, the L1 point is on a mathematical surface called the Roche Lobe. As a star swells up due to its evolution, its outer atmosphere will finally reach this Roche Lobe, as illustrated there. Then, gas will start to spill across the L1 point onto the other star.

When both stars in a binary system are smaller than their respective Roche lobes, no mass transfer will occur. Such systems are called detached binary systems. But when one star fills its Roche Lobe and begins to pour matter onto the other star, the system is called a semi-detached binary system. Finally, it is possible that the second star will also swell up so that it also fills its Roche Lobe. In that case, the stars merge into a common-envelope binary system, which has a single dumbell-shaped outer surface and two cores. In such systems, the cores may eventually merge to form a single star. But in some cases the system may expel its common envelope and turn back into a detached or semi-detached binary. Figure 20.21 of your text illustrates these various possibilities.

The "Algol Paradox": We have already discussed Algol -- a binary star system that was known as the "Demon Star" because it was seen by the ancients to become dim periodically as the red star eclipsed the brighter blue star. The red star is bigger than the blue star. Evidently, it is nearing the end of its lifetime and becoming a red giant. But the red star is less massive than its blue companion, which appears to be a main sequence star. Here's the "paradox": we believe that both stars in a binary system are formed at the same time. We also believe that more massive stars evolve faster than less massive stars. So, how can it be that the less massive red star is more evolved than the more massive blue star in Algol?

This process is illustrated very nicely in John Blondin's Algol page. Be sure to hit the answer. That shows a realistic supercomputer simulation of how the gas passes from the evolved star to the less evolved star. In particular, the simulation shows how an accretion disk is formed due to the rotational motion of the gas as it is transferred in a binary system.

We began with the three basic physical principles needed to describe the structure of main sequence stars. The first principle, hydrostatic equilibrium, told us how to infer the central temperatures of stars from their observed mass-radius diagram. The result showed us that all main sequence stars had very similar central temperatures, ranging from 10⁷ K for the least luminous red dwarf stars to 4 x 10⁷ K for the most luminous blue giants. The second principle, energy transport, describes how the radiation leaks out from the center of the star. This principle explains the observed mass-luminosity relationship of main sequence stars. The third principle, energy generation, describes how the energy that leaks out as radiation from the star is replenished by thermonuclear reactions in the core of the star. With the theory of nuclear reactions, it explains why the central temperatures of all main sequence stars should lie in such a narrow range as inferred from the first principle.

Armed with an understanding of how stars work, we then proceeded to describe what happens to a star after it exhausts the hydrogen fuel in its core. We described how the interior structure and external appearance of the star changes as it evolves from a main sequence (M-S) star, to a red giant (RG), to a horizontal branch (H-B) star, to an asymptotic giant branch (AGB) star, and finally to a white dwarf (WD), surrounded in its initial stages by a planetary nebula. You should review the diagrams of the interior structure of each of these kinds of stars.

White dwarf stars are dying embers of stars, in which no nuclear reactions are taking place. They do not collapse because they are supported by degeneracy pressure, a kind of atomic pressure that does not depend on heat. Degeneracy pressure can prevent the cores of stars from collapsing, but only if they have mass less than the Chandrasekhar limit of 1.4 solar masses.

We can see all these kinds of stars in star clusters. In young clusters, most of the stars are main sequence stars. After ten million years or so, the most massive stars in the cluster have changed from blue giants to red giants. In clusters more than 10 billion years old, the only stars remaining on the main sequence have mass and luminosity less than the Sun, but the cluster contains many red giants, horizontal branch stars, and white dwarfs. By measuring the position of the main sequence turnoff, astronomers can infer the ages of star clusters. They find that some globular clusters must be older than 10 billion years.

Roughly half the stars in the sky belong to close binary systems. In such systems, the more massive star can transfer mass to the less massive one as it evolves to become a red giant. This mass transfer process solves the Algol Paradox, a puzzle about the eclipsing binary system Algol. It also gives rise to a variety of spectacular phenomena that we will discuss in the next lesson.

(Return to course home page)