19-1 During a star’s main-sequence lifetime, it expands and becomes more luminous

Over the past 4.56 billion years, thermonuclear reactions have caused an accumulation of helium in our Sun’s core

In their cores, main-sequence stars are all fundamentally alike. As we saw in Section 18-4, it is in their cores that all such stars convert hydrogen into helium by thermonuclear reactions. This process is called core hydrogen fusion. The total time that a star will spend fusing hydrogen into helium in its core, and thus the total time that it will spend as a main-sequence star, is called its main-sequence lifetime. For a star of a given mass, its main-sequence lifetime can be estimated by modeling the nuclear reactions within its core. For our Sun, the main-sequence lifetime is about 12 billion (1.2 × 1010) years. Hydrogen fusion has been going on in the Sun’s core for the past 4.56 billion (4.56 × 109) years, so our Sun is less than halfway through its main-sequence lifetime.

What happens to a star like the Sun after the core hydrogen has been used up, so that it is no longer a main-sequence star? As we will see, it expands dramatically to become a red giant. To understand why this happens, it is useful to first look at how a star evolves during its main-sequence lifetime. The nature of that evolution depends on whether its mass is less than or greater than about 0.4 M⊙.

Main-Sequence Stars of 0.4 M or Greater: Consuming Core Hydrogen

A protostar becomes a main-sequence star when steady hydrogen fusion begins in its core and it achieves hydrostatic equilibrium—a balance between the inward force of gravity and the outward pressure produced by hydrogen fusion (see Section 16-2 and Section 18-4). Such a freshly formed main-sequence star is called a zero-age main-sequence star.

We make the distinction between “main-sequence” and “zero-age main-sequence” because a star undergoes noticeable changes in luminosity, surface temperature, and radius during its main-sequence lifetime. These changes are a result of core hydrogen fusion, which alters the chemical composition of the core. As an example, when our Sun first formed, its composition was the same at all points throughout its volume: by mass, about 74% hydrogen, 25% helium, and 1% heavy elements. But as Figure 19-1 shows, the Sun’s core now contains a greater mass of helium than of hydrogen. (There is still enough hydrogen in the Sun’s core for another 7 billion years or so of core hydrogen fusion.)

Figure 19-1: Changes in the Sun’s Chemical Composition These graphs show the percentage by mass of (a) hydrogen and (b) helium at different points within the Sun’s interior. The dashed horizontal lines show that these percentages were the same throughout the Sun’s volume when it first formed. As the solid curves show, over the past 4.56 × 109 years, thermonuclear reactions at the core have depleted hydrogen in the core and increased the amount of helium in the core.

CAUTION!

Although the outer layers of the Sun are also predominantly hydrogen, there are two reasons why this hydrogen cannot undergo fusion. The first reason is that while the temperature and pressure in the core are high enough for thermonuclear reactions to take place, the temperatures and pressure in the outer layers are not. The second reason is that there is no flow of material between the Sun’s core and outer layers, so the hydrogen in the outer layers cannot move into the hot, high-pressure core to undergo fusion. The same is true for main-sequence stars with masses of about 0.4 M⊙ or greater. (We will see below that the outer layers can undergo fusion in main-sequence stars with a mass less than about 0.4 M⊙.)

Thanks to core hydrogen fusion, the total number of atomic nuclei in a star’s core decreases with time: In each reaction, four hydrogen nuclei are converted to a single helium nucleus (see the Cosmic Connections figure in Section 16-1, as well as Box 16-1). With fewer particles bouncing around to provide the core’s internal pressure, the core contracts slightly under the weight of the star’s outer layers. Compression makes the core denser and increases its temperature. (Box 19-1 gives some everyday examples of how the temperature of a gas changes when it compresses or expands.) As a result of these changes in density and temperature, the pressure in the compressed core is actually higher than before.

ASTRONOMY DOWN TO EARTH

Compressing and Expanding Gases

As a star evolves, various parts of the star either contract or expand. When this happens, the gases behave in much the same way as gases here on Earth when they are forced to compress or allowed to expand.

When a gas is compressed, its temperature rises. You know this by personal experience if you have ever had to inflate a bicycle tire with a hand pump. As you pump, the compressed air gets warm and makes the pump warm to the touch. The same effect happens on a larger scale in southern California during Santa Ana winds or downwind from the Rocky Mountains when there are Chinook winds. Both of these strong winds blow from the mountains down to the lowlands. Even though the mountain air is cold, the winds that reach low elevations can be very hot. (Chinook winds have been known to raise the temperature by as much as 27°C, or 49°F, in only 2 minutes!) The explanation is compression. Air blown downhill by the winds is compressed by the greater air pressure at lower altitudes, and this compression raises the temperature of the air.

On the other hand, expanding gases tend to drop in temperature. When you open a bottle of carbonated beverage, the gases trapped in the bottle expand and cool down. The cooling can be so great that a little cloud forms within the neck of the bottle. Clouds form in the atmosphere in the same way. Rising air cools as it goes to higher altitudes, where the pressure is lower, and the cooling makes water in the air condense into droplets.

Here’s an experiment you can do to feel the cooling of expanding gases. Your breath is actually quite warm, as you can feel if you open your mouth wide, hold the back of your hand next to your mouth, and exhale. But if you bring your lips together to form a small “o” and again blow on your hand, your breath feels cool. In this second case, your exhaled breath has to expand as it passes between your lips to the outside, which makes its temperature drop.

535

As the star’s core shrinks, its outer layers expand and shine more brightly. Here’s why: As the core’s density and temperature increase, hydrogen nuclei in the core collide with one another more frequently, and the rate of core hydrogen fusion increases. Hence, the star’s release of energy increases, which increases the star’s luminosity. The radius of the star as a whole also increases slightly, because increased core pressure pushes outward on the star’s outer layers. The star’s surface temperature changes as well, because it is related to the luminosity and radius (see Section 17-6 and Box 17-4). As an example, theoretical calculations indicate that over the past 4.56 × 109 years, our Sun has become 40% more luminous, grown in radius by 6%, and increased in surface temperature by 300 K (Figure 19-2).

Figure 19-2: The Zero-Age Sun and Today’s Sun Over the past 4.56 × 109 years, about half of the hydrogen in the Sun’s core has been converted into helium, the core has contracted a bit, and the Sun’s luminosity has gone up by about 40%. These changes in the core have made the Sun’s outer layers expand in radius by 6% and increased the surface temperature from 5500 K to 5800 K.

Main-Sequence Stars of Less than 0.4 M: Consuming All Their Hydrogen

Figure 19-3: A Fully Convective Red Dwarf In a red dwarf—a main-sequence star with less than about 0.4 solar masses—helium (He) created in the core by thermonuclear reactions is carried to the star’s outer layers by convection. Convection also brings fresh hydrogen (H) from the outer layers into the core. This process continues until the entire star is helium.

The story is somewhat different for the least massive main-sequence stars, with masses between 0.08 M⊙ (the minimum mass for sustained thermonuclear reactions to take place in a star’s core) and about 0.4 M⊙. These stars, of spectral class M, are called red dwarfs because they are small in size and have a red color due to their low surface temperature. They are also very numerous; about 85% of all stars in the Milky Way Galaxy are red dwarfs.

In a red dwarf, helium does not accumulate in the core to the same extent as in the Sun’s core. The reason is that in a red dwarf there are convection cells of rising and falling gas that extend throughout the star’s volume and penetrate into the core (see Figure 18-12c). These convection cells drag helium outward from the core and replace it with hydrogen from the outer layers (Figure 19-3). The fresh hydrogen can undergo thermonuclear fusion that releases energy and makes additional helium. This helium is then dragged out of the core by convection and replaced by even more hydrogen from the red dwarf’s outer layers.

536

As a consequence, over a red dwarf’s main-sequence lifetime essentially all of the star’s hydrogen can be consumed and converted to helium. The core temperature and pressure in a red dwarf is less than in the Sun, so thermonuclear reactions happen more slowly than in our Sun. Does it take a long time for a red dwarf to burn all its hydrogen? Indeed it does! Calculations indicate that it takes hundreds of billions of years for a red dwarf to convert all of its hydrogen completely to helium. The present age of the universe is only 13.7 billion years, so there has not yet been time for any red dwarfs—not one—to become pure helium.

A Star’s Mass Determines Its Main-Sequence Lifetime

The main-sequence lifetime of a star depends critically on its mass. As Table 19-1 shows, massive stars have short main-sequence lifetimes. The greater core pressure created by a larger total mass causes nuclear reactions to occur more rapidly, and burn through the available hydrogen fuel more quickly. Hence, even though a massive main-sequence star contains much more hydrogen fuel in its core than is in the entire volume of a red dwarf, a massive star exhausts its hydrogen much sooner. More rapid burning also produces a greater luminosity so that massive stars are more luminous (see Section 17-9, and particularly the Cosmic Connections figure for Chapter 17). Thus, a main-sequence star’s mass determines not only its luminosity, but also how long it can remain a main-sequence star (see Box 19-2 for details).

Table 19-1: Approximate Main-Sequence Lifetimes
Mass (M⊙) Surface temperature (K) Spectral class Luminosity (L⊙) Main-sequence lifetime (106 years)
   25 35,000 O 80,000            4
   15 30,000 B 10,000          15
     3 11,000 A       60        800
  1.5    7000 F         5      4500
  1.0    6000 G         1   12,000
0.75    5000 K       0.5   25,000
0.50    4000 M     0.03 700,000
The main-sequence lifetimes were estimated using the relationship t∝ 1/M2.5 (see Box 19-2).

We saw in Section 18-4 how more-massive stars evolve more quickly through the protostar phase to become main-sequence stars (see Figure 18-10). In general, the more massive the star, the more rapidly it goes through all the phases of its life. Still, most of the stars we are able to detect are in their main-sequence phase, because this phase lasts so much longer than other luminous phases.

In the remainder of this chapter we will look at the luminous phases that can take place after the end of a star’s main-sequence lifetime. (In Chapters 20 and 21 we will explore the final phases of a star’s existence, when it ceases to have an appreciable luminosity.)

537

TOOLS OF THE ASTRONOMER’S TRADE

Main-Sequence Lifetimes

Hydrogen fusion converts a portion of a star’s mass into energy. We can use Einstein’s famous equation relating mass and energy to calculate how long a star will remain on the main sequence.

Suppose that M is the mass of a star and f is the fraction of the star’s mass that is converted into energy by hydrogen fusion. During its main-sequence lifetime, the total energy E supplied by the hydrogen fusion can be expressed as

E = fMc2

In this equation c is the speed of light.

This energy from hydrogen fusion is released gradually over millions or billions of years. If L is the star’s luminosity (energy released per unit time) and t is the star’s main-sequence lifetime (the total time over which the hydrogen fusion occurs), then we can write

(Actually, this equation is only an approximation. A star’s luminosity is not quite constant over its entire main-sequence lifetime. But the variations are not important for our purposes.) We can rewrite this equation as

E = Lt

From this equation and E = fMc2, we see that

Lt = fMc2

We can rearrange this equation as

Thus, a star’s lifetime on the main sequence is proportional to its mass (M) divided by its luminosity (L). Using the symbol ∝ to denote “is proportional to,” we write

We can carry this analysis further by recalling that main-sequence stars obey the mass-luminosity relation (see Section 17-9, especially the Cosmic Connections figure). The distribution of data on the graph in the Cosmic Connections figure in Section 17-9 tells us that a star’s luminosity is roughly proportional to the 3.5 power of its mass:

L ∝ M3.5

Substituting this relationship into the previous proportionality, we find that

This approximate relationship can be used to obtain rough estimates of how long a star will remain on the main sequence. It is often convenient to relate these estimates to the Sun (a typical 1-M⊙ star), which will spend 1.2 × 1010 years on the main sequence.

EXAMPLE: How long will a star whose mass is 4 M⊙ remain on the main sequence?

Situation: Given the mass of a star, we are asked to determine its main-sequence lifetime.

Tools: We use the relationship t ∝ 1/M2.5.

Answer: The star has 4 times the mass of the Sun, so it will be on the main sequence for approximately

times the Sun’s main-sequence lifetime

Thus, a 4-M⊙ main-sequence star will fuse hydrogen in its core for about (1/32) × 1.2 × 1010 years, or about 4 × 108 (400 million) years.

Review: Our result makes sense: A star more massive than the Sun must have a shorter main-sequence lifetime.

CONCEPT CHECK 19-1

Why are there no red dwarfs that have already left the main sequence?