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# 13.13 Nuclear Reactions in Main Sequence Stars

Studies of our own main sequence star, the Sun, reveal that its energy comes from a series of nuclear reactions called the proton-proton chain. This reaction has great importance for stellar evolution:

1H + 1H → 2H + e+ + neutrino

2H + 1H → 3He + photon

3He + 3He → 4He + 1H + 1H + photon

Schematic of the proton-proton chain nuclear fusion reaction. Click here for original source URL

It takes a typical proton tens of million of years to take the first step in this chain. The second step occurs very quickly. The final step takes several million years. The timescale for any nuclear reaction chain is set by its slowest step. Image the process of growing vegetables: planting the seeds takes a few minutes, waiting for the vegetables to grow can take weeks or months, and harvesting them can take minutes. Even though some steps in the process are fast, in the end we would say it takes weeks or months to grow vegetables. Each passage through the proton-proton chain liberates a tiny amount of mass-energy since the helium nucleus weighs less than the sum of the four protons. Energy emerges in three forms: neutrinos that flee the Sun's core at the speed of light, protons that participate in a new round of fusion, and photons that percolate through the Sun (still at the speed of light) until they leave the Sun's surface as visible light.

The proton-proton chain is the primary energy-producing process not only inside the Sun but also inside all main sequence stars less massive than about 1.5 solar masses. This reaction dominates if the central temperatures are less than about 15 million K. Since most main sequence stars have low mass, we see that the great majority of stars are using the lightest element to build the second-lightest element. This accounts for the highest peak in the cosmic abundance of elements.

Overview of the CNO cycle. Click here for original source URL.

In main-sequence stars more massive than about 1.5 solar masses, where interior temperatures are higher than about 15 million Kelvin, another reaction series dominates in producing energy. This is the carbon cycle, sometimes also called the CNO cycle to reflect the involvement of carbon, nitrogen, and oxygen. The reactions are as follows:

12C + 1H → 13N + photon

13N → 13C + e+ + neutrino

13C + 1H → 14N + photon

14N + 1H → 15O + photon

15O → 15N + e+ + neutrino

15N + 1H → 12C + 4He

Although it looks much more complicated, the net result is that hydrogen atoms are consumed to produce helium-4 atoms with an associated release of energy. In a sense, carbon acts as a catalyst (a stimulant of change), because carbon-12 reappears at the end of the cycle to be used again in the first reaction of a subsequent cycle. Since carbon has six protons, it offers a fierce electrical resistance to fusion with another proton. Consequently, the carbon cycle requires a high temperature to operate, which can only be found in the core of a more massive star. For stars like the Sun, the carbon cycle provides less than 10% of the energy release.

In main-sequence stars both the proton-proton chain and the CNO cycle are active, but depending on the core temperature one or the other of these reactions consumes the majority of the hydrogen in the core. Major structural changes occur as the hydrogen is used up. How long do stars live? If we think of hydrogen as fuel, then the simplest estimate of lifetime is to look at the size of the fuel tank. We might start by assuming that stars use their fuel at the same rate. The most massive stars have 100 times the mass of the Sun so they should last 100 times as long on the main sequence. The least massive stars have 1/10 the mass of the Sun, so they should last 1/10 the time.

Stars do not use their fuel at the same rate. Since the energy comes from converting mass into energy, luminosity is a measure of how fast a star uses its fuel. So a better estimate of lifetime is the amount of fuel (M) divided by the rate at which it is consumed (L). We assume only that the fuel is used at a roughly constant rate. The results are surprising. A star at the top of the main sequence (hot and high luminosity) has a mass of about 100 solar masses and a luminosity of about 10times solar luminosity. It has 100 times the fuel of the Sun’s but uses it a million times faster. The estimated main sequence lifetime is M/L = 100/106 = 10-4 times that of the Sun, or only a million years. This is actually a slight underestimate because massive stars use a larger fraction of their total hydrogen fuel than low-mass stars. A star on the lower main sequence (cool and low luminosity) has a mass of only 0.1 solar mass and a luminosity of 10-3 times solar luminosity. The estimated main sequence lifetime in this case is M/L = 0.1/10-3 = 100 times that of the Sun, or 1012 years. This is a hundred times Astronomers have been searching for good examples of brown dwarfs, which would help us understand the relations between planets and stars. Brown dwarfs, of course, are hard to detect because of their faintness. A number of brown dwarf candidates have been reported, including one with luminosity only 0.0004 that of the Sun — the lowest luminosity object yet found outside the solar system. There have also been many false alarms, and it is hard to prove these objects are in the 13 to 80 MJupiter mass range needed to call them true brown dwarfs. The best example is an object called Gliese 229B, the spectrum of which shows methane and water vapor. The atmosphere of Gliese 229B has a temperature of 1000 K, too hot to be a planet but too cool to be a star. Recently, using infrared detectors sensitive to cool objects, astronomers have begun to discover brown dwarfs in increasing numbers. Despite their dim appearance, brown dwarfs are important in the census of stars — about 10% of the mass in the solar neighborhood is in the form of objects too cool to be fusing hydrogen into helium.longer than the age of the universe!

What are the implications of the calculation of stellar ages from the fuel consumption rate? Since low-mass stars live so long, none of them has ever left the main sequence. By contrast, high-mass stars evolve quickly and die. The extra amount of fuel is more than compensated for by the fast rate of consumption. Imagine a big armor-plated stretch limousine with a 30-gallon tank that could only get 5 miles to the gallon — a consumption rate of 0.2 gallons per mile. The limousine would travel 30/0.2 = 150 miles before stopping. A small economy car with a 10-gallon tank might get 50 miles to the gallon — a consumption rate of 0.02 gallons per mile. The economy car would travel 10/0.02 = 500 miles before stopping. If a group of stars forms together, the low-mass stars will be shining feebly long after the massive stars have blazed and quit.