Thermonuclear reactions in stars

Thermonuclear reactions are the way stars generate energy for most of their lives. The various reactions are related to the changes stars undergo. They explain how the Sun has continued to shine for 4.5 billion years, and predict how the Sun will change in the future.

Overview

Our Sun is a main sequence star, about halfway through its estimated 10 billion-year lifespan. As such, it is the closest stellar “laboratory” for research into the astrophysics and astrochemistry believed to take place in similar stars throughout the universe. The nature of the Sun’s energy source baffled scientists for centuries. In 1905, however, while Albert Einstein was developing his special theory of relativity, an equation unexpectedly emerged: E = mc², where E is energy, m is mass, and c is the speed of light. This equation indicates that energy and matter are equivalent and can be transformed into each other. Another implication is that a small amount of mass is equivalent to a large amount of energy; for example, a 1-kilogram mass, if converted totally into energy, would yield 9 10¹⁶ joules. A joule is the energy needed to lift that 1-kilogram mass about 10 centimeters. The idea that mass could be converted into energy became the basis for thermonuclear weapons and for uranium-fueled fission reactors. It also is the mechanism that produces the energy of the Sun and other stars for most of their lives.

The Sun’s power output, the rate at which it produces energy, is 3.83 10²⁶ joules per second. One second of the Sun’s energy output is equivalent to 10 million times the annual electricity consumed within the United States. To radiate at this rate, the Sun converts 4.25 billion kilograms of matter into energy every second.

The solar system, with its Sun, planets, satellites, and associated objects, developed from a cloud of gas and dust several light-years in diameter. That slowly spinning cloud started to contract under its gravitational attraction. As it shrank, the cloud rotated faster and spun off an equatorial disk. At the center, a ball of gas called the proto-Sun continued to contract and grow hotter. It was not yet a full-fledged star, since it was not converting mass into energy. Its rising temperature was caused by the gravitational contraction of the gas and the conversion of its gravitational potential energy into thermal energy. As the contraction continued, temperature, density, and pressure in the interior of the proto-Sun rose. When these became high enough, nuclear fusion ignited, and the proto-Sun became a star.

Since one cannot physically measure conditions within the interior of a star, astronomers mathematically model the Sun’s internal structure. Using known conditions at the surface and the Sun’s total mass, diameter, and energy output, computers can be used to calculate the changing conditions from the surface into the interior. The model’s values for mass and energy output must match those of the Sun for the model to hold validity. Calculations suggest that the core temperature is 15 million kelvins, pressure is as high as 200 billion atmospheres, and the density reaches 150 grams per cubic centimeter. It is under these conditions of high temperature, high pressure, and high particle density that the fusion of four protons (each a nucleus of ordinary hydrogen) into helium 4 occurs. It is a series of thermonuclear reactions, known as the proton-proton chain, in the 350,000-kilometer-diameter core that powers the Sun.

Reactions start with the collision of two high-speed protons. Normally, as two protons approach each other, their positive electrical charge produces a mutual repulsion, and they slow down, stop, and then move away in a nearly elastic collision. If they are moving sufficiently fast, however, they can come close enough for the short-range strong nuclear force to allow the protons to attract each other. At this point, one of the protons is converted into a neutron, and a deuterium nucleus (hydrogen 2, an Isotope of hydrogen with one Proton and one neutron) results. Since electrical charge must be conserved, the change of a proton into a Neutron results in the emission of a positron, a positively charged electron, which is the antimatter form of the electron. When a Positron and an ordinary electron collide, they mutually annihilate each other, and their mass is completely converted into 1.6 10^-13 joules, as described by Einstein’s mass-energy equation. The energy is released as gamma rays, a high-frequency, high-energy form of electromagnetic radiation. A neutrino, a neutral particle with negligible mass, is also emitted by the nuclear fusion reaction to conserve energy and momentum simultaneously.

Next, a fast-moving proton collides with the deuterium nucleus, yielding a helium-3 nucleus and a gamma ray. The helium-3 nucleus has two protons and one neutron. Finally, two helium-3 nuclei collide to produce one helium-4 nucleus (with two protons and two neutrons) and two free protons. The net result of the interactions is that four protons fuse to yield one helium-4 nucleus, two positrons (which are annihilated when they encounter two free electrons), two neutrinos, and gamma rays.

The mass of four protons is 6.6942 10^-27 kilograms, while the mass of a helium-4 nucleus is 6.6466 10^-27 kilograms. That mass difference of 0.0476 10^-27 kilograms is converted into 4.28 10^-12 joules of energy according to Einstein’s mass-energy equation. The amount of matter converted per second by the Sun into energy is equivalent to 4 billion kilograms, or the mass of about 2 million automobiles. Each second, 8.9 10³⁷ nuclear reactions transform 610 billion kilograms of hydrogen into 606 billion kilograms of helium and 3.8 10²⁶ joules of energy.

The gamma-ray photons resulting from the reactions are absorbed and reradiated many times during their journey to the Sun’s photosphere. This energy requires from 100,000 to 1 million years to make the 700,000-kilometer trip from the core of the Sun to its “surface,” where most of the energy departs as photons of ultraviolet, visible, and infrared electromagnetic radiation.

The released radiative energy also prevents the Sun from collapsing. The gravitational attraction of the Sun’s mass produces an inward pull, but the radiative energy of the Sun heats the interior gases. This produces an outward gas pressure to counter the inward pull of gravity. Fortunately for life on Earth, this has resulted in a star that has been stable for the past 4.5 billion years. With the remaining hydrogen in the core, our Sun should last for about another 5 billion years. At the end of that time, the core’s hydrogen supply will be depleted, and the Sun’s helium “ash” core will start to contract.

This core contraction stage will raise the internal temperature to the point at which the hydrogen in a shell around the contracting helium core becomes hot enough to fuse into helium 4. The Sun’s outer layers will expand because of this increased energy production, and the Sun will enter a red giant stage in which it will engulf the inner planets of the Solar system at least out to Venus and perhaps as far out as Mars. Later, as the core temperature increases to more than 100 million kelvins, two of the core’s helium-4 nuclei will fuse into a beryllium-8 nucleus. Another helium-4 nucleus will fuse with the beryllium to produce carbon 12. In each case, a gamma-ray Photon is emitted. Core reactions will cease as the helium 4 is depleted, but hydrogen and helium fusion in shells around the carbon core will continue. This ultimately will lead to the expulsion of the Sun’s outer layers to form a planetary nebula.

The Sun’s carbon core will shrink to become a white dwarf about the size of the Earth. The Sun as a white dwarf will radiate only its residual heat, since no thermonuclear reactions will occur within it. It will slowly cool to its final black dwarf phase, a burnt-out star with less than the original mass of the Sun.

Stars that form with masses greater than about 1.4 solar masses also fuse hydrogen into helium in their cores, but with a series of thermonuclear reactions known as the carbon-nitrogen-oxygen (CNO) cycle, rather than the proton-proton chain, due to core temperature, pressure, and particle density higher than those of a 1-solar-mass star. These conditions are caused by the larger mass of the star producing a greater gravitational pull inward. Because of higher temperatures and densities, the reactions will proceed at a faster rate. The star has a larger mass but consumes itself more rapidly. This results in a shorter lifetime for such a star. As the core depletes its hydrogen, it contracts and heats, and helium fusion commences. Hydrogen-to-helium fusion is initiated in the hydrogen shell around the core.

When the core’s helium is expended, the core contracts again, with a resulting increase in temperature and density, and a new fusion reaction is ignited. This cycle of fusion, depletion, contraction, and new fusion continues until the nuclei in the core are ultimately converted into iron nuclei and the core is surrounded by several concentric shells of different fusing nuclei. The outer shell consists of hydrogen fusing to helium. Iron is the core’s end product, since further fusion requires input of additional energy rather than its exothermic release. With no source of thermal energy to prevent the further gravitational collapse of the star, it implodes. Material bounces off the core, and the star violently explodes into a core-collapse supernova.

Energy is so abundant during the supernova explosion that nuclei are fused into elements with atomic numbers higher than iron, elements even as heavy as uranium and thorium. The remnant of the star collapses to as little as 10 kilometers in diameter and becomes a neutron star, because at such high pressures the electrons and protons of the star combine into neutrons. If a supernova leaves a stellar remnant with more than 3 solar masses, an even more dramatic end state occurs. The collapse continues beyond the neutron-star stage into a black hole. Nothing can stop the gravitational attraction that collapses the star to a size from which not even light can escape.

Applications

The study of thermonuclear reactions is applicable to humanity’s most urgent need: energy. The standard of living is governed by the availability of easily obtained energy, and humanity’s present control of energy permits people to perform feats that previously were thought impossible. Human and animal muscle power, wind and water, coal, natural gas, petroleum, and nuclear fission have each proved to be inadequate, either because it is insufficient in amounts or intensity, or because it produces harmful waste. Humanity needs a reliable source of energy, and thermonuclear reactions, either directly or indirectly, may well be that source.

Solar energy is the product of a natural thermonuclear reactor 150 million kilometers from Earth, and humans use solar energy in many forms without knowing it. In fact, except for tidal and geothermal energy, all energy sources are implicitly solar-related. Coal, oil, and natural gas, for example, are forms of “fossilized” energy-based originally on solar illumination of the planet. The direct application of solar energy is difficult, since solar energy is a diffuse energy resource. Large arrays of solar panels are needed to supply the required energy, and they operate only when the Sun shines.

One method of avoiding the earthbound problems of solar power is to construct in geostationary orbit around the Earth large solar power satellites. One such station, 5 kilometers by 15 kilometers in area, could supply the needs of New York City with plenty of energy to spare. Its solar cells would convert light into electricity, which would produce microwave radiation. That would be beamed to Earth, where a receiving antenna would convert the Microwaves back into electrical power. One of these satellites would provide six to ten times the energy that an array of the same size on Earth would provide. Several hundred of these satellites would supply a large portion of humanity’s energy needs.

Another possibility is to design machines that would permit the control of fusion reactions on Earth. For decades, engineers and scientists have struggled with the problem of obtaining sufficiently high particle densities, temperatures, and pressures. The main problem is the design of a vessel that will constrain deuterium long enough for the fusion to occur. Magnetic fields and lasers have been employed to initiate the fusion process, but the goal of a net energy output on an economic scale remains elusive. When practical fusion reactors are developed, hydrogen from the water of the Earth’s oceans will provide an abundant source of fuel.

Scientists who construct models of the thermonuclear reactions and other processes that occur in the cores of stars use the Sun as their test case. Certain predictions about the Sun can be made from the model. One of these involves the number of neutrinos emitted by the reactions in the Sun’s core. Neutrinos are particles with little mass and no electrical charge, traveling at speeds close to that of light. They do not interact readily with matter; a neutrino can easily pass through a light-year-thick layer of lead shielding. The model predicts the number of solar neutrinos that should be counted by experiments designed to detect them. The first measurements in the 1970s detected only one-third of the expected number of neutrinos. Scientists questioned if the models of the Sun’s interior processes were incorrect, or if there was something unknown happening in the Sun’s interior that the models had not taken into account. After decades of uncertainty over the Sun’s missing neutrino flux, it has been determined that solar neutrinos change into other forms or flavors during their trip between the Sun and Earth, and the original apparent discrepancy seems to have been resolved.

Since neutrinos make the trip from the core to the detectors at nearly the speed of light, they provide information about what is occurring now in the Sun’s core. The electromagnetic radiation from the Sun’s photosphere discloses what occurred in the core 100,000 or more years ago. Further theoretical and experimental work should produce an answer on determining if the Sun’s energy output varies substantially over time. Indeed, many scientists think of the Sun as a variable star, but fortunately, that level of variability is quite low compared to other “traditional” classes of variable stars.

Despite the need for more research to establish the Sun's output, scientists theorize that the Sun's energy output is on the rise, perhaps as much as 30% since the Sun first began emitting energy. One theory is that over the next 1 billion years, the sun will become approximately 10 percent more luminous, growing to 40 percent brighter over the next 3.5 billion years. The effect on the Earth is projected to be catastrophic, as the planet will become far to hot to sustain life.

Context

In the eighteenth century, Immanuel Kant estimated that if the Sun were composed of coal, it would burn only for several thousand years. In the nineteenth century, Hermann von Helmholtz and Lord Kelvin independently reasoned that gravitational contraction of the Sun could provide its energy by converting gravitational potential energy into thermal energy. This theory allowed the Sun’s age to be increased to 20 to 50 million years, but it did not satisfy geologists and the biologists, who argued that hundreds of millions to billions of years were necessary for the evolution of life and the geophysical development of the Earth.

In the nineteenth century, physicists conducted experiments to determine how the speed of light changed as the speed of the medium through which it traveled was varied. The conclusion was that there is no medium through which light moves. It was also observed that the speed of light is constant, no matter how fast its source moves. Classical physics was unable to accept this seemingly nonsensical result for the speed of light. Einstein, however, stated that the speed of light is a constant in any inertial frame of reference, and went on to investigate the consequences of this postulate. While making those derivations, his famous mass-energy equivalence emerged. In 1939, Hans Albrecht Bethe and Carl von Weizsäcker hypothesized that nuclear reactions could generate the Sun’s energy. They suggested that four protons could fuse into helium 4, and that the mass difference was converted into energy.

In the early 1950s, the thermonuclear or hydrogen bomb was developed, where the ignition of a fission bomb trigger produces high temperatures that lead to the fusion of deuterium and the release of even more energy. This produces the hydrogen bomb’s greater destructive power and in a sense replicates in an uncontrolled fashion the tremendous power of the Sun’s fusion process.

Bibliography:

Chaisson, Eric, and Steven McMillan. Astronomy Today. 6th ed. New York: Addision-Wesley, 2008.

Dinwiddie, Robert, et al. Universe. New York: DK Adult, 2005.

Fraknoi, Andrew, David Morrison, and Sidney Wolff. Voyages to the Stars and Galaxies. Belmont, Calif.: Brooks/Cole-Thomson Learning, 2006.

Freedman, Roger A., and William J. Kaufmann III. Universe. 8th ed. New York: W. H. Freeman, 2008.

Hansen, Carl, Steven Kawaler, and Virginia Trimble. Stellar Interiors: Physical Principles, Structure, and Evolution. 2d ed. New York: Springer, 2004.

Prialnik, Dina. An Introduction to the Theory of Stellar Structure and Evolution.

Schneider, Stephen E., and Thomas T. Arny. Pathways to Astronomy. 2d ed. New York: McGraw-Hill, 2008.

"The Life Cycle of a Sun-like Star (Annotated)." NASA Exoplanet Exploration, 15 Dec. 2022, exoplanets.nasa.gov/resources/165/the-life-cycle-of-a-sun-like-star-annotated. Accessed 28 Sept. 2023.

Thornton, Stephen T., and Andrew Rex. Modern Physics for Students and Engineers. 3d ed. New York: Brooks/Cole, 2005.

Todd, Ian. "The Life Cycle of a Star: How Will our Solar System End?" BBC Science Focus, 21 Oct. 2022, www.sciencefocus.com/space/the-life-cycle-of-a-star-how-will-our-solar-system-end. Accessed 28 Sept. 2023.

Young, Hugh D., and Roger A. Freedman. University Physics with Modern Physics. 11th ed. New York: Addison-Wesley, 2003.