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Nuclear Fusion

Nuclear fusion separates stars and brown dwarfs from Jupiter-like objects. Nuclear fusion is the process of light nuclei combining to form heavier nuclei. For elements lighter than iron, this process liberates energy. The fusion of elements heavier than iron takes energy rather than gives energy. Stars are therefore powered by the fusion of elements lighter than iron, particularly of hydrogen.

Recall how an atom is constituted: an atom has a nucleus composed of protons and neutrons, collectively known as nucleons, around which electrons orbit. In nuclear fusion, the total number of protons and neutrons is conserved, but some protons are converted into neutrons in the process. A proton becomes a neutron by emitting a positron, the antiparticle of the electron, and a neutrino in an exothermic process that releases 0.8 MeV of energy.1 A neutron becomes a proton by emitting an electron and a neutrino in an endothermic process.

Available Energy and Abundances

How much energy can be released through fusion? This is found by looking at the mass per nucleon in an atom. The energy released in nuclear fusion is substantial enough that it appears in the atom's rest mass. An examination of the excess rest mass energy per nucleon for all isotopes shows that the nucleus with the greatest available energy is hydrogen, which has an 7.3 MeV energy excess relative to carbon-12. Helium, on the other hand, has only 0.6 MeV energy excess relative to carbon-12, so fusion that takes hydrogen to helium releases a total of 26.7 MeV for each helium nucleus that is created. Helium itself fuses to create carbon, but this process releases a more modest 7.2 MeV per created carbon atom. Other stable atoms that are created through nuclear fusion are oxygen-16, with an energy excess of -0.3 MeV per nucleon relative to carbon, neon-20, with -0.4 MeV, and magnesium, with -0.6 MeV. The lowest value of excess energy per nucleon is found for iron-56 at -1.1 MeV relative to carbon. Most of a star's nuclear energy is therefore released in the conversion of hydrogen into helium.

The impact of stellar nuclear fusion on the elements in our universe is apparent. The most abundant elements after hydrogen and helium, which were created in the early expanding universe, are oxygen, an end product of helium fusion, neon, another end product of helium fusion, nitrogen, an element created from oxygen and carbon during hydrogen fusion, and carbon, the initial end product of helium fusion. This observational result is a consequence of the expulsion of gases by stars as they evolve. These fusion products mixed into the interstellar gasses influences the subsequent evolution of stars created from these gases. In this way, each generation of star influences the next generation. The fusion products created by the first stellar generation prepared the universe for life by creating carbon and oxygen.

Thermal Equilibrium in Nuclear Fusion

A star's core is badly out of thermal equilibrium. For a fixed temperature the constituents of an isolated system are determined solely by the temperature. For the relatively low temperature at the core of a solar mass star, a cool 15 million degrees (1 keV), the equilibrium state is composed primarily of iron-56 and other elements with similar numbers of nucleons it their nuclei. An equilibrium state that is predominately hydrogen requires a much higher temperature, one of order 100 billion degrees (9 MeV). A stellar core therefore attempts through nuclear fusion to bring its elemental composition into thermal equilibrium. The stumbling block to this is the low reaction rates for the various nuclear reactions. The reactions that occur first are the reactions that become most rapid at low temperatures, and rapid in this case means rapid enough to replenish the energy lost by the star's core through the propagation of radiation and the conduction of heat.

The first reactions that can counteract the transport of radiation out of the star as gravitational collapse raises the core's temperature and density are the reactions that convert hydrogen into helium. The next set of reactions that can occur as the temperature rises converts helium into carbon and other light elements with nuclei composed of multiple helium nuclei. Further increases in temperature initiate carbon fusion, and then oxygen fusion. These reactions continue until either the core becomes gravitationally stable from the degeneracy pressure of electrons, after which it is a cooling degenerate dwarf star, or until the core collapses into either a neutron star, which has a high-density equilibrium configuration radically different from the iron equilibrium that characterizes lower-density cores, or a black hole, in which the core configuration is unobservable and the subject of very speculative and esoteric research.

Uncertainty in Nuclear Fusion

Nuclear fusion rates are expressed in terms of cross sections. The cross section can be thought of as a disk of a particular area centered on one of the two nuclei involved in a fusion process. If the other nuclei moving in a strait line perpendicular to this disk hits the disk, then a reaction occurs. If it does not, then no reaction occurs. This is a convenient abstract representation of the interactions among nuclei that allow a simple calculation of the reaction rate for a given density and temperature. The cross section for a reaction is a function of both temperature and density.

Determining the cross section of the nuclear reactions occurring in stars is done in the laboratory. This work is difficult, and involves some extrapolation, because the density found in stars cannot be achieved in the laboratory. The electrons at the core of a star shield a nucleus's electric field. They act as match makers to the nuclei, enabling two nuclei to approach without immediately repulsing each other. This increases the cross section from what it is in the low-density laboratory environment.

The other difficulty is that despite the massive amounts of energy released by the stars, reaction rates necessary to produce these luminosities are very small. In the laboratory, cross sections are measured at the energies for which they are large, and then they are extrapolated to the energies characteristic of the sun.

1 The unit of energy MeV stands for mega electron volt, or one million electron volts. The photons that appear as visible light have an energy of one electron volt. One MeV corresponds to 1.6 10-6 ergs of energy and to 11.6 billion degrees of temperature.

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