Energy Sources in Stars
Two Energy sources are available to stars: Gravitational Contraction (Kelvin-Helmholtz) and Nuclear Fusion in the hot core.
Kelvin-Helmholtz
Kelvin and Helmholtz argued that gravitational contraction would cause the sun’s gases to become hot enough to radiate heat energy into space. This process, called the Kelvin-Helmholtz contraction, does in fact happen in the protostar phase of stellar formation.
However, the Kelvin-Helmholtz contraction cannot be the main source of stellar energy since, in the case of the Sun, calculations show that in order to produce the solar luminosity we see today, the sun would have had to contract from a size larger than the earth’s orbit only 25 million years ago.
E = mc2
A clue to the source of stellar energy was provided by Albert Einstein (1879-1955). In 1905, while developing his special theory of relativity, Einstein showed that mass can be converted into energy and vice-versa. These quantities are related by the mass-energy relation, E = mc2.
Thus, the total conversion of 1kg of matter yields an equivalent of 1x(3×108)2 = 9×1016 joules – this is approximately the energy output of a 200 MW power station running for 14 years!
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Nuclear Fusion in Stars
Nuclear fusion is the principal source of energy in stars and fusion can happen if each nucleus has sufficient kinetic energy to enable them to overcome their mutual repulsion, be captured by the strong nuclear force, and stick together. In nuclear fusion, energy is released when two light nuclei are fused together to form a heavier nucleus.
In star formation, the kinetic energy to do this comes from the conversion of gravitational energy into thermal energy by the Kelvin-Helmholtz contraction. In the case of stars like the Sun, fusion can occur when the temperature of the contracting cloud reaches about 8×106K.
It is because of the high temperatures which are needed to give the protons sufficient kinetic energy, that these nuclear reactions are also known as thermonuclear fusion reactions. It is the fusion of hydrogen nuclei by thermonuclear fusion reactions with a release of binding energy that is the primary source of energy.
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Hydrogen in Stars
Hydrogen is converted to helium and the binding energy liberated is responsible for the star’s tremendous energy output generation in stars.
A hydrogen nucleus consists of a single proton whereas helium nuclei have two protons and two neutrons. Although the fusion process involves several stages, it can be summarized as 4H → He + energy released. We know that mass of 4 H atoms = 4 x 1.008 = 4.032 amu, the mass of 1 He atom = 4.003 amu, therefore the mass loss = 4.032 – 4.003 = 0.029 amu, where amu is the atomic mass unit.
Using the mass-energy relation, the mass converted into energy is = (0.029×1.66×10-27) kg x (3×108m/s)2 = 4.33×10-12J or equivalently, 27 MeV.
Nuclear Reaction Pathways in Stars
There are two principal nuclear reaction pathways in which fusion occurs in the sun or any star. They are the proton-proton chain reaction and the CNO (carbon-nitrogen-oxygen) cycle.
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The p-p chain reaction and CNO cycle
The p-p chain reactions dominate at low core temperatures (TC<18MK). In the CNO cycle, carbon acts as a catalyst and this reaction dominate at high core Temperatures (TC>18MK). The temperature in the interior of a star determines which of these reaction pathways takes place.
For stars that have masses not exceeding that of the sun, the core temperature does not get higher than about 16 MK and hydrogen burning occurs via the proton-proton chain reaction. In stars with masses greater than that of the Sun, the core temperatures exceed 16 MK, and hydrogen burning proceeds through the CNO cycle.
In the cores of lower-mass main-sequence stars such as the Sun, the dominant energy production process is the proton-proton chain reaction. Hydrogen fuses into helium via a three-step nuclear reaction chain.
The first step in all the branches is the fusion of two protons into deuterium. As the protons fuse, one of them undergoes beta plus decay, converting into a neutron by emitting a positron and an electron neutrino.
p + p → 2H + e+ + νe
The positron will probably annihilate with an electron from the environment into two gamma rays. Including this annihilation, the whole reaction has a Q value (released energy) of 1.442 MeV. This reaction is extremely slow due to it being initiated by a weak nuclear force.
The average proton in the core of the Sun waits 9 billion years before it successfully fuses with another proton. It has not been possible to measure the cross-section of this reaction experimentally because of these longer time scales.
After it is formed, the deuterium produced in the first stage can fuse with another proton to produce the light isotope of helium, 3He
2H + p → 3He + γ + 5.49 MeV
This process, mediated by the strong nuclear force rather than the weak force, is extremely fast in comparison to the first step. It is estimated that, under the conditions in the Sun’s core, each newly created deuterium nucleus exists for only about four seconds before it is converted to helium-3.
In the Sun, each helium-3 nucleus produced in these reactions exists for only about 400 years before it is converted into helium-4. Once, the helium-3 has been produced, there are four possible paths to generate 4He.
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The p–p I branch
3He + 3He → 4He + 2 1H + 12.86 MeV
The complete p–p I chain reaction releases net energy of 26.732 MeV. Two percent of this energy is lost to the neutrinos that are produced. The p–p I branch is dominant at temperatures of 10 to 14 MK. Below 10 MK, the p–p chain does not produce much 4He.
The p–p II branch
3He + 4He → 7Be + γ
7Be + e− → 7Li− + νe + 0.861 MeV / 0.383 MeV
7Li + 1H → 2 4He
The p–p II branch is dominant at temperatures of 14 to 23 MK. Note that the energies in the equation above are not the energy released by the reaction. Rather, they are the energies of the neutrinos that are produced by the reaction.
90% of the neutrinos produced in the reaction of 7Be to 7Li carry an energy of 0.861 MeV, while the remaining 10% carry 0.383 MeV. The difference is whether the lithium-7 produced is in the ground state or an excited (metastable) state, respectively.
The p–p III branch
3He + 4He → 7Be + γ
7Be + 1H → 8B + γ
8B → 8Be + e+ + νe
8Be → 2 4He
The p–p III chain is dominant if the temperature exceeds 23 MK. The p–p III chain is not a major source of energy in the Sun (only 0.11%), but it was very important in the solar neutrino problem because it generates very high energy neutrinos (up to 14.06 MeV).
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The p–p IV branch
This reaction is predicted theoretically, but it has never been observed due to its rarity (about 0.3 ppm in the Sun). In this reaction, helium-3 captures a proton directly to give helium-4, with an even higher possible neutrino energy (up to 18.8 MeV).
3He + 1H → 4He + e+ + νe + 18.8 MeV
Comparing the mass of the final helium-4 atom with the masses of the four protons reveals that 0.7% of the mass of the original protons has been lost. This mass has been converted into energy, in the form of gamma rays and neutrinos released during each of the individual reactions. The total energy yield of one whole chain is 26.73 MeV.
The energy released as gamma rays will interact with electrons and protons and heat the interior of the Sun. Also, the kinetic energy of fusion products (e.g. of the two protons and the 4He from the p–p I reaction increases the temperature of the plasma in the Sun. This heating supports the Sun and prevents it from collapsing under its own weight.
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Neutrinos do not interact significantly with matter and therefore do not help support the Sun against gravitational collapse. Their energy is lost: the neutrinos in the p–p I, p–p II, and p–p III chains carry away 2.0%, 4.0%, and 28.3% of the energy in those reactions, respectively.
While the sun burns hydrogen by the proton-proton cycle, this is not the case for more massive stars greater than 2 solar masses, where hydrogen burning is achieved by the CNO cycle. The CNO cycle occurs in stars where the central temperatures exceed about 2.0×107K and the pressure of carbon plays an important role in converting hydrogen to helium.
The CNO cycle
In the CNO cycle, four protons fuse, using carbon, nitrogen, and oxygen isotopes as catalysts, to produce one alpha particle, two positrons, and two-electron neutrinos. It involves the following steps,
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