Stellar nucleosynthesis

right|upright=1.5|thumb|[[Logarithmic scales plot of the relative energy output () of the following fusion processes at different temperatures ():

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In astrophysics, stellar nucleosynthesis is the creation of chemical elements by nuclear fusion reactions within stars. Stellar nucleosynthesis has occurred since the original creation of hydrogen, helium and lithium during the Big Bang. As a predictive theory, it yields accurate estimates of the observed abundances of the elements. It explains why the observed abundances of elements change over time and why some elements and their isotopes are much more abundant than others. The theory was initially proposed by Fred Hoyle in 1946, Further advances were made, especially to nucleosynthesis by neutron capture of the elements heavier than iron, by Margaret and Geoffrey Burbidge, William Alfred Fowler and Fred Hoyle in their famous 1957 B2FH paper, which became one of the most heavily cited papers in astrophysics history.

Stars evolve because of changes in their composition (the abundance of their constituent elements) over their lifespans, first by burning hydrogen (main sequence star), then helium (horizontal branch star), and progressively burning higher elements. However, this does not by itself significantly alter the abundances of elements in the universe as the elements are contained within the star. Later in its life, a low-mass star will slowly eject its atmosphere via stellar wind, forming a planetary nebula, while a higher–mass star will eject mass via a sudden catastrophic event called a supernova. The term supernova nucleosynthesis is used to describe the creation of elements during the explosion of a massive star or white dwarf.

The advanced sequence of burning fuels is driven by gravitational collapse and its associated heating, resulting in the subsequent burning of carbon, oxygen and silicon. However, most of the nucleosynthesis in the mass range (from silicon to nickel) is actually caused by the upper layers of the star collapsing onto the core, creating a compressional shock wave rebounding outward. The shock front briefly raises temperatures by roughly 50%, thereby causing furious burning for about a second. This final burning in massive stars, called explosive nucleosynthesis or supernova nucleosynthesis, is the final epoch of stellar nucleosynthesis.

A stimulus to the development of the theory of nucleosynthesis was the discovery early in the 20th century of regular variations in the abundances of elements found in the universe according to the Oddo–Harkins rule. The need for a physical description was already inspired by the relative abundances of the chemical elements in the Solar System. Those abundances, when plotted on a graph as a function of the atomic number of the element, have a jagged sawtooth shape that varies by factors of tens of millions (see history of nucleosynthesis theory). This suggested a natural process that is not random. A second stimulus to understanding the processes of stellar nucleosynthesis occurred later in the same century, when it was realized by Arthur Eddington and expanded on by George Gamow and then Hans Bethe that the energy released from nuclear fusion reactions accounted for the longevity of the Sun as a source of heat and light.

History

250px|right|thumb|In 1920, [[Arthur Eddington proposed that stars obtained their energy from nuclear fusion of hydrogen to form helium and also raised the possibility that the heavier elements are produced in stars.]]

In 1920, Arthur Eddington, on the basis of the precise measurements of atomic masses by F.W. Aston and a preliminary suggestion by Jean Perrin, proposed that stars obtained their energy from nuclear fusion of hydrogen to form helium and raised the possibility that the heavier elements are produced in stars. This was a preliminary step toward the idea of stellar nucleosynthesis. In 1928 George Gamow derived what is now called the Gamow factor, a quantum-mechanical formula yielding the probability for two contiguous nuclei to overcome the electrostatic Coulomb barrier between them and approach each other closely enough to undergo nuclear reaction due to the strong nuclear force which is effective only at very short distances. In the following decade the Gamow factor was used by Robert d'Escourt Atkinson and Fritz Houtermans and later by Edward Teller and Gamow himself to derive the rate at which nuclear reactions would occur at the high temperatures believed to exist in stellar interiors.

In a 1939 paper entitled "Energy Production in Stars", Hans Bethe analyzed the different possibilities for reactions by which hydrogen is fused into helium. He defined two processes that he believed to be the sources of energy in stars. The first one, the proton–proton chain reaction, is the dominant energy source in stars with masses up to about the mass of the Sun. The second process, the carbon–nitrogen–oxygen cycle, which was also considered by Carl Friedrich von Weizsäcker in 1938, is more important in more massive main-sequence stars. These works concerned the energy generation capable of keeping stars hot. Bethe's two papers did not address the creation of heavier nuclei, however. That theory was begun by Fred Hoyle in 1946 with his argument that a collection of very hot nuclei would assemble thermodynamically into iron. Hoyle followed that in 1954 with a paper describing how advanced fusion stages within massive stars would synthesize the elements from carbon to iron in mass.

Hoyle's theory was extended to other processes, beginning with the publication of the 1957 review paper "Synthesis of the Elements in Stars" by Margaret Burbidge, Geoffrey Burbidge, William Alfred Fowler and Fred Hoyle, more commonly referred to as the B2FH paper.. In 1957 Alastair G. W. Cameron, working independently showed the Oddo–Harkins even-odd abundance rule would follow from processes outlined by the B2FH paper. Clayton calculated the first time-dependent models of the s-process in 1961 and of the r-process in 1965, as well as of the burning of silicon into the abundant alpha-particle nuclei and iron-group elements in 1968, and discovered radiogenic chronologies for determining the age of the elements.

thumb|Cross section of a [[supergiant showing nucleosynthesis and elements formed.]]

Key reactions

thumb|right|500px|A version of the periodic table indicating the origins – including stellar nucleosynthesis – of the elements. All elements above 103 are omitted from this table and they are produced only by human synthesis.

The most important reactions in stellar nucleosynthesis:

Hydrogen fusion:
Deuterium fusion
The proton–proton chain
The carbon–nitrogen–oxygen cycle
Helium fusion:
The triple-alpha process
The alpha process
Fusion of heavier elements:
Lithium burning: a process found most commonly in brown dwarfs
Carbon-burning process
Neon-burning process
Oxygen-burning process
Silicon-burning process
Production of elements heavier than iron:
Neutron capture:
The r-process
The s-process
Proton capture:
The rp-process
The p-process
Photodisintegration

Hydrogen fusion

Hydrogen fusion (nuclear fusion of four protons to form a helium-4 nucleus

In the cores of lower-mass main-sequence stars such as the Sun, the dominant energy production process is the proton–proton chain reaction. This creates a helium-4 nucleus through a sequence of reactions that begin with the fusion of two protons to form a deuterium nucleus (one proton plus one neutron) along with an ejected positron and neutrino.

The type of hydrogen fusion process that dominates in a star is determined by the temperature dependency differences between the two reactions. The proton–proton chain reaction starts at temperatures about , As a main-sequence star ages, the core temperature will rise, resulting in a steadily increasing contribution from its CNO cycle. Despite the name, stars on a blue loop from the red giant branch are typically not blue in colour but are rather yellow giants, possibly Cepheid variables. They fuse helium until the core is largely carbon and oxygen. The most massive stars become supergiants when they leave the main sequence and quickly start helium fusion as they become red supergiants. After the helium is exhausted in the core of a star, helium fusion will continue in a shell around the carbon–oxygen core. This can then form oxygen, neon, and heavier elements via the alpha process. In this way, the alpha process preferentially produces elements with even numbers of protons by the capture of helium nuclei. Elements with odd numbers of protons are formed by other fusion pathways.

Reaction rate

The reaction rate density between species A and B, having number densities nA,B, is given by:<math display="block">r=n_A\,n_B\,k_r </math>where kr is the reaction rate constant of each single elementary binary reaction composing the nuclear fusion process;<math display="block">k_r=\langle\sigma(v)\,v\rangle</math>where σ(v) is the cross-section at relative velocity v, and averaging is performed over all velocities.

Semi-classically, the cross section is proportional to <math display="inline">\pi\,\lambda^2</math>, where <math display="inline">\lambda =h/p</math> is the de Broglie wavelength. Thus semi-classically the cross section is proportional to <math display="inline">\frac{E}{m} =c^{2}</math>.

However, since the reaction involves quantum tunneling, there is an exponential damping at low energies that depends on Gamow factor EG, given by an Arrhenius-type equation:<math display="block">\sigma(E) = \frac{S(E)}{E} e^{-\sqrt{\frac{E_\text{G{E}.</math>Here astrophysical S-factor S(E) depends on the details of the nuclear interaction, and has the dimension of an energy multiplied by a cross section.

One then integrates over all energies to get the total reaction rate, using the Maxwell–Boltzmann distribution and the relation:<math display="block">\frac{r}{V}=n_A n_B \int_0^{\infty}\Bigl(\frac{S(E)}{E}\, e^{-\sqrt{\frac{E_\text{G{E} \cdot2\sqrt{\frac{E}{\pi(kT)^3\, e^{-\frac{E}{kT \,\cdot\sqrt{\frac{2E}{m_\text{R}\Bigr)dE</math>where k = 86,17 μeV/K, <math>m_\text{R} =\frac{m_Am_B}{m_A+m_B}</math> is the reduced mass. The integrand equals

Since this integration of f(E, constant T) has an exponential damping at high energies of the form <math display="inline">\sim e^{-\frac{E}{kT</math> and at low energies from the Gamow factor, the integral almost vanishes everywhere except around the peak at E0, called Gamow peak. There:<math display="block">-\frac{\partial}{\partial E} \left(\sqrt{\frac{E_\text{G{E+\frac{E}{kT}\right)\,=\, 0</math>

Thus:

<math>E_0 = \left(\frac{1}{2}kT \sqrt{E_\text{G\right)^\frac{2}{3}</math> and <math>\sqrt{E_\text{G=E_0^\frac{3}{2}/\frac{1}{2}kT </math>

The exponent can then be approximated around E0 as:<math display="block">e^{-(\frac{E}{kT}+\sqrt{\frac{E_\text{G{E)}\approx e^{-\frac{3E_0}{kTe^{\bigl(-\frac{3(E-E_0)^2}{4E_0kT}\bigr)}=e^{-\frac{3E_0}{kT}\bigl(1+(\frac{E-E_0}{2E_0})^2\bigr)}=e^{-\frac{3E_0}{kT}\bigl(1+(E/E_0-1)^2/4\bigr)}</math>

And the reaction rate is approximated as:<math display="block">\frac{r}{V} \approx n_A \,n_B \,\frac{4\sqrt(2/3)}{ \sqrt{m_\text{R} \,\sqrt{E_0}\frac{S(E_0)}{kT} \, e^{-\frac{3E_0}{kT </math>

Values of S(E0) are typically , but are damped by a huge factor when involving a beta decay, due to the relation between the intermediate bound state (e.g. diproton) half-life and the beta decay half-life, as in the proton–proton chain reaction. Note that typical core temperatures in main-sequence stars (the Sun) give kT of the order of 1 keV: <math display="inline">\log_{10}kT=-16+\log_{10}2.17</math>.

Thus, the limiting reaction in the CNO cycle, proton capture by , has S(E0) ~ S(0) = 3.5keV·b, while the limiting reaction in the proton–proton chain reaction, the creation of deuterium from two protons, has a much lower S(E0) ~ S(0) = 4×10−22keV·b. Incidentally, since the former reaction has a much higher Gamow factor, and due to the relative abundance of elements in typical stars, the two reaction rates are equal at a temperature value that is within the core temperature ranges of main-sequence stars.

References

Notes

Citations

External links

"How the Sun Shines", by John N. Bahcall (Nobel Prize site, accessed 6 January 2020)
Nucleosynthesis in NASA's Cosmicopia. Archived on 1999-01-29.