Stellar nucleosynthesis

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. showing nucleosynthesis and elements formed. ==Key reactions==

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:r=n_A\,n_B\,k_r where kr is the reaction rate constant of each single elementary binary reaction composing the nuclear fusion process;k_r=\langle\sigma(v)\,v\ranglewhere 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 \pi\,\lambda^2, where \lambda =h/p is the de Broglie wavelength. Thus semi-classically the cross section is proportional to \frac{E}{m} =c^{2}. 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:\sigma(E) = \frac{S(E)}{E} e^{-\sqrt{\frac{E_\text{G}}{E}}}.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:\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)dEwhere k = 86,17 μeV/K, m_\text{R} =\frac{m_Am_B}{m_A+m_B} is the reduced mass. The integrand equals S(E)\,e^{-\sqrt{\frac{E_\text{G}}{E}}}\cdot2\sqrt{2/\pi}(kT)^{-3/2}\, e^{-\frac{E}{kT}}\,/\sqrt{{m_\text{R}}}. Since this integration of f(E, constant T) has an exponential damping at high energies of the form \sim e^{-\frac{E}{kT}} and at low energies from the Gamow factor, the integral almost vanishes everywhere except around the peak at E0, called Gamow peak. There:-\frac{\partial}{\partial E} \left(\sqrt{\frac{E_\text{G}}{E}}+\frac{E}{kT}\right)\,=\, 0 Thus: E_0 = \left(\frac{1}{2}kT \sqrt{E_\text{G}}\right)^\frac{2}{3} and \sqrt{E_\text{G}}=E_0^\frac{3}{2}/\frac{1}{2}kT The exponent can then be approximated around E0 as:e^{-(\frac{E}{kT}+\sqrt{\frac{E_\text{G}}{E}})}\approx e^{-\frac{3E_0}{kT}}e^{\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)} And the reaction rate is approximated as:\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}} 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: \log_{10}kT=-16+\log_{10}2.17. 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==