Stellar nucleosynthesis

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,^[1] who later refined it in 1954.^[2] 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,^[3] which became one of the most heavily cited papers in astrophysics history.

In 1920,

Hoyle's theory was extended to other processes, beginning with the publication of the 1957 review paper "Synthesis of the Elements in Stars" by Burbidge, Burbidge, Fowler and Hoyle, more commonly referred to as the B²FH paper.^[3] This review paper collected and refined earlier research into a heavily cited picture that gave promise of accounting for the observed relative abundances of the elements; but it did not itself enlarge Hoyle's 1954 picture for the origin of primary nuclei as much as many assumed, except in the understanding of nucleosynthesis of those elements heavier than iron by neutron capture. Significant improvements were made by Alastair G. W. Cameron and by Donald D. Clayton. In 1957 Cameron presented his own independent approach to nucleosynthesis,^[14] informed by Hoyle's example, and introduced computers into time-dependent calculations of evolution of nuclear systems. Clayton calculated the first time-dependent models of the s-process in 1961^[15] and of the r-process in 1965,^[16] as well as of the burning of silicon into the abundant alpha-particle nuclei and iron-group elements in 1968,^[17][18] and discovered radiogenic chronologies^[19] for determining the age of the elements.

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

Proton–proton chain reaction

CNO-I cycle
The helium nucleus is released at the top-left step.

Hydrogen fusion (nuclear fusion of four protons to form a

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 4×10⁶ K,^[30] making it the dominant fusion mechanism in smaller stars. A self-maintaining CNO chain requires a higher temperature of approximately 1.6×10⁷ K, but thereafter it increases more rapidly in efficiency as the temperature rises, than does the proton–proton reaction.^[31] Above approximately 1.7×10⁷ K, the CNO cycle becomes the dominant source of energy. This temperature is achieved in the cores of main-sequence stars with at least 1.3 times the mass of the Sun.^[32] The Sun itself has a core temperature of about 1.57×10⁷ K.^[33]: 5 As a main-sequence star ages, the core temperature will rise, resulting in a steadily increasing contribution from its CNO cycle.^[25]

Helium fusion

Main sequence stars accumulate helium in their cores as a result of hydrogen fusion, but the core does not become hot enough to initiate helium fusion. Helium fusion first begins when a star leaves the

In all cases, helium is fused to carbon via the triple-alpha process, i.e., three helium nuclei are transformed into carbon via ⁸Be.^[35]: 30 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.^[36]: 398

Reaction rate

The reaction rate density between species A and B, having number densities n_A,B, is given by:

$${\displaystyle r=n_{A}\,n_{B}\,k}$$

$${\displaystyle k=\langle \sigma (v)\,v\rangle }$$

Semi-classically, the cross section is proportional to ${\displaystyle \pi \,\lambda ^{2}}$, where ${\displaystyle \lambda =h/p}$ is the de Broglie wavelength. Thus semi-classically the cross section is proportional to ${\textstyle {\frac {m}{E}}}$.

However, since the reaction involves

:

$${\displaystyle \sigma (E)={\frac {S(E)}{E}}e^{-{\sqrt {\frac {E_{\text{G}}}{E}}}}}$$

where S(E) depends on the details of the nuclear interaction, and has the dimension of an energy multiplied for a cross section.

One then integrates over all energies to get the total reaction rate, using the Maxwell–Boltzmann distribution and the relation:

$${\displaystyle {\frac {r}{V}}=n_{A}n_{B}\int _{0}^{\infty }{\frac {S(E)}{E}}\,e^{-{\sqrt {\frac {E_{\text{G}}}{E}}}}2{\sqrt {\frac {E}{\pi (kT)^{3}}}}e^{-{\frac {E}{kT}}}\,{\sqrt {\frac {2E}{m_{\text{R}}}}}dE}$$

${\displaystyle m_{\text{R}}={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}}$

Since this integration has an exponential damping at high energies of the form ${\displaystyle \sim e^{-{\frac {E}{kT}}}}$ and at low energies from the Gamow factor, the integral almost vanished everywhere except around the peak, called Gamow peak,^[37]: 185 at E₀, where:

$${\displaystyle {\frac {\partial }{\partial E}}\left(-{\sqrt {\frac {E_{\text{G}}}{E}}}-{\frac {E}{kT}}\right)\,=\,0}$$

Thus:

$${\displaystyle E_{0}=\left({\frac {1}{2}}kT{\sqrt {E_{\text{G}}}}\right)^{\frac {2}{3}}}$$

The exponent can then be approximated around E₀ as:

$${\displaystyle e^{-{\frac {E}{kT}}-{\sqrt {\frac {E_{\text{G}}}{E}}}}\approx e^{-{\frac {3E_{0}}{kT}}}\exp \left(-{\frac {(E-E_{0})^{2}}{{\frac {4}{3}}E_{0}kT}}\right)}$$

And the reaction rate is approximated as:^[38]

$${\displaystyle {\frac {r}{V}}\approx n_{A}\,n_{B}\,{\frac {4{\sqrt {2}}}{\sqrt {3m_{\text{R}}}}}\,{\sqrt {E_{0}}}{\frac {S(E_{0})}{kT}}e^{-{\frac {3E_{0}}{kT}}}}$$

Values of S(E₀) are typically 10⁻³ – 10³

Citations