stellar nucleosynthesis

File:Arthur Stanley Eddington.jpg 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.

{{cite journal

|last1=Eddington |first1=A. S.

|year=1920

|title=The internal constitution of the stars

|journal=The Observatory

|volume=43 |issue=1341

|pages=341–358

|doi=10.1126/science.52.1341.233

|pmid=17747682

|bibcode=1920Obs....43..341E

|url=https://zenodo.org/record/1429642

}}

{{cite journal

|last1=Eddington |first1=A. S.

|year=1920

|title=The Internal Constitution of the Stars

|journal=Nature

|volume=106 |issue=2653 |pages=233–240

|bibcode=1920Natur.106...14E

|doi=10.1038/106014a0

|pmid=17747682

|url=https://zenodo.org/record/1429642

|doi-access=free

}}

{{cite web

|last=Selle |first=D.

|date=October 2012

|title=Why the Stars Shine

|url=https://www.astronomyhouston.org/sites/default/files/guidestar/2012October.pdf

|work=Guidestar

|pages=6–8

|publisher=Houston Astronomical Society

|archive-url=https://web.archive.org/web/20131203024638/http://www.astronomyhouston.org/sites/default/files/guidestar/2012October.pdf

|archive-date=2013-12-03

|url-status=live

}} 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.Krane, K. S., Modern Physics (Hoboken, NJ: Wiley, 1983), [https://books.google.com/books?id=-x-VDwAAQBAJ&pg=PA410&redir_esc=y#v=onepage&q&f=false p. 410].{{rp|410}} 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 1939, in a Nobel lecture entitled "Energy Production in Stars", Hans Bethe analyzed the different possibilities for reactions by which hydrogen is fused into helium.

{{cite journal

|last1=Bethe |first1=H. A.

|year=1939

|title=Energy Production in Stars

|journal=Physical Review

|volume=55 |issue=5 |pages=434–456

|bibcode=1939PhRv...55..434B

|doi=10.1103/PhysRev.55.434

|pmid=17835673

|doi-access=free

}} 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.{{cite book|last=Lang|first=K. R.|title=The Life and Death of Stars|year=2013|publisher=Cambridge University Press|isbn=978-1-107-01638-5|page=[https://books.google.com/books?id=MN-UCkUK9pcC&pg=PA167&redir_esc=y#v=onepage&q&f=false 167]}}.{{rp|167}} These works concerned the energy generation capable of keeping stars hot. A clear physical description of the proton–proton chain and of the CNO cycle appears in a 1968 textbook.Clayton, D. D. (1968). Principles of Stellar Evolution and Nucleosynthesis. University of Chicago Press. [https://books.google.com/books?id=8HSGFThnbvkC&pg=PT365&redir_esc=y#v=onepage&q&f=false p. 365].{{rp|365}} 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.

{{cite journal

|last=Hoyle |first=F.

|year=1946

|title=The synthesis of the elements from hydrogen

|journal=Monthly Notices of the Royal Astronomical Society

|volume=106 |issue=5 |pages=343–383

|bibcode=1946MNRAS.106..343H

|doi=10.1093/mnras/106.5.343

|doi-access=free

}} 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.

{{cite journal

|last1=Clayton |first1=D. D.

|title=History of Science: Hoyle's Equation

|journal=Science

|year=2007

|volume=318 |issue=5858 |pages=1876–1877

|doi=10.1126/science.1151167

|pmid=18096793

|s2cid=118423007

}}

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. 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,{{cite report

|last=Cameron |first=A. G. W.

|year=1957

|title=Stellar Evolution, Nuclear Astrophysics, and Nucleogenesis

|url=https://fas.org/sgp/eprint/CRL-41.pdf

|publisher=Atomic Energy of Canada Limited

|id=Report CRL-41

}} 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

{{cite journal

|last1=Clayton |first1=D. D.

|last2=Fowler |first2=W. A.

|last3=Hull |first3=T. E.

|last4=Zimmerman |first4=B. A.

|year=1961

|title=Neutron capture chains in heavy element synthesis

|journal=Annals of Physics

|volume=12 |issue=3 |pages=331–408

|bibcode=1961AnPhy..12..331C

|doi=10.1016/0003-4916(61)90067-7

}} and of the r-process in 1965,

{{cite journal

|last1=Seeger |first1=P. A.

|last2=Fowler |first2=W. A.

|last3=Clayton |first3=D. D.

|year=1965

|title=Nucleosynthesis of Heavy Elements by Neutron Capture

|journal=The Astrophysical Journal Supplement Series

|volume=11 |pages=121–126

|bibcode=1965ApJS...11..121S

|doi=10.1086/190111

|url=https://tigerprints.clemson.edu/cgi/viewcontent.cgi?article=1307&context=physastro_pubs

}} as well as of the burning of silicon into the abundant alpha-particle nuclei and iron-group elements in 1968,

{{cite journal

|last1=Bodansky |first1=D.

|last2=Clayton |first2=D. D.

|last3=Fowler |first3=W. A.

|year=1968

|title=Nucleosynthesis During Silicon Burning

|journal=Physical Review Letters

|volume=20 |issue=4 |pages=161–164

|bibcode=1968PhRvL..20..161B

|doi=10.1103/PhysRevLett.20.161

|url=https://tigerprints.clemson.edu/cgi/viewcontent.cgi?article=1393&context=physastro_pubs

|url-access=subscription

}}

{{cite journal

|last1=Bodansky |first1=D.

|last2=Clayton |first2=D. D.

|last3=Fowler |first3=W. A.

|year=1968

|title=Nuclear Quasi-Equilibrium during Silicon Burning

|journal=The Astrophysical Journal Supplement Series

|volume=16 |page=299

|bibcode=1968ApJS...16..299B

|doi=10.1086/190176

|url=https://tigerprints.clemson.edu/physastro_pubs/312

|url-access=subscription

}} and discovered radiogenic chronologies

{{cite journal

|last1=Clayton |first1=D. D.

|year=1964

|title=Cosmoradiogenic Chronologies of Nucleosynthesis

|journal=The Astrophysical Journal

|volume=139 |page=637

|bibcode=1964ApJ...139..637C

|doi=10.1086/147791

|url=https://tigerprints.clemson.edu/cgi/viewcontent.cgi?article=1305&context=physastro_pubs

|url-access=subscription

}} for determining the age of the elements.

File:Nucleosynthesis in a star.gif showing nucleosynthesis and elements formed.]]

File:Nucleosynthesis periodic table.svg

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

{{Redirect|Hydrogen burning|the combustion of hydrogen gas|Hydrogen#Combustion}}

{{Main|Proton–proton chain reaction|CNO cycle|Deuterium fusion}}

{{multiple image

| align = right

| direction = vertical

| width = 300

| image1 = Fusion in the Sun.svg

| caption1 = Proton–proton chain reaction

| image2 = CNO Cycle.svg

| caption2 = CNO-I cycle
The helium nucleus is released at the top-left step.

}}

Hydrogen fusion (nuclear fusion of four protons to form a helium-4 nucleus) is the dominant process that generates energy in the cores of main-sequence stars. It is also called "hydrogen burning", which should not be confused with the chemical combustion of hydrogen in an oxidizing atmosphere. There are two predominant processes by which stellar hydrogen fusion occurs: proton–proton chain and the carbon–nitrogen–oxygen (CNO) cycle. Ninety percent of all stars, with the exception of white dwarfs, are fusing hydrogen by these two processes.Seeds, M. A., Foundations of Astronomy (Belmont, CA: Wadsworth Publishing Company, 1986), p. 245.{{rp|245}}

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. In each complete fusion cycle, the proton–proton chain reaction releases about 26.2 MeV. Proton-proton chain with a dependence of approximately T{{sup|4}}, meaning the reaction cycle is highly sensitive to temperature; a 10% rise of temperature would increase energy production by this method by 46%, hence, this hydrogen fusion process can occur in up to a third of the star's radius and occupy half the star's mass. For stars above 35% of the Sun's mass, the energy flux toward the surface is sufficiently low and energy transfer from the core region remains by radiative heat transfer, rather than by convective heat transfer. As a result, there is little mixing of fresh hydrogen into the core or fusion products outward.

In higher-mass stars, the dominant energy production process is the CNO cycle, which is a catalytic cycle that uses nuclei of carbon, nitrogen and oxygen as intermediaries and in the end produces a helium nucleus as with the proton–proton chain. During a complete CNO cycle, 25.0 MeV of energy is released. The difference in energy production of this cycle, compared to the proton–proton chain reaction, is accounted for by the energy lost through neutrino emission. CNO cycle is highly sensitive to temperature, with rates proportional to the 16th to 20th power of the temperature; a 10% increase in temperature would result in a 350% increase in energy production. About 90% of the CNO cycle energy generation occurs within the inner 15% of the star's mass, hence it is strongly concentrated at the core. This results in such an intense outward energy flux that convective energy transfer becomes more important than does radiative transfer. As a result, the core region becomes a convection zone, which stirs the hydrogen fusion region and keeps it well mixed with the surrounding proton-rich region. This core convection occurs in stars where the CNO cycle contributes more than 20% of the total energy. As the star ages and the core temperature increases, the region occupied by the convection zone slowly shrinks from 20% of the mass down to the inner 8% of the mass. The Sun produces on the order of 1% of its energy from the CNO cycle.{{Cite web|title=Neutrinos yield first experimental evidence of catalyzed fusion dominant in many stars|url=https://phys.org/news/2020-11-neutrinos-yield-experimental-evidence-catalyzed.html|access-date=2020-11-26|website=phys.org|language=en}}{{efn|In the [https://www.nature.com/articles/s41586-020-2934-0 November 2020] issue of Nature, particle physicist Andrea Pocar points out, "Confirmation of CNO burning in our sun, where it operates at only one percent, reinforces our confidence that we understand how stars work."}}Choppin, G. R., Liljenzin, J.-O., Rydberg, J., & Ekberg, C., Radiochemistry and Nuclear Chemistry (Cambridge, MA: Academic Press, 2013), [https://books.google.com/books?id=CN88gBPtiucC&pg=PA357&redir_esc=y#v=onepage&q&f=false p. 357].{{rp|357}}{{Cite journal|last1=Agostini|first1=M.|last2=Altenmüller|first2=K.|last3=Appel|first3=S.|last4=Atroshchenko|first4=V.|last5=Bagdasarian|first5=Z.|last6=Basilico|first6=D.|last7=Bellini|first7=G.|last8=Benziger|first8=J.|last9=Biondi|first9=R.|last10=Bravo|first10=D.|last11=Caccianiga|first11=B.|date=25 November 2020|title=Experimental evidence of neutrinos produced in the CNO fusion cycle in the Sun|url=https://www.nature.com/articles/s41586-020-2934-0|journal=Nature|language=en|volume=587|issue=7835|pages=577–582|doi=10.1038/s41586-020-2934-0|pmid=33239797|issn=1476-4687|arxiv=2006.15115|bibcode=2020Natur.587..577B|s2cid=227174644}}{{efn|"This result therefore paves the way toward a direct measurement of the solar metallicity using CNO neutrinos. Our findings quantify the relative contribution of CNO fusion in the Sun to be of the order of 1 per cent."—M. Agostini, et al.}}

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 {{val|4|e=6|ul=K}}, making it the dominant fusion mechanism in smaller stars. A self-maintaining CNO chain requires a higher temperature of approximately {{val|1.6|e=7|u=K}}, but thereafter it increases more rapidly in efficiency as the temperature rises, than does the proton–proton reaction. Above approximately {{val|1.7|e=7|u=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. The Sun itself has a core temperature of about {{val|1.57|e=7|u=K}}.Wolf, E. L., Physics and Technology of Sustainable Energy (Oxford, Oxford University Press, 2018), [https://books.google.com/books?id=BP9eDwAAQBAJ&pg=PA5&redir_esc=y#v=onepage&q&f=false p. 5].{{rp|5}} As a main-sequence star ages, the core temperature will rise, resulting in a steadily increasing contribution from its CNO cycle.

{{Main|Triple-alpha process|Alpha process}}

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 red giant branch after accumulating sufficient helium in its core to ignite it. In stars around the mass of the Sun, this begins at the tip of the red giant branch with a helium flash from a degenerate helium core, and the star moves to the horizontal branch where it burns helium in its core. More massive stars ignite helium in their core without a flash and execute a blue loop before reaching the asymptotic giant branch. Such a star initially moves away from the AGB toward bluer colours, then loops back again to what is called the Hayashi track. An important consequence of blue loops is that they give rise to classical Cepheid variables, of central importance in determining distances in the Milky Way and to nearby galaxies.Karttunen, H., Kröger, P., Oja, H., Poutanen, M., & Donner, K. J., eds., Fundamental Astronomy (Berlin/Heidelberg: Springer, 1987), [https://books.google.com/books?id=DjeVdb0sLEAC&pg=PA250&redir_esc=y#v=onepage&q&f=false p. 250].{{rp|250}} 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.

In all cases, helium is fused to carbon via the triple-alpha process, i.e., three helium nuclei are transformed into carbon via ⁸Be.Rehder, D., Chemistry in Space: From Interstellar Matter to the Origin of Life (Weinheim: Wiley-VCH, 2010), [https://books.google.com/books?id=baI91e8lgm0C&pg=PT30&redir_esc=y#v=onepage&q&f=false p. 30].{{rp|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.Perryman, M., The Exoplanet Handbook (Cambridge: Cambridge University Press, 2011), [https://books.google.com/books?id=ngtmDwAAQBAJ&pg=PA398&redir_esc=y#v=onepage&q&f=false p. 398].{{rp|398}}

The reaction rate density between species A and B, having number densities n_A,B, is given by:$$r=n\_A\backslash ,n\_B\backslash ,k\_r$$where k_r is the reaction rate constant of each single elementary binary reaction composing the nuclear fusion process;$$k\_r=\backslash langle\backslash sigma(v)\backslash ,v\backslash rangle$$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 $\backslash pi\backslash ,\backslash lambda^2$, where $\backslash lambda\; =h/p$ is the de Broglie wavelength. Thus semi-classically the cross section is proportional to $\backslash frac\{E\}\{m\}\; =c^\{2\}$.

However, since the reaction involves quantum tunneling, there is an exponential damping at low energies that depends on Gamow factor E_G, given by an Arrhenius-type equation:$$\backslash sigma(E)\; =\; \backslash frac\{S(E)\}\{E\}\; e^\{-\backslash sqrt\{\backslash frac\{E\_\backslash text\{G\}\}\{E\}\}\}.$$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:$$\backslash frac\{r\}\{V\}=n\_A\; n\_B\; \backslash int\_0^\{\backslash infty\}\backslash Bigl(\backslash frac\{S(E)\}\{E\}\backslash ,\; e^\{-\backslash sqrt\{\backslash frac\{E\_\backslash text\{G\}\}\{E\}\}\}\; \backslash cdot2\backslash sqrt\{\backslash frac\{E\}\{\backslash pi(kT)^3\}\}\backslash ,\; e^\{-\backslash frac\{E\}\{kT\}\}\; \backslash ,\backslash cdot\backslash sqrt\{\backslash frac\{2E\}\{m\_\backslash text\{R\}\}\}\backslash Bigr)dE$$where k = 86,17 μeV/K, $m\_\backslash text\{R\}\; =\backslash frac\{m\_Am\_B\}\{m\_A+m\_B\}$ is the reduced mass. The integrand equals

$S(E)\backslash ,e^\{-\backslash sqrt\{\backslash frac\{E\_\backslash text\{G\}\}\{E\}\}\}\backslash cdot2\backslash sqrt\{2/\backslash pi\}(kT)^\{-3/2\}\backslash ,\; e^\{-\backslash frac\{E\}\{kT\}\}\backslash ,/\backslash sqrt\{\{m\_\backslash text\{R\}\}\}.$

Since this integration of f(E, constant T) has an exponential damping at high energies of the form $\backslash sim\; e^\{-\backslash frac\{E\}\{kT\}\}$ and at low energies from the Gamow factor, the integral almost vanishes everywhere except around the peak at E₀, called Gamow peak.Iliadis, C., Nuclear Physics of Stars (Weinheim: Wiley-VCH, 2015), [https://books.google.com/books?id=kLZNCAAAQBAJ&pg=PA185&redir_esc=y#v=onepage&q&f=false p. 185].{{rp|185}} There:$$-\backslash frac\{\backslash partial\}\{\backslash partial\; E\}\; \backslash left(\backslash sqrt\{\backslash frac\{E\_\backslash text\{G\}\}\{E\}\}+\backslash frac\{E\}\{kT\}\backslash right)\backslash ,=\backslash ,\; 0$$

Thus:

$E\_0\; =\; \backslash left(\backslash frac\{1\}\{2\}kT\; \backslash sqrt\{E\_\backslash text\{G\}\}\backslash right)^\backslash frac\{2\}\{3\}$ and $\backslash sqrt\{E\_\backslash text\{G\}\}=E\_0^\backslash frac\{3\}\{2\}/\backslash frac\{1\}\{2\}kT$

The exponent can then be approximated around E₀ as:$$e^\{-(\backslash frac\{E\}\{kT\}+\backslash sqrt\{\backslash frac\{E\_\backslash text\{G\}\}\{E\}\})\}\backslash approx\; e^\{-\backslash frac\{3E\_0\}\{kT\}\}e^\{\backslash bigl(-\backslash frac\{3(E-E\_0)^2\}\{4E\_0kT\}\backslash bigr)\}=e^\{-\backslash frac\{3E\_0\}\{kT\}\backslash bigl(1+(\backslash frac\{E-E\_0\}\{2E\_0\})^2\backslash bigr)\}=e^\{-\backslash frac\{3E\_0\}\{kT\}\backslash bigl(1+(E/E\_0-1)^2/4\backslash bigr)\}$$

And the reaction rate is approximated as:{{Cite web |title=University College London astrophysics course: lecture 7 – Stars |url=https://zuserver2.star.ucl.ac.uk/~idh/PHAS2112/Lectures/Current/Part7.pdf |url-status=dead |archive-url=https://web.archive.org/web/20170115214447/https://zuserver2.star.ucl.ac.uk/~idh/PHAS2112/Lectures/Current/Part7.pdf |archive-date=January 15, 2017 |access-date=May 8, 2020}}$$\backslash frac\{r\}\{V\}\; \backslash approx\; n\_A\; \backslash ,n\_B\; \backslash ,\backslash frac\{4\backslash sqrt(2/3)\}\{\; \backslash sqrt\{m\_\backslash text\{R\}\}\}\; \backslash ,\backslash sqrt\{E\_0\}\backslash frac\{S(E\_0)\}\{kT\}\; \backslash ,\; e^\{-\backslash frac\{3E\_0\}\{kT\}\}$$

Values of S(E₀) are typically {{nowrap|10⁻³ – 10³ keV·b}}, 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:$\backslash log\_\{10\}k\; =(-16-7)+\backslash log\_\{10\}1.3806$ and $\backslash log\_\{10\}T=\; 7+\backslash log\_\{10\}1.57$: kT = 0.217 fJ = 0.135 keV $\backslash log\_\{10\}kT=-16+\backslash log\_\{10\}2.17$.Maoz, D., Astrophysics in a Nutshell (Princeton: Princeton University Press, 2007), [https://assets.press.princeton.edu/chapters/s3-10772.pdf ch. 3].{{rp|ch. 3}}

Thus, the limiting reaction in the CNO cycle, proton capture by {{nuclide|nitrogen|14|link=yes}}, has S(E₀) ~ S(0) = 3.5{{nbsp}}keV·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) = 4×10⁻²²{{nbsp}}keV·b.{{Cite journal|last1=Adelberger|first1=Eric G.|author1-link=Eric G. Adelberger|last2=Austin|first2=Sam M.|last3=Bahcall|first3=John N.|author3-link=John N. Bahcall|last4=Balantekin|first4=A. B.|author4-link=A. Baha Balantekin|last5=Bogaert|first5=Gilles|last6=Brown|first6=Lowell S.|author6-link=Lowell S. Brown|last7=Buchmann|first7=Lothar|last8=Cecil|first8=F. Edward|last9=Champagne|first9=Arthur E.|last10=de Braeckeleer|first10=Ludwig|last11=Duba|first11=Charles A.|date=1998-10-01|title=Solar fusion cross sections|journal=Reviews of Modern Physics|language=en|volume=70|issue=4|pages=1265–1291|doi=10.1103/RevModPhys.70.1265|arxiv=astro-ph/9805121|bibcode=1998RvMP...70.1265A|s2cid=16061677|issn=0034-6861}}{{Cite journal |last1=Adelberger |first1=E. G. |year=2011 |title=Solar fusion cross sections. II. The pp chain and CNO cycles |journal=Reviews of Modern Physics |volume=83 |issue=1 |pages=195–245 |arxiv=1004.2318 |bibcode=2011RvMP...83..195A |doi=10.1103/RevModPhys.83.195 |s2cid=119117147}} 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.Goupil, M., Belkacem, K., Neiner, C., Lignières, F., & Green, J. J., eds., Studying Stellar Rotation and Convection: Theoretical Background and Seismic Diagnostics (Berlin/Heidelberg: Springer, 2013), [https://books.google.com/books?id=ovO5BQAAQBAJ&pg=PA211&redir_esc=y#v=onepage&q&f=false p. 211].

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