Editing Stellar nucleosynthesis (section)

==Reaction rate==
The reaction rate density between species ''A'' and ''B'', having number densities ''n''<sub>''A'',''B''</sub>, is given by:<math display="block">r=n_A\,n_B\,k_r </math>where ''k<sub>r</sub>'' 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 [[matter wave|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]] ''E''<sub>G</sub>, given by an [[Arrhenius equation|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''-factor]] ''S''(''E'') depends on the details of the nuclear interaction, and has the dimension of an energy multiplied by a [[Barn (unit)|cross section]].

One then integrates over all energies to get the total reaction rate, using the [[Maxwell–Boltzmann distribution#Distribution for the energy|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

<math>S(E)\,e^{-\sqrt{\frac{E_\text{G}}{E}}}\cdot2\sqrt{2/\pi}(kT)^{-3/2}\, e^{-\frac{E}{kT}}\,/\sqrt{{m_\text{R}}}.</math>

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 E<sub>0</sub>, called '''Gamow peak.'''<ref>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].</ref>{{rp|185}} 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 ''E''<sub>0</sub> as:<math display="block">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)}</math>

And the reaction rate is approximated as:<ref>{{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}}</ref><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''(''E''<sub>0</sub>) are typically {{nowrap|10<sup>−3</sup> – 10<sup>3</sup> [[keV]]·[[barn (unit)|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 star]]s (the Sun) give ''kT'' of the order of 1 keV:<ref><math>\log_{10}k =(-16-7)+\log_{10}1.3806</math> and <math>\log_{10}T= 7+\log_{10}1.57</math>: ''kT'' = 0.217 fJ = 0.135 keV</ref> <math display="inline">\log_{10}kT=-16+\log_{10}2.17</math>.<ref>Maoz, D., ''Astrophysics in a Nutshell'' ([[Princeton, New Jersey|Princeton]]: [[Princeton University Press]], 2007), [https://assets.press.princeton.edu/chapters/s3-10772.pdf ch. 3].</ref>{{rp|ch. 3}}

Thus, the limiting reaction in the [[CNO cycle]], proton capture by {{nuclide|nitrogen|14|link=yes}}, has ''S''(''E''<sub>0</sub>) ~ ''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''(''E''<sub>0</sub>) ~ ''S''(0) = 4×10<sup>−22</sup>{{nbsp}}keV·b.<ref>{{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}}</ref><ref>{{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}}</ref> 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.<ref>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].</ref>