= Astrophysical S factor
{title2=$S(E)=E\sigma(E)\exp[\sqrt{E_G/E}]$}
The astrophysical S factor separates the dominant <Coulomb barrier> suppression from a charged-particle <nuclear reaction cross-section> $\sigma(E)$:
$$
\sigma(E)=\frac{S(E)}{E}e^{-\sqrt{E_G/E}},\qquad E_G=2\mu c^2(\pi\alpha Z_1Z_2)^2.
$$
Here $E$ is relative <kinetic energy>, $\mu$ is <reduced mass>, $\alpha$ is the <fine-structure constant>, and $Z_1,Z_2$ are <atomic numbers>. The astrophysical S factor has units of <energy> times area. Factoring out a barrier does not remove all nuclear physics: nuclear levels and reaction mechanisms can still make $S(E)$ vary strongly.
For a dilute nonrelativistic <Maxwell-Boltzmann velocity distribution> of relative motion, multiplying the relative-energy density by speed and <nuclear reaction cross-section> cancels the explicit factors of $E$, giving
$$
\langle\sigma v\rangle=\sqrt{\frac{8}{\pi\mu}}(k_BT)^{-3/2}\int_0^\infty S(E)e^{-\Phi(E)}\,dE,\qquad \Phi(E)=\frac{E}{k_BT}+\sqrt{\frac{E_G}{E}}.
$$
Thus the <Gamow peak> describes a rate-integral kernel, not a histogram of reacting <stars> or even the unweighted particle population. <Plasma screening> and departures from the assumed <Maxwell-Boltzmann velocity distribution> require additional physics.
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