Astrophysical S factor ($S(E)=E\sigma (E)\exp [\sqrt{E_G/E}]$)

The astrophysical S factor separates the dominant Coulomb barrier suppression from a charged-particle nuclear reaction cross-section $\sigma (E)$: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 $E$, givingThus 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.