 Gaussian approximation to a nonresonant thermonuclear rate

For a slowly varying nuclear factor and penetration factor $\exp [-2\sqrt{E_B/E}]$, the thermal integral is ${\int }_0^{\infty }{e}^{-\Phi (E)}dE$, where $\Phi (E)=E/(k_{B}T)+2\sqrt{E_B/E}$. The Gamow peak occurs at $E_0=[E_B(k_{B}T{)}^2{]}^{1/3}$, with $\Phi (E_0)=3E_0/(k_{B}T)$ and ${\Phi }^{\prime \prime }(E_0)=3/(2k_{B}T{E}_0)$. The Gaussian integral gives $\sqrt{4\pi k_{B}T{E}_0/3}{e}^{-3E_0/(k_{B}T)}$. Combined with the Maxwellian prefactor $(k_{B}T{)}^{-3/2}$ this produces the displayed thermonuclear reaction rate at fixed reactant density. The approximation assumes $E_B\gg k_{B}T$ and excludes strong resonances or rapidly varying nuclear factors.