= Gaussian approximation to a nonresonant thermonuclear rate
{c}
{title2=$R\propto T^{-2/3}\exp[-3(E_B/k_BT)^{1/3}]$}
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 $\Phi(E)=E/(k_BT)+2\sqrt{E_B/E}$. The <Gamow peak> occurs at $E_0=[E_B(k_BT)^2]^{1/3}$, with $\Phi(E_0)=3E_0/(k_BT)$ and $\Phi''(E_0)=3/(2k_BTE_0)$. The <Gaussian integral> gives $\sqrt{4\pi k_BTE_0/3}e^{-3E_0/(k_BT)}$. Combined with the Maxwellian prefactor $(k_BT)^{-3/2}$ this produces the displayed <thermonuclear reaction rate> at fixed reactant density. The approximation assumes $E_B\gg k_BT$ and excludes strong resonances or rapidly varying nuclear factors.
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