= Nuclear S-factor variation across a Gamow window
{title2=$\delta E\simeq\sigma_G^2\,d\log S/dE$}
The <Gaussian approximation to a nonresonant thermonuclear rate> already gives the saddle and curvature. In its notation $E_G=4E_B$, so the same result reads
$$
E_0=\left[\frac{E_G(k_BT)^2}{4}\right]^{1/3},\qquad \sigma_G^2=\frac{2E_0k_BT}{3},\qquad \Delta_{1/e}=2\sqrt2\,\sigma_G=\sqrt{\frac{16E_0k_BT}{3}}.
$$
The full width $\Delta_{1/e}$ joins the two points at $e^{-1}$ of the peak of the approximating <Gaussian function>; it is neither its <standard deviation> $\sigma_G$ nor its full width at half maximum $2\sqrt{2\log2}\,\sigma_G$. These are local <Gaussian approximation> widths, not exact equal-height endpoints of the asymmetric kernel.
Why keep the <astrophysical S factor> inside the integral? If $S(E)$ is positive and smooth, stationarity of $\log S(E)-\Phi(E)$ gives a leading peak displacement $\delta E\simeq\sigma_G^2(d\log S/dE)_{E_0}$. The slow-variation test is therefore $|d\log S/dE|\sigma_G\ll1$, together with small curvature corrections. More generally any energy-dependent prefactor shifts the saddle; a <temperature>-dependent prefactor also changes the <local temperature exponent of a thermonuclear reaction>. A sharp nuclear feature can invalidate this expansion altogether.
Choose $T=15\,\mathrm{MK}$ and $E_G=493\,\mathrm{keV}$ for an illustrative <proton> pair. Then $k_BT=1.293\,\mathrm{keV}$, $E_0=5.905\,\mathrm{keV}$ and $\Delta_{1/e}=6.380\,\mathrm{keV}$. The relative <Maxwell-Boltzmann velocity distribution> has energy-density maximum $k_BT/2=0.646\,\mathrm{keV}$, far below the reaction kernel's maximum. Now choose the deliberately artificial enhancement
$$
\frac{S(E)}{S_0}=1+5\exp\left[-\frac{(E-10\,\mathrm{keV})^2}{2(0.7\,\mathrm{keV})^2}\right].
$$
Numerical integration raises the rate by about $54.2\%$ at the same <temperature>, even though the enhancement is away from $E_0$. This is a sensitivity experiment, not a claimed <proton>–<proton> nuclear level or a physical nuclear line-shape model. It illustrates why extrapolating $S(E_0)$ without checking nearby structure can fail.
\Image[/stellar-gamow-window-s-factor.png]
{title=Particle energies, the Gamow kernel, and a chosen S-factor enhancement}
{description=Original analytic curves at 15 MK. Left: separately peak-normalized relative Maxwell energy density and constant-S reaction kernel; shading denotes the approximate full Gaussian e-minus-one width. Right: both integrands use the constant-S peak as their common denominator, so the extra area represents a rate increase. The localized S-factor feature is an artificial sensitivity test.}
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