= Resonant triple-alpha temperature sensitivity
{title2=$r_{3\alpha}\propto n_\alpha^3T^{-3}e^{-E_r/(k_BT)}$}
In a dilute thermal <helium> gas with rapid formation–decay balance of <beryllium-8>, write its small equilibrium abundance schematically as $n_8\propto n_\alpha^2T^{-3/2}\exp[-\Delta_8/(k_BT)]$. A narrow isolated <Hoyle state> gives a second-step coefficient proportional to $T^{-3/2}\exp[-\Delta_C/(k_BT)]$. Multiplication yields
$$
r_{3\alpha}\propto n_\alpha^3T^{-3}\exp\left[-\frac{E_r}{k_BT}\right],\qquad E_r=\Delta_8+\Delta_C,
$$
with the stabilizing decay strength absorbed into the coefficient. Here $k_B$ is the <Boltzmann constant>. The two positive offsets count <energy> relative to their respective entrance thresholds, not relative to the <carbon-12> <ground state>. A resonance enhances capture at the relevant <energy>; the <Boltzmann factor> still selects how many encounters reach it. This is a sequential narrow-level approximation with fixed nuclear widths, excluding <plasma screening> and changes of composition or <mass density>.
<William Alfred Fowler> lists the <Hoyle state> $7.654\,\mathrm{MeV}$ above the <carbon-12> <ground state>, $0.2875\,\mathrm{MeV}$ above <beryllium-8> plus an <alpha particle>, and $0.3794\,\mathrm{MeV}$ above three <alpha particles>.<Experimental and Theoretical Nuclear Astrophysics; the Quest for the Origin of the Elements by William Alfred Fowler>[II, p. 179]{ref} Subtraction gives $\Delta_8=0.0919\,\mathrm{MeV}$ for the diagram.
Normalize the calculated rate at $T_0=10^8\,\mathrm K$ and $E_r=0.3794\,\mathrm{MeV}$. Holding <alpha particle> <number density> and the decay strength fixed gives the original sensitivity calculation
$$
\frac{r(T,E_r+\delta E)}{r(T_0,E_r)}
=\left(\frac{T}{T_0}\right)^{-3}
\exp\left[-\frac{E_r+\delta E}{k_BT}+\frac{E_r}{k_BT_0}\right].
$$
At $T_0$, a deliberately chosen shift $\delta E=+100\,\mathrm{keV}$ multiplies the rate by $9.12\times10^{-6}$; $-100\,\mathrm{keV}$ gives $1.10\times10^5$. These offsets are hypothetical perturbations, not alternative measured nuclear levels. The reference curve has <local temperature exponent of a thermonuclear reaction> $-3+E_r/(k_BT_0)=41.03$: raising <temperature> by $10\%$ gives a factor $41.12$ in this fixed-environment calculation. A real <star> changes its <stellar structure> and composition as it burns, so these factors do not directly predict final <carbon> abundance or a constraint on life.
\Image[/triple-alpha-resonance-temperature-selection.png]
{title=Triple-alpha energy thresholds and calculated resonance sensitivity}
{description=Original energy bookkeeping using Fowler's rounded offsets (1983 lecture, p. 179). The normalized narrow-resonance rate calculation holds alpha density and decay strength fixed. The plus/minus 100 keV shifts are deliberately chosen sensitivity tests. No plotted curve is laboratory data. Bound carbon levels lie below the diagram's energy window; the downward arrow indicates stabilization rather than a measured lifetime or branching fraction.}
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