E]$ )

For a nonresonant charged-particle thermonuclear reaction rate, the thermal high-energy tail favors low

E

while the Coulomb barrier penetration factor favors high

E

. Their product peaks in an intermediate energy window. Resonant reactions such as the Triple-alpha process require an additional resonance treatment.

Table of contents
- Gaussian approximation to a nonresonant thermonuclear rate Gamow peak
- Nuclear S-factor variation across a Gamow window Gamow peak

Gaussian approximation to a nonresonant thermonuclear rate ( $R \propto T^{- 2/3} exp [- 3 (E_{B} / k_{B} T)^{1/3}]$ )

 0  0

For a slowly varying nuclear factor and penetration factor

exp [- 2 E_{B} / E]

, the thermal integral is

\int_{0}^{\infty} e^{- Φ (E)} d E

, where

Φ (E) = E / (k_{B} T) + 2 E_{B} / E

. The Gamow peak occurs at

E_{0} = [E_{B} (k_{B} T)^{2}]^{1/3}

, with

Φ (E_{0}) = 3 E_{0} / (k_{B} T)

and

Φ^{''} (E_{0}) = 3/ (2 k_{B} T E_{0})

. The Gaussian integral gives

4 π k_{B} T E_{0} /3 e^{- 3 E_{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} ≫ k_{B} T

and excludes strong resonances or rapidly varying nuclear factors.

Nuclear S-factor variation across a Gamow window ( $δ E ≃ σ_{G}^{2} d lo g S / d E$ )

 0  0

The Gaussian approximation to a nonresonant thermonuclear rate already gives the saddle and curvature. In its notation

E_{G} = 4 E_{B}

, so the same result reads

E_{0} = [\frac{E _{G} ( k _{B} T ) ^{2}}{4}]^{1/3}, σ_{G}^{2} = \frac{2 E _{0} k _{B} T}{3}, Δ_{1/ e} = 22 σ_{G} = \frac{16 E _{0} k _{B} T}{3} .

(1)

The full width

Δ_{1/ e}

joins the two points at

e^{- 1}

of the peak of the approximating Gaussian function; it is neither its standard deviation

σ_{G}

nor its full width at half maximum

2 2 lo g 2 σ_{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

lo g S (E) - Φ (E)

gives a leading peak displacement

δ E ≃ σ_{G}^{2} (d lo g S / d E)_{E_{0}}

. The slow-variation test is therefore

∣ d lo g S / d E ∣ σ_{G} ≪ 1

, 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 MK

and

E_{G} = 493 keV

for an illustrative proton pair. Then

k_{B} T = 1.293 keV

E_{0} = 5.905 keV

and

Δ_{1/ e} = 6.380 keV

. The relative Maxwell-Boltzmann velocity distribution has energy-density maximum

k_{B} T /2 = 0.646 keV

, far below the reaction kernel's maximum. Now choose the deliberately artificial enhancement

\frac{S ( E )}{S _{0}} = 1 + 5 exp [- \frac{( E - 10 keV ) ^{2}}{2 ( 0.7 keV ) ^{2}}] .

(2)

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