Two colliding helium-4 nuclei (α particles) can briefly fuse into beryllium-8, but ⁸Be is not bound — it decays back to 2α in about 8.2 × 10⁻¹⁷ seconds. Carbon is only made if a third α particle strikes that ⁸Be nucleus before it falls apart:
⁴He + ⁴He ⇌ ⁸Be (endothermic, unstable)
⁸Be + ⁴He → ¹²C* (7.65 MeV) (Hoyle state — resonant capture)
¹²C* → ¹²C + γ + γ (releases 7.275 MeV net)
Fred Hoyle predicted the 7.65 MeV excited state of ¹²C in 1954 purely from the fact that carbon exists in the universe: without a resonance whose energy nearly matches ⁸Be + α, the capture cross-section would be far too small for stars to build carbon during their helium-burning phase in any reasonable time. The excited state was found experimentally soon after, confirming the prediction.
Because the reaction needs three particles to meet with the right total energy, its rate is ferociously temperature-sensitive. The standard energy-generation-rate formula for helium burning is
ε₃α ≈ 5.1×10⁸ · ρ² Y⁴ · T₈⁻³ · exp(−44.0 / T₈) erg g⁻¹ s⁻¹
T₈ = T / 10⁸ K, ρ = density, Y = helium mass fraction
— roughly ε ∝ T⁴⁰ near 10⁸ K, which is why helium burning barely happens below ~10⁸ K and runs away quickly above ~2–3 × 10⁸ K, exactly the temperature range the slider covers.
- Core temperature — sets thermal speed (∝ √T) and the fusion/capture probability per close approach, following the steep T-dependence above.
- Plasma density — how many α particles are active in the box; more particles means more frequent triple encounters.
- Hoyle Resonance toggle — turning it off drops the ⁸Be + α → ¹²C capture probability by roughly 20×, showing why the resonance is what makes stellar carbon possible at all.
Scale note: real ⁸Be lifetimes and nuclear collision times are femtoseconds; this simulation compresses them by roughly 15 orders of magnitude so the resonance window is visible, while keeping the relative physics — steep T-dependence, ⁸Be instability, and the resonance's outsized effect — intact.