This is the 2D-native counterpart to the 3D triple-alpha particle-collision simulator. Instead of tracking individual α/⁸Be/¹²C spheres bouncing in a 3D box, it evolves the mean-field reaction-rate equation for the whole plasma directly — the same equation stellar-structure codes actually integrate:
dY/dt = -3·k(T,ρ)·Y³ (Y = He-4 mass fraction, 0..1)
k(T,ρ) = A·ρ²·T₈⁻³·exp(-44.0/T₈)·R_Hoyle
with R_Hoyle = 1 (resonance ON) or ≈0.05 (OFF, "~20× slower")
The cubic Y³ term is not cosmetic: forging carbon needs three He-4 nuclei to meet, so — exactly like any termolecular mass-action rate law — the reaction rate is third order in the reactant's own concentration. Each frame this stiff ODE is advanced with its exact analytic solution (a Bernoulli equation, Y(t)=Y₀/√(1+2·k·Y₀²·t)) rather than a fixed-step Euler/RK integrator, so the update stays numerically stable even at the temperatures where k(T) spans more than ten orders of magnitude — the huge dynamic range that makes the 3D per-collision Monte-Carlo approach so sensitive to its particle count.
The middle panel plots log₁₀(rate) against T₈ directly, the steep-slope diagnostic astrophysicists use to see the reaction "switch on" — this simulation's ε₃α ∝ T₈⁻³exp(-44.0/T₈) law is the standard helium-burning energy-generation rate (Kippenhahn & Weigert; Clayton), reused here as the actual integrator rather than just prose.
The bottom panel is a real nuclear energy-level ladder (in MeV, referenced to the ¹²C ground state): the 3-alpha threshold sits 7.275 MeV above ¹²C ground (the net Q-value), ⁸Be+α sits a further 0.092 MeV up (⁸Be's real endothermic excess over 2 free α), and the Hoyle resonance sits at 7.65 MeV — only ≈0.28 MeV above the 3-alpha threshold, which is why a resonance exists there at all. Every real ¹²C produced by the rate equation above triggers one climb-and-drop animation along this ladder, so the flash frequency you see is not decorative — it is the actual dC/dt of the ODE.
- Core temperature — sets T₈ in the exponential; the log-rate panel shows exactly how violent this dependence is.
- Plasma density — scales k(T,ρ) ∝ ρ² and sets the He-4 reservoir size N₀ used to convert the dimensionless Y, C into fuel/product counts.
- Hoyle Resonance toggle — multiplies k by R_Hoyle; switching it off reproduces the "~20× slower" capture rate quoted for the 3D model, visible immediately as a much flatter log-rate curve and far rarer ladder flashes.
Note on ⁸Be: since ⁸Be's true lifetime (~8×10⁻¹⁷ s) makes its real abundance too small to plot on any sane linear scale, the "relative equilibrium abundance" shown is a Saha-type Boltzmann suppression factor Y²·exp(-Q_Be/(k_B·T)) (Q_Be = 0.092 MeV) — physically the right shape (it tracks how thermally accessible the endothermic ⁸Be step is) but in arbitrary units, not a literal particle count.