eqburn.f

EQBURN gives explicit hydrogen and helium burning rates for one shell or one convection zone, assuming that $^3$He and the CN cycle are in equilibrium. Its comment says the result is "an initial guess at the burning rates at the start of the time step," which a fully implicit non-equilibrium calculation can supplement later.

• Who calls it: MIX (twice, to use the explicit hydrogen burning rate) and XTIME, which turns DXDT into a nuclear time step limit.
• Inputs: the reaction-rate arrays HR1…HR12, which both callers compute with RATES just before the call. KEMCOM's header gives their units: rates "per gigayear per amu," to be multiplied by the reactants' mass fractions and the product's atomic weight; so DXDT and the other outputs are changes of mass fraction per Gyr. Also: the shell masses HS2, the temperatures HT ($\log T$), and the range IBEGIN..IEND. The abundances come from the previous model, HCOMPP in OLDMOD_MOD.
• Outputs: the rates DXDT, DYDT, DCDT, DODT (hydrogen, helium, $^{12}$C, $^{16}$O) and the zone's mean X, Z, XC12, XO16.
• Modules: NMLPAR_MOD gives the temperature cutoffs TCUT(1..4) (cutoffs in $\log T$, settings read in PARMIN; the defaults are 6.5, 6.5, 6.82, 7.7; formerly /CTLIM/); OLDMOD_MOD (formerly /OLDMOD/) holds the previous model, written by the main program and read by many routines (listed on the modules page).

1. Below TCUT(1) at the zone's first shell, all rates are zero.
2. For a convection zone (IBEGIN ≠ IEND), abundances $X_j$ and rates $R_k$ are mass-weighted means over the zone, $\bar q = \sum_i q_i m_i / \sum_i m_i$ with $m_i$ = HS2.

3. pp chains (above TCUT(2), if $X > 10^{-10}$). The equilibrium $^3$He abundance solves $2R_{33}x^2 + R_{34}Y x - R_{11}X^2 = 0$: $$x_{3,\mathrm{eq}} = \frac{\sqrt{(R_{34}Y)^2 + 8R_{33}R_{11}X^2} - R_{34}Y}{4R_{33}},$$ with $R_{11}$ = GR1, $R_{33}$ = GR2, $R_{34}$ = GR3. The code takes $x_3 = \min(x_3, x_{3,\mathrm{eq}})$ "to avoid spuriously high burning rates in outer layers," and $$\dot X = -3R_{11}X^2 + 2R_{33}x_3^2 - R_{34}Y x_3 .$$

4. CN cycle (above TCUT(3), and only if step 3 ran). In equilibrium, $X_{12} = X_{14}R_{14}/R_{12}$ and $X_{13} = X_{14}R_{14}/R_{13}$, with $R_{12}, R_{13}, R_{14}$ = GR4, GR5, GR6, and the number of CN nuclei $S = X_{12}/12 + X_{13}/13 + X_{14}/14$ is conserved, so $$X_{14} = \frac{S}{R_{14}/(12R_{12}) + R_{14}/(13R_{13}) + 1/14}.$$ Then $\dot X \mathrel{-}= X\,(R_{12}X_{12} + R_{13}X_{13} + 2R_{14}X_{14} + 2R_{16}X_{16})$, with $R_{16}$ = GR7 ($^{16}$O + p). Two protons are counted for $^{16}$O + p as for $^{14}$N + p, since each rate stands for a pair of reactions: KEMCOM's comments name them "N14,P + N15,P" and "O16,P + O17,P" (kemcom.f, lines 161–164↗), and its network counts them the same way (settled from the code, Nov 9, 2026).

   MIX expects the equilibrium CN abundances. XC12 is overwritten by its CN-equilibrium value in step 4 (line 199↗) and returned to the caller; MIX (at ITLVL = 1) sets HCOMP(5,·) to XC12 + DCDT*DT from it where helium burns (mix.f, lines 311–314↗; mix.f, lines 337–346↗), and XTIME uses only DXDT. That is consistent with MIX's comment that its explicit step solves for the rates "ASSUMING EQUILIBRIUM HE3 AND CN CYCLE ABUNDANCES" (mix.f, lines 286–289↗), though the comment is about the rates and does not say that the equilibrium ¹²C should be stored. The helium-burning step also uses this equilibrium XC12 for $^{12}$C + $\alpha$ (line 208↗); where $X \le 10^{-10}$ or $T$ is below TCUT(2), no CN cycle runs (line 169↗) and the actual abundance passes through. 5. Helium burning (above TCUT(4), if $Y > 10^{-10}$): with $r_{3\alpha} = R_{3\alpha}Y^3$ (GR12) and $r_{12\alpha} = R_{12\alpha}YX_{12}$ (GR10), $$\dot Y = -4(r_{12\alpha} + 3r_{3\alpha}),\quad \dot X_{12} = 12(r_{3\alpha} - r_{12\alpha}),\quad \dot X_{16} = 16\,r_{12\alpha}.$$

Physics Options lists TCUT(2) and TCUT(3) under "Unused parameters," as the former thresholds for pp-chain and CNO-cycle equilibrium; TCUT(1) (6.5) is the threshold for computing $\epsilon_\mathrm{nuc}$.