CNO cycles and catalytic burning

The elementary CNO-I loop begins with carbon twelve capturing a proton to form nitrogen thirteen. A beta-plus decay produces carbon thirteen, which captures a proton to form nitrogen fourteen. Another capture makes oxygen fifteen; beta-plus decay makes nitrogen fifteen; proton capture with alpha emission finally regenerates carbon twelve. Four protons have been consumed and one helium-four nucleus produced. Two beta decays emit positrons and electron neutrinos, completing the same net weak conversion required in other hydrogen-burning routes.

The carbon nucleus appearing at both ends is part of a changing catalyst population rather than a newly created carbon yield. During transients, individual carbon, nitrogen, and oxygen isotope abundances can change substantially as nuclei travel around the cycle. If the loop remains closed and all included reactions carry one catalyst nucleus into another, their total number stays fixed. Side branches, breakout reactions, transport, and destruction of catalysts break that restricted conservation statement and must be included when they are important.

For a homogeneous parcel with fixed proton reservoir and temperature, each capture can be represented by a first-order destruction rate per catalyst nucleus. The density dependence sits inside that rate through the proton density times the thermal reactivity. Let each intermediate's mean residence time be the inverse of its effective outgoing rate. In a one-way closed loop at stationary throughput, every link carries the same number of catalyst transformations per second. A slow link therefore requires a larger inventory in its input state.

Summing the intermediate populations gives a useful expression: total catalyst count equals common throughput times the sum of residence times. This is a renewal description of sequential transformations, not an assumption that every individual nucleus completes a cycle in exactly the mean time. At ordinary CNO burning conditions the nitrogen-fourteen capture often supplies the dominant residence time, but that claim is conditional on temperature, density, and the available branches. It is not a universal statement about every explosive hydrogen-burning network.

Charged-particle capture rates can rise steeply with temperature. If they become faster than the needed beta decays, increasing them further produces little improvement in complete-cycle throughput: the weak residence times dominate the sum. This beta-limited regime can therefore be less temperature sensitive than a capture-controlled cycle. The simple cold CNO-I route may also be modified by additional proton captures on unstable nuclei, so an actual hot-CNO calculation must check its full branch topology rather than merely speed up every arrow of the original diagram.

Beta-limited flow is not beta equilibrium. The decays carry net directed flow and do not require inverse weak reactions to balance them. Similarly, a stationary distribution around a driven catalytic cycle is not thermodynamic equilibrium: hydrogen is being consumed and energy and neutrinos leave the system. Mixing can replenish fuel or redistribute catalysts, while breakout can change the conserved inventory. These effects separate the useful local residence-time model from a complete star, nova, or burst calculation.