We construct a two-zone model to describe hydrogen and helium burning on an accreting neutron star and use it to study the triggering of type Ⅰ X-ray bursts. Although highly simplified, the model reproduces all of the bursting regimes seen in the more complete global stability analysis of Narayan & Heyl, including the delayed mixed burst regime. The results are also consistent with observations. At accretion rates M/M_(Edd) approx< 0.1, helium burning via the well-known thin-shell thermal instability triggers bursts. As M increases, however, the triggering mechanism evolves from the fast thermal instability to a slowly growing overstability involving both hydrogen and helium burning. The competition between nuclear heating via the β-limited CNO cycle and the triple-α process on the one hand, and radiative cooling via photon diffusion and emission on the other, drives oscillations with a period approximately equal to the hydrogen burning timescale. If these oscillations grow, the gradually rising temperature in the helium layer will eventually provoke a thin-shell thermal instability and hence a delayed mixed burst. For M/M_(Edd) approx> 0.25, nuclear burning is stable and there are no bursts. Nearly all other theoretical models, including detailed time-dependent multizone calculations, predict that bursts should occur for all M/M_(Edd) approx< 1, in conflict both with our results and with observations. We suggest that this discrepancy arises from the assumed strength of the hot CNO cycle breakout reaction ~(15)O(α, γ)~(19)Ne in these other models. That observations agree much better with the results of Narayan & Heyl and our two-zone model, both of which neglect breakout reactions, may imply that the true ~(15)O(α, γ)~(19)Ne rate is smaller than assumed in previous investigations.
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