Approach to Equilibrium in the CNO Bi-Cycle

Abstract

Differential equations of the CNO bi-cycle are solved to obtain the abundances of carbon, nitrogen, and oxygen nuclei as functions of protons consumed per initial nucleus for hydrogen burning in the temperature range 10-50 x 106 K. General features of the curves of abundance ratios for C12, C'3, N14, N'2, and 016 nuclei as functions of protons consumed are found to be insensitive to the temperature at which the reactions take place. The resonance at 65 keV in the O' (p, a) N'4 reaction is responsible for the temperature dependence of ratios involving O' nuclei. Analysis is made of the effect of convection with microscopic mixing on CN material passing through a hydrogen-burning zone. If a simple linear relationship or a sinusoidal relationship is assumed for the time dependence of temperature, the ratios of abundances attained at the surface in a given time are the same whether the time is spent in a single transit to the surface or in several transits back and forth from the bottom of the hydrogen-burning zone to the surface of the star. It is shown that the C"-C" ratio observed in typical carbon stars can be explained on the basis of the CNO bi-cycle if the time spent in the process is at least one-half the mean lifetime of C" for proton capture, but that, in order to explain the ratio of nitrogen to oxygen suggested by astronomers for these stars, it is necessary to assume that the time spent by the nuclei at the site of hydrogen burning must be no longer than the mean lifetime of C" for proton capture. In the Appendix a table is given of contributions of several resonant levels in the compound nuclei involved in the proton-capture reactions of C", C", and N'4. These contributions are presented in the form of equations for the temperature dependence of the mean lifetimes of the reactions.