The Astrophysics Spectator
The Astrophysics Spectator
The reactions of the CNO 1 cycle are as a whole much faster than the reactions involving oxygen in the CNO 2 cycle. This appears as two sets of equilibria: the first occurring when carbon-12 and 13 plateau at a low value when nitrogen-14 reaches a peak value; and the second occurring when oxygen-16 reaches a peak value, while nitrogen-14 falls from its peak, and carbon-12 and 13 fall to lower values. Because the isotopes of the CNO 1 achieve equilibrium faster than the isotopes of the CNO 2, the CNO 1 cycle dominates the conversion of hydrogen into helium and the liberation of nuclear energy. Both of these points are clear in the ?Power? and ?Processes? plots.
For a temperature of 25 million degrees and the default compositions, the isotopes of the CNO 1 cycle are in equilibrium at just over 105 years. A carbon-12 nucleus therefore completes one CNO 1 cycle and converts four protons into a helium-4 nucleus on this time scale. Because the number of carbon-12 nuclei started at about 10-4/12, each nucleus must complete about 105 cycles to completely convert the hydrogen into helium. This occurs at just over 1010 years. From the composition plot, one sees that this occurs.
The isotopes in the CNO 2 cycle achieve equilibrium at 25 million degrees at about 1010, just before the hydrogen is destroyed through the CNO 1 process. This new equilibrium lowers the number densities of the isotopes of the CNO 1 cycle, which suppresses the rate of energy generation and the rate of conversion of hydrogen into helium. This can be seen very clearly in the ?Power? plot, where the rate of power generation drops-off at 109 years.
From these results for the default composition at 25 million degrees Kelvin, we expect the CNO 2 cycle to never achieve equilibrium if the isotopes of the CNO 1 cycle are at least ten times more abundant . For instance, setting the initial nucleon part of carbon-12 to 0.001 while keeping all other nucleon parts at their default values produces a full conversion of hydrogen into helium just as oxygen-16 is reaching equilibrium. Setting carbon-12 to 0.01 gives a plot with oxygen-16 far from equilibrium when the hydrogen is consumed; this is clear, because the abundance of nitrogen-14 remains at its peak. At the other extreme, a value of 0.00001 causes the oxygen-16 to reach equilibrium long before the hydrogen is exhausted.
Once the CNO isotopes reach equilibrium, the rate at which hydrogen is proportional to the amount of hydrogen. This means that the amount of hydrogen falls exponentially with time. The CNO cycles therefore deplete hydrogen to completion much more rapidly than the PP processes, because the hydrogen density under these latter processes false inversely with time. The exponential fall-off of hydrogen is apparent in the simulations for a sufficiently-high temperature.
Keeping the composition fixed and varying the temperature primarily affects the time scale of all processes by roughly the same factor. The time scales change dramatically with temperature, so, while the hydrogen burns to helium on a time scale of 1011 years at 25 million degrees Kelvin, it burns on a time scale of 109 year at a temperature of 35 million degrees Kelvin. This strong behavior is not unexpected if one looks at the reaction rates for these processes. All of the reaction rates rise dramatically with temperature, much more rapidly than the rates of the PP processes. The rates rise by about the same factor for every process in the CNO cycles, so the equilibrium time scales change by the same fact as the time scale for the complete burning of hydrogen.
The temperature has a influence on the final nitrogen and oxygen composition of the gas. For a temperature of 20 million degrees Kelvin, with the composition set at the default levels, the equilibrium levels of nitrogen-14 and oxygen-16 in a simulation are equal, while the nucleon density of carbon-12 is about 500 times less. Raise the temperature to 25 million degrees Kelvin in the simulation, and one finds that the nucleon density of oxygen-16 is about half that of nitrogen-14, while the nucleon density of carbon-12 is about 100 times less than nitrogen-14. Increase the temperature again to 40 million degrees Kelvin, and one sees that the nucleon density of oxygen-16 drops to about a third of the nucleon density of nitrogen-14, and the nucleon density of carbon-12 rises to about 50 times less than the nucleon density of nitrogen-14.
The power curves show a common behavior through the evolution of the gas: the power created by the gas is massive until the isotopes of the CNO 1 cycle reach equilibrium, after which the power is emitted at a constant rate until either the hydrogen is exhausted or the equilibrium of the CNO 1 isotopes is modified by the equilibrium with the CNO 2 isotopes. When the isotopes of CNO 2 reach equilibrium, the power is generated at a lower rate than previously.
