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Hydrogen Fusion Simulator Results

Temperature

For the default abundances, one sees that the PP processes dominate the generation of helium-4 for temperatures below about 19 million degrees Kelvin, which is only slightly above the solar core temperature. The rise in the dominance of the CNO 1 cycle with temperature is very dramatic, going from producing less than 2% of the helium-4 at 15 million °K, to producing about 50% of the helium at 20 million °K, to producing over 90% of the helium-4 at 25 million °K. At a temperature of 33 million °K, the dominant PP chain, PP 3, has fallen below the CNO 2 cycle in its production of helium-4, which itself is providing less than 0.2% of the helium-4 that the CNO 1 cycle is producing.

Setting the nucleon parts to one-tenth their default values, we expect the cross-over temperature from PP domination to CNO domination to rise. If you ran simulations for the default values of the abundances, you may have found that changing the temperature from 20 to 25 million degrees Kelvin increases the rate at which the CNO 1 process creates energy by a factor of about 30. Because the rate of energy generation in the CNO1 process in equilibrium is proportional to the density of carbon, nitrogen, and oxygen, one expects the temperature for parity between the PP processes and the CNO processes at abundances of one-tenth the default values to be below 25 million degrees Kelvin. With the simulator, one finds that this equality occurs around 24 million degrees Kelvin.

Decrease the abundances of carbon, nitrogen, and oxygen to 1% of their default values, and one finds that the CNO cycle becomes dominate at 30 million degrees Kelvin. This suggests that even early in the history of our universe, before the stars had time to create large amounts of carbon and oxygen, that the CNO cycle dominated energy generation in the most massive stars.

Composition and Equilibrium

The pages covering the results of the PP and the CNO simulators discuss the equilibrium that various isotopes achieved before hydrogen is fully converted into helium. For the PP chains, deuterium, lithium-7, and beryllium-7 rapidly come to their equilibrium values, while the helium-3 comes to equilibrium over longer times. For the CNO cycles, the isotopes of the CNO 1 cycle—carbon-12, carbon-13, nitrogen-14, and nitrogen-15—always reached equilibrium, while the isotopes of the slower CNO 2 cycle reached equilibrium for sufficiently-low abundances of carbon and nitrogen, but not for high carbon abundances.

With competition from the PP chains in creating helium-4, one does not expect the isotopes of the CNO 1 cycle to always reach equilibrium. Using the simulator, one finds that equilibrium is achieved when the temperature is 11 million degrees Kelvin or greater. This result is independent of the abundances of carbon, nitrogen, and oxygen. This result shows that the CNO 1 cycle can change the density of carbon-12 and nitrogen-14 at the core of a star that is powered by the PP chains.

For initial abundances of carbon, nitrogen, and oxygen that are a factor of ten lower than the default values, the CNO 2 cycle has time to alter the density of oxygen before the CNO 1 cycle fully consumes the hydrogen in a star. For abundances set to 1% of their default values, the abundances of the oxygen isotopes are driven to their equilibrium values.

Power

The power produced by hydrogen fusion increases rapidly with temperature. With the default values for the composition, the total power increases by slightly less than a decade in going from 10 to 15 million ° K, with the power in neutrinos increasing by a comparable factor. Increasing the temperature by another 5 million °K raises the total energy production by another a decade, and it raises the neutrino emission by more than this, by about a factor of 30, as the PP 3 chain becomes the dominant PP chain. The truly striking increase in energy production, however, occurs when the temperature rises from 20 to 25 million ° K, where the CNO 1 cycle dominates the energy production; the power increases by a factor of 30. The rate of increase in power then slows as the temperature rises to 40 million degrees Kelvin.

This rapid rise in power as the temperature rises from the PP regime to the CNO regime has an equivalent effect on the time required to burn a substantial fraction of hydrogen.

At the default composition, the loss of energy to neutrinos is never great, because the PP 3 chain, which is the principle source of neutrino power, never becomes the dominant energy producing mechanism; at the high temperatures where PP 3 dominates PP 1 and PP 2, the CNO 1 cycle dominates all three PP chains. For smaller abundances of carbon, nitrogen, and oxygen, however, one can open a range of temperatures where PP 3 dominates energy production. If the abundances of carbon, nitrogen, and oxygen are set to one-tenth of their default values in the simulator, one finds that the fraction of energy carried by neutrinos peaks at around 20% for 22 million °K. For very high temperatures, where the CNO cycles dominates, the fraction of the energy carried by neutrinos drops below 10%. For very low temperatures, where the PP 1 chain dominates, the energy carried by neutrinos drops close to 2%. As the abundances of carbon, nitrogen, and oxygen fall, the upper temperature limit for the domination of PP 3 in the production of energy rises.

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