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PP Fusion Simulator Results

Composition and Equilibrium

Three of the six isotopes involved in PP binary processes come into equilibrium very rapidly. Deuterium, which is destroyed at a faster rate than any other isotope in the PP processes, comes into equilibrium almost immediately. Soon after, lithium-7 comes into equilibrium, followed somewhat later by beryllium-7. The deuterium and lithium-7 come into equilibrium in much less than a year, while the time scale for beryllium-7 to come into equilibrium is on order several hundreds years. This reflects the high high reaction rates for destroying these constituents. In the PP fusion simulator, the deuterium comes into equilibrium too fast to be seen on any plot. The equilibrium values of lithium-7 and beryllium-7 fall below the lower limit of the “Composition” plot, but their effect can be seen in the “Processes” plot. In this plot, the equilibrium of lithium-7 appears in the flattening of the PP 2 curve. The equilibrium of beryllium-7 occurs when the “PP 23 Loss” curve falls below the PP 3 curve; at this point, the rate of beryllium-7 creation is offset by its destruction in the PP 3 process. This behavior persists at all temperatures, although the times at which equilibrium are achieved go down as the temperature rises. For example, in a simulation with a temperature of 30 million degrees, the beryllium-7 reaches equilibrium in a little over 1 year.

Helium-3 takes many years to reach its equilibrium value, and if the temperature is low enough, the time for equilibrium may be longer than a trillion years. With the default values for the composition, the helium-3 reaches an equilibrium after 107 years for a simulation of a 15 million degrees Kelvin gas, and it reaches an equilibrium after 104 years for a simulation of a 30 million degrees Kelvin gas, but helium-3 does not reach an equilibrium in the 1012 years covered by the simulator for a 5 million degrees Kelvin gas.

The time scale for converting hydrogen into helium is generally greater than 100 million years, and is often over 10 billion years, because the fusion rate for converting protons into deuterium is very low. Once massive conversion of hydrogen into helium commences, the rate of helium-3 production falls as the square of the hydrogen density. As a consequence, the hydrogen density falls inversely with time. Setting the temperature in the simulator to its maximum value, one sees this inverse-time behavior in the “Composition” plot, with the nucleon fraction of hydrogen falling from near unity to 0.01 as time increases from 1010 years to 1012 years.

Effects of Initial Composition

Helium is always present in a star. Because helium-3 is present in tiny amounts at early times in the simulation, it interacts preferentially with helium-4 to produce beryllium-7 for the default helium-4 abundance. The PP 1 chain is therefore weak at early times in a simulation, and during these times, helium-4 is primarily created through the PP 2 chain; this is shown in the “Processes” plot. The creation of beryllium-7 causes a loss of helium-4 that is not counterbalanced by it creation through the fusion of beryllium-7 until beryllium-7 reaches its equilibrium value, which is apparent for the simulator default values. As the density of helium-3 builds up, the PP 1 chain becomes important.

This role of helium-4 in driving fusion into the PP 2 and PP 3 chains can be seen by dropping the nucleon ratio of helium to a minimum. For example, set this value to 0.00001 for helium and 1 for hydrogen and run the simulator for the default temperature. The “Processes” plot shows that the loss of helium-4 through the creation of beryllium-7 becomes negligible, and the PP 1 chain rapidly becomes the dominant helium-4 creation mechanism.

Effects of Temperature

At low temperatures, helium-4 is created predominately through the PP 1 chain. As the temperature rises, the PP 2 and PP 3 chains become more important. For the default composition values, one sees in the “Processes” plot that the three chains produce roughly the same amount of helium-4 at about 20 million degrees Kelvin once the helium-3 reaches its equilibrium value. At higher temperatures, the PP 3 chain becomes the dominant process producing helium-4. With the temperature set to 25 million degrees Kelvin in the simulator, the PP 3 chain produces helium-4 at ten times the rate that it is produced in the PP 1 chain.

Power

The energy emission reaches a peak when the helium-3 production reaches a peak. Some of the energy is lost to neutrinos that escape immediately from a star. The neutrino energy loss is most severe with the PP 3 chain, where a neutrino created in the decay of B8 into Be8 carries away 7.2 MeV, which is 28% of the 26.73 MeV available in the conversion of four protons into a single helium-4 nucleus. In contrast, a neutrino carries away 0.26 MeV in the PP 1 chain, and a neutrino carries away 0.80 MeV in the PP 2 chain.

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