Stars

Hydrogen Fusion Rates

The reaction rates for the numerous binary reactions that constitute the PP and CNO hydrogen fusion processes are shown in the figure below as functions of temperature. These rates are given in units that remove the dependence on density. To attain a reaction rate per unit volume, the rate that is given must be multiplied by the density of the two constituents involved in each reaction. The primary feature of the nuclear reactions in these figures is their strong dependence on temperature. With the exception of the interaction of lithium-7 with the electron, all rates rise dramatically with temperature. For the cores of main-sequence stars, the relevant temperatures are between 10 and 30 million degrees. The nuclear decays shown on the hydrogen fusion page are incorporated into the binary reaction rates.

The PP Process

Of the reaction rates in the PP process, the reaction rate for the production of deuterium from protons is dramatically smaller than any other reaction rate for temperatures above 10 million degrees. This rate rises more slowly with temperature than any of the other nucleon-nucleon rates, although it does still rise by a factor of a thousand when the temperature increases from 10 million degrees to 100 million degrees.

Although deuterium is created at a very slow rate, it is destroyed at an extremely high rate, as its reaction rate is the highest of the reactions in the PP chain. Therefore, as deuterium is created, it is immediately converted into helium-3. This keeps the abundance of deuterium in the core of a star at a negligible value.

The fate of helium-3 is more complex, because two factors come into play: the higher reaction rate of the He³ + He³ reaction over the He³ + He⁴ reaction, and the the higher density of He⁴ over He³. In general, for temperatures below 15 million degrees, the He³ + He³ reaction is dominant, while for higher temperatures, the He⁴ + He³ reaction is dominant. The reason is that as the temperature rises the reaction rates for destroying He³ increase more rapidly than the reaction rate for creating He³; as a consequence, the equilibrium abundance of He³ falls as the temperature rises, decreasing the rate of the He³ + He³ reaction relative to the He⁴ + He³ reaction.

Of the two PP branches that pass through the Li⁷ stage, the dominant branch is delineated by the crossing point of the Li⁷ + e^- and the Li⁷ + H¹ reaction rates; the former is dominant below 20 million degrees, and the latter is dominant above this temperature.

The CNO Process

The CNO process has reaction rates that rise dramatically with temperature. Of the two cycles, the cycle that converts N¹⁵ to C¹² is the fastest. It has a much faster rate than the reaction that creates deuterium for temperatures above 10 million degrees. The issue is then the abundance of elements such as C¹². These elements are not present early in the universe, so the first stars could not utilize this process. With time, as older stars created and threw carbon and oxygen into space, stars on the main sequence were born with sufficient carbon and oxygen to permit the CNO process. Because the CNO process is a pair of cycles, and because the reaction rates in these cycles are similar, the cycle produces an equilibrium abundance of carbon, nitrogen, and oxygen from the initial abundances. Only one of these elements need have at a high abundance for the CNO cycle to commence. About 0.1% of the Sun's atoms are carbon, nitrogen, and oxygen. With these abundances, the higher-rate CNO cycle rapidly outstrips the PP process as temperatures rise above about 15 million degrees. For this reason, the CNO cycle is the dominant mechanism for the nuclear fusion of hydrogen in massive main-sequence stars, which have the highest core temperatures.

A Note on Notation

In the figure a compact notation for nuclear reactions is used. The general form is A(b,c)D, which is equivalent to A + b gives c + D. So the reaction for creating Deuterium is written as H¹(H¹,)H², which means H¹ + H¹ gives H² plus a gamma-ray.

Also note that, depending on the computer fonts available on your computer, the symbol for a neutrino, ν, and the symbol for a gamma-ray, , may look very similar. The key for distinguishing them is that the neutrino is involved only in reactions that involve an electron (e^-) or a positron (e⁺).

A Comment on Reaction Rates

The rates given in the figure are based on formulae given in Astrophysical Formulae by Lang.¹