approximate form for energy release
We have seen that the probabililty of fusion is related to the energies of
the particles involved and hence to the temperature of the plasma.
Using experimental values for the rates at which the
PP chain, CNO cycle and triple-alpha reactions occur, it is possible
to plot the energy release of each reaction as a function of
temperature, as shown in figure 20.
Rate of energy release from hydrogen and helium burning as a function
of temperature. indicates the gradient of the
straight-line fit to the curve.
The energy released by the PP chain and CNO cycle are smooth functions
of temperature. In a limited temperature range we can replace the true
dependence on temperature by power law fits to the curves in
figure 20. These fits have the following
where is the energy release per unit
mass per unit time, is the
density of the stellar material, T is the temperature of the
stellar material and 0 is a
constant which depends on the chemical composition of the stellar
The above expressions for reflect the
fact that the rate of fusion is a very sensitive function of
temperature, because the probability of tunnelling depends on kinetic
energy, which in turn depends on temperature. Furthermore, it can be
seen that fusion reactions involving successively heavier elements (in
ascending order: the PP chain, the CNO cycle and the triple-alpha
reaction) become even more temperature dependent, and require higher
temperatures to operate, in order to overcome the larger Coulomb
barrier due to the heavier (and hence more positively charged)
nuclei. As well as depending on temperature,
also depends on the density of the stellar material.
For two-particle reactions such as
PP chain and CNO cycle reactions, the dependence on density is linear,
whereas for three-particle reactions such as the triple-alpha process,
the dependence is quadratic.
Clearly, the true laws of energy release are not power
laws, but the expressions given are good approximations as the energy
release increases very rapidly with temperature and the range of
temperatures in which significant release occurs is small. We shall see
in the next part of the course how the use of the above approximate
expressions for the laws of energy production enables us
to obtain useful qualitative
information about the structure of stars.
©Vik Dhillon, 3rd December 2013