In the two preceding activities, we looked at a very simple model by which conversion of hydrogen to helium might power the Sun. We considered turning four protons into an alpha particle without worrying about the intervening processes that would first turn two of the protons into neutrons. Since we were only interested in checking on the general viability of nuclear reactions to power the Sun (and other stars, of course), this was fine. However, in a real theory of stellar energy production we have to worry about the details. Those details require quite a lot of advanced knowledge however, so we will merely lay them out for you.
The theory of stellar energy generation by nuclear reactions was first proposed by the German physicist Hans Bethe (1906–2005). Bethe proposed two mechanisms for hydrogen burning. This “hydrogen burning” is not the sort of burning we are used to seeing in chemical processes. Those involve atomic electrons (See Numerical Activity: The Chemical Lifetime of the Sun). Hydrogen burning refers to nuclear reactions that combine protons to make helium. The first, called the p–p chain (for proton–proton) is what powers small stars, those slightly more massive than the Sun or smaller. One branch of the p–p chain is shown below:
\begin{equation} p +p \rightarrow \ ^2H + e^{+} + \nu_{e}\end{equation} \begin{equation}\ ^2H + p \rightarrow \ ^3He + \gamma\end{equation} \begin{equation}\ ^3He + \ ^3He \rightarrow \ ^4He + 2p\end{equation}
In the first step, two protons (p) are converted to a deuteron (2H), an isotope of hydrogen with a neutron as well as a proton. Also produced are a positron (e+) and an electron neutrino (νe). The second step combines a deuteron with a proton to form the mass-3 (two protons plus one neutron) isotope of helium (3He). It also liberates some energy, which is carried away by a photon (γ). In the final step, two 3He nuclei combine to form 4He and liberate two protons, which are then free to continue reacting. The net result of this reaction chain is to convert four protons into an alpha particle (4He), with the proton-to-neutron conversion happening in the first step. All the other intermediate nuclei are consumed in subsequent steps of the chain. And, of course, energy is liberated in the form of photons, the positron (e+) and neutrino (νe) emitted in step one. There are two other versions of the p–p chain that compete with this one, though we do not show them for brevity. All have the net result of converting hydrogen nuclei to 4He nuclei.
For more massive stars, above about twice the mass of the Sun, there is a different nuclear reaction chain that occurs. This is called the CNO cycle. In this process, carbon, nitrogen, and oxygen act as catalysts. They participate in the reaction, but are not consumed. One version of the CNO cycle is shown below.
\begin{equation} \ ^{12}C + p \rightarrow \ ^{13}N\end{equation} \begin{equation}\ ^{13}N \rightarrow \ ^{13}C + e^{+} + \nu_{e}\end{equation} \begin{equation}\ ^{13}C + p \rightarrow \ ^{14}N + \gamma\end{equation} \begin{equation}\ ^{14}N + p \rightarrow \ ^{15}O + \gamma\end{equation} \begin{equation}\ ^{15}O \rightarrow \ ^{15}N + e^{+} + \nu_{e}\end{equation} \begin{equation}\ ^{15}N + p \rightarrow \ ^{12}C + \ ^{4}He\end{equation}
Here again, by inspecting the right- and left-hand sides of the equations, we see that the net result is conversion of four protons into an alpha particle and some additional particles (photons and neutrinos). The carbon and nitrogen are neither consumed nor produced in the reaction chain; they merely catalyze it.
An interesting aspect of the CNO cycle is that each step in the chain has a characteristic timescale. The proton capture timescales depend on the temperature and the proton density. They can typically go quite rapidly, requiring only fractions of a second. The other steps, which involve beta decays, require much longer times, each over 100 seconds. As a result, the products on the left-hand side for the slow steps tend to pile up, whereas the nuclei on the left-hand side of the fast steps are depleted. So the CNO cycle (which, like the p–p chain involves another branch, not shown for brevity) sets the relative abundances of the various isotopes of carbon, nitrogen, and oxygen. This provides an observational check on the theory, which turns out to be confirmed.