vik dhillon: phy213 - the physics of stellar interiors

hydrogen and helium burning

We turn now to look at the most important nuclear reactions which occur in stars.

hydrogen burning reactions

The most important series of fusion reactions are those converting hydrogen to helium in a process known as hydrogen burning. The chances of four protons fusing together to form helium in one go are completely negligible. Instead, the reaction must proceed through a series of steps. There are many possibilities here, but we will be looking at the two main hydrogen-burning reaction chains: the proton-proton (PP) chain and the carbon-nitrogen (CNO) cycle. The PP chain divides into three main branches, which are called the PPI, PPII and PPIII chains. The first reaction is the interaction of two protons (p or ¹H) to form a nucleus of heavy hydrogen (deuteron, d, or ²H), consisting of one proton and one neutron, with the emission of a positron (e⁺) and a neutrino (

_e). The deuteron then captures another proton and forms the light isotope of helium with the emission of a -ray. The ³He nucleus can then either interact with another ³He nucleus or with a nucleus of ⁴He (an

particle), which has either already been formed or has been present since the birth of the star. The former case is the last reaction of the PPI chain, whereas the latter reaction leads into either the PPII or the PPIII chain, as shown below:

PPI chain	PPII chain	PPIII chain
	this starts with reactions 1 and 2	this starts with reactions 1, 2 and 3^'
1 p + p --> d + e⁺ + _e	3^' ³He + ⁴He --> ⁷Be +	4^'' ⁷Be + p --> ⁸B +
2 d + p --> ³He +	4^' ⁷Be + e^- --> ⁷Li + _e	5^'' ⁸B --> ⁸Be + e⁺ + _e
3 ³He + ³He --> ⁴He + p + p	5^' ⁷Li + p --> ⁴He + ⁴He	6^'' ⁸Be --> ⁴He + ⁴He

It can be seen that there is another choice in the chain when ⁷Be either captures an electron to form ⁷Li in the PPII chain or captures another proton to form ⁸B in the PPIII chain. At the end of the PPIII chain, the unstable nucleus of ⁸Be breaks up to form two ⁴He nuclei. The PP chain reactions are summarized pictorially in figure 17.

Figure 17:

The proton-proton chain.

The reaction rate of the PP chain is set by the rate of the slowest step, which is the fusion of two protons to produce a deuteron. What actually happens in this reaction is that the two protons fuse to form a highly unstable nucleus composed of two protons (a diproton), which immediately decays back into two protons. On rare occasions, however, one of the protons in the diproton will undergo a ⁺ decay,

p --> n + e⁺ +

_e,

forming a deuteron, which is stable. This reaction occurs via the weak nuclear force and the average proton in the Sun will undergo such a reaction approximately once in the lifetime of the Sun, i.e. once every 10¹⁰ years. The subsequent reactions occur much more quickly, with the second step of the PP chain taking approximately 6 seconds and the third step approximately 10⁶ years in the Sun. Note that the neutrino produced by the ⁺ decay will most probably leave the star without interacting again, whereas the positron will annihilate with an electron to produce a photon. Note also that the fusion of a free proton with a free neutron to form a deuteron directly is not an important energy source in stars due to the fact that free neutrons decay into protons with a half-life of only 15 minutes.

The relative importance of the PPI and PPII chains depend on the relative importance of the reactions of ³He with ³He in PPI as compared to the reactions of ³He with ⁴He in PPII. For temperatures in excess of 1.4x10⁷K, ³He prefers to react with ⁴He. At lower temperatures, the PPI chain is more important. The PPIII chain is never very important for energy generation, but it does generate abundant high energy neutrinos.

The other hydrogen burning reaction of importance is the CNO cycle:

CNO cycle

5 ¹⁵O --> ¹⁵N + e⁺ +

6 ¹⁵N + p --> ¹²C + ⁴He

The reaction starts with a carbon nucleus, to which are added four protons successively. In two cases the proton addition is followed immediately by a ⁺ decay, with the emission of a positron and a neutrino, and at the end of the cycle a helium nucleus is emitted and a nucleus of carbon remains. The reactions of the CNO cycle are shown pictorially in figure 18.

Figure 18:

The CNO cycle.

Note that there are less important side reactions of the CNO cycle which are not listed here. Carbon is sometimes described as a catalyst in the above reaction because it is not destroyed by its operation and it must be present in the original material of the star for the CNO cycle to operate. When the cycle is working in equilibrium, the rates of all of the reactions in the chain must be the same. In order for this to be so, the abundances of the isotopes must take up values so that those isotopes which react more slowly have higher abundances. It can be seen from figure 18 that the slowest reaction in the CNO cycle is the capture of a proton by ¹⁴N. As a result, most of the ¹²C is converted to ¹⁴N before the cycle reaches equilibrium and this is the source of most of the nitrogen in the Universe.

helium burning reactions

When there is no longer any hydrogen left to burn in the central regions of a star, gravity compresses the core until the temperature reaches the point where helium burning reactions become possible. In such reactions, two ⁴He nuclei fuse to form a ⁸Be nucleus, but this is very unstable to fission and rapidly decays to two ⁴He nuclei again. Very rarely, however, a third helium nucleus can be added to ⁸Be before it decays, forming ¹²C by the so-called triple-alpha reaction:

⁴He + ⁴He --> ⁸Be

⁸Be + ⁴He --> ¹²C +

The triple-alpha reaction is shown pictorially in figure 19. It can be seen that the reaction leaps from helium to carbon in one go, by-passing lithium, beryllium and boron. It is no coincidence, therefore, that these three elements are over 10⁵ times less abundant by number than carbon in the Universe.

Figure 19:

The triple alpha reaction.

Once helium is used up in the central regions of a star, further contraction and heating may occur, and that may lead to additional nuclear reactions such as the burning of carbon and heavier elements. We will not discuss these reactions here as the majority of the possible energy release by nuclear fusion reactions has occurred by the time that hydrogen and helium have been burnt.