Stellar Energy Sources

When the equations of stellar structure were derived, the character of the source of stellar energy was left unspecified. Knowing a typical stellar luminosity, one can calculate how long different energy sources would last. For instance, normal chemical burning could produce energy for only a few thousand years. The energy released by the contraction of a star would last slightly longer, but after a few million years this energy source would also run out.

Terrestrial biological and geological evidence shows that the solar luminosity has remained fairly constant for at least a few thousand million years. Since the age of the Earth is about 5000 million years, the Sun has presumably existed at least for that time. Since the solar luminosity is 4 x 1026 W, it has radiated about 6 x 1043 J in 5 x 109 years. The Sun's mass is 2 x 1030 kg; thus it must be able to produce at least 3 x 1013 J/kg.

The general conditions in the solar interior are known, regardless of the exact energy source. Thus, in Example 10.5, it will be estimated that the temperature at half the radius is about 5 million degrees. The central temperature must be about ten million kelvins, which is

Fig. 10.4. The nuclear binding energy per nucleon as a function of the atomic weight. Among isotopes with the same atomic weight the one with the largest binding energy is shown. The points correspond to nuclei with even proton and neutron numbers, the crosses to nuclei with odd mass numbers. Preston, M.A. (1962): Physics of the Nucleus (Addison-Wesley Publishing Company, Inc., Reading, Mass.)

high enough for thermonuclear fusion reactions to take place.

In fusion reactions light elements are transformed into heavier ones. The final reaction products have a smaller total mass than the initial nuclei. This mass difference is released as energy according to Einstein's relation E = mc2. Thermonuclear reactions are commonly referred to as burning, although they have no relation to the chemical burning of ordinary fuels.

The atomic nucleus consists of protons and neutrons, together referred to as nucleons. We define mp = proton mass , mn = neutron mass ,

Z = nuclear charge = atomic number,

N = neutron number ,

A = Z + N = atomic weight, m( Z, N) = mass of the nucleus .

The mass of the nucleus is smaller than the sum of the masses of all its nucleons. The difference is called the binding energy. The binding energy per nucleon is

It turns out that Q increases towards heavier elements up to iron (Z = 26). Beyond iron the binding energy again begins to decrease (Fig. 10.4).

It is known that the stars consist mostly of hydrogen. Let us consider how much energy would be released by the fusion of four hydrogen nuclei into a helium nucleus. The mass of a proton is 1.672 x 10-27 kg and that of a helium nucleus is 6.644 x 10-27 kg. The mass difference, 4.6 x 10-29 kg, corresponds to an energy difference E = 4.1 x 10-12 J. Thus 0.7% of the mass is turned into energy in the reaction, corresponding to an energy release of 6.4 x 1014 J per one kilogram of hydrogen. This should be compared with our previous estimate that 3 x 1013 J/kg is needed.

Already in the 1930's it was generally accepted that stellar energy had to be produced by nuclear fusion. In 1938 Hans Bethe and independently Carl Friedrich von Weizs├Ącker put forward the first detailed mechanism for energy production in the stars, the carbon-nitrogen-oxygen (CNO) cycle. The other important energy generation processes (the proton-proton chain and the triple-alpha reaction) were not proposed until the 1950's.

The Proton-Proton Chain (Fig. 10.5). In stars with masses of about that of the Sun or smaller, the energy

Fig. 10.5. The protonproton chain. In the ppI branch, four protons are transformed into one helium nucleus, two positrons, two neutrinos and radiation. The relative weights of the reactions are given for conditions in the Sun. The pp chain is the most important energy source in stars with mass below 1.5 Me

0.1 % He very temperature sensitive

is produced by the proton-proton (pp) chain. It consists of the following steps:

For each reaction (3) the reactions (1) and (2) have to take place twice. The first reaction step has a very small probability, which has not been measured in the laboratory. At the central density and temperature of the Sun, the expected time for a proton to collide with another one to form a deuteron is 1010 years on the average. It is only thanks to the slowness of this reaction that the Sun is still shining. If it were faster, the Sun would have burnt out long ago. The neutrino produced in the reaction (1) can escape freely from the star and carries away some of the energy released. The positron e+ is immediately annihilated together with an electron, giving rise to two gamma quanta.

The second reaction, where a deuteron and a proton unite to form the helium isotope 3He, is very fast

compared to the preceding one. Thus the abundance of deuterons inside stars is very small.

The last step in the pp chain can take three different forms. The ppl chain shown above is the most probable one. In the Sun 91% of the energy is produced by the ppl chain. It is also possible for 3He nuclei to unite into 4He nuclei in two additional branches of the pp chain.

ppII:

3He + 4He ^ 7Be + y , 7Be + e- ^ 7Li + Ve, 7Li + JH ^ 4He + 4He ,

The Carbon Cycle (Fig. 10.6). At temperatures below 20 million degrees the pp chain is the main energy production mechanism. At higher temperatures corresponding to stars with masses above 1.5 Me, the carbon (CNO) cycle becomes dominant, because its reaction rate increases more rapidly with temperature. In the CNO cycle carbon, oxygen and nitrogen act as catalysts. The reaction cycle is the following:

(1)

12C +

1H

13N + Y,

(2)

13n

13C + e+ + Ve

(3)

13C +

1H

14N + Y ,

(4)

14n +

1H

15O + y ,

(5)

15O

15N + Y + Ve ,

(6)

15n +

1H

12C + 4He .

Reaction (4) is the slowest, and thus determines the rate of the CNO cycle. At a temperature of 20 million degrees the reaction time for the reaction (4) is a million years.

The fraction of energy released as radiation in the CNO cycle is slightly smaller than in the pp chain, because more energy is carried away by neutrinos.

The Triple Alpha Reaction. As a result of the preceding reactions, the abundance of helium in the stellar interior increases. At a temperature above 108 degrees

Helium Burning Stellar Energy

Proton in

Fig. 10.6. The CNO cycle is catalysed by 12C. It transforms four protons into a helium nucleus, two positrons, two neutrinos and radiation. It is the dominant energy source for stars more massive than 1.5 Mq

Proton in

Fig. 10.6. The CNO cycle is catalysed by 12C. It transforms four protons into a helium nucleus, two positrons, two neutrinos and radiation. It is the dominant energy source for stars more massive than 1.5 Mq the helium can be transformed into carbon in the triple alpha reaction:

Here 8Be is unstable and decays into two helium nuclei or alpha particles in 2.6 x 10-16 seconds. The production of carbon thus requires the almost simultaneous collision of three particles. The reaction is often written

Once helium burning has been completed, at higher temperatures other reactions become possible, in which heavier elements up to iron and nickel are built up. Examples of such reactions are various alpha reactions and oxygen, carbon and silicon burning.

Alpha Reactions. During helium burning some of the carbon nuclei produced react with helium nuclei to form

oxygen, which in turn reacts to form neon, etc. These reactions are fairly rare and thus are not important as stellar energy sources. Examples are

Carbon Burning. After the helium is exhausted, carbon burning sets in at the temperature (5-8) x 1010 K:

12C + 12C ^ 24Mg + y ^ 23Na + ^ 20Ne + 4He ^ 23Mg + n ^ 16O + 24He .

Oxygen Burning. Oxygen is consumed at slightly higher temperatures in the reactions

16O + 16O ^ 32 S + Y ^ 31P + ^ 28Si + 4He ^ 31S + n ^ 24Mg + 24He .

Silicon Burning. After several intermediate steps the burning of silicon produces nickel and iron. The total process may be expressed as

56-n

The rates of the reactions presented above can be determined by laboratory experiments or by theoretical calculations. Knowing them, one can calculate the rate at which energy is released per unit mass and time as a function of the density, temperature and chemical composition:

When the temperature becomes higher than about 109 K, the energy of the photons becomes large enough to destroy certain nuclei. Such reactions are called photonuclear reactions or photodissociations.

The production of elements heavier than iron requires an input of energy, and therefore such elements cannot be produced by thermonuclear reactions. Elements heavier than iron are almost exclusively produced by neutron capture during the final violent stages of stellar evolution (Sect. 11.5).

In reality the relative abundance of each of the heavier nuclei needs to be known, not just their total abundance Z.

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