Energy Generation in Stars

(NB. Blue links are links within the current page. Purple links are external links to other web sites.)
Where Does a Star's Energy Come From?
The Nuclear Atom - Fission and Fusion
Energy Generation in the Sun
                      The Proton - Proton Cycle
                      The CNO Cycle
Energy Transport
                      Radiative Diffusion
The Solar Neutrino Problem

Where Does a Star's Energy Come From?

Knowledge of the atom and of nuclear theory have enabled astrophysicists to deduce the source of stellar energy. The huge amounts of energy generated by stars are due to nuclear fusion reactions occurring deep within their interiors. Using physical assumptions and observational data, an interior model of a star can be made which describes how energy is transported from it into space.

Before astrophysicists deduced that the sun (and other stars) shine due to nuclear energy, other ideas were put forward to explain their enormous output of energy. The source of the sun's energy could not be chemical because if it was made out of coal, for instance, which produces heat and light from chemical energy, then it would have only lasted for about 300 years.

In the mid 1800s two physicists, Lord Kelvin (1824-1907) and Hermann von Helmholtz (1821-94), put forward the idea that the huge weight of the sun's outer layers should cause the sun to gradually contract. As it does so the gases in its interior become compressed and when a gas is compressed its temperature increases.

Kelvin and Helmholtz argued that gravitational contraction would cause the sun's gases to become hot enough to radiate heat energy into space. This process, called the Kelvin-Helmholtz contraction, does in fact happen in the protostar phase of stellar formation.

However, the Kelvin-Helmholtz contraction cannot be the main source of stellar energy since, in the case of the sun, calculations show that in order to produce the solar luminosity we see today, the sun would have had to contract from a size larger than the earth's orbit only 25 million years ago. A clue to the source of stellar energy was provided by Albert Einstein (1879-1955). In 1905, while developing his special theory of relativity, Einstein showed that mass can be converted into energy and vice-versa. These quantities are related by the mass-energy relation Albert Einstein (1879-1955)

E = mc2

where E is the energy released in joules from the conversion of a mass m in kg, and c is the speed of light

Thus the total conversion of 1kg of matter yields an equivalent of 1 x (3x108)2 = 9x 1016 joules - this is approximately the energy output of a 200 MW power station running for 14 years!

To understand how stars can shine brightly, we must examine the structure of the atom and in particular, the properties of the nucleus.

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Biography of Lord Kelvin from the excellent St Andrews site

Biography of his life and work including a picture

STARBASE information on Einstein

For information on a selection of famous physicists click here to goto our 'People' page

The Nuclear Atom - Fission and Fusion

For nuclei to be stable there must exist a strong nuclear force between the nucleons that is short range, attractive, and can overcome the coulomb repulsion of the protons. Now suppose we assemble a nucleus of N neutrons and Z protons. There will be an increase in the electric potential energy due to the electrostatic forces between the protons trying to push the nucleus apart but there is a greater decrease of potential energy due to the strong nuclear force acting between the nucleons and attracting them to one another.

Mass Defect
                                         The Mass Defect

As a consequence, the nucleus has an overall net decrease in its potential energy. This decrease in potential energy is called the nuclear binding energy and the decrease per nucleon is called the binding energy per nucleon. The loss of this energy is, by the mass-energy relation, equivalent to a loss of mass called the mass defect. So how is energy released in stars? This can be explained by a graph of the binding energy per nucleon against atomic mass number A (see below graph).

The variation of binding energy per nucleon with atomic mass number
The variation of binding energy per nucleon with atomic mass number

The curve reaches a maximum at iron which, because of its high binding energy per nucleon, indicates that the protons and neutrons are very tightly bound and iron is therefore a very stable nucleus. Beyond iron, the binding energy per nucleon falls slightly as A increases towards the more massive nuclei. Two processes can release energy from the nucleus of an atom. They are nuclear fission and nuclear fusion.

In nuclear fission a massive nucleus such as uranium splits in two to form two lighter nuclei of approximately equal mass. This happens on the falling part of the curve so that mass is lost and binding energy released when very heavy elements fission to nuclei of smaller mass number. Nuclear fission is responsible for the release of energy in nuclear reactors and atomic bombs.

In nuclear fusion, energy is released when two light nuclei are fused together to form a heavier nucleus. This happens on the rising part of the graph. Nuclear fusion is the principal source of energy in stars and fusion can happen if each nucleus has sufficient kinetic energy to enable them to overcome their mutual repulsion, be captured by the strong nuclear force and stick together. In star formation, the kinetic energy to do this comes from the conversion of gravitational energy into thermal energy by the Kelvin Helmholtz contraction. In the case of stars like the sun, fusion can occur when the temperature of the contracting cloud reaches about 8 x 106 K. It is because of the high temperatures which are needed to give the protons sufficient kinetic energy, that these nuclear reactions are also known as thermonuclear fusion reactions.

It is fusion of hydrogen nuclei by thermonuclear fusion reactions with a release of binding energy that is the primary source of energy generation in stars.

This is a very important process in the astrophysics of stars and is called hydrogen burning (although nothing is 'burnt' in the ordinary sense of the word). Hydrogen is converted to helium and the binding energy liberated is responsible for the star's tremendous energy output. A hydrogen nucleus consists of a single proton whereas helium nuclei have two protons and two neutrons . Although the fusion process involves several stages, it can be summarised as...

4H --> He + energy released

How much energy is released in the process?

mass of 4 H atoms = 4 x 1.008 = 4.032 amu

- mass of 1 He atom = 4.003 amu

therefore... mass loss = 4.032 - 4.003 = 0.029 amu

Where amu is the atomic mass unit. Using the mass-energy relation, the mass converted into energy is

= (0.029 x 1.66 x 10-27)kg x (3 x 108)2

= 4.33 x 10-12 J or, equivalently, 27 MeV.

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Information on Nuclear Fission and Fusion including nice animations

A good general nuclear physics page including nuclear fission

Description of nuclear fission and half-lives with equations

Some nuclear fusion basics from JET

Why we want to use nuclear fusion power and what it is including a diagram

Excellent site about nuclear fusion in the Sun

Energy Generation in the Sun

We have seen that the fusion of 4 hydrogen atoms into a single He nucleus releases 4.33 x 1012 J and this is quite a small amount of energy. However, this also means that 1kg of hydrogen yields 6 x 1014 J which is a very large amount of energy. We know that the luminosity of the sun is 3.90 x 1026 W (J/s) and since the conversion of 1 kg of H into He yields 6 x 1014 J of energy, then approximately...

  of H must be converted into He every second. This may seem like a huge consumption of hydrogen, but we need not worry as the sun has sufficient reserves of H for it to go on shining for at least another 5 thousand million years!

There are two principal nuclear reaction pathways in which hydrogen burning occurs in the sun or any star. These are the proton-proton chain and the Carbon-Nitrogen-Oxygen or CNO cycle. In each of these reactions, four protons combine by nuclear fusion to form a single He nucleus with a small loss of mass which, by the mass-energy relation, is released as energy.

The temperature in the interior of a star determines which of these reaction pathways takes place. For stars which have masses not exceeding that of the sun, the core temperature does not get higher than about 16 million K and hydrogen burning occurs via the proton-proton chain. In stars with masses greater than the sun, the core temperatures exceed 16 million K and hydrogen burning proceeds through the CNO cycle.

The Proton - Proton Cycle

Step 1

Step 2

Step 3

where e+ is an antielectron called a positron, v (greek letter nu) is an elementary particle with zero charge (and may have a small mass) called a neutrino and is a gamma ray.
It is important to understand that very high temperatures and densities are needed for hydrogen burning to occur. The temperature in the core of the sun; that is the inner quarter of its radius, must be at least 8 x 106 K with densities ranging from 1.6 x 105 kg m-3 in the centre to about 2 x 104 kg m-3 at the core perimeter.

The CNO Cycle

While the sun burns hydrogen by the proton-proton cycle, this is not the case for more massive stars (>than 2 solar masses), where hydrogen burning is achieved by the CNO cycle. The CNO cycle occurs in stars where the central temperatures exceed about 2.0 x 107 K and the pressure of carbon plays an important role in converting hydrogen to helium. The CNO cycle has 6 steps...

Step 1
A carbon atom and a proton fuse to form an isotope of nitrogen and a ray:

Step 2
The isotope of nitrogen decays to an isotope of carbon, emitting a positron (e+) and a neutrino in the process:

Step 3
The isotope of carbon fuses with another proton to form stable nitrogen and a ray:

Step 4
Stable nitrogen fuses with a third proton to form an isotope of oxygen and a ray:

Step 5
The isotope of oxygen decays to form an isotope of nitrogen, a positron and a neutrino:

Step 6
Finally, a helium nucleus is created when the isotope of nitrogen fuses with a fourth proton and carbon 12 is restored

Notice that, as in the p-p chain, the CNO cycle takes four hydrogen nuclei (protons) and converts them into a single helium nucleus together with positrons, neutrinos and some high energy gamma rays. the nucleus acts as a catalyst for the reaction and while consumed in step 1, is replaced at step 6, so that in the CNO reaction chain carbon is not used up.

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The proton-proton cycle designed for A-level students

The proton-proton cycle site from Cornwallis School in Kent

Explanation of the CNO Cycle and reactions of it...including diagram

The CNO Fusion cycle in stars including animations

Energy Transport

How does the energy produced by hydrogen burning in the sun's interior find its way to the surface of the sun and escape as sunlight?
Heat energy is transported by three processes: conduction, convection and radiation. Heat is conducted by the collision of atoms or molecules with those that have more kinetic energy transferring some kinetic energy. By using theoretical models, astrophysicists have found that the conditions inside the sun do not particularly favour conduction as an efficient energy transport process mainly because the solar material is not very dense (although it is important in small, very dense stars like white dwarfs).

Sun's Corona and Solar Turbulence :

In the sun, energy transport is mainly by convection and radiative diffusion. Convection in the sun occurs when hot gases rise towards its surface and cooler gases sink back down. As a result, circulation currents are set up in which heat energy is transferred to the outer layers of the sun from its interior.

In radiative diffusion photons are absorbed and re-emitted when they interact with atoms and electrons in the solar interior. The net motion through is towards the cooler, outer layers of the sun where they escape into space. This photon migration towards the surface can take tens of thousands of years and in this fashion photons carry energy from the interior to the outside.

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Definitions of convection, conduction and radiation including tables and diagrams

The inner turbulence of the sun is explained including 3-D simulations

Solar convection and magneto convection simulations in 3-D

The Solar Neutrino Problem

A puzzling feature of the sun's stellar model, is the Solar Neutrino Problem. Astrophysicists are interested in detecting neutrinos because they provide a window into the thermonuclear reactions occurring in the sun's core. Various neutrino detectors have been constructed such as one located deep underground in a gold mine in Homestake, South Dakota.

A Solar Neutrino Detector (Japan) :

The curious result is that only one third of the number of solar neutrinos as predicted by the stellar model are found. Other detectors (such as the Kamiokande detector in Japan) seem to confirm this finding. Perhaps our knowledge of the nuclear physics is deficient in some way, or perhaps our understanding of the neutrino as a particle is lacking. One explanation for the mystery is that the neutrinos may actually possess a small mass. If so, electron neutrinos can convert to muon neutrinos on their way to the earth. Recent experiments suggest that thisis indeed the case, that the neutrino has a tiny mass of about 3 eV, and that the neutrino problem may have been solved.

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Answers to many solar neutrino problem questions

Useful summary of the solar neutrino problem

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