The goal of this exercise is to familiarize young people with the basic facts concerning the production of energy in the Sun and the stars and to demonstrate that many important parameters can be deduced from a few elementary data and simple physical principles.

Also, a word of caution: in what follows, several quantities are expressed with numbers either too big or too small, so an exponential notation is necessary. All exponents are indicated using the caret symbol (^). For example: the gravitational constant G = 6.67 x 10^-11 N m^2 kgr^-2 , which means it is equal to 6.67 times ten to the power (-11) Newtons times square meters per kilogram squared.

The Sun is our nearest star and the major energy source for all processes on Earth. It provides the energy needed for the atmospheric and ocean circulation and the development of plant life (therefore, of life in general). It is important for astronomy, also, because many phenomena, which can only be studied indirectly in other stars, can be directly observed in the Sun.

Our present picture of the Sun, and of stars in general, has evolved through a long series of observational and theoretical arguments. The fiery nature of the Sun was appreciated from ancient times. For instance, the Pythagorean philosophers in ancient Greece (most notably Aristarchos of Samos) described the Sun as 'the central fire' , around which the Earth is orbiting. Anaxagoras proposed that the Sun is a big mass of 'red-hot metal' , whereas Aristotle maintained that it is 'pure fire'. These views prevailed for almost 2,000 years, until the scientific revolution of the 17th century. Then, it became gradually accepted that the Sun is a massive object of burning gases but its source of energy was a mystery.

This picture of the Sun was taken in May 1996 by the SOHO spacecraft. It depicts the Sun as seen in the light emitted by ionized helium in the chromosphere. For more details, you my want to visit the ESA/NASA archives

The first decent theory about the energy source of the Sun was proposed around 1860 by Helmholtz and Lord Kelvin. They postulated that the gravitational attraction of the Sun causes a gradual contraction, thus liberating dynamical energy in the form of heating and radiation. This mechanism implied a reduction in Sun's radius of about 30 meters per year, which is undetectable, and also an age of the Sun of several million years. This theory was refined by Lane (1870) and Emden (1907), who suggested an age of 22 million years. By 1920, however, the evidence from fossils and old rocks implied an age for Earth greater than a billion (10^9) years and, clearly, the Sun could not be younger than Earth.

The foundation of our present theory about the energy source of the Sun was laid by Eddington in 1926. He proposed that the energy source can only be the conversion of some fraction of Sun's mass to energy, according to Einstein's famous formula :

In addition, he suggested that the energy comes during the transformation of hydrogen to helium but he was unable to propose any specific mechanism. The details of the energy production became known after the development of quantum mechanics, as will be exposed in later section.

A great deal of information about the Sun can be deduced from a few and well-known data of the Earth and some elementary physics.

To start with, one can estimate the main properties of the Sun using only :

- The distance **a** between Sun and Earth, which is 1 AU
(astronomical unit) = 1.5 x 10^11 m.

- The duration **T** of one complete orbit of Earth around the
Sun (tropical year) = 365.256 days

- The apparent radius **r** of Sun's disk, as seen from Earth,
which is about 16' (minutes of arc). This quantity can be measured
very easily, e.g. by projecting the Sun through a small hole and
dividing the radius of the bright image by its distance from the
hole.

- Finally, the value of the Gravitational constant G = 6.67 x 10^-11 N m^2 kgr^-2.

1. The radius **R** of the Sun is determined from its apparent
radius **r** and the Earth-Sun distance **a**.

2. Once the radius is known, the surface area **S** and the
volume **V** of the Sun are computed.

3. Approximating the Earth's orbit with a circle, its linear
velocity **v** is computed from the length of the orbit and the
duration of the year (one day has 86,400 sec).

4. From elementary mechanics, the centripetal force on Earth is
equal to the gravitational attraction of the Sun. This equation
provides the solar mass **M**.

In order to deduce other, energy-related, quantities of the Sun, an
additional piece of information is needed: it is the amount of solar
energy reaching Earth. This is described by the solar constant
**h** , which gives the amount of energy reaching the top of
Earth's atmosphere, per unit time and per unit area, perpendicular to
the direction to the Sun. The value of **h** is about 1.4 x 10^3 W
m^-2.

5. From the solar constant **h**, the total solar power at the
Earth's distance **a** is computed. Assuming no energy losses in
the interplanetary medium, this is equal to the total flux at the
surface of the Sun (luminocity **L**). Therefore, the brightness
**F** (flux per area) of the solar surface (photosphere) can be
estimated.

6. It is observed that radiation from most stars, and the Sun, can be adequately described as a black-body radiation. The characteristics of this radiation were described in the 19th century, even before Planck gave an explanation by proposing the light quanta (photons). In the present context, two characteristics are useful : the relation between brightness and temperature of the black-body (Stefan - Boltzmann law) and the relation between temperature and wavelength of greatest emissivity (Wien law).

The graph shows the intensity distribution, vs. wavelength, of the radiation of a black body at a temperature of 5800 K. The area under the curve gives the brightness of the source and is proportional to the fourth power of the temperature (Stefan-Boltzmann law). The peak of the emission occurs at a wavelength lamda(max) which is inversely proportional to the temperature (Wien law)./p>

- Stefan - Boltzmann law: F = s T^4 , where s = 5.67 x 10^-8 W m^-2 K^-4. From this law, the (effective) temperature of the Sun's surface is computed.

- Wien law: (lamda)max T = const. = 2.9 x 10^-3 m K. The wavelength (lamda)max where the Sun radiates more intensely is computed. Due to the exponential character of Planck's function, which describes the black-body radiation, most of the energy (above 95%) is confined in the region from 0.1(lamda)max to 4(lamda)max.

A black body is a kind of ideal radiator, who absorbs (hence its name) and re-emits the electromagnetic radiation completely. Therefore, the black body is in thermodynamic equilibrium with its radiation, which depends only on the (absolute) temperature of the body. In reality, only the general behaviour of the radiation from the stars can be approximated by the black body radiation. The absorption occuring at specific wavelengths, due to atomic transitions in the stellar atmosphere, creates thousands of absorption lines in the radiation distribution (spectrum). In addition, the condition of the thermodynamic equilibrium is not valid throughout the star but may hold locally.

From the value of the total energy generated in the Sun, as computed above, it is apparent that no ordinary mechanism can supply the vast amounts of energy needed. The only way to produce this energy is by thermonuclear reactions, taking place at the central parts of the Sun (and all other stars, for that matter). Although there are still some unresolved points (e.g. neutrino deficit), the standard model describing the generation of energy in low-mass stars, like the Sun, is fairly well understood.

The picture shows schematically the basic parameters of the solar structure: temperature (shades of green), pressure (shades of blue) and energy transfer mechanisms. The core of the Sun (white) contains about half of its total mass and all energy production occurs there. Throughout the core and the radiative envelope, energy is slowly transferred by radiation (no mass movements). In the outer convective envelope, the conditions are such that most of the energy is transferred by bulk mass motion (convection cells).

Let us first describe briefly the internal structure of the Sun. Whereas at the photosphere the temperature is of order 10^4 K and the pressure 10^5 atm., these critical parameters increase sharply towards the centre. At about 25% of the radius from the centre, i.e. in the core of the star, the temperature has reached 10^7 K and the pressure 10^10 atm. At these conditions, the material (mostly H and He) is completely ionised and behaves like an ideal gas mixture of protons, neutrons, electrons, He nuclei and traces of heavier nuclei. Due to the high temperature, the protons have enough kinetic energy to overcome the repulsive electrical forces and to bind together or to other nuclei, in a series of nuclear reactions similar to an hydrogen bomb, in order to form He nuclei. During this fusion process, energy is released in the form of gamma-ray photons and neutrinos. The neutrinos (which carry away only one millionth of the released energy) leave the star almost unobstracted but the gamma-ray photons start a difficult journey, from the centre to the surface, which lasts about 1 million years! During this time, they collide with particles (mostly free electrons) on average every 10^-12 sec and, at each collision, they loose a small amount of energy, until they finally emerge from the photosphere as visible photons. Most of their energy has been transformed to kinetic energy of the particles, enabling them to resist the enormous gravitational attraction. Since the stars remain in this 'hydrogen-burning' phase for the biggest part of their lives, a dynamical equilibrium is maintained and the amount of energy generated in their cores equals the amount radiated from their surface.

The detailed mechanisms of energy production in the stars were first proposed in 1938 by Bethe and, independently, by Weizsaecker. In the 'carbon-nitrogen-oxygen' cycle, protons are combined with carbon nuclei, temporarily forming nitrogen and oxygen isotopes, until a He nucleus is formed and the carbon nucleus remains intact. In effect, carbon acts as a catalyst. It is now believed that the CNO cycle is operating in massive stars (more than 1.5 times the mass of the Sun). In lighter stars, like the Sun, another mechanism is working: the proton-proton chain. In a series of 4 reactions, four protons combine to form a He nucleus (with temporary creation of hydrogen and helium isotopes).

The energy generated during these reactions comes from the smaller mass of the He nucleus: four protons have a total mass m = 6.69000 x 10^-27 kgr , whereas a He nucleus has a mass m(He) = 6.64408 x 10^-27 kgr. The mass difference (4.59 x 10^-29 kgr) is transformed into energy, according to the equivalence of mass and energy proposed by Einstein and is carried away by photons and neutrinos.

The neutrinos are elementary particles with (almost) zero rest mass and travelling at the speed of light. They were initially suggested in 1931 by Pauli, who tried to explain the variable 'missing' energy observed in the radioactive decay of several nuclei (beta-decay). Fermi worked out the theory of the processes involving neutrinos and also coined their name. Neutrinos were for the first time detected in 1956. Their detection is exceedingly difficult because they interact very rarely with other particles and so they are able to pass right through objects like the Sun or the planets as if there was nothing there!

Several important parameters, related to the energy production in the Sun, can be estimated. The only additional information needed is the H content of the Sun, which is X = 0.70 (i.e. 70% of the mass of the Sun is hydrogen).

7. From the masses of protons and He, discussed above, compute the fraction of H mass converted to energy during the thermonuclear reactions (conversion efficiency).

8. Using Einstein's formula, compute the mass conversion rate needed to maintain the luminosity of the Sun, determined earlier in step 5. For the speed of light use the value: c = 3 x 10^8 m/sec.

9. Combine the results of steps 7 and 8 and compute the rate of H mass converted to He.

10. From the value of X and the solar mass, estimate the total H mass of Sun. Then, using the conversion rate found above, compute the maximum time span of the hydrogen-burning phase. It is to be noted, however, that less than 15% of the total H mass can actually be converted before the next, more rapid evolutionary stage takes place.

11. Estimate the percentage of Sun's mass radiated during his whole H-burning phase.

12. The estimated age of Earth is about 4.5 x 10^9 years. Compare this with the time span of H burning and estimate how far in the H-burning process is the Sun (assuming that all bodies in the Solar system were created at the same time).