For a long time there was a belief that the Sun shone by actually burning its material, like a fire, i.e., through combustion. However Rutherford's radioactive dating increased the life-time of the Earth from a few thousand years as previously believed to a few billion years. Even before that geologists had suggested that a few hundred thousand years was needed for the important features on the Earth's surface to form. The Sun would need inconceivable amounts of fuel to keep burning that long. Lord Kelvin and Helmholtz had worked out that Sun could generate energy by its own gravitational collapse. But this would generate energy only for 20 million years. In fact Lord Kelvin argued against Darwin's theory of evolution because there wasn't enough time for evolution theory to work. Radio active dating however showed that he was in the error. It wasn't until the physics of the nucleus of atoms was understood that we had the solution to this problem.
Remember Rutherford's experiments where he scattered 
particles off a gold film? He was led to the conclusion that most of
atoms was empty space with the nucleus occupying a tiny volume of
or 
cm in radius at the center of the atom. Now
the mass of the lighter atoms is 
gms, which means that the
density of the nucleus (as the electron has hardly any mass nearly all
the atom's mass resides in the nucleus), is a staggering 
gm/cm
(the density of water is 1 gm/cm). From Becquerel's
discovery that some atoms decay by radiating and 
particles led to suggestions that atomic nuclei were made of elementary
constituents. Prout suggested that all atomic nuclei are built up of
hydrogen nuclei. This led to expressing weights of atomic nuclei in
terms of the weight of hydrogen nuclei. This is the atomic
weight of any element. So hydrogen as an atomic weight of 1, helium
4, oxygen 16, carbon 12, etc. Prout's model ran into some problems
when improved measuring techniques showed that chlorine's atomic
weight was a fraction, 35.46. In the meantime Mendeleev had shown that
one could construct a table grouping elements according to their
weights and chemical properties. He believed that atomic weights of
the elements was the crucial feature that placed an element in a
specific place in his table. It was later realized that there was
another quantity, the atomic number that had to be assigned according
to the elements place in the Mendeleev table that better correlated
with the element's chemical property. This was later found to be the
number of electrons around the nucleus in that element. So it was the
positive charge of the nucleus in units of electron charge. It was
found that some elements have species of different atomic masses but
the same atomic number and chemical properties. These species, called
isotopes of the element helped resurrect Prout's
hypothesis. Because now elements with fractional atomic weights could
be understood as having two or more isotopes with varying atomic
weights. Chlorine for example has two isotopes with atomic weights
34.98 and 35.98, resulting in an average atomic weight of 35.46. The
atomic weights of each isotope however is awfully close to a whole
number. But each isotope is not equally abundant. Some isotopes
are much more prevalent than others. Hydrogen with atomic weight 1 is
much more prevalent that its isotope deuterium, with atomic weight 2,
or tritium with atomic weight 3. The modern notation to denote an
element X, with atomic number N and atomic weight A, is
.
One of the earlier suggestions was that the constituents of the
nucleus were protons and electrons. If an element has an atomic weight
of A and an atomic number Z then, it was suggested, it had A
protons and A-Z electrons. This would result in the nucleus having a
positive charge of Z as required and a weight of A as needed. This
however ran into some problems. The expected energy of an electron
inside a nucleus is huge. The electrons radiated in radio-active decay
of elements however have much smaller energies. Rutherford suggested
that perhaps the electron and proton when pushed together so close
form a new elementary particle, the neutron, a chargeless
particle with nearly the same mass as the proton. These were
discovered finally in 1930. The radiation of particles
observed in Uranium and other elements then needed an explanation. A
particle is an electron, so it was hypothesized that a neutron
could break into a proton, electron and a neutrino, radiating the
electron and the neutrino. The neutrino was yet another elementary
particle that has no charge and very little mass. It was postulated to
explain the radiation and has since been seen.
Radioactivity is the propensity of an element to release either an
particle (a helium nucleus) or a particle (an
electron) and change its atomic number and/or weight. This is a
statistical process, that is we can't say when an individual unstable
atom will decay, but for a number of atoms we can tell by what time
half of the atoms will have decayed and half of that remaining half
will decay and so on.
This spontaneous decay led to the speculation that ordinary stable
nuclei could be induced to transmute (change its atomic number) by
bombardment with or particles. Rutherford bombarded
Nitrogen with particles and found that it was transmuted. It
was further seen by Blackett that the particle disappeared,
so the conclusion was that the nitrogen nucleus absorbed the
particle, became a new nucleus that was unstable and that decayed into
two new stable nuclei. In modern notation the equation that occurred
is,
In this equation, the two nuclei on the left are reacting to produce
the unstable nuclei inside the square brackets which decays to the two
daughter nuclei on the right. This is often abbreviated as
. In this abbreviated notation p refers to a
proton. One of the issues that is of great significance in these
nuclear reaction is conservation of mass and energy. Imagine a
reaction where a target X is hit by an incident particle x,
producing the nucleus Y and the particle y. So,
Then the total energies on the right hand side must equal the energy on the left hand side. Energies now refer to not only the energy in motion (kinetic energy) of the particles and nuclei but the energy in the mass of the particles and nuclei as well. If the total mass on the left hand side exceeds the total mass on the right hand side then this excess mass is converted to energy of the resulting particle and nucleus. Such reactions that produce energy are called exothermic. If the total mass on the left hand side is less than that of the right hand side, energy has to be imparted through the incident particle for the reaction to proceed. Such reactions are called endothermic reactions. Some of the energy goes into producing the excess mass of the daughter products.
Remember the issue is that a star like our Sun has to survive billions of years, generating huge amounts of energy. In any star there is constant conflict going on between two disparate forces. The stars are gaseous, and hot gases try to expand. At the same time the enormous gravitational pull of the inner material on any layer of gas is trying to pull the gas in itself. For the star to be stable these two forces must be in equilibrium. Of course the heat in the star leaks out, that is why we can see it. The degree to which the star can block this leakage of heat is called opacity. Imagine the star being constructed of many layers of gas. Each layer is held up by the pressure of the lower layers and held down by the weight of the upper layers. The star now has to regulate its opacity such that the heat slowly leaks out through the layers so each layer can stay at just the heat needed to keep it up. This means that if we know the mass, radius and composition of the star (and provided we can characterize the opacity of the star) we can say what the central temperature and density will be of the star. Then if nuclear reactions can carry on at this temperature at the sufficient rate then the star will be stable. The way this works is that rate of generation of energy is acutely sensitive of the changing environment through the density and temperature. A slight change in temperature or density causes drastic changes in the rate of energy production. Imagine a star contracts a little. This increases opacity causing the central density and temperature to go up. The energy production rate follows up, heating the stars a little and allowing it re-expand to its original size. Similarly if it expands a little the opacity goes down, the star loses heat easily, the central temperature and density goes down, with the energy production following down. The star contracts back to its original size. This balance keeps the stars stable for a long time. These considerations allow us to calculate that the central density of the Sun is 150 grams per cubic cm and temperature is 15 million degrees Celsius. Remember the density and temperature at the surface of the Sun is only 1 grams per cubic cm and 5500 degree Celsius.
Cecile Payne was a british graduate student working in Harvard University following in the footsteps of Henrietta Leavitt (whose desk she got). Her Ph.D. was described by Otto Struve as the greatest Ph.D. in astronomy. She was the first woman to hold a position in Harvard that wasn't specifically designed for women. She discovered that most of the mass in stars was in form of hydrogen. Following her discovery it was expected that nuclear reactions involving hydrogen should dominate the energy producing reactions in the stars. The two people responsible for the discovery of the actual equation responsible for the energy production in the Sun had curiously divergent paths. Hans Bethe, after his work on the energy production in stars became a US citizen in 1941, headed the theoretical physics division of the Manhattan project and proceeded to become a vocal activist for nuclear disarmament and responsibility. Carl Weizsacker on the other hand stayed back in Germany and with Ernst Heisenberg and Otto Hahn was heavily involved in the development of the Nazi atomic bomb. There has been considerable controversy over his role in it. He has claimed that he deliberated sabotaged the research which helped delay the Nazi atomic bomb.
The problem of the nuclear reaction of hydrogen was that it was not
possible to combine two hydrogen nuclei to form a helium nucleus. The
reaction, 
energy doesn't occur because there is no observed isotope of Helium
of atomic weight two, presumably because this is too unstable. Instead
the reaction of the hydrogen must proceed circuitously. First the
hydrogen nucleus form a deuterium nucleus, a heavier isotope of
hydrogen,
where, 
is a positron, which is identical to an electron except
for a positive charge instead of negative and 
refers to a
neutrino. This deuterium nucleus reacts with a hydrogen nucleus to
produce a helium nucleus of atomic weight 3,
In the above 
is a photon which is how the extra energy
is released. Two of these helium nuclei can react with each other to
produce a helium atom of the atomic weight 4, the common isotope of
helium,
This chain of reactions is called the p-p chain. Under the
conditions that occur in the center of the Sun the first reaction
would take on average 
years, the second 4 seconds and
the third 
years. So clearly the speed is set by the
first reaction. Even though it is so slow, because the density of
hydrogen in the center is so high, enough number of reactions occur to
produce the amount of energy needed to keep the Sun stable.
One of the most important issues that faced astronomy was how were
elements that were heavier than helium formed. George Gamov was a
Russian astrophycist who formulated the model of the Universe as we
understand it today, the Big bang model. In a paper written with Ralph
Alpher (to which Hans Bethe's name was attached so it became a paper
written by Alpher, Bethe and Gamov!) he suggested that all elements
were constructed out of hydrogen in the very early Universe. A later
paper by Burbridge, Burbridge, Fowler and Hoyle (referred to as
B
FH) showed that only the lightest elements were formed in the Big
bang, hydrogen, helium, deuterium and lithium. All the remaining
elements were formed in the heavy stars. In the center of the heavy
star gradually ashes of previous burnings act as fuels as the cores of
the stars get hotter and denser. Thus three helium nuclei combine to
produce carbon, carbon produces neon, neon turns into
magnesium. Carbon also produces oxygen which in its turn produces
sulphur and silicon. Finally silicon burns into iron. Iron is the last
element that can be produced this way. Regardless how hot or dense the
core of the star gets iron refuses to burn. And eventually the gravity
of the core can overcome the pressure and induce the core to
collapse. And the star eventually explodes in supernova leaving behind
a neutron star.
rays
Nuclear forces that hold the nucleons (constituents of nuclei)
together are very strong (compared to gravity or electromagnetic
forces). This means that a lot of energy is released in nuclear
reactions. Remember that the energy of a photon is proportional to its
frequency. So if a photon has to carry away all that energy it must
have a very high frequency. In fact they are rays, with
wavelengths about meters (frequencies around 
cycles per second)! Remember visible light has frequencies around
meters and frequencies around 
cycles per
second. Clearly the sun isn't just emitting radiation in
rays. If it were we would be at least different from what we are
today, if not all quite dead. The reason is the huge ball of gas that
is around the core producing the radiation. This gas constantly
absorbs and re-emits the radiation. Sometimes the atoms of gas absorb
a photon and re-emits two or more photons of lower energy. This
happens many many times. The radiation forgets that it was first
generated as rays in the center of the Sun. It only remember
the temperature of the last patch of gas that it left. So the light
that we receive, from the surface of the Sun, only depends on the
temperature of the surface layers. This is why the light from the Sun,
and other stars, is so close to a blackbody of the same temperature as
the surface of the Sun. This is also why the internal density and
temperature of the Sun is not determined by the nuclear processes that
occur in it. The light leaving surface of the Sun reflects the
self-regulatory mechanism of the opacity and the structure of the Sun
(the run of density and temperature at different depths) constantly
adjusting to keep the Sun in the same state.
Although the light we receive from the Sun only tells us about the surface, it having long since forgotten where it originally came from, the neutrinos however reach us directly from the center of the Sun. Neutrinos are very weakly interacting particles. They streak through the Sun unimpeded. They streak through the Earth nearly unimpeded as well. With huge tanks of fluid however we can stop a few per day. They induce changes in the elements in the tank according to their energies. From their number and energies we can test the theories of nuclear generation of energy in the Sun. Most recent observations point to some possible discrepancies between the observations and the expected numbers from the Sun. There are two possibilities, that the solar model is incorrect, or that our understanding of neutrinos is incorrect. Future experiments that can tell these two apart are expected to give better answers. There are already competing theories for neutrinos that could explain the discrepancy.