Thermonuclear Fusion
The Sun, like all other stable stars, produces its energy by means of thermonuclear fusion. In a fusion reaction, hydrogen is heated and compressed to extreme temperatures and pressures. This is the basis for hydrogen bombs. But that is a rather uncontrolled reaction. Scientists here on Earth are actively engaged in research that would allow us to utilize this very efficient method of producing energy. Recently, they seemed to have made significant progress.
In the hot, dense core of the Sun the process is specifically called the proton--proton chain takes place. The element hydrogen is converted into the element helium. Here is a nice interactive demonstration of the process.
The by-products of this chain of reactions are some subatomic particles, and quite a bit of energy. Perhaps the most interesting of the subatomic particles is a massless, neutral, particle called the neutrino. Whereas the energy produced by the Sun takes over million years to reach the surface, neutrinos pass through the entire mass of the Sun without impediment to reach the Earth in little over eight minutes.
There are a number of neutrino telescopes, such as the Super Kamiokande, which hope to detect these elusive particles. Here is a good summary of these experiments. This Youtube video highlights one such solar neutrino telescope installation, Borexino.
One of the main issues in solar physics is something called the Solar Neutrino Problem. It seems that we are not detecting the number of neutrinos predicted by our models. One solution to this 'problem' is that one neutrino may oscillate and change into another type of neutrino.
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Updated March 1 2011