Why the sun is so hot

Perhaps the most important contribution Einstein made to our understanding of the nature of reality was the explanation of the relationship that matter has with energy. He expressed this with the equation E=mc², or, the amount of energy represented by a given amount of matter can be given by the product of its mass times the constant that is the speed of light squared. Many see this only as an abstraction to illustrate the tremendous amount of energy bound within the atom but in fact the equation is valid. By way of example, the fusion of two Hydrogen atoms to form one Helium atom is accomplished by combining two Hydrogen atoms with two neutrons, producing a new nucleus with two protons and two neutrons, but this reaction is not accomplished without a change of mass. In fact, the mass of the new Helium atom is only 4.002 atomic mass units (amu), versus 4.0333 amu if the reaction were to take place without a change in mass. There is a loss of mass of 0.031 amu and thus this represents a strong exothermic reaction. The equation E=mc² can be used to calculate the amount of energy released in this reaction.

We find that the amount of energy released in the fusion of a single Helium atom from two Hydrogen atoms and two neutrons equals 28,891,200 electron volts. In ergs this is 4.63 x 10^-5 erg per atom (there are 6.24 x 10^11 electron volts in 1 erg). To appreciate the scope of this amount of force, the erg is a unit of work, or energy, and is equivalent to the work done by a force of 1 dyne moving through a distance of 1 cm, where 1 dyne is the force required to accelerate a mass of 1 gram at the rate of 1 cm/sec².

This does not seem like much until we increase the amount of mass. If we were to create just one gram of helium through the fusion process we would release a tremendous amount of energy because there are 6.023 x 10^23 atoms in the gram-atomic weight of any element (Avogadro's number) and since the gram-atomic weight of Helium is 4.0026 the number of atoms in one gram of Helium is 1.505 x 10^23. The energy released in the formation of 1 gram of Helium is therefore 4.38 x 10^30 electron volts. Since the force of one dyne is equivalent to 6.24 x 10^11 electron volts then the formation of Helium produces a force of 6.968 x 10^18 dynes, or in other words, the work done in ergs is 6.957 x 10^18 ergs.

Now hold on to your hats. A fusion reaction of this type using enough Hydrogen and enough neutrons to yield just 1 gram of Helium would generate enough force to accelerate a one million kilogram mass at the rate of ~7 million kilometres/sec² !

This is of course the ideal case and this type of perfection is not found in nature. This is not the type of fusion reaction in a Hydrogen bomb. In the Hydrogen bomb, a Deuterium ion and a Tritium ion, the positive isotopes of the heavy isotopes of Hydrogen, undergo fusion to form a Helium nucleus and a neutron. This releases energy which is ideally about 40% less by calculation than that of or example - still considerable, but actually the energy produced is much less than that since the conversion is far from perfect. The energy inside the sun is primarily generated by the fusion of four Hydrogen nuclei into one Helium nucleus. The energy released in this process is only about 7% less than that of our hypothetical example. That's why it's so hot.