For other shaped objects, the gravitational potential energy is generally proportional to —GM2/R, where R is some average length. The constant of proportionality is generally close to unity.
The thermal energy is (3/2)kT per particle. The total thermal energy is then
where N is the total number of particles in the cloud. If the mass per particle is m, and the total mass of the cloud is M, then
(0.6)(6.67 X 10—8 dyn cm2/g2)(2 X 1033g)2 (7 X 1010 cm)
This gives a lifetime, called the Kelvin time,
This is a lifetime of 20 million years. It may sound like a long time. However, we know from geological evidence that the Earth has been around for over four billion years. This means that the Sun must be at least that old. Therefore, the Sun (and presumably other stars) cannot exist in a stable configuration on stored gravitational energy.
An alternative source of energy could be chemical reactions. After all, we use chemical reactions to make automobiles go on Earth. We can estimate the amount of energy stored in the Sun, capable of being released in chemical reactions. Typical energies of these chemical reactions should be equivalent to some fraction of the binding energy of molecules that might be formed and destroyed in these reactions. This means that we might expect something like 1 eV per atom in the Sun. The total chemical energy is then 1 eV multiplied by the total number of atoms in the Sun, M/m. The energy available is then
(1 eV)(1.6 X 10~12 erg/eV)(2 X 1033 g) (1.67 X 10~24 g)
This is much less than is stored in gravitational potential energy, so chemical reactions clearly cannot provide a longer term energy source for the Sun.
You might wonder if we have not been too casual in our dismissal of chemical reactions as a possible energy source for the Sun. After all, we didn't even say what chemical reactions might be involved. The important point is that any chemical reaction involves moving electrons from atom to atom, and the energies associated with this are a few eV, independent of what the reaction is. Suppose there were some very energetic reaction that produced 10 eV per atom, that would increase the chemical energy by a factor of ten but it would still be many orders of magnitude short of the required amount. If our estimate had shown that chemical reactions might work, then we would have to worry about the details, figure out what reactions were important and then do a more accurate calculation of the stored energy. However, our estimate tells us that it is not worth wasting our time on the details. These types of calculations, called order of magnitude calculations, are very important in astronomy. They help us eliminate processes that obviously don't work and allow us to focus our attention on possibilities that might.
The answer to the problem of stellar energy sources is nuclear reactions. The typical energies available in nuclear reactions are about 1 MeV per atom, instead of 1 eV. This is an improvement of a factor of 106. To see how nuclear reactions provide energy for the Sun, we look at some of the elements of nuclear physics.
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