The source of the Sun's power was for a long time a mystery. The nature of energy itself was not clear. It was imagined that something was being burnt, but the only things that were known of that could be burnt to produce a lot of energy were wood, coal, oil and gas. Simple calculations showed that none of these could power the Sun for more than a few thousand years. When it was discovered in the nineteenth century that the Earth wa very very old indeed, it soon became clear that these sources of energy could not possibly be the Sun's energy source.
One other possible candidate for the energy source of the Sun was gravitational contraction. When a gas cloud like the cloud that forms the Sun contracts, it heats up. If it contracts enough, it could heat up far enough to ignite. We can estimate how long the Sun could be powered by this source of energy. We can picture the Sun as a sphere of radius r and mass M. If a point mass of mass dm is brought close to it, the gravitational energy released is
If the mass
is in fact in the form of a spherical shell, of radius
and thickness
then
where
is the density of the shell. If we assume the Sun to have uniform densitythen
hence
Now integrate to find
The density %rho of the Sun is
and the radius is
The luminosity of the Sun is
so the energy of gravitational contraction could power the Sun for
In a year there are
so gravitational contraction could power the Sun for less than a year. Some other source of energy is needed.