The Kelvin-Helmholtz Timescale

The Sun contains a great deal of gravitational potential energy. You can compute the amount of this energy: it is given by the expression E_g = GM²/R

Quasi-Static Contraction

Suppose the Sun were not in equilibrium: there were no forces opposing gravitational collapse. In order for the collapse to proceed, the pressure supporting the Sun against collapse, provided by the hot gas, would have to be lost. As the gas radiates its energy away, pressure support would decrease, and the Sun would contract. But the release of gravitational potential energy would reheat the Sun (at a slightly smaller radius).

The Sun would slowly contract, radiating away all of its gravitational potential energy. If it does so at its present luminosity, the time it takes to radiate away its energy is given by the total energy divided by the rate the energy is lost (which is the luminosity). This time,

t_KH = GM²/RL is called the Kelvin-Helmholtz Time. It is the time neded to radiate away a significant fraction of the thermal energy of a star. For today's Sun, this timescale is about 30 million years.

Consequences of the Kelvin-Helmholtz Timescale

The Kelvin-Helmholtz Timescale sets the time for

protostars to collapse to the main sequence
main sequence stars to evolve into giants

Note that if the Sun were to cease burning nuclear fuel now, it would continue to shine for tens of millions of years.