Kelvin–Helmholtz mechanism

Description

The Kelvin–Helmholtz mechanism is an astronomical process that occurs when the surface of a star or a planet cools. The cooling causes the pressure to drop, and the star or planet shrinks as a result. This compression, in turn, heats the core of the star/planet. This mechanism is evident in T Tauri stars on their evolutionary path to the main sequence.

The mechanism was originally proposed by Kelvin and Helmholtz in the late 19th century to explain the source of energy of the Sun. By the mid-19th century, conservation of energy had been accepted, and one consequence of this law of physics is that the Sun must have some energy source to continue to shine. Because nuclear reactions were unknown, the main candidate for the source of solar energy was gravitational contraction.

However, it soon was recognized by Sir Arthur Eddington and others that the total amount of energy available through this mechanism only allowed the Sun to shine for millions of years rather than the billions of years that the geological and biological evidence suggested for the age of the Earth. (Kelvin himself had argued that the Earth was millions, not billions, of years old.) The true source of the Sun’s energy remained uncertain until the 1930s, in which it was shown by Hans Bethe to be nuclear fusion.

The total radiated energy generated by a Kelvin–Helmholtz contraction is shown.

It was theorised that the gravitational potential energy from the contraction of the Sun could be its source of power. To calculate the total amount of energy that would be released by the Sun in such a mechanism (assuming uniform density), it was approximated to a perfect sphere made up of concentric shells. The gravitational potential energy could then be found as the integral over all the shells from the centre to its outer radius.

While uniform density is not correct, one can get a rough order of magnitude estimate of the expected age of our star by inserting known values for the mass and radius of the Sun, and then dividing by the known luminosity of the Sun (note that this will involve another approximation, as the power output of the Sun has not always been constant). ~8 900 00 years.

While giving enough power for considerably longer than many other physical methods, such as chemical energy, this value was clearly still not long enough due to geological and biological evidence that the Earth was billions of years old. It was eventually discovered that thermonuclear energy was responsible for the power output and long lifetimes of stars.

Related formulas

Variables

Urtotal radiated energy (joule)
GNewtonian constant of gravitation
Mmass of the sphere (kg)
Router radius of the sphere (m)