The Kelvin-Helmholtz mechanism is an astronomical event that occurs when the surface of a star or a planet cools. As a result of this cooling, the pressure drops, and the star or planet compresses to compensate. This compression, in turn, heats up the core of the star/planet. This mechanism is evident on Jupiter and Saturn.

The mechanism was originally proposed by Kelvin and Helmholtz in the late 1800s to explain the source of energy of the sun. As we now know, the amount of energy generated by the Kelvin-Helmholtz mechanism is far too low to power the sun.

Power generated by a Kelvin-Helmholtz contraction

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.

Gravitational potential energy from Newtonian mechanics is defined as:

$U\; =\; -\backslash frac\{Gm\_1m\_2\}\{r\}$

Where G is the gravitational constant, and the two masses in this case are that of the thin shells of width dr, and the contained mass within radius r as one integrates between zero and the radius of the total sphere. This gives:

$U\; =\; -G\backslash int\_\{0\}^\{R\}\; \backslash frac\{m(r)\; 4\; \backslash pi\; r^2\; \backslash rho\}\{r\}\backslash ,\; dr$

Where R is the outer radius of the sphere, and m(r) is the mass contained within the radius r. Changing m(r) into a product of volume and density to satisfy the integral:

$U\; =\; -G\backslash int\_\{0\}^\{R\}\; \backslash frac\{4\; \backslash pi\; r^3\; \backslash rho\; 4\; \backslash pi\; r^2\; \backslash rho\}\{3r\}\backslash ,\; dr\; =\; -\backslash frac\{16\}\{15\}G\; \backslash pi^2\; \backslash rho^2\; R^5$

Dividing this by the total mass of the sphere as a product of volume and density gives the final answer:

$U\; =\; -\backslash frac\{3M^2G\}\{5R\}$

While uniform density is not correct, one can get a rough order of magnitude estimate of the expected lifetime of our star by putting in known values for the mass and radius of the sun, and then dividing by the known luminosity of the sun. Note this will involve another approximation, as the power output of the sun has not always been constant.

$\backslash frac\{U\}\{L\_\backslash bigodot\}\; \backslash approx\; \backslash frac\{2.3\; \backslash times\; 10^\{41\}\}\{4\; \backslash times\; 10^\{26\}\}\; \backslash approx\; 18,220,650\backslash \; yrs$

Where L is the luminosity of the sun. While giving enough power for considerably longer than many other physical methods, such as electrochemical energy, this value was clearly still not long enough due to evidence to the contrary. It was eventually discovered that thermonuclear energy was responsible for the power output and long lifetimes of stars.

Astronomy

Kelvin-Helmholtz-Mechanismus | Mécanisme de Kelvin-Helmholtz | Meccanismo di Kelvin-Helmholtz | Mechanizm Kelvina-Helmholtza