Kelvin-Helmholtz mechanism

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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. It is estimated that Jupiter and Saturn each radiate more energy through this mechanism than each receives from the Sun.

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.

[edit] 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 = -\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\int_{0}^{R} \frac{m(r) 4 \pi r^2 \rho}{r}\, 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\int_{0}^{R} \frac{4 \pi r^3 \rho 4 \pi r^2 \rho}{3r}\, dr = -\frac{16}{15}G \pi^2 \rho^2 R^5

Recasting in terms of the mass of the sphere gives the final answer:

U = -\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 inserting 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.

\frac{U}{L_\bigodot} \approx \frac{2.3 \times 10^{41}}{4 \times 10^{26}} \approx 18,220,650\ 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.

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