Kelvin–Helmholtz mechanism
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Theoretical concepts |
The Kelvin–Helmholtz mechanism is an astronomical process that occurs when the surface of a star or a planet cools. The cooling causes the internal 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 on Jupiter and Saturn and on brown dwarfs whose central temperatures are not high enough to undergo hydrogen fusion. It is estimated that Jupiter radiates more energy through this mechanism than it receives from the Sun, but Saturn might not. Jupiter has been estimated to shrink at a rate of approximately 1 mm/year by this process,[1] corresponding to an internal flux of 7.485 W/m2.[2]
The mechanism was originally proposed by
However, it soon was recognized by Sir
Power generated by a Kelvin–Helmholtz contraction
It was theorised that the
Gravitational potential energy from
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:[3]
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,[3]
Recasting in terms of the mass of the sphere gives the total gravitational potential energy as[3]
According to the
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):[3]
where is the luminosity of the Sun. 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.[4]
The flux of internal heat for Jupiter is given by the derivative according to the time of the total energy
With a shrinking of , one gets
dividing by the whole area of Jupiter, i.e. , one gets
Of course, one usually calculates this equation in the other direction: the experimental figure of the specific flux of internal heat, 7.485 W/m2, was given from the direct measures made on the spot by the
References
- ISBN 978-3-642-09888-8.
- PMID 30213944.
- ^ ISBN 978-0-8053-0402-2. Archived from the originalon 2015-12-22.
- ^ Pogge, Richard (2006-01-15). "The Kelvin-Helmholtz Mechanism". Lecture 12: As Long as the Sun Shines. Ohio State University. Retrieved 2009-11-05.