The Kelvin–Helmholtz mechanism is an
astronomical
Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest include ...
process that occurs when the surface of a
star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
or a
planet
A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
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
Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...
and
Saturn
Saturn is the sixth planet from the Sun and the second largest in the Solar System, after Jupiter. It is a gas giant, with an average radius of about 9 times that of Earth. It has an eighth the average density of Earth, but is over 95 tim ...
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, corresponding to an internal flux of 7.485 W/m
2.
The mechanism was originally proposed by
Kelvin and
Helmholtz in the late nineteenth century to explain the source of energy of the
Sun. By the mid-nineteenth century,
conservation of energy
The law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be Conservation law, ''conserved'' over time. In the case of a Closed system#In thermodynamics, closed system, the principle s ...
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
The age of Earth is estimated to be 4.54 ± 0.05 billion years. This age may represent the age of Earth's accretion (astrophysics), accretion, or Internal structure of Earth, core formation, or of the material from which Earth formed. This dating ...
. (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, when it was shown by
Hans Bethe to be
nuclear fusion
Nuclear fusion is a nuclear reaction, reaction in which two or more atomic nuclei combine to form a larger nuclei, nuclei/neutrons, neutron by-products. The difference in mass between the reactants and products is manifested as either the rele ...
.
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
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
), 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
Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows:
# A body r ...
is defined as:
:
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:
[
:
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,][
:
Recasting in terms of the mass of the sphere gives the total gravitational potential energy as][
:
According to the Virial Theorem, the total energy for gravitationally bound systems in equilibrium is one half of the time-averaged potential energy,
:
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
Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
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):[
:
where is the luminosity of the Sun. While giving enough power for considerably longer than many other physical methods, such as ]chemical energy
Chemical energy is the energy of chemical substances that is released when the substances undergo a chemical reaction and transform into other substances. Some examples of storage media of chemical energy include batteries, Schmidt-Rohr, K. (20 ...
, 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.[
]
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 Cassini probe during its flyby on 30 December 2000 and one gets the amount of the shrinking, ~1 mm/year, a minute figure below the boundaries of practical measurement.
References
{{DEFAULTSORT:Kelvin-Helmholtz Mechanism
Concepts in astrophysics
Effects of gravity
Mechanism
Stellar evolution
William Thomson, 1st Baron Kelvin