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Kelvin–Helmholtz mechanism

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Process of energy release of a contracting star or planet
Not to be confused withKelvin–Helmholtz instability orGravitational collapse.
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TheKelvin–Helmholtz mechanism is anastronomical process that occurs when the surface of astar or aplanet 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 onJupiter andSaturn and onbrown dwarfs whose central temperatures are not high enough to undergohydrogen 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 byKelvin andHelmholtz in the late nineteenth century to explain the source of energy of theSun. By the mid-nineteenth 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 SirArthur 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 theage 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, when it was shown byHans Bethe to benuclear fusion.

Power generated by a Kelvin–Helmholtz contraction

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It was theorised that thegravitational 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 uniformdensity), it was approximated to a perfect sphere made up ofconcentric 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 fromNewtonian mechanics is defined as:[3]

U=−Gm1m2r,{\displaystyle U=-{\frac {Gm_{1}m_{2}}{r}},}

whereG is thegravitational constant, and the two masses in this case are that of the thin shells of widthdr, and the contained mass within radiusr as one integrates between zero and the radius of the total sphere. This gives:[3]

U=−G∫0Rm(r)4πr2ρrdr,{\displaystyle U=-G\int _{0}^{R}{\frac {m(r)4\pi r^{2}\rho }{r}}\,dr,}

whereR is the outer radius of the sphere, andm(r) is the mass contained within the radiusr. Changingm(r) into a product of volume and density to satisfy the integral,[3]

U=−G∫0R4πr3ρ4πr2ρ3rdr=−1615Gπ2ρ2R5.{\displaystyle 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 total gravitational potential energy as[3]

U=−3GM25R.{\displaystyle U=-{\frac {3GM^{2}}{5R}}.}

According to theVirial Theorem, the total energy for gravitationally bound systems in equilibrium is one half of the time-averaged potential energy,

Ur=|⟨U⟩|2=3GM210R.{\displaystyle U_{r}={\frac {|\langle U\rangle |}{2}}={\frac {3GM^{2}}{10R}}.}

While uniform density is not correct, one can get a roughorder of magnitude estimate of the expected age of our star by inserting known values for themass andradius of the Sun, and then dividing by the knownluminosity of the Sun (note that this will involve another approximation, as the power output of the Sun has not always been constant):[3]

UrL⊙≈1.1×1041 J3.828×1026 W=2.874×1014 s≈8900000 years,{\displaystyle {\frac {U_{\text{r}}}{L_{\odot }}}\approx {\frac {1.1\times 10^{41}~{\text{J}}}{3.828\times 10^{26}~{\text{W}}}}=2.874\times 10^{14}~\mathrm {s} \,\approx 8\,900\,000~{\text{years}},}

whereL⊙{\displaystyle L_{\odot }} is the luminosity of the Sun. While giving enough power for considerably longer than many other physical methods, such aschemical 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 thatthermonuclear energy was responsible for the power output and long lifetimes of stars.[4]

The flux ofinternal heat for Jupiter is given by the derivative according to the time of the total energy

dUrdt=−3GM210R2dRdt=−1.46×1028 [J/m] ×dRdt [m/s].{\displaystyle {\frac {dU_{r}}{dt}}={\frac {-3GM^{2}}{10R^{2}}}{\frac {dR}{dt}}=-1.46\times 10^{28}~{\text{[J/m]}}~\times {\frac {dR}{dt}}~{\text{[m/s]}}.}

With a shrinking of−1 mmyr=−0.001 myr=−3.17×10−11 ms{\textstyle -1\mathrm {\frac {~mm}{yr}} =-0.001\mathrm {\frac {~m}{yr}} =-3.17\times 10^{-11}~\mathrm {\frac {m}{s}} }, one gets

dUrdt=4.63×1017 W,{\displaystyle {\frac {dU_{r}}{dt}}=4.63\times 10^{17}~{\text{W}},}

dividing by the whole area of Jupiter, i.e.S=6.14×1016 m2{\displaystyle S=6.14\times 10^{16}~\mathrm {m^{2}} }, one gets

1SdUrdt=7.5 Wm2.{\displaystyle {\frac {1}{S}}{\frac {dU_{r}}{dt}}=7.5~\mathrm {\frac {W}{m^{2}}} .}

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 theCassini 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

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  1. ^Patrick G. J. Irwin (2009).Giant Planets of Our Solar System: Atmospheres, Composition, and Structure 2nd edition. Springer. pp. 4–5.ISBN 978-3-642-09888-8.
  2. ^Liming, Li; et al. (2018)."Less absorbed solar energy and more internal heat for Jupiter".Nature Communications.9 (3709):1–10.Bibcode:2018NatCo...9.3709L.doi:10.1038/s41467-018-06107-2.PMC 6137063.PMID 30213944.
  3. ^abcdeCarroll, Bradley W.; Ostlie, Dale A. (2007).An Introduction to Modern Astrophysics (2nd ed.). Pearson Addison Wesley. pp. 296–298.ISBN 978-0-8053-0402-2. Archived fromthe original on 2015-12-22.
  4. ^Pogge, Richard (2006-01-15)."The Kelvin-Helmholtz Mechanism".Lecture 12: As Long as the Sun Shines.Ohio State University. Retrieved2009-11-05.
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