Tidal heating

Orbital and friction heating on a planet or moon oceans, or interior

Tidal heating (also known as tidal working or tidal flexing) occurs through the tidal friction processes: orbital and rotational energy is dissipated as heat in either (or both) the surface ocean or interior of a planet or satellite. When an object is in an elliptical orbit, the tidal forces acting on it are stronger near periapsis than near apoapsis. Thus the deformation of the body due to tidal forces (i.e. the tidal bulge) varies over the course of its orbit, generating internal friction which heats its interior. This energy gained by the object comes from its orbital energy and/or rotational energy, so over time in a two-body system, the initial elliptical orbit decays into a circular orbit (tidal circularization) and the rotational periods of the two bodies adjust towards matching the orbital period (tidal locking). Sustained tidal heating occurs when the elliptical orbit is prevented from circularizing due to additional gravitational forces from other bodies that keep tugging the object back into an elliptical orbit. In this more complex system, orbital and rotational energy still is being converted to thermal energy; however, now the orbit's semimajor axis would shrink rather than its eccentricity.

## Moons of giant planets

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Main article: Tidal heating of Io

Tidal heating is responsible for the geologic activity of the most volcanically active body in the Solar System: Io, a moon of Jupiter. Io's eccentricity persists as the result of its orbital resonances with the Galilean moons Europa and Ganymede.[1] The same mechanism has provided the energy to melt the lower layers of the ice surrounding the rocky mantle of Jupiter's next-closest large moon, Europa. However, the heating of the latter is weaker, because of reduced flexing—Europa has half Io's orbital frequency and a 14% smaller radius; also, while Europa's orbit is about twice as eccentric as Io's, tidal force falls off with the cube of distance and is only a quarter as strong at Europa. Jupiter maintains the moons' orbits via tides they raise on it and thus its rotational energy ultimately powers the system.[1] Saturn's moon Enceladus is similarly thought to have a liquid water ocean beneath its icy crust, due to tidal heating related to its resonance with Dione. The water vapor geysers which eject material from Enceladus are thought to be powered by friction generated within its interior.[2]

## Earth

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Munk & Wunsch (1998) estimated that Earth experiences 3.7 TW (0.0073 W/m2) of tidal heating, of which 95% (3.5 TW or 0.0069 W/m2) is associated with ocean tides and 5% (0.2 TW or 0.0004 W/m2) is associated with Earth tides, with 3.2 TW being due to tidal interactions with the Moon and 0.5 TW being due to tidal interactions with the Sun.[3] Egbert & Ray (2001) confirmed that overall estimate, writing "the total amount of tidal energy dissipated in the Earth-Moon-Sun system is now well-determined. The methods of space geodesy—altimetry, satellite laser ranging, lunar laser ranging—have converged to 3.7 TW ..."[4]

Heller et al. (2021) estimated that shortly after the Moon was formed, when the Moon orbited 10-15 times closer to Earth than it does now, tidal heating might have contributed ~10 W/m2 of heating. This heating happened over perhaps 100 million years, and could have accounted for a temperature increase of up to 5°C on the early Earth.[5][6]

## Moon

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Harada et al. (2014) proposed that tidal heating may have created a molten layer at the core-mantle boundary within Earth's Moon.[7]

## Formula

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The tidal heating rate,  E ˙ Tidal {\displaystyle {\dot {E}}_{\text{Tidal}}} , in a satellite that is spin-synchronous, coplanar ( I = 0 {\displaystyle I=0} ), and has an eccentric orbit is given by:  E ˙ Tidal = − Im ⁡ ( k 2 ) 21 2 G M h 2 R 5 n e 2 a 6 {\displaystyle {\dot {E}}_{\text{Tidal}}=-\operatorname {Im} (k_{2}){\frac {21}{2}}{\frac {GM_{h}^{2}R^{5}ne^{2}}{a^{6}}}} where  R {\displaystyle R} ,  n {\displaystyle n} ,  a {\displaystyle a} , and  e {\displaystyle e} are respectively the satellite's mean radius, mean orbital motion, orbital distance, and eccentricity.[8] M h {\displaystyle M_{h}} is the host (or central) body's mass and  Im ⁡ ( k 2 ) {\displaystyle \operatorname {Im} (k_{2})} represents the imaginary portion of the second-order Love Number which measures the efficiency at which the satellite dissipates tidal energy into frictional heat. This imaginary portion is defined by interplay of the body's rheology and self-gravitation. It, therefore, is a function of the body's radius, density, and rheological parameters (the shear modulus, viscosity, and others – dependent upon the rheological model).[9][10] The rheological parameters' values, in turn, depend upon the temperature and the concentration of partial melt in the body's interior.[11]

The tidally dissipated power in a nonsynchronised rotator is given by a more complex expression.[12]

## See also

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* Cryovolcano
* Tidal acceleration
* Tidal locking
* Io Volcano Observer
* Planetary differentiation

## References

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1. ^ a b Peale, S.J.; Cassen, P.; Reynolds, R.T. (1979). "Melting of Io by Tidal Dissipation". Science. 203 (4383): 892–894\. Bibcode:1979Sci...203..892P. doi:10.1126/science.203.4383.892. JSTOR 1747884. PMID 17771724. S2CID 21271617.
2. ^ Peale, S.J. (2003). "Tidally induced volcanism". Celestial Mechanics and Dynamical Astronomy 87, 129–155.
3. ^ Munk, Walter; Wunsch, Carl (1998). "Abyssal recipes II: energetics of tidal and wind mixing" (PDF). Deep Sea Research Part I: Oceanographic Research Papers. 45 (12): 1977–2010\. Bibcode:1998DSRI...45.1977M. doi:10.1016/S0967-0637(98)00070-3. Retrieved 26 March 2023.
4. ^ Egbert, Gary D.; Ray, Richard D. (October 15, 2001). "Estimatesof tidal energy dissipationfrom TOPEX/Poseidon altimeter data". Journal of Geophysical Research. 106 (C10): 22475–22502\. Bibcode:2001JGR...10622475E. doi:10.1029/2000JC000699.
5. ^ Heller, R; Duda, JP; Winkler, M; Reitner, J; Gizon, L (2021). "Habitability of the early Earth: liquid water under a faint young Sun facilitated by strong tidal heating due to a closer Moon". PalZ. 95 (4): 563–575\. arXiv:2007.03423. Bibcode:2021PalZ...95..563H. doi:10.1007/s12542-021-00582-7. S2CID 244532427.
6. ^ Jure Japelj (11 January 2022). "How Much Did the Moon Heat Young Earth?". EOS. 103. doi:10.1029/2022EO220017. Retrieved 26 March 2023.
7. ^ Harada, Y; Goosens, S; Matsumoto, K; Yan, J; Ping, J; Noda, H; Harayama, J (27 July 2014). "Strong tidal heating in an ultralow-viscosity zone at the core–mantle boundary of the Moon". Nature Geoscience. 7 (8): 569–572\. Bibcode:2014NatGe...7..569H. doi:10.1038/ngeo2211.
8. ^ Segatz, M.; Spohn, T.; Ross, M.N.; Schubert, G. (August 1988). "Tidal dissipation, surface heat flow, and figure of viscoelastic models of Io". Icarus. 75 (2): 187–206\. Bibcode:1988Icar...75..187S. doi:10.1016/0019-1035(88)90001-2.
9. ^ Henning, Wade G. (2009). "Tidally Heated Terrestrial Exoplanets: Viscoelastic Response Models". The Astrophysical Journal. 707 (2): 1000–1015\. arXiv:0912.1907. Bibcode:2009ApJ...707.1000H. doi:10.1088/0004-637X/707/2/1000. S2CID 119286375.
10. ^ Renaud, Joe P.; Henning, Wade G. (2018). "Increased Tidal Dissipation Using Advanced Rheological Models: Implications for Io and Tidally Active Exoplanets". The Astrophysical Journal. 857 (2): 98. arXiv:1707.06701. Bibcode:2018ApJ...857...98R. doi:10.3847/1538-4357/aab784.
11. ^ Efroimsky, Michael (2012). "Tidal Dissipation Compared to Seismic Dissipation: In Small Bodies, in Earths, and in Superearths". The Astrophysical Journal. 746: 150. arXiv:1105.3936. doi:10.1088/0004-637X/746/2/150.
12. ^ Efroimsky, Michael; Makarov, Valeri V. (2014). "Tidal Dissipation in a Homogeneous Spherical Body. I. Methods". The Astrophysical Journal. 795 (1): 6. arXiv:1406.2376. Bibcode:2014ApJ...795....6E. doi:10.1088/0004-637X/795/1/6.