Tidal heating

Last updated January 28, 2024

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 Gas Giants

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

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 over perhaps 100 million years, and that this could have accounted for a temperature increase of up to 5°C on the early Earth.^[5]^[6]

Moon

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

Io

Jupiter's closest moon Io experiences considerable tidal heating.

Formula

The tidal heating rate, ${\dot {E}}_{\text{Tidal}}$ , in a satellite that is spin-synchronous, coplanar ( $I=0$ ), and has an eccentric orbit is given by:

{\dot {E}}_{\text{Tidal}}=-\operatorname {Im} (k_{2}){\frac {21}{2}}{\frac {GM_{h}^{2}R^{5}ne^{2}}{a^{6}}}

where $R$ , $n$ , $a$ , and $e$ are respectively the satellite's mean radius, mean orbital motion, orbital distance, and eccentricity.^[8] $M_{h}$ is the host (or central) body's mass and $\operatorname {Im} (k_{2})$ 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]

Related Research Articles

The Galilean moons, or Galilean satellites, are the four largest moons of Jupiter: Io, Europa, Ganymede, and Callisto. They are the most readily visible Solar System objects after the unaided visible Saturn, the dimmest of the classical planets, allowing observation with common binoculars, even under night sky conditions of high light pollution. The invention of the telescope enabled the discovery of the moons in 1610. Through this, they became the first Solar System objects discovered since humans have started tracking the classical planets, and the first objects to be found to orbit any planet beyond Earth.

In celestial mechanics, orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers. Most commonly, this relationship is found between a pair of objects. The physical principle behind orbital resonance is similar in concept to pushing a child on a swing, whereby the orbit and the swing both have a natural frequency, and the body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be self-correcting and thus stable. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa and Io, and the 2:3 resonance between Neptune and Pluto. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance between bodies with similar orbital radii causes large planetary system bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood, an effect that is used in the current definition of a planet.

Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite and the primary planet that it orbits. The acceleration causes a gradual recession of a satellite in a prograde orbit, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking, usually of the smaller body first, and later the larger body. The Earth–Moon system is the best-studied case.

Europa, or Jupiter II, is the smallest of the four Galilean moons orbiting Jupiter, and the sixth-closest to the planet of all the 95 known moons of Jupiter. It is also the sixth-largest moon in the Solar System. Europa was discovered independently by Simon Marius and Galileo Galilei and was named after Europa, the Phoenician mother of King Minos of Crete and lover of Zeus.

<span class="mw-page-title-main">Natural satellite</span> Astronomical body that orbits a planet

A natural satellite is, in the most common usage, an astronomical body that orbits a planet, dwarf planet, or small Solar System body. Natural satellites are colloquially referred to as moons, a derivation from the Moon of Earth.

Ganymede, or Jupiter III, is the largest and most massive natural satellite of Jupiter as well as the largest in the Solar System, being a planetary-mass moon. It is the largest Solar System object without an atmosphere, despite being the only moon in the Solar System with a substantial magnetic field. Like Titan, Saturn's largest moon, it is larger than the planet Mercury, but has somewhat less surface gravity than Mercury, Io or the Moon due to its lower density compared to the three.

Tidal locking between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit. In the case where a tidally locked body possesses synchronous rotation, the object takes just as long to rotate around its own axis as it does to revolve around its partner. For example, the same side of the Moon always faces the Earth, although there is some variability because the Moon's orbit is not perfectly circular. Usually, only the satellite is tidally locked to the larger body. However, if both the difference in mass between the two bodies and the distance between them are relatively small, each may be tidally locked to the other; this is the case for Pluto and Charon, as well as for Eris and Dysnomia. Alternative names for the tidal locking process are gravitational locking, captured rotation, and spin–orbit locking.

Hot Jupiters are a class of gas giant exoplanets that are inferred to be physically similar to Jupiter but that have very short orbital periods. The close proximity to their stars and high surface-atmosphere temperatures resulted in their informal name "hot Jupiters".

Io, or Jupiter I, is the innermost and third-largest of the four Galilean moons of the planet Jupiter. Slightly larger than Earth's moon, Io is the fourth-largest moon in the Solar System, has the highest density of any moon, the strongest surface gravity of any moon, and the lowest amount of water of any known astronomical object in the Solar System. It was discovered in 1610 by Galileo Galilei and was named after the mythological character Io, a priestess of Hera who became one of Zeus's lovers.

An exomoon or extrasolar moon is a natural satellite that orbits an exoplanet or other non-stellar extrasolar body.

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<span class="mw-page-title-main">Habitability of natural satellites</span> Measure of the potential of natural satellites to have environments hospitable to life

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The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have historically been stable as observed, and will be in the "short" term, their weak gravitational effects on one another can add up in ways that are not predictable by any simple means.

In astronomy, a regular moon or a regular satellite is a natural satellite following a relatively close, stable, and circular orbit which is generally aligned to its primary's equator. They formed within discs of debris that surround their primary, usually the aftermath of a large collision or material accumulated from the protoplanetary disc. These debris discs then accrete into regular moons, as opposed to irregular moons, which were captured.

<span class="mw-page-title-main">Retrograde and prograde motion</span> Relative directions of orbit or rotation

Retrograde motion in astronomy is, in general, orbital or rotational motion of an object in the direction opposite the rotation of its primary, that is, the central object. It may also describe other motions such as precession or nutation of an object's rotational axis. Prograde or direct motion is more normal motion in the same direction as the primary rotates. However, "retrograde" and "prograde" can also refer to an object other than the primary if so described. The direction of rotation is determined by an inertial frame of reference, such as distant fixed stars.

A lava planet is a type of terrestrial planet, with a surface mostly or entirely covered by molten lava. Situations where such planets could exist include a young terrestrial planet just after its formation, a planet that has recently suffered a large collision event, or a planet orbiting very close to its star, causing intense irradiation and tidal forces.

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References

1 2 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.
↑ Peale, S.J. (2003). "Tidally induced volcanism". Celestial Mechanics and Dynamical Astronomy 87, 129–155.
↑ 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.
↑ 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 .
↑ 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.
↑ Jure Japelj (11 January 2022). "How Much Did the Moon Heat Young Earth?". EOS. 103. doi: 10.1029/2022EO220017 . Retrieved 26 March 2023.
↑ 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.
↑ 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. doi:10.1016/0019-1035(88)90001-2.
↑ 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.
↑ 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 .
↑ 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 .
↑ 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 .

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[Peale1979-1] 1 2 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.

[munkwunsch98-3] 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.

[egbert01-4] 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 .

[Heller21-5] 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.

[harada14-7] 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. doi:10.1016/0019-1035(88)90001-2.

[Henning2009-9] 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.

[RenaudHenning2018-10] 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 .

[E2012-11] 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 .

[EM2014-12] 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 .

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