tidal heating

{{Short description|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

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.{{cite journal |title=Melting of Io by Tidal Dissipation |jstor=1747884 |year=1979 |journal=Science |pages=892–894 |volume=203 |issue=4383 |last1=Peale |first1=S.J. |last2=Cassen |first2=P. |last3=Reynolds |first3=R.T. |doi=10.1126/science.203.4383.892 |pmid=17771724 |bibcode=1979Sci...203..892P|s2cid=21271617 }} 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. 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.Peale, S.J. (2003). "Tidally induced volcanism". Celestial Mechanics and Dynamical Astronomy 87, 129–155.

Earth

Munk & Wunsch (1998) estimated that Earth experiences 3.7 TW (0.0073 W/m{{sup|2}}) of tidal heating, of which 95% (3.5 TW or 0.0069 W/m{{sup|2}}) is associated with ocean tides and 5% (0.2 TW or 0.0004 W/m{{sup|2}}) 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.{{cite journal |last1=Munk |first1=Walter |last2=Wunsch |first2=Carl |title=Abyssal recipes II: energetics of tidal and wind mixing |journal= Deep Sea Research Part I: Oceanographic Research Papers|date=1998 |volume=45 |issue=12 |pages=1977–2010 |doi=10.1016/S0967-0637(98)00070-3 |bibcode=1998DSRI...45.1977M |url=https://www.whoi.edu/cms/files/Munk_Wunsch_DSR_1998_32129.pdf |access-date=26 March 2023}} 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{{nbsp}}..."{{cite journal |last1=Egbert |first1=Gary D. |last2=Ray |first2=Richard D. |title=Estimatesof tidal energy dissipationfrom TOPEX/Poseidon altimeter data |journal=Journal of Geophysical Research |date=October 15, 2001 |volume=106 |issue=C10 |pages=22475–22502 |doi=10.1029/2000JC000699 |bibcode=2001JGR...10622475E |doi-access=free }}

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/m{{sup|2}} 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.{{cite journal |last1=Heller |first1=R |last2=Duda |first2=JP |last3=Winkler |first3=M |last4=Reitner |first4=J |last5=Gizon |first5=L |title=Habitability of the early Earth: liquid water under a faint young Sun facilitated by strong tidal heating due to a closer Moon |journal=PalZ |date=2021 |volume=95 |issue=4 |pages=563–575 |doi=10.1007/s12542-021-00582-7 |arxiv=2007.03423 |bibcode=2021PalZ...95..563H |s2cid=244532427 |url=https://link.springer.com/article/10.1007/s12542-021-00582-7}}{{cite journal |author1=Jure Japelj |title=How Much Did the Moon Heat Young Earth? |url=https://eos.org/articles/how-much-did-the-moon-heat-young-earth |journal=EOS |access-date=26 March 2023 |doi=10.1029/2022EO220017 |date=11 January 2022|volume=103 |doi-access=free }}

Moon

Harada et al. (2014) proposed that tidal heating may have created a molten layer at the core-mantle boundary within Earth's Moon.{{cite journal |last1=Harada |first1=Y |last2=Goosens |first2=S |last3=Matsumoto |first3=K |last4=Yan |first4=J |last5=Ping |first5=J |last6=Noda |first6=H |last7=Harayama |first7=J |title=Strong tidal heating in an ultralow-viscosity zone at the core–mantle boundary of the Moon |journal=Nature Geoscience |date=27 July 2014 |volume=7 |issue=8 |pages=569–572 |doi=10.1038/ngeo2211 |bibcode=2014NatGe...7..569H |url=https://www.researchgate.net/publication/273434039}}

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: $\dot{E}_\text{Tidal} = -\operatorname{Im}(k_2)\frac{21}{2} \frac{G M_h^2 R^5 n e^2}{a^6}$ where $R$ , $n$ , $a$ , and $e$ are respectively the satellite's mean radius, mean orbital motion, orbital distance, and eccentricity.{{cite journal |last1=Segatz |first1=M. |last2=Spohn |first2=T. |last3=Ross |first3=M.N. |last4=Schubert |first4=G. |title=Tidal dissipation, surface heat flow, and figure of viscoelastic models of Io |journal=Icarus |date=August 1988 |volume=75 |issue=2 |pages=187–206 |doi=10.1016/0019-1035(88)90001-2|bibcode=1988Icar...75..187S }} $M_{h}$ 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).{{cite journal |title=Tidally Heated Terrestrial Exoplanets: Viscoelastic Response Models |year=2009 |journal=The Astrophysical Journal |pages=1000–1015 |volume=707 |last1=Henning |first1=Wade G. |issue=2 |doi=10.1088/0004-637X/707/2/1000 |arxiv=0912.1907 |bibcode=2009ApJ...707.1000H|s2cid=119286375 }}{{cite journal |title=Increased Tidal Dissipation Using Advanced Rheological Models: Implications for Io and Tidally Active Exoplanets |year=2018 |journal=The Astrophysical Journal |pages=98 |volume=857 |last1=Renaud |first1=Joe P. |last2=Henning |first2=Wade G. |issue=2 |doi=10.3847/1538-4357/aab784 |arxiv=1707.06701 |bibcode=2018ApJ...857...98R |doi-access=free}} The rheological parameters' values, in turn, depend upon the temperature and the concentration of partial melt in the body's interior.{{cite journal |title=Tidal Dissipation Compared to Seismic Dissipation: In Small Bodies, in Earths, and in Superearths |year=2012 |journal=The Astrophysical Journal |pages=150 |volume=746 |last1=Efroimsky |first1=Michael |doi=10.1088/0004-637X/746/2/150 |doi-access=free|arxiv=1105.3936 }}

The tidally dissipated power in a nonsynchronised rotator is given by a more complex expression.{{cite journal |title=Tidal Dissipation in a Homogeneous Spherical Body. I. Methods |year=2014 |journal=The Astrophysical Journal |pages=6 |volume=795 |last1=Efroimsky |first1=Michael |last2=Makarov |first2=Valeri V. |issue=1 |doi=10.1088/0004-637X/795/1/6 |arxiv=1406.2376 |bibcode=2014ApJ...795....6E |doi-access=free}}

References

Category:Concepts in astrophysics

Category:Planetary science

Category:Tidal forces