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Tidal locking between a pair of co-
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
ing 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 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 bo ...
, 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 The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
always faces the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
, although there is some variability because the Moon's orbit is not perfectly circular. Usually, only the
satellite A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioi ...
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 Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the Sun. It is the largest ...
and Charon. Alternative names for the tidal locking process are gravitational locking, captured rotation, and spin–orbit locking. The effect arises between two bodies when their
gravitational interaction In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
slows a body's rotation until it becomes tidally locked. Over many millions of years, the interaction forces changes to their orbits and rotation rates as a result of energy exchange and heat
dissipation In thermodynamics, dissipation is the result of an irreversible process that takes place in homogeneous thermodynamic systems. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to ...
. When one of the bodies reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit, it is said to be tidally locked. The object tends to stay in this state because leaving it would require adding energy back into the system. The object's orbit may migrate over time so as to undo the tidal lock, for example, if a giant planet perturbs the object. Not every case of tidal locking involves synchronous rotation. With Mercury, for example, this tidally locked planet completes three rotations for every two revolutions around the Sun, a 3:2 spin–orbit resonance. In the special case where an orbit is nearly circular and the body's rotation axis is not significantly tilted, such as the Moon, tidal locking results in the same hemisphere of the revolving object constantly facing its partner. However, in this case the exact same portion of the body does not always face the partner on all orbits. There can be some shifting due to variations in the locked body's orbital velocity and the inclination of its rotation axis.


Mechanism

Consider a pair of co-orbiting objects, A and B. The change in rotation rate necessary to tidally lock body B to the larger body A is caused by the
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
applied by A's
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
on bulges it has induced on B by
tidal force The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...
s. The gravitational force from object A upon B will vary with distance, being greatest at the nearest surface to A and least at the most distant. This creates a gravitational
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
across object B that will distort its equilibrium shape slightly. The body of object B will become elongated along the axis oriented toward A, and conversely, slightly reduced in dimension in directions orthogonal to this axis. The elongated distortions are known as
tidal bulge The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...
s. (For the solid Earth, these bulges can reach displacements of up to around .) When B is not yet tidally locked, the bulges travel over its surface due to orbital motions, with one of the two "high" tidal bulges traveling close to the point where body A is overhead. For large astronomical bodies that are nearly
spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
due to self-gravitation, the tidal distortion produces a slightly
prolate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ci ...
, i.e. an axially symmetric ellipsoid that is elongated along its major axis. Smaller bodies also experience distortion, but this distortion is less regular. The material of B exerts resistance to this periodic reshaping caused by the tidal force. In effect, some time is required to reshape B to the gravitational equilibrium shape, by which time the forming bulges have already been carried some distance away from the A–B axis by B's rotation. Seen from a vantage point in space, the points of maximum bulge extension are displaced from the axis oriented toward A. If B's rotation period is shorter than its orbital period, the bulges are carried forward of the axis oriented toward A in the direction of rotation, whereas if B's rotation period is longer, the bulges instead lag behind. Because the bulges are now displaced from the A–B axis, A's gravitational pull on the mass in them exerts a torque on B. The torque on the A-facing bulge acts to bring B's rotation in line with its orbital period, whereas the "back" bulge, which faces away from A, acts in the opposite sense. However, the bulge on the A-facing side is closer to A than the back bulge by a distance of approximately B's diameter, and so experiences a slightly stronger gravitational force and torque. The net resulting torque from both bulges, then, is always in the direction that acts to synchronize B's rotation with its orbital period, leading eventually to tidal locking.


Orbital changes

The
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
of the whole A–B system is conserved in this process, so that when B slows down and loses rotational angular momentum, its ''orbital'' angular momentum is boosted by a similar amount (there are also some smaller effects on A's rotation). This results in a raising of B's orbit about A in tandem with its rotational slowdown. For the other case where B starts off rotating too slowly, tidal locking both speeds up its rotation, and ''lowers'' its orbit.


Locking of the larger body

The tidal locking effect is also experienced by the larger body A, but at a slower rate because B's gravitational effect is weaker due to B's smaller mass. For example, Earth's rotation is gradually being slowed by the Moon, by an amount that becomes noticeable over geological time as revealed in the fossil record. Current estimations are that this (together with the tidal influence of the Sun) has helped lengthen the Earth day from about 6 hours to the current 24 hours (over ≈ ⁠4½ billion years). Currently,
atomic clock An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwe ...
s show that Earth's day lengthens, on average, by about 2.3 milliseconds per century. Given enough time, this would create a mutual tidal locking between Earth and the Moon. The length of the Earth's day would increase and the length of a
lunar month In lunar calendars, a lunar month is the time between two successive syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month. Variations In Shona, Middle Eastern, and Eur ...
would also increase. The Earth's sidereal day would eventually have the same length as the Moon's orbital period, about 47 times the length of the Earth's day at present. However, Earth is not expected to become tidally locked to the Moon before the Sun becomes a red giant and engulfs Earth and the Moon. For bodies of similar size the effect may be of comparable size for both, and both may become tidally locked to each other on a much shorter timescale. An example is the
dwarf planet A dwarf planet is a small planetary-mass object that is in direct orbit of the Sun, smaller than any of the eight classical planets but still a world in its own right. The prototypical dwarf planet is Pluto. The interest of dwarf planets to ...
Pluto Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the Sun. It is the largest ...
and its satellite Charon. They have already reached a state where Charon is visible from only one hemisphere of Pluto and vice versa.


Eccentric orbits

For orbits that do not have an eccentricity close to zero, the rotation rate tends to become locked with the
orbital speed In gravitationally bound systems, the orbital speed of an astronomical body or object (e.g. planet, moon, artificial satellite, spacecraft, or star) is the speed at which it orbits around either the barycenter or, if one body is much more mas ...
when the body is at
periapsis An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any elli ...
, which is the point of strongest tidal interaction between the two objects. If the orbiting object has a companion, this third body can cause the rotation rate of the parent object to vary in an oscillatory manner. This interaction can also drive an increase in orbital eccentricity of the orbiting object around the primary – an effect known as eccentricity pumping. In some cases where the orbit is eccentric and the tidal effect is relatively weak, the smaller body may end up in a so-called ''spin–orbit resonance'', rather than being tidally locked. Here, the ratio of the rotation period of a body to its own orbital period is some simple fraction different from 1:1. A well known case is the rotation of Mercury, which is locked to its own orbit around the Sun in a 3:2 resonance. This results in the rotation speed roughly matching the orbital speed around perihelion. Many exoplanets (especially the close-in ones) are expected to be in spin–orbit resonances higher than 1:1. A Mercury-like terrestrial planet can, for example, become captured in a 3:2, 2:1, or 5:2 spin–orbit resonance, with the probability of each being dependent on the orbital eccentricity.


Occurrence


Moons

All twenty known moons in the
Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
that are large enough to be round are tidally locked with their primaries, because they orbit very closely and tidal force increases rapidly (as a
cubic function In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
) with decreasing distance. On the other hand, the irregular outer satellites of the
gas giant A gas giant is a giant planet composed mainly of hydrogen and helium. Gas giants are also called failed stars because they contain the same basic elements as a star. Jupiter and Saturn are the gas giants of the Solar System. The term "gas giant" ...
s (e.g. Phoebe), which orbit much farther away than the large well-known moons, are not tidally locked.
Pluto Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the Sun. It is the largest ...
and Charon are an extreme example of a tidal lock. Charon is a relatively large moon in comparison to its primary and also has a very close
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
. This results in Pluto and Charon being mutually tidally locked. Pluto's other moons are not tidally locked; Styx, Nix, Kerberos, and Hydra all rotate chaotically due to the influence of Charon. Similarly, and Dysnomia are mutually tidally locked. The tidal locking situation for asteroid moons is largely unknown, but closely orbiting binaries are expected to be tidally locked, as well as contact binaries.


Earth's Moon

Earth's Moon's rotation and orbital periods are tidally locked with each other, so no matter when the Moon is observed from Earth, the same hemisphere of the Moon is always seen. The
far side of the Moon The far side of the Moon is the lunar hemisphere that always faces away from Earth, opposite to the near side, because of synchronous rotation in the Moon's orbit. Compared to the near side, the far side's terrain is rugged, with a multitu ...
was not seen until 1959, when photographs of most of the far side were transmitted from the
Soviet The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nation ...
spacecraft ''
Luna 3 Luna 3, or E-2A No.1 ( rus, Луна 3}) was a Soviet spacecraft launched in 1959 as part of the Luna programme. It was the first mission to photograph the far side of the Moon and the third Soviet space probe to be sent to the neighborhood of th ...
''. When the Earth is observed from the Moon, the Earth does not appear to move across the sky. It remains in the same place while showing nearly all its surface as it rotates on its axis. Despite the Moon's rotational and orbital periods being exactly locked, about 59 percent of the Moon's total surface may be seen with repeated observations from Earth, due to the phenomena of libration and parallax. Librations are primarily caused by the Moon's varying orbital speed due to the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-Centre (geometry), center, in geometry * Eccentricity (g ...
of its orbit: this allows up to about 6° more along its perimeter to be seen from Earth. Parallax is a geometric effect: at the surface of Earth observers are offset from the line through the centers of Earth and Moon, and because of this about 1° more can be seen around the side of the Moon when it is on the local horizon.


Planets

It was thought for some time that Mercury was in synchronous rotation with the Sun. This was because whenever Mercury was best placed for observation, the same side faced inward. Radar observations in 1965 demonstrated instead that Mercury has a 3:2 spin–orbit resonance, rotating three times for every two revolutions around the Sun, which results in the same positioning at those observation points. Modeling has demonstrated that Mercury was captured into the 3:2 spin–orbit state very early in its history, probably within 10–20 million years after its formation. The 583.92-day interval between successive close approaches of
Venus Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never f ...
to Earth is equal to 5.001444 Venusian solar days, making approximately the same face visible from Earth at each close approach. Whether this relationship arose by chance or is the result of some kind of tidal locking with Earth is unknown. The exoplanet
Proxima Centauri b Proxima Centauri b (or Proxima b), sometimes referred to as Alpha Centauri Cb, is an exoplanet orbiting in the habitable zone of the red dwarf star Proxima Centauri, which is the closest star to the Sun and part of the triple star system ...
, discovered in 2016 that orbits around Proxima Centauri, is almost assuredly tidally locked, expressing either synchronized rotation or a 3:2 spin–orbit resonance like that of Mercury. One form of hypothetical tidally locked exoplanets are eyeball planets, which in turn are divided into "hot" and "cold" eyeball planets.


Stars

Close binary stars throughout the universe are expected to be tidally locked with each other, and
extrasolar planet An exoplanet or extrasolar planet is a planet outside the Solar System. The first possible evidence of an exoplanet was noted in 1917 but was not recognized as such. The first confirmation of detection occurred in 1992. A different planet, init ...
s that have been found to orbit their primaries extremely closely are also thought to be tidally locked to them. An unusual example, confirmed by
MOST Most or Möst or ''variation'', may refer to: Places * Most, Kardzhali Province, a village in Bulgaria * Most (city), a city in the Czech Republic ** Most District, a district surrounding the city ** Most Basin, a lowland named after the city ** A ...
, may be
Tau Boötis Tau Boötis, Latinised from τ Boötis, is an F-type main-sequence star approximately 51 light-years away in the constellation of Boötes. It is a binary star system, with the secondary star being a red dwarf. As of 1999, an extrasol ...
, a star that is probably tidally locked by its planet
Tau Boötis b Tau Boötis b, or more precisely Tau Boötis Ab, is an extrasolar planet approximately 51 light-years away. The planet and its host star is one of the planetary systems selected by the International Astronomical Union as part of NameExoWorlds, th ...
. If so, the tidal locking is almost certainly mutual.


Timescale

An estimate of the time for a body to become tidally locked can be obtained using the following formula: : t_ \approx \frac where * \omega\, is the initial spin rate expressed in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s per second, * a\, is the semi-major axis of the motion of the satellite around the planet (given by the average of the
periapsis An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any elli ...
and
apoapsis An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any elli ...
distances), * I\, \approx 0.4\; m_s R^2 is the moment of inertia of the satellite, where m_s is the mass of the satellite and R is the mean radius of the satellite, * Q\, is the dissipation function of the satellite, * G\, is the gravitational constant, * m_p\, is the mass of the planet (i.e., the object being orbited), and * k_2\, is the tidal Love number of the satellite. Q and k_2 are generally very poorly known except for the Moon, which has k_2/Q=0.0011. For a really rough estimate it is common to take Q \approx 100 (perhaps conservatively, giving overestimated locking times), and : k_2 \approx \frac, where * \rho\, is the density of the satellite * g\approx Gm_s/R^2 is the surface gravity of the satellite * \mu\, is the rigidity of the satellite. This can be roughly taken as 3 N·m−2 for rocky objects and 4 N·m−2 for icy ones. Even knowing the size and density of the satellite leaves many parameters that must be estimated (especially ''ω'', ''Q'', and ''μ''), so that any calculated locking times obtained are expected to be inaccurate, even to factors of ten. Further, during the tidal locking phase the semi-major axis a may have been significantly different from that observed nowadays due to subsequent tidal acceleration, and the locking time is extremely sensitive to this value. Because the uncertainty is so high, the above formulas can be simplified to give a somewhat less cumbersome one. By assuming that the satellite is spherical, k_2\ll1\, , Q = 100, and it is sensible to guess one revolution every 12 hours in the initial non-locked state (most asteroids have rotational periods between about 2 hours and about 2 days) : t_ \approx 6\ \frac \times 10^\ \text, with masses in kilograms, distances in meters, and \mu in newtons per meter squared; \mu can be roughly taken as 3 N·m−2 for rocky objects and 4 N·m−2 for icy ones. There is an extremely strong dependence on semi-major axis a. For the locking of a primary body to its satellite as in the case of Pluto, the satellite and primary body parameters can be swapped. One conclusion is that, ''other things being equal'' (such as Q and \mu), a large moon will lock faster than a smaller moon at the same orbital distance from the planet because m_s\, grows as the cube of the satellite radius R. A possible example of this is in the Saturn system, where Hyperion is not tidally locked, whereas the larger Iapetus, which orbits at a greater distance, is. However, this is not clear cut because Hyperion also experiences strong driving from the nearby Titan, which forces its rotation to be chaotic. The above formulae for the timescale of locking may be off by orders of magnitude, because they ignore the frequency dependence of k_2/Q. More importantly, they may be inapplicable to viscous binaries (double stars, or double asteroids that are rubble), because the spin–orbit dynamics of such bodies is defined mainly by their viscosity, not rigidity.


List of known tidally locked bodies


Solar System


Extra-solar

* The most successful detection methods of exoplanets (transits and radial velocities) suffer from a clear observational bias favoring the detection of planets near the star; thus, 85% of the exoplanets detected are inside the tidal locking zone, which makes it difficult to estimate the true incidence of this phenomenon.
Tau Boötis Tau Boötis, Latinised from τ Boötis, is an F-type main-sequence star approximately 51 light-years away in the constellation of Boötes. It is a binary star system, with the secondary star being a red dwarf. As of 1999, an extrasol ...
is known to be locked to the close-orbiting
giant planet The giant planets constitute a diverse type of planet much larger than Earth. They are usually primarily composed of low-boiling-point materials (volatiles), rather than rock or other solid matter, but massive solid planets can also exist. The ...
Tau Boötis b Tau Boötis b, or more precisely Tau Boötis Ab, is an extrasolar planet approximately 51 light-years away. The planet and its host star is one of the planetary systems selected by the International Astronomical Union as part of NameExoWorlds, th ...
.


Bodies likely to be locked


Solar System

Based on comparison between the likely time needed to lock a body to its primary, and the time it has been in its present orbit (comparable with the age of the Solar System for most planetary moons), a number of moons are thought to be locked. However their rotations are not known or not known enough. These are:


Probably locked to Saturn

*
Daphnis In Greek mythology, Daphnis (; grc, Δάφνις, from , ''daphne'', "Bay Laurel") was a Sicilian shepherd who was said to be the inventor of pastoral poetry. Family According to tradition, he was the son of Hermes and a nymph, despite which ...
* Aegaeon * Methone * Anthe * Pallene * Helene * Polydeuces


Probably locked to Uranus

*
Cordelia Cordelia is a feminine given name. It was borne by the tragic heroine of Shakespeare's ''King Lear'' (1606), a character based on the List of legendary kings of Britain, legendary queen Cordelia of Britain, Cordelia. The name is of uncertain origi ...
*
Ophelia Ophelia () is a character in William Shakespeare's drama '' Hamlet'' (1599–1601). She is a young noblewoman of Denmark, the daughter of Polonius, sister of Laertes and potential wife of Prince Hamlet, who, due to Hamlet's actions, ends u ...
*
Bianca Bianca is a feminine given name. It means "white" and is an Italian cognate of Blanche. Variants * Blanche: French * Bianca: Italian * Bianka ( Polish, Hungarian, Slovak, German, English, French, Icelandic, Finnish, Dutch, Norwegian, C ...
*
Cressida Cressida (; also Criseida, Cresseid or Criseyde) is a character who appears in many Medieval and Renaissance retellings of the story of the Trojan War. She is a Trojan woman, the daughter of Calchas, a Greek seer. She falls in love with Troilus, ...
*
Desdemona Desdemona () is a character in William Shakespeare's play ''Othello'' (c. 1601–1604). Shakespeare's Desdemona is a Venetian beauty who enrages and disappoints her father, a Venetian senator, when she elopes with Othello, a Moorish Venetian ...
* Juliet * Portia * Rosalind * Cupid *
Belinda Belinda is a feminine given name of unknown origin, apparently coined from Italian ''bella'', meaning "beautiful". Alternatively it may be derived from the Old High German name ''Betlinde'', which possibly meant "bright serpent" or "bright linde ...
* Perdita * Puck * Mab


Probably locked to Neptune

* Naiad * Thalassa * Despina *
Galatea Galatea is an ancient Greek name meaning "she who is milk-white". Galatea, Galathea or Gallathea may refer to: In mythology * Galatea (Greek myth), three different mythological figures In the arts * ''Aci, Galatea e Polifemo'', cantata by H ...
* Larissa


Extrasolar

* Gliese 581c,
Gliese 581g Gliese 581g , unofficially known as Zarmina (or Zarmina's World), was a candidate exoplanet postulated to orbit within the Gliese 581 system, twenty light-years from Earth. It was discovered by the Lick–Carnegie Exoplanet Survey, and was the si ...
, Gliese 581b, and Gliese 581e may be tidally locked to their parent star Gliese 581. Gliese 581d is almost certainly captured either into the 2:1 or the 3:2 spin–orbit resonance with the same star. * All planets in the TRAPPIST-1 system are likely to be tidally locked.


See also

* * Earth tide#Effects * * * * * * *


References

{{DEFAULTSORT:Tidal Locking Celestial mechanics Orbits Locking