Tidal locking

Tidal locking results in the Moon rotating about its axis in about the same time it takes to orbit Earth. Except for libration effects, this results in the Moon keeping the same face turned towards Earth, as seen in the figure on the left. (The Moon is shown in polar view, and is not drawn to scale.) If the Moon were not rotating at all, it would alternately show its near and far sides to Earth, while moving around Earth in orbit, as shown in the figure on the right.

Tidal locking (also called gravitational locking or captured rotation) occurs when the gravitational gradient makes one side of an astronomical body always face another, an effect known as synchronous rotation. For example, the same side of the Moon always faces the Earth. A tidally locked body takes just as long to rotate around its own axis as it does to revolve around its partner. This causes one hemisphere constantly to face the partner body. Usually, at any given time only the satellite is tidally locked around the larger body, because this is the first stage that occurs and it takes a lot more time to also lock the heavier main body to the partner body. But given enough time, eventually each may be tidally locked to the other. This will occur more quickly if the difference in mass between the two bodies and their physical separation is small; which probably helped the fact that this is already the case with Pluto and its moon Charon.

This effect is employed to stabilize some artificial satellites.

Mechanism

The change in rotation rate necessary to tidally lock a body B to a larger body A is caused by the torque applied by A's gravity on bulges it has induced on B by tidal forces.

Tidal bulges

A's gravity produces a tidal force on B that distorts its gravitational equilibrium shape slightly so that it becomes elongated along the axis oriented toward A, and conversely, is slightly reduced in dimension in directions perpendicular to this axis. These distortions are known as tidal bulges. When B is not yet tidally locked, the bulges travel over its surface, with one of the two "high" tidal bulges traveling close to the point where body A is overhead. For large astronomical bodies that are near-spherical due to self-gravitation, the tidal distortion produces a slightly prolate spheroid - i.e., an axially symmetric ellipsoid that is elongated along its major axis. Smaller bodies also experience distortion, but this distortion is less regular.

Bulge dragging

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 towards A. If B's rotation period is shorter than its orbital period, the bulges are carried forward of the axis oriented towards A in the direction of rotation, whereas if B's rotation period is longer the bulges lag behind instead.

Resulting torque

If the tidal bulges of a body are misaligned with the major axis, the tidal forces exert a net torque on that body that twists the body towards the direction of realignment.

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

If rotational frequency is larger than orbital frequency, a small torque counteracting the rotation arises, eventually locking the frequencies (situation depicted in green)

The angular momentum 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

See also: Synchronous orbit

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 size. For example, Earth's rotation is gradually slowing down because of the Moon, by an amount that becomes noticeable over geological time in some fossils. The process is still happening and has significantly slowed down the rotation of Earth since the beginning of its existence. Current estimations are that this has helped lengthen the Earth day (together with the tidal influence of the Sun) from about 6 hours to the current 24 hours. And at this moment, atomic clocks show that Earth's day lengthens by about 15 microseconds every year. Given enough time, this would create a mutual tidal locking between Earth and the Moon, where the length of a day has lengthened and the length of a lunar month would have shortened to a point where they are the same length. But it is happening so slowly that 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. The dwarf planet Pluto and its satellite Charon are good examples of this. They have already reached a state where Charon is only visible from one hemisphere of Pluto and vice versa, while Pluto is by estimation roughly the same age as Earth and the other planets in the solar system.

Rotation–orbit resonance

Finally, 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 tidally locked. Here the ratio of the rotation period of a body to its own orbital period is some well-defined fraction different from 1:1. A well known case is the rotation of Mercury—locked to its own orbit around the Sun in a 3:2 resonance.

Many exoplanets (especially the close-in ones) are expected to be in spin–orbit resonances higher than 1:1. For example, the super-Earth Gliese 581 d would most probably be in a spin–orbit resonance of 2:1, rotating twice for each orbit of its host star.

Occurrence

Moons

Due to tidal locking, the inhabitants of the central body will never be able to see the satellite's green area.

Most significant moons in the Solar System are tidally locked with their primaries, because they orbit very closely and tidal force increases rapidly (as a cubic) with decreasing distance. Notable exceptions are the irregular outer satellites of the gas giants, which orbit much farther away than the large well-known moons.

Pluto 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. This has made Pluto also tidally locked to Charon. In effect, these two celestial bodies revolve around each other (their barycenter lies outside of Pluto) as if joined with a rod connecting two opposite points on their surfaces.

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.

The Moon

Because the Moon is 1:1 tidally locked, only one side is visible from Earth.

The 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 was not seen in its entirety until 1959, when photographs were transmitted from the Soviet spacecraft Luna 3.

Despite the Moon's rotational and orbital periods being exactly locked, about 59% 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 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 we are offset from the line through the centers of Earth and Moon, and because of this we can observe a bit (about 1°) more around the side of the Moon when it is on our local horizon.

Planets

It was thought for some time that Mercury was tidally locked 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. The eccentricity of Mercury's orbit makes this 3:2 resonance stable.

Venus's 583.92-day interval between successive close approaches 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.

A planet that is tidally locked to its star has one side that is in perpetual starlight and another that is in perpetual darkness.

Stars

Close binary stars throughout the universe are expected to be tidally locked with each other, and extrasolar planets 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, is Tau Boötis, a star tidally locked by a planet. 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:

$t_{\textrm{lock}} \approx \frac{w a^6 I Q}{3 G m_p^2 k_2 R^5}$

where

• $w\,$ is the initial spin rate (radians per second)
• $a\,$ is the semi-major axis of the motion of the satellite around the planet (given by average of perigee and apogee distances)
• $I\,$ $\approx 0.4 m_s R^2$ is the moment of inertia of the satellite.
• $Q\,$ is the dissipation function of the satellite.
• $G\,$ is the gravitational constant
• $m_p\,$ is the mass of the planet
• $m_s\,$ is the mass of the satellite
• $k_2\,$ is the tidal Love number of the satellite
• $R\,$ is the mean radius 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 $Q$≈100 (perhaps conservatively, giving overestimated locking times), and

$k_2 \approx \frac{1.5}{1+\frac{19\mu}{2\rho g R}},$

where

• $\rho\,$ is the density of the satellite
• $g\approx Gm_s/R^2$ is the surface gravity of the satellite
• $\mu\,$ is rigidity of the satellite. This can be roughly taken as 3×10¹⁰ Nm⁻² for rocky objects and 4×10⁹ Nm⁻² for icy ones.

As can be seen, even knowing the size and density of the satellite leaves many parameters that must be estimated (especially w, Q, and $\mu\,$), so that any calculated locking times obtained are expected to be inaccurate, to even factors of ten. Further, during the tidal locking phase the orbital radius 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)

$t_{\textrm{lock}}\quad \approx\quad 6\ \frac{a^6R\mu}{m_sm_p^2}\quad \times 10^{10}\ \textrm{ years},$

with masses in kg, distances in meters, and μ in Nm⁻². μ can be roughly taken as 3×10¹⁰ Nm⁻² for rocky objects and 4×10⁹ Nm⁻² for icy ones.

Note the extremely strong dependence on orbital radius 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 interchanged.

One conclusion is that other things being equal (such as Q and μ), a large moon will lock faster than a smaller moon at the same orbital radius 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 $k_2/Q$.

List of known tidally locked bodies

Solar System

Locked to the Earth

• Moon

Locked to Mars

• Phobos
• Deimos

Locked to Jupiter

• Metis
• Adrastea
• Amalthea
• Thebe
• Io
• Europa
• Ganymede
• Callisto

Locked to Saturn

• Pan
• Atlas
• Prometheus
• Pandora
• Epimetheus
• Janus
• Mimas
• Enceladus
• Telesto
• Tethys
• Calypso
• Dione
• Rhea
• Titan
• Iapetus

Locked to Uranus

• Miranda
• Ariel
• Umbriel
• Titania
• Oberon

Locked to Neptune

• Proteus
• Triton

Locked to Pluto

• Charon (Pluto is itself locked to Charon)

Extra-solar

• Tau Boötis is known to be locked to the close-orbiting giant planet Tau Boötis b.

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
• Methone
• Pallene
• Helene
• Polydeuces

Probably locked to Uranus

• Cordelia
• Ophelia
• Bianca
• Cressida
• Desdemona
• Juliet
• Portia
• Rosalind
• Cupid
• Belinda
• Perdita
• Puck
• Mab
• Oberon

Probably locked to Neptune

• Naiad
• Thalassa
• Despina
• Galatea
• Larissa

Extrasolar

• Alpha Centauri Bb may be tidally locked to its parent star Alpha Centauri.
• Gliese 581 c, Gliese 581 g, Gliese 581 b, and Gliese 581 e may be tidally locked to their parent star Gliese 581. Gliese 581 d is almost certainly locked either into the 2:1 or the 3:2 spin–orbit resonance with the same star.

See also

• Tidal acceleration
• Gravity-gradient stabilization
• Orbital resonance
• Synchronous orbit