Definitions

# Tidal locking

A separate article treats the phenomenon of tidal resonance in oceanography.
See the article tidal acceleration for a more quantitative description of the Earth-Moon system.

Tidal locking occurs when the gravitational gradient makes one side of an astronomical body always face another; for example, one side of the Earth's 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 synchronous rotation causes one hemisphere constantly to face the partner body. Usually, only the satellite becomes tidally locked around the larger planet, but if the difference in mass between the two bodies and their physical separation is small, both may become tidally locked to the other, as is the case between Pluto and 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 which distorts its gravitational equilibrium shape slightly so that it becomes stretched along the axis oriented toward A, and conversely, is slightly compressed in the two perpendicular directions. 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 which are near-spherical due to self-gravitation, the tidal distortion produces a slightly prolate spheroid or ellipsoid. 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 orbital period is shorter the bulges lag behind instead.

Resulting torque: Since 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, while 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 which acts to synchronize B's rotation with its orbital period, leading eventually to tidal locking.

Orbital changes: 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: 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, the Earth's rotation is gradually slowing down because of the Moon, by an amount that becomes noticeable over geological time in some fossils. For similar sized bodies the effect may be of comparable size for both, and both may become tidally locked to each other. The dwarf planet Pluto and its satellite Charon are good examples of this—Charon is only visible from one hemisphere of Pluto and vice versa.

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

## Final configuration

There is a tendency for a moon to orient itself in the lowest energy configuration, with the heavy side facing the planet. Irregularly shaped bodies will align their long axis to point towards the planet. Both cases are analogous to how a rounded floating object will orient itself with its heavy end downwards. In many cases this planet-facing hemisphere is visibly different from the rest of the moon's surface.

The orientation of the Earth's moon might be related to this process. The lunar maria are composed of basalt, which is heavier than the surrounding highland crust, and were formed on the side of the moon on which the crust is markedly thinner. The Earth-facing hemisphere contains all the large maria. The simple picture of the moon stabilising with its heavy side towards the Earth is incorrect, however, because the tidal locking occurred over a very short timescale of a thousand years or less, while the maria formed much later.

## Occurrence

### Earth's Moon

The Moon's rotation and orbital periods are both just under four weeks, so no matter when the Moon is observed from the 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, we may actually observe about 59% of the moon's total surface with repeated observations from earth due to the phenomena of librations and parallax. Librations are primarily caused by the Moon's varying orbital speed due to the eccentricity of its orbit: this allows us to see up to about 6° more along its perimeter. Parallax is a geometric effect: at the surface of the 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.

### Moons

Most significant moons in the Solar System are tidally locked with their primaries, since 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 giant planets, which orbit much further 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 mass center 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, obviously, contact binaries.

### Planets

Until radar observations in 1965 proved otherwise, it was thought that Mercury was tidally locked with the Sun. Instead, it turned out that Mercury has a 3:2 spin-orbit resonance, rotating three times for every two revolutions around the Sun; the eccentricity of Mercury's orbit makes this resonance stable. The original reason astronomers thought it was tidally locked was because whenever Mercury was best placed for observation, it was always at the same point in its 3:2 resonance, so showing the same face, which would be also the case if it were tidally locked.

A curious aspect of Venus' orbit and rotation periods is that the 583.92-day interval between successive close approaches to the Earth is almost exactly equal to 5 Venusian solar days (precisely, 5.001444 of these), 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 the Earth is unknown .

### 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 (revolutions per second)
• $a,$ is the semi-major axis of the motion of the satellite around the planet
• $Iapprox 0.4 m_s R^2$ 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 radius of the satellite.

Q and $k_2$ are generally very poorly known except for the Earth's Moon which has $k_2/Q=0.0011$. However, for a really rough estimate one can take Q≈100 (perhaps conservatively, giving overestimated locking times), and


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

• $rho,$ is the density of the satellite
• $gapprox Gm_s/R^2$ is the surface gravity of the satellite
• $mu,$ is rigidity of the satellite. This can be roughly taken as 3 Nm-2 for rocky objects and 4 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.

Since 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, $k_2ll1,$, 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)



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

Note the extremely strong dependence on orbital radius a.

For the locking of a primary body to its moon as in the case of Pluto, 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 much faster with satellite radius than $R$. A possible example of this is in the Saturn system, where Hyperion is not tidally locked, while the larger Iapetus, which orbits at a greater distance, is. This is not clear cut because Hyperion also experiences strong driving from the nearby Titan, which forces its rotation to be chaotic.

## List of known tidally locked bodies

### Solar System

Locked to the Sun

• Mercury (in a 3:2 rotation:orbit resonance)

Locked to the Earth

Locked to Mars

Locked to Jupiter

Locked to Saturn

Locked to Uranus

Locked to Neptune

Locked to Pluto

• Charon (Pluto being itself locked to Charon)

## 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

Probably locked to Uranus

Probably locked to Neptune

Probably locked to other dwarf planets and minor planets
Numerous asteroid and TNO moons are expected to be locked to their primaries. However, in the absence of direct observation reliable candidates are difficult to verify. While locking timescales can be estimated, the age of the primary+satellite system is difficult to gauge; most are thought to be the results of collisions in the last few hundred million years.