### Synchronous Rotation

Tidal 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 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 causes one hemisphere constantly to face the partner body. Usually, at any given time only the satellite is tidally locked around the larger body, but if the difference in mass between the two bodies and their physical separation is small, each may be 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 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 which 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

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 bodies of similar size 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.

## Occurrence

### 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 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

The Moon's rotation and orbital periods are tidally locked with each other, 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, 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 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.

### 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 the 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 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 (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 Earth's 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.

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, $k_2\ll1\,$, 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)



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 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 as the cube of the satellite radius,$R$.[contradictory] 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. 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

Locked to Mars

Locked to Jupiter

Locked to Saturn

Locked to Uranus

Locked to Neptune

Locked to Pluto

• Charon (Pluto is 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