Tidal locking

Occurrence

Moons

All nineteen known moons in the Solar System 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) with decreasing distance. On the other hand, most of the irregular outer satellites of the giant planets (e.g. Phoebe), which orbit much farther away than the large well-known moons, are not tidally locked.

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 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, Eris and Dysnomia are mutually tidally locked, as are Salacia and Actaea. Orcus and Vanth might also be mutually tidally locked, but the data is not conclusive.

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. Most of the far side of the Moon was not seen until 1959, when photographs of most of the far side were transmitted from the Soviet spacecraft Luna 3.

When Earth is observed from the Moon, Earth does not appear to move across the sky. It remains in nearly 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 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; this accounts for about a 1° difference in the Moon's surface which can be seen around the sides of the Moon when comparing observations made during moonrise and moonset.

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 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 discovered in 2016 which orbits around Proxima Centauri, is almost certainly 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 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, may be Tau Boötis, a star that is probably tidally locked by its planet Tau Boötis b. 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:

${\displaystyle t_{\text{lock}}\approx \frac{\omega a^{6}IQ}{3G{m}_p^2{k}_2{R}^5}}$

where

• ${\displaystyle \omega }$ is the initial spin rate expressed in radians per unit time,
• ${\displaystyle a}$ is the semi-major axis of the motion of the satellite around the planet (given by the average of the periapsis and apoapsis distances),
• ${\displaystyle I}$ ${\displaystyle \approx 0.4m_s{R}^2}$ is the moment of inertia of the satellite, where
• ${\displaystyle m_s}$ is the mass of the satellite and
• ${\displaystyle R}$ is the mean radius of the satellite,
• ${\displaystyle Q}$ is the dissipation function of the satellite,
• ${\displaystyle G}$ is the gravitational constant,
• ${\displaystyle m_p}$ is the mass of the planet (i.e., the object being orbited), and
• ${\displaystyle k_2}$ is the tidal Love number of the satellite.

${\displaystyle Q}$ and ${\displaystyle k_2}$ are generally very poorly known except for the Moon, which has ${\displaystyle k_2/Q=0.0011}$. For a really rough estimate it is common to take ${\displaystyle Q\approx 100}$ (perhaps conservatively, giving overestimated locking times), and

${\displaystyle k_2\approx \frac{1.5}{1+\frac{19\mu }{2\rho gR}},}$

where

• ${\displaystyle \rho }$ is the density of the satellite
• ${\displaystyle g\approx G{m}_s/R^2}$ is the surface gravity of the satellite
• ${\displaystyle \mu }$ is the rigidity of the satellite. This can be roughly taken as 3×10¹⁰ N/m² for rocky objects and 4×10⁹ N/m² 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 ${\displaystyle a}$ 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, ${\displaystyle k_2\ll 1,Q=100}$, 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)

${\displaystyle t_{\text{lock}}\approx 6\frac{a^{6}R\mu }{m_s{m}_p^2}\times {10}^{10}\text{years},}$

with masses in kilograms, distances in meters, and ${\displaystyle \mu }$ in newtons per meter squared; ${\displaystyle \mu }$ can be roughly taken as 3×10¹⁰ N/m² for rocky objects and 4×10⁹ N/m² for icy ones.

There is an extremely strong dependence on semi-major axis ${\displaystyle a}$.

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 ${\displaystyle Q}$ and ${\displaystyle \mu }$), a large moon will lock faster than a smaller moon at the same orbital distance from the planet because ${\displaystyle m_s}$ grows as the cube of the satellite radius ${\displaystyle R}$. 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 ${\displaystyle k_2/Q}$. 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

All the bodies below are tidally locked, and all but Mercury are moreover in synchronous rotation. (Mercury is tidally locked, but not in synchronous rotation.)

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 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
• Aegaeon
• Methone
• Anthe
• Pallene
• Helene
• Polydeuces

Probably locked to Uranus

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

Probably locked to Neptune

• Naiad
• Thalassa
• Despina
• Galatea
• Larissa

Probably mutually tidally locked

• Orcus and Vanth

Extrasolar

• Gliese 581c, Gliese 581b, and Gliese 581e may be tidally locked to their parent star Gliese 581.
• All planets in the TRAPPIST-1 system are likely to be tidally locked.