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 makes one side of an astronomical body always face another, like the Moon facing 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 only one constant side or hemisphere to face the partner body. The most common situation in our solar system is for a satellite to face its planet (like our Moon). However, if the difference in mass between the two bodies and their separation is small, both may become tidally locked to each other. The best-known example of this is between Pluto and Charon.

Earth's Moon

In the case of the Moon, both its rotation and orbital period are just over 4 weeks. The effect is that no matter where you are on the Earth you always see the same face of the Moon. The entirety of the far side of the Moon was not seen until 1959, when photographs were transmitted from the Soviet spacecraft Luna 3.

More precisely, despite the Moon's rotational and orbital periods being exactly locked, we may actually see about 59% of the moon's total surface with repeated observations 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 to see up to about 6° more along its perimeter. Parallax is a geometric effect because at the surface of the Earth we are offset from the line through the centers of Earth and Moon: we can look a bit (~1°) around the side of the Moon when it is just on our local horizon.

Mechanism

Gravitational attraction between two bodies produces a tidal force on each of them, stretching each body along the axis oriented towards its partner and compressing it along the other two perpendicular axes. This distorts the orbiting bodies' shapes slightly. Larger astronomical bodies which are near-spherical due to self-gravitation become slightly prolate (ovoid), small bodies are similarly distorted but on a much smaller scale than their structural irregularity.

If either of the two orbiting bodies is rotating relative to the other, the bulge's position on its surface is not stable. The rotation of the body will cause the bulge to move out of alignment with the other body, and the tidal force will have to reshape the shape to restore the situation. The tidal bulges move around the body as it rotates to stay in alignment with the other object. This is most clearly seen on Earth by how the ocean tides rise and fall with the rising and setting of the Moon.

As the material in the body rotates in the distorted gravity field it is forced to deviate from a circular path, and push against neighbouring material. This causes a frictional slowing force, and the whole body slows down in response. The lost kinetic energy is radiated as heat.

The angular momentum lost due to the slowdown is balanced by a speeding up along the orbit caused by the other body pulling on the bulge. Since it takes a small but nonzero amount of time for the bulge to shift position, it is always located slightly away from the nearest point to the other object. For the case of a rotation period shorter than the orbital period, this bulge is located in the direction of the rotation, pointing slightly backwards to the direction of orbital movement. Because of this misalignment a component of the gravitational attraction by the other body on the bulge is then directed along the direction of orbital motion. This steady but significant pull on the bulge accelerates the first body along its orbit, boosting its orbital angular momentum. In the opposite case of a rotation period longer than the orbital period, the moving bulge drives the body while holding its orbital movement back, and the rate of rotation is increased at the expense of orbital momentum instead.

The tidal locking effect is also experienced by the "planet", but at a vastly slower rate. 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. Pluto and Charon are good examples of this - you can only see Charon from one hemisphere of Pluto.

Tidal locking is caused by gravity, and does not require the bodies to be planets and moons. For example, it is thought that many binary stars are mutually tidally locked.

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. Irregular 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

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 its moon Charon are an extreme example of a tidal lock. Charon is the biggest moon in the Solar System in comparison to its planet 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 totally locked.

As mentioned above, Pluto is mutually locked to its moon Charon.

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 Gold T., Soter S. (1969), Atmospheric tides and the resonant rotation of Venus, Icarus, v. 11, p 356-366.

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 (See pages 169-170 of this article. Formula (9) is quoted here, which comes from S.J. Peale, Rotation histories of the natural satellites, in ):

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
$I\approx 0.4 m_s R^2$ 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 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{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 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, there are numerous poorly known parameters (especially w, Q, and $\mu\,$ ), so that any locking times obtained are expected to be inaccurate to even factors of ten. Note also that during the tidal locking phase, the orbital radius a may have been significantly different to that observed nowadays due to subsequent tidal acceleration, while the locking time is extremely sensitive to this input.

Since the uncertainty is so high anyway, a hatchet job can be made on the above formulas to get a less cumbersome one: The additional assumtions are: spherical satellite, $k_2\ll1\,$ , Q = 100, and let's guess one revolution per 12 hours in the initial non-locked state (e.g. most asteroids have rotational periods from ~2 hours to ~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 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 planet to its moon as in the case of Pluto, satellite and planet parameters can be interchanged.

An interesting conclusion from the above considerations 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. This is because $m_s\,$ grows much faster with satellite radius than $R$ . A possible example of this might be in the Saturn system, where Hyperion is not tidally locked, while the larger Iapetus is locked despite orbiting further out. This case is not, however, clear cut because Hyperion also experiences strong driving from the nearby Titan which forces its rotation to be chaotic.