Tidal locking
Tidal locking between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit. In the case where a tidally locked body possesses synchronous rotation, the object takes just as long to rotate around its own axis as it does to revolve around its partner. For example, the same side of the Moon always faces Earth, although there is some variability because the Moon's orbit is not perfectly circular. Usually, only the satellite is tidally locked to the larger body.[1] However, if both the difference in mass between the two bodies and the distance between them are relatively small, each may be tidally locked to the other; this is the case for Pluto and Charon, as well as for Eris and Dysnomia. Alternative names for the tidal locking process are gravitational locking,[2] captured rotation, and spin–orbit locking.
The effect arises between two bodies when their gravitational interaction slows a body's rotation until it becomes tidally locked. Over many millions of years, the interaction forces changes to their orbits and rotation rates as a result of energy exchange and heat dissipation. When one of the bodies reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit, it is said to be tidally locked.[3] The object tends to stay in this state because leaving it would require adding energy back into the system. The object's orbit may migrate over time so as to undo the tidal lock, for example, if a giant planet perturbs the object.
There is ambiguity in the use of the terms 'tidally locked' and 'tidal locking', in that some scientific sources use it to refer exclusively to 1:1 synchronous rotation (e.g. the Moon), while others include non-synchronous orbital resonances in which there is no further transfer of angular momentum over the course of one orbit (e.g. Mercury).[4] In Mercury's case, the planet completes three rotations for every two revolutions around the Sun, a 3:2 spin–orbit resonance. In the special case where an orbit is nearly circular and the body's rotation axis is not significantly tilted, such as the Moon, tidal locking results in the same hemisphere of the revolving object constantly facing its partner.[3][4][5]
Regardless of which definition of tidal locking is used, the exact same portion of the body will not necessarily always face the partner. There can be minor differences due to variations in the locked body's orbital velocity and the inclination of its rotation axis over time.
is the of the satellite, where is the mass of the satellite and is the mean radius of the satellite,
moment of inertia
is the of the satellite,
dissipation function
is the ,
gravitational constant
is the mass of the planet (i.e., the object being orbited), and
is the tidal of the satellite.
Love number
An estimate of the time for a body to become tidally locked can be obtained using the following formula:[31]
where
and are generally very poorly known except for the Moon, which has . For a really rough estimate it is common to take (perhaps conservatively, giving overestimated locking times), and
where
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 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, , 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 kilograms, distances in meters, and in newtons per meter squared; 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 .
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 and ), a large moon will lock faster than a smaller moon at the same orbital distance from the planet because grows as the cube of the satellite radius . 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 . 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.[33]
Bodies likely to be locked[edit]
Solar System[edit]
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: