Tidal Locking - Timescale

Timescale

An estimate of the time for a body to become tidally locked can be obtained using the following formula:

where

• is the initial spin rate (radians per second)
• is the semi-major axis of the motion of the satellite around the planet
• is the moment of inertia of the satellite.
• is the dissipation function of the satellite.
• is the gravitational constant
• is the mass of the planet
• is the mass of the satellite
• is the tidal Love number of the satellite
• is the radius of the satellite.

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

where

• is the density of the satellite
• is the surface gravity of the satellite
• 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 ), 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, 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 grows much faster with satellite radius than . 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.