# Tidal lock

Tidal locking is the process by which two bodies in an orbital relationship will, over time, come to show the same face to one another. In other words, the period of rotation about the axis of the body will come to match the body's orbital period.

## Mechanism

Considering two bodies, one larger (A) than the other (B): Body A will exert a gravitational force on B, causing B to be bulged towards body A.

As body B continues to rotate, a different part of the surface will be pulled towards A. In effect, the bulge will "move". However, the material of body B will resist being warped in this way, resulting in a torque that will affect the rotation of B.

This torque acts in such a way that eventually, the rotation of B is such that the same face always points towards A.

This process will also alter the rotation of body A, but if A is much larger, then the process will take a greater amount of time.

## Example

One example of tidal locking is the Earth/Moon system. The moon has become tidally locked, and so continually shows the same face to inhabitants of planet Earth.

The earth's rotation is itself slowing. Given enough time, the earth could lock itself to face the Moon.

## Time scale

The time required for tidal lock is difficult to estimate, primarily because it depends on many factors, unique to any given satellite or primary, that are difficult to measure. The formula for the time required for any moon to enter tidal lock with its primary is:[1]

$t_{\textrm{lock}} = \frac{16 \rho \omega a^6 Q}{45 G m_p^2 k_2}$

where

• $\rho\,$ is the density of the moon
• $\omega\,$ is the initial rotation rate in rad s⁻¹
• $a\,$ is the semi-major axis of the moon's orbit.
• $Q\,$ is the dissipation function of the moon (not to be confused with its apoapsis).
• $G\,$ is the gravitational constant.
• $m_p\,$ is the mass of the primary.
• $k_2\,$ is the tidal second-order Love number of the moon.

Thus the time required is very sensitive to orbital distance and somewhat less sensitive to the mass of the primary. Thus dense moons relatively close to their primaries are more likely to be in tidal lock.

## Problems for uniformitarianism posed by tidal lock

Tidal locking is as likely to happen to the primary as to the moon. Indeed Pluto and Charon are mutually locked. But tidal lock has its most profound implications for the Earth-Moon system. Though the presence of tidal locking might appear to militate in favor of a great age for the solar system, the dynamics of tidal lock suggest youth, not age.

As earth's rotation decreases, the moon must recede from the earth, or else angular momentum is not conserved (see above). Therefore the rate of deceleration of Earth's rotation must itself decelerate over time. For that reason alone, the Earth-Moon system cannot be more than 1.2 billion years old, because at such a time the Earth would have been rotating dangerously fast, and the Moon would have been touching the Earth.

## References

1. Gladman B., Quinn D. D., Nicholson P., and Rand R. "Synchronous Locking of Tidally Evolving Satellites." Icarus 122(1):166-192, July 1996. doi:10.1006/icar.1996.0117 Accessed July 4, 2008.