Tides and
Gravitational Locking

We have introduced tides in our earlier discussion of the Moon's observational characteristics through the effect of the Moon on the Earth's oceans, but the effect is much more general, and has a number of important consequences.

## Tidal Coupling and Gravitational Locking

Some important consequences of tidal forces in the Solar System include:
1. Tidal forces will distort any body experiencing differential gravitational forces. This will normally occur for bodies of finite extent in gravitational fields because of the strong distance dependence of the gravitational force. Thus, not only the oceans, but the body of the Earth is distorted by the Lunar gravity. However, because the Earth is rigid compared with the oceans, the "tides" in the body of the Earth are much smaller than in the oceans.

2. There is a limiting radius for the orbit of one body around another, inside of which the tidal forces are so large that no large solid objects can exist that are held together only by gravitational forces. This radius is called the Roche Limit. Thus, solid objects put into orbit inside the Roche limit will be torn apart by tidal forces, and conversely, solid objects cannot grow by accreting into larger objects if they orbit inside the Roche limit. A famous example is the rings of Saturn: because they lie inside the Roche limit for Saturn, they cannot be solid objects held together by gravitation and must be composed of many small particles.

Obviously solid objects can exist inside the Roche limit (for example, spacecraft) but they must be held together by forces other than gravity. This is true of a spacecraft, where chemical forces between the atoms and molecules are much larger than the gravitational forces.

3. The tidal forces are reciprocal. Not only will the Moon induce tides in the body of the Earth and the Earth's oceans, but by the same argument the gravitational field of the Earth will induce differential forces and therefore tides in the body of the Moon. Again, because the body of the Moon is quite rigid these Lunar tides will be very small, but they occur.

4. This reciprocal induction of tides in the body of the Earth and the Moon leads to a complicated coupling of the rotational and orbital motions of the two objects. These tidal forces and associated couplings have the following general effects:

• The interior of the Earth and Moon are heated by the tides in their bodies, just as a paper clip is heated by constant bending. This effect is very small for the Earth and Moon, but we shall see that it can be dramatic for other objects that experience much larger differential gravitational forces and therefore much larger tidal forces. For example, we shall see that the tidal forces exerted by Jupiter on its moon Io are so large that the solid surface of Io is raised and lowered by hundreds of meters twice in each rotational period. This motion so heats the interior of Io that it is probably mostly molten; as a consequence, Io is covered with active volcanos and is the geologically most active object in the Solar System.

• The tidal coupling of the orbital and rotational motion tends to synchronize them. In the simplest instance, the period of rotation for the two bodies and the orbital period eventually become exactly equal because of this tidal coupling (and as a result, the size of the orbit is changed in such a way as to conserve angular momentum for the entire system). This is called gravitational (or tidal) locking, because as the two objects revolve around their common center of mass each keeps the same side turned toward the other.

## Tidal Coupling in the Earth-Moon System

Thus, the fact that the rotational period of the Moon and the orbital period of the Earth-Moon system are of the same length is not an accident. Presumably this was not always true, but over billions of years the tidal coupling of the Earth and the Moon has led to this synchronization. In the case of the Earth-Moon system the synchronization is not yet complete. The Earth is slowly decreasing its rotational period and eventually the Earth and Moon will have exactly the same rotational period, and these will also exactly equal the orbital period. At the same time, the separation between the Earth and Moon will slowly increase in just such a way as to conserve angular momentum for the entire system.

Thus, billions of years from now the Earth will always keep the same face turned toward the Moon, just as the Moon already always keeps the same face turned toward the Earth. We will encounter other examples of such tidal locking in other pairs of objects in the Solar System.