Consequences of Tidal Forces

Let us summarize the most important consequences of tidal forces in the Solar System:

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. 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.

3. 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 called spin-orbit coupling that has the following general effects: As noted previously, the tidal coupling of the orbital and rotational motion tends to synchronize them. 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 perhaps hundreds of meters 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 youngest object in the Solar System. The rotation-orbit tidal coupling tends to make elliptical orbits circular.

4. 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 held together solely by gravitational forces can exist. This radius is called the Roche Limit. Solid objects put into orbit inside the Roche Limit may be torn apart by tidal forces and, conversely, solid objects cannot grow by accreting gravitationally 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 and must be composed of many small particles. Conversely, the tidal forces associated with the rings being inside the Roche Limit keep the ring particles from condensing to form a moon. The box below gives formulas defining the Roche limiting radius and here is an animation illustrating the effect of the Roche Limit. As shown in the box, for simple estimates we may take the Roche Limit of a body to be 2.4 times its radius.

As a consequence of tidal interactions with the Moon, the Earth is slowly decreasing its rotational rate. Eventually the Earth and Moon will have exactly the same rotational period, and these will also exactly equal the orbital period. Presently, the Earth is spinning about 0.0016 seconds slower each century and the Moon is spiraling outward about 3-4 centimeters per year. Calculations indicate that in several billion years the length of the day will be about 47 present days and the Earth and Moon will be completely tidally locked with the Moon in a larger circular orbit. 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. A more detailed introduction to the consequences of tidal coupling between a planet and a moon may be found in this animation.

The Roche Limit The following diagram illustrates the calculation of the Roche Limit for an object in the gravitational field of a massive spherical object. Notice that if we can consider the densities of the two bodies to be equivalent the Roche Limit takes a simple form. In that case, it is just 2.4 times the radius of the massive object. One very often uses this simplified form of the Roche Limit equation to estimate the Roche limiting radius.