Celestial Mechanics

A tidal effect is produced when the gravitational pull of one body on a second one is appreciably greater on the nearer part of the second one than on its center, and in turn, the first body's pull on the second one's center is greater than its pull on the second one's most distant part. Unlike the gravitational force F_g, which varies as the inverse square of the distance r between the centers of the two bodies (1/r₂); see Equation 1, the tidal effect varies as the inverse cube (1/r³) of the distance between their centers. Both the Moon and the Sun raise tides in Earth's oceans, atmosphere, and solid body. The lag of the tides raised in the oceans behind the Moon's crossings of the celestial meridian causes a gravitational interaction between Earth and Moon which slows the earth's rotation and moves the Moon's orbit further from Earth.

Tides raised in the Moon's solid body by Earth have slowed its rotation until it has become tidally locked to Earth (the Moon keeps the same hemisphere turned towards Earth, and its periods of rotation and revolution around Earth are the same, 27.32 mean solar days). Eventually Earth's rotation will be slowed to where Earth will be tidally locked to the Moon, and the durations of the sidereal day and sidereal month will both equal about 47 present mean solar days.

Tidal evolution has forced most planetary satellites to become tidally locked to their planets. This includes all of Jupiter's Galilean satellites and its four small satellites closer to Jupiter than Io, probably most of Saturn's satellites out to Iasetus (Titan, Saturn's largest satellite, is probably tidally locked to Saturn), the satellites of Uranus and Neptune, and the Pluto-Charon double planet. Tidal action in Io's interior produced by Jupiter (and to a lesser degree by its next satellite out Europa) powers volcanism on Io, making it the most volcanically active body in the solar systems. Tidal effects also may have powered volcanic activity on Europa and Ganymede, Saturn's satellite Enceladus, and Uranus satellites Miranda and Ariel; all of them show some evidence of resurfacing. Some of Earth's internal heat may have been produced by the Moon's tidal action.

The Sun's tidal action on Mercury at perihelion has tidally locked Mercury's rotation to its angular velocity near perihelion, which is 1.5 times Mercury's average orbital angular velocity; Mercury's rotation period is 58.6 days, 2/3 of its 87.9 day period of revolution around the Sun.

When two bodies are very close together, tidal forces tending to disrupt a body can equal or exceed the attractive gravitational forces holding it together. If the tidal stresses exceed the yield limits of the body's material, the body will gradually disintegrate into many smaller bodies. The mathematician E. Roche (1820-1885) studied the limiting separation of two bodies where the tidal and gravitational forces are equal; it usually between 2-3 times the radius of the more massive body and depends on the relative densities of the bodies and their state of motion. If two bodies approach closer than this Roche limit, one (usually the smaller, less massive body) or both bodies will begin to disintegrate. The rings of some of the Jovian planets may have formed from the tidal disintegration of one or more of their close satellites. Theory predicts that after Earth's and Moon's rotations become tidally locked (see above), the Sun's tides raised on Earth will cause the Moon to approach Earth. If this effect lasts long enough, the Moon may get closer to Earth than its Roche limit, be disintegrated by Earth's tidal forces, and form a ring of small bodies which orbit Earth.

Tidal effects act on close double stars, distorting their shapes, changing their orbits, and sometimes tidally locking their rotations. In some cases, tidal effects cause streams of gas to flow in a double star system and can transfer matter from one star to the other or allow it to escape into interstellar space. Tidal effects even seem to act between galaxies, with one galaxy distorting the form of its neighbor.