Couplings and resonances In planetary orbits

Type of physical science: Astronomy; Astrophysics

Field of study: Planetary systems

Forced by gravitational attraction, resonances between two or more celestial bodies occur whenever the ratio of one period of motion makes a simple fraction with another period. Several structures and patterns in the solar system are caused by resonances, but the details of these patterns require sophisticated analysis that may help in understanding the structure of galaxies.

Overview

Patterns or resonances in the orbital and spin characteristics of planets and satellites became increasingly apparent as exploration of the solar system proceeded in the last half of the twentieth century. Resonances and the couplings that cause them are recognized as major features of the structure of the solar system. Among the resonance patterns, two types can be identified. One type involves two bodies such as the earth and the Moon or the sun and Mercury and is sometimes called spin, or spin-orbit resonance. The second type of resonance, called orbital resonance, is a simple pattern between the orbital periods of two or more smaller bodies as they orbit around a much larger body. In both cases, gravity is believed to be the major force coupling the objects involved.

Spin-orbit resonance is a simple pattern between the period a smaller object takes to spin on its axis and to orbit around a larger body. Often, the time to spin and to orbit are the same for the earth and the Moon. Consequently, spin-orbit resonance always keeps one side of the Moon facing the earth. In the case of the sun and Mercury's spin-orbit resonance, Mercury spins exactly three times for every two orbits it makes around the sun. The result of this resonance is that the same side of Mercury either points directly toward or away from the sun whenever the planet is closest to the sun at perihelion.

Orbital resonances occur when two objects orbit around a very massive third body, with orbital periods that are simple fractions of each other, such as 2:1, 3:1, 3:2, and 1:1. (When the ratio is 1:1, the two objects are said to be coorbital.) For a 2:1 ratio, for example, every two orbits of the inner body take exactly the same amount of time as one orbit of the outer body.

Such a relationship would put the resonant objects in alignment more often than other objects so that their influence on each other is unusually strong.

The way two masses influence each other was described by Sir Isaac Newton in 1686.

Newton's law of universal gravitation and his laws of motion provide a fundamental understanding of planetary orbits. Gravity provides the coupling mechanism between two or three bodies that results in resonances. In its simplest form, Newton's law of gravity describes the influence on each other of masses at two separate points. This gravitational influence increases the closer the mass points are and decreases quickly for points farther away. In fact, the force is proportional to the inverse of the square of the separating distance.

The gravitational force's dependence on distance provides the coupling mechanism that leads to spin-orbit resonances. Large objects, such as the Moon, will experience uneven gravitational forces over their surfaces because of the varying distances between points on its surface and Earth. Those points on the Moon that are closest to Earth feel the greatest force, and the weakest force is on the opposite side of the Moon. This unevenness of force--sometimes called the differential gravitational force--is responsible for tides. The effect of the Moon's differential gravitational force on Earth is, consequently, to produce two tidal bulges--one where the force is strongest and the other where the force is weakest. Friction between tidal bulges and the surface slows the rotation of a planet or moon until resonance is established. The gravitational force would also couple two (or more) satellites orbiting a much more massive body. Satellites in resonant orbits align more frequently than other satellites would. The inner satellite would consequently feel a gravitational force from the outer satellite. This frequent and regular tugging on the inner satellite would tend to move it away from its resonant orbit. If the outer satellite were massive enough, which is often the case, it would be essentially unaffected.

The result is that no appreciable satellite remains for long in an orbit that resonates with a large outer satellite. Gaps will therefore be produced and maintained in a belt or ring of material at the resonant orbits of a large outer satellite. Such gaps or divisions are observed in Saturn's rings and the asteroid belt. Details of these divisions suggest that the gravitational influence of the rest of the material in the ring also plays a part. The main driving mechanism for the presence and location of these divisions, however, is orbital resonance.

A special case of orbital resonance occurs when two or more satellites have the same orbital period and are coorbital. A careful study of Newton's laws by Joseph-Louis Lagrange in 1772 predicted two locations for stable orbits of a very small body coincident with the orbit of an intermediate body around a very massive body. Referred to as the leading and trailing Lagrange points, these stable points form the vertices of equilateral triangles, with the line between the very massive body and the intermediate body forming a common side for both triangles.

Consequently, the leading Lagrange point is 60 degrees (as measured from the very massive body) ahead in the orbit of the intermediate body, and the trailing Lagrange point is 60 degrees behind. The planet Jupiter has similar coorbital companions as do some of the moons of Saturn.

Applications

Several types of resonance manifest themselves in the solar system. In all cases, gravity provides the coupling force, although the way gravity is applied to cause the resonance varies. In the cases of spin-orbit resonance, gravity-produced tides cause the resonances. In cases of orbital resonance, the gravitational force from two bodies combine to produce either resonant gaps or stable, coorbital points where small particles accumulate. Examples of all these types of resonance can be found in the solar system. The most familiar example of spin-orbit resonance is the motion of the Earth's moon. Tidal stresses on the Moon from Earth have locked the Moon in its spin-orbit resonance so that only one side faces Earth. Earth dwellers are inclined to think that the Moon does not spin or rotate, but if viewed from far out in space, the Moon would be seen to spin once for every orbit it makes of the earth. As an illustration, the Moon has phases because sunlight reaches all points of the Moon. This indicates that, as viewed from the sun, the Moon spins once a month, which is exactly the same time that it takes to orbit the earth, and is in a 1:1, or synchronous spin-orbit resonance, with the earth. The Moon is often said to be tidally locked to the earth because of this resonance and its cause.

The Moon is not the only secondary satellite in the solar system to exhibit synchronous rotation. Tidal locking appears to be the rule for all moons close to a planet. In fact, the moons Phobos and Deimos are in synchronous rotation around Mars. The four Galilean moons of Jupiter, which are some of the largest moons in the solar system, also exhibit a 1:1 spin-orbit resonance. Of the twelve other moons of Jupiter, only two have had their rotational periods measured, and one of those, the closest to Jupiter, is synchronous. A similar situation exists for the moons of Saturn, where eight moons are known to have synchronous rotations, eight moons (including the giant, cloud-covered moon Titan) have not been measured, and two moons are nonsynchronous. The largest moons of Uranus and Neptune are also in synchronous rotation, and the smaller ones have not been measured. Pluto's moon Charon not only is in synchronous rotation around Pluto but also is large and close enough to have caused Pluto's rotation to be synchronous with Charon's orbit. Thus, all planets with moons have examples of synchronously rotating moons.

Mercury, although lacking any moon, also exhibits spin-orbit resonance. Mercury spins three times for every two orbits it makes of the sun. This 3:2 spin-orbit resonance is related to the unusually elongated, elliptical orbit of Mercury. As a result of its resonance condition, whenever Mercury is at perihelion, the same point on the planet is either facing directly toward or away from the sun. The Mariner 10 probe identified a huge crater, the Caloris Basin, at this point on the surface and the strange Weird Terrain on the planet's opposite side. These discoveries suggest that a huge, ancient impact that nearly tore the planet apart made one side of the planet heavier than the other and probably elongated and tilted the orbit. Tidal effects over the years have slowed the rotation of the planet so that whenever Mercury is at perihelion, its heavy side points either toward or away from the sun as a tidal bulge would. The fact that the spin is not synchronous with the orbit, as it is for Earth's moon, is most likely the result of Mercury's large mass and elongated orbit, which brings it considerably closer to the sun at perihelion than at aphelion.

Evidence of orbital resonance was discovered in the asteroid belt between Mars and Jupiter in 1866 by Daniel Kirkwood. As he studied the orbits of the asteroids, Kirkwood discovered gaps in an otherwise congested region of space. Since Kirkwood's original discovery of gaps at 2:1, 3:1, and 4:1 resonances with Jupiter, five other gaps have been identified. It is apparent that the strong and repeated pull of Jupiter destabilized orbits of asteroids with these periods and opened up the Kirkwood gaps.

Divisions in Saturn's rings have a cause similar to the Kirkwood gaps. Gian Domenico Cassini first observed the largest gap--the Cassini Division--in 1675; in 1867, Kirkwood discovered that the Cassini Division has a 2:1 orbital resonance with the moon Mimas. Kirkwood also showed that the Cassini Division was in a 3:1 resonance with the moon Enceladus, a 4:1 resonance with the moon Tethys, and a 6:1 resonance with the moon Dione, although the Mimas resonance is probably more significant because that coupling is stronger and more frequent. In addition to the Cassini Division, there are gaps in the A ring at resonances with the moons Janus (S10) and Epimetheus (S11). Moreover, the edges of the A and B rings, which are very well defined, occur at resonance locations.

There are still some mysteries to be found in the asteroid belt and Saturn's rings. At the Jupiter 3:2 resonance location in the asteroid belt, there is an accumulation of material instead of a gap. In the Cassini Division, there are ringlets, which may be spiral density waves excited by resonances with the moon Iapetus. Other details of the structure and shape of the divisions are still not well understood and may be aspects of density waves and chaotic behavior.

Nevertheless, these are details, and the main features must be caused by the simple resonances.

Jupiter and Saturn also have several examples of coorbital satellites. In Jupiter's orbit around the sun, there are clumps of asteroids one-sixth of an orbit ahead and behind Jupiter at the Lagrange points. These coorbital asteroids are called Trojan asteroids. In the Saturnian system, the moon Tethys has two Lagrangian coorbital satellites--Telesto (following Tethys) and Calypso (leading Tethys). In addition, the moon Dione has a coorbital satellite named Helene at the leading Lagrange point. Other examples are expected to exist, but they have not yet been observed.

Two of Saturn's satellites, Janus and Epimetheus, are also coorbital, but in a different way from that of the Lagrangian coorbitals. Janus and Epimetheus have orbits that are so close together that their gravitational attraction for each other is sufficient for them to interchange orbits without colliding. The differences in this case and the Lagrangian coorbital satellites results from the fact that Janus and Epimetheus are nearly the same size and that one satellite does not always lead the other.

Context

The Moon's spin-orbit resonance has been known ever since humans became aware of the world around them. Yet, the mechanism for understanding why such a resonance would occur had not been discovered until Newton formulated his laws. In fact, tides were explained in his PHILOSOPHIAE NATURALIS PRINCIPIA MATHEMATICA (1687; NEWTON'S PRINCIPIA: THE MATHEMATICAL PRINCIPALS OF NATURAL PHILOSOPHY, 1846), in which he first used his laws publicly to explain the orbits of planets and Johannes Kepler's laws of planetary motion. It was thought that most moons would have similar resonances with their planets, but this extrapolation needed to be confirmed. Planetary probes have visited all planets except Pluto; with a few notable exceptions, the moons studied did exhibit spin-orbit resonance. Unfortunately, all the moons were not able to be studied thoroughly to determine their spin rates; as a result, knowing which moons are tidally locked to their planet will not be answered fully until additional probes or improved technology become available.

Interestingly, Mercury generally was thought to be tidally locked to the sun ever since Giovanni Schiaparelli made crude maps of the surface in the 1880's. This opinion seemed to be confirmed by later Earth-based observations that were carried out up to the early 1960's. It was not until Doppler radar techniques were applied to Mercury in 1965 that the 3:2 spin-orbit resonance was discovered. In this study, radar signals were sent from the 300-meter Arecibo radio telescope in Puerto Rico and bounced off Mercury. The change in the signal's frequency (the Doppler effect) proved that Mercury rotates once in 58.65 Earth days instead of the 88 days that it takes to orbit the sun. It was not until the three Mariner 10 flybys of Mercury in 1974 and early 1975 that the Caloris Basin and the Weird Terrain were discovered and the orbital resonance was confirmed.

The discovery of the Cassini Division in 1675 also predates Newton's laws, which are essential to explain the division. It was not until Kirkwood's discovery of resonance conditions in 1866 and 1867 that a reasonable explanation for the formation of this division was offered.

Kirkwood's model for the Cassini Division and the gaps in the asteroid belt was thought to be adequate until the Voyager data became available in 1981. The images of the Cassini Division from the Voyager probes revealed a number of unexpected details. These details require a more sophisticated application of Newton's laws and provide a testing ground for density wave theories and theories of chaotic behavior. These theories could explain the structure of galaxies.

In contrast to the previous cases, Lagrange predicted in 1772 the location of stable, coorbital companions to Jupiter and other planets. In 1906, Jupiter's coorbital companions were found. The search continues with nearly two hundred such asteroids discovered to date and perhaps ten times that number orbiting at Jupiter's Lagrange points. The coorbital satellites in Saturn's system were undiscovered until the Voyager flybys in 1980 and 1981. In fact, Lagrangian coorbital satellites may exist for Earth, Mars, and other large bodies.

Principal terms

ASTEROID BELT: the region between the orbits of Mars and Jupiter, which is filled with stony or metallic bodies called asteroids

COORBITAL SATELLITES: bodies that share the same orbit; these bodies are in a 1:1 orbital resonance

DIFFERENTIAL GRAVITATIONAL FORCE: the unevenness in gravitational pull on a body because of its own size; tides result from this effect

GRAVITY: a fundamental force of nature by which all masses attract one another

LAGRANGE POINTS: a stable point in the orbit of an intermediate body around a larger body where small particles may accumulate; these are sometimes referred to as L4 and L5

ORBIT: the motion of a small body around a larger body such as that of Earth around the sun, also called revolution

RESONANCE: the situation when the periods of at least two periodic motions have a ratio of simple whole numbers such as 3:2

SPIN: the rotating of a body about an axis through itself

SYNCHRONOUS ROTATION: when a satellite rotates and revolves at the same rate so that only one side faces its primary body; this is a 1:1 spin-orbit resonance and is also called tidal locking

Bibliography

Abell, George O., David Morrison, and Sidney C. Wolff. EXPLORATION OF THE UNIVERSE. 5th ed. Philadelphia, Pa.: Saunders College Publishing, 1987. A thorough and detailed discussion of astronomy and its history, this textbook has been used in introductory courses for decades and is considered a classic. It was updated in 1987 before Supernova 1987A was discovered and the Voyager 2 flyby of Neptune.

Editors of Time-Life Books. COMETS, ASTEROIDS, AND METEORITES. Alexandria, Va.: Time-Life Books, 1990. This volume discusses the Trojan asteroids orbiting with Jupiter and the theories about the formation of the Kirkwood gaps. Very readable and well illustrated.

Editors of Time-Life Books. THE FAR PLANETS. Alexandria, Va.: Time-Life Books, 1988. Notable for its pictures and informative illustrations of Saturn's ring system. Published before the August, 1989, flyby of Neptune by Voyager 2.

Editors of Time-Life Books. THE NEAR PLANETS. Alexandria, Va.: Time-Life Books, 1989. This volume provides a very readable summary of the exploration of Mercury and its spin-orbit resonance. A good source of scientifically accurate illustrations.

Elliot, James, and Richard Kerr. RINGS: DISCOVERIES FROM GALILEO TO VOYAGER. Cambridge, Mass.: MIT Press, 1984. A detailed discussion of the structure and formation of planetary rings. Discusses the connection between the Cassini Division, spiral density waves, and galaxies.

Hartmann, William K. ASTRONOMY: THE COSMIC JOURNEY. Belmont, Calif.: Wadsworth, 1991. An extremely well-illustrated and well-written general astronomy text by a noted astronomer and artist. Table 8.1 summarizes data on planets, moons, and major asteroids.

Moore, Patrick, and Garry Hunt. THE ATLAS OF THE SOLAR SYSTEM. New York: Crescent Books, 1990. This truly encyclopedic source for information about the solar system is filled with illustrations, maps, and photographs. Contains valuable information on resonances and includes Voyager 2 data from Neptune.

Morrison, David, and Tobias Owen. THE PLANETARY SYSTEM. Reading, Mass.: Addison-Wesley, 1987. An interesting and detailed discussion of the solar system by two prominent members of the Voyager and Galileo missions. Published before the Voyager 2 encounter with Neptune.

Shu, Frank H. THE PHYSICAL UNIVERSE: AN INTRODUCTION TO ASTRONOMY. Mill Valley, Calif.: University Science Books, 1982. Although somewhat dated and more mathematical than the other references, this book contains an extensive and enlightening discussion of resonances in the solar system. Also noteworthy is the discussion of spiral density waves and its application to the Cassini Division by one of the pioneers of that theory.

Wagner, Jeffrey K. INTRODUCTION TO THE SOLAR SYSTEM. Philadelphia, Pa.: Saunders College Publishing, 1991. A well-written and up-to-date discussion of all aspects of the solar system, especially spin-orbit coupling.

Ring Systems of Planets