Planetary orbits: Couplings and resonances

Orbital or rotational (spin) motions of objects are said to be “coupled” or “in resonance” when the relationships between the periods of such motions can be expressed as ratios of small integers such as 1:1, 1:2, 1:3, 2:3, or 3:4. This usually occurs as the result of the gravitational interaction of the objects. Many examples of couplings and resonances occur in the orbital and rotational motions of solar system objects.

Overview

The orbital or rotational (spin) periods of many solar-system objects have been found to be related by ratios of small integers, such as 1:1, 1:2, 1:3, 2:3, or 3:4. Such relationships in the motions are called couplings or resonances. Usually, they have developed over time as the result of gravitational interactions between the objects. Because of their ubiquity, couplings and resonances are thought to have played a major role in shaping the structure of the solar system.

The many couplings and resonances in the solar system can be categorized into two main types. One type, called spin-orbit resonance, is manifested by a simple ratio between an object’s period of spinning (rotating) on its axis and its period of orbiting (revolving) around a more massive body. For example, the time it takes the Moon to rotate on its axis exactly equals the time it takes to revolve around Earth, a ratio of 1:1. As a result, the Moon always keeps the same side facing Earth. (Because the Moon’s orbit around Earth is not precisely circular, its orbital speed varies slightly. Consequently, it appears to us on Earth as if the Moon rocks a bit from side to side—a motion called libration—so we end up seeing slightly more than half the Moon’s surface during one of its orbits.) Another example is that Mercury’s period of rotating on its axis is exactly two-thirds of its period of revolving around the Sun, a ratio of 2:3. Thus, Mercury spins three times on its axis during two orbits around the Sun.

The other type of resonance, called orbital resonance, involves small-integer ratios between the orbital periods of two or more small-mass bodies orbiting around a much more massive object. Such relationships reinforce the gravitational interactions between the resonant objects. Suppose one of the orbiting small-mass bodies is much less massive than the other small-mass body. (Examples include an asteroid and Jupiter as both orbit the Sun and a ring particle and a satellite as both orbit Saturn.) The repeated gravitational tugs of the more massive orbiting body (Jupiter or the satellite) on the less massive orbiting body (the asteroid or the ring particle) will tend to pull the less massive body away from its resonant orbit, while the more massive orbiting body is little affected. This clearing of resonant orbits produces gaps in belt and ring systems. Such gaps, or divisions, are observed in the asteroid belt (the Kirkwood gaps), located mainly at the resonances with Jupiter and in Saturn’s ring system as a result of resonances with some of Saturn’s satellites.

When two or more objects have exactly the same orbital period around a more massive object, they are called coorbital, a special case of orbital resonance. In 1772, Joseph-Louis Lagrange mathematically discovered five points at which coorbital bodies could exist in equilibrium. These points are called the Lagrangian points and are labeled L1 through L5. Three of the points (L1 through L3) are unstable in that an object displaced slightly from the point will drift farther away. However, the L4 and L5 points are stable in that an object displaced slightly from the point will remain nearby and oscillate around the equilibrium position. These two stable points are located 60 degrees ahead (L4) and 60 degrees behind (L5), the second-largest body along its orbit around the largest body. The Trojan asteroids (so-called because they have been named after the heroes, both Trojan and Greek, of the Trojan War) oscillate around the L4, and L5 points 60 degrees ahead of and behind Jupiter along its orbit around the Sun.

Methods of Study

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, gravitationally produced tides cause the resonances. In cases of orbital resonance, the gravitational forces 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 within the solar system.

The most familiar example of spin-orbit resonance is the motion of Earth’s only natural satellite, the 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 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 Earth, and is, therefore, in a 1:1, or synchronous spin-orbit, resonance with Earth. The Moon is often said to be “tidally locked” to 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 satellites close to a planet. In fact, Phobos and Deimos are in synchronous rotation around Mars. The four Galilean moons of Jupiter, which are some of the largest satellites in the solar system, also exhibit a 1:1 spin-orbit resonance. Of the many other satellites of Jupiter, four others exhibit synchronous rotation. A similar situation exists for the satellites of Saturn, where numerous are known to have synchronous rotations. At least two Saturnian satellites are nonsynchronous. The largest satellites of Uranus and Neptune are also in synchronous rotation, and the smaller ones have not yet been measured. Pluto’s companion Charon is 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 satellites 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 10spacecraft identified a huge feature, the Caloris Basin, at this point on the surface and some strange surface features, dubbed 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, at least five other gaps have been identified. It is apparent that the strong and repeated pull of Jupiter destabilized the 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 satellite Mimas. Kirkwood also showed that the Cassini division was in a 3:1 resonance with the satellite Enceladus, a 4:1 resonance with the moon Tethys, and a 6:1 resonance with the satellite 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 satellites 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. Surprisingly, at the Jupiter 3:2 resonance location in the asteroid belt, there is accumulated material instead of an empty gap. In the Cassini division, there are ringlets, which may be spiral density waves excited by resonances with the satellite 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 Lagrangian points. These coorbital asteroids are called Trojan asteroids. In the Saturnian system, the satellite Tethys has two Lagrangian coorbital satellites, Telesto (following Tethys) and Calypso (leading Tethys). In addition, the satellite Dione has a coorbital satellite named Helene at the leading Lagrangian point.

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 difference between this case and that of 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 result of the Moon’s spin-orbit resonance has been known ever since humans became aware of the world around them, but the mechanism for understanding why such a resonance would occur was not discovered until Sir Isaac 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 Johannes Kepler’s laws of planetary motion. It was thought that most satellites would have similar resonances with their planets, but this extrapolation needed to be confirmed. Planetary probes have visited all planets, and the New Horizons spacecraft completed a flyby of the Pluto-Charon system in 2015. With a few notable exceptions, many satellites have exhibited spin-orbit resonance. Unfortunately, all the satellites were not able to be studied thoroughly to determine their spin rates; as a result, a complete understanding of which moons are tidally locked to their planets and how will not be within reach until additional probes or improved technology becomes available. The Galileo probe improved the understanding of Jupiter’s satellites, and Cassini added to what is known about Saturn’s.

Interestingly, Mercury generally was thought to be tidally locked to the Sun ever since Giovanni Schiaparelli made crude maps of the surface in the 1880s. This opinion seemed to be confirmed by later Earth-based observations that were carried out prior to the early 1960s. 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 three hundred–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 eighty-eight 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.

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 gaps in the asteroid belt was thought to be adequate until the Voyager data became available in 1981. Images of the Cassini division from the two Voyager spacecrafts revealed a number of unexpected details. These details require a more sophisticated application of Newtonian mechanics and provide a testing ground for density wave theories and theories of chaotic behavior. These theories could help explain the structure of galaxies.

In contrast to 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 coorbital satellites in Saturn’s system were undiscovered until the Voyager flybys in 1980 and 1981. In 2011, NASA's Wide-field Infrared Survey Explorer (WISE) found a Trojan asteroid, 2010 TK₇, that is orbiting the Sun with the Earth.

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