In celestial mechanics, orbital resonance occurs when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers. Most commonly, this relationship is found between a pair of objects (binary resonance). The physical principle behind orbital resonance is similar in concept to pushing a child on a swing, whereby the orbit and the swing both have a natural frequency, and the body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies (i.e., their ability to alter or constrain each other's orbits). In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be self-correcting and thus stable. Examples are the near 28:14:7:3 resonance of Jupiter's moons Io, Europa, Ganymede and Callisto, and the 3:2 resonance between Neptune and Pluto. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance between bodies with similar orbital radii causes large planetary system bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood, an effect that is used in the current definition of a planet.
A binary resonance ratio in this article should be interpreted as the ratio of number of orbits completed in the same time interval, rather than as the ratio of orbital periods, which would be the inverse ratio. Thus, the 2:3 ratio above means that Pluto completes two orbits in the time it takes Neptune to complete three. In the case of resonance relationships among three or more bodies, either type of ratio may be used (whereby the smallest whole-integer ratio sequences are not necessarily reversals of each other), and the type of ratio will be specified.
History Since the discovery of Newton's law of universal gravitation in the 17th century, the stability of the Solar System has preoccupied many mathematicians, starting with Pierre-Simon Laplace. The stable orbits that arise in a two-body approximation ignore the influence of other bodies. The effect of these added interactions on the stability of the Solar System is very small, but at first it was not known whether they might add up over longer periods to significantly change the orbital parameters and lead to a completely different configuration, or whether some other stabilising effects might maintain the configuration of the orbits of the planets.
It was Laplace who found the first answers explaining the linked orbits of the Galilean moons (see below). Before Newton, there was also consideration of ratios and proportions in orbital motions, in what was called "the music of the spheres", or musica universalis.
The article on resonant interactions describes resonance in the general modern setting. A primary result from the study of dynamical systems is the discovery and description of a highly simplified model of mode-locking; this is an oscillator that receives periodic kicks via a weak coupling to some driving motor. The analog here would be that a more massive body provides a periodic gravitational kick to a smaller body, as it passes by. The mode-locking regions are named Arnold tongues.
Types
In general, an orbital resonance may involve one or any combination of the orbit parameters (e.g. eccentricity versus semimajor axis, or eccentricity versus inclination). act on any time scale from short term, commensurable with the orbit periods, to secular, measured in 104 to 106 years. lead to either long-term stabilization of the orbits or be the cause of their destabilization.
Mean motion orbital resonance A mean motion orbital resonance (MMR) occurs when multiple bodies have orbital periods or mean motions (orbital frequencies) that are simple integer ratios of each other.
Two-body mean motion resonance The simplest cases of MMRs involve only two bodies. The ratio of the periods needs to be near a rational number, but not exactly a rational number, even averaged over a long period, because there is a dependence on the motion of the pericenter. For example, in the case of Pluto and Neptune (see below), the true equation says that the average rate of change of is exactly zero, where is the longitude of Pluto, is the longitude of Neptune, and is the longitude of Pluto's perihelion. Since the rate of motion of the latter is about degrees per year, the ratio of periods is actually 1.503 in the long term.
Depending on the details, two-body MMRs can either stabilize or destabilize the orbit of one of the resonant bodies. Stabilization may occur when the two bodies move in such a synchronised fashion that they never closely approach. For instance: The orbits of Pluto and the plutinos are stable, despite crossing that of the much larger Neptune, because they are in a 2:3 resonance with it. The resonance ensures that, when they approach perihelion and Neptune's orbit, Neptune is consistently distant (averaging a quarter of its orbit away). Other (much more numerous) Neptune-crossing bodies that were not in resonance were ejected from that region by strong perturbations due to Neptune. There are also smaller but significant groups of resonant trans-Neptunian objects occupying the 1:1 (Neptune trojans), 3:5, 4:7, 1:2 (twotinos) and 2:5 resonances, among others, with respect to Neptune. In the asteroid belt beyond 3.5 AU from the Sun, the 3:2, 4:3 and 1:1 resonances with Jupiter are populated by clumps of asteroids (the Hilda family, the few Thule asteroids, and the numerous Trojan asteroids, respectively).
MMRs can also destabilize one of the orbits. This process can be exploited to find energy-efficient ways of deorbiting spacecraft. For small bodies, destabilization is actually far more likely. For instance: In the asteroid belt within 3.5 AU from the Sun, the major MMRs with Jupiter are locations of gaps in the asteroid distribution, the Kirkwood gaps (most notably at the 4:1, 3:1, 5:2, 7:3 and 2:1 resonances). Asteroids have been ejected from these almost empty lanes by repeated perturbations. However, there are still populations of asteroids temporarily present in or near these resonances. For example, asteroids of the Alinda family are in or close to the 3:1 resonance, with their orbital eccentricity steadily increased by interactions with Jupiter until they eventually have a close encounter with an inner planet that ejects them from the resonance. In the rings of Saturn, the Cassini Division is a gap between the inner B Ring and the outer A Ring that has been cleared by a 2:1 resonance with the moon Mimas. (More specifically, the site of the resonance is the Huygens Gap, which bounds the outer edge of the B Ring.) In the rings of Saturn, the Encke and Keeler gaps within the A Ring are cleared by 1:1 resonances with the embedded moonlets Pan and Daphnis, respectively. The A Ring's outer edge is maintained by a destabilizing 7:6 resonance with the moon Janus.
Most bodies that are in two-body MMRs orbit in the same direction; however, the retrograde asteroid 514107 Kaʻepaokaʻāwela appears to be in a stable (for a period of at least a million years) 1:−1 resonance with Jupiter. In addition, a few retrograde damocloids have been found that are temporarily captured in MMR with Jupiter or Saturn. Such orbital interactions are weaker than the corresponding interactions between bodies orbiting in the same direction. The trans-Neptunian object 471325 Taowu has an orbital inclination of 110° with respect to the planets' orbital plane and is currently in a 7:9 polar resonance with Neptune.
N-body mean motion resonance MMRs involving more than two bodies have been observed in the Solar System. For example, there are three-body MMRs involving Jupiter, Saturn, and some main-belt asteroids. These three-body MMRs are unstable and main-belt asteroids involved in these three-body MMRs have chaotic orbital evolutions.
A Laplace resonance is a three-body MMR with a 1:2:4 orbital period ratio (equivalent to a 4:2:1 ratio of orbits). The term arose because Pierre-Simon Laplace discovered that such a resonance governed the motions of Jupiter's moons Io, Europa, and Ganymede. It is now also often applied to other 3-body resonances with the same ratios, such as that between the extrasolar planets Gliese 876 c, b, and e. Three-body resonances involving other simple integer ratios have been termed "Laplace-like" or "Laplace-type".
Lindblad resonance A Lindblad resonance drives spiral density waves both in galaxies (where stars are subject to forcing by the spiral arms themselves) and in Saturn's rings (where ring particles are subject to forcing by Saturn's moons).
Secular resonance A secular resonance occurs when the precession of two orbits is synchronised (usually a precession of the perihelion or ascending node). A small body in secular resonance with a much larger one (e.g. a planet) will precess at the same rate as the large body. Over long times (a million years, or so) a secular resonance will change the eccentricity and inclination of the small body.
Several prominent examples of secular resonance involve Saturn. There is a near-resonance between the precession of Saturn's rotational axis and that of Neptune's orbital axis (both of which have periods of about 1.87 million years), which has been identified as the likely source of Saturn's large axial tilt (26.7°). Initially, Saturn probably had a tilt closer to that of Jupiter (3.1°). The gradual depletion of the Kuiper belt would have decreased the precession rate of Neptune's orbit; eventually, the frequencies matched, and Saturn's axial precession was captured into a spin-orbit resonance, leading to an increase in Saturn's obliquity. (The angular momentum of Neptune's orbit is 104 times that of Saturn's rotation rate, and thus dominates the interaction.) However, it seems that the resonance no longer exists. Detailed analysis of data from the Cassini spacecraft gives a value of the moment of inertia of Saturn that is just outside the range for the resonance to exist, meaning that the spin axis does not stay in phase with Neptune's orbital inclination in the long term, as it apparently did in the past. One theory for why the resonance came to an end is that there was another moon around Saturn whose orbit destabilized about 100 million years ago, perturbing Saturn.
The perihelion secular resonance between asteroids and Saturn helps shape the asteroid belt. This involves a rate of change of angle called ν6, defined as g − g6 where g is the rate of change of the longitude of perihelion of the asteroid and g is that of Saturn (as the sixth planet from the Sun). The value of ν depends on the semi-major axis (or period) of the asteroid and its inclination. Asteroids which approach a state where ν goes to zero have their eccentricity slowly increased until they become Mars-crossers, at which point they are usually ejected from the asteroid belt by a close pass to Mars. This resonance forms the inner and "side" boundaries of the asteroid belt around 2 AU, and at inclinations of about 20°.
Numerical simulations have suggested that the eventual formation of a perihelion secular resonance between Mercury and Jupiter (g1 = g5, where again g is the rate of change of the longitude of perihelion) has the potential to greatly increase Mercury's eccentricity and destabilize the inner Solar System several billion years from now.
The Titan Ringlet within Saturn's C Ring represents another type of resonance in which the rate of apsidal precession of one orbit exactly matches the speed of revolution of another. The outer end of this eccentric ringlet always points towards Saturn's major moon Titan.