In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Celestial mechanics is the branch of Astrophysics that deals with the motions of Celestial objects The field applies principles of Physics, historically In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star Orbital resonances greatly enhance the mutual gravitational influence of the bodies. 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 stable and self correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa, and Io, and the 2:3 resonance between Pluto and Neptune. TemplateInfobox Planet.--> Ganymede (ˈgænɨmiːd, or as Greek TemplateInfobox Planet.--> Europa (jʊˈroʊpə; or as TemplateInfobox Planet.--> Io (ˈaɪoʊ, or as Greek Neptune ( English|AmE] ] is the eighth and farthest Planet from the Sun in the Solar System. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. Saturn has the most extensive Planetary ring system of any planet in the Solar System. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to clear the neighborhood around their orbits by ejecting nearly everything else around them; this effect is used in the current definition of a planet. From its beginnings denoting the "wandering stars" of the classical world the definition of " Planet " has been fraught with ambiguity

NOTE: In this article (except as noted in the Laplace resonance figure), a resonance ratio 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). The 2:3 ratio above means Pluto completes 2 orbits in the time it takes Neptune to complete 3.

History

Ever since the discovery of Newton's law of universal gravitation in the 17th century, the stability of planetary orbits has preoccupied many mathematicians, starting with Laplace. Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass The stable orbits that arise in a two-body approximation ignore the influence of other bodies. The n -body problem is the problem of finding given the initial positions masses and velocities of n bodies their subsequent motions as determined by 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. The Solar System consists of the Sun and those celestial objects bound to it by Gravity.

It was Laplace who found the first answers explaining the remarkable dance of the Galilean moons (see below). It is fair to say that this general field of study has remained very active since then, with plenty more yet to be understood (e. g. how interactions of moonlets with particles of the rings of giant planets result in maintaining the rings).

Types of resonance

In general, an orbital resonance may

• involve one or any combination of the orbit parameters (e. g. eccentricity versus semimajor axis, or eccentricity versus orbit inclination). In Astrodynamics, under standard assumptions, any Orbit must be of Conic section shape In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae
• 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 stabilisation of the orbits or be the cause of their destabilization.

A mean motion orbital resonance occurs when two bodies have periods of revolution that are a simple integer ratio of each other. Depending on the details, this can either stabilize or destabilize the orbit. Stabilization occurs when the two bodies move in such a synchronised fashion that they never closely approach. For instance:

• Pluto and the plutinos are in stable orbits, despite crossing the orbit of the much larger Neptune. In Astronomy, a plutino is a Trans-Neptunian object in 23 mean motion resonance with Neptune. Neptune ( English|AmE] ] is the eighth and farthest Planet from the Sun in the Solar System. This is because a 2:3 resonance keeps them always at a large distance from it. 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), 1:2 (twotinos) and 2:5 resonances with respect to Neptune. In Astronomy, a resonant trans-Neptunian object is a Trans-Neptunian object (TNO in mean motion Orbital resonance with Neptune As of May 2008, there are six known Neptune trojans (named by analogy to the Trojan asteroids which have the same orbital period as Neptune. In Astronomy, a resonant trans-Neptunian object is a Trans-Neptunian object (TNO in mean motion Orbital resonance with Neptune
• In the asteroid belt beyond 3. The asteroid belt is the region of the Solar System located roughly between the orbits of the Planets Mars and Jupiter. 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, 279 Thule, and the Trojan asteroids, respectively). The Hilda family of asteroids consists of Asteroids with a Semi-major axis between 3 279 Thule (ˈθjuːli thew'-lee) is a very large Main belt Asteroid.
• The extrasolar planets Gliese 876b and Gliese 876c are in a 1:2 orbital resonance

Orbital resonances can also destabilize one of the orbits. An extrasolar planet, or exoplanet, is a Planet beyond the Solar System, orbiting around other Stars As of September 2008 312 Gliese 876, also catalogued as IL Aquarii, is a Red dwarf Star approximately 15 Light-years away in the Constellation of Gliese 876, also catalogued as IL Aquarii, is a Red dwarf Star approximately 15 Light-years away in the Constellation of For small bodies, destabilization is actually far more likely. For instance:

• In the asteroid belt within 3. Asteroids, sometimes called Minor planets or planetoids', are bodies—primarily of the inner Solar System —that are smaller than planets but 5 AU from the sun, the major mean-motion resonances with Jupiter are locations of gaps in the asteroid distribution, the Kirkwood gaps (most notably at the 3:1, 5:2, 7:3, and 2:1 resonances). Kirkwood gaps are gaps or dips in the distribution of Main belt Asteroids with Semi-major axis (or equivalently their Orbital period) as seen 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 increasing due to interactions with Jupiter until they eventually have a close encounter with an inner planet that ejects them from the resonance. The Alinda asteroids are a group of Asteroids with a Semi-major axis of about 2
• 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. Saturn has the most extensive Planetary ring system of any planet in the Solar System. Saturn has the most extensive Planetary ring system of any planet in the Solar System. Saturn has the most extensive Planetary ring system of any planet in the Solar System. Saturn has the most extensive Planetary ring system of any planet in the Solar System. TemplateInfobox Planet. --> Mimas (ˈmaɪməs, or as Greek (More specifically, the site of the resonance is the Huygens Gap, which bounds the outer edge of the B Ring. Saturn has the most extensive Planetary ring system of any planet in the Solar System. Saturn has the most extensive Planetary ring system of any planet in the Solar System. )
• In the rings of Saturn, the A Ring's outer edge is maintained by a destabilizing 7:6 resonance with the moon Janus. Saturn X redirects here For the spurious moon reported in 1905 see Themis (moon

A Laplace resonance occurs when three or more orbiting bodies have a simple integer ratio between their orbital periods. For example, Jupiter's moons Ganymede, Europa, and Io are in a 1:2:4 orbital resonance. TemplateInfobox Planet.--> Ganymede (ˈgænɨmiːd, or as Greek TemplateInfobox Planet.--> Europa (jʊˈroʊpə; or as TemplateInfobox Planet.--> Io (ˈaɪoʊ, or as Greek

A Secular resonance occurs when the precession of two orbits is synchronised (usually a precession of the perihelion or ascending node). A secular resonance is a type of Orbital resonance. Secular resonances occur when the precession of two orbits is synchronised (a precession of the Perihelion Precession refers to a change in the direction of the axis of a rotating object In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from An orbital node is one of the two points where an Orbit crosses a Plane of reference which it is inclined to 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. A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is Over long times (a million years, or so) a secular resonance will change the eccentricity and inclination of the small body. In Astrodynamics, under standard assumptions, any Orbit must be of Conic section shape Inclination in general is the Angle between a Reference plane and another plane or axis of direction A prominent example is the ν₆ secular resonance between asteroids and Saturn. Asteroids, sometimes called Minor planets or planetoids', are bodies—primarily of the inner Solar System —that are smaller than planets but Asteroids which approach it have their eccentricity slowly increased until they become Mars-crossers, at which point they are usually ejected from the asteroid belt due to a close pass to Mars. The asteroid belt is the region of the Solar System located roughly between the orbits of the Planets Mars and Jupiter. This resonance forms the inner and "side" boundaries of the main asteroid belt around 2 AU, and at inclinations of about 20°. The asteroid belt is the region of the Solar System located roughly between the orbits of the Planets Mars and Jupiter. The astronomical unit ( AU or au or au or sometimes ua) is a unit of Length based on the distance from the Earth to the Inclination in general is the Angle between a Reference plane and another plane or axis of direction

The Titan Ringlet within Saturn's C Ring exemplifies another type of resonance in which the rate of apsidal precession of one orbit exactly matches the speed of revolution of another. Saturn has the most extensive Planetary ring system of any planet in the Solar System. Saturn has the most extensive Planetary ring system of any planet in the Solar System. Precession refers to a change in the direction of the axis of a rotating object The outer end of this eccentric ringlet always points towards Saturn's major moon Titan. TemplateInfobox Planet.--> Titan (ˈtaɪtən, or as

A Kozai resonance occurs when the inclination and eccentricity of a perturbed orbit oscillate synchronously (increasing eccentricity while decreasing inclination and vice versa). In Celestial mechanics, the Kozai mechanism is a secular perturbative effect on certain orbits This article describes perturbation theory as a general mathematical method This resonance applies only to bodies on highly inclined orbits; as a consequence, such orbits tend to be unstable, since the growing eccentricity would result in small pericenters, typically leading to a collision or (for large moons) destruction by tidal forces. In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from The tidal force is a secondary effect of the Force of Gravity and is responsible for the Tides It arises because the gravitational acceleration experienced

Mean motion resonances in the Solar System

The Laplace resonance exhibited by three of the Galilean moons. The ratios in the figure are of orbital periods.

There are only a few known mean motion resonances in the Solar System involving planets or larger satellites (a much greater number involve asteroids, Kuiper belt objects, planetary rings and moonlets). The Solar System consists of the Sun and those celestial objects bound to it by Gravity. A natural satellite or moon is a Celestial body that Orbits a Planet or smaller body which is called the primary. Asteroids, sometimes called Minor planets or planetoids', are bodies—primarily of the inner Solar System —that are smaller than planets but The Kuiper belt (ˈkaɪpɚ to rhyme with "viper" sometimes called the Edgeworth-Kuiper belt, is a region of the Solar System beyond the planets extending A planetary ring is a ring of Cosmic dust and other small particles Orbiting around a Planet in a flat disc-shaped region In Astronomy, an inner moon is a Natural satellite following a prograde, low inclination Orbit inwards of the large satellites of

• 2:3 Pluto-Neptune
• 2:4 Tethys-Mimas (Saturn’s moons)
• 1:2 Dione-Enceladus (Saturn’s moons)
• 3:4 Hyperion-Titan (Saturn's moons)
• 1:2:4 Ganymede-Europa-Io (Jupiter’s moons); the only Laplace resonance

The simple integer ratios between periods are a convenient simplification hiding more complex relations:

• the point of conjunction can oscillate (librate) around an equilibrium point defined by the resonance. Neptune ( English|AmE] ] is the eighth and farthest Planet from the Sun in the Solar System. TemplateInfobox Planet.--> Tethys (ˈtiːθɨs, /ˈtɛθɨs/, or TemplateInfobox Planet. --> Mimas (ˈmaɪməs, or as Greek TemplateInfobox Planet.--> Dione (daɪˈoʊni, or as in Greek TemplateInfobox Planet.--> This article is about the moon of TemplateInfobox Planet.--> Hyperion (haɪˈpɪəriən, or as in TemplateInfobox Planet.--> Titan (ˈtaɪtən, or as TemplateInfobox Planet.--> Ganymede (ˈgænɨmiːd, or as Greek TemplateInfobox Planet.--> Europa (jʊˈroʊpə; or as TemplateInfobox Planet.--> Io (ˈaɪoʊ, or as Greek Conjunction is a term used in Positional astronomy and Astrology. In Astronomy libration (from the Latin verb librare "to balance to sway" cf
• given non-zero eccentricities, the nodes or periapsides can drift (a resonance related, short period, not secular precession). In Astrodynamics, under standard assumptions, any Orbit must be of Conic section shape An orbital node is one of the two points where an Orbit crosses a Plane of reference which it is inclined to In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from

As illustration of the latter, consider the well known 2:1 resonance of Io-Europa. If the orbiting periods were in this relation, the mean motions (inverse of periods, often expressed in degrees per day) would satisfy the following

Substituting the data (from Wikipedia) one will get −0. Mean motion, n\\! is a measure of how fast a Satellite progresses around its Orbit. 7395° day⁻¹, a value substantially different from zero!

Actually, the resonance is perfect but it involves also the precession of perijove (the point closest to Jupiter) The correct equation (part of the Laplace equations) is:

In other words, the mean motion of Io is indeed double of that of Europa taking into account the precession of the perijove. In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from An observer sitting on the (drifting) perijove will see the moons coming into conjunction in the same place (elongation). The other pairs listed above satisfy the same type of equation with the exception of Mimas-Tethys resonance. In this case, the resonance satisfies the equation

The point of conjunctions librates around the midpoint between the nodes of the two moons. An orbital node is one of the two points where an Orbit crosses a Plane of reference which it is inclined to

The Laplace resonance

Illustration of Io-Europa-Ganymede resonance. From the centre outwards: Io (yellow), Europa (gray) and Ganymede (dark).

The most remarkable resonance involving Io-Europa-Ganymede includes the following relation locking the orbital phase of the moons:

Φ_L

where λ are mean longitudes of the moons. In Astrodynamics or Celestial dynamics mean longitude is the Longitude at which an orbiting body could be found if its orbit were circular and This relation makes a triple conjunction impossible. The graph illustrates the positions of the moons after 1, 2 and 3 Io periods.

Pluto resonances

Pluto is following an orbit trapped in a web of resonances with Neptune. Neptune ( English|AmE] ] is the eighth and farthest Planet from the Sun in the Solar System. The resonances include:

• A mean motion resonance of 2:3
• The resonance of the perihelion (libration around 90°), keeping the perihelion above the ecliptic
• The resonance of the longitude of the perihelion in relation to that of Neptune

One consequence of these resonances is that a separation of at least 30 AU is maintained when Pluto crosses Neptune's orbit. In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from The ecliptic is the apparent path that the Sun traces out in the sky during the year The minimum separation between the two bodies overall is 17 AU, while the minimum separation between Pluto and Uranus is just 11 AU^[1] (see Pluto's orbit for detailed explanation and graphs).

Coincidental 'near' ratios of mean motion

A number of near-integer-ratio relationships between the orbital frequencies of the planets or major moons are sometimes pointed out (see list below). The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French However, these have no dynamical significance because there is no appropriate precession of perihelion or other libration to make the resonance perfect (see the detailed discussion in the Mean-motion resonances in the Solar System section, above). In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from

Such near-resonances are dynamically insignificant even if the mismatch is quite small because (unlike a true resonance), after each cycle the relative position of the bodies shifts. When averaged over astronomically short timescales, their relative position is random, just like bodies which are nowhere near resonance.

For example, consider the orbits of Earth and Venus, which arrive at almost the same configuration after 8 Earth orbits and 13 Venus orbits. The actual ratio is 0. 61518624, which is only 0. 032% away from exactly 8:13. The mismatch after 8 years is only 1. 5° of Venus' orbital movement. Still, this is enough that Venus and Earth find themselves in the opposite relative orientation to the original every 120 such cycles, which is 960 years. Therefore, on timescales of thousands of years or more (still tiny by astronomical standards), their relative position is effectively random.

The presence of a near resonance may reflect that a perfect resonance existed in the past, or that the system is evolving towards one in the future.

Some orbital frequency coincidences that have been pointed out include:

The most remarkable (least probable) orbital correlation in the list is that between Callisto and Ganymede, followed in second place by that between Hydra and Charon.

The two near resonances listed for Earth and Venus are reflected in the timing of transits of Venus, which occur in pairs 8 years apart, in a cycle that repeats every 243 years. A transit of Venus across the Sun takes place when the Planet Venus passes directly between the Sun and Earth, obscuring a small portion of the

The near 1:12 resonance between Jupiter and Earth causes the Alinda asteroids, which occupy (or are close to) the 3:1 resonance with Jupiter, to be close to a 1:4 resonance with Earth. The Alinda asteroids are a group of Asteroids with a Semi-major axis of about 2

Possible past mean motion resonances

While Dione and Tethys are not close to an exact resonance now, they may have been in a 2:3 resonance early in the Solar System's history. This would have led to orbital eccentricity and tidal heating that may have warmed Tethys' interior enough to form a subsurface ocean. Tidal acceleration is an effect of the Tidal forces between an orbiting Natural satellite ( i Subsequent freezing of the ocean after the moons escaped from the resonance may have generated the extensional stresses that created the enormous graben system of Ithaca Chasma on Tethys. A graben is a depressed block of land bordered by parallel faults Graben is German for ditch. Ithaca Chasma is a valley on Saturn 's moon Tethys. It is 100 km wide 3 to 5 km deep and 2000 km long running approximately three-fourths of the way around Tethys' ^[6]

The satellite system of Uranus is notably different from those of Jupiter and Saturn in that it lacks precise resonances among the larger moons, while the majority of the larger moons of Jupiter (3 of the 4 largest) and of Saturn (6 of the 8 largest) are in mean motion resonances. In all three satellite systems, moons were likely captured into mean motion resonances in the past as their orbits shifted due to tidal dissipation (a process by which satellites gain orbital energy at the expense of the primary's rotational energy, affecting inner moons disproportionately). Tidal acceleration is an effect of the Tidal forces between an orbiting Natural satellite ( i In the Uranus System, however, due to the planet's lesser degree of oblateness, and the larger relative size of its satellites, escape from a mean motion resonance is much easier. An oblate Spheroid is a rotationally symmetric Ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane Lower oblateness of the primary alters its gravitational field in such a way that different possible resonances are spaced more closely together. A larger relative satellite size increases the strength of their interactions. Both factors lead to more chaotic orbital behavior at or near mean motion resonances. Escape from a resonance may be associated with capture into a secondary resonance, and/or tidal evolution-driven increases in orbital eccentricity or inclination. In Astrodynamics, under standard assumptions, any Orbit must be of Conic section shape Inclination in general is the Angle between a Reference plane and another plane or axis of direction

Mean motion resonances that probably once existed in the Uranus System include (3:5) Ariel-Miranda, (1:3) Umbriel-Miranda, (3:5) Umbriel-Ariel, and (1:4) Titania-Ariel. ^[7][8] Evidence for such past resonances includes the relatively high eccentricities of the orbits of Uranus' inner satellites, and the anomalously high orbital inclination of Miranda. High past orbital eccentricities associated with the (1:3) Umbriel-Miranda and (1:4) Titania-Ariel resonances may have led to tidal heating of the interiors of Miranda and Ariel,^[9] respectively. Tidal acceleration is an effect of the Tidal forces between an orbiting Natural satellite ( i Miranda probably escaped from its resonance with Umbriel via a secondary resonance, and the mechanism of this escape is believed to explain why its orbital inclination is more than 10 times those of the other nonirregular Uranian moons (see Uranus' natural satellites). Uranus has twenty-seven named moons. Five of them are massive enough to have achieved Hydrostatic equilibrium and so would be considered Dwarf planets ^[10][11]

In the case of Pluto's satellites, it has been proposed that the present near resonances are relics of a previous precise resonance that was disrupted by tidal damping of the eccentricity of Charon's orbit (see Pluto's natural satellites for details). Pluto has three known moons. The largest Charon, is proportionally larger compared to its primary than any other satellite of a known planet or dwarf planet in The near resonances may be maintained by a 15% local fluctuation in the Pluto-Charon gravitational field. Thus, these near resonances may not be coincidental.

See also

• Resonant trans-Neptunian object
• Commensurability (astronomy)
• Secular resonance
• Kozai resonance
• Dermott's Law
• Lagrangian points
• Mercury, which has a 3:2 spin-orbit resonance. In Astronomy, a resonant trans-Neptunian object is a Trans-Neptunian object (TNO in mean motion Orbital resonance with Neptune In Astronomy, two orbiting objects such as Planets satellites or Asteroids are said to be in exhibit commensurability if their orbital A secular resonance is a type of Orbital resonance. Secular resonances occur when the precession of two orbits is synchronised (a precession of the Perihelion In Celestial mechanics, the Kozai mechanism is a secular perturbative effect on certain orbits Dermott's Law is an empirical formula for the Sidereal period of major satellites orbiting planets in the Solar system.
• Tidal locking
• Tidal resonance
• Titius-Bode law
• Kirkwood gap
• Musica universalis ("music of the spheres")

References and Notes

• C. D. Murray, S. F. Dermott (1999). Solar System Dynamics, Cambridge University Press, ISBN 0-521-57597-4
• Renu Malhotra Orbital Resonances and Chaos in the Solar System. In Solar System Formation and Evolution, ASP Conference Series, 149 (1998) preprint
• Renu Malhotra, The Origin of Pluto's Orbit: Implications for the Solar System Beyond Neptune, The Astronomical Journal, 110 (1995), p. 420 Preprint.

External links

• Locations of Solar System Planetary Mean-Motion Resonances. Web calculator that plots distributions of the semimajor axes (or in one case the perihelion distances) of the minor planets in relation to mean motion resonances of the planets (website maintained by M. A. Murison).