Galilean moon Laplace resonance animation 2

celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...

, orbital resonance occurs when

orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...

ing bodies exert regular, periodic gravitational influence on each other, usually because their

orbital period The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets ...

s are related by a ratio of small

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s. 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 Swing or swinging may refer to: Apparatus * Swing (seat), a hanging seat that swings back and forth * Pendulum, an object that swings * Russian swing, a swing-like circus apparatus * Sex swing, a type of harness for sexual intercourse * Swing rid ...

, 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 In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...

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 1:2:4 resonance of

Jupiter Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousandt ...

's moons Ganymede, Europa and Io, and the 2:3 resonance between

Pluto Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the Sun. It is the largest k ...

and

Neptune Neptune is the eighth planet from the Sun and the farthest known planet in the Solar System. It is the fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 time ...

. Unstable resonances with

Saturn Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with an average radius of about nine and a half times that of Earth. It has only one-eighth the average density of Earth; ...

'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

solar system The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...

bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of

clearing the neighbourhood "Clearing the neighbourhood" (or dynamical dominance) around a celestial body's orbit describes the body becoming gravitationally dominant such that there are no other bodies of comparable size other than its natural satellites or those otherwise ...

, an effect that is used in the current

definition of a planet The definition of ''planet'', since the word was coined by the ancient Greeks, has included within its scope a wide range of celestial bodies. Greek astronomy, Greek astronomers employed the term (), 'wandering stars', for star-like objects wh ...

. 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 Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...

. 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 The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...

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 system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

s 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 tongue In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynam ...

Types of resonance

PIA10452 - Saturn A ring spiral density waves

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 Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a Plane of reference, reference plane and the orbital plane or Axis of rotation, axis of direction of the orbiting object ...

). *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. A ''mean-motion orbital resonance'' occurs when two bodies have periods of

revolution In political science, a revolution (Latin: ''revolutio'', "a turn around") is a fundamental and relatively sudden change in political power and political organization which occurs when the population revolts against the government, typically due ...

that are a simple integer ratio of each other. Depending on the details, this can either stabilize or destabilize the orbit. ''Stabilization'' may occur when the two bodies move in such a synchronised fashion that they never closely approach. For instance: *The orbits of

and the plutinos are stable, despite crossing that of the much larger

, 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 Perturbation or perturb may refer to: * Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly * Perturbation (geology), changes in the nature of alluvial deposits over time * Perturbatio ...

due to Neptune. There are also smaller but significant groups of

resonant trans-Neptunian object In astronomy, a resonant trans-Neptunian object is a trans-Neptunian object (TNO) in mean-motion orbital resonance with Neptune. The orbital periods of the resonant objects are in a simple integer relations with the period of Neptune, e.g. 1:2, 2 ...

s 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

are populated by ''clumps'' of asteroids (the Hilda family, the few Thule asteroids, and the numerous Trojan asteroids, respectively). Orbital resonances 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 mean-motion resonances with

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).

Asteroid An asteroid is a minor planet of the Solar System#Inner solar system, inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic o ...

s 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 The Alinda asteroids are a dynamical group of asteroids with a semi-major axis of about 2.5 AU and an orbital eccentricity approximately between 0.4 and 0.65.rings of Saturn, the

Cassini Division The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirel ...

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 Mimas may refer to: *Mimas (Giant), son of Gaia in Greek mythology, one of the Gigantes * Mimas (''Aeneid''), a son of Amycus and Theono, born the same night as Paris, who escorted Aeneas to Italy *Karaburun, a town and district in Turkey, formerl ...

. (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 In ancient Roman religion and myth, Janus ( ; la, Ianvs ) is the god of beginnings, gates, transitions, time, duality, doorways, passages, frames, and endings. He is usually depicted as having two faces. The month of January is named for Janu ...

. Most bodies that are in resonance orbit in the same direction; however, the retrograde asteroid

514107 Kaʻepaokaʻawela 514107 Kaʻepaokaʻāwela (), provisionally designated and nicknamed Bee-Zed, is a small asteroid, approximately in diameter, in a resonant, co-orbital motion with Jupiter. It is an unusual minor planet in that its orbit is retrograde, which ...

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 mean-motion resonance with

. Such orbital interactions are weaker than the corresponding interactions between bodies orbiting in the same direction. A ''Laplace resonance'' is a three-body resonance with a 1:2:4 orbital period ratio (equivalent to a 4:2:1 ratio of orbits). The term arose because

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 planet An exoplanet or extrasolar planet is a planet outside the Solar System. The first possible evidence of an exoplanet was noted in 1917 but was not recognized as such. The first confirmation of detection occurred in 1992. A different planet, init ...

s Gliese 876 c, b, and e. Three-body resonances involving other simple integer ratios have been termed "Laplace-like" or "Laplace-type". A ''

Lindblad resonance A Lindblad resonance, named for the Swedish galactic astronomer Bertil Lindblad, is an orbital resonance in which an object's epicyclic frequency (the rate at which one periapse follows another) is a simple multiple of some forcing frequency. Res ...

'' drives spiral density waves both in galaxies (where stars are subject to

forcing Forcing may refer to: Mathematics and science * Forcing (mathematics), a technique for obtaining independence proofs for set theory *Forcing (recursion theory), a modification of Paul Cohen's original set theoretic technique of forcing to deal with ...

by the spiral arms themselves) and in Saturn's rings (where ring particles are subject to forcing by Saturn's moons). A '' secular resonance'' occurs when the

precession Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In o ...

of two orbits is synchronised (usually a precession of the

perihelion An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any e ...

or ascending node). A small body in secular resonance with a much larger one (e.g. a

planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a ...

) 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

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 In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orb ...

(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

asteroid An asteroid is a minor planet of the Solar System#Inner solar system, inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic o ...

s and

(''ν₆'' = ''g'' − ''g₆'') helps shape the asteroid belt (the subscript "6" identifies Saturn as the sixth planet from the Sun). 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 by a close pass to

Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin atmos ...

. 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'') has the potential to greatly increase Mercury's eccentricity and possibly 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 Titan most often refers to: * Titan (moon), the largest moon of Saturn * Titans, a race of deities in Greek mythology Titan or Titans may also refer to: Arts and entertainment Fictional entities Fictional locations * Titan in fiction, fictiona ...

. A '' Kozai resonance'' occurs when the inclination and eccentricity of a perturbed orbit oscillate synchronously (increasing eccentricity while decreasing inclination and vice versa). 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 an example of another type of resonance involving orbital eccentricity, the eccentricities of Ganymede and Callisto vary with a common period of 181 years, although with opposite phases.

Mean-motion resonances in the Solar System

There are only a few known mean-motion resonances (MMR) in the

involving planets,

dwarf planet A dwarf planet is a small planetary-mass object that is in direct orbit of the Sun, smaller than any of the eight classical planets but still a world in its own right. The prototypical dwarf planet is Pluto. The interest of dwarf planets to ...

s or larger

satellites A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisotop ...

(a much greater number involve

s, planetary rings,

moonlets A moonlet, minor moon, minor natural satellite, or minor satellite is a particularly small natural satellite orbiting a planet, dwarf planet, or other minor planet. Up until 1995, moonlets were only hypothetical components of Saturn's F-ring ...

and smaller

Kuiper belt The Kuiper belt () is a circumstellar disc in the outer Solar System, extending from the orbit of Neptune at 30 astronomical units (AU) to approximately 50 AU from the Sun. It is similar to the asteroid belt, but is far larger—20 t ...

objects, including many possible dwarf planets). * 2:3

–

(also and other plutinos) * 2:4 Tethys–

(Saturn's moons). Not simplified, because the libration of the nodes must be taken into account. * 1:2

Dione Dione may refer to: Astronomy *106 Dione, a large main belt asteroid *Dione (moon), a moon of Saturn *Helene (moon), a moon of Saturn sometimes referred to as "Dione B" Mythology *Dione (Titaness), a Titaness in Greek mythology *Dione (mythology) ...

–

Enceladus Enceladus is the sixth-largest moon of Saturn (19th largest in the Solar System). It is about in diameter, about a tenth of that of Saturn's largest moon, Titan. Enceladus is mostly covered by fresh, clean ice, making it one of the most refle ...

(Saturn's moons) * 3:4

Hyperion Hyperion may refer to: Greek mythology * Hyperion (Titan), one of the twelve Titans * ''Hyperion'', a byname of the Sun, Helios * Hyperion of Troy or Yperion, son of King Priam Science * Hyperion (moon), a moon of the planet Saturn * ''Hyp ...

–

(Saturn's moons) * 1:2:4 Ganymede– Europa– Io (Jupiter's moons, ratio of ''orbits''). Additionally, Haumea is thought to be in a 7:12 resonance with Neptune, and is thought to be in a 3:10 resonance with Neptune. The simple integer ratios between periods hide more complex relations: *the point of conjunction can oscillate ( librate) around an equilibrium point defined by the resonance. *given non-zero eccentricities, the nodes or periapsides can drift (a resonance related, short period, not secular precession). 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

n\,\!

(inverse of periods, often expressed in degrees per day) would satisfy the following :

n_ - 2\cdot n_=0

Substituting the data (from Wikipedia) one will get −0.7395° day⁻¹, a value substantially different from zero. Actually, the resonance perfect, but it involves also the precession of

perijove An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any elli ...

(the point closest to Jupiter),

\dot\omega

. The correct equation (part of the Laplace equations) is: :

n_ - 2\cdot n_ + \dot\omega_=0

In other words, the mean motion of Io is indeed double of that of Europa taking into account the precession of the perijove. 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 :

4\cdot n_ - 2\cdot n_ - \dot\Omega_- \dot\Omega_=0

The point of conjunctions librates around the midpoint between the nodes of the two moons.

Laplace resonance

The Laplace resonance involving Io–Europa–Ganymede includes the following relation locking the ''orbital phase'' of the moons: :

\Phi_L=\lambda_ - 3\cdot\lambda_ + 2\cdot\lambda_=180^\circ

where

\lambda

are mean longitudes of the moons (the second equals sign ignores libration). This relation makes a triple conjunction impossible. (A Laplace resonance in the Gliese 876 system, in contrast, is associated with one triple conjunction per orbit of the outermost planet, ignoring libration.) The graph illustrates the positions of the moons after 1, 2 and 3 Io periods.

\Phi_L

librates about 180° with an amplitude of 0.03°. Another "Laplace-like" resonance involves the moons Styx, Nix and

Hydra Hydra generally refers to: * Lernaean Hydra, a many-headed serpent in Greek mythology * ''Hydra'' (genus), a genus of simple freshwater animals belonging to the phylum Cnidaria Hydra or The Hydra may also refer to: Astronomy * Hydra (constel ...

of Pluto: :

\Phi=3\cdot\lambda_ - 5\cdot\lambda_ + 2\cdot\lambda_=180^\circ

This reflects orbital periods for Styx, Nix and Hydra, respectively, that are close to a ratio of 18:22:33 (or, in terms of the near resonances with Charon's period, 3+3/11:4:6; see

below Below may refer to: *Earth * Ground (disambiguation) * Soil * Floor * Bottom (disambiguation) * Less than *Temperatures below freezing * Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fr ...

); the respective ratio of orbits is 11:9:6. Based on the ratios of synodic periods, there are 5 conjunctions of Styx and Hydra and 3 conjunctions of Nix and Hydra for every 2 conjunctions of Styx and Nix. As with the Galilean satellite resonance, triple conjunctions are forbidden.

\Phi

librates about 180° with an amplitude of at least 10°.

Plutino resonances

The dwarf planet

is following an orbit trapped in a web of resonances with

. The resonances include: *A mean-motion resonance of 2:3 *The resonance of the

(

libration In lunar astronomy, libration is the wagging or wavering of the Moon perceived by Earth-bound observers and caused by changes in their perspective. It permits an observer to see slightly different hemispheres of the surface at different tim ...

around 90°), keeping the perihelion above the

ecliptic The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic agai ...

*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. The minimum separation between the two bodies overall is 17 AU, while the minimum separation between Pluto and

Uranus Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus (Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars), grandfather of Zeus (Jupiter) and father of Cronu ...

is just 11 AU (see Pluto's orbit for detailed explanation and graphs). The next largest body in a similar 2:3 resonance with Neptune, called a '' plutino'', is the probable dwarf planet Orcus. Orcus has an orbit similar in inclination and eccentricity to Pluto's. However, the two are constrained by their mutual resonance with Neptune to always be in opposite phases of their orbits; Orcus is thus sometimes described as the "anti-Pluto". Naiad-Thalassa 73-69 orbital resonance

Naiad:Thalassa 73:69 resonance

Neptune's innermost moon,

Naiad In Greek mythology, the naiads (; grc-gre, ναϊάδες, naïádes) are a type of female spirit, or nymph, presiding over fountains, wells, springs, streams, brooks and other bodies of fresh water. They are distinct from river gods, who ...

, is in a 73:69 fourth-order resonance with the next outward moon,

Thalassa Thalassa (; grc-gre, Θάλασσα, Thálassa, sea; Attic Greek: , ''Thálatta'') was the general word for 'sea' and for its divine female personification in Greek mythology. The word may have been of Pre-Greek origin. Mythology According to ...

. As it orbits Neptune, the more inclined Naiad successively passes Thalassa twice from above and then twice from below, in a cycle that repeats every ~21.5 Earth days. The two moons are about 3540 km apart when they pass each other. Although their orbital radii differ by only 1850 km, Naiad swings ~2800 km above or below Thalassa's orbital plane at closest approach. As is common, this resonance stabilizes the orbits by maximizing separation at conjunction, but it is unusual for the role played by orbital inclination in facilitating this avoidance in a case where eccentricities are minimal.

Mean-motion resonances among extrasolar planets

While most

ary systems discovered have not been found to have planets in mean-motion resonances, chains of up to five resonant planets and up to seven at least near resonant planets have been uncovered. Simulations have shown that during planetary system formation, the appearance of resonant chains of planetary embryos is favored by the presence of the primordial gas disc. Once that gas dissipates, 90–95% of those chains must then become unstable to match the low frequency of resonant chains observed. *As mentioned above, Gliese 876 e, b and c are in a Laplace resonance, with a 4:2:1 ratio of periods (124.3, 61.1 and 30.0 days). In this case,

\Phi_L

librates with an amplitude of 40° ± 13° and the resonance follows the time-averaged relation: :

\Phi_L=\lambda_ - 3\cdot\lambda_ + 2\cdot\lambda_=0^\circ

Kepler-223 Kepler-223 (KOI-730, KIC 10227020) is a G5V star with an extrasolar planetary system discovered by the Kepler mission. Studies indicate that the Kepler-223 star system consists of 4 planets orbiting the star. Planetary system ...

has four planets in a resonance with an 8:6:4:3 orbit ratio, and a 3:4:6:8 ratio of periods (7.3845, 9.8456, 14.7887 and 19.7257 days). This represents the first confirmed 4-body orbital resonance. The librations within this system are such that close encounters between two planets occur only when the other planets are in distant parts of their orbits. Simulations indicate that this system of resonances must have formed via planetary migration. *

Kepler-80 Kepler-80, also known as KOI-500, is a red dwarf star of the spectral type M0V. This stellar classification places Kepler-80 among the very common, cool, class M stars that are still within their main evolutionary stage, known as the main ...

d, e, b, c and g have periods in a ~ 1.000: 1.512: 2.296: 3.100: 4.767 ratio (3.0722, 4.6449, 7.0525, 9.5236 and 14.6456 days). However, in a frame of reference that rotates with the conjunctions, this reduces to a period ratio of 4:6:9:12:18 (an orbit ratio of 9:6:4:3:2). Conjunctions of d and e, e and b, b and c, and c and g occur at relative intervals of 2:3:6:6 (9.07, 13.61 and 27.21 days) in a pattern that repeats about every 190.5 days (seven full cycles in the rotating frame) in the inertial or nonrotating frame (equivalent to a 62:41:27:20:13 orbit ratio resonance in the nonrotating frame, because the conjunctions circulate in the direction opposite orbital motion). Librations of possible three-body resonances have amplitudes of only about 3 degrees, and modeling indicates the resonant system is stable to perturbations. Triple conjunctions do not occur. *

TOI-178 TOI-178 is a planetary system in the constellation Sculptor,Requête spécifique à TOI-178sur VizieR. which appears to have at least five, and possibly six, planets in a chain of Laplace resonances, which constitute one of the longest c ...

has 6 confirmed planets, of which the outer 5 planets form a similar resonant chain in a rotating frame of reference, which can be expressed as 2:4:6:9:12 in period ratios, or as 18:9:6:4:3 in orbit ratios. In addition, the innermost planet b with period of 1.91d orbits close to where it would also be part of the same Laplace resonance chain, as a 3:5 resonance with the planet c would be fulfilled at period of ~1.95d, implying that it might have evolved there but pulled out of resonance, possibly by tidal forces. * TRAPPIST-1's seven approximately Earth-sized planets are in a chain of near resonances (the longest such chain known), having an orbit ratio of approximately 24, 15, 9, 6, 4, 3 and 2, or nearest-neighbor period ratios (proceeding outward) of about 8/5, 5/3, 3/2, 3/2, 4/3 and 3/2 (1.603, 1.672, 1.506, 1.509, 1.342 and 1.519). They are also configured such that each triple of adjacent planets is in a Laplace resonance (i.e., b, c and d in one such Laplace configuration; c, d and e in another, etc.). The resonant configuration is expected to be stable on a time scale of billions of years, assuming it arose during planetary migration. A musical interpretation of the resonance has been provided. *

Kepler-29 Kepler-29 is a Sun-like star in the northern constellation of Cygnus. It is located at the celestial coordinates: Right Ascension , Declination . With an apparent visual magnitude of 15.456, this star is too faint to be seen with the nak ...

has a pair of planets in a 7:9 resonance (ratio of 1/1.28587). * Kepler-36 has a pair of planets close to a 6:7 resonance. * Kepler-37 d, c and b are within one percent of a resonance with an 8:15:24 orbit ratio and a 15:8:5 ratio of periods (39.792187, 21.301886 and 13.367308 days). :*And *Of

Kepler-90 Kepler-90, also designated 2MASS J18574403+4918185, is an F-type star located about from Earth in the constellation of Draco (constellation), Draco. It is notable for possessing a planetary system that has the same number of observed pla ...

's eight known planets, the period ratios b:c, c:i and i:d are close to 4:5, 3:5 and 1:4, respectively (4:4.977, 3:4.97 and 1:4.13) and d, e, f, g and h are close to a 2:3:4:7:11 period ratio (2: 3.078: 4.182: 7.051: 11.102; also 7: 11.021). f, g and h are also close to a 3:5:8 period ratio (3: 5.058: 7.964). Relevant to systems like this and that of Kepler-36, calculations suggest that the presence of an outer gas giant planet facilitates the formation of closely packed resonances among inner super-Earths. *

HD 41248 HD 41248 is a star in the constellation Pictor. It is a star very similar to the Sun, with the same spectral type (G2V). It has 68% of the Sun's luminosity, 92% of its mass and 78% of its diameter. It is estimated to be around 2 billion yea ...

has a pair of super-Earths within 0.3% of a 5:7 resonance (ratio of 1/1.39718). * K2-138 has 5 confirmed planets in an unbroken near-3:2 resonance chain (with periods of 2.353, 3.560, 5.405, 8.261 and 12.758 days). The system was discovered in the

citizen science Citizen science (CS) (similar to community science, crowd science, crowd-sourced science, civic science, participatory monitoring, or volunteer monitoring) is scientific research conducted with participation from the public (who are sometimes re ...

project Exoplanet Explorers, using K2 data. K2-138 could host co-orbital bodies (in a 1:1 mean-motion resonance). Resonant chain systems can stabilize co-orbital bodies and a dedicated analysis of the K2 light curve and radial-velocity from

HARPS The High Accuracy Radial Velocity Planet Searcher (HARPS) is a high-precision echelle planet-finding spectrograph installed in 2002 on the ESO's 3.6m telescope at La Silla Observatory in Chile. The first light was achieved in February 2003 ...

might reveal them. Follow-up observations with the Spitzer Space Telescope suggest a sixth planet continuing the 3:2 resonance chain, while leaving two gaps in the chain (its period is 41.97 days). These gaps could be filled by smaller non-transiting planets. Future observations with CHEOPS will measure transit-timing variations of the system to further analyse the mass of the planets and could potentially find other planetary bodies in the system. *

K2-32 K2-32 is a G9-type main sequence star slightly smaller and less massive than the sun. Four confirmed transiting exoplanets are known to orbit this star. A study of atmospheric escape from the planet K2-32b caused by high-energy stellar i ...

has four planets in a near 1:2:5:7 resonance (with periods of 4.34, 8.99, 20.66 and 31.71 days). Planet e has a radius almost identical to that of the Earth. The other planets have a size between Neptune and Saturn. *

V1298 Tauri V1298 Tauri is a young (23±4 Myr) weakly-lined T Tauri star that is part of the Taurus-Auriga association in the Taurus Molecular Cloud. Alternatively it is part of a proposed moving group, called Group 29 (stellar association), Group 2 ...

has four confirmed planets of which planets c, d and b are near a 1:2:3 resonance (with periods of 8.25, 12.40 and 24.14 days). Planet e only shows a single transit in the K2 light curve and has a period larger than 36 days. Planet e might be in a low-order resonance (of 2:3, 3:5, 1:2, or 1:3) with planet b. The system is very young (23±4

Myr The abbreviation Myr, "million years", is a unit of a quantity of (i.e. ) years, or 31.556926 teraseconds. Usage Myr (million years) is in common use in fields such as Earth science and cosmology. Myr is also used with Mya (million years ago). ...

) and might be a precursor of a compact multiplanet system. The 2:3 resonance suggests that some close-in planets may either form in resonances or evolve into them on timescales of less than 10 Myr. The planets in the system have a size between Neptune and Saturn. Only planet b has a size similar to Jupiter. *

HD 158259 HD 158259 is a main sequence star located 88 light years away in the constellation Draco, discovered by the SOPHIE échelle spectrograph using the radial velocity method. Characteristics HD 158259 is a G0 star with a rotation period of d ...

contains four planets in a 3:2 near resonance chain (with periods of 3.432, 5.198, 7.954 and 12.03 days, or period ratios of 1.51, 1.53 and 1.51, respectively), with a possible fifth planet also near a 3:2 resonance (with a period of 17.4 days). The exoplanets were found with the SOPHIE échelle spectrograph, using the

radial velocity The radial velocity or line-of-sight velocity, also known as radial speed or range rate, of a target with respect to an observer is the rate of change of the distance or range between the two points. It is equivalent to the vector projection ...

method. *

Kepler-1649 Kepler-1649 is a red dwarf star of spectral type M5V with a radius , a mass , and a metallicity of -0.15 e/H Planetary system Two confirmed planets orbit the star: Kepler-1649b and Kepler-1649c. Kepler-1649b is similar to Venus, where ...

contains two Earth-size planets close to a 9:4 resonance (with periods of 19.53527 and 8.689099 days, or a period ratio of 2.24825), including one ( "c") in the habitable zone. An undetected planet with a 13.0-day period would create a 3:2 resonance chain. * Kepler-88 has a pair of inner planets close to a 1:2 resonance (period ratio of 2.0396), with a mass ratio of ~22.5, producing very large transit timing variations of ~0.5 days for the innermost planet. There is a yet more massive outer planet in a ~1400 day orbit. Cases of extrasolar planets close to a 1:2 mean-motion resonance are fairly common. Sixteen percent of systems found by the transit method are reported to have an example of this (with period ratios in the range 1.83–2.18), as well as one sixth of planetary systems characterized by

Doppler spectroscopy Doppler spectroscopy (also known as the radial-velocity method, or colloquially, the wobble method) is an indirect method for finding extrasolar planets and brown dwarfs from radial-velocity measurements via observation of Doppler shifts in t ...

(with in this case a narrower period ratio range). Due to incomplete knowledge of the systems, the actual proportions are likely to be higher. Overall, about a third of radial velocity characterized systems appear to have a pair of planets close to a commensurability. It is much more common for pairs of planets to have orbital period ratios a few percent larger than a mean-motion resonance ratio than a few percent smaller (particularly in the case of first order resonances, in which the integers in the ratio differ by one). This was predicted to be true in cases where tidal interactions with the star are significant.

Coincidental 'near' ratios of mean motion

A number of near-

-ratio relationships between the orbital frequencies of the planets or major moons are sometimes pointed out (see list below). However, these have no dynamical significance because there is no appropriate precession of

or other libration to make the resonance perfect (see the detailed discussion in the section above). 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 that 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 include: The least probable orbital correlation in the list is that between Io and Metis, followed by those between Rosalind and Cordelia, Pallas and Ceres, Jupiter and Pallas, Callisto and Ganymede, and Hydra and Charon, respectively.

Possible past mean-motion resonances

A past resonance between Jupiter and Saturn may have played a dramatic role in early Solar System history. A 2004 computer model by Alessandro Morbidelli of the Observatoire de la Côte d'Azur in

Nice Nice ( , ; Niçard: , classical norm, or , nonstandard, ; it, Nizza ; lij, Nissa; grc, Νίκαια; la, Nicaea) is the prefecture of the Alpes-Maritimes department in France. The Nice agglomeration extends far beyond the administrative c ...

suggested that the formation of a 1:2 resonance between Jupiter and Saturn (due to interactions with planetesimals that caused them to migrate inward and outward, respectively) created a gravitational push that propelled both Uranus and Neptune into higher orbits, and in some scenarios caused them to switch places, which would have doubled Neptune's distance from the Sun. The resultant expulsion of objects from the proto-Kuiper belt as Neptune moved outwards could explain the Late Heavy Bombardment 600 million years after the Solar System's formation and the origin of Jupiter's Trojan asteroids. An outward migration of Neptune could also explain the current occupancy of some of its resonances (particularly the 2:5 resonance) within the Kuiper belt. While Saturn's mid-sized moons 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. Subsequent freezing of the ocean after the moons escaped from the resonance may have generated the extensional stresses that created the enormous

graben In geology, a graben () is a depressed block of the crust of a planet or moon, bordered by parallel normal faults. Etymology ''Graben'' is a loan word from German, meaning 'ditch' or 'trench'. The word was first used in the geologic contex ...

system of Ithaca Chasma on Tethys. 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). In the Uranian system, however, due to the planet's lesser degree of

oblateness Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is ...

, and the larger relative size of its satellites, escape from a mean-motion resonance is much easier. 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 In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values b ...

. 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. 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, respectively. 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

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

Uranian moons (see Uranus' natural satellites). Similar to the case of Miranda, the present inclinations of Jupiter's moonlets Amalthea and

Thebe Thebe may refer to: * Any of several female characters in Greek mythology - see List of mythological figures named Thebe * Thebe (moon), a moon of Jupiter * Thebe (currency), 1/100 of a Botswana pula * Thebe, an Amazon * Thebe, alternate name ...

are thought to be indications of past passage through the 3:1 and 4:2 resonances with Io, respectively. Neptune's regular moons Proteus and Larissa are thought to have passed through a 1:2 resonance a few hundred million years ago; the moons have drifted away from each other since then because Proteus is outside a synchronous orbit and Larissa is within one. Passage through the resonance is thought to have excited both moons' eccentricities to a degree that has not since been entirely damped out. In the case of

'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 The dwarf planet Pluto has five natural satellites. In order of distance from Pluto, they are Charon, Styx, Nix, Kerberos, and Hydra. Charon, the largest, is mutually tidally locked with Pluto, and is massive enough that Pluto–Charon is somet ...

for details). 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. The smaller inner moon of the

Haumea, Namaka, is one tenth the mass of the larger outer moon, Hiʻiaka. Namaka revolves around Haumea in 18 days in an eccentric, non-Keplerian orbit, and as of 2008 is inclined 13° from Hiʻiaka. Over the timescale of the system, it should have been tidally damped into a more circular orbit. It appears that it has been disturbed by resonances with the more massive Hiʻiaka, due to converging orbits as it moved outward from Haumea because of tidal dissipation. The moons may have been caught in and then escaped from orbital resonance several times. They probably passed through the 3:1 resonance relatively recently, and currently are in or at least close to an 8:3 resonance. Namaka's orbit is strongly perturbed, with a current precession of about −6.5° per year.

History

Types of resonance

Mean-motion resonances in the Solar System

Laplace resonance

Plutino resonances

Naiad:Thalassa 73:69 resonance

Mean-motion resonances among extrasolar planets

Coincidental 'near' ratios of mean motion

Possible past mean-motion resonances

See also

Notes

References

External links