Editing Orbital resonance

{{Short description|Regular and periodic mutual gravitational influence of orbiting bodies}}
{{for|the science fiction novel|Orbital Resonance (novel)}}
{{Use dmy dates|date=September 2020}}
[[File:Galilean moon Laplace resonance animation 2.gif|thumb|275px|The three-body Laplace resonance exhibited by three of Jupiter's [[Galilean moons]]. [[Conjunction (astronomy)|Conjunctions]] are highlighted by brief color changes. There are two Io-Europa conjunctions (green) and three Io-Ganymede conjunctions (grey) for each Europa-Ganymede conjunction (magenta). This diagram is not to scale.]]

In [[celestial mechanics]], '''orbital resonance''' occurs when [[orbit]]ing bodies exert regular, [[Periodic function#Real number examples|periodic]] [[gravitational]] influence on each other, usually because their [[orbital period]]s are related by a ratio of small [[integer]]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 (seat)|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 ''[[Orbital stability|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 1:2:4 resonance of [[Jupiter]]'s moons [[Ganymede (moon)|Ganymede]], [[Europa (moon)|Europa]] and [[Io (moon)|Io]], and the 2:3 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 planet|definition of a planet]].<ref>{{cite news|url=http://www.iau.org/static/resolutions/Resolution_GA26-5-6.pdf|title=IAU 2006 General Assembly: Resolutions 5 and 6|date=2006-08-24|publisher=IAU|access-date=2009-06-23}}</ref>

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 [[N-body problem|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 moon]]s (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 interaction]]s describes resonance in the general modern setting. A primary result from the study of [[dynamical system]]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]]s.

==Types of resonance==<!-- This section is linked from [[Pandora (moon)]] -->

[[File:TheKuiperBelt 75AU All.svg|thumb|300px|The [[semimajor axis|semimajor axes]] of [[resonant trans-Neptunian object]]s (red) are clumped at locations of low-integer resonances with [[Neptune]] (vertical red bars near top), in contrast to those of [[cubewano]]s (blue) and nonresonant (or not known to be resonant) [[scattered disk|scattered objects]] (grey).]]
[[File:Kirkwood Gaps.svg|300px|thumb|A chart of the distribution of [[asteroid]] semimajor axes, showing the [[Kirkwood gap]]s where orbits are destabilized by resonances with [[Jupiter]]]]
[[File:PIA10452 - Saturn A ring spiral density waves.jpg|300px|thumb|[[Spiral density wave]]s in [[Rings of Saturn#A Ring|Saturn's A Ring]] excited by resonances with [[Moons of Saturn#Ring shepherds|inner moons]]. Such waves propagate away from the planet (towards upper left). The large set of waves just below center is due to the 6:5 resonance with [[Janus (moon)|Janus]].]]
[[File:PIA17173 Titan resonances in Saturn's C ring.jpg|200px|thumb|The eccentric [[Rings of Saturn#Colombo Gap and Titan Ringlet|Titan Ringlet]]<ref name="Porco1984" /> in the Columbo Gap of Saturn's [[Rings of Saturn#C Ring|C Ring]] (center) and the inclined orbits of resonant particles in the bending wave<ref name="Rosen1988">{{Cite journal |last1=Rosen |first1=P. A. |last2=Lissauer |first2=J. J. |author-link2=Jack J. Lissauer |year=1988 |title=The Titan −1:0 Nodal Bending Wave in Saturn's Ring C |journal=[[Science (journal)|Science]] |volume=241 |issue=4866 |pages=690–694 |bibcode=1988Sci...241..690R |doi=10.1126/science.241.4866.690 |pmid=17839081|s2cid=32938282 }}</ref><ref name="Chakrabarti2001">{{Cite journal |last1=Chakrabarti |first1=S. K. |last2=Bhattacharyya |first2=A. |year=2001 |title=Constraints on the C ring parameters of Saturn at the Titan -1:0 resonance |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=326 |issue=2 |pages=L23 |bibcode=2001MNRAS.326L..23C |doi=10.1046/j.1365-8711.2001.04813.x|doi-access=free }}</ref> just inside it have [[Apsidal precession|apsidal]] and [[Nodal precession|nodal]] precessions, respectively, commensurate with [[Titan (moon)|Titan]]'s mean motion.]]

In general, an orbital resonance may
*involve one or any combination of the orbit parameters (e.g. [[Orbital eccentricity|eccentricity]] versus [[semimajor axis]], or eccentricity versus [[orbital inclination|inclination]]).
*act on any time scale from short term, commensurable with the orbit periods, to [[Secular phenomena|secular]], measured in 10<sup>4</sup> to 10<sup>6</sup> 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 period]]s or [[mean motion]]s (orbital frequencies) that are simple integer ratios of each other.

==== Two-body mean motion resonance ====
The simplest cases of MMRs involve only two bodies. It does not depend only on the existence of such a ratio, and more precisely the ratio of periods is not exactly a rational number, even averaged over a long period. For example, in the case of [[Pluto]] and [[Neptune]] (see below), the true equation says that the average rate of change of <math>3\alpha_P-2\alpha_N-\varpi_P</math> is exactly zero, where <math>\alpha_P</math> is the longitude of Pluto, <math>\alpha_N</math> is the longitude of Neptune, and <math>\varpi_P</math> is the longitude of Pluto's [[perihelion]]. Since the rate of motion of the latter is about {{value|0.97e-4}} degrees per year, the ratio of periods is actually 1.503 in the long term.<ref name="williams71">{{cite journal
| title = Resonances in the Neptune-Pluto System
| first1 = James G.
| last1 = Williams
| first2 = G. S.
| last2 = Benson
| journal = Astronomical Journal
| volume = 76
| page = 167
| date = 1971
| bibcode = 1971AJ.....76..167W
| doi = 10.1086/111100
| s2cid = 120122522
| doi-access = free
}}</ref>

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 [[plutino]]s 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 [[perturbation (astronomy)|perturbations]] due to Neptune. There are also smaller but significant groups of [[resonant trans-Neptunian object]]s occupying the 1:1 ([[Neptune trojan]]s), [[resonant Kuiper belt object#3:5 resonance (period ~275 years)|3:5]], [[resonant Kuiper belt object#4:7 resonance (period ~290 years)|4:7]], 1:2 ([[resonant Kuiper belt object#1:2 resonance ("twotinos", period ~330 years)|twotinos]]) and [[resonant Kuiper belt object#2:5 resonance (period ~410 years)|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 asteroid]]s, and the numerous [[Jupiter trojan|Trojan asteroids]], respectively).

MMRs can also ''destabilize'' one of the orbits. This process can be exploited to find energy-efficient ways of [[deorbit]]ing spacecraft.<ref name="Witze2018">{{cite journal |last1=Witze |first1=A. |title=The quest to conquer Earth's space junk problem |journal=Nature |volume=561 |issue=7721 |date=5 September 2018 |pages=24–26 |doi=10.1038/d41586-018-06170-1|pmid=30185967 |bibcode=2018Natur.561...24W |doi-access=free }}</ref><ref name="Daquin2016">{{cite journal |last1=Daquin |first1=J. |last2=Rosengren |first2=A. J. |last3=Alessi |first3=E. M. |last4=Deleflie |first4=F. |last5=Valsecchi |first5=G. B. |last6=Rossi |first6=A. |title=The dynamical structure of the MEO region: long-term stability, chaos, and transport |journal=Celestial Mechanics and Dynamical Astronomy |volume=124 |issue=4 |year=2016 |pages=335–366 |doi=10.1007/s10569-015-9665-9|arxiv=1507.06170 |bibcode=2016CeMDA.124..335D |s2cid=119183742 }}</ref> 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 gap]]s (most notably at the 4:1, 3:1, 5:2, 7:3 and 2:1 resonances). [[Asteroid]]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]] 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 [[Rings of Saturn#Cassini Division|Cassini Division]] is a gap between the inner [[Rings of Saturn#B Ring|B Ring]] and the outer [[Rings of Saturn#A Ring|A Ring]] that has been cleared by a 2:1 resonance with the moon [[Mimas (moon)|Mimas]]. (More specifically, the site of the resonance is the [[Rings of Saturn#Huygens Gap|Huygens Gap]], which bounds the outer edge of the [[Rings of Saturn#B Ring|B Ring]].)
*In the rings of Saturn, the [[Rings of Saturn#Encke Gap|Encke]] and [[Rings of Saturn#Keeler Gap|Keeler]] gaps within the A Ring are cleared by 1:1 resonances with the embedded moonlets [[Pan (moon)|Pan]] and [[Daphnis (moon)|Daphnis]], respectively. The A Ring's outer edge is maintained by a destabilizing 7:6 resonance with the moon [[Janus (moon)|Janus]].

Most bodies that are in two-body MMRs orbit in the same direction; however, the [[Retrograde motion|retrograde]] asteroid [[514107 Kaʻepaokaʻawela]] appears to be in a stable (for a period of at least a million years) 1:−1 resonance with Jupiter.<ref name="Wieger2017">{{cite journal |last1=Wiegert |first1=P. |last2=Connors |first2=M. |last3=Veillet |first3=C. |title=A retrograde co-orbital asteroid of Jupiter |journal=Nature |volume=543 |issue=7647 |date=30 March 2017 |pages=687–689 |doi=10.1038/nature22029 |pmid=28358083 |bibcode=2017Natur.543..687W |s2cid=205255113 }}</ref> In addition, a few retrograde [[Damocloid asteroid|damocloids]] have been found that are temporarily captured in MMR with [[Jupiter]] or [[Saturn]].<ref name="Morais_2013">{{cite journal |last1=Morais |first1=M. H. M. |last2=Namouni |first2=F. |date=21 September 2013 |title=Asteroids in retrograde resonance with Jupiter and Saturn |journal=[[Monthly Notices of the Royal Astronomical Society Letters]] |arxiv=1308.0216 |bibcode=2013MNRAS.436L..30M |doi=10.1093/mnrasl/slt106 |volume=436 |issue=1 |pages=L30–L34 |doi-access=free |s2cid=119263066 }}</ref> Such orbital interactions are weaker than the corresponding interactions between bodies orbiting in the same direction.<ref name="Morais_2013" /><ref name="Morais2013cmda">{{Cite journal |first1=Maria Helena Moreira |last1=Morais |first2=Fathi |last2=Namouni |date=12 October 2013 |title=Retrograde resonance in the planar three-body problem |journal=Celestial Mechanics and Dynamical Astronomy |volume=117 |issue=4 |pages=405–421 |bibcode=2013CeMDA.117..405M |doi=10.1007/s10569-013-9519-2 |arxiv=1305.0016 |s2cid=254379849 |issn=1572-9478}}</ref>
The [[trans-Neptunian object]] [[471325 Taowu]] has an orbital inclination of 110[[Degree (angle)|°]] with respect to the planets' [[orbital plane]] and is currently in a 7:9 polar resonance with Neptune.<ref name="Morais_Namouni_2017">{{cite journal |last1=Morais |first1=M. H. M. |last2=Nomouni |first2=F. |title=First transneptunian object in polar resonance with Neptune |date=2017 |arxiv=1708.00346 |doi=10.1093/mnrasl/slx125 |journal=Monthly Notices of the Royal Astronomical Society |volume=472 |issue=1 |pages=L1–L4 |doi-access=free |department=Letters |bibcode=2017MNRAS.472L...1M }}</ref>

==== N-body mean motion resonance ====
MMRs involving more than two bodies have been observed in the Solar System. For example, there are [[three-body problem|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 [[Chaos theory|chaotic]] orbital evolutions.<ref name="Nesvorny1998"/> 

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 (moon)|Io]], [[Europa (moon)|Europa]], and [[Ganymede (moon)|Ganymede]]. It is now also often applied to other 3-body resonances with the same ratios,<ref name="Gargaud2011">{{cite book |last1=Barnes |first1=R. |year=2011 |chapter=Laplace Resonance |editor-last=Gargaud |editor-first=M. |title=Encyclopedia of Astrobiology |chapter-url=https://books.google.com/books?id=oEq1y9GIcr0C&pg=PA905 |pages=905–906 |publisher=[[Springer Science+Business Media]] |isbn=978-3-642-11271-3 |doi=10.1007/978-3-642-11274-4_864}}</ref> such as that between the [[extrasolar planet]]s [[Gliese 876]] c, b, and e.<ref name="rivera2010" /><ref>{{cite journal |last1=Nelson |first1=B. E. |last2=Robertson |first2=P. M. |last3=Payne |first3=M. J. |last4=Pritchard |first4=S. M. |last5=Deck |first5=K. M. |last6=Ford |first6=E. B. |last7=Wright |first7=J. T. |last8=Isaacson |first8=H. T. |date=2015 |title=An empirically derived three-dimensional Laplace resonance in the Gliese 876 planetary system |journal=Monthly Notices of the Royal Astronomical Society |volume=455 |issue=3 |pages=2484–2499 |doi=10.1093/mnras/stv2367 |doi-access=free |arxiv=1504.07995 }}</ref><ref name="MartiGiuppone2013">{{cite journal |last1=Marti |first1=J. G. |last2=Giuppone |first2=C. A. |last3=Beauge |first3=C. |year=2013 |title=Dynamical analysis of the Gliese-876 Laplace resonance |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=433 |issue=2 |pages=928–934 |arxiv=1305.6768 |bibcode=2013MNRAS.433..928M |doi=10.1093/mnras/stt765|doi-access=free |s2cid=118643833 }}</ref> Three-body resonances involving other simple integer ratios have been termed "Laplace-like"<ref name="ShowalterHamilton2015" /> or "Laplace-type".<ref name="MurrayDermott1999">{{cite book |last1=Murray |first1=C. D. |last2=Dermott |first2=S. F. |year=1999 |title=Solar System Dynamics |url=https://books.google.com/books?id=aU6vcy5L8GAC&pg=PA17 |page=17 |publisher=[[Cambridge University Press]] |isbn=978-0-521-57597-3}}</ref>

=== Lindblad resonance ===
A ''[[Lindblad resonance]]'' drives [[Density wave theory|spiral density waves]] both in [[galaxies]] (where stars are subject to [[Harmonic oscillator|forcing]] by the spiral arms themselves) and in [[Rings of Saturn|Saturn's rings]] (where ring particles are subject to forcing by [[Moons of Saturn|Saturn's moons]]).

=== Secular resonance ===
A ''[[secular resonance]]'' occurs when the [[precession#Astronomy|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 (orbit)|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°).<ref>{{cite web |last=Beatty |first=J. K. |title=Why Is Saturn Tipsy? |url=http://www.skyandtelescope.com/news/3306806.html?page=1&c=y |work=[[Sky & Telescope]] |date=23 July 2003 |access-date=25 February 2009 |archive-url=https://web.archive.org/web/20090903170550/http://www.skyandtelescope.com/news/3306806.html?page=1&c=y |archive-date=3 September 2009 |url-status=dead }}</ref><ref>{{cite journal |last1=Ward |first1=W. R. |last2=Hamilton |first2=D. P. |year=2004 |title=Tilting Saturn. I. Analytic Model |journal=[[The Astronomical Journal]] |volume=128 |issue=5 |pages=2501–2509 |bibcode=2004AJ....128.2501W |doi=10.1086/424533|doi-access=free }}</ref><ref>{{cite journal |last1=Hamilton |first1=D. P. |last2=Ward |first2=W. R. |year=2004 |title=Tilting Saturn. II. Numerical Model |journal=[[The Astronomical Journal]] |volume=128 |issue=5 |pages=2510–2517 |bibcode=2004AJ....128.2510H |doi=10.1086/424534|s2cid=33083447 }}</ref> 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 10<sup>4</sup> 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.<ref>{{cite journal |last1=Maryame El Moutamid |title=How Saturn got its tilt and its rings |journal=Science |date=Sep 15, 2022 |volume=377 |issue=6612 |pages=1264–1265 |doi=10.1126/science.abq3184|pmid=36108002 |bibcode=2022Sci...377.1264E |s2cid=252309068 }}</ref><ref>{{cite journal|display-authors=etal |last1=Jack Wisdom |title=Loss of a satellite could explain Saturn's obliquity and young rings |journal=Science |date=Sep 15, 2022 |volume=377 |issue=6612 |pages=1285–1289 |doi=10.1126/science.abn1234|pmid=36107998 |bibcode=2022Sci...377.1285W |s2cid=252310492 |hdl=1721.1/148216 |hdl-access=free }}</ref>

The [[secular resonance#?6 resonance|perihelion secular resonance]] between [[asteroid]]s and [[Saturn]] (''ν<sub>6</sub>'' = ''g'' − ''g<sub>6</sub>'') 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-crossing asteroid|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 [[astronomical unit|AU]], and at inclinations of about 20°.

Numerical simulations have suggested that the eventual formation of a perihelion secular resonance between [[Mercury (planet)|Mercury]] and Jupiter (''g<sub>1</sub>'' = ''g<sub>5</sub>'') has the potential to greatly increase Mercury's eccentricity and possibly destabilize the inner Solar System several billion years from now.<ref name="Laskar2008">{{cite journal |last=Laskar |first=J. |year=2008 |title=Chaotic diffusion in the Solar System |journal=[[Icarus (journal)|Icarus]] |volume=196 |issue=1 |pages=1–15 |arxiv=0802.3371 |bibcode=2008Icar..196....1L |doi=10.1016/j.icarus.2008.02.017|s2cid=11586168 }}</ref><ref name="Laskar2009">{{cite journal |last1=Laskar |first1=J. |last2=Gastineau |first2=M. |year=2009 |title=Existence of collisional trajectories of Mercury, Mars and Venus with the Earth |journal=[[Nature (journal)|Nature]] |volume=459 |issue=7248 |pages=817–819 |bibcode=2009Natur.459..817L |doi=10.1038/nature08096 |pmid=19516336|s2cid=4416436 }}</ref>

The [[Rings of Saturn#Colombo Gap and Titan Ringlet|Titan Ringlet]] within Saturn's [[Rings of Saturn#C Ring|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 (moon)|Titan]].<ref name="Porco1984">{{cite journal |last1=Porco |first1=C. |author-link=Carolyn Porco |last2=Nicholson |first2=P. D. |author-link2=Phil Nicholson |last3=Borderies |first3=N. |last4=Danielson |first4=G. E. |last5=Goldreich |first5=P. |author-link5=Peter Goldreich |last6=Holdberg |first6=J. B. |last7=Lane |first7=A. L. |year=1984 |title=The eccentric Saturnian ringlets at 1.29R<sub>s</sub> and 1.45R<sub>s</sub> |journal=[[Icarus (journal)|Icarus]] |volume=60 |issue=1 |pages=1–16 |bibcode=1984Icar...60....1P |doi=10.1016/0019-1035(84)90134-9}}</ref>

A ''[[Kozai resonance]]'' occurs when the inclination and eccentricity of a [[perturbation theory|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 [[Apsis|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.<ref name=Musotto2002>{{cite journal |last1=Musotto |first1=S. |last2=Varad |first2=F. |last3=Moore |first3=W. |last4=Schubert |first4=G. |year=2002 |title=Numerical Simulations of the Orbits of the Galilean Satellites |journal=[[Icarus (journal)|Icarus]] |volume=159 |issue=2 |pages=500–504 |doi=10.1006/icar.2002.6939 |bibcode=2002Icar..159..500M}}</ref>

== Mean-motion resonances in the Solar System ==
[[File:Haumea.GIF|thumb|300px|right|Depiction of [[Haumea (dwarf planet)|Haumea]]'s presumed 7:12 resonance with [[Neptune]] in a [[rotating frame]], with Neptune (blue dot at lower right) held stationary. Haumea's shifting orbital alignment relative to Neptune periodically reverses ([[libration|librates]]), preserving the resonance.]]

There are only a few known mean-motion resonances (MMR) in the [[Solar System]] involving planets, [[dwarf planet]]s or larger [[natural satellite|satellites]] (a much greater number involve [[asteroid]]s, [[planetary ring]]s, [[Inner satellite|moonlets]] and smaller [[Kuiper belt]] objects, including many [[possible dwarf planets]]).
* 2:3 [[Pluto]]–[[Neptune]] (also {{dp|Orcus}} and other [[plutino]]s)
* 2:4 [[Tethys (moon)|Tethys]]–[[Mimas (moon)|Mimas]] (Saturn's moons). Not simplified, because the libration of the nodes must be taken into account.
* 1:2 [[Dione (moon)|Dione]]–[[Enceladus]] (Saturn's moons)
* 3:4 [[Hyperion (moon)|Hyperion]]–[[Titan (moon)|Titan]] (Saturn's moons)
* 1:2:4 [[Ganymede (moon)|Ganymede]]–[[Europa (moon)|Europa]]–[[Io (moon)|Io]] (Jupiter's moons, ratio of ''orbits'').

Additionally, [[Haumea]] is thought to be in a 7:12 resonance with Neptune,<ref name="Brown_2007">{{cite journal |last1=Brown |first1=M. E. |author-link=Michael E. Brown |last2=Barkume |first2=K. M. |last3=Ragozzine |first3=D. |last4=Schaller |first4=E. L. |year=2007 |title=A collisional family of icy objects in the Kuiper belt |journal=[[Nature (journal)|Nature]] |volume=446 |issue=7133 |pages=294–296 |bibcode=2007Natur.446..294B |doi=10.1038/nature05619 |pmid=17361177|s2cid=4430027 |url=https://authors.library.caltech.edu/34346/2/nature05619-s1.pdf }}</ref><ref name="Ragozzine">{{cite journal |last1=Ragozzine |first1=D. |last2=Brown |first2=M. E. |year=2007 |title=Candidate members and age estimate of the family of Kuiper Belt object 2003 EL61 |journal=[[The Astronomical Journal]] |volume=134 |issue=6 |pages=2160–2167 |arxiv=0709.0328 |bibcode=2007AJ....134.2160R |doi=10.1086/522334|s2cid=8387493 }}</ref> and {{dp|Gonggong}} is thought to be in a 3:10 resonance with Neptune.<ref name="Buie">{{cite web |last=Buie |first=M. W. |author-link=Marc William Buie |date=24 October 2011 |title=Orbit Fit and Astrometric record for 225088 |publisher=SwRI (Space Science Department) |url=http://www.boulder.swri.edu/~buie/kbo/astrom/225088.html |access-date=14 November 2014}}</ref>

The simple integer ratios between periods hide more complex relations:
*the point of [[astronomical conjunction|conjunction]] can oscillate ([[libration|librate]]) around an equilibrium point defined by the resonance.
*given non-zero [[Eccentricity (orbit)|eccentricities]], the [[orbital node|nodes]] or [[perihelion|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 motion]]s <math>n\,\!</math> (inverse of periods, often expressed in degrees per day) would satisfy the following

: <math>n_{\rm Io} - 2\cdot n_{\rm Eu}=0 </math>

Substituting the data (from Wikipedia) one will get −0.7395° day<sup>−1</sup>, a value substantially different from zero.

Actually, the resonance {{em|is}} perfect, but it involves also the precession of [[Perihelion|perijove]] (the point closest to Jupiter), <math>\dot\omega</math>. The correct equation (part of the Laplace equations) is:

: <math>n_{\rm Io} - 2\cdot n_{\rm Eu} + \dot\omega_{\rm Io}=0 </math>

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

: <math>4\cdot n_{\rm Te} - 2\cdot n_{\rm Mi} - \dot\Omega_{\rm Te}- \dot\Omega_{\rm Mi}=0</math>

The point of conjunctions librates around the midpoint between the [[orbital node|nodes]] of the two moons.

=== Laplace resonance ===
[[File:TheLaplaceResonance2.png|thumb|300px|Illustration of Io–Europa–Ganymede resonance. From the centre outwards: Io (yellow), Europa (gray), and Ganymede (dark)]]
The Laplace resonance involving Io–Europa–Ganymede includes the following relation locking the ''orbital phase'' of the moons:

:<math>\Phi_L=\lambda_{\rm Io} - 3\cdot\lambda_{\rm Eu} + 2\cdot\lambda_{\rm Ga}=180^\circ</math>

where <math>\lambda</math> are [[mean longitude]]s 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. <math>\Phi_L</math> librates about 180° with an amplitude of 0.03°.<ref name="Sinclair1975">{{cite journal |last1=Sinclair |first1=A. T. |year=1975 |title=The Orbital Resonance Amongst the Galilean Satellites of Jupiter |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=171 |issue=1 |pages=59–72 |bibcode=1975MNRAS.171...59S |doi=10.1093/mnras/171.1.59|doi-access=free }}</ref>

Another "Laplace-like" resonance involves the [[Moons of Pluto|moons]] [[Styx (moon)|Styx]], [[Nix (moon)|Nix]], and [[Hydra (moon)|Hydra]] of Pluto:<ref name="ShowalterHamilton2015">{{cite journal |last1=Showalter |first1=M. R. |author1-link=Mark R. Showalter |last2=Hamilton |first2=D. P. |year=2015 |title=Resonant interactions and chaotic rotation of Pluto's small moons |journal=[[Nature (journal)|Nature]] |volume=522 |issue=7554 |pages=45–49 |bibcode=2015Natur.522...45S |doi=10.1038/nature14469 |pmid=26040889|s2cid=205243819 }}</ref>

:<math>\Phi=3\cdot\lambda_{\rm S} - 5\cdot\lambda_{\rm N} + 2\cdot\lambda_{\rm H}=180^\circ</math>

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 [[Orbital resonance#Coincidental 'near' ratios of mean motion|below]]); the respective ratio of orbits is 11:9:6. Based on the ratios of [[synodic period]]s, there are 5 conjunctions of Styx and Hydra and 3 conjunctions of Nix and Hydra for every 2 conjunctions of Styx and Nix.<ref name="ShowalterHamilton2015" /><ref name="Witze2015">{{cite journal |last1=Witze |first1=A. |date=3 June 2015 |title=Pluto's moons move in synchrony |journal=[[Nature News]] |doi=10.1038/nature.2015.17681|s2cid=134519717 }}</ref> As with the Galilean satellite resonance, triple conjunctions are forbidden. <math>\Phi</math> librates about 180° with an amplitude of at least 10°.<ref name="ShowalterHamilton2015" />
{{center|
{{Annotated image
 |image=Hydra, Nix, Styx conjunctions cycle.png
 |image-width=625
 |width=625
 |height=124<!-- to crop the lower part of the image -->
 |float=center
 |annotations=<!-- this parameter must be there, empty or not! -->
 |caption=Sequence of conjunctions of Hydra (blue), Nix (red), and Styx (black) over one third of their resonance cycle. Movements are counterclockwise and orbits completed are tallied at upper right of diagrams (click on image to see the whole cycle).}}
}}

=== Plutino resonances ===
The dwarf planet [[Pluto]] is following an orbit trapped in a web of resonances with [[Neptune]]. 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. The minimum separation between the two bodies overall is 17 AU, while the minimum separation between Pluto and [[Uranus]] is just 11 AU<ref>{{cite web |last=Malhotra |first=R. |author-link=Renu Malhotra |date=1997 |title=Pluto's Orbit |url=http://www.nineplanets.org/plutodyn.html |access-date=26 March 2007}}</ref> (see [[Pluto#Orbit|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 [[90482 Orcus|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".<ref name=MBP>{{cite web |last=Brown |first=M. E. |author-link=Michael E. Brown |date=23 March 2009 |title=S/2005 (90482) 1 needs your help |work=[[Mike Brown's Planets]] |url=http://www.mikebrownsplanets.com/2009/03/s1-90482-2005-needs-your-help.html |access-date=25 March 2009}}</ref>

[[File:Naiad-Thalassa 73-69 orbital resonance.jpg|thumb|300px|Depiction of the resonance between Neptune's moons [[Naiad (moon)|Naiad]] (whose orbital motion is shown in red) and [[Thalassa (moon)|Thalassa]], in a view that co-rotates with the latter]]

=== Naiad:Thalassa 73:69 resonance ===
Neptune's innermost moon, [[Naiad (moon)|Naiad]], is in a 73:69 fourth-order resonance with the next outward moon, [[Thalassa (moon)|Thalassa]]. 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&nbsp;km apart when they pass each other. Although their orbital radii differ by only 1850&nbsp;km, Naiad swings ~2800&nbsp;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.<ref name="JPLnews2019">{{cite web |url=https://www.jpl.nasa.gov/news/news.php?feature=7540 |title=NASA Finds Neptune Moons Locked in 'Dance of Avoidance' |date=14 November 2019 |website=Jet Propulsion Laboratory |access-date=15 November 2019}}</ref><ref name="Brozovic2019">{{cite journal |last1=Brozović |first1=M. |last2=Showalter |first2=M. R. |last3=Jacobson |first3=R. A. |last4=French |first4=R. S. |last5=Lissauer |first5=J. J. |last6=de Pater |first6=I. |title=Orbits and resonances of the regular moons of Neptune |date=31 October 2019 |journal=Icarus |volume=338 |issue=2 |pages=113462 |arxiv=1910.13612 |doi=10.1016/j.icarus.2019.113462 |bibcode=2020Icar..33813462B |s2cid=204960799}}</ref>{{NoteTag |The nature of this resonance (ignoring subtleties like libration and precession) can be crudely obtained from the orbital periods as follows. From Showalter ''et al.'', 2019,<ref name="Showalter2019">{{cite journal |last1=Showalter |first1=M. R. |last2=de Pater |first2=I. |last3=Lissauer |first3=J. J. |last4=French |first4=R. S. |url=https://www.spacetelescope.org/static/archives/releases/science_papers/heic1904/heic1904a.pdf |title=The seventh inner moon of Neptune |journal=Nature |volume=566 |issue=7744 |year=2019 |pages=350–353 |doi=10.1038/s41586-019-0909-9 |pmc=6424524 |pmid=30787452 |bibcode=2019Natur.566..350S}}</ref> the periods of Naiad (Pn) and Thalassa (Pt) are 0.294396 and 0.311484 days, respectively. From these, the period between conjunctions can be calculated as 5.366 days (1/[1/Pn – 1/Pt]), which is 18.23 (≈ 18.25) orbits of Naiad and 17.23 (≈ 17.25) orbits of Thalassa. Thus, after four conjunction periods, 73 orbits of Naiad and 69 orbits of Thalassa have elapsed, and the original configuration will be restored.}}

== Mean-motion resonances among extrasolar planets ==
[[File:Resonant planetary system.gif|thumb|Resonant planetary system of two planets with a 1:2 orbit ratio]]
While most [[extrasolar planet]]ary systems discovered have not been found to have planets in mean-motion resonances, chains of up to five resonant planets<ref name="Shale2017" /> and up to seven at least near resonant planets<ref name="Luger_etal_2017" /> have been uncovered. Simulations have shown that during [[Nebular hypothesis|planetary system formation]], the appearance of resonant chains of planetary embryos is favored by the presence of the [[Protoplanetary disk|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.<ref name="Izidoro2017">{{cite journal |last1=Izidoro |first1=A. |last2=Ogihara |first2=M. |last3=Raymond |first3=S. N. |last4=Morbidelli |first4=A. |last5=Pierens |first5=A. |last6=Bitsch |first6=B. |last7=Cossou |first7=C. |last8=Hersant |first8=F. |title=Breaking the chains: hot super-Earth systems from migration and disruption of compact resonant chains |journal=Monthly Notices of the Royal Astronomical Society |volume=470 |issue=2 |year=2017 |pages=1750–1770 |doi=10.1093/mnras/stx1232|doi-access=free |arxiv=1703.03634 |bibcode=2017MNRAS.470.1750I |s2cid=119493483 }}</ref>
*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).<ref name="rivera2010">{{cite journal |last1=Rivera |first1=E. J. |last2=Laughlin |first2=G. |last3=Butler |first3=R. P. |last4=Vogt |first4=S. S. |last5=Haghighipour |first5=N. |last6=Meschiari |first6=S. |year=2010 |title=The Lick-Carnegie Exoplanet Survey: A Uranus-mass Fourth Planet for GJ 876 in an Extrasolar Laplace Configuration |journal=[[The Astrophysical Journal]] |volume=719 |issue=1 |pages=890–899 |arxiv=1006.4244 |bibcode=2010ApJ...719..890R |doi=10.1088/0004-637X/719/1/890|s2cid=118707953 }}</ref><ref name="Laughlin2013">{{cite web |last=Laughlin |first=G. |date=23 June 2010 |title=A second Laplace resonance |url=http://oklo.org/2010/06/23/a-second-laplace-resonance/ |work=Systemic: Characterizing Planets |access-date=30 June 2015 |archive-url=https://web.archive.org/web/20131229124449/http://oklo.org/2010/06/23/a-second-laplace-resonance/ |archive-date=29 December 2013}}</ref><ref name="Marcy_2001">{{cite journal |last1=Marcy |first1=Ge. W. |last2=Butler |first2=R. P. |last3=Fischer |first3=D. |last4=Vogt |first4=S. S. |last5=Lissauer |first5=J. J. |last6=Rivera |first6=E. J. |year=2001 |title=A Pair of Resonant Planets Orbiting GJ 876 |journal=[[The Astrophysical Journal]] |volume=556 |issue=1 |pages=296–301 |bibcode=2001ApJ...556..296M |doi=10.1086/321552|doi-access=free }}</ref> In this case, <math>\Phi_L</math> librates with an amplitude of 40° ± 13° and the resonance follows the time-averaged relation:<ref name="rivera2010" />
:<math>\Phi_L=\lambda_{\rm c} - 3\cdot\lambda_{\rm d} + 2\cdot\lambda_{\rm e}=0^\circ</math>
*[[Kepler-223]] 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).<ref name=EPE-KOI730>{{cite encyclopedia |title=Planet Kepler-223 b |url=http://exoplanet.eu/catalog/kepler-223_b/ |access-date=21 January 2018 |archive-date=22 January 2018 |archive-url=https://web.archive.org/web/20180122072529/http://exoplanet.eu/catalog/kepler-223_b/ |encyclopedia=[[Extrasolar Planets Encyclopaedia]] |url-status=dead }}</ref><ref name="Beatty">{{cite web |last=Beatty |first=K. |date=5 March 2011 |title=Kepler Finds Planets in Tight Dance |url=http://www.skyandtelescope.com/astronomy-news/kepler-finds-planets-in-tight-dance/ |work=[[Sky and Telescope]] |access-date=16 October 2012}}</ref><ref name="Lissauer_2011">{{cite journal |last1=Lissauer |first1=J. J. |author1-link=Jack J. Lissauer |last2=Ragozzine |first2=D. |last3=Fabrycky |first3=D. C. |last4=Steffen |first4=J. H. |last5=Ford |first5=E. B. |last6=Jenkins |first6=J. M. |last7=Shporer |first7=A. |last8=Holman |first8=M. J. |last9=Rowe |first9=J. F. |last10=Quintana |first10=E. V. |last11=Batalha |first11=N. M. |last12=Borucki |first12=W. J. |last13=Bryson |first13=S. T. |last14=Caldwell |first14=D. A. |last15=Carter |first15=J. A. |last16=Ciardi |first16=D. |last17=Dunham |first17=E. W. |last18=Fortney |first18=J. J. |last19=Gautier, III |first19=T. N. |last20=Howell |first20=S. B. |last21=Koch |first21=D. G. |last22=Latham |first22=D. W. |last23=Marcy |first23=G. W. |last24=Morehead |first24=R. C. |last25=Sasselov |first25=D. |display-authors=1 |year=2011 |title=Architecture and dynamics of Kepler's candidate multiple transiting planet systems |journal=[[The Astrophysical Journal Supplement Series]] |volume=197 |issue=1 |pages=1–26 |arxiv=1102.0543 |bibcode=2011ApJS..197....8L |doi=10.1088/0067-0049/197/1/8|s2cid=43095783 }}</ref><ref name="Mills2016">{{cite journal |last1=Mills |first1=S. M. |last2=Fabrycky |first2=D. C. |last3=Migaszewski |first3=C. |last4=Ford |first4=E. B. |last5=Petigura |first5=E. |last6=Isaacson |first6=H. |title=A resonant chain of four transiting, sub-Neptune planets |journal=Nature |date=11 May 2016 |doi=10.1038/nature17445 |volume=533 |issue=7604 |pages=509–512 |pmid=27225123 |arxiv=1612.07376 |bibcode=2016Natur.533..509M |s2cid=205248546 }}</ref> This represents the first confirmed 4-body orbital resonance.<ref name="Koppes2016">{{cite web |url=http://www.jpl.nasa.gov/news/news.php?feature=6515 |title=Kepler-223 System: Clues to Planetary Migration |last=Koppes |first=S. |date=17 May 2016 |website=[[Jet Propulsion Lab]] |access-date=18 May 2016}}</ref> 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]].<ref name="Mills2016" />
*[[Kepler-80]] 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.<ref name="MacDonald2016">{{Cite journal |last1=MacDonald |first1=M. G. |last2=Ragozzine |first2=D. |last3=Fabrycky |first3=D. C. |last4=Ford |first4=E. B. |last5=Holman |first5=M. J. |last6=Isaacson |first6=H. T. |last7=Lissauer |first7=J. J. |last8=Lopez |first8=E. D. |last9=Mazeh |first9=T. |date=1 January 2016 |title=A Dynamical Analysis of the Kepler-80 System of Five Transiting Planets |journal=The Astronomical Journal |volume=152 |issue=4 |pages=105 |doi=10.3847/0004-6256/152/4/105 |arxiv=1607.07540 |bibcode=2016AJ....152..105M|s2cid=119265122 |doi-access=free }}</ref><ref name="Shale2017">{{cite journal |last1=Shale |first1=C. J. |last2=Vanderburg |first2=A. |title=Identifying Exoplanets With Deep Learning: A Five Planet Resonant Chain Around Kepler-80 And An Eighth Planet Around Kepler-90 |journal=[[The Astrophysical Journal]] |volume= 155|issue= 2|pages= 94|date=2017 |url=https://www.cfa.harvard.edu/~avanderb/kepler90i.pdf |doi=10.3847/1538-3881/aa9e09 |access-date=15 December 2017 |arxiv=1712.05044 |bibcode=2018AJ....155...94S|s2cid=4535051 |doi-access=free }}</ref>
*[[TOI-178]] 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.<ref>{{Cite journal|last1=Leleu |first1=A. |last2=Alibert |first2=Y. |last3=Hara |first3=N. C. |last4=Hooton |first4=M. J. |last5=Wilson |first5=T. G. |last6=Robutel |first6=P. |last7=Delisle |first7=J. -B. |last8=Laskar |first8=J. |last9=Hoyer |first9=S. |last10=Lovis |first10=C. |last11=Bryant |first11=E. M. |last12=Ducrot |first12=E. |last13=Cabrera |first13=J. |last14=Delrez |first14=L. |last15=Acton |first15=J. 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* [[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.).<ref name="Gillon2016">{{Cite journal |last1=Gillon |first1=M. |last2=Triaud |first2=A. H. M. J. |last3=Demory |first3=B.-O. |last4=Jehin |first4=E. |last5=Agol |first5=E. |last6=Deck |first6=K. M. |last7=Lederer |first7=S. M. |last8=de Wit |first8=J. |last9=Burdanov |first9=A. |title=Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1 |journal=Nature |volume=542 |issue=7642 |pages=456–460 |doi=10.1038/nature21360 |date=22 February 2017 |pmid=28230125 |pmc=5330437 |arxiv=1703.01424 |bibcode=2017Natur.542..456G}}</ref><ref name="Luger_etal_2017">{{cite journal |title=A seven-planet resonant chain in TRAPPIST-1 |first1=R. |last1=Luger |first2=M. |last2=Sestovic |first3=E. |last3=Kruse |first4=S. L. |last4=Grimm |first5=B.-O. |last5=Demory |last6=Agol |first6=E. |last7=Bolmont |first7=E. |last8=Fabrycky |first8=D. |last9=Fernandes |first9=C. S. |last10=Van Grootel|first10= V. |last11=Burgasser |first11=A. |last12=Gillon |first12=M. |last13=Ingalls |first13=J. G. |last14=Jehin |first14=E. |last15=Raymond |first15=S. N. |last16=Selsis |first16=F. |last17=Triaud |first17=A. H. M. J. |last18=Barclay |first18=T. |last19=Barentsen |first19=G. |last20=Delrez |first20=L. |last21=de Wit |first21=J. |last22=Foreman-Mackey |first22=D. |last23=Holdsworth |first23=D. L. |last24=Leconte |first24=J. |last25=Lederer |first25=S. |last26=Turbet |first26=M. |last27=Almleaky |first27=Y. |last28=Benkhaldoun |first28=Z. |last29=Magain |first29=P. |last30=Morris |first30=B. |date=22 May 2017 |journal=Nature Astronomy |volume=1 |issue=6 |pages=0129 |doi=10.1038/s41550-017-0129 |arxiv=1703.04166 |bibcode=2017NatAs...1E.129L|s2cid=54770728 }}</ref> The resonant configuration is expected to be stable on a time scale of billions of years, assuming it arose during planetary migration.<ref name="Tamayo2017">{{cite journal |last1=Tamayo |first1=D. |last2=Rein |first2=H. |last3=Petrovich |first3=C. |last4=Murray |first4=N. |title=Convergent Migration Renders TRAPPIST-1 Long-lived |journal=The Astrophysical Journal |volume=840 |issue=2 |date=10 May 2017 |pages=L19 |doi=10.3847/2041-8213/aa70ea |arxiv=1704.02957 |bibcode=2017ApJ...840L..19T|s2cid=119336960 |doi-access=free }}</ref><ref name="NYT-20170510">{{cite news |url=https://www.nytimes.com/2017/05/10/science/trappist-earth-size-planets-orbits-music.html |title=The Harmony That Keeps Trappist-1's 7 Earth-size Worlds From Colliding |work=[[The New York Times]] |last=Chang |first=K. |date=10 May 2017 |access-date=26 June 2017}}</ref> A musical interpretation of the resonance has been provided.<ref name="NYT-20170510" />
*[[Kepler-29]] has a pair of planets in a 7:9 resonance (ratio of 1/1.28587).<ref name="Lissauer_2011" />
*[[Kepler-36]] has a pair of planets close to a 6:7 resonance.<ref name=Carter2012>{{cite journal |author1=Carter, J. A. |author2=Agol, E. |author3=Chaplin, W. J. |author4=Basu, S. |author5=Bedding, T. R. |author6=Buchhave, L. A. |author7=Christensen-Dalsgaard, J. |author8=Deck, K. M. |author9=Elsworth, Y. |author10=Fabrycky, D. C. |author11=Ford, E. B. |author12=Fortney, J. J. |author13=Hale, S. J. |author14=Handberg, R. |author15=Hekker, S. |author16=Holman, M. J. |author17=Huber, D. |author18=Karoff, C. |author19=Kawaler, S. D. |author20=Kjeldsen, H. |author21=Lissauer, J. J. |author22=Lopez, E. D. |author23=Lund, M. N. |author24=Lundkvist, M. |author25=Metcalfe, T. S. |author26=Miglio, A. |author27=Rogers, L. A. |author28=Stello, D. |author29=Borucki, W. J. |author30=Bryson, S. |author31=Christiansen, J. L. |author32=Cochran, W. D. |author33=Geary, J. C. |author34=Gilliland, R. L. |author35=Haas, M. R. |author36=Hall, J. |author37=Howard, A. W. |author38=Jenkins, J. M. |author39=Klaus, T. |author40=Koch, D. G. |author41=Latham, D. W. |author42=MacQueen, P. J. |author43=Sasselov, D. |author44=Steffen, J. H. |author45=Twicken, J. D. |author46=Winn, J. N. |display-authors=3 |title=Kepler-36: A Pair of Planets with Neighboring Orbits and Dissimilar Densities |journal=Science |date=21 June 2012 |doi=10.1126/science.1223269 |arxiv=1206.4718 |bibcode=2012Sci...337..556C |volume=337 |issue=6094 |pages=556–559 |pmid=22722249|s2cid=40245894 }}</ref>
*[[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).<ref name="BarclayRowe2013">{{cite journal |last1=Barclay |first1=T. |last2=Rowe |first2=J. F. |last3=Lissauer |first3=J. J. |last4=Huber |first4=D. |last5=Fressin |first5=F. |last6=Howell |first6=S. B. |last7=Bryson |first7=S. T. |last8=Chaplin |first8=W. J. |last9=Désert |first9=J.-M. |last10=Lopez |first10=E. D. |last11=Marcy |first11=G. W. |last12=Mullally |first12=F. |last13=Ragozzine |first13=D. |last14=Torres |first14=G. |last15=Adams |first15=E. R. |last16=Agol |first16=E. |last17=Barrado |first17=D. |last18=Basu |first18=S. |last19=Bedding |first19=T. R. |last20=Buchhave |first20=L. A. |last21=Charbonneau |first21=D. |last22=Christiansen |first22=J. L. |last23=Christensen-Dalsgaard |first23=J. |last24=Ciardi |first24=D. |last25=Cochran |first25=W. D. |last26=Dupree |first26=A. K. |last27=Elsworth |first27=Y. |last28=Everett |first28=M. |last29=Fischer |first29=D. A. |last30=Ford |first30=E. B. |last31=Fortney |first31=J. J. |last32=Geary |first32=J. C. |last33=Haas |first33=M. R. |last34=Handberg |first34=R. |last35=Hekker |first35=S. |last36=Henze |first36=C. E. |last37=Horch |first37=E. |last38=Howard |first38=A. W. |last39=Hunter |first39=R. C. |last40=Isaacson |first40=H. |last41=Jenkins |first41=J. M. |last42=Karoff |first42=C. |last43=Kawaler |first43=S. D. |last44=Kjeldsen |first44=H. |last45=Klaus |first45=T. C. |last46=Latham |first46=D. W. |last47=Li |first47=J. |last48=Lillo-Box |first48=J. |last49=Lund |first49=M. N. |last50=Lundkvist |first50=M. |last51=Metcalfe |first51=T. S. |last52=Miglio |first52=A. |last53=Morris |first53=R. L. |last54=Quintana |first54=E. V. |last55=Stello |first55=D. |last56=Smith |first56=J. C. |last57=Still |first57=M. |last58=Thompson |first58=S. E. |display-authors=1 |year=2013 |title=A sub-Mercury-sized exoplanet |journal=[[Nature (journal)|Nature]] |volume=494 |issue=7438 |pages=452–454 |arxiv=1305.5587 |bibcode=2013Natur.494..452B |doi=10.1038/nature11914 |pmid=23426260|s2cid=205232792 }}
:*And {{cite journal |title=Erratum: A sub-Mercury-sized exoplanet |year=2013 |journal=Nature |volume=496 |issue=7444 |pages=252 |bibcode=2013Natur.496..252B |doi=10.1038/nature12067|doi-access=free |last1=Barclay |first1=Thomas |last2=Rowe |first2=Jason F. |last3=Lissauer |first3=Jack J. |last4=Huber |first4=Daniel |last5=Fressin |first5=François |last6=Howell |first6=Steve B. |last7=Bryson |first7=Stephen T. |last8=Chaplin |first8=William J. |last9=Désert |first9=Jean-Michel |last10=Lopez |first10=Eric D. |last11=Marcy |first11=Geoffrey W. |last12=Mullally |first12=Fergal |last13=Ragozzine |first13=Darin |last14=Torres |first14=Guillermo |last15=Adams |first15=Elisabeth R. |last16=Agol |first16=Eric |last17=Barrado |first17=David |last18=Basu |first18=Sarbani |last19=Bedding |first19=Timothy R. |last20=Buchhave |first20=Lars A. |last21=Charbonneau |first21=David |last22=Christiansen |first22=Jessie L. |last23=Christensen-Dalsgaard |first23=Jørgen |last24=Ciardi |first24=David |last25=Cochran |first25=William D. |last26=Dupree |first26=Andrea K. |last27=Elsworth |first27=Yvonne |last28=Everett |first28=Mark |last29=Fischer |first29=Debra A. |last30=Ford |first30=Eric B. |display-authors=1 }}</ref>
*Of [[Kepler-90]]'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).<ref>{{Cite journal |arxiv=1402.6352 |title=Validation of Kepler's Multiple Planet Candidates. II: Refined Statistical Framework and Descriptions of Systems of Special Interest |journal=The Astrophysical Journal |volume=784 |issue=1 |pages=44 |date=25 February 2014 |last1=Lissauer |first1=J. J. |last2=Marcy |first2=G. W. |last3=Bryson |first3=S. T. |last4=Rowe |first4=J. F. |last5=Jontof-Hutter |first5=D. |last6=Agol |first6=E. |last7=Borucki |first7=W. J. |last8=Carter |first8=J. A. |last9=Ford |first9=E. B.|last10= Gilliland|first10= R. L. |last11=Kolbl |first11=R. |last12=Star |first12=K. M. |last13=Steffen |first13=J. H. |last14=Torres |first14=G. |doi=10.1088/0004-637X/784/1/44 |bibcode=2014ApJ...784...44L|s2cid=119108651 }}</ref><ref name="Shale2017" /> f, g and h are also close to a 3:5:8 period ratio (3: 5.058: 7.964).<ref name="Cabrera2013">{{cite journal |last1=Cabrera |first1=J. |last2=Csizmadia |first2=Sz. |last3=Lehmann |first3=H. |last4=Dvorak |first4=R. |last5=Gandolfi |first5=D. |last6=Rauer |first6=H. |last7=Erikson |first7=A. |last8=Dreyer |first8=C. |last9=Eigmüller |first9=Ph.|last10= Hatzes|first10= A. |title=The Planetary System to KIC 11442793: A Compact Analogue to the Solar System |journal=The Astrophysical Journal |volume=781 |issue=1 |date=31 December 2013 |page=18 |doi=10.1088/0004-637X/781/1/18 |arxiv=1310.6248 |bibcode=2014ApJ...781...18C|s2cid=118875825 }}</ref> 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.<ref name="Hands2016">{{cite journal |last1=Hands |first1=T. O. |last2=Alexander |first2=R. D. |title=There might be giants: unseen Jupiter-mass planets as sculptors of tightly packed planetary systems |journal=Monthly Notices of the Royal Astronomical Society |volume=456 |issue=4 |date=13 January 2016 |pages=4121–4127 |doi=10.1093/mnras/stv2897 |doi-access=free |arxiv=1512.02649 |bibcode=2016MNRAS.456.4121H|s2cid=55175754 }}</ref>
*[[HD 41248]] has a pair of [[super-Earth]]s within 0.3% of a 5:7 resonance (ratio of 1/1.39718).<ref name=Jenkins13>{{cite journal |last1=Jenkins |first1=J. S. |last2=Tuomi |first2=M. |last3=Brasser |first3=R. |last4=Ivanyuk |first4=O. |last5=Murgas |first5=F. |year=2013 |title=Two Super-Earths Orbiting the Solar Analog HD 41248 on the Edge of a 7:5 Mean Motion Resonance |journal=[[The Astrophysical Journal]] |volume=771 |issue=1 |page=41 |arxiv=1304.7374 |bibcode=2013ApJ...771...41J |doi=10.1088/0004-637X/771/1/41|s2cid=14827197 }}</ref>
*[[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]] project Exoplanet Explorers, using K2 data.<ref>{{Cite journal|last1=Christiansen|first1=Jessie L.|last2=Crossfield|first2=Ian J. M.|last3= Barentsen|first3= G.|last4= Lintott|first4=C. J.|last5= Barclay|first5= T.|last6= Simmons|first6=B. D.|last7= Petigura|first7= E.|last8= Schlieder|first8=J. E.|last9= Dressing|first9=C. D.|last10= Vanderburg|first10= A.|last11= Allen|first11= C.|date= 2018-01-11|title=The K2-138 System: A Near-resonant Chain of Five Sub-Neptune Planets Discovered by Citizen Scientists|journal= The Astronomical Journal|volume= 155|issue= 2|pages= 57|arxiv= 1801.03874|doi= 10.3847/1538-3881/aa9be0|bibcode=2018AJ....155...57C|s2cid=52971376 |doi-access=free }}</ref> K2-138 could host [[Co-orbital configuration|co-orbital bodies]] (in a 1:1 mean-motion resonance).<ref name=":0">{{Cite journal|last1=Lopez |first1=T. A. |last2=Barros |first2=S. C. C. |last3=Santerne |first3=A. |last4=Deleuil |first4=M. |last5=Adibekyan |first5=V. |last6=Almenara |first6=J.-M. |last7=Armstrong |first7=D. J. |last8=Brugger |first8=B. |last9=Barrado |first9=D. |last10=Bayliss |first10=D. |last11=Boisse |first11=I. |last12=Bonomo |first12=A. S. |last13=Bouchy |first13=F. |last14=Brown |first14=D. J. A. |last15=Carli |first15=E. |last16=Demangeon |first16=O. |last17=Dumusque |first17=X. |last18=Díaz |first18=R. F. |last19=Faria |first19=J. P. |last20=Figueira |first20=P. |last21=Foxell |first21=E. |last22=Giles |first22=H. |last23=Hébrard |first23=G. |last24=Hojjatpanah |first24=S. |last25=Kirk |first25=J. |last26=Lillo-Box |first26=J. |last27=Lovis |first27=C. |last28=Mousis |first28=O. |last29=da Nóbrega |first29=H. J. |last30=Nielsen |first30=L. D. |last31=Neal |first31=J. J. |last32=Osborn |first32=H. P. |last33=Pepe |first33=F. |last34=Pollacco |first34=D. |last35=Santos |first35=N. C. |last36=Sousa |first36=S. G. |last37=Udry |first37=S. |last38=Vigan |first38=A. |last39=Wheatley |first39=P. J.|date= 2019-11-01|title= Exoplanet characterisation in the longest known resonant chain: the K2-138 system seen by HARPS|journal=Astronomy & Astrophysics|language=en|volume=631|pages=A90|doi=10.1051/0004-6361/201936267|bibcode=2019A&A...631A..90L |arxiv=1909.13527 |s2cid=203593804 }}</ref> Resonant chain systems can stabilize co-orbital bodies<ref>{{Cite journal|last1=Leleu|first1=Adrien|last2=Coleman|first2=Gavin A. L.|last3= Ataiee|first3= S.|date= 2019-11-01|title= Stability of the co-orbital resonance under dissipation – Application to its evolution in protoplanetary discs|journal=Astronomy & Astrophysics|language=en|volume=631|pages=A6|arxiv=1901.07640|doi=10.1051/0004-6361/201834486|bibcode=2019A&A...631A...6L|s2cid=219840769}}</ref> and a dedicated analysis of the K2 light curve and radial-velocity from [[High Accuracy Radial Velocity Planet Searcher|HARPS]] might reveal them.<ref name=":0" /> 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.<ref>{{Cite web|url=https://www.jpl.nasa.gov/spaceimages/details.php?id=PIA23003|title=K2-138 System Diagram|website=jpl.nasa.gov|access-date=2019-11-20}}</ref><ref>{{Cite journal|last1= Hardegree-Ullman|first1= K.|last2= Christiansen|first2= J.|date= January 2019|title=K2-138 g: Spitzer Spots a Sixth Sub-Neptune for the Citizen Science System|journal= American Astronomical Society Meeting Abstracts #233|language=en|volume=233|pages=164.07|bibcode= 2019AAS...23316407H}}</ref> Future observations with [[CHEOPS]] will measure [[transit-timing variation]]s of the system to further analyse the mass of the planets and could potentially find other planetary bodies in the system.<ref>{{Cite web|url=https://www.cosmos.esa.int/web/cheops-guest-observers-programme/ao-1-programmes|title=AO-1 Programmes – CHEOPS Guest Observers Programme – Cosmos |website=cosmos.esa.int|access-date=2019-11-20}}</ref>
*[[K2-32]] 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.<ref>{{Cite journal|last1=Heller|first1=René|last2=Rodenbeck|first2=Kai|last3=Hippke|first3=Michael|date=2019-05-01|title=Transit least-squares survey – I. Discovery and validation of an Earth-sized planet in the four-planet system K2-32 near the 1:2:5:7 resonance|journal=Astronomy & Astrophysics|language=en|volume=625|pages=A31|doi=10.1051/0004-6361/201935276|issn=0004-6361|bibcode=2019A&A...625A..31H|arxiv=1904.00651|s2cid=90259349}}</ref>
*[[V1298 Tauri]] 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]]) 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.<ref>{{Cite journal|last1=David|first1=Trevor J.|last2=Petigura|first2=Erik A.|last3=Luger|first3=Rodrigo|last4=Foreman-Mackey|first4=Daniel|last5=Livingston|first5=John H.|last6=Mamajek|first6=Eric E.|last7=Hillenbrand|first7=Lynne A.|date=2019-10-29|title=Four Newborn Planets Transiting the Young Solar Analog V1298 Tau|journal=The Astrophysical Journal|volume=885|issue=1|pages=L12|arxiv=1910.04563|doi=10.3847/2041-8213/ab4c99|issn=2041-8213|bibcode=2019ApJ...885L..12D|s2cid=204008446 |doi-access=free }}</ref>
*[[HD 158259]] 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 [[Doppler spectroscopy|radial velocity]] method.<ref>{{cite journal|last1=Hara|first1=N. C.|last2=Bouchy|first2=F.|last3=Stalport|first3=M.|last4=Boisse|first4=I.|last5=Rodrigues|first5=J.|last6=Delisle|first6=J.- B.|last7=Santerne|first7=A.|last8=Henry|first8=G. W.|last9=Arnold|first9=L.|last10=Astudillo-Defru|first10=N.|last11=Borgniet|first11=S.|title=The SOPHIE search for northern extrasolar planets. XVII. A compact planetary system in a near 3:2 mean motion resonance chain|journal=Astronomy & Astrophysics|year=2020|language=en|volume=636|pages=L6|doi=10.1051/0004-6361/201937254|arxiv=1911.13296|bibcode=2020A&A...636L...6H |s2cid=208512859}}</ref>
*[[Kepler-1649]] 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 ([[Kepler-1649c|"c"]]) in the habitable zone. An undetected planet with a 13.0-day period would create a 3:2 resonance chain.<ref name="Vanderburg2020">{{cite journal|last1= Vanderburg|first1= A.|last2= Rowden|first2= P.|last3= Bryson|first3= S.|last4= Coughlin|first4= J.|last5= Batalha|first5= N.|last6= Collins|first6= K.A.|last7= Latham|first7= D.W.|last8= Mullally|first8= S.E.|last9= Colón|first9= K.D.|last10= Henze|first10= C.|last11= Huang|first11= C.X.|last12= Quinn|first12= S.N.|title=A Habitable-zone Earth-sized Planet Rescued from False Positive Status |journal=The Astrophysical Journal |volume= 893|issue= 1|year= 2020|pages= L27|doi= 10.3847/2041-8213/ab84e5|arxiv= 2004.06725|bibcode= 2020ApJ...893L..27V|s2cid= 215768850|doi-access= free}}</ref>
*[[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 variation]]s of ~0.5 days for the innermost planet. There is a yet more massive outer planet in a ~1400 day orbit.<ref name="Weiss2020">{{cite journal|last1= Weiss|first1= L.M.|last2= Fabrycky|first2= D.C.|last3= Agol|first3= E.|last4= Mills|first4= S.M.|last5= Howard|first5= A.W.|last6= Isaacson|first6= H.|last7= Petigura|first7= E.A.|last8= Fulton|first8= B.|last9= Hirsch|first9= L.|last10= Sinukoff|first10= E.|title=The Discovery of the Long-Period, Eccentric Planet Kepler-88 d and System Characterization with Radial Velocities and Photodynamical Analysis |journal=The Astronomical Journal |volume= 159|issue= 5|year= 2020|pages= 242|doi= 10.3847/1538-3881/ab88ca |arxiv=1909.02427|bibcode= 2020AJ....159..242W|s2cid= 202539420|url= https://authors.library.caltech.edu/99954/1/1909.02427.pdf|doi-access= free}}</ref>
* [[HD 110067]] has six known planets, in a 54:36:24:16:12:9 resonance ratio.<ref>{{Cite web |last=Klesman |first=Alison |date=2023-11-29 |title='Shocked and delighted': Astronomers find six planets orbiting in resonance |url=https://www.astronomy.com/science/astronomers-find-six-planets-orbiting-in-resonance/ |access-date=2023-12-23 |website=Astronomy Magazine |language=en-US}}</ref>

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),<ref name="Lissauer_2011" /> as well as one sixth of planetary systems characterized by [[Doppler spectroscopy]] (with in this case a narrower period ratio range).<ref name="Wright_2011" /> Due to incomplete knowledge of the systems, the actual proportions are likely to be higher.<ref name="Lissauer_2011" /> Overall, about a third of radial velocity characterized systems appear to have a pair of planets close to a [[Commensurability (astronomy)|commensurability]].<ref name="Lissauer_2011" /><ref name="Wright_2011">{{cite journal |last1=Wright |first1=J. T. |last2=Fakhouri |first2=O. |last3=Marcy |first3=G. W. |last4=Han |first4=E. |last5=Feng |first5=Y. |last6=Johnson |first6=J. A. |last7=Howard |first7=A. W. |last8=Fischer |first8=D. A. |last9=Valenti |first9=J. A. |last10=Anderson |first10=J. |last11=Piskunov |first11=N. |year=2011 |title=The Exoplanet Orbit Database |journal=[[Publications of the Astronomical Society of the Pacific]] |volume=123 |issue=902 |pages=412–42 |arxiv=1012.5676 |bibcode=2011PASP..123..412W |doi=10.1086/659427|s2cid=51769219 }}</ref> 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).<ref name="Lissauer_2011" /> This was predicted to be true in cases where [[Tidal acceleration|tidal interactions]] with the star are significant.<ref name="Terquem_2007">{{cite journal |last1=Terquem |first1=C. |last2=Papaloizou |first2=J. C. B. |year=2007 |title=Migration and the Formation of Systems of Hot Super-Earths and Neptunes |journal=[[The Astrophysical Journal]] |volume=654 |issue=2 |pages=1110–1120 |arxiv=astro-ph/0609779 |bibcode=2007ApJ...654.1110T |doi=10.1086/509497|s2cid=14034512 }}</ref>

== Coincidental 'near' ratios of mean motion ==
[[File:PallasJupiter.GIF|300px|thumb|Depiction of asteroid [[2 Pallas|Pallas']] 18:7 near resonance with Jupiter in a rotating frame (''click for animation''). Jupiter (pink loop at upper left) is held nearly stationary. The shift in Pallas' orbital alignment relative to Jupiter increases steadily over time; it never reverses course (i.e., there is no libration).]]
[[File:Venus pentagram.png|300px|thumb|Depiction of the [[Earth]]:[[Venus]] 8:13 near resonance. With Earth held stationary at the center of a nonrotating frame, the successive [[inferior conjunction]]s of Venus over eight Earth years trace a [[pentagram]]mic pattern (reflecting the difference between the numbers in the ratio).]]
[[File:Moons of Pluto.png|thumb|300px|Diagram of the orbits of [[Pluto]]'s small outer four moons, which follow a 3:4:5:6 sequence of near resonances relative to the period of its large inner satellite [[Charon (moon)|Charon]]. The moons Styx, Nix and Hydra are also involved in a true [[Orbital resonance#Laplace resonance|3-body resonance]].]]

A number of near-[[integer]]-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 [[perihelion]] or other libration to make the resonance perfect (see the detailed discussion in the [[Orbital resonance#Mean-motion resonances in the Solar System|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:

{| class="wikitable" style="vertical-align:center;text-align:center;" 
|+ Table of some orbital frequency coincidences in the Solar system
|-
! Ratio
! Bodies
! Mismatch<br/>after one<br/>cycle{{efn|
Mismatch in orbital longitude of the inner body, as compared to its position at the beginning of the cycle (with the cycle defined as {{mvar|n}} orbits of the outer body – see below). Circular orbits are assumed (i.e., precession is ignored).
}}
! Randmztn.<br/>time{{efn|
The ''randomization time'' is the amount of time needed for the mismatch from the initial relative longitudinal orbital positions of the bodies to grow to 180°. The listed number is rounded to the nearest first [[significant digit]].
}}
! Probability{{efn|
Estimated [[probability]] of obtaining by chance an orbital coincidence of equal or smaller mismatch, at least once in {{mvar|n}} attempts, where {{mvar|n}} is the integer number of orbits of the outer body per cycle, and the mismatch is assumed to randomly vary between 0° and 180°. The value is calculated as {{nobr| 1 − ( 1 − {{small|{{sfrac| mismatch | 180° }} }} ){{sup| {{mvar|n}} }} .}} This is a crude calculation that only attempts to give a rough idea of relative probabilities.
}}{{efn|
Smaller is better: The smaller the probability of an apparently resonant relationship arising as a mere chance alignment of random numbers, the more credible the proposal that gravitational interaction causes persistence of the relationship, or prolongs it / delays its ultimate dissolution by other, disruptive perturbations.
}}
|-
!colspan="5"| {{big|''Trans-planetary resonances''}}
|-
| 9:23 || [[Venus]]–[[Mercury (planet)|Mercury]] || 4.0° || 200 [[year|y]] || 19%
|-
|1:4
|Earth-Mercury
|54.8°
|3 y
|0.3%
|-
| 8:13 || [[Earth]]–[[Venus]]<ref name=Langford/><ref name=Bazsó>
{{cite journal |last1=Bazsó |first1=A. |last2=Eybl |first2=V. |last3=Dvorak |first3=R. |last4=Pilat-Lohinger |first4=E. |last5=Lhotka |first5=C. |year=2010 |title=A survey of near-mean-motion resonances between Venus and Earth |journal=[[Celestial Mechanics and Dynamical Astronomy]] |volume=107 |issue=1 |pages=63–76 |arxiv=0911.2357 |bibcode=2010CeMDA.107...63B |doi=10.1007/s10569-010-9266-6 |s2cid=117795811 }}
</ref>{{efn|
The two near [[Commensurability (astronomy)|commensurabilities]] listed for Earth and Venus are reflected in the timing of [[transit of Venus|transits of Venus]], which occur in pairs 8&nbsp;years apart, in a cycle that repeats every 243&nbsp;years.<ref name=Langford>
{{cite web |last=Langford |first=P.M. |date=12 March 2012 |title=Transits of Venus |publisher=Astronomical Society of the Channel Island of Guernsey |url=http://www.astronomy.org.gg/venustransitsb.htm |url-status=dead |access-date=15 January 2016 |archive-url=https://web.archive.org/web/20120111153545/http://www.astronomy.org.gg/venustransitsb.htm |archive-date=11 January 2012 }}
</ref><ref name=Shortt>
{{cite web |last=Shortt |first=D. |date=22 May 2012 |title=Some Details About Transits of Venus |publisher=[[The Planetary Society]] |url=http://www.planetary.org/blogs/guest-blogs/Some-Details-About-Transits-of-Venus.html |access-date=22 May 2012}}
</ref>
}}
| 1.5° || 1000 [[year|y]] || 6.5%
|-
| 243:395 || [[Earth]]–[[Venus]]<ref name=Langford/><ref name=Shortt/> || 0.8° || 50,000 [[year|y]] || 68%
|-
| 1:3 || [[Mars]]–[[Venus]] || 20.6° || 20 y || 11%
|-
| 1:2 || [[Mars]]–[[Earth]] || 42.9° || 8 y || 24%
|-
|193:363
|Mars-Earth
|0.9°
|70,000 y
|0.6%
|-
| 1:12 || [[Jupiter]]–[[Earth]]{{efn|
The near 1:12 resonance between Jupiter and Earth has the coincidental side-effect of making the [[Alinda family|Alinda asteroids]], which occupy (or are close to) the 3:1 resonance with Jupiter, to be close to a 1:4 resonance with Earth.
}}
| 49.1° || 40 y || 28%
|-
|3:19
|Jupiter-Mars
|28.7°
|200 y
|0.4%
|-
| 2:5 || [[Saturn]]–[[Jupiter]]{{efn|
The long-known near resonance between Jupiter and Saturn has traditionally been called the ''[[Great Inequality]]''. It was first described by [[Pierre-Simon Laplace|Laplace]] in a series of papers published 1784–1789.
}}
| 12.8° || 800 y || 13%
|-
| 1:7 || [[Uranus]]–[[Jupiter]] || 31.1° || 500 y || 18%
|-
| 7:20 || [[Uranus]]–[[Saturn]] || 5.7° || 20,000 y || 20%
|-
| 5:28 || [[Neptune]]–[[Saturn]] || 1.9° || 80,000 y || 5.2%
|-
| 1:2 || [[Neptune]]–[[Uranus]] || 14.0° || 2000 y || 7.8%
|-
!colspan="5"| {{big|''Mars' satellite  system''}}
|-
| 1:4 || [[Deimos (moon)|Deimos]]–[[Phobos (moon)|Phobos]]{{efn|
Resonances with a now-vanished inner moon are likely to have been involved in the formation of Phobos and Deimos.<ref name=Rosenblatt2016>
{{cite journal |last1=Rosenblatt |first1=P. |last2=Charnoz |first2=S. |last3=Dunseath |first3=K.M. |last4=Terao-Dunseath |first4=M. |last5=Trinh |first5=A. |last6=Hyodo |first6=R. |last7=Genda |first7=H. |last8=Toupin |first8=S. |display-authors=6 |date=4 July 2016 |title=Accretion of Phobos and Deimos in an extended debris disc stirred by transient moons |journal=Nature Geoscience |doi=10.1038/ngeo2742 |volume=9 |issue=8 |pages=581–583 |bibcode=2016NatGe...9..581R|s2cid=133174714 |url=https://hal.archives-ouvertes.fr/hal-01350105/file/Letter.pdf }}
</ref>
}}
| 14.9° || 0.04 y || 8.3%
|-
!colspan="5"| {{big|''Major asteroids' resonances''}}
|-
| 1:1 || [[2 Pallas|Pallas]]–[[Ceres (dwarf planet)|Ceres]]<ref name=Goffin2001>
{{cite journal |last=Goffin |first=E. |year=2001 |title=New determination of the mass of Pallas |journal=[[Astronomy and Astrophysics]] |volume=365 |issue=3 |pages=627–630 |bibcode=2001A&A...365..627G |doi=10.1051/0004-6361:20000023 |doi-access=free }}
</ref><ref name=Kovacevic>
{{cite journal |last=Kovacevic |first=A.B. |year=2012 |title=Determination of the mass of Ceres based on the most gravitationally efficient close encounters |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=419 |issue=3 |pages=2725–2736 |arxiv=1109.6455 |bibcode=2012MNRAS.419.2725K |doi=10.1111/j.1365-2966.2011.19919.x|doi-access=free }}
</ref>
| 0.7° || 1000 y || 0.39%{{efn|
Based on the [[Proper orbital elements|proper orbital periods]], 1684.869 and 1681.601 days, for Pallas and Ceres, respectively.
}}
|-
| 7:18 || [[Jupiter]]–[[2 Pallas|Pallas]]<ref name=Taylor1982>
{{cite journal |last=Taylor |first=D. B. |year=1982 |title=The secular motion of Pallas |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=199 |issue=2 |pages=255–265 |bibcode=1982MNRAS.199..255T |doi=10.1093/mnras/199.2.255|doi-access=free }}
</ref>
| 0.10° || 100,000 y || 0.4%{{efn|
Based on the [[proper orbital elements|"proper" orbital period]] of Pallas, 1684.869&nbsp;days, and 4332.59&nbsp;days for Jupiter.
}}
|-
!colspan="5"| {{big|''[[87 Sylvia]]'s satellite  system''}}{{efn|
[[87 Sylvia]] is the first asteroid discovered to have more than one moon.
}}
|-
| 17:45 || [[Romulus (moon)|Romulus]]–[[Remus (moon)|Remus]] || 0.7° || 40 y || 6.7%
|-
!colspan="5"| {{big|''Jupiter's satellite  system''}}
|-
| 1:6 || [[Io (moon)|Io]]–[[Metis (moon)|Metis]] || 0.6° || 2 y || 0.31%
|-
| 3:5 || [[Amalthea (moon)|Amalthea]]–[[Adrastea (moon)|Adrastea]] ||3.9° ||0.2 y || 6.4%
|-
| 3:7 || [[Callisto (moon)|Callisto]]–[[Ganymede (moon)|Ganymede]]<ref name=Goldreich_1965>
{{cite journal |last=Goldreich |first=P. |author-link=Peter Goldreich |year=1965 |title=An explanation of the frequent occurrence of commensurable mean motions in the solar system |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=130 |issue=3 |pages=159–181 |bibcode=1965MNRAS.130..159G |doi=10.1093/mnras/130.3.159|doi-access=free }}
</ref>
| 0.7° || 30 y || 1.2%
|-
!colspan="5"| {{big|''Saturn's satellite  system''}}
|-
| 2:3 || [[Enceladus]]–[[Mimas (moon)|Mimas]] || 33.2° || 0.04 y || 33%
|-
| 2:3 || [[Dione (moon)|Dione]]–[[Tethys (moon)|Tethys]]{{efn|
This resonance may have been occupied in the past.<ref name=Chen2008/>
}}
| 36.2° || 0.07 y || 36%
|-
| 3:5 || [[Rhea (moon)|Rhea]]–[[Dione (moon)|Dione]] || 17.1° || 0.4 y || 26%
|-
| 2:7 || [[Titan (moon)|Titan]]–[[Rhea (moon)|Rhea]] || 21.0° || 0.7 y || 22%
|-
| 1:5 || [[Iapetus (moon)|Iapetus]]–[[Titan (moon)|Titan]] || 9.2° || 4 y || 5.1%
|-
!colspan="5"| {{big|''Major [[centaur (minor planet)|centaurs]]' resonances''}}{{efn|
Some [[Centaur (minor planet)#Classification|definitions of centaurs]] require that they not be resonant.
}}
|-
| 3:4 || [[Uranus]]–[[10199 Chariklo|Chariklo]] || 4.5° || 10,000 y || 7.3%
|-
!colspan="5"| {{big|''Uranus' satellite  system''}}
|-
| 3:5 || [[Rosalind (moon)|Rosalind]]–[[Cordelia (moon)|Cordelia]]<ref name=Murray_1990>
{{cite journal |last1=Murray |first1=C.D. |last2=Thompson |first2=R.P. |year=1990 |title=Orbits of shepherd satellites deduced from the structure of the rings of Uranus |journal=[[Nature (journal)|Nature]] |volume=348 |issue=6301 |pages=499–502 |bibcode=1990Natur.348..499M |doi=10.1038/348499a0 |s2cid=4320268 }}
</ref>
| 0.22° || 4 y || 0.37%
|-
| 1:3 || [[Umbriel]]–[[Miranda (moon)|Miranda]]{{efn|
This resonance may have been occupied in the past.<ref name="Tittemore Wisdom 1990"/>
}}
| 24.5° || 0.08 y || 14%
|-
| 3:5 || [[Umbriel]]–[[Ariel (moon)|Ariel]]{{efn|
This resonance may have been occupied in the past.<ref name=Tittemore1988/>
}}
| 24.2° || 0.3 y || 35%
|-
| 1:2 || [[Titania (moon)|Titania]]–[[Umbriel]] || 36.3° || 0.1 y || 20%
|-
| 2:3 || [[Oberon (moon)|Oberon]]–[[Titania (moon)|Titania]] || 33.4° || 0.4 y || 34%
|-
!colspan="5"| {{big|''Neptune's satellite  system''}}
|-
| 1:20 || [[Triton (moon)|Triton]]–[[Naiad (moon)|Naiad]] || 13.5° || 0.2 y || 7.5%
|-
| 1:2 || [[Proteus (moon)|Proteus]]–[[Larissa (moon)|Larissa]]<ref name=ZhangHamilton2007>
{{cite journal |last1=Zhang |first1=K. |last2=Hamilton |first2=D.P. |year=2007 |title=Orbital resonances in the inner Neptunian system: I. The 2:1 Proteus–Larissa mean-motion resonance |journal=[[Icarus (journal)|Icarus]] |volume=188 |issue=2 |pages=386–399 |bibcode=2007Icar..188..386Z |doi=10.1016/j.icarus.2006.12.002}}
</ref><ref name=ZhangHamilton2008>
{{cite journal |last1=Zhang |first1=K. |last2=Hamilton |first2=D.P. |year=2008 |title=Orbital resonances in the inner Neptunian system: II. Resonant history of Proteus, Larissa, Galatea, and Despina |journal=[[Icarus (journal)|Icarus]] |volume=193 |issue=1 |pages=267–282 |bibcode=2008Icar..193..267Z |doi=10.1016/j.icarus.2007.08.024}}
</ref>
| 8.4° || 0.07 y || 4.7%
|-
| 5:6 || [[Proteus (moon)|Proteus]]–[[Hippocamp (moon)|Hippocamp]] || 2.1° || 1 y || 5.7%
|-
!colspan="5"| {{big|''Pluto's satellite  system''}}
|-
| 1:3 || [[Styx (moon)|Styx]]–[[Charon (moon)|Charon]]<ref name=Matson>
{{cite news |last=Matson |first=J. |date=11 July 2012 |title=New moon for Pluto: Hubble Telescope spots a 5th Plutonian satellite |magazine=[[Scientific American]] |url=http://www.scientificamerican.com/article.cfm?id=pluto-moon-p5 |access-date=12 July 2012}}
</ref>
| 58.5° || 0.2 y || 33%
|-
| 1:4 || [[Nix (moon)|Nix]]–[[Charon (moon)|Charon]]<ref name=Matson/><ref name=WardCanup2006>
{{cite journal |last1=Ward |first1=W.R. |last2=Canup |first2=R.M. |author2-link=Robin Canup |year=2006 |title=Forced resonant migration of Pluto's outer satellites by Charon |journal=[[Science (journal)|Science]] |volume=313 |issue=5790 |pages=1107–1109 |bibcode=2006Sci...313.1107W |doi=10.1126/science.1127293 |pmid=16825533 |s2cid=36703085 }}
</ref>
| 39.1° || 0.3 y || 22%
|-
| 1:5 || [[Kerberos (moon)|Kerberos]]–[[Charon (moon)|Charon]]<ref name=Matson/>
| 9.2° || 2 y || 5%
|-
| 1:6 || [[Hydra (moon)|Hydra]]–[[Charon (moon)|Charon]]<ref name=Matson/><ref name=WardCanup2006/>
| 6.6° || 3 y || 3.7%
|-
!colspan="5"| {{big|''Haumea's satellite system''}}
|-
| 3:8 || [[Hiʻiaka (moon)|Hiʻiaka]]–[[Namaka (moon)|Namaka]]{{efn|
The results for the Haumea system aren't very meaningful because, contrary to the assumptions implicit in the calculations, Namaka has an eccentric, [[Osculating orbit|non-Keplerian]] orbit that precesses rapidly (see below). Hiʻiaka and Namaka are much closer to a 3:8 resonance than indicated, and may actually be in it.<ref name="Ragozzine&Brown2009">
{{cite journal |last1=Ragozzine |first1=D. |last2=Brown |first2=M.E. |year=2009 |title=Orbits and masses of the satellites of the dwarf planet Haumea {{=}} 2003&nbsp;EL{{sub|61}} |journal=[[The Astronomical Journal]] |volume=137 |issue=6 |pages=4766–4776 |arxiv=0903.4213 |bibcode=2009AJ....137.4766R |doi=10.1088/0004-6256/137/6/4766 |s2cid=15310444 }}
</ref>
}}
| 42.5° || 2 y || 55%
|}

{{notelist}}

The least probable orbital correlation in the list – meaning the relationship that seems most likely to have not just be by random chance – 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 [[Nice model|computer model]] by [[Alessandro Morbidelli (astronomer)|Alessandro Morbidelli]] of the [[Côte d'Azur Observatory|Observatoire de la Côte d'Azur]] in [[Nice]] suggested the formation of a 1:2 resonance between Jupiter and Saturn due to interactions with [[planetesimal]]s that caused them to migrate inward and outward, respectively. In the model, this 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&nbsp;million years after the Solar System's formation and the origin of Jupiter's [[Trojan asteroid]]s.<ref>{{cite web |last=Hansen |first=K. |date=7 June 2004 |title=Orbital shuffle for early solar system |url=http://www.geotimes.org/june05/WebExtra060705.html |work=[[Geotimes]] |access-date=26 August 2007}}</ref> 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]] system of [[Ithaca Chasma]] on Tethys.<ref name="Chen2008">{{cite conference |last1=Chen |first1=E. M. A. |last2=Nimmo |first2=F. |year=2008 |title=Thermal and Orbital Evolution of Tethys as Constrained by Surface Observations |url=http://www.lpi.usra.edu/meetings/lpsc2008/pdf/1968.pdf |book-title=Lunar and Planetary Science XXXIX |publisher=[[Lunar and Planetary Institute]] |id=#1968 |access-date=14 March 2008}}</ref>

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 acceleration|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 [[Oblate spheroid|oblateness]], 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]] or [[inclination]].

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.<ref name="Tittemore1988">{{cite journal |last1=Tittemore |first1=W. C. |last2=Wisdom |first2=J. |year=1988 |title=Tidal Evolution of the Uranian Satellites I. Passage of Ariel and Umbriel through the 5:3 Mean-Motion Commensurability |journal=[[Icarus (journal)|Icarus]] |volume=74 |issue=2 |pages=172–230 |bibcode=1988Icar...74..172T |doi=10.1016/0019-1035(88)90038-3|hdl=1721.1/57632 |hdl-access=free }}</ref><ref name="Tittemore Wisdom 1990">{{cite journal |last1=Tittemore |first1=W. C. |last2=Wisdom |first2=J. |year=1990 |title=Tidal evolution of the Uranian satellites: III. Evolution through the Miranda-Umbriel 3:1, Miranda-Ariel 5:3, and Ariel-Umbriel 2:1 mean-motion commensurabilities |journal=[[Icarus (journal)|Icarus]] |volume=85 |issue=2 |pages=394–443 |bibcode=1990Icar...85..394T |doi=10.1016/0019-1035(90)90125-S |hdl=1721.1/57632 |hdl-access=free }}</ref> 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,<ref name="Tittemore 1990">{{cite journal |last=Tittemore |first=W. C. |year=1990 |title=Tidal heating of Ariel |journal=[[Icarus (journal)|Icarus]] |volume=87 |issue=1 |pages=110–139 |bibcode=1990Icar...87..110T |doi=10.1016/0019-1035(90)90024-4 }}</ref> 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 moon|regular]] Uranian moons (see [[Uranus' natural satellites]]).<ref name="Tittemore1989">{{cite journal |last1=Tittemore |first1=W. C. |last2=Wisdom |first2=J. |year=1989 |title=Tidal Evolution of the Uranian Satellites II. An Explanation of the Anomalously High Orbital Inclination of Miranda |journal=[[Icarus (journal)|Icarus]] |volume=78 |issue=1 |pages=63–89 |bibcode=1989Icar...78...63T |doi=10.1016/0019-1035(89)90070-5 |url=http://dspace.mit.edu/bitstream/1721.1/57632/2/19834233-MIT.pdf|hdl=1721.1/57632 |hdl-access=free }}</ref><ref>{{cite journal |last1=Malhotra |first1=R. |last2=Dermott |first2=S. F |year=1990 |title=The Role of Secondary Resonances in the Orbital History of Miranda |journal=[[Icarus (journal)|Icarus]] |volume=85 |issue=2 |pages=444–480 |bibcode=1990Icar...85..444M |doi=10.1016/0019-1035(90)90126-T|doi-access=free }}</ref>

Similar to the case of Miranda, the present inclinations of Jupiter's moonlets Amalthea and [[Thebe (moon)|Thebe]] are thought to be indications of past passage through the 3:1 and 4:2 resonances with Io, respectively.<ref name="Burns2004">{{cite book |last1=Burns |first1=J. A. |last2=Simonelli |first2=D. P. |last3=Showalter |first3=M. R. |last4=Hamilton |first4=D. P. |last5=Porco |first5=Carolyn C. |last6=Esposito |first6=L. W. |last7=Throop |first7=H. |year=2004 |chapter=Jupiter's Ring-Moon System |chapter-url=http://www.astro.umd.edu/~hamilton/research/preprints/BurSimSho03.pdf |editor=Bagenal, Fran |editor2=Dowling, Timothy E. |editor3=McKinnon, William B. |title=Jupiter: The Planet, Satellites and Magnetosphere |url=http://google.com/books?id=aMERHqj9ivcC&printsec=frontcover |publisher=[[Cambridge University Press]] |isbn=978-0-521-03545-3}}</ref>

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.<ref name="ZhangHamilton2007" /><ref name="ZhangHamilton2008" />

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). 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 [[dwarf planet]] [[Haumea (dwarf planet)|Haumea]], [[Namaka (moon)|Namaka]], is one tenth the mass of the larger outer moon, [[Hiʻiaka (moon)|Hiʻiaka]]. Namaka revolves around Haumea in 18 days in an eccentric, [[Osculating orbit|non-Keplerian]] orbit, and as of 2008 is inclined 13° from Hiʻiaka.<ref name="Ragozzine&Brown2009" /> 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 [[Perturbation (astronomy)|perturbed]], with a current precession of about −6.5° per year.<ref name="Ragozzine&Brown2009" />

== See also ==
{{Div col|colwidth=20em|small=no}}
*[[1685 Toro]], an asteroid in 5:8 resonance with the Earth
*[[3753 Cruithne]], an asteroid in 1:1 resonance with the Earth
*[[Arnold tongue]]
*[[Commensurability (astronomy)]]
*[[Dermott's law]]
*[[Horseshoe orbit]], followed by an object in another type of 1:1 resonance
*[[Kozai mechanism|Kozai resonance]]
*[[Lagrange point]]s
*[[Mercury (planet)#Spin-orbit resonance|Mercury]], which has a 3:2 spin-orbit resonance
*[[Musica universalis]] ("music of the spheres")
*[[Resonant interaction]]
*[[Resonant trans-Neptunian object]]
*[[Tidal locking]]
*[[Tidal resonance]]
*[[Titius–Bode law]]
*[[Transfer operator]]
*[[Trojan (celestial body)]], a body in a type of 1:1 resonance
*[[Venus#Orbit and rotation|Venus]], whose Earth conjunction period (584 Earth days) is close to 5 times its [[solar day]] (116.75 days)
{{Div col end}}

== Notes ==
{{NoteFoot}}

== References ==
{{Reflist|refs=

<ref name="Nesvorny1998">{{cite journal
  |first1 = D. |last1 = Nesvorný 
  |first2 = A. |last2 = Morbidelli
  |title = Three-Body Mean Motion Resonances and the Chaotic Structure of the Asteroid Belt
  |journal = The Astronomical Journal
  |date = December 1998
  |volume = 116
  |issue = 6
  |pages = 3029–3037
  |doi-access = free
  |doi = 10.1086/300632
  |bibcode = 1998AJ....116.3029N
  |s2cid = 51779089}}</ref>

}}

* {{cite book |first1=C. D. |last1=Murray |first2=S. F. |last2=Dermott |date=1999 |title=Solar System Dynamics |publisher=Cambridge University Press |isbn=978-0-521-57597-3}}
* {{cite journal |first1=Renu |last1=Malhotra |first2=Matthew |last2=Holman |first3=Takashi |last3=Ito |title=Orbital Resonances and Chaos in the Solar System |journal= Proceedings of the National Academy of Sciences of the United States of America|volume=98 |issue=22 |pages=12342–12343 |date=23 October 2001 |doi=10.1073/pnas.231384098 |doi-access=free |pmid=11606772 |pmc=60054 }}
* {{cite journal |first=Renu |last=Malhotra |title=The Origin of Pluto's Orbit: Implications for the Solar System Beyond Neptune |journal=The Astronomical Journal |volume=110 |date=1995 |pages=420 |arxiv=astro-ph/9504036 |doi=10.1086/117532 |bibcode=1995AJ....110..420M|s2cid=10622344 }}
*{{Cite book |last=Lemaître |first=A. |author-link= Anne Lemaître |title=Dynamics of Small Solar System Bodies and Exoplanets |editor-last=Souchay |editor-first=J. |editor2-last=Dvorak |editor2-first=R. |publisher=[[Springer Science+Business Media|Springer]] |date=2010 |volume=790 |series=Lecture Notes in Physics |chapter=Resonances: Models and Captures |pages=1–62 |doi=10.1007/978-3-642-04458-8 |isbn=978-3-642-04457-1|chapter-url=https://cds.cern.ch/record/1339552 }}

== External links ==
* {{Commons category-inline|Orbital resonance}}
{{Portal bar|Astronomy|Stars|Spaceflight|Outer space|Solar System}}
[[Category:Orbital resonance| ]]