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{{for|the science fiction novel|Orbital Resonance (novel)}}
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In [[celestial mechanics]], an '''orbital resonance''' occurs when two [[orbit]]ing bodies exert a regular, periodic [[gravitational]] influence on each other, usually due to their [[orbital period]]s being related by a ratio of two small [[integer]]s. The physics principle behind orbital resonance is similar in concept to pushing a child on a swing, where the orbit and the swing both have a [[natural frequency]], and the other body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies, i.e., their ability to alter or constrain each other's orbits.  In most cases, this results in an ''unstable'' interaction, in which the bodies exchange [[momentum]] and shift orbits until the resonance no longer exists.  Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance.  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 [[Pluto]] and [[Neptune]].  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 [[Solar 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]].
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Except as noted in the Laplace resonance figure (below), a 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). The 2:3 ratio above means Pluto completes two orbits in the time it takes Neptune to complete three.
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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|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.
 
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It was Laplace who found the first answers explaining the remarkable dance of the [[Galilean moon]]s (see below). It is fair to say that this general field of study has remained very active since then, with plenty more yet to be understood (e.g., how interactions of moonlets with particles of the rings of giant planets result in maintaining the rings).
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[[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).]]
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[[Image: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]].]]
[[Image: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]].]]
[[Image: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 doi| 10.1126/science.241.4866.690}}</ref><ref name="Chakrabarti2001">{{Cite doi| 10.1046/j.1365-8711.2001.04813.x}}</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
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*involve one or any combination of the orbit parameters (e.g. [[eccentricity (orbit)|eccentricity]] versus [[semimajor axis]], or eccentricity versus orbital inclination).
 
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A '''mean-motion orbital resonance''' occurs when two bodies have periods of [[orbit|revolution]] that are a simple integer ratio of each other. Depending on the details, this can either stabilize or destabilize the orbit.
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''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)|perturbation]]s 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]], [[279 Thule]], and the [[Trojan asteroid]]s, respectively).


Orbital resonances can also ''destabilize'' one of the orbits. For small bodies, destabilization is actually far more likely. For instance:
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*In the [[asteroid belt]] within 3.5 AU from the Sun, the major mean-motion resonances with [[Jupiter]] are locations of ''gaps'' in the asteroid distribution, the [[Kirkwood gap]]s (most notably at the 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]].)
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*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]].
 
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Most bodies that are in resonance orbit in the same direction; however, a few [[Retrograde motion|retrograde]] [[Damocloid asteroid|damocloids]] have been found that are temporarily captured in mean-motion resonance with [[Jupiter]] or [[Saturn]].<ref name = "Morais_2013">{{cite journal
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  | last = Morais  | first = M. H. M. | coauthors = Namouni, F.
  | title =  Asteroids in retrograde resonance with Jupiter and Saturn
  | journal = Monthly Notices of the Royal Astronomical Society Letters (in press)
  | arxiv = arXiv:1308.0216 | url = http://arxiv.org/abs/1308.0216}}</ref> Such orbital interactions are weaker than the corresponding interactions between bodies orbiting in the same direction.<ref name = "Morais_2013"/>


A '''Laplace resonance''' occurs when three or more orbiting bodies have a simple integer ratio between their orbital periods. For example, Jupiter's moons [[Ganymede (moon)|Ganymede]], [[Europa (moon)|Europa]] and [[Io (moon)|Io]] are in a 1:2:4 orbital resonance. The [[extrasolar planet]]s [[Gliese 876]] e, b and c are also in a 1:2:4 orbital resonance.<ref name="rivera2010"/>
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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]]).
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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.
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Several prominent examples of secular resonance involve Saturn. A 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) has been identified as the likely source of Saturn's large [[axial tilt]] (26.7°).<ref>{{cite web
== 1 Nike Jordan 3 ==
  | last = Beatty
  | first = J. K.
  | authorlink =
  | coauthors =
  | title = Why Is Saturn Tipsy?
  | work = SkyAndTelescope.Com
  | publisher =
  | date = 2003-07-23
  | url = http://www.skyandtelescope.com/news/3306806.html?page=1&c=y
  | doi =
  | accessdate = 2009-02-25}}</ref><ref>{{cite journal
  | last = Ward
  | first = W. R.
  | authorlink =
  | coauthors = Hamilton, D. P.
  | title = Tilting Saturn. I. Analytic Model
  | journal = [[Astronomical Journal]]
  | volume = 128
  | issue = 5
  | pages = 2501–2509
  | publisher = [[American Astronomical Society]]
  | date = November 2004
  | url = http://www.iop.org/EJ/abstract/1538-3881/128/5/2501/
  | doi = 10.1086/424533
  | accessdate = 2009-02-25
| bibcode=2004AJ....128.2501W}}</ref><ref>{{cite journal
  | last = Hamilton
  | first = D. P.
  | authorlink =
  | coauthors = Ward, W. R.
  | title = Tilting Saturn. II. Numerical Model
  | journal = [[Astronomical Journal]]
  | volume = 128
  | issue = 5
  | pages = 2510–2517
  | date = November 2004
  | url = http://www.iop.org/EJ/abstract/1538-3881/128/5/2510
  | doi = 10.1086/424534
  | accessdate = 2009-02-25
| bibcode=2004AJ....128.2510H}}</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 the 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 spin, and thus dominates the interaction.)


The [[secular resonance#ν6 resonance|perihelion secular resonance]] between [[asteroid]]s and [[Saturn]] (''ν<sub>6</sub>'' = ''g -g<sub>6</sub>'') helps shape the asteroid belt. 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 [[inclination]]s of about 20°.
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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
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| author= Laskar, J. | date= 2008-03-18
| title= Chaotic diffusion in the Solar System
| journal= [[Icarus (journal)|Icarus]] | volume=196 | issue=1 |pages =1–15
| bibcode=  2008Icar..196....1L
| doi= 10.1016/j.icarus.2008.02.017|arxiv = 0802.3371 }}</ref><ref name = "Laskar2009">{{cite journal
| author= Laskar, J.; Gastineau, M. | date= 2009-06-11
| title= Existence of collisional trajectories of Mercury, Mars and Venus with the Earth
| journal= [[Nature (journal)|Nature]] | volume=459 | issue=7248 | pages=817–819
| doi=10.1038/nature08096 | bibcode=2009Natur.459..817L | pmid=19516336 }}</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|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
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| last = Porco | first = C. | authorlink = Carolyn Porco
 
| coauthors = [[Phil Nicholson|Nicholson, P. D.]]; Borderies, N.; Danielson, G. E.; [[Peter Goldreich|Goldreich, P.]]; Holdberg, J. B.; Lane, A. L.
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| title = The eccentric Saturnian ringlets at 1.29R<sub>s</sub> and 1.45R<sub>s</sub>
 
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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]].
 
 
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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|last=Musotto|first=S.|coauthors=Varadi, F.; Moore, W.; Schubert, G.|title=Numerical Simulations of the Orbits of the Galilean Satellites|year=2002|volume=159|issue=2|pages=500–504|doi=10.1006/icar.2002.6939| bibcode=2002Icar..159..500M | journal = [[Icarus (journal)|Icarus]]}}</ref>
 
 
  </ul>
==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.]]
 
[[Image:Galilean moon Laplace resonance animation.gif|thumb|right|365px|The Laplace resonance exhibited by three of the [[Galilean moon]]s. The ratios in the figure are of ''orbital periods''.]]
 
There are only a few known mean-motion resonances in the [[Solar System]] involving planets, [[dwarf planet]]s or larger [[natural satellite|satellite]]s (a much greater number involve [[asteroid]]s, [[planetary ring]]s, [[Inner satellite|moonlet]]s and smaller [[Kuiper belt]] objects, including many [[possible dwarf planets]]).
* 2:3 [[Pluto]]–[[Neptune]]
* 2:4 [[Tethys (moon)|Tethys]]–[[Mimas (moon)|Mimas]] (Saturn’s moons)
* 1:2 [[Dione (moon)|Dione]]–[[Enceladus (moon)|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).
 
Additionally, [[Haumea (dwarf planet)|Haumea]] is believed to be in a 7:12 resonance with Neptune,<ref name = "Brown_2007">{{cite journal
| last = Brown | first = M. E. | authorlink = Michael E. Brown
| coauthors = Barkume, K. M.; Ragozzine, D.; Schaller, E. L.
| title = A collisional family of icy objects in the Kuiper belt
| journal = [[Nature (journal)|Nature]]
| volume = 446 | issue = 7133| pages = 294–296
| publisher = | date = 2007-03-15 | url = http://www.nature.com/nature/journal/v446/n7133/abs/nature05619.html
| doi = 10.1038/nature05619 | pmid = 17361177 | bibcode = 2007Natur.446..294B
| accessdate = 2012-02-18}}</ref><ref name = "Ragozzine">{{cite journal
| last = Ragozzine | first = D. | authorlink = | coauthors = Brown, M. E.
| 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
| publisher =  | date = 2007-10-18 | url = http://iopscience.iop.org/1538-3881/134/6/2160
| doi = 10.1086/522334 | bibcode = 2007AJ....134.2160R | arxiv = 0709.0328
  | accessdate = 2012-02-18}}</ref> and [[Eris (dwarf planet)|Eris]] and [[Makemake (dwarf planet)|Makemake]] may be in 5:17 and 6:11 resonances with Neptune, respectively.<ref name = "10th Planet">{{cite web
| last = Dunn | first = T. | authorlink =
| title = The 10th Planet
| work = [http://www.orbitsimulator.com/gravity/articles/what.html GravitySimulator.Com]
| publisher = | date =
| url = http://www.orbitsimulator.com/gravity/articles/newtno.html
| accessdate = 2012-02-18}}</ref>
 
The ''simple integer ratios'' between periods are a convenient simplification hiding 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|node]]s or [[perihelion|periapside]]s 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 '''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.
 
===The Laplace resonance===
[[Image:TheLaplaceResonance2.png|right|thumb|300px|Illustration of Io-Europa-Ganymede resonance. From the centre outwards: Io (yellow), Europa (gray) and Ganymede (dark)]]
The most remarkable resonance involving Io-Europa-Ganymede includes the following relation locking the ''orbital phase'' of the moons:
 
:<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.
This relation makes a triple conjunction impossible. The graph illustrates the positions of the moons after 1, 2 and 3 Io periods. (The Laplace resonance in the [[Gliese 876]] system, in contrast, is associated with one triple conjunction per orbit of the outermost planet.<ref name="rivera2010"/>)
 
===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
|title = Pluto's Orbit
|author = [[Renu Malhotra]]
|url = http://www.nineplanets.org/plutodyn.html
|year = 1997
|accessdate = 2007-03-26
}}</ref> (see [[Pluto#Orbit and rotation|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
  |date=2009-03-23 |title=S/2005 (90482) 1 needs your help
  |publisher=Mike Brown's Planets (blog) |author=Michael E. Brown
  |url=http://www.mikebrownsplanets.com/2009/03/s1-90482-2005-needs-your-help.html
  |accessdate=2009-03-25
  |authorlink=Michael E. Brown}}</ref>
 
==Mean-motion resonances among extrasolar planets==
While most [[extrasolar planet]]ary systems discovered have not been found to have planets in mean-motion resonances, some remarkable examples have been uncovered:
*As mentioned above, [[Gliese 876]] e, b and c are in a 1:2:4 orbital resonance, with periods of 124.3, 61.1 and 30.0 days.<ref name="rivera2010">{{cite arXiv
| eprint=1006.4244 | version=v1
| title=The Lick-Carnegie Exoplanet Survey: A Uranus-mass Fourth Planet for GJ 876 in an Extrasolar Laplace Configuration
| last1=Rivera | first1=Eugenio J. | last2=Laughlin | first2=Gregory
| last3=Butler | first3=R. Paul | last4=Vogt | first4=Steven S.
| last5=Haghighipour | first5=Nader | last6=Meschiari | first6=Stefano
| month=June | year=2010 | class=astro-ph.EP}}</ref><ref name = "Marcy_2001">{{cite journal | url= | last=Marcy | first = Geoffrey W. | title=A Pair of Resonant Planets Orbiting GJ 876 | journal=[[The Astrophysical Journal]] | volume=556 | issue=1 | pages=296–301| year=2001 | doi=10.1086/321552 | last2=Butler | first2=R. Paul | last3=Fischer | first3=Debra | last4=Vogt | first4=Steven S. | last5=Lissauer | first5=Jack J. | last6=Rivera | first6=Eugenio J. | bibcode=2001ApJ...556..296M}}</ref>
*[[KOI-730]] d, b, c and e appear to be in a 3:4:6:8 resonance, with periods of 19.72, 14.79, 9.85 and 7.38 days.<ref name=EPE-KOI730>Extrasolar Planets Encyclopaedia, [http://exoplanet.eu/star.php?st=KOI-730 KOI-730]</ref><ref name = "Beatty">{{cite web
| last = Beatty | first = Kelley
| title = Kepler Finds Planets in Tight Dance
| work = | publisher = [[Sky and Telescope]]
| date = 2011-03-05
| url = http://www.skyandtelescope.com/news/117467488.html
| accessdate = 2012-10-16}}</ref><ref name = "Lissauer_2011">{{cite journal
| last =  Lissauer | first = J. | authorlink = Jack J. Lissauer
| coauthors = Ragozzine, D.;  Fabrycky, D. C.; ''et al.''
| 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
| date = 2011-10-13 | url = http://iopscience.iop.org/0067-0049/197/1/8
| doi = 10.1088/0067-0049/197/1/8 | accessdate = 2012-10-16|arxiv = 1102.0543 |bibcode = 2011ApJS..197....8L }}</ref>
*[[KOI-500]] c, b, e, d and f appear to be in or close to a 20:27:41:62:193 resonance, with periods of 9.522, 7.053, 4.645, 3.072 and 0.9868 days.<ref name = "Lissauer_2011"/><ref name=EPE-KOI500>Extrasolar Planets Encyclopaedia, [http://exoplanet.eu/star.php?st=KOI-500 KOI-500]</ref><ref name = "Choi_2012">{{cite web
| last = Choi | first = Charles Q.
| title = Tiniest Alien Solar System Discovered: 5 Packed Planets
| work = [http://www.space.com Space.Com web site] | publisher = TechMediaNetwork.com | date = 2012-10-15
| url = http://www.space.com/18073-tiny-alien-solar-system-exoplanets.html
| accessdate = 2012-10-16}}</ref>
*Both [[KOI-738]] and [[KOI-787]] appear to have pairs of planets in a 7:9 resonance (ratios of 1/1.285871 and 1/1.284008, respectively).<ref name = "Lissauer_2011"/>
*[[Kepler-37]] d, c and b are within one percent of a 5:8:15 resonance, with periods of 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.|title=A sub-Mercury-sized exoplanet|url=http://www.nature.com/nature/journal/vaop/ncurrent/full/nature11914.html| journal=Nature|date=2013-02-20|issn=0028-0836|doi=10.1038/nature11914|accessdate=2013-02-21|arxiv = 1305.5587 |bibcode = 2013Natur.494..452B }}</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
| last = Wright | first = J. T.
| coauthors = Fakhouri, O.; Marcy, G. W.; Han, E.; Feng, Y.; Johnson, J. A.; Howard, A. W.; Fischer, D. A.; Valenti, J. A.; Anderson, J.; Piskunov, N.
| title = The Exoplanet Orbit Database
| journal = [[Publications of the Astronomical Society of the Pacific]]
| volume = 123 | issue = 902 | pages = 412–42 | date = April 2011
| url = http://www.jstor.org/stable/info/10.1086/659427
| doi = 10.1086/659427 | accessdate = 2012-11-07|arxiv = 1012.5676 |bibcode = 2011PASP..123..412W }}</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
| last = Terquem | first = C. | coauthors = Papaloizou, J. C. B.
| title = Migration and the Formation of Systems of Hot Super-Earths and Neptunes
| journal = [[The Astrophysical Journal]]
| volume = 654 | issue = 2 | pages = 1110–1120 | date = 2007-01-10
| url = http://iopscience.iop.org/0004-637X/654/2/1110/
| doi = 10.1086/509497 | accessdate = 2012-11-08|arxiv = astro-ph/0609779 |bibcode = 2007ApJ...654.1110T }}</ref>
 
==Coincidental 'near' ratios of mean motion==
[[File:PallasJupiter.GIF|300px|thumb|right|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|right|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 remarkable 3:4:5:6 sequence of near resonances relative to the period of its large inner satellite [[Charon (moon)|Charon]].]]
 
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:
 
{| align="center" class="wikitable"
|-
! (Ratio) and Bodies !! Mismatch after one cycle{{refn | Mismatch in orbital longitude of the inner body, as compared to its position at the beginning of the cycle (with the cycle defined as ''n'' orbits of the outer body – see below). Circular orbits are assumed (i.e., precession is ignored). | group = lower-alpha }}!! Randomization time{{refn | The time needed for the mismatch from the initial relative longitudinal orbital positions of the bodies to grow to 180°, rounded to the nearest first [[significant digit]].| group = lower-alpha }}!! Probability{{refn | The [[probability]] of obtaining an orbital coincidence of equal or smaller mismatch by chance at least once in ''n'' attempts, where ''n'' is the integer number of orbits of the outer body per cycle, and the mismatch is assumed to vary between 0° and 180° at random. The value is calculated as 1- (1- mismatch/180°)^''n''. The smaller the probability, the more remarkable the coincidence. This is a crude calculation that only attempts to give a rough idea of relative probabilities.| group = lower-alpha }}
|-
!colspan="4" | ''Planets''
|-
| (9:23) [[Venus]]−[[Mercury (planet)|Mercury]] || 4.0° || 200 [[year|y]] || 0.19
|-
| (8:13) [[Earth]]−[[Venus]]<ref name = "Langford"/><ref name = "Bazsó">{{cite journal
| last = Bazsó | first = A. | authorlink =
| coauthors =Eybl, V.; Dvorak, R.; Pilat-Lohinger, E.; Lhotka, C.
| 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
| publisher = [[Springer Science+Business Media|Springer]] | date = 2010-04-07 | url = http://www.springerlink.com/content/tp6n6652976308m8/
| doi = 10.1007/s10569-010-9266-6 | arxiv = 0911.2357v1 | accessdate = 2012-03-01|bibcode = 2010CeMDA.107...63B }}</ref>{{refn | 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 years apart, in a cycle that repeats every 243 years.<ref name = "Langford">{{cite web | last = Langford | first = Peter M.  | authorlink =
| title = Transits of Venus
| work = [http://www.astronomy.org.gg/ La Société Guernesiaise Astronomy Section web site]
| publisher = Astronomical Society of the Channel Island of Guernsey
| date = September 1998  | url = http://www.astronomy.org.gg/venustransitsb.htm
| accessdate = 2012-03-01}}</ref><ref name = "Shortt">{{cite web |last=Shortt |first=David |title=Some Details About Transits of Venus |work=Planetary Society web site |publisher=[[The Planetary Society]] |date=22 May 2012 |url=http://www.planetary.org/blogs/guest-blogs/Some-Details-About-Transits-of-Venus.html |accessdate=22 May 2012}}</ref> | group = lower-alpha }} || 1.5° || 1000 [[year|y]] || 0.065
|-
| (243:395) [[Earth]]−[[Venus]]<ref name = "Langford"/><ref name = "Shortt"/> || 0.8° || 50,000 [[year|y]] || 0.68
|-
| (1:3) [[Mars]]−[[Venus]] || 20.6° || 20 y || 0.11
|-
| (1:2) [[Mars]]−[[Earth]] || 42.9° || 8 y || 0.24
|-
| (1:12) [[Jupiter]]−[[Earth]]{{refn | The near 1:12 resonance between Jupiter and Earth causes 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. | group = lower-alpha }} || 49.1° || 40 y || 0.28
|-
| (2:5) [[Saturn]]–[[Jupiter]]{{refn | This near resonance has been termed the ''[[Great Inequality]]''. It was first described by [[Pierre-Simon Laplace|Laplace]] in a series of papers published 1784–1789. | group = lower-alpha }}|| 12.8° || 800 y || 0.13
|-
| (1:7) [[Uranus]]−[[Jupiter]] || 31.1° || 500 y || 0.18
|-
| (7:20) [[Uranus]]−[[Saturn]] || 5.7° || 20,000 y || 0.20
|-
| (5:28) [[Neptune]]−[[Saturn]] || 1.9° || 80,000 y || 0.052
|-
| (1:2) [[Neptune]]−[[Uranus]] || 14.0° || 2000 y || 0.078
|-
!colspan="4" | ''Mars system''
|-
| (1:4) [[Deimos (moon)|Deimos]]−[[Phobos (moon)|Phobos]] || 14.9° || 0.04 y || 0.083
|-
!colspan="4" | ''Major asteroids''
|-
| (1:1) [[2 Pallas|Pallas]] − [[Ceres (dwarf planet)|Ceres]]<ref name="Goffin2001">{{cite journal
  |last=Goffin |first=E.
  |title=New determination of the mass of Pallas |journal=Astronomy and Astrophysics |year=2001 |volume=365
  |issue=3 |pages=627–630
  |bibcode=2001A&A...365..627G
  |doi=10.1051/0004-6361:20000023 }}</ref><ref name="Kovačević">{{cite journal
  |last= Kovačević |first= A. B.
  |title= Determination of the mass of Ceres based on the most gravitationally efficient close encounters
  |journal= [[Monthly Notices of the Royal Astronomical Society]] |date= 2011-12-05
  | volume=419 | issue=3 | pages=2725–2736
  | bibcode=2012MNRAS.419.2725K
  | doi=10.1111/j.1365-2966.2011.19919.x |arxiv = 1109.6455 }}</ref> || 1.2° || 700 y || 0.0066
|-
| (7:18) [[Jupiter]] − [[2 Pallas|Pallas]]<ref name=Taylor1982>{{cite journal
  |last=Taylor |first=D. B.
  |title=The secular motion of Pallas
  |journal=Royal Astronomical Society
  |volume=199 |pages=255–265 |year=1982
  |bibcode=1982MNRAS.199..255T }}</ref> || 4.1° || 4000 y || 0.15
|-
!colspan="4" | ''[[87 Sylvia]] system''{{refn | [[87 Sylvia]] is the first asteroid discovered to have more than one moon. | group = lower-alpha }}
|-
| (17:45) [[Romulus (moon)|Romulus]]−[[Remus (moon)|Remus]] || 0.7° || 40 y || 0.067
|-
!colspan="4" | ''Jupiter system''
|-
| (1:6) [[Io (moon)|Io]]−[[Metis (moon)|Metis]] || 0.6° || 2 y || 0.0031
|-
| (3:5) [[Amalthea (moon)|Amalthea]]−[[Adrastea (moon)|Adrastea]] || 3.9° || 0.2 y || 0.064
|-
| (3:7) [[Callisto (moon)|Callisto]]−[[Ganymede (moon)|Ganymede]]<ref name = "Goldreich_1965">{{cite journal
| last = Goldreich | first = P. | authorlink = Peter Goldreich
| 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
| year = 1965
| url = http://adsabs.harvard.edu/abs/1965MNRAS.130..159G
| accessdate = 2012-11-07|bibcode = 1965MNRAS.130..159G }}</ref> || 0.7° || 30 y || 0.012
|-
!colspan="4" | ''Saturn system''
|-
| (2:3) [[Enceladus (moon)|Enceladus]]−[[Mimas (moon)|Mimas]] || 33.2° || 0.04 y || 0.33
|-
| (2:3) [[Dione (moon)|Dione]]−[[Tethys (moon)|Tethys]]{{refn | This resonance may have been occupied in the past.<ref name = "Chen2008"/> | group = lower-alpha }} || 36.2° || 0.07 y || 0.36
|-
| (3:5) [[Rhea (moon)|Rhea]]−[[Dione (moon)|Dione]] || 17.1° || 0.4 y || 0.26
|-
| (2:7) [[Titan (moon)|Titan]]−[[Rhea (moon)|Rhea]] || 21.0° || 0.7 y || 0.22
|-
| (1:5) [[Iapetus (moon)|Iapetus]]−[[Titan (moon)|Titan]] || 9.2° || 4 y || 0.051
|-
!colspan="4" | ''Major [[Centaur (minor planet)|centaurs]]''{{refn | Some [[Centaur (minor planet)#Classification|definitions of centaurs]] stipulate that they are nonresonant bodies. | group = lower-alpha }}
|-
| (3:4) [[Uranus]]−[[10199 Chariklo|Chariklo]] || 4.5° || 10,000 y || 0.073
|-
!colspan="4" | ''Uranus system''
|-
| (3:5) [[Rosalind (moon)|Rosalind]]−[[Cordelia (moon)|Cordelia]]<ref name = "Murray_1990">{{cite journal
| last = Murray | first = C. D.  | authorlink = | coauthors = Thompson, R. P.
| 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
| publisher = | date = 1990-12-06 | url = http://www.nature.com/nature/journal/v348/n6301/abs/348499a0.html
| doi = 10.1038/348499a0
| accessdate = 2012-02-20|bibcode = 1990Natur.348..499M }}</ref> || 0.22° || 4 y || 0.0037
|-
| (1:3) [[Umbriel (moon)|Umbriel]]−[[Miranda (moon)|Miranda]]{{refn | This resonance may have been occupied in the past.<ref name = "Tittemore Wisdom 1990"/> | group = lower-alpha }} || 24.5° || 0.08 y || 0.14
|-
| (3:5) [[Umbriel (moon)|Umbriel]]−[[Ariel (moon)|Ariel]]{{refn | This resonance may have been occupied in the past.<ref name = "Tittemore1988"/> | group = lower-alpha }} || 24.2° || 0.3 y || 0.35
|-
| (1:2) [[Titania (moon)|Titania]]−[[Umbriel (moon)|Umbriel]] || 36.3° || 0.1 y || 0.20
|-
| (2:3) [[Oberon (moon)|Oberon]]−[[Titania (moon)|Titania]] || 33.4° || 0.4 y || 0.34
|-
!colspan="4" | ''Neptune system''
|-
| (1:20) [[Triton (moon)|Triton]]−[[Naiad (moon)|Naiad]] || 13.5° || 0.2 y || 0.075
|-
| (1:2) [[Proteus (moon)|Proteus]]−[[Larissa (moon)|Larissa]]<ref name="ZhangHamilton2007">{{cite journal|last1=Zhang|first1=K.|last2=Hamilton|first2=D. P.|title=Orbital resonances in the inner neptunian system: I. The 2:1 Proteus–Larissa mean-motion resonance|journal=Icarus|volume= 188|issue=2|date=June 2007|pages= 386–399|issn= 00191035|doi= 10.1016/j.icarus.2006.12.002|bibcode = 2007Icar..188..386Z }}</ref><ref name="ZhangHamilton2008">{{cite journal|last1=Zhang|first1= K.|last2=Hamilton|first2=D. P.|title=Orbital resonances in the inner neptunian system: II. Resonant history of Proteus, Larissa, Galatea, and Despina|journal=Icarus|volume= 193|issue=1|date= January 2008|pages=267–282|issn= 00191035|doi= 10.1016/j.icarus.2007.08.024|bibcode = 2008Icar..193..267Z }}</ref> || 8.4° || 0.07 y || 0.047
|-
| (5:6) [[Proteus (moon)|Proteus]]−[[S/2004 N 1]] || 2.1° || 1 y || 0.057
|-
!colspan="4" | ''Pluto system''
|-
| (1:3) [[Styx (moon)|Styx]]−[[Charon (moon)|Charon]]<ref name="Matson">{{cite web
| last = Matson | first = J. | authorlink = | date = 11 July 2012
| title = New Moon for Pluto: Hubble Telescope Spots a 5th Plutonian Satellite
| work = [[Scientific American]] web site | publisher =
| url = http://www.scientificamerican.com/article.cfm?id=pluto-moon-p5 | accessdate = 12 July 2012
}}</ref> || 58.5° || 0.2 y || 0.33
|-
| (1:4) [[Nix (moon)|Nix]]−[[Charon (moon)|Charon]]<ref name="Matson"/><ref name="WardCanup2006">{{cite journal
| first1 = William R.| last1 = Ward| first2 = Robin M.| last2 = Canup
| year = 2006 | title = Forced Resonant Migration of Pluto's Outer Satellites by Charon
| journal = Science | volume = 313 | issue = 5790 | pages = 1107–1109
| doi = 10.1126/science.1127293 | pmid = 16825533 | bibcode = 2006Sci...313.1107W}}</ref> || 39.1° || 0.3 y || 0.22
|-
| (1:5) [[Kerberos (moon)|Kerberos]]−[[Charon (moon)|Charon]]<ref name="Matson"/> || 9.2° || 2 y || 0.05
|-
| (1:6) [[Hydra (moon)|Hydra]]−[[Charon (moon)|Charon]]<ref name="Matson"/><ref name="WardCanup2006"/> || 6.6° || 3 y || 0.037
|-
!colspan="4" | ''Haumea system''
|-
| (3:8) [[Hi'iaka (moon)|Hi{{okina}}iaka]]−[[Namaka (moon)|Namaka]]{{refn | 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
|last=Ragozzine |first=D. |coauthors=Brown, M.E.
|year=2009 |title=Orbits and Masses of the Satellites of the Dwarf Planet Haumea = 2003 EL61
|journal=The Astronomical Journal
|arxiv=0903.4213 |bibcode = 2009AJ....137.4766R
|doi = 10.1088/0004-6256/137/6/4766
|volume=137 |issue=6 |pages=4766–4776
}}</ref> | group = lower-alpha }} || 42.5° || 2 y || 0.55
|-
|}
{{reflist
| group = lower-alpha
}}
 
The most remarkable (least probable) orbital correlation in the list is that between Io and Metis, followed by those between Rosalind and Cordelia, Pallas and Ceres, 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 that the formation of a 1:2 resonance between Jupiter and Saturn (due to interactions with [[planetesimals]] that caused them to migrate inward and outward, respectively) created a gravitational push that propelled both Uranus and Neptune into higher orbits, and in some scenarios caused them to switch places, which would have doubled Neptune's distance from the Sun. The resultant expulsion of objects from the proto-Kuiper belt as Neptune moved outwards could explain the [[Late Heavy Bombardment]] 600&nbsp;million years after the Solar System's formation and the origin of Jupiter's [[Trojan asteroid]]s.<ref>{{cite web
| title=Orbital shuffle for early solar system
| first=Kathryn | last= Hansen | publisher=Geotimes
| url=http://www.geotimes.org/june05/WebExtra060705.html
| date=June 7, 2005 | accessdate=2007-08-26}}</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 acceleration#Tidal heating|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
  | first = E. M. A. | last = Chen | authorlink =
  | coauthors = Nimmo, F.
  | title = Thermal and Orbital Evolution of Tethys as Constrained by Surface Observations
  | booktitle = Lunar and Planetary Science XXXIX (2008)
  | pages = | publisher =
  | date = March 2008 | location =
  | url = http://www.lpi.usra.edu/meetings/lpsc2008/pdf/1968.pdf
  |format=PDF| doi = | accessdate = 2008-03-14}}</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 Uranus System, however, due to the planet's lesser degree of [[Oblate spheroid|oblate]]ness, 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 [[Orbital eccentricity|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
| title=Tidal Evolution of the Uranian Satellites I. Passage of Ariel and Umbriel through the 5:3 Mean-Motion Commensurability
| author= Tittemore, W. C., Wisdom, J.
| journal= [[Icarus (journal)|Icarus]] | volume=74 | issue=2 | pages=172–230 | year=1988
|doi= 10.1016/0019-1035(88)90038-3
| bibcode=1988Icar...74..172T
}}</ref><ref name = "Tittemore Wisdom 1990">{{cite doi | 10.1016/0019-1035(90)90125-S }}</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 acceleration#Tidal heating|tidal heating]] of the interiors of Miranda and Ariel,<ref name="Tittemore 1990">
{{cite 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
| title=Tidal Evolution of the Uranian Satellites II. An Explanation of the Anomalously High Orbital Inclination of Miranda
| author= Tittemore, W. C., Wisdom, J.
| journal=[[Icarus (journal)|Icarus]] | volume=78 | issue=1 | pages=63–89 | year=1989
|doi= 10.1016/0019-1035(89)90070-5
| bibcode=1989Icar...78...63T}}</ref><ref>{{cite journal
| title=The Role of Secondary Resonances in the Orbital History of Miranda
| author= Malhotra, R., Dermott, S. F.
| journal= [[Icarus (journal)|Icarus]] | volume=85 | issue=2 | pages=444–480 | year=1990
| doi= 10.1016/0019-1035(90)90126-T | bibcode=1990Icar...85..444M}}</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
| last = Burns | first = J. A.
| coauthors = Simonelli, D. P.; Showalter, M. R.; Hamilton, D. P.; Porco, C. C.; Esposito, L. W.; Throop, H.
| chapter = Jupiter’s Ring-Moon System
| title = Jupiter: The Planet, Satellites and Magnetosphere
| year = 2004 | publisher = [[Cambridge University Press]]
| editors = Bagenal, F.; Dowling, T. E.; McKinnon, W. B.
| url = http://google.com/books?id=aMERHqj9ivcC&printsec=frontcover
| chapterurl = http://www.astro.umd.edu/~hamilton/research/preprints/BurSimSho03.pdf
}}</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, [[Hi'iaka (moon)|Hi{{okina}}iaka]]. Namaka revolves around Haumea in 18 days in an eccentric, [[Osculating orbit|non-Keplerian]] orbit, and as of 2008 is inclined 13° from Hi{{okina}}iaka.<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 Hi{{okina}}iaka, 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==
*[[1685 Toro]], an asteroid in 5:8 resonance with the Earth
*[[3753 Cruithne]], an asteroid in 1:1 resonance with the Earth
*[[Commensurability (astronomy)]]
*[[Dermott's Law]]
*[[Horseshoe orbit]], followed by an object in another type of 1:1 resonance
*[[Kozai mechanism|Kozai resonance]]
*[[Lagrangian point]]s
*[[Mercury (planet)#Spin–orbit resonance|Mercury]], which has a 3:2 spin–orbit resonance
*[[Musica universalis]] ("music of the spheres")
*[[Resonant trans-Neptunian object]]
*[[Tidal locking]]
*[[Tidal resonance]]
*[[Titius–Bode law]]
*[[Trojan (astronomy)|Trojan object]], a body in a type of 1:1 resonance
 
==References==
{{reflist|2}}
 
* C. D. Murray, S. F. Dermott (1999). ''Solar System Dynamics'', Cambridge University Press, ISBN 0-521-57597-4.
*Renu Malhotra ''Orbital Resonances and Chaos in the Solar System''. In ''Solar System Formation and Evolution'', ASP Conference Series, '''149''' (1998) [http://www.lpl.arizona.edu/people/faculty/malhotra_preprints/rio97.pdf preprint].
*Renu Malhotra, ''The Origin of Pluto's Orbit: Implications for the Solar System Beyond Neptune'', The Astronomical Journal, '''110''' (1995), p.&nbsp;420 [http://arxiv.org/abs/astro-ph/9504036 Preprint].
*{{Cite book
  | last = Lemaître | first = A. | author-link =
  | title = Dynamics of Small Solar System Bodies and Exoplanets
  | editor-last = Souchay | editor-first = J.
  | editor2-last = Dvorak | editor2-first = R.
  | place =  | publisher = [[Springer Science+Business Media|Springer]]
  | year = 2010 | month= | volume = 790
  | series = Lecture Notes in Physics | chapter = Resonances: Models and Captures
  | chapterurl = | pages = 1–62 | url =
  | archiveurl = | archivedate = | doi = 10.1007/978-3-642-04458-8
  | isbn = 978-3-642-04457-1}}
 
==External links==
* [http://www.alpheratz.net/murison/asteroids/resonances/ Locations of Solar System Planetary Mean-Motion Resonances]. Web calculator that plots distributions of the semimajor axes (or in one case the perihelion distances) of the minor planets in relation to mean-motion resonances of the planets (website maintained by M.A. Murison).
 
{{DEFAULTSORT:Orbital Resonance}}
[[Category:Orbits]]

Latest revision as of 04:47, 1 January 2015

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