Editing Orbital resonance (section)

== 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.