Editing Orbital resonance (section)

== 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
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 |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.}}