Editing Orbital resonance (section)

==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>