Editing Orbital resonance (section)

== Mean-motion resonances among extrasolar planets ==
[[File:Resonant planetary system.gif|thumb|Resonant planetary system of two planets with a 1:2 orbit ratio]]
While most [[extrasolar planet]]ary systems discovered have not been found to have planets in mean-motion resonances, chains of up to five resonant planets<ref name="Shale2017" /> and up to seven at least near resonant planets<ref name="Luger_etal_2017" /> have been uncovered. Simulations have shown that during [[Nebular hypothesis|planetary system formation]], the appearance of resonant chains of planetary embryos is favored by the presence of the [[Protoplanetary disk|primordial gas disc]]. Once that gas dissipates, 90–95% of those chains must then become unstable to match the low frequency of resonant chains observed.<ref name="Izidoro2017">{{cite journal |last1=Izidoro |first1=A. |last2=Ogihara |first2=M. |last3=Raymond |first3=S. N. |last4=Morbidelli |first4=A. |last5=Pierens |first5=A. |last6=Bitsch |first6=B. |last7=Cossou |first7=C. |last8=Hersant |first8=F. |title=Breaking the chains: hot super-Earth systems from migration and disruption of compact resonant chains |journal=Monthly Notices of the Royal Astronomical Society |volume=470 |issue=2 |year=2017 |pages=1750–1770 |doi=10.1093/mnras/stx1232|doi-access=free |arxiv=1703.03634 |bibcode=2017MNRAS.470.1750I |s2cid=119493483 }}</ref>
*As mentioned above, [[Gliese 876]] e, b and c are in a Laplace resonance, with a 4:2:1 ratio of periods (124.3, 61.1 and 30.0 days).<ref name="rivera2010">{{cite journal |last1=Rivera |first1=E. J. |last2=Laughlin |first2=G. |last3=Butler |first3=R. P. |last4=Vogt |first4=S. S. |last5=Haghighipour |first5=N. |last6=Meschiari |first6=S. |year=2010 |title=The Lick-Carnegie Exoplanet Survey: A Uranus-mass Fourth Planet for GJ 876 in an Extrasolar Laplace Configuration |journal=[[The Astrophysical Journal]] |volume=719 |issue=1 |pages=890–899 |arxiv=1006.4244 |bibcode=2010ApJ...719..890R |doi=10.1088/0004-637X/719/1/890|s2cid=118707953 }}</ref><ref name="Laughlin2013">{{cite web |last=Laughlin |first=G. |date=23 June 2010 |title=A second Laplace resonance |url=http://oklo.org/2010/06/23/a-second-laplace-resonance/ |work=Systemic: Characterizing Planets |access-date=30 June 2015 |archive-url=https://web.archive.org/web/20131229124449/http://oklo.org/2010/06/23/a-second-laplace-resonance/ |archive-date=29 December 2013}}</ref><ref name="Marcy_2001">{{cite journal |last1=Marcy |first1=Ge. W. |last2=Butler |first2=R. P. |last3=Fischer |first3=D. |last4=Vogt |first4=S. S. |last5=Lissauer |first5=J. J. |last6=Rivera |first6=E. J. |year=2001 |title=A Pair of Resonant Planets Orbiting GJ 876 |journal=[[The Astrophysical Journal]] |volume=556 |issue=1 |pages=296–301 |bibcode=2001ApJ...556..296M |doi=10.1086/321552|doi-access=free }}</ref> In this case, <math>\Phi_L</math> librates with an amplitude of 40° ± 13° and the resonance follows the time-averaged relation:<ref name="rivera2010" />
:<math>\Phi_L=\lambda_{\rm c} - 3\cdot\lambda_{\rm d} + 2\cdot\lambda_{\rm e}=0^\circ</math>
*[[Kepler-223]] has four planets in a resonance with an 8:6:4:3 orbit ratio, and a 3:4:6:8 ratio of periods (7.3845, 9.8456, 14.7887 and 19.7257 days).<ref name=EPE-KOI730>{{cite encyclopedia |title=Planet Kepler-223 b |url=http://exoplanet.eu/catalog/kepler-223_b/ |access-date=21 January 2018 |archive-date=22 January 2018 |archive-url=https://web.archive.org/web/20180122072529/http://exoplanet.eu/catalog/kepler-223_b/ |encyclopedia=[[Extrasolar Planets Encyclopaedia]] |url-status=dead }}</ref><ref name="Beatty">{{cite web |last=Beatty |first=K. |date=5 March 2011 |title=Kepler Finds Planets in Tight Dance |url=http://www.skyandtelescope.com/astronomy-news/kepler-finds-planets-in-tight-dance/ |work=[[Sky and Telescope]] |access-date=16 October 2012}}</ref><ref name="Lissauer_2011">{{cite journal |last1=Lissauer |first1=J. J. |author1-link=Jack J. Lissauer |last2=Ragozzine |first2=D. |last3=Fabrycky |first3=D. C. |last4=Steffen |first4=J. H. |last5=Ford |first5=E. B. |last6=Jenkins |first6=J. M. |last7=Shporer |first7=A. |last8=Holman |first8=M. J. |last9=Rowe |first9=J. F. |last10=Quintana |first10=E. V. |last11=Batalha |first11=N. M. |last12=Borucki |first12=W. J. |last13=Bryson |first13=S. T. |last14=Caldwell |first14=D. A. |last15=Carter |first15=J. A. |last16=Ciardi |first16=D. |last17=Dunham |first17=E. W. |last18=Fortney |first18=J. J. |last19=Gautier, III |first19=T. N. |last20=Howell |first20=S. B. |last21=Koch |first21=D. G. |last22=Latham |first22=D. W. |last23=Marcy |first23=G. W. |last24=Morehead |first24=R. C. |last25=Sasselov |first25=D. |display-authors=1 |year=2011 |title=Architecture and dynamics of Kepler's candidate multiple transiting planet systems |journal=[[The Astrophysical Journal Supplement Series]] |volume=197 |issue=1 |pages=1–26 |arxiv=1102.0543 |bibcode=2011ApJS..197....8L |doi=10.1088/0067-0049/197/1/8|s2cid=43095783 }}</ref><ref name="Mills2016">{{cite journal |last1=Mills |first1=S. M. |last2=Fabrycky |first2=D. C. |last3=Migaszewski |first3=C. |last4=Ford |first4=E. B. |last5=Petigura |first5=E. |last6=Isaacson |first6=H. |title=A resonant chain of four transiting, sub-Neptune planets |journal=Nature |date=11 May 2016 |doi=10.1038/nature17445 |volume=533 |issue=7604 |pages=509–512 |pmid=27225123 |arxiv=1612.07376 |bibcode=2016Natur.533..509M |s2cid=205248546 }}</ref> This represents the first confirmed 4-body orbital resonance.<ref name="Koppes2016">{{cite web |url=http://www.jpl.nasa.gov/news/news.php?feature=6515 |title=Kepler-223 System: Clues to Planetary Migration |last=Koppes |first=S. |date=17 May 2016 |website=[[Jet Propulsion Lab]] |access-date=18 May 2016}}</ref> The librations within this system are such that close encounters between two planets occur only when the other planets are in distant parts of their orbits. Simulations indicate that this system of resonances must have formed via [[planetary migration]].<ref name="Mills2016" />
*[[Kepler-80]] d, e, b, c and g have periods in a ~ 1.000: 1.512: 2.296: 3.100: 4.767 ratio (3.0722, 4.6449, 7.0525, 9.5236 and 14.6456 days). However, in a frame of reference that rotates with the conjunctions, this reduces to a period ratio of 4:6:9:12:18 (an orbit ratio of 9:6:4:3:2). Conjunctions of d and e, e and b, b and c, and c and g occur at relative intervals of 2:3:6:6 (9.07, 13.61 and 27.21 days) in a pattern that repeats about every 190.5 days (seven full cycles in the rotating frame) in the inertial or nonrotating frame (equivalent to a 62:41:27:20:13 orbit ratio resonance in the nonrotating frame, because the conjunctions circulate in the direction opposite orbital motion). Librations of possible three-body resonances have amplitudes of only about 3 degrees, and modeling indicates the resonant system is stable to perturbations. Triple conjunctions do not occur.<ref name="MacDonald2016">{{Cite journal |last1=MacDonald |first1=M. G. |last2=Ragozzine |first2=D. |last3=Fabrycky |first3=D. C. |last4=Ford |first4=E. B. |last5=Holman |first5=M. J. |last6=Isaacson |first6=H. T. |last7=Lissauer |first7=J. J. |last8=Lopez |first8=E. D. |last9=Mazeh |first9=T. |date=1 January 2016 |title=A Dynamical Analysis of the Kepler-80 System of Five Transiting Planets |journal=The Astronomical Journal |volume=152 |issue=4 |pages=105 |doi=10.3847/0004-6256/152/4/105 |arxiv=1607.07540 |bibcode=2016AJ....152..105M|s2cid=119265122 |doi-access=free }}</ref><ref name="Shale2017">{{cite journal |last1=Shale |first1=C. J. |last2=Vanderburg |first2=A. |title=Identifying Exoplanets With Deep Learning: A Five Planet Resonant Chain Around Kepler-80 And An Eighth Planet Around Kepler-90 |journal=[[The Astrophysical Journal]] |volume= 155|issue= 2|pages= 94|date=2017 |url=https://www.cfa.harvard.edu/~avanderb/kepler90i.pdf |doi=10.3847/1538-3881/aa9e09 |access-date=15 December 2017 |arxiv=1712.05044 |bibcode=2018AJ....155...94S|s2cid=4535051 |doi-access=free }}</ref>
*[[TOI-178]] has 6 confirmed planets, of which the outer 5 planets form a similar resonant chain in a rotating frame of reference, which can be expressed as 2:4:6:9:12 in period ratios, or as 18:9:6:4:3 in orbit ratios. In addition, the innermost planet b with period of 1.91d orbits close to where it would also be part of the same Laplace resonance chain, as a 3:5 resonance with the planet c would be fulfilled at period of ~1.95d, implying that it might have evolved there but pulled out of resonance, possibly by tidal forces.<ref>{{Cite journal|last1=Leleu |first1=A. |last2=Alibert |first2=Y. |last3=Hara |first3=N. C. |last4=Hooton |first4=M. J. |last5=Wilson |first5=T. G. |last6=Robutel |first6=P. |last7=Delisle |first7=J. -B. |last8=Laskar |first8=J. |last9=Hoyer |first9=S. |last10=Lovis |first10=C. |last11=Bryant |first11=E. M. |last12=Ducrot |first12=E. |last13=Cabrera |first13=J. |last14=Delrez |first14=L. |last15=Acton |first15=J. S. |last16=Adibekyan |first16=V. |last17=Allart |first17=R. |last18=Prieto |first18=Allende |last19=Alonso |first19=R. |last20=Alves |first20=D. |last21=Anderson |first21=D. R. |last22=Angerhausen |first22=D. |last23=Escudé |first23=Anglada |last24=Asquier |first24=J. |last25=Barrado |first25=D. |last26=Barros |first26=S. C. C. |last27=Baumjohann |first27=W. |last28=Bayliss |first28=D. |last29=Beck |first29=M. |last30=Beck |first30=T. |last31=Bekkelien |first31=A. |last32=Benz |first32=W. |last33=Billot |first33=N. |last34=Bonfanti |first34=A. |last35=Bonfils |first35=X. |last36=Bouchy |first36=F. |last37=Bourrier |first37=V. |last38=Boué |first38=G. |last39=Brandeker |first39=A. |last40=Broeg |first40=C. |last41=Buder |first41=M. |last42=Burdanov |first42=A. |last43=Burleigh |first43=M. R. |last44=Bárczy |first44=T. |last45=Cameron |first45=A. C. |last46=Chamberlain |first46=S. |last47=Charnoz |first47=S. |last48=Cooke |first48=B. F. |last49=Damme |first49=Corral Van |last50=Correia |first50=A. C. M. |last51=Cristiani |first51=S. |last52=Damasso |first52=M. |last53=Davies |first53=M. B. |last54=Deleuil |first54=M. |last55=Demangeon |first55=O. D. S. |last56=Demory |first56=B. -O. |last57=Marcantonio |first57=Di |last58=Persio |first58=Di |last59=Dumusque |first59=X. |last60=Ehrenreich |first60=D. |last61=Erikson |first61=A. |last62=Figueira |first62=P. |last63=Fortier |first63=A. |last64=Fossati |first64=L. |last65=Fridlund |first65=M. |last66=Futyan |first66=D. |last67=Gandolfi |first67=D. |last68=Muñoz |first68=García |last69=Garcia |first69=L. J. |last70=Gill |first70=S. |last71=Gillen |first71=E. |last72=Gillon |first72=M. |last73=Goad |first73=M. R. |last74=Hernández |first74=González |last75=I. |first75=J. |last76=Guedel |first76=M. |last77=Günther |first77=M. N. |last78=Haldemann |first78=J. |last79=Henderson |first79=B. |last80=Heng |first80=K. |last81=Hogan |first81=A. E. |last82=Isaak |first82=K. |last83=Jehin |first83=E. |last84=Jenkins |first84=J. S. |last85=Jordán |first85=A. |last86=Kiss |first86=L. |last87=Kristiansen |first87=M. H. |last88=Lam |first88=K. |last89=Lavie |first89=B. |last90=Etangs |first90=Lecavelier des |last91=Lendl |first91=M. |last92=Lillo-Box |first92=J. |last93=Curto |first93=Lo |last94=Magrin |first94=D. |last95=Martins |first95=C. J. A. P. |last96=Maxted |first96=P. F. L. |last97=McCormac |first97=J. |last98=Mehner |first98=A. |last99=Micela |first99=G. |last100=Molaro |first100=P. |last101=Moyano |first101=M. |last102=Murray |first102=C. A. |last103=Nascimbeni |first103=V. |last104=Nunes |first104=N. J. |last105=Olofsson |first105=G. |last106=Osborn |first106=H. P. |last107=Oshagh |first107=M. |last108=Ottensamer |first108=R. |last109=Pagano |first109=I. |last110=Pallé |first110=E. |last111=Pedersen |first111=P. P. |last112=Pepe |first112=F. A. |last113=Persson |first113=C. M. |last114=Peter |first114=G. |last115=Piotto |first115=G. |last116=Polenta |first116=G. |last117=Pollacco |first117=D. |last118=Poretti |first118=E. |last119=Pozuelos |first119=F. J. |last120=Queloz |first120=D. |last121=Ragazzoni |first121=R. |last122=Rando |first122=N. |last123=Ratti |first123=F. |last124=Rauer |first124=H. |last125=Raynard |first125=L. |last126=Rebolo |first126=R. |last127=Reimers |first127=C. |last128=Ribas |first128=I. |last129=Santos |first129=N. C. |last130=Scandariato |first130=G. |last131=Schneider |first131=J. |last132=Sebastian |first132=D. |last133=Sestovic |first133=M. |last134=Simon |first134=A. E. |last135=Smith |first135=A. M. S. |last136=Sousa |first136=S. G. |last137=Sozzetti |first137=A. |last138=Steller |first138=M. |last139=Mascareño |first139=Suárez |last140=Szabó |first140=Gy. M. |last141=Ségransan |first141=D. |last142=Thomas |first142=N. |last143=Thompson |first143=S. |last144=Tilbrook |first144=R. H. |last145=Triaud |first145=A. |last146=Turner |first146=O. |last147=Udry |first147=S. |last148=Grootel |first148=Van |last149=Venus |first149=H. |last150=Verrecchia |first150=F. |last151=Vines |first151=J. I. |last152=Walton |first152=N. A. |last153=West |first153=R. G. |last154=Wheatley |first154=P. J. |last155=Wolter |first155=D. |last156=Osorio |first156=Zapatero |last157=R. |first157=M. |display-authors=20 |date=2021-01-20|title=Six transiting planets and a chain of Laplace resonances in TOI-178|journal=Astronomy & Astrophysics|volume=649 |pages=A26 |language=en|arxiv=2101.09260|doi=10.1051/0004-6361/202039767|bibcode=2021A&A...649A..26L |issn=0004-6361|s2cid=231693292 }}</ref>
* [[TRAPPIST-1]]'s seven approximately Earth-sized planets are in a chain of near resonances (the longest such chain known), having an orbit ratio of approximately 24, 15, 9, 6, 4, 3 and 2, or nearest-neighbor period ratios (proceeding outward) of about 8/5, 5/3, 3/2, 3/2, 4/3 and 3/2 (1.603, 1.672, 1.506, 1.509, 1.342 and 1.519). They are also configured such that each triple of adjacent planets is in a Laplace resonance (i.e., b, c and d in one such Laplace configuration; c, d and e in another, etc.).<ref name="Gillon2016">{{Cite journal |last1=Gillon |first1=M. |last2=Triaud |first2=A. H. M. J. |last3=Demory |first3=B.-O. |last4=Jehin |first4=E. |last5=Agol |first5=E. |last6=Deck |first6=K. M. |last7=Lederer |first7=S. M. |last8=de Wit |first8=J. |last9=Burdanov |first9=A. |title=Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1 |journal=Nature |volume=542 |issue=7642 |pages=456–460 |doi=10.1038/nature21360 |date=22 February 2017 |pmid=28230125 |pmc=5330437 |arxiv=1703.01424 |bibcode=2017Natur.542..456G}}</ref><ref name="Luger_etal_2017">{{cite journal |title=A seven-planet resonant chain in TRAPPIST-1 |first1=R. |last1=Luger |first2=M. |last2=Sestovic |first3=E. |last3=Kruse |first4=S. L. |last4=Grimm |first5=B.-O. |last5=Demory |last6=Agol |first6=E. |last7=Bolmont |first7=E. |last8=Fabrycky |first8=D. |last9=Fernandes |first9=C. S. |last10=Van Grootel|first10= V. |last11=Burgasser |first11=A. |last12=Gillon |first12=M. |last13=Ingalls |first13=J. G. |last14=Jehin |first14=E. |last15=Raymond |first15=S. N. |last16=Selsis |first16=F. |last17=Triaud |first17=A. H. M. J. |last18=Barclay |first18=T. |last19=Barentsen |first19=G. |last20=Delrez |first20=L. |last21=de Wit |first21=J. |last22=Foreman-Mackey |first22=D. |last23=Holdsworth |first23=D. L. |last24=Leconte |first24=J. |last25=Lederer |first25=S. |last26=Turbet |first26=M. |last27=Almleaky |first27=Y. |last28=Benkhaldoun |first28=Z. |last29=Magain |first29=P. |last30=Morris |first30=B. |date=22 May 2017 |journal=Nature Astronomy |volume=1 |issue=6 |pages=0129 |doi=10.1038/s41550-017-0129 |arxiv=1703.04166 |bibcode=2017NatAs...1E.129L|s2cid=54770728 }}</ref> The resonant configuration is expected to be stable on a time scale of billions of years, assuming it arose during planetary migration.<ref name="Tamayo2017">{{cite journal |last1=Tamayo |first1=D. |last2=Rein |first2=H. |last3=Petrovich |first3=C. |last4=Murray |first4=N. |title=Convergent Migration Renders TRAPPIST-1 Long-lived |journal=The Astrophysical Journal |volume=840 |issue=2 |date=10 May 2017 |pages=L19 |doi=10.3847/2041-8213/aa70ea |arxiv=1704.02957 |bibcode=2017ApJ...840L..19T|s2cid=119336960 |doi-access=free }}</ref><ref name="NYT-20170510">{{cite news |url=https://www.nytimes.com/2017/05/10/science/trappist-earth-size-planets-orbits-music.html |title=The Harmony That Keeps Trappist-1's 7 Earth-size Worlds From Colliding |work=[[The New York Times]] |last=Chang |first=K. |date=10 May 2017 |access-date=26 June 2017}}</ref> A musical interpretation of the resonance has been provided.<ref name="NYT-20170510" />
*[[Kepler-29]] has a pair of planets in a 7:9 resonance (ratio of 1/1.28587).<ref name="Lissauer_2011" />
*[[Kepler-36]] has a pair of planets close to a 6:7 resonance.<ref name=Carter2012>{{cite journal |author1=Carter, J. A. |author2=Agol, E. |author3=Chaplin, W. J. |author4=Basu, S. |author5=Bedding, T. R. |author6=Buchhave, L. A. |author7=Christensen-Dalsgaard, J. |author8=Deck, K. M. |author9=Elsworth, Y. |author10=Fabrycky, D. C. |author11=Ford, E. B. |author12=Fortney, J. J. |author13=Hale, S. J. |author14=Handberg, R. |author15=Hekker, S. |author16=Holman, M. J. |author17=Huber, D. |author18=Karoff, C. |author19=Kawaler, S. D. |author20=Kjeldsen, H. |author21=Lissauer, J. J. |author22=Lopez, E. D. |author23=Lund, M. N. |author24=Lundkvist, M. |author25=Metcalfe, T. S. |author26=Miglio, A. |author27=Rogers, L. A. |author28=Stello, D. |author29=Borucki, W. J. |author30=Bryson, S. |author31=Christiansen, J. L. |author32=Cochran, W. D. |author33=Geary, J. C. |author34=Gilliland, R. L. |author35=Haas, M. R. |author36=Hall, J. |author37=Howard, A. W. |author38=Jenkins, J. M. |author39=Klaus, T. |author40=Koch, D. G. |author41=Latham, D. W. |author42=MacQueen, P. J. |author43=Sasselov, D. |author44=Steffen, J. H. |author45=Twicken, J. D. |author46=Winn, J. N. |display-authors=3 |title=Kepler-36: A Pair of Planets with Neighboring Orbits and Dissimilar Densities |journal=Science |date=21 June 2012 |doi=10.1126/science.1223269 |arxiv=1206.4718 |bibcode=2012Sci...337..556C |volume=337 |issue=6094 |pages=556–559 |pmid=22722249|s2cid=40245894 }}</ref>
*[[Kepler-37]] d, c and b are within one percent of a resonance with an 8:15:24 orbit ratio and a 15:8:5 ratio of periods (39.792187, 21.301886 and 13.367308 days).<ref name="BarclayRowe2013">{{cite journal |last1=Barclay |first1=T. |last2=Rowe |first2=J. F. |last3=Lissauer |first3=J. J. |last4=Huber |first4=D. |last5=Fressin |first5=F. |last6=Howell |first6=S. B. |last7=Bryson |first7=S. T. |last8=Chaplin |first8=W. J. |last9=Désert |first9=J.-M. |last10=Lopez |first10=E. D. |last11=Marcy |first11=G. W. |last12=Mullally |first12=F. |last13=Ragozzine |first13=D. |last14=Torres |first14=G. |last15=Adams |first15=E. R. |last16=Agol |first16=E. |last17=Barrado |first17=D. |last18=Basu |first18=S. |last19=Bedding |first19=T. R. |last20=Buchhave |first20=L. A. |last21=Charbonneau |first21=D. |last22=Christiansen |first22=J. L. |last23=Christensen-Dalsgaard |first23=J. |last24=Ciardi |first24=D. |last25=Cochran |first25=W. D. |last26=Dupree |first26=A. K. |last27=Elsworth |first27=Y. |last28=Everett |first28=M. |last29=Fischer |first29=D. A. |last30=Ford |first30=E. B. |last31=Fortney |first31=J. J. |last32=Geary |first32=J. C. |last33=Haas |first33=M. R. |last34=Handberg |first34=R. |last35=Hekker |first35=S. |last36=Henze |first36=C. E. |last37=Horch |first37=E. |last38=Howard |first38=A. W. |last39=Hunter |first39=R. C. |last40=Isaacson |first40=H. |last41=Jenkins |first41=J. M. |last42=Karoff |first42=C. |last43=Kawaler |first43=S. D. |last44=Kjeldsen |first44=H. |last45=Klaus |first45=T. C. |last46=Latham |first46=D. W. |last47=Li |first47=J. |last48=Lillo-Box |first48=J. |last49=Lund |first49=M. N. |last50=Lundkvist |first50=M. |last51=Metcalfe |first51=T. S. |last52=Miglio |first52=A. |last53=Morris |first53=R. L. |last54=Quintana |first54=E. V. |last55=Stello |first55=D. |last56=Smith |first56=J. C. |last57=Still |first57=M. |last58=Thompson |first58=S. E. |display-authors=1 |year=2013 |title=A sub-Mercury-sized exoplanet |journal=[[Nature (journal)|Nature]] |volume=494 |issue=7438 |pages=452–454 |arxiv=1305.5587 |bibcode=2013Natur.494..452B |doi=10.1038/nature11914 |pmid=23426260|s2cid=205232792 }}
:*And {{cite journal |title=Erratum: A sub-Mercury-sized exoplanet |year=2013 |journal=Nature |volume=496 |issue=7444 |pages=252 |bibcode=2013Natur.496..252B |doi=10.1038/nature12067|doi-access=free |last1=Barclay |first1=Thomas |last2=Rowe |first2=Jason F. |last3=Lissauer |first3=Jack J. |last4=Huber |first4=Daniel |last5=Fressin |first5=François |last6=Howell |first6=Steve B. |last7=Bryson |first7=Stephen T. |last8=Chaplin |first8=William J. |last9=Désert |first9=Jean-Michel |last10=Lopez |first10=Eric D. |last11=Marcy |first11=Geoffrey W. |last12=Mullally |first12=Fergal |last13=Ragozzine |first13=Darin |last14=Torres |first14=Guillermo |last15=Adams |first15=Elisabeth R. |last16=Agol |first16=Eric |last17=Barrado |first17=David |last18=Basu |first18=Sarbani |last19=Bedding |first19=Timothy R. |last20=Buchhave |first20=Lars A. |last21=Charbonneau |first21=David |last22=Christiansen |first22=Jessie L. |last23=Christensen-Dalsgaard |first23=Jørgen |last24=Ciardi |first24=David |last25=Cochran |first25=William D. |last26=Dupree |first26=Andrea K. |last27=Elsworth |first27=Yvonne |last28=Everett |first28=Mark |last29=Fischer |first29=Debra A. |last30=Ford |first30=Eric B. |display-authors=1 }}</ref>
*Of [[Kepler-90]]'s eight known planets, the period ratios b:c, c:i and i:d are close to 4:5, 3:5 and 1:4, respectively (4:4.977, 3:4.97 and 1:4.13) and d, e, f, g and h are close to a 2:3:4:7:11 period ratio (2: 3.078: 4.182: 7.051: 11.102; also 7: 11.021).<ref>{{Cite journal |arxiv=1402.6352 |title=Validation of Kepler's Multiple Planet Candidates. II: Refined Statistical Framework and Descriptions of Systems of Special Interest |journal=The Astrophysical Journal |volume=784 |issue=1 |pages=44 |date=25 February 2014 |last1=Lissauer |first1=J. J. |last2=Marcy |first2=G. W. |last3=Bryson |first3=S. T. |last4=Rowe |first4=J. F. |last5=Jontof-Hutter |first5=D. |last6=Agol |first6=E. |last7=Borucki |first7=W. J. |last8=Carter |first8=J. A. |last9=Ford |first9=E. B.|last10= Gilliland|first10= R. L. |last11=Kolbl |first11=R. |last12=Star |first12=K. M. |last13=Steffen |first13=J. H. |last14=Torres |first14=G. |doi=10.1088/0004-637X/784/1/44 |bibcode=2014ApJ...784...44L|s2cid=119108651 }}</ref><ref name="Shale2017" /> f, g and h are also close to a 3:5:8 period ratio (3: 5.058: 7.964).<ref name="Cabrera2013">{{cite journal |last1=Cabrera |first1=J. |last2=Csizmadia |first2=Sz. |last3=Lehmann |first3=H. |last4=Dvorak |first4=R. |last5=Gandolfi |first5=D. |last6=Rauer |first6=H. |last7=Erikson |first7=A. |last8=Dreyer |first8=C. |last9=Eigmüller |first9=Ph.|last10= Hatzes|first10= A. |title=The Planetary System to KIC 11442793: A Compact Analogue to the Solar System |journal=The Astrophysical Journal |volume=781 |issue=1 |date=31 December 2013 |page=18 |doi=10.1088/0004-637X/781/1/18 |arxiv=1310.6248 |bibcode=2014ApJ...781...18C|s2cid=118875825 }}</ref> Relevant to systems like this and that of [[Kepler-36]], calculations suggest that the presence of an outer gas giant planet facilitates the formation of closely packed resonances among inner super-Earths.<ref name="Hands2016">{{cite journal |last1=Hands |first1=T. O. |last2=Alexander |first2=R. D. |title=There might be giants: unseen Jupiter-mass planets as sculptors of tightly packed planetary systems |journal=Monthly Notices of the Royal Astronomical Society |volume=456 |issue=4 |date=13 January 2016 |pages=4121–4127 |doi=10.1093/mnras/stv2897 |doi-access=free |arxiv=1512.02649 |bibcode=2016MNRAS.456.4121H|s2cid=55175754 }}</ref>
*[[HD 41248]] has a pair of [[super-Earth]]s within 0.3% of a 5:7 resonance (ratio of 1/1.39718).<ref name=Jenkins13>{{cite journal |last1=Jenkins |first1=J. S. |last2=Tuomi |first2=M. |last3=Brasser |first3=R. |last4=Ivanyuk |first4=O. |last5=Murgas |first5=F. |year=2013 |title=Two Super-Earths Orbiting the Solar Analog HD 41248 on the Edge of a 7:5 Mean Motion Resonance |journal=[[The Astrophysical Journal]] |volume=771 |issue=1 |page=41 |arxiv=1304.7374 |bibcode=2013ApJ...771...41J |doi=10.1088/0004-637X/771/1/41|s2cid=14827197 }}</ref>
*[[K2-138]] has 5 confirmed planets in an unbroken near-3:2 resonance chain (with periods of 2.353, 3.560, 5.405, 8.261 and 12.758 days). The system was discovered in the [[citizen science]] project Exoplanet Explorers, using K2 data.<ref>{{Cite journal|last1=Christiansen|first1=Jessie L.|last2=Crossfield|first2=Ian J. M.|last3= Barentsen|first3= G.|last4= Lintott|first4=C. J.|last5= Barclay|first5= T.|last6= Simmons|first6=B. D.|last7= Petigura|first7= E.|last8= Schlieder|first8=J. E.|last9= Dressing|first9=C. D.|last10= Vanderburg|first10= A.|last11= Allen|first11= C.|date= 2018-01-11|title=The K2-138 System: A Near-resonant Chain of Five Sub-Neptune Planets Discovered by Citizen Scientists|journal= The Astronomical Journal|volume= 155|issue= 2|pages= 57|arxiv= 1801.03874|doi= 10.3847/1538-3881/aa9be0|bibcode=2018AJ....155...57C|s2cid=52971376 |doi-access=free }}</ref> K2-138 could host [[Co-orbital configuration|co-orbital bodies]] (in a 1:1 mean-motion resonance).<ref name=":0">{{Cite journal|last1=Lopez |first1=T. A. |last2=Barros |first2=S. C. C. |last3=Santerne |first3=A. |last4=Deleuil |first4=M. |last5=Adibekyan |first5=V. |last6=Almenara |first6=J.-M. |last7=Armstrong |first7=D. J. |last8=Brugger |first8=B. |last9=Barrado |first9=D. |last10=Bayliss |first10=D. |last11=Boisse |first11=I. |last12=Bonomo |first12=A. S. |last13=Bouchy |first13=F. |last14=Brown |first14=D. J. A. |last15=Carli |first15=E. |last16=Demangeon |first16=O. |last17=Dumusque |first17=X. |last18=Díaz |first18=R. F. |last19=Faria |first19=J. P. |last20=Figueira |first20=P. |last21=Foxell |first21=E. |last22=Giles |first22=H. |last23=Hébrard |first23=G. |last24=Hojjatpanah |first24=S. |last25=Kirk |first25=J. |last26=Lillo-Box |first26=J. |last27=Lovis |first27=C. |last28=Mousis |first28=O. |last29=da Nóbrega |first29=H. J. |last30=Nielsen |first30=L. D. |last31=Neal |first31=J. J. |last32=Osborn |first32=H. P. |last33=Pepe |first33=F. |last34=Pollacco |first34=D. |last35=Santos |first35=N. C. |last36=Sousa |first36=S. G. |last37=Udry |first37=S. |last38=Vigan |first38=A. |last39=Wheatley |first39=P. J.|date= 2019-11-01|title= Exoplanet characterisation in the longest known resonant chain: the K2-138 system seen by HARPS|journal=Astronomy & Astrophysics|language=en|volume=631|pages=A90|doi=10.1051/0004-6361/201936267|bibcode=2019A&A...631A..90L |arxiv=1909.13527 |s2cid=203593804 }}</ref> Resonant chain systems can stabilize co-orbital bodies<ref>{{Cite journal|last1=Leleu|first1=Adrien|last2=Coleman|first2=Gavin A. L.|last3= Ataiee|first3= S.|date= 2019-11-01|title= Stability of the co-orbital resonance under dissipation – Application to its evolution in protoplanetary discs|journal=Astronomy & Astrophysics|language=en|volume=631|pages=A6|arxiv=1901.07640|doi=10.1051/0004-6361/201834486|bibcode=2019A&A...631A...6L|s2cid=219840769}}</ref> and a dedicated analysis of the K2 light curve and radial-velocity from [[High Accuracy Radial Velocity Planet Searcher|HARPS]] might reveal them.<ref name=":0" /> Follow-up observations with the [[Spitzer Space Telescope]] suggest a sixth planet continuing the 3:2 resonance chain, while leaving two gaps in the chain (its period is 41.97 days). These gaps could be filled by smaller non-transiting planets.<ref>{{Cite web|url=https://www.jpl.nasa.gov/spaceimages/details.php?id=PIA23003|title=K2-138 System Diagram|website=jpl.nasa.gov|access-date=2019-11-20}}</ref><ref>{{Cite journal|last1= Hardegree-Ullman|first1= K.|last2= Christiansen|first2= J.|date= January 2019|title=K2-138 g: Spitzer Spots a Sixth Sub-Neptune for the Citizen Science System|journal= American Astronomical Society Meeting Abstracts #233|language=en|volume=233|pages=164.07|bibcode= 2019AAS...23316407H}}</ref> Future observations with [[CHEOPS]] will measure [[transit-timing variation]]s of the system to further analyse the mass of the planets and could potentially find other planetary bodies in the system.<ref>{{Cite web|url=https://www.cosmos.esa.int/web/cheops-guest-observers-programme/ao-1-programmes|title=AO-1 Programmes – CHEOPS Guest Observers Programme – Cosmos |website=cosmos.esa.int|access-date=2019-11-20}}</ref>
*[[K2-32]] has four planets in a near 1:2:5:7 resonance (with periods of 4.34, 8.99, 20.66 and 31.71 days). Planet e has a radius almost identical to that of the Earth. The other planets have a size between Neptune and Saturn.<ref>{{Cite journal|last1=Heller|first1=René|last2=Rodenbeck|first2=Kai|last3=Hippke|first3=Michael|date=2019-05-01|title=Transit least-squares survey – I. Discovery and validation of an Earth-sized planet in the four-planet system K2-32 near the 1:2:5:7 resonance|journal=Astronomy & Astrophysics|language=en|volume=625|pages=A31|doi=10.1051/0004-6361/201935276|issn=0004-6361|bibcode=2019A&A...625A..31H|arxiv=1904.00651|s2cid=90259349}}</ref>
*[[V1298 Tauri]] has four confirmed planets of which planets c, d and b are near a 1:2:3 resonance (with periods of 8.25, 12.40 and 24.14 days). Planet e only shows a single transit in the K2 light curve and has a period larger than 36 days. Planet e might be in a low-order resonance (of 2:3, 3:5, 1:2, or 1:3) with planet b. The system is very young (23±4 [[Myr]]) and might be a precursor of a compact multiplanet system. The 2:3 resonance suggests that some close-in planets may either form in resonances or evolve into them on timescales of less than 10 Myr. The planets in the system have a size between Neptune and Saturn. Only planet b has a size similar to Jupiter.<ref>{{Cite journal|last1=David|first1=Trevor J.|last2=Petigura|first2=Erik A.|last3=Luger|first3=Rodrigo|last4=Foreman-Mackey|first4=Daniel|last5=Livingston|first5=John H.|last6=Mamajek|first6=Eric E.|last7=Hillenbrand|first7=Lynne A.|date=2019-10-29|title=Four Newborn Planets Transiting the Young Solar Analog V1298 Tau|journal=The Astrophysical Journal|volume=885|issue=1|pages=L12|arxiv=1910.04563|doi=10.3847/2041-8213/ab4c99|issn=2041-8213|bibcode=2019ApJ...885L..12D|s2cid=204008446 |doi-access=free }}</ref>
*[[HD 158259]] contains four planets in a 3:2 near resonance chain (with periods of 3.432, 5.198, 7.954 and 12.03 days, or period ratios of 1.51, 1.53 and 1.51, respectively), with a possible fifth planet also near a 3:2 resonance (with a period of 17.4 days). The exoplanets were found with the [[SOPHIE échelle spectrograph]], using the [[Doppler spectroscopy|radial velocity]] method.<ref>{{cite journal|last1=Hara|first1=N. C.|last2=Bouchy|first2=F.|last3=Stalport|first3=M.|last4=Boisse|first4=I.|last5=Rodrigues|first5=J.|last6=Delisle|first6=J.- B.|last7=Santerne|first7=A.|last8=Henry|first8=G. W.|last9=Arnold|first9=L.|last10=Astudillo-Defru|first10=N.|last11=Borgniet|first11=S.|title=The SOPHIE search for northern extrasolar planets. XVII. A compact planetary system in a near 3:2 mean motion resonance chain|journal=Astronomy & Astrophysics|year=2020|language=en|volume=636|pages=L6|doi=10.1051/0004-6361/201937254|arxiv=1911.13296|bibcode=2020A&A...636L...6H |s2cid=208512859}}</ref>
*[[Kepler-1649]] contains two Earth-size planets close to a 9:4 resonance (with periods of 19.53527 and 8.689099 days, or a period ratio of 2.24825), including one ([[Kepler-1649c|"c"]]) in the habitable zone. An undetected planet with a 13.0-day period would create a 3:2 resonance chain.<ref name="Vanderburg2020">{{cite journal|last1= Vanderburg|first1= A.|last2= Rowden|first2= P.|last3= Bryson|first3= S.|last4= Coughlin|first4= J.|last5= Batalha|first5= N.|last6= Collins|first6= K.A.|last7= Latham|first7= D.W.|last8= Mullally|first8= S.E.|last9= Colón|first9= K.D.|last10= Henze|first10= C.|last11= Huang|first11= C.X.|last12= Quinn|first12= S.N.|title=A Habitable-zone Earth-sized Planet Rescued from False Positive Status |journal=The Astrophysical Journal |volume= 893|issue= 1|year= 2020|pages= L27|doi= 10.3847/2041-8213/ab84e5|arxiv= 2004.06725|bibcode= 2020ApJ...893L..27V|s2cid= 215768850|doi-access= free}}</ref>
*[[Kepler-88]] has a pair of inner planets close to a 1:2 resonance (period ratio of 2.0396), with a mass ratio of ~22.5, producing very large [[transit timing variation]]s of ~0.5 days for the innermost planet. There is a yet more massive outer planet in a ~1400 day orbit.<ref name="Weiss2020">{{cite journal|last1= Weiss|first1= L.M.|last2= Fabrycky|first2= D.C.|last3= Agol|first3= E.|last4= Mills|first4= S.M.|last5= Howard|first5= A.W.|last6= Isaacson|first6= H.|last7= Petigura|first7= E.A.|last8= Fulton|first8= B.|last9= Hirsch|first9= L.|last10= Sinukoff|first10= E.|title=The Discovery of the Long-Period, Eccentric Planet Kepler-88 d and System Characterization with Radial Velocities and Photodynamical Analysis |journal=The Astronomical Journal |volume= 159|issue= 5|year= 2020|pages= 242|doi= 10.3847/1538-3881/ab88ca |arxiv=1909.02427|bibcode= 2020AJ....159..242W|s2cid= 202539420|url= https://authors.library.caltech.edu/99954/1/1909.02427.pdf|doi-access= free}}</ref>
* [[HD 110067]] has six known planets, in a 54:36:24:16:12:9 resonance ratio.<ref>{{Cite web |last=Klesman |first=Alison |date=2023-11-29 |title='Shocked and delighted': Astronomers find six planets orbiting in resonance |url=https://www.astronomy.com/science/astronomers-find-six-planets-orbiting-in-resonance/ |access-date=2023-12-23 |website=Astronomy Magazine |language=en-US}}</ref>

Cases of extrasolar planets close to a 1:2 mean-motion resonance are fairly common. Sixteen percent of systems found by the [[transit method]] are reported to have an example of this (with period ratios in the range 1.83–2.18),<ref name="Lissauer_2011" /> as well as one sixth of planetary systems characterized by [[Doppler spectroscopy]] (with in this case a narrower period ratio range).<ref name="Wright_2011" /> Due to incomplete knowledge of the systems, the actual proportions are likely to be higher.<ref name="Lissauer_2011" /> Overall, about a third of radial velocity characterized systems appear to have a pair of planets close to a [[Commensurability (astronomy)|commensurability]].<ref name="Lissauer_2011" /><ref name="Wright_2011">{{cite journal |last1=Wright |first1=J. T. |last2=Fakhouri |first2=O. |last3=Marcy |first3=G. W. |last4=Han |first4=E. |last5=Feng |first5=Y. |last6=Johnson |first6=J. A. |last7=Howard |first7=A. W. |last8=Fischer |first8=D. A. |last9=Valenti |first9=J. A. |last10=Anderson |first10=J. |last11=Piskunov |first11=N. |year=2011 |title=The Exoplanet Orbit Database |journal=[[Publications of the Astronomical Society of the Pacific]] |volume=123 |issue=902 |pages=412–42 |arxiv=1012.5676 |bibcode=2011PASP..123..412W |doi=10.1086/659427|s2cid=51769219 }}</ref> It is much more common for pairs of planets to have orbital period ratios a few percent larger than a mean-motion resonance ratio than a few percent smaller (particularly in the case of first order resonances, in which the integers in the ratio differ by one).<ref name="Lissauer_2011" /> This was predicted to be true in cases where [[Tidal acceleration|tidal interactions]] with the star are significant.<ref name="Terquem_2007">{{cite journal |last1=Terquem |first1=C. |last2=Papaloizou |first2=J. C. B. |year=2007 |title=Migration and the Formation of Systems of Hot Super-Earths and Neptunes |journal=[[The Astrophysical Journal]] |volume=654 |issue=2 |pages=1110–1120 |arxiv=astro-ph/0609779 |bibcode=2007ApJ...654.1110T |doi=10.1086/509497|s2cid=14034512 }}</ref>