Editing Orbital resonance (section)

==== 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>