COSMIC Orbital resonanceIn celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of the bodies, i.e., their ability to alter or constrain each others' orbits. In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood, an effect that is used in the current definition of a planet.

Except as noted in the Laplace resonance figure (below), a resonance ratio in this article should be interpreted as the ratio of number of orbits completed in the same time interval, rather than as the ratio of orbital periods (which would be the inverse ratio). The 2:3 ratio above means Pluto completes 2 orbits in the time it takes Neptune to complete 3.

Contents

 [hide]  1 History 2 Types of resonance 3 Mean motion resonances in the Solar System

o 3.1 The Laplace resonance o 3.2 Pluto resonances

4 Coincidental 'near' ratios of mean motion 5 Possible past mean motion resonances 6 See also 7 References and notes

8 External links

[edit] History

Since the discovery of Newton's law of universal gravitation in the 17th century, the stability of the solar system has preoccupied many mathematicians, starting with Laplace. The stable orbits that arise in a two-body approximation ignore the influence of other bodies. The effect of these added interactions on the stability of the Solar System is very

small, but at first it was not known whether they might add up over longer periods to significantly change the orbital parameters and lead to a completely different configuration, or whether some other stabilising effects might maintain the configuration of the orbits of the planets.

It was Laplace who found the first answers explaining the remarkable dance of the Galilean moons (see below). It is fair to say that this general field of study has remained very active since then, with plenty more yet to be understood (e.g. how interactions of moonlets with particles of the rings of giant planets result in maintaining the rings).

[edit] Types of resonance

In general, an orbital resonance may

involve one or any combination of the orbit parameters (e.g. eccentricity versus semimajor axis, or eccentricity versus orbit inclination).

act on any time scale from short term, commensurable with the orbit periods, to secular, measured in 104 to 106 years.

lead to either long term stabilization of the orbits or be the cause of their destabilization.

A mean motion orbital resonance occurs when two bodies have periods of revolution that are a simple integer ratio of each other. Depending on the details, this can either stabilize or destabilize the orbit. Stabilization occurs when the two bodies move in such a synchronised fashion that they never closely approach. For instance:

The orbits of Pluto and the plutinos are stable, despite crossing that of 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 perturbations due to Neptune. There are also smaller but significant groups of resonant trans-Neptunian objects occupying the 1:1 (Neptune trojans), 3:5, 4:7, 1:2 (twotinos) and 2:5 resonances 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, 279 Thule, and the Trojan asteroids, respectively).

Orbital resonances can also destabilize one of the orbits. For small bodies, destabilization is actually far more likely. For instance:

In the asteroid belt within 3.5 AU from the Sun, the major mean-motion resonances with Jupiter are locations of gaps in the asteroid distribution, the Kirkwood gaps (most notably at the 3:1, 5:2, 7:3 and 2:1 resonances). Asteroids 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 Cassini Division is a gap between the inner B Ring and the outer A Ring that has been cleared by a 2:1 resonance with the moon Mimas. (More specifically, the site of the resonance is the Huygens Gap, which bounds the outer edge of the B Ring.)

In the rings of Saturn, the Encke and Keeler gaps within the A Ring are cleared by 1:1 resonances with the embedded moonlets Pan and Daphnis, respectively. The A Ring's outer edge is maintained by a destabilizing 7:6 resonance with the moon Janus.

A Laplace resonance occurs when three or more orbiting bodies have a simple integer ratio between their orbital periods. For example, Jupiter's moons Ganymede, Europa and Io are in a 1:2:4 orbital resonance. The extrasolar planets Gliese 876e, Gliese 876b and Gliese 876c are also in a 1:2:4 orbital resonance (with periods of 124.3, 61.1 and 30.0 days).[1][2]

A Lindblad resonance drives spiral density waves both in galaxies (where stars are subject to forcing by the spiral arms themselves) and in Saturn's rings (where ring particles are subject to forcing by Saturn's moons).

A secular resonance occurs when the 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 and inclination of the small body.

Several prominent examples of secular resonance involve Saturn. A 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) has been identified as the likely source of Saturn's large axial tilt (26.7°).[3][4][5] 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 the spin-orbit resonance, leading to an increase in Saturn's obliquity. (The angular momentum of Neptune's orbit is 104 times that of that of Saturn's spin, and thus dominates the interaction.)

The ν6[clarification needed] secular resonance between asteroids and Saturn helps shape the

asteroid belt. Asteroids which approach it have their eccentricity slowly increased until they become 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 main asteroid belt around 2 AU, and at inclinations of about 20°.

The Titan Ringlet within Saturn's C Ring exemplifies 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.

A Kozai resonance occurs when the inclination and eccentricity of a 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 pericenters, typically leading to a collision or (for large moons) destruction by tidal forces.

[edit] Mean motion resonances in the Solar System

The Laplace resonance exhibited by three of the Galilean moons. The ratios in the figure are of orbital periods.

There are only a few known mean motion resonances in the Solar System involving planets, dwarf planets or larger satellites (a much greater number involve asteroids, smaller Kuiper belt objects, planetary rings and moonlets).

2:3 Pluto-Neptune 2:4 Tethys-Mimas (Saturn’s moons) 1:2 Dione-Enceladus (Saturn’s moons) 3:4 Hyperion-Titan (Saturn's moons) 1:2:4 Ganymede-Europa-Io (Jupiter’s moons).

The simple integer ratios between periods are a convenient simplification hiding more complex relations:

the point of conjunction can oscillate (librate) around an equilibrium point defined by the resonance.

given non-zero eccentricities, the nodes or 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 motions (inverse of periods, often expressed in degrees per day) would satisfy the following

Substituting the data (from Wikipedia) one will get −0.7395° day−1, a value substantially different from zero!

Actually, the resonance is perfect but it involves also the precession of perijove (the point closest to Jupiter), . The correct equation (part of the Laplace equations) is:

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

The point of conjunctions librates around the midpoint between the nodes of the two moons.

[edit] The Laplace resonance

Illustration of Io-Europa-Ganymede resonance. From the centre outwards: Io (yellow), Europa (gray) and Ganymede (dark)

The most remarkable resonance involving Io-Europa-Ganymede includes the following relation locking the orbital phase of the moons:

where λ are mean longitudes of the moons. This relation makes a triple conjunction impossible. The graph illustrates the positions of the moons after 1, 2 and 3 Io periods.

(The Laplace resonance in the Gliese 876 system, in contrast, is associated with one triple conjunction per orbit of the outermost planet.[2])

[edit] Pluto 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[6] (see Pluto's orbit for detailed explanation and graphs).

The next largest body in a similar 2:3 resonance with Neptune is the candidate dwarf planet 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".[7]

[edit] Coincidental 'near' ratios of mean motion

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 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:

(Ratio) and BodiesMismatch after one

cycle[8]Randomization

time[9] Probability[10]

Planets(9:23) Venus − Mercury 4.0° 200 y 0.19(8:13) Earth − Venus 1.5° 1000 y 0.065(243:395) Earth − Venus 0.8° 50,000 y 0.68(1:3) Mars − Venus 20.6° 20 y 0.11(1:2) Mars − Earth 42.9° 8 y 0.24(1:12) Jupiter − Earth 49.1° 40 y 0.27(2:5) Saturn − Jupiter [11] 12.8° 800 y 0.14(1:7) Uranus − Jupiter 31.1° 500 y 0.17(7:20) Uranus − Saturn 5.7° 20,000 y 0.20(5:28) Neptune − Saturn 1.9° 80,000 y 0.052(1:2) Neptune − Uranus 14.0° 2000 y 0.078

Mars System(1:4) Deimos − Phobos 14.9° 0.04 y 0.083

Jupiter System(3:7) Callisto − Ganymede

0.7° 30 y 0.012

Saturn System(2:3) Enceladus − Mimas

33.2° 0.04 y 0.33

(2:3) Dione − Tethys 36.2° 0.07 y 0.36(3:5) Rhea − Dione 17.1° 0.4 y 0.26(2:7) Titan − Rhea 21.0° 0.7 y 0.22(1:5) Iapetus − Titan 9.2° 4.0 y 0.051

Uranus System(1:3) Umbriel − Miranda 24.5° 0.08 y 0.14(3:5) Umbriel − Ariel 24.2° 0.3 y 0.35(1:2) Titania − Umbriel 36.3° 0.1 y 0.20(2:3) Oberon − Titania 33.4° 0.4 y 0.34

Pluto System(1:4) Nix − Charon 39.1° 0.3 y 0.22(1:5) P4 − Charon  ?  ?  ?(1:6) Hydra − Charon 6.6° 3.0 y 0.037

The most remarkable (least probable) orbital correlation in the list is that between Callisto and Ganymede, followed in second place by that between Hydra and Charon.

The two near resonances listed for Earth and Venus are reflected in the timing of transits of Venus, which occur in pairs 8 years apart, in a cycle that repeats every 243 years.

The near 1:12 resonance between Jupiter and Earth causes the Alinda asteroids, which occupy (or are close to) the 3:1 resonance with Jupiter, to be close to a 1:4 resonance with Earth.

[edit] Possible past mean motion resonances

A past resonance between Jupiter and Saturn may have played a dramatic role in early Solar System history. A 2004 computer model by Alessandro Morbidelli of the Observatoire de la Côte d'Azur in Nice suggested that the formation of a 1:2 resonance between Jupiter and Saturn (due to interactions with planetesimals that caused them to migrate inward and outward, respectively) created a gravitational push that propelled both Uranus and Neptune into higher orbits, and in some scenarios caused them to switch places, which would have doubled Neptune's distance from the Sun. The resultant expulsion of objects from the proto-Kuiper belt as Neptune moved outwards could explain the Late Heavy Bombardment 600 million years after the Solar System's formation and the origin of Jupiter's Trojan asteroids.[12] An outward migration of Neptune could also explain the current occupancy of some of its resonances (particularly the 2:5 resonance) within the Kuiper belt.

While Saturn's mid-sized moons Dione and Tethys are not close to an exact resonance now, they may have been in a 2:3 resonance early in the Solar System's history. This would have led to orbital eccentricity and tidal heating that may have warmed Tethys' interior enough to form a subsurface ocean. Subsequent freezing of the ocean after the moons escaped from the resonance may have generated the extensional stresses that created the enormous graben system of Ithaca Chasma on Tethys.[13]

The satellite system of Uranus is notably different from those of Jupiter and Saturn in that it lacks precise resonances among the larger moons, while the majority of the larger moons of Jupiter (3 of the 4 largest) and of Saturn (6 of the 8 largest) are in mean motion resonances. In all three satellite systems, moons were likely captured into mean motion resonances in the past as their orbits shifted due to tidal dissipation (a process by which satellites gain orbital energy at the expense of the primary's rotational energy, affecting inner moons disproportionately). In the Uranus System, however, due to the planet's lesser degree of oblateness, and the larger relative size of its satellites, escape from a mean motion resonance is much easier. Lower oblateness of the primary alters its gravitational field in such a way that different possible resonances are spaced more closely together. A larger relative satellite size increases the strength of their interactions. Both factors lead to more chaotic orbital behavior at or near mean motion resonances. Escape from a resonance may be associated with capture into a secondary resonance, and/or tidal evolution-driven increases in orbital eccentricity or inclination.

Mean motion resonances that probably once existed in the Uranus System include (3:5) Ariel-Miranda, (1:3) Umbriel-Miranda, (3:5) Umbriel-Ariel, and (1:4) Titania-Ariel.[14][15] Evidence for such past resonances includes the relatively high eccentricities of the orbits of Uranus' inner satellites, and the anomalously high orbital inclination of Miranda. High past orbital eccentricities associated with the (1:3) Umbriel-Miranda and (1:4) Titania-

Ariel resonances may have led to tidal heating of the interiors of Miranda and Ariel,[16] respectively. Miranda probably escaped from its resonance with Umbriel via a secondary resonance, and the mechanism of this escape is believed to explain why its orbital inclination is more than 10 times those of the other regular Uranian moons (see Uranus' natural satellites).[17][18]

In the case of Pluto's satellites, it has been proposed that the present near resonances are relics of a previous precise resonance that was disrupted by tidal damping of the eccentricity of Charon's orbit (see Pluto's natural satellites for details). The near resonances may be maintained by a 15% local fluctuation in the Pluto-Charon gravitational field. Thus, these near resonances may not be coincidental.

Plutoid Haumea's smaller inner moon Namaka is a tenth the mass of Hi ̒ iaka , orbiting Haumea in 18 days in a highly elliptical, non-Keplerian orbit, and as of 2008 is inclined 13° from the larger moon, which perturbs its orbit.[19] Over the timescale of the system, it should have been tidally damped into a more circular orbit. It appears that it has been disturbed by resonances with the more massive Hiʻiaka, due to converging orbits as it moved outward from Haumea because of tidal dissipation. The moons may have been caught in and then escaped from orbital resonance several times; they currently are in or at least close to an 8:3 resonance. This strongly perturbs Namaka's orbit, with a current precession of ~20°.

[edit] See also

3753 Cruithne , an asteroid in approximate 1:1 orbital resonance with the Earth Resonant trans-Neptunian object Commensurability (astronomy) Kozai resonance Dermott's Law Lagrangian points Mercury , which has a 3:2 spin-orbit resonance Tidal locking Tidal resonance Titius-Bode law Trojan object , a body in a type of 1:1 resonance Horseshoe orbit , followed by an object in another type of 1:1 resonance Musica universalis ("music of the spheres")

Resonant trans-Neptunian objectFrom Wikipedia, the free encyclopedia

Types of Distant Minor Planets

Cis-Neptunian objects

o Centaurs

o Neptune trojans

Trans-Neptunian objects (TNOs)‡

o Kuiper-belt objects (KBOs)

Classical KBOs (Cubewanos)

Resonant KBOs

Plutinos (2:3 Resonance)

o Scattered-disc objects (SDOs)

Resonant SDOs

o Detached objects

o Oort cloud objects (OCOs)

‡ Trans–Neptunian dwarf planets are "plutoids"

v · d · e

In astronomy, a resonant trans-Neptunian object is a trans-Neptunian object (TNO) in mean motion orbital resonance with Neptune. The orbital periods of the resonant objects are in a simple integer relations with the period of Neptune e.g. 1:2, 2:3 etc. Resonant TNOs can be either part of the main Kuiper belt population, or the more distant scattered disc population.[1]

Contents

 [hide]  1 Distribution 2 Origin 3 Known populations

o 3.1 2:3 resonance ("plutinos", period ~250 years) o 3.2 3:5 resonance (period ~275 years) o 3.3 4:7 resonance (period ~290 years) o 3.4 1:2 resonance ("twotinos", period ~330 years) o 3.5 2:5 resonance (period ~410 years) o 3.6 Other resonances o 3.7 1:1 resonance (Neptune trojans, period ~165 years)

4 Coincidental vs true resonances 5 Toward a formal definition 6 Classification methods

7 References

8 Further reading

[edit] Distribution

Distribution of trans-Neptunian Objects. Resonant objects in red.

The diagram illustrates the distribution of the known trans-Neptunian objects (up to 70 AU) in relation to the orbits of the planets together with centaurs for reference. Resonant objects are plotted in red. Orbital resonances with Neptune are marked with vertical bars; 1:1 marks the position of Neptune’s orbit (and its trojans), 2:3 marks the orbit of Pluto and plutinos, 1:2, 2:5 etc. a number of smaller families).

The designation 2:3 or 3:2 refer both to the same resonance for TNOs. There’s no confusion possible as TNOs, by definition, have periods longer than Neptune. The usage depends on the author and the field of research. The statement "Pluto is in 2:3 resonance to Neptune" appears better to capture the meaning: Pluto completes 2 orbits for every 3 orbits of Neptune.

[edit] Origin

For details of the evolution of Neptune's orbit, see Nice model.

Detailed analytical and numerical studies[2][3] of Neptune’s resonances have shown that they are quite "narrow" (i.e. the objects must have a relatively precise range of energy). If the object semi-major axis is outside these narrow ranges, the orbit becomes chaotic, with widely changing orbital elements.

As TNOs were discovered, a substantial (more than 10%) proportion were found to be in 2:3 resonances, far from a random distribution. It is now believed that the objects have

been collected from wider distances by sweeping resonances during the migration of Neptune.[4]

Well before the discovery of the first TNO, it was suggested that interaction between giant planets and a massive disk of small particles would, via momentum transfer, make Jupiter migrate inwards while Saturn, Uranus and especially Neptune would migrate outwards. During this relatively short period of time, Neptune’s resonances would be sweeping the space, trapping objects on initially-varying heliocentric orbits into resonance.[5]

[edit] Known populations

[edit] 2:3 resonance ("plutinos", period ~250 years)

Main article: Plutino

The motion of Orcus in a rotating frame with a period equal to Neptune's orbital period. (Neptune is held stationary.)

Pluto and its moons (top) compared in size, albedo and colour with Orcus and Ixion.

The 2:3 resonance at 39.4 AU is by far the dominant category among the resonant objects, with 92 confirmed and 104 possible member bodies.[6] The objects following orbits in this resonance are named plutinos after Pluto, the first such body discovered. Large, numbered plutinos include:[7]

90482 Orcus (84922) 2003 VS 2

2003 AZ 84

28978 Ixion 38628 Huya

[edit] 3:5 resonance (period ~275 years)

A population of 10 objects at 42.3 AU as of October, 2008, including:[7]

(126154) 2001 YH140

(15809) 1994 JS (143751) 2003 US292

[edit] 4:7 resonance (period ~290 years)

Another important population of objects (20 identified as of October 2008) is orbiting the Sun at 43.7 AU (in the midst of the classical objects). The objects are rather small (with a single exception, H>6) and most of them follow orbits close to the ecliptic. Objects with well established orbits include:[7]

1999 CD 158, the largest (119956) 2002 PA149

(119070) 2001 KP 77

(118378) 1999 HT 11

(118698) 2000 OY51

[edit] 1:2 resonance ("twotinos", period ~330 years)

This resonance at 47.8 AU is often considered as the outer "edge" of the Kuiper Belt and the objects in this resonance are sometimes referred to as twotinos. Twotinos have inclinations less than 15 degrees and generally moderate eccentricities (0.1 < e < 0.3).[8] An unknown number of the 2:1 resonants likely did not originate in a planetesimal disk that was swept by the resonance during Neptune's migration.[9]

There are far fewer objects in this resonance (a total of 14 as of October, 2008) than plutinos. Long-term orbital integration shows that the 1:2 resonance is less stable than 2:3 resonance; only 15% of the objects in 1:2 resonance were found to survive 4 Gyr as compared with 28% of the plutinos.[8] Consequently it might be that twotinos were originally as numerous as plutinos, but their population has dropped significantly below that of plutinos since.[8]

Objects with well established orbits include (in order of the absolute magnitude):[7]

(119979) 2002 WC 19

(26308) 1998 SM 165

(137295) 1999 RB 216

(20161) 1996 TR 66

(130391) 2000 JG 81

Selected resonant objects (in red).

[edit] 2:5 resonance (period ~410 years)

Objects with well established orbits at 55.4 AU include:[7]

(84522) 2002 TC 302, a large dwarf-planet candidate (143707) 2003UY117

(119068) 2001 KC 77

(135571) 2002GG32

(69988) 1998 WA 31

In total, the orbits of 11 objects are classified as 2:5 as of October, 2008.

[edit] Other resonances

The nominal 7:12 libration of Haumea in a rotating frame. Where red turns to green is where it crosses the ecliptic.

So called higher-order resonances are known for a limited number of objects, including the following numbered objects[7]

4:5 (35 AU, ~205 years) (131697) 2001 XH255

3:4 (36.5 AU, ~220 years) (143685) 2003 SS317, (15836) 1995 DA2

5:9 (44.5 AU, ~295 years) 2002 GD32[10]

4:9 (52 AU, ~370 years) (42301) 2001 UR163, (182397) 2001 QW297[11]

3:7 (53 AU, ~385 years) (131696) 2001 XT254, (95625) 2002 GX32, (183964) 2004 DJ71, (181867) 1999 CV118

5:12 (55 AU, ~395 years) (79978) 1999 CC158, (119878) 2001 CY224[12] (84%

probability according to Emel’yanenko) 3:8 (57 AU, ~440 years) (82075) 2000 YW134

[13] (84% probability according to Emel’yanenko)

2:7 (70 AU, ~580 years) 2006 HX122[14] (The preliminary orbit suggests a weak

2:7 resonance. Further observations will be required.)

A few objects are known on simple, distant resonances[7]

1:3 (62.5 AU, ~495 years) (136120) 2003 LG7

1:4 (76 AU, ~660 years) 2003 LA7[15]

1:5 (88 AU, ~820 years) 2003 YQ179 (likely coincidental)[16]

Some notable unproven (they could be coincidental) dwarf planet resonances include:

7:12 (43 AU, ~283 years) Haumea [17] (nominal orbit very likely in resonance) 6:11 (45 AU, ~302 years) Makemake [18] ((182294) 2001 KU76 appears to be in the

6:11 resonance) 3:10 (67 AU, ~549 years) (225088) 2007 OR10 (based on a very preliminary orbit) 5:17 (67 AU, ~560 years) Eris [18] (2007 OR10 has a similar orbit)

[edit] 1:1 resonance (Neptune trojans, period ~165 years)

Main article: Neptune trojan

A few objects have been discovered following orbits with semi-major axes similar to that of Neptune, near the Sun–Neptune Lagrangian points. These Neptune trojan objects, named by analogy to the Trojan asteroids, are in 1:1 resonance with Neptune. Eight are known as of November 2011:

2001 QR 322

2004 UP 10

2005 TN 53

2005 TO 74

2006 RJ 103

2007 VL 305

2008 LC 18

2004 KV 18

Only the last two objects are near Neptune's L5 Lagrangian point; the others are located in Neptune's L4 region.[19]

[edit] Coincidental vs true resonances

One of the concerns is that weak resonances may exist and would be difficult to prove due to the current lack of accuracy in the orbits of these distant objects. Many objects have orbital periods of more than 300 years and most have only been observed over a short observation arc of a couple years. Due to their great distance and slow movement against background stars, it may be decades before many of these distant orbits are determined well enough to confidently confirm whether a resonance is true or merely coincidental. A true resonance will smoothly oscillate while a coincidental near resonance will circulate. (See Toward a formal definition)

Simulations by Emel’yanenko and Kiseleva in 2007 show that (131696) 2001 XT254 is librating in a 7:3 resonance with Neptune.[20] This libration can be stable for less than 100 million to billions of years.[20]

The orbital period of 2001 XT254 around the 7:3 (2.333) resonance of Neptune.

Emel’yanenko and Kiseleva also show that (48639) 1995 TL8 appears to have less than a 1% probability of being in a 7:3 resonance with Neptune, but it does execute circulations near this resonance.[20]

The orbital period of 1995 TL8 missing the 7:3 (2.333) resonance of Neptune.

[edit] Toward a formal definition

The classes of TNO have no universally agreed precise definitions, the boundaries are often unclear and the notion of resonance is not defined precisely. The Deep Ecliptic Survey introduced formally defined dynamical classes based on long-term forward integration of orbits under the combined perturbations from all four giant planets. (see also formal definition of classical KBO)

In general, the mean motion resonance may involve not only orbital periods of the form

where p and q are small integers, λ and λN are respectively the mean longitudes of the object and Neptune, but can also involve the longitude of the perihelion and the longitudes of the nodes (see orbital resonance, for elementary examples)

An object is a Resonant if for some small integers p,q,n,m,r,s, the argument (angle) defined below is librating (i.e. is bounded)[21]

where the are the longitudes of perihelia and the Ω are the longitudes of the ascending nodes, for Neptune (with subscripts "N") and the resonant object (no subscripts).

The term libration denotes here periodic oscillation of the angle around some value and is opposed to circulation where the angle can take all values from 0 to 360°. For example, in the case of Pluto, the resonant angle ϕ librates around 180° with an amplitude of around 82° degrees, i.e. the angle changes periodically from 180°-82° to 180°+82°.

All new plutinos discovered during the Deep Ecliptic Survey proved to be of the type

similar to Pluto's mean motion resonance.

More generally, this 2:3 resonance is an example of the resonances p:(p+1) (example 1:2, 2:3, 3:4 etc.) that have proved to lead to stable orbits.[4] Their resonant angle is

In this case, the importance of the resonant angle can be understood by noting that when the object is at perihelion i.e. then

i.e. gives a measure of the distance of the object's perihelion from Neptune.[4] The object is protected from the perturbation by keeping its perihelion far from Neptune provided librates around an angle far from 0°.

[edit] Classification methods

As the orbital elements are known with a limited precision, the uncertainties may lead to false positives (i.e. classification as resonant of an orbit which is not).

A recent approach[22] considers not only the current best-fit orbit but also two additional orbits corresponding to the uncertainties of the observational data. In simple terms, the algorithm determines whether the object would be still classified as resonant if its actual orbit differed from the best fit orbit, as the result of the errors in the observations.

The three orbits are numerically integrated over a period of 10 million years. If all three orbits remain resonant (i.e. the argument of the resonance is librating, see formal definion), the classification as a resonant object is considered secure.[22]

If only two out of the three orbits are librating the object is classified as probably in resonance. Finally, if only one orbit passes the test, the vicinity of the resonance is noted to encourage further observations to improve the data.[22]

The two extreme values of the semi-major axis used in the algorithm are determined to correspond to uncertainties of the data of at most 3 standard deviations. Such range of semi-axis values should, with a number of assumptions, reduce the probability that the actual orbit is beyond this range to less than 0.3%.

The method is applicable to objects with observations spanning at least 3 oppositions.[22]