orbital resonance - wikipedia, the free encyclopedia

8/12/2019 Orbital Resonance - Wikipedia, The Free Encyclopedia

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rbital resonancem Wikipedia, the free encyclopedia

elestial mechanics, an orbital resonanceoccurs when two orbiting bodies exert a regular, periodic

vitational influence on each other, usually due to their orbital periods being related by a ratio of two small

gers. The physics principle behind orbital resonance is similar in concept to pushing a child on a swing,

ere the orbit and the swing both have a natural frequency, and the other body doing the "pushing" will act

eriodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the

tual gravitational influence of the bodies, i.e., their ability to alter or constrain each other's orbits. In mostes, this results in an unstableinteraction, in which the bodies exchange momentum and shift orbits until

resonance no longer exists. Under some circumstances, a resonant system can be stable and self-

recting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons

nymede, Europa and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances with

urn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between

ies with similar orbital radii) causes large Solar System bodies to eject most other bodies sharing their

its; this is partof the much more extensive process of clearing the neighbourhood, an effect that is used

he current definition of a planet.

ceptas noted in the Laplace resonance figure (below), a resonance ratio in this article should be

rpreted asthe ratio of number of orbitscompleted in the same time interval, rather than as the ratio of

ital periods(which would be the inverse ratio). The 2:3 ratio above means Pluto completes two orbits in

time it takes Neptune to complete three.

ontents

1 History2 Types of resonance3 Mean-motion resonances in the Solar System

3.1 The Laplace resonance3.2 Plutino resonances

4 Mean-motion resonances among extrasolar planets5 Coincidental 'near' ratios of mean motion6 Possible past mean-motion resonances7 See also8 References9 External links

istory

ce the discovery of Newton's law of universal gravitation in the 17th century, the stability of the Solar

tem has preoccupied many mathematicians, starting withLaplace. The stable orbits that arise in a two-

y approximation ignore the influence of other bodies. The effect of these added interactions on the

bility of the Solar System is very small, but at first it was not known whether they might add up overger periods to significantly change the orbital parameters and lead toa completely different configuration,

whether some other stabilising effects might maintain the configuration of the orbits of the planets.
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The semimajor axes of resonant trans-Neptunian

objects (red) are clumped at locations of low-integer

resonances with Neptune (vertical red bars near

top), in contrast to those of cubewanos (blue) and

nonresonant (or not known to be resonant) scattered

objects (grey).

A chart of the distribution of asteroid semimajor

axes, showing the Kirkwood gaps where orbits are

destabilized by resonances with Jupiter.

was Laplace who found the first answers explaining the remarkable dance of the Galilean moons (see

ow). It is fair to say that this general field of study has remained very active since then, with plenty more

to be understood (e.g., how interactions of moonlets with particles of the rings of giant planets result in

ntaining the rings).

ypes of resonance

general, an orbital resonance may

involve one or any combination of the orbitparameters (e.g. eccentricity versus semimajoraxis, or eccentricity versus orbital inclination).act on any time scale from short term,commensurable with the orbit periods, to secular,

measured in 104to 106years.lead to either long term stabilization of the orbitsor be the cause of their destabilization.

mean-motion orbital resonanceoccurs when twoies have periods of revolution that are a simple

ger ratio of each other. Depending on the details, this

either stabilize or destabilize the orbit. Stabilization

y occur when the two bodies move in such a

chronised fashion that they never closely approach.

instance:

The orbits of Pluto and the plutinos are stable,

despite crossing that of the much larger Neptune,because they are in a 2:3 resonance with it. Theresonance ensures that, when they approachperihelion and Neptune's orbit, Neptune isconsistently distant (averaging a quarter of itsorbit away). Other (much more numerous)Neptune-crossing bodies that were not inresonance were ejected from that region by strongperturbations due to Neptune. There are alsosmaller but significant groups of resonant trans-

Neptunian objects occupying the 1:1 (Neptunetrojans), 3:5, 4:7, 1:2 (twotinos) and 2:5resonances, among others, with respect toNeptune.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 clumpsof asteroids (the Hilda family,279 Thule, and the Trojan asteroids, respectively).

bital resonances can also destabilizeone of the orbits. For small bodies, destabilization is actually far

re likely. For instance:

In the asteroid belt within 3.5 AU from the Sun, the major mean-motion resonances with Jupiter arelocations of gapsin the asteroid distribution, the Kirkwood gaps (most notably at the 3:1, 5:2, 7:3 and2:1 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. Forexample, asteroids of the Alinda family are in or close to the 3:1 resonance, with their orbital
http://en.wikipedia.org/wiki/Alinda_familyhttp://en.wikipedia.org/wiki/Asteroidhttp://en.wikipedia.org/wiki/Kirkwood_gaphttp://en.wikipedia.org/wiki/Jupiterhttp://en.wikipedia.org/wiki/Asteroid_belthttp://en.wikipedia.org/wiki/Trojan_asteroidhttp://en.wikipedia.org/wiki/279_Thulehttp://en.wikipedia.org/wiki/Hilda_familyhttp://en.wikipedia.org/wiki/Jupiterhttp://en.wikipedia.org/wiki/Asteroid_belthttp://en.wikipedia.org/wiki/Resonant_Kuiper_belt_object#2:5_resonance_.28period_.7E410_years.29http://en.wikipedia.org/wiki/Resonant_Kuiper_belt_object#1:2_resonance_.28.22twotinos.22.2C_period_.7E330_years.29http://en.wikipedia.org/wiki/Resonant_Kuiper_belt_object#4:7_resonance_.28period_.7E290_years.29http://en.wikipedia.org/wiki/Resonant_Kuiper_belt_object#3:5_resonance_.28period_.7E275_years.29http://en.wikipedia.org/wiki/Neptune_trojanhttp://en.wikipedia.org/wiki/Resonant_trans-Neptunian_objecthttp://en.wikipedia.org/wiki/Perturbation_(astronomy)http://en.wikipedia.org/wiki/Neptunehttp://en.wikipedia.org/wiki/Plutinohttp://en.wikipedia.org/wiki/Plutohttp://en.wikipedia.org/wiki/Orbithttp://en.wikipedia.org/wiki/Secular_phenomenahttp://en.wikipedia.org/wiki/Semimajor_axishttp://en.wikipedia.org/wiki/Eccentricity_(orbit)http://en.wikipedia.org/wiki/Galilean_moonhttp://en.wikipedia.org/wiki/Jupiterhttp://en.wikipedia.org/wiki/Kirkwood_gaphttp://en.wikipedia.org/wiki/Asteroidhttp://en.wikipedia.org/wiki/File:Kirkwood_Gaps.svghttp://en.wikipedia.org/wiki/Scattered_diskhttp://en.wikipedia.org/wiki/Cubewanohttp://en.wikipedia.org/wiki/Neptunehttp://en.wikipedia.org/wiki/Resonant_trans-Neptunian_objecthttp://en.wikipedia.org/wiki/Semimajor_axishttp://en.wikipedia.org/wiki/File:TheKuiperBelt_75AU_All.svg


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Spiral density waves in Saturn's A Ring excited by

resonances with 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.

eccentricity steadily increased by interactions with Jupiter until they eventually have a close encounterwith 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 Ringthat has been cleared by a 2:1 resonance with the moon Mimas. (More specifically, the site of theresonance 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 withthe embedded moonlets Pan and Daphnis, respectively. The A Ring's outer edge is maintained by adestabilizing 7:6 resonance with the moon Janus.

st bodies that are in resonance orbit in the same direction; however, a few retrograde damocloids haven found that are temporarily captured in mean-motion resonance with Jupiter or Saturn.[4]Such orbital

ractions are weaker than the corresponding

ractions between bodies orbiting in the same

ection.[4]

Laplace resonanceoccurs when three or more

iting bodies have a simple integer ratio between their

ital periods. For example, Jupiter's moons Ganymede,

opa and Io are in a 1:2:4 orbital resonance. Therasolar planets Gliese 876 e, b and c are also in a

4 orbital resonance.[5]

Lindblad resonancedrives spiral density waves both

alaxies (where stars are subject to forcing by the

al arms themselves) and in Saturn's rings (where ring

ticles are subject to forcing by Saturn's moons).

ecular resonanceoccurs when the precession of two

its is synchronised (usually a precession of the

ihelion or ascending node). A small body in secular

onance with a much larger one (e.g. a planet) will

cess at the same rate as the large body. Over long

es (a million years, or so) a secular resonance will

nge the eccentricity and inclination of the small body.

eral prominent examples of secular resonance involve Saturn. A resonance between the precession of

urn's rotational axis and that of Neptune's orbital axis (both of which have periods of about 1.87 million

rs) has been identified as the likely source of Saturn's large axial tilt (26.7).[6][7][8]Initially, Saturnbably had a tilt closer to that of Jupiter (3.1). The gradual depletion of the Kuiper belt would have

reased the precession rate of Neptune's orbit; eventually, the frequencies matched, and Saturn's axial

cession was captured into the spin-orbit resonance, leading to an increase in Saturn's obliquity. (The

ular momentum of Neptune's orbit is 104times that of Saturn's spin, and thus dominates the interaction.)

e perihelion secular resonance between asteroids and Saturn (!6= g -g6) helps shape the asteroid belt.

eroids which approach it have their eccentricity slowly increased until they become Mars-crossers, at

ch point they are usually ejected from the asteroid belt by a close pass to Mars. This resonance forms the

er and "side" boundaries of the asteroid belt around 2 AU, and at inclinations of about 20.

merical simulations have suggested that the eventual formation of a perihelion secular resonance between

rcury and Jupiter (g1=g5) has the potential to greatly increase Mercury's eccentricity and possibly

tabilize the inner Solar System several billion years from now.[9][10]
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The eccentric Titan Ringlet[1]in

the Columbo Gap of Saturn's C

Ring (center) and the inclined

orbits of resonant particles in the

bending wave[2][3]just inside it

have apsidal and nodal

precessions, respectively,commensurate with Titan's mean

motion.

Depiction of Haumea's presumed 7:12 resonancewith Neptune in a rotating frame, with Neptune

(blue dot at lower right) held stationary. Haumea's

shifting orbital alignment relative to Neptune

periodically reverses (librates), preserving the

resonance.

e Titan Ringlet within Saturn's C Ring represents another type of

onance in which the rate of apsidal precession of one orbit exactly

ches the speed of revolution of another. The outer end of this

entric ringlet always points towards Saturn's major moon Titan.[1]

Kozai resonanceoccurs when the inclination and eccentricity of a

turbed orbit oscillate synchronously (increasing eccentricity while

reasing inclination and vice versa). This resonance applies only to

ies on highly inclined orbits; as a consequence, such orbits tend to

unstable, since the growing eccentricity would result in small

icenters, typically leading to a collision or (for large moons)

truction by tidal forces.

n example of another type of resonance involving orbital

entricity, the eccentricities of Ganymede and Callisto vary with a

mmon period of 181 years, although with opposite phases.[11]

ean-motion resonances in the Solar Systemere are only a few known mean-motion resonances in the Solar

tem involving planets, dwarf planets or larger satellites (a much

ater number involve asteroids, planetary rings,

onlets and smaller Kuiper belt objects, including

ny possible dwarf planets).

2:3 PlutoNeptune2:4 TethysMimas (Saturns moons)

1:2 DioneEnceladus (Saturns moons)3:4 HyperionTitan (Saturn's moons)1:2:4 GanymedeEuropaIo (Jupiters moons).

ditionally, Haumea is believed to be in a 7:12

onance with Neptune,[12][13]and Eris and Makemake

y be in 5:17 and 6:11 resonances with Neptune,

pectively.[14]

e simple integer ratiosbetween periods are avenient simplification hiding more complex

tions:

the point of conjunction can oscillate (librate)around an equilibrium point defined by theresonance.given non-zero eccentricities, the nodes orperiapsides can drift (a resonance related, short period, not secular precession).

illustration of the latter, consider the well known 2:1 resonance of Io-Europa. If the orbiting periods werehis relation, the mean motions (inverse of periods, often expressed in degrees per day) would satisfy

following
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The Laplace resonance exhibited by three of the Galilean

moons. The ratios in the figure are of orbital periods.

Illustration of Io-Europa-Ganymede resonance.

From the centre outwards: Io (yellow), Europa

(gray) and Ganymede (dark)

bstituting the data (from Wikipedia) one will get !0.7395 day!1, a value substantially different from

o!

ually, the resonance isperfect but it involves also the precession of perijove (the point closest to Jupiter),

The correct equation (part of the Laplace equations) is:

other words, the mean motion of Io is

eed double of that of Europa taking intoount the precession of the perijove. An

erver sitting on the (drifting) perijove will

the moons coming into conjunction in the

me place (elongation). The other pairs listed

ve satisfy the same type of equation with

exception of Mimas-Tethys resonance. In

case, the resonance satisfies the equation

e point of conjunctions librates around the midpoint between the nodesof the two moons.

e Laplace resonance

e most remarkable resonance involving Io-Europa-

nymede includes the following relation locking the

ital phaseof the moons:

ere are mean longitudes of the moons. This relationkes a triple conjunction impossible. The graph

strates the positions of the moons after 1, 2 and 3 Io

iods. (The Laplace resonance in the Gliese 876

tem, in contrast, is associated with one triple

junction per orbit of the outermost planet.[5])

utino resonances

e dwarf planet Pluto is following an orbit trapped in a web of resonances with Neptune. The resonances

ude:

A mean-motion resonance of 2:3The resonance of the perihelion (libration around 90), keeping the perihelion above the eclipticThe resonance of the longitude of the perihelion in relation to that of Neptune
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e consequence of these resonances is that a separation of at least 30 AU is maintained when Pluto crosses

ptune's orbit. The minimum separation between the two bodies overall is 17 AU, while the minimum

aration between Pluto and Uranus is just 11 AU[15](see Pluto's orbit for detailed explanation and

phs).

e next largest body in a similar 2:3 resonance with Neptune, called aplutino, is the probable dwarf planet

us. Orcus has an orbit similar in inclination and eccentricity to Pluto's. However, the two are constrained

heir mutual resonance with Neptune to always be in opposite phases of their orbits; Orcus is thus

metimes described as the "anti-Pluto".[16]

ean-motion resonances among extrasolar planets

ile most extrasolar planetary systems discovered have not been found to have planets in mean-motion

onances, some remarkable examples have been uncovered:

As mentioned above, Gliese 876 e, b and c are in a 1:2:4 orbital resonance, with periods of 124.3, 61.1

and 30.0 days.[5][17]

KOI-730 d, b, c and e appear to be in a 3:4:6:8 resonance, with periods of 19.72, 14.79, 9.85 and 7.38days.[18][19][20]

KOI-500 c, b, e, d and f appear to be in or close to a 20:27:41:62:193 resonance, with periods of

9.522, 7.053, 4.645, 3.072 and 0.9868 days.[20][21][22]

Both KOI-738 and KOI-787 appear to have pairs of planets in a 7:9 resonance (ratios of 1/1.285871

and 1/1.284008, respectively).[20]

Kepler-37 d, c and b are within one percent of a 5:8:15 resonance, with periods of 39.792187,

21.301886 and 13.367308 days.[23]

es of extrasolar planets close to a 1:2 mean-motion resonance are fairly common. Sixteen percent of

tems found by the transit method are reported to have an example of this (with period ratios in the range

3-2.18),[20]as well as one sixth of planetary systems characterized by Doppler spectroscopy (with in this

e a narrower period ratio range).[24]Due to incomplete knowledge of the systems, the actual proportions

likely to be higher.[20]Overall, about a third of radial velocity characterized systems appear to have a

r of planets close to a commensurability.[20][24]It is much more common for pairs of planets to have

ital period ratios a few percent larger than a mean-motion resonance ratio than a few percent smaller

rticularly in the case of first order resonances, in which the integers in the ratio differ by one).[20]This

predicted to be true in cases where tidal interactions with the star are significant.[25]

oincidental 'near' ratios of mean motion

umber of near-integer-ratio relationships between the orbital frequencies of the planets or major moons

sometimes pointed out (see list below). However, these have no dynamical significance because there is

appropriate precession of perihelion or other libration to make the resonance perfect (see the detailed

cussion in the section above). Such near resonances are dynamically insignificant even if the mismatch is

te small because (unlike a true resonance), after each cycle the relative position of the bodies shifts.

en averaged over astronomically short timescales, their relative position is random, just like bodies thatnowhere near resonance. For example, consider the orbits of Earth and Venus, which arrive at almost the

me configuration after 8 Earth orbits and 13 Venus orbits. The actual ratio is 0.61518624, which is only

32% away from exactly 8:13. The mismatch after 8 years is only 1.5 of Venus' orbital movement. Still,

is enough that Venus and Earth find themselves in the opposite relative orientation to the original every

such cycles, which is 960 years. Therefore, on timescales of thousands of years or more (still tiny by
http://en.wikipedia.org/wiki/Orbital_resonance#Mean-motion_resonances_in_the_Solar_Systemhttp://en.wikipedia.org/wiki/Perihelionhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Terquem_2007-25http://en.wikipedia.org/wiki/Tidal_accelerationhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Lissauer_2011-20http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Wright_2011-24http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Lissauer_2011-20http://en.wikipedia.org/wiki/Commensurability_(astronomy)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Lissauer_2011-20http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Wright_2011-24http://en.wikipedia.org/wiki/Doppler_spectroscopyhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Lissauer_2011-20http://en.wikipedia.org/wiki/Transit_methodhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-BarclayRowe2013-23http://en.wikipedia.org/wiki/Kepler-37http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Lissauer_2011-20http://en.wikipedia.org/w/index.php?title=KOI-787&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=KOI-738&action=edit&redlink=1http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Choi_2012-22http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-EPE-KOI500-21http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Lissauer_2011-20http://en.wikipedia.org/wiki/KOI-500http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Lissauer_2011-20http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Beatty-19http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-EPE-KOI730-18http://en.wikipedia.org/wiki/KOI-730http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Marcy_2001-17http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-rivera2010-5http://en.wikipedia.org/wiki/Gliese_876http://en.wikipedia.org/wiki/Extrasolar_planethttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-MBP-16http://en.wikipedia.org/wiki/90482_Orcushttp://en.wikipedia.org/wiki/Plutinohttp://en.wikipedia.org/wiki/Pluto#Orbit_and_rotationhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-15http://en.wikipedia.org/wiki/Uranus


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Depiction of asteroid Pallas' 18:7 near resonance

with Jupiter in a rotating frame (click for

animation). Jupiter (pink loop at upper left) is held

nearly stationary. The shift in Pallas' orbital

alignment relative to Jupiter increases steadily over

time; it never reverses course (i.e., there is no

libration).

onomical standards), their relative position is effectively random.

e presence of a near resonance may reflect that a perfect resonance existed in the past, or that the system

volving towards one in the future.

me orbital frequency coincidences include:
http://en.wikipedia.org/wiki/2_Pallashttp://en.wikipedia.org/wiki/File:PallasJupiter.GIF


8/15

Depiction of the Earth:Venus 8:13 near resonance.

With Earth held stationary at the center of a

nonrotating frame, the successive inferiorconjunctions of Venus over eight Earth years trace a

pentagrammic pattern (reflecting the difference

between the numbers in the ratio).

Diagram of the orbits of Pluto's small outer fourmoons, which follow a remarkable 3:4:5:6 sequence

of near resonances relative to the period of its large

inner satellite Charon.

(Ratio) and Bodies Mismatch after one cycle[a] Randomization time[b] Probability[c]

Planets

23) Venus!Mercury 4.0 200 y 0.19

13) Earth!Venus[26][27][d] 1.5 1000 y 0.065

43:395) Earth!Venus[26][28] 0.8 50,000 y 0.68

3) Mars!Venus 20.6 20 y 0.11

2) Mars!Earth 42.9 8 y 0.24
http://en.wikipedia.org/wiki/Earthhttp://en.wikipedia.org/wiki/Marshttp://en.wikipedia.org/wiki/Venushttp://en.wikipedia.org/wiki/Marshttp://en.wikipedia.org/wiki/Yearhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Shortt-31http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Langford-29http://en.wikipedia.org/wiki/Venushttp://en.wikipedia.org/wiki/Earthhttp://en.wikipedia.org/wiki/Yearhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-32http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Bazs.C3.B3-30http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Langford-29http://en.wikipedia.org/wiki/Venushttp://en.wikipedia.org/wiki/Earthhttp://en.wikipedia.org/wiki/Yearhttp://en.wikipedia.org/wiki/Mercury_(planet)http://en.wikipedia.org/wiki/Venushttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-28http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-27http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-26http://en.wikipedia.org/wiki/Charon_(moon)http://en.wikipedia.org/wiki/Plutohttp://en.wikipedia.org/wiki/File:Moons_of_Pluto.pnghttp://en.wikipedia.org/wiki/Pentagramhttp://en.wikipedia.org/wiki/Inferior_conjunctionhttp://en.wikipedia.org/wiki/Venushttp://en.wikipedia.org/wiki/Earthhttp://en.wikipedia.org/wiki/File:Venus_pentagram.png


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12) Jupiter!Earth[e] 49.1 40 y 0.28

5) SaturnJupiter[f] 12.8 800 y 0.13

7) Uranus!Jupiter 31.1 500 y 0.18

20) Uranus!Saturn 5.7 20,000 y 0.20

28) Neptune!Saturn 1.9 80,000 y 0.052

2) Neptune!Uranus 14.0 2000 y 0.078

Mars system

4) Deimos!Phobos 14.9 0.04 y 0.083

Major asteroids

1) Pallas !Ceres[29][30] 1.2 700 y 0.0066

18) Jupiter !Pallas[31] 4.1 4000 y 0.15

87 Sylvia system[g]

7:45) Romulus!Remus 0.7 40 y 0.067

Jupiter system

6) Io!Metis 0.6 2 y 0.0031

5) Amalthea!Adrastea 3.9 0.2 y 0.064

7) Callisto!Ganymede[32] 0.7 30 y 0.012

Saturn system

3) Enceladus!Mimas 33.2 0.04 y 0.33

3) Dione!Tethys[h] 36.2 0.07 y 0.36

5) Rhea!Dione 17.1 0.4 y 0.26

7) Titan!Rhea 21.0 0.7 y 0.22

5) Iapetus!Titan 9.2 4 y 0.051

Major centaurs[i]

4) Uranus!Chariklo 4.5 10,000 y 0.073

Uranus system

5) Rosalind!Cordelia[34] 0.22 4 y 0.0037

3) Umbriel!Miranda[j] 24.5 0.08 y 0.14

5) Umbriel!Ariel[k] 24.2 0.3 y 0.35

2) Titania!Umbriel 36.3 0.1 y 0.20

3) Oberon!Titania 33.4 0.4 y 0.34

Neptune system

20) Triton!Naiad 13.5 0.2 y 0.075

2) Proteus!Larissa[37][38] 8.4 0.07 y 0.047

6) Proteus!S/2004 N 1 2.1 1 y 0.057

Pluto system

3) Styx!Charon[39] 58.5 0.2 y 0.33
http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Matson-50http://en.wikipedia.org/wiki/Charon_(moon)http://en.wikipedia.org/wiki/Styx_(moon)http://en.wikipedia.org/wiki/S/2004_N_1http://en.wikipedia.org/wiki/Proteus_(moon)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-ZhangHamilton2008-49http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-ZhangHamilton2007-48http://en.wikipedia.org/wiki/Larissa_(moon)http://en.wikipedia.org/wiki/Proteus_(moon)http://en.wikipedia.org/wiki/Naiad_(moon)http://en.wikipedia.org/wiki/Triton_(moon)http://en.wikipedia.org/wiki/Titania_(moon)http://en.wikipedia.org/wiki/Oberon_(moon)http://en.wikipedia.org/wiki/Umbriel_(moon)http://en.wikipedia.org/wiki/Titania_(moon)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-47http://en.wikipedia.org/wiki/Ariel_(moon)http://en.wikipedia.org/wiki/Umbriel_(moon)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-45http://en.wikipedia.org/wiki/Miranda_(moon)http://en.wikipedia.org/wiki/Umbriel_(moon)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Murray_1990-43http://en.wikipedia.org/wiki/Cordelia_(moon)http://en.wikipedia.org/wiki/Rosalind_(moon)http://en.wikipedia.org/wiki/10199_Chariklohttp://en.wikipedia.org/wiki/Uranushttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-42http://en.wikipedia.org/wiki/Centaur_(minor_planet)http://en.wikipedia.org/wiki/Titan_(moon)http://en.wikipedia.org/wiki/Iapetus_(moon)http://en.wikipedia.org/wiki/Rhea_(moon)http://en.wikipedia.org/wiki/Titan_(moon)http://en.wikipedia.org/wiki/Dione_(moon)http://en.wikipedia.org/wiki/Rhea_(moon)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-41http://en.wikipedia.org/wiki/Tethys_(moon)http://en.wikipedia.org/wiki/Dione_(moon)http://en.wikipedia.org/wiki/Mimas_(moon)http://en.wikipedia.org/wiki/Enceladus_(moon)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Goldreich_1965-39http://en.wikipedia.org/wiki/Ganymede_(moon)http://en.wikipedia.org/wiki/Callisto_(moon)http://en.wikipedia.org/wiki/Adrastea_(moon)http://en.wikipedia.org/wiki/Amalthea_(moon)http://en.wikipedia.org/wiki/Metis_(moon)http://en.wikipedia.org/wiki/Io_(moon)http://en.wikipedia.org/wiki/Remus_(moon)http://en.wikipedia.org/wiki/Romulus_(moon)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-38http://en.wikipedia.org/wiki/87_Sylviahttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Taylor1982-37http://en.wikipedia.org/wiki/2_Pallashttp://en.wikipedia.org/wiki/Jupiterhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Kova.C4.8Devi.C4.87-36http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Goffin2001-35http://en.wikipedia.org/wiki/Ceres_(dwarf_planet)http://en.wikipedia.org/wiki/2_Pallashttp://en.wikipedia.org/wiki/Phobos_(moon)http://en.wikipedia.org/wiki/Deimos_(moon)http://en.wikipedia.org/wiki/Uranushttp://en.wikipedia.org/wiki/Neptunehttp://en.wikipedia.org/wiki/Saturnhttp://en.wikipedia.org/wiki/Neptunehttp://en.wikipedia.org/wiki/Saturnhttp://en.wikipedia.org/wiki/Uranushttp://en.wikipedia.org/wiki/Jupiterhttp://en.wikipedia.org/wiki/Uranushttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-34http://en.wikipedia.org/wiki/Jupiterhttp://en.wikipedia.org/wiki/Saturnhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-33http://en.wikipedia.org/wiki/Earthhttp://en.wikipedia.org/wiki/Jupiter


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4) Nix!Charon[39][40] 39.1 0.3 y 0.22

5) Kerberos!Charon[39] 9.2 2 y 0.05

6) Hydra!Charon[39][40] 6.6 3 y 0.037

Haumea system

8) Hi"iaka!Namaka[l] 42.5 2 y 0.55

a. ^Mismatch in orbital longitude of the inner body, as compared to its position at the beginning of the cycle (with the

cycle defined as norbits of the outer body see below). Circular orbits are assumed (i.e., precession is ignored).b. ^The time needed for the mismatch from the initial relative longitudinal orbital positions of the bodies to grow to

180, rounded to the nearest first significant digit.

c. ^The probability of obtaining an orbital coincidence of equal or smaller mismatch by chance at least once in n

attempts, where nis the integer number of orbits of the outer body per cycle, and the mismatch is assumed to vary

between 0 and 180 at random. The value is calculated as 1- (1- mismatch/180)^n. The smaller the probability, the

more remarkable the coincidence. This is a crude calculation that only attempts to give a rough idea of relative

probabilities.

d. ^The two near commensurabilities 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.[26][28]

e. ^The near 1:12 resonance between Jupiter and Earth causes the Alinda asteroids, which occupy (or are close to) the3:1 resonance with Jupiter, to be close to a 1:4 resonance with Earth.

f. ^This near resonance has been termed the Great Inequality. It was first described by Laplace in a series of papers

published 17841789.

g. ^87 Sylvia is the first asteroid discovered to have more than one moon.

h. ^This resonance may have been occupied in the past.[33]

i. ^Some definitions of centaurs stipulate that they are nonresonant bodies.

j. ^This resonance may have been occupied in the past.[35]

k. ^This resonance may have been occupied in the past.[36]

l. ^The results for the Haumea system aren't very meaningful because, contrary to the assumptions implicit in the

calculations, Namaka has an eccentric, non-Keplerian orbit that precesses rapidly (see below). Hi"iaka and Namakaare much closer to a 3:8 resonance than indicated, and may actually be in it.[41]

e most remarkable (least probable) orbital correlation in the list is that between Io and Metis, followed by

se between Rosalind and Cordelia, Pallas and Ceres, Callisto and Ganymede, and Hydra and Charon,

pectively.

ossible past mean-motion resonances

ast resonance between Jupiter and Saturn may have played a dramatic role in early Solar System history.004 computer model by Alessandro Morbidelli of the Observatoire de la Cte d'Azur in Nice suggested

the formation of a 1:2 resonance between Jupiter and Saturn (due to interactions with planetesimals that

sed them to migrate inward and outward, respectively) created a gravitational push that propelled both

nus and Neptune into higher orbits, and in some scenarios caused them to switch places, which would

e doubled Neptune's distance from the Sun. The resultant expulsion of objects from the proto-Kuiper belt

Neptune moved outwards could explain the Late Heavy Bombardment 600 million years after the Solar

tem's formation and the origin of Jupiter's Trojan asteroids.[42]An outward migration of Neptune could

o explain the current occupancy of some of its resonances (particularly the 2:5 resonance) within the

per belt.

ile Saturn's mid-sized moons Dione and Tethys are not close to an exact resonance now, they may have

n in a 2:3 resonance early in the Solar System's history. This would have led to orbital eccentricity and

l heating that may have warmed Tethys' interior enough to form a subsurface ocean. Subsequent freezing
http://en.wikipedia.org/wiki/Tidal_acceleration#Tidal_heatinghttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-54http://en.wikipedia.org/wiki/Trojan_asteroidhttp://en.wikipedia.org/wiki/Late_Heavy_Bombardmenthttp://en.wikipedia.org/wiki/Planetesimalshttp://en.wikipedia.org/wiki/Nicehttp://en.wikipedia.org/wiki/C%C3%B4te_d%27Azur_Observatoryhttp://en.wikipedia.org/wiki/Alessandro_Morbidelli_(astronomer)http://en.wikipedia.org/wiki/Nice_modelhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Ragozzine.26Brown2009-52http://en.wikipedia.org/wiki/Osculating_orbithttp://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-53http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Tittemore1988-46http://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-47http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Tittemore_Wisdom_1990-44http://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-45http://en.wikipedia.org/wiki/Centaur_(minor_planet)#Classificationhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-42http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Chen2008-40http://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-41http://en.wikipedia.org/wiki/87_Sylviahttp://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-38http://en.wikipedia.org/wiki/Pierre-Simon_Laplacehttp://en.wikipedia.org/wiki/Great_Inequalityhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-34http://en.wikipedia.org/wiki/Alinda_familyhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-33http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Shortt-31http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Langford-29http://en.wikipedia.org/wiki/Transit_of_Venushttp://en.wikipedia.org/wiki/Commensurability_(astronomy)http://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-32http://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-28http://en.wikipedia.org/wiki/Significant_digithttp://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-27http://en.wikipedia.org/wiki/Orbital_resonance#cite_ref-26http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-53http://en.wikipedia.org/wiki/Namaka_(moon)http://en.wikipedia.org/wiki/Hi%27iaka_(moon)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-WardCanup2006-51http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Matson-50http://en.wikipedia.org/wiki/Charon_(moon)http://en.wikipedia.org/wiki/Hydra_(moon)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Matson-50http://en.wikipedia.org/wiki/Charon_(moon)http://en.wikipedia.org/wiki/Kerberos_(moon)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-WardCanup2006-51http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Matson-50http://en.wikipedia.org/wiki/Charon_(moon)http://en.wikipedia.org/wiki/Nix_(moon)


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he ocean after the moons escaped from the resonance may have generated the extensional stresses that

ated the enormous graben system of Ithaca Chasma on Tethys.[33]

e satellite system of Uranus is notably different from those of Jupiter and Saturn in that it lacks precise

onances among the larger moons, while the majority of the larger moons of Jupiter (3 of the 4 largest) and

Saturn (6 of the 8 largest) are in mean-motion resonances. In all three satellite systems, moons were likely

tured into mean-motion resonances in the past as their orbits shifted due to tidal dissipation (a process by

ch satellites gain orbital energy at the expense of the primary's rotational energy, affecting inner moons

proportionately). In the Uranus System, however, due to the planet's lesser degree of oblateness, and the

ger relative size of its satellites, escape from a mean-motion resonance is much easier. Lower oblateness

he primary alters its gravitational field in such a way that different possible resonances are spaced more

sely together. A larger relative satellite size increases the strength of their interactions. Both factors lead

more chaotic orbital behavior at or near mean-motion resonances. Escape from a resonance may be

ociated with capture into a secondary resonance, and/or tidal evolution-driven increases in orbital

entricity or inclination.

an-motion resonances that probably once existed in the Uranus System include (3:5) Ariel-Miranda, (1:3)

briel-Miranda, (3:5) Umbriel-Ariel, and (1:4) Titania-Ariel.[36][35]Evidence for such past resonances

udes the relatively high eccentricities of the orbits of Uranus' inner satellites, and the anomalously highital inclination of Miranda. High past orbital eccentricities associated with the (1:3) Umbriel-Miranda

(1:4) Titania-Ariel resonances may have led to tidal heating of the interiors of Miranda and Ariel,[43]

pectively. Miranda probably escaped from its resonance with Umbriel via a secondary resonance, and the

chanism of this escape is believed to explain why its orbital inclination is more than 10 times those of the

er regular Uranian moons (see Uranus' natural satellites).[44][45]

milar to the case of Miranda, the present inclinations of Jupiter's moonlets Amalthea and Thebe are

ught to be indications of past passage through the 3:1 and 4:2 resonances with Io, respectively.[46]

ptune's regular moons Proteus and Larissa are thought to have passed through a 1:2 resonance a few

dred million years ago; the moons have drifted away from each other since then because Proteus is

side a synchronous orbit and Larissa is within one. Passage through the resonance is thought to have

ited both moons' eccentricities to a degree that has not since been entirely damped out.[37][38]

he case of Pluto's satellites, it has been proposed that the present near resonances are relics of a previous

cise resonance that was disrupted by tidal damping of the eccentricity of Charon's orbit (see Pluto's

ural satellites for details). The near resonances may be maintained by a 15% local fluctuation in the Pluto-

aron gravitational field. Thus, these near resonances may not be coincidental.

e smaller inner moon of the dwarf planet Haumea, Namaka, is one tenth the mass of the larger outer

on, Hi"iaka. Namaka revolves around Haumea in 18 days in an eccentric, non-Keplerian orbit, and as of

8 is inclined 13 from Hi"iaka.[41]Over the timescale of the system, it should have been tidally damped

o a more circular orbit. It appears that it has been disturbed by resonances with the more massive Hi"iaka,

to converging orbits as it moved outward from Haumea because of tidal dissipation. The moons may

e been caught in and then escaped from orbital resonance several times. They probably passed through

3:1 resonance relatively recently, and currently are in or at least close to an 8:3 resonance. Namaka's

it is strongly perturbed, with a current precession of about !6.5 per year.[41]

e also

1685 Toro, an asteroid in 5:8 resonance with the Earth
http://en.wikipedia.org/wiki/1685_Torohttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Ragozzine.26Brown2009-52http://en.wikipedia.org/wiki/Perturbation_(astronomy)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Ragozzine.26Brown2009-52http://en.wikipedia.org/wiki/Osculating_orbithttp://en.wikipedia.org/wiki/Hi%27iaka_(moon)http://en.wikipedia.org/wiki/Namaka_(moon)http://en.wikipedia.org/wiki/Haumea_(dwarf_planet)http://en.wikipedia.org/wiki/Dwarf_planethttp://en.wikipedia.org/wiki/Pluto%27s_natural_satelliteshttp://en.wikipedia.org/wiki/Plutohttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-ZhangHamilton2008-49http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-ZhangHamilton2007-48http://en.wikipedia.org/wiki/Synchronous_orbithttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Burns2004-58http://en.wikipedia.org/wiki/Thebe_(moon)http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-57http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Tittemore1989-56http://en.wikipedia.org/wiki/Uranus%27_natural_satelliteshttp://en.wikipedia.org/wiki/Regular_moonhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Tittemore_1990-55http://en.wikipedia.org/wiki/Tidal_acceleration#Tidal_heatinghttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Tittemore_Wisdom_1990-44http://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Tittemore1988-46http://en.wikipedia.org/wiki/Inclinationhttp://en.wikipedia.org/wiki/Orbital_eccentricityhttp://en.wikipedia.org/wiki/Oblate_spheroidhttp://en.wikipedia.org/wiki/Tidal_accelerationhttp://en.wikipedia.org/wiki/Orbital_resonance#cite_note-Chen2008-40http://en.wikipedia.org/wiki/Ithaca_Chasmahttp://en.wikipedia.org/wiki/Graben


12/15

3753 Cruithne, an asteroid in 1:1 resonance with the EarthCommensurability (astronomy)Dermott's LawHorseshoe orbit, followed by an object in another type of 1:1 resonanceKozai resonanceLagrangian pointsMercury, which has a 3:2 spinorbit resonanceMusica universalis ("music of the spheres")Resonant trans-Neptunian object

Tidal lockingTidal resonanceTitiusBode lawTrojan object, a body in a type of 1:1 resonance

eferences

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10). "The eccentric Saturnian ringlets at 1.29Rsand 1.45Rs"

(http://www.sciencedirect.com/science/article/pii/0019103584901349).Icarus60(1): 116. Retrieved 2014-01-09.2. ^Rosen, P. A.; Lissauer, J. J. (1988-08-05). "The Titan -1:0 Nodal Bending Wave in Saturn's Ring C". Science241

(4866): 690694. doi:10.1126/science.241.4866.690 (http://dx.doi.org/10.1126%2Fscience.241.4866.690).

PMID 17839081 (//www.ncbi.nlm.nih.gov/pubmed/17839081).

3. ^Chakrabarti, S. K.; Bhattacharyya, A. (2001). "Constraints on the C ring parameters of Saturn at the Titan !1:0

resonance".Monthly Notices of the Royal Astronomical Society326(2): L23. doi:10.1046/j.1365-8711.2001.04813.x

(http://dx.doi.org/10.1046%2Fj.1365-8711.2001.04813.x).

4. ^ abMorais, M. H. M.; Namouni, F. "Asteroids in retrograde resonance with Jupiter and Saturn"

(http://arxiv.org/abs/1308.0216).Monthly Notices of the Royal Astronomical Society Letters (in press).

arXiv:arXiv:1308.0216 (//arxiv.org/abs/arXiv:1308.0216).

5. ^abcRivera, Eugenio J.; Laughlin, Gregory; Butler, R. Paul; Vogt, Steven S.; Haghighipour, Nader; Meschiari,

Stefano (June 2010). "The Lick-Carnegie Exoplanet Survey: A Uranus-mass Fourth Planet for GJ 876 in an

Extrasolar Laplace Configuration". arXiv:1006.4244v1 (http://arxiv.org/abs/1006.4244v1) [astro-ph.EP

(http://arxiv.org/archive/astro-ph.EP)].

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Small Solar System Bodies and Exoplanets. Lecture Notes in Physics 790. Springer. pp. 162.doi:10.1007/978-3-642-04458-8 (http://dx.doi.org/10.1007%2F978-3-642-04458-8). ISBN 978-3-642-04457-1.

xternal links

Locations of Solar System Planetary Mean-Motion Resonances(http://www.alpheratz.net/murison/asteroids/resonances/). Web calculator that plots distributions of thesemimajor axes (or in one case the perihelion distances) of the minor planets in relation to mean-

motion resonances of the planets (website maintained by M.A. Murison).

rieved from "http://en.wikipedia.org/w/index.php?title=Orbital_resonance&oldid=594406400"

egories: Orbits

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