Orbital resonance

For the science fiction novel, see Orbital Resonance (novel).

In 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. The physics principle behind orbital resonance is similar in concept to pushing a child on a swing, where the orbit and the swing both have a natural frequency, and the other body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies, i.e., their ability to alter or constrain each other's 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. Thus the 2:3 ratio above means Pluto completes two orbits in the time it takes Neptune to complete three.

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).

Types of resonances

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.
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.
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.

In general, an orbital resonance may

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 may occur when the two bodies move in such a synchronised fashion that they never closely approach. For instance:

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

Most bodies that are in resonance orbit in the same direction; however, a few retrograde damocloids have been found that are temporarily captured in mean-motion resonance with Jupiter or Saturn.[4] Such orbital interactions are weaker than the corresponding interactions between bodies orbiting in the same direction.[4]

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 876 e, b and c are also in a 1:2:4 orbital resonance.[5]

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°).[6][7][8] 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 Saturn's spin, and thus dominates the interaction.)

The perihelion secular resonance between asteroids and Saturn (ν6 = g − g6) 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 asteroid belt around 2 AU, and at inclinations of about 20°.

Numerical simulations have suggested that the eventual formation of a perihelion secular resonance between Mercury and Jupiter (g1 = g5) has the potential to greatly increase Mercury's eccentricity and possibly destabilize the inner Solar System several billion years from now.[9][10]

The Titan Ringlet within Saturn's C Ring represents another type of resonance in which the rate of apsidal precession of one orbit exactly matches the speed of revolution of another. The outer end of this eccentric ringlet always points towards Saturn's major moon Titan.[1]

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.

In an example of another type of resonance involving orbital eccentricity, the eccentricities of Ganymede and Callisto vary with a common period of 181 years, although with opposite phases.[11]

Mean-motion resonances in the Solar System

Depiction of Haumea's presumed 7:12 resonance with 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.
The Laplace resonance exhibited by three of the Galilean moons. The ratios in the figure are of orbital periods. Conjunctions are highlighted by brief color changes.

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, planetary rings, moonlets and smaller Kuiper belt objects, including many possible dwarf planets).

Additionally, Haumea is believed to be in a 7:12 resonance with Neptune,[12][13] and Eris and Makemake may be in 5:17 and 6:11 resonances with Neptune, respectively.[14]

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

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 n\,\! (inverse of periods, often expressed in degrees per day) would satisfy the following

n_{\rm Io} - 2\cdot n_{\rm Eu} = 0

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), \dot\omega. The correct equation (part of the Laplace equations) is:

n_{\rm Io} - 2\cdot n_{\rm Eu} + \dot\omega_{\rm Io} = 0

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

4\cdot n_{\rm Te} - 2\cdot n_{\rm Mi} - \dot\Omega_{\rm Te}- \dot\Omega_{\rm Mi}= 0

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

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:

\Phi_L = \lambda_{\rm Io} - 3\cdot\lambda_{\rm Eu} + 2\cdot\lambda_{\rm Ga} = 180^\circ

where \lambda 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.[5])

Plutino resonances

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

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

The next largest body in a similar 2:3 resonance with Neptune, called a plutino, is the probable 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".[16]

Mean-motion resonances among extrasolar planets

While most extrasolar planetary systems discovered have not been found to have planets in mean-motion resonances, some remarkable examples have been uncovered:

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),[20] as well as one sixth of planetary systems characterized by Doppler spectroscopy (with in this case a narrower period ratio range).[25] Due to incomplete knowledge of the systems, the actual proportions are likely to be higher.[20] Overall, about a third of radial velocity characterized systems appear to have a pair of planets close to a commensurability.[20][25] 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).[20] This was predicted to be true in cases where tidal interactions with the star are significant.[26]

Coincidental 'near' ratios of mean motion

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).
Depiction of the Earth:Venus 8:13 near resonance. With Earth held stationary at the center of a nonrotating frame, the successive inferior conjunctions 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 four moons, which follow a remarkable 3:4:5:6 sequence of near resonances relative to the period of its large inner satellite Charon.

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 Bodies Mismatch after one cycle[lower-alpha 1] Randomization time[lower-alpha 2] Probability[lower-alpha 3]
Planets
(9:23) Venus−Mercury 4.0° 200 y 0.19
(8:13) Earth−Venus[27][28][lower-alpha 4] 1.5° 1000 y 0.065
(243:395) Earth−Venus[27][29] 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[lower-alpha 5] 49.1° 40 y 0.28
(2:5) Saturn–Jupiter[lower-alpha 6] 12.8° 800 y 0.13
(1:7) Uranus−Jupiter 31.1° 500 y 0.18
(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
Major asteroids
(1:1) Pallas − Ceres[30][31] 0.7° 1000 y 0.0039[lower-alpha 7]
(7:18) Jupiter − Pallas[32] 0.10° 100,000 y 0.0040[lower-alpha 8]
87 Sylvia system[lower-alpha 9]
(17:45) Romulus−Remus 0.7° 40 y 0.067
Jupiter system
(1:6) Io−Metis 0.6° 2 y 0.0031
(3:5) Amalthea−Adrastea 3.9° 0.2 y 0.064
(3:7) Callisto−Ganymede[33] 0.7° 30 y 0.012
Saturn system
(2:3) Enceladus−Mimas 33.2° 0.04 y 0.33
(2:3) Dione−Tethys[lower-alpha 10] 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 y 0.051
Major centaurs[lower-alpha 11]
(3:4) Uranus−Chariklo 4.5° 10,000 y 0.073
Uranus system
(3:5) Rosalind−Cordelia[35] 0.22° 4 y 0.0037
(1:3) Umbriel−Miranda[lower-alpha 12] 24.5° 0.08 y 0.14
(3:5) Umbriel−Ariel[lower-alpha 13] 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
Neptune system
(1:20) Triton−Naiad 13.5° 0.2 y 0.075
(1:2) Proteus−Larissa[38][39] 8.4° 0.07 y 0.047
(5:6) Proteus−S/2004 N 1 2.1° 1 y 0.057
Pluto system
(1:3) Styx−Charon[40] 58.5° 0.2 y 0.33
(1:4) Nix−Charon[40][41] 39.1° 0.3 y 0.22
(1:5) Kerberos−Charon[40] 9.2° 2 y 0.05
(1:6) Hydra−Charon[40][41] 6.6° 3 y 0.037
Haumea system
(3:8) Hiʻiaka−Namaka[lower-alpha 14] 42.5° 2 y 0.55
  1. ↑ Mismatch in orbital longitude of the inner body, as compared to its position at the beginning of the cycle (with the cycle defined as n orbits of the outer body – see below). Circular orbits are assumed (i.e., precession is ignored).
  2. ↑ 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.
  3. ↑ The probability of obtaining an orbital coincidence of equal or smaller mismatch by chance at least once in n attempts, where n is 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.
  4. ↑ 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.[27][29]
  5. ↑ 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.
  6. ↑ This near resonance has been termed the Great Inequality. It was first described by Laplace in a series of papers published 1784–1789.
  7. ↑ Based on the proper orbital periods, 1684.869 and 1681.601 days, for Pallas and Ceres, respectively.
  8. ↑ Based on the proper orbital period of Pallas, 1684.869 days, and 4332.59 days for Jupiter.
  9. ↑ 87 Sylvia is the first asteroid discovered to have more than one moon.
  10. ↑ This resonance may have been occupied in the past.[34]
  11. ↑ Some definitions of centaurs stipulate that they are nonresonant bodies.
  12. ↑ This resonance may have been occupied in the past.[36]
  13. ↑ This resonance may have been occupied in the past.[37]
  14. ↑ 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). Hiʻiaka and Namaka are much closer to a 3:8 resonance than indicated, and may actually be in it.[42]

The most remarkable (least probable) orbital correlation in the list is that between Io and Metis, followed by those between Rosalind and Cordelia, Pallas and Ceres, Jupiter and Pallas, Callisto and Ganymede, and Hydra and Charon, respectively.

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.[43] 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.[34]

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.[37][36] 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,[44] 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).[45][46]

Similar to the case of Miranda, the present inclinations of Jupiter's moonlets Amalthea and Thebe are thought to be indications of past passage through the 3:1 and 4:2 resonances with Io, respectively.[47]

Neptune's regular moons Proteus and Larissa are thought to have passed through a 1:2 resonance a few hundred million years ago; the moons have drifted away from each other since then because Proteus is outside a synchronous orbit and Larissa is within one. Passage through the resonance is thought to have excited both moons' eccentricities to a degree that has not since been entirely damped out.[38][39]

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.

The smaller inner moon of the dwarf planet Haumea, Namaka, is one tenth the mass of the larger outer moon, Hiʻiaka. Namaka revolves around Haumea in 18 days in an eccentric, non-Keplerian orbit, and as of 2008 is inclined 13° from Hiʻiaka.[42] 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 probably passed through the 3:1 resonance relatively recently, and currently are in or at least close to an 8:3 resonance. Namaka's orbit is strongly perturbed, with a current precession of about −6.5° per year.[42]

See also

References

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