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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. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. 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 "clear out" the region around their orbits by ejecting nearly everything else around them; this effect is used in the current definition of a planet.

History

Ever since the discovery of Newton's law of universal gravitation in the 17th century, the stability of planetary orbits has preoccupied many mathematicians, starting with Laplace. The stable orbits that arise in a two-body approximation ignore the influence of other bodies. These added interactions, even when very small, might add up over longer periods to significantly change the orbital parameters and leading to a completely different configuration of the Solar System. Or, it was thought, some other stabilising mechanisms might be there. 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 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 stabilisation 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:

  • Pluto and the Plutinos are in stable orbits, despite crossing the orbit of the much larger Neptune. This is because a 2:3 resonance keeps them always at a large distance from it. 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, 1:2 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).
  • The extrasolar planets Gliese 876b and Gliese 876c are in a 1:2 orbital resonance

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.
  • 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.)

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.

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. A prominent example is the ν6 secular resonance between asteroids and Saturn. 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 due to 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 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. One of the consequences of this resonance is the lack of bodies on highly inclined orbits, as the growing eccentricity would result in small pericenters, typically leading to a collision or destruction by tidal forces for large moons.

Mean motion resonances in the Solar System

Galilean moon Laplace resonance animation

The Laplace resonance exhibited by three of the Galilean moons.

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

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 Th} - 2\cdot n_{\rm Mi} - \Omega_{\rm Th}- \Omega_{\rm Mi}= 0

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

The Laplace resonance

TheLaplaceResonance2

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.

Pluto resonances

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

  • Mean motion resonance 2:3 (2 orbits of Pluto for 3 orbits of Neptune)
  • 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 19 AU, about twice the minimum separation between Pluto and Uranus (see Pluto's orbit for detailed explanation and graphs).

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 Mean-motion resonances in the Solar system 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 which 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.

Some orbital frequency coincidences that have been pointed out include:

(Ratio) and Bodies Mismatch after one cycle[1] Randomization time[2]
Planets
(9:23) VenusMercury 4.0° 200 y
(8:13) EarthVenus 1.5° 1000 y
(1:3) MarsVenus 20.6° 20 y
(8:15) MarsEarth 16.6° 200 y
(2:5) SaturnJupiter 12.8° 800 y
(1:7) UranusJupiter 31.1° 500 y
(7:20) UranusSaturn 5.7° 20,000 y
(5:28) NeptuneSaturn 1.9° 80,000 y
(1:2) NeptuneUranus 14.0° 2000 y
Mars system
(1:4) DeimosPhobos 14.9° 0.04 y
Jupiter system
(3:7) CallistoGanymede 0.7° 30 y
Saturn system
(3:5) RheaDione 17.1° 0.4 y
(2:7) TitanRhea 21.0° 0.7 y
(1:5) IapetusTitan 9.2° 4.0 y
Uranus system
(1:3) UmbrielMiranda 24.6° 0.1 y
(3:5) UmbrielAriel 24.0° 0.3 y
(1:2) TitaniaUmbriel 36.3° 0.1 y
(2:3) OberonTitania 33.4° 0.4 y
Pluto system
(1:4) NixCharon 39.1° 0.3 y
(1:6) HydraCharon 6.6° 3.0 y

No explanation has been found for the contrast between the satellite systems of Jupiter and Saturn, in both of which the majority of the major moons (3 of Jupiter's 4 largest, and 6 of Saturn's 8 largest) are involved in mean motion resonances, and the satellite system of Uranus, in which there are no precise resonances among the larger moons.

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.

See also

References

  1. Mismatch in orbital longitude of the inner body, as compared to its position at the beginning of the cycle.
  2. The time needed for the mismatch from the initial relative longitudinal orbital positions of the bodies to grow to 180°, rounded to nearest first significant digit.
  • Murray, Dermott Solar System Dynamics, Cambridge University Press, ISBN 0-521-57597-4
  • Renu Malhotra Orbital Resonances and Chaos in the Solar System. In Solar system Formation and Evolution, ASP Conference Series, 149 (1998) preprint
  • Renu Malhotra, The Origin of Pluto's Orbit: Implications for the Solar System Beyond Neptune, The Astronomical Journal, 110 (1995), p. 420 Preprint.

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