Orbital resonance

In most cases, this results in an unstable interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists.

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 Pierre-Simon Laplace.

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.

Before Newton, there was also consideration of ratios and proportions in orbital motions, in what was called "the music of the spheres", or musica universalis.

A primary result from the study of dynamical systems is the discovery and description of a highly simplified model of mode-locking; this is an oscillator that receives periodic kicks via a weak coupling to some driving motor.

The term arose because Pierre-Simon Laplace discovered that such a resonance governed the motions of Jupiter's moons Io, Europa, and Ganymede.

Over long times (a million years, or so) a secular resonance will change the eccentricity and inclination of the small body.

(The angular momentum of Neptune's orbit is 104 times that of Saturn's rotation rate, and thus dominates the interaction.)

Detailed analysis of data from the Cassini spacecraft gives a value of the moment of inertia of Saturn that is just outside the range for the resonance to exist, meaning that the spin axis does not stay in phase with Neptune's orbital inclination in the long term, as it apparently did in the past.

[23][24] 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.

[28] The simple integer ratios between periods hide more complex relations: As illustration of the latter, consider the well-known 2:1 resonance of Io-Europa.

(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.

(A Laplace resonance in the Gliese 876 system, in contrast, is associated with one triple conjunction per orbit of the outermost planet, ignoring libration.)

[29] Another "Laplace-like" resonance involves the moons Styx, Nix, and Hydra of Pluto:[16] This reflects orbital periods for Styx, Nix, and Hydra, respectively, that are close to a ratio of 18:22:33 (or, in terms of the near resonances with Charon's period, 3+3/11:4:6; see below); the respective ratio of orbits is 11:9:6.

The next largest body in a similar 2:3 resonance with Neptune, called a plutino, is the probable dwarf planet Orcus.

As it orbits Neptune, the more inclined Naiad successively passes Thalassa twice from above and then twice from below, in a cycle that repeats every ~21.5 Earth days.

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).

Some orbital frequency coincidences include: The least probable orbital correlation in the list – meaning the relationship that seems most likely to have not just be by random chance – 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.

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 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.

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.

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.

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 Uranian 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.

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.

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,[89] respectively.

[90][91] 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.

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 three-body Laplace resonance exhibited by three of Jupiter's Galilean moons . Conjunctions are highlighted by brief color changes. There are two Io-Europa conjunctions (green) and three Io-Ganymede conjunctions (grey) for each Europa-Ganymede conjunction (magenta). This diagram is not to scale.
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 [ 2 ] in the Columbo Gap of Saturn's C Ring (center) and the inclined orbits of resonant particles in the bending wave [ 3 ] [ 4 ] just inside it have apsidal and nodal precessions, respectively, commensurate with Titan 's mean motion.
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.
Illustration of Io–Europa–Ganymede resonance. From the centre outwards: Io (yellow), Europa (gray), and Ganymede (dark)
Depiction of the resonance between Neptune's moons Naiad (whose orbital motion is shown in red) and Thalassa , in a view that co-rotates with the latter
Resonant planetary system of two planets with a 1:2 orbit ratio
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 3:4:5:6 sequence of near resonances relative to the period of its large inner satellite Charon . The moons Styx, Nix and Hydra are also involved in a true 3-body resonance .