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. Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... Adjectives: Jovian Atmosphere Surface pressure: 70 kPa Composition: ~86% Hydrogen ~14% Helium 0. ... Atmospheric characteristics Atmospheric pressure trace Oxygen 100% Ganymede (gan-É™-meed, IPA: ; Greek Γανυμήδης) is Jupiters largest moon, and indeed the largest moon in the entire solar system; it is larger in diameter than Mercury but only about half its mass. ... Atmospheric characteristics Atmospheric pressure 1 µPa Oxygen 100% Europa (ew-roe-pÉ™, IPA:  ; Greek Ευρώπη) is a moon of the planet Jupiter. ... Atmospheric characteristics Atmospheric pressure trace Sulfur dioxide 90% Io (eye-oe, IPA: , Greek Ῑώ) is the innermost of the four Galilean moons of Jupiter. ... Adjectives: Plutonian Atmosphere Surface pressure: 0. ... Note: This article contains special characters. ... Adjective Saturnian Atmospheric characteristics Atmospheric pressure 140 kPa Hydrogen >93% Helium >5% Methane 0. ... The full set of rings The rings of Saturn are a series of planetary rings that orbit the planet Saturn. ... For the astrological concept, see Planets in astrology. ...

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). Isaac Newtons theory of universal gravitation states the following: Every single point mass attracts every other point mass by a force heading along the line combining the two. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i. ... Major features of the Solar System (not to scale, from left to right): Pluto, Neptune, Uranus, Saturn, a comet, Jupiter, Ceres which lies in the asteroid belt, the Sun, Mercury, Venus, Earth & Moon, and Mars. ...

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: In astrodynamics, under standard assumptions any orbit must be of conic section shape. ... In geometry, the semi-major axis (also semimajor axis) a applies to ellipses and hyperbolas. ...

• 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: Atmospheric characteristics Atmospheric pressure 0. ... In astronomy, a plutino is a trans-Neptunian object that has a 3:2 orbital resonance with Neptune. ... Note: This article contains special characters. ... In astronomy, a resonant Trans-Neptunian Object is a Trans-Neptunian Object (TNO) in mean motion orbital resonance with Neptune. ... Image of the main asteroid belt between the orbits of Mars and Jupiter. ... Adjectives: Jovian Atmosphere Surface pressure: 70 kPa Composition: ~86% Hydrogen ~14% Helium 0. ... Hilda asteroids are asteroids with a semi-major axis between 3. ... 279 Thule is a very large Main belt asteroid. ... Image of the Trojan asteroids in front of and behind Jupiter along its orbital path. ... An extrasolar planet, or exoplanet, is a planet beyond the Solar System. ... Gliese 876 (Gl 876) is a red dwarf star (spectral type M4V) 15 light years from Earth in the constellation of Aquarius. ... Gliese 876 (Gl 876) is a red dwarf star (spectral type M4V) 15 light years from Earth in the constellation of Aquarius. ...

• 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. 253 Mathilde, a C-type asteroid. ... Adjectives: Jovian Atmosphere Surface pressure: 70 kPa Composition: ~86% Hydrogen ~14% Helium 0. ... Kirkwood gaps are gaps that appear in a graph if we classify the asteroids according to their periods, which is proportional to their mean radius from the Sun. ... The full set of rings The rings of Saturn are a series of planetary rings that orbit the planet Saturn. ... The full set of rings The rings of Saturn are a series of planetary rings that orbit the planet Saturn. ... The full set of rings The rings of Saturn are a series of planetary rings that orbit the planet Saturn. ... The full set of rings The rings of Saturn are a series of planetary rings that orbit the planet Saturn. ... Atmosphere none Mimas (mye-mÉ™s, IPA , Greek Μίμᾱς, rarely Μίμανς) is a moon of Saturn that was discovered in 1789 by William Herschel. ... The full set of rings The rings of Saturn are a series of planetary rings that orbit the planet Saturn. ... The full set of rings The rings of Saturn are a series of planetary rings that orbit the planet Saturn. ... Atmospheric characteristics Atmospheric pressure trace Oxygen 100% Ganymede (gan-É™-meed, IPA: ; Greek Γανυμήδης) is Jupiters largest moon, and indeed the largest moon in the entire solar system; it is larger in diameter than Mercury but only about half its mass. ... Atmospheric characteristics Atmospheric pressure 1 µPa Oxygen 100% Europa (ew-roe-pÉ™, IPA:  ; Greek Ευρώπη) is a moon of the planet Jupiter. ... Atmospheric characteristics Atmospheric pressure trace Sulfur dioxide 90% Io (eye-oe, IPA: , Greek Ῑώ) is the innermost of the four Galilean moons of Jupiter. ...

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°. A secular resonance is a type of orbital resonance. ... Precession refers to a change in the direction of the axis of a rotating object. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... The ascending node is one of the orbital nodes, a point in the orbit of an object where it crosses the plane of the ecliptic from the south celestial hemisphere to the north celestial hemisphere in the direction of motion. ... For the astrological concept, see Planets in astrology. ... In astrodynamics, under standard assumptions any orbit must be of conic section shape. ... Inclination in general is the angle between a reference plane and another plane or axis of direction. ... 253 Mathilde, a C-type asteroid. ... Adjective Saturnian Atmospheric characteristics Atmospheric pressure 140 kPa Hydrogen >93% Helium >5% Methane 0. ... A Mars-crosser asteroid is an asteroid whose orbit crosses that of Mars. ... Image of the main asteroid belt between the orbits of Mars and Jupiter. ... Note: This article contains special characters. ... Image of the main asteroid belt between the orbits of Mars and Jupiter. ... The astronomical unit (AU or au or a. ... Inclination in general is the angle between a reference plane and another plane or axis of direction. ...

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. The full set of rings The rings of Saturn are a series of planetary rings that orbit the planet Saturn. ... The full set of rings The rings of Saturn are a series of planetary rings that orbit the planet Saturn. ... Titan (tye-tən, IPA: , Greek Τιτάν) or Saturn VI is the largest moon of Saturn and the second largest moon in the solar system,[3] after Jupiters moon Ganymede. ...

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. Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... The tidal force is a secondary effect of the force of gravity and is responsible for the tides. ...

Mean motion resonances in the Solar System

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). Moons of the Solar System scaled to Earths Moon A natural satellite is an object that orbits a planet or other body larger than itself and which is not man-made. ... 253 Mathilde, a C-type asteroid. ... Artists rendering of the Kuiper Belt and hypothetical more distant Oort cloud. ... A planetary ring is a ring of dust and other small particles orbiting around a planet in a flat disc-shaped region. ...

• 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 only Laplace resonance

The simple integer ratios between periods are a convenient simplification hiding more complex relations: Adjectives: Plutonian Atmosphere Surface pressure: 0. ... Note: This article contains special characters. ... Atmosphere none Tethys (tee-thÉ™s or teth-É™s, IPA , Greek Τηθύς) is a moon of Saturn that was discovered by Giovanni Domenico Cassini in 1684. ... Atmosphere none Mimas (mye-mÉ™s, IPA , Greek Μίμᾱς, rarely Μίμανς) is a moon of Saturn that was discovered in 1789 by William Herschel. ... Atmosphere none Dione (dye-oe-nee, Greek Διώνη) is a moon of Saturn discovered by Giovanni Cassini in 1684. ... [5] Atmospheric characteristics Pressure trace, significant spatial variability [6], [7] Water vapour 91% [8] Carbon dioxide 3. ... Hyperion (hye-peer-ee-É™n, IPA , Greek Ὑπερίων) is a moon of Saturn discovered by William Cranch Bond, George Phillips Bond and William Lassell in 1848. ... Titan (tye-tÉ™n, IPA: , Greek Τιτάν) or Saturn VI is the largest moon of Saturn and the second largest moon in the solar system,[3] after Jupiters moon Ganymede. ... Atmospheric characteristics Atmospheric pressure trace Oxygen 100% Ganymede (gan-É™-meed, IPA: ; Greek Γανυμήδης) is Jupiters largest moon, and indeed the largest moon in the entire solar system; it is larger in diameter than Mercury but only about half its mass. ... Atmospheric characteristics Atmospheric pressure 1 µPa Oxygen 100% Europa (ew-roe-pÉ™, IPA:  ; Greek Ευρώπη) is a moon of the planet Jupiter. ... Atmospheric characteristics Atmospheric pressure trace Sulfur dioxide 90% Io (eye-oe, IPA: , Greek Ῑώ) is the innermost of the four Galilean moons of Jupiter. ...

• 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 The animation shows a set of simulated views of the Moon over one month, like a picture taken at the same time each day. ... In astrodynamics, under standard assumptions any orbit must be of conic section shape. ... An orbital node is one of the two points where an inclined orbit crosses a plane of reference (e. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... Mean Motion, , is a measure of how far a satellite has progressed around its orbit, from perigee. ...

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: This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ...

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. An orbital node is one of the two points where an inclined orbit crosses a plane of reference (e. ...

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: Image File history File links TheLaplaceResonance2. ... Image File history File links TheLaplaceResonance2. ...

Φ_L

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. In astrodynamics or celestial dynamics mean longitude of an orbiting body is . ...

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). Adjectives: Plutonian Atmosphere Surface pressure: 0. ... Note: This article contains special characters. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... The plane of the ecliptic is well seen in this picture from the 1994 lunar prospecting Clementine spacecraft. ... Adjective Uranian Atmospheric characteristics Atmospheric pressure 120 kPa (at the cloud level) Hydrogen 83% Helium 15% Methane 1. ... Adjectives: Plutonian Atmosphere Surface pressure: 0. ...

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). The integers are commonly denoted by the above symbol. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ...

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:

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. Adjectives: Venusian or (rarely) Cytherean Atmosphere Surface pressure: 9. ... Note: This article contains special characters. ... A year is the time between two recurrences of an event related to the orbit of the Earth around the Sun. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... Adjectives: Venusian or (rarely) Cytherean Atmosphere Surface pressure: 9. ... A year is the time between two recurrences of an event related to the orbit of the Earth around the Sun. ... Note: This article contains special characters. ... Adjectives: Venusian or (rarely) Cytherean Atmosphere Surface pressure: 9. ... Note: This article contains special characters. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... Adjective Saturnian Atmospheric characteristics Atmospheric pressure 140 kPa Hydrogen >93% Helium >5% Methane 0. ... Adjectives: Jovian Atmosphere Surface pressure: 70 kPa Composition: ~86% Hydrogen ~14% Helium 0. ... Adjective Uranian Atmospheric characteristics Atmospheric pressure 120 kPa (at the cloud level) Hydrogen 83% Helium 15% Methane 1. ... Adjectives: Jovian Atmosphere Surface pressure: 70 kPa Composition: ~86% Hydrogen ~14% Helium 0. ... Adjective Uranian Atmospheric characteristics Atmospheric pressure 120 kPa (at the cloud level) Hydrogen 83% Helium 15% Methane 1. ... Adjective Saturnian Atmospheric characteristics Atmospheric pressure 140 kPa Hydrogen >93% Helium >5% Methane 0. ... Note: This article contains special characters. ... Adjective Saturnian Atmospheric characteristics Atmospheric pressure 140 kPa Hydrogen >93% Helium >5% Methane 0. ... Note: This article contains special characters. ... Adjective Uranian Atmospheric characteristics Atmospheric pressure 120 kPa (at the cloud level) Hydrogen 83% Helium 15% Methane 1. ... Deimos (IPA or ; Greek Δείμος: Dread), is the smaller and outermost of Mars’ two moons, named after Deimos from Greek Mythology. ... Phobos (IPA or , Greek Φόβος: Fright), is the larger and innermost of Mars two moons (the other being Deimos), and is named after Phobos, son of Ares (Mars) from Greek Mythology. ... This article or section does not cite its references or sources. ... Atmospheric characteristics Atmospheric pressure trace Oxygen 100% Ganymede (gan-É™-meed, IPA: ; Greek Γανυμήδης) is Jupiters largest moon, and indeed the largest moon in the entire solar system; it is larger in diameter than Mercury but only about half its mass. ... Atmosphere none Rhea (ree-a, Greek ‘Ρέα) is the second largest moon of Saturn and was discovered in 1672 by Giovanni Domenico Cassini. ... Atmosphere none Dione (dye-oe-nee, Greek Διώνη) is a moon of Saturn discovered by Giovanni Cassini in 1684. ... Titan (tye-tÉ™n, IPA: , Greek Τιτάν) or Saturn VI is the largest moon of Saturn and the second largest moon in the solar system,[3] after Jupiters moon Ganymede. ... Atmosphere none Rhea (ree-a, Greek ‘Ρέα) is the second largest moon of Saturn and was discovered in 1672 by Giovanni Domenico Cassini. ... Iapetus (eye-ap-É™-tÉ™s, IPA , Greek Ιαπετός) is the third-largest moon of Saturn, discovered by Giovanni Domenico Cassini in 1671. ... Titan (tye-tÉ™n, IPA: , Greek Τιτάν) or Saturn VI is the largest moon of Saturn and the second largest moon in the solar system,[3] after Jupiters moon Ganymede. ... Atmospheric pressure 0 kPa Umbriel (um-bree-É™l, IPA ) is a moon of Uranus discovered on 1851-10-24 by William Lassell. ... Atmospheric pressure 0 kPa Miranda (mÉ™r-an-dÉ™, IPA: ) is the smallest and innermost of Uranus major moons. ... Atmospheric pressure 0 kPa Umbriel (um-bree-É™l, IPA ) is a moon of Uranus discovered on 1851-10-24 by William Lassell. ... Atmospheric pressure 0 kPa Ariel (air-ee-É™l, IPA ) is a moon of Uranus discovered on 24 October 1851 by William Lassell. ... Atmospheric pressure   Titania (ti-taan-ee-É™ or tye-tan-ee-É™) is the largest moon of Uranus. ... Atmospheric pressure 0 kPa Umbriel (um-bree-É™l, IPA ) is a moon of Uranus discovered on 1851-10-24 by William Lassell. ... Atmospheric pressure 0 kPa Oberon (oe-bur-on) is the outermost of the major moons of the planet Uranus. ... Atmospheric pressure   Titania (ti-taan-ee-É™ or tye-tan-ee-É™) is the largest moon of Uranus. ... Nix (formerly known as S/2005 P 2), is a natural satellite of Pluto. ... Media:Example. ... Hydra (formerly known as S/2005 P 1) is a natural satellite of Pluto. ... Media:Example. ...

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. Adjectives: Plutonian Atmosphere Surface pressure: 0. ... The planet Pluto has three known moons. ...

See also

• Commensurability (astronomy)
• Secular resonance
• Kozai resonance
• Dermott's Law
• Lagrangian points
• Mercury, which has a 3:2 spin-orbit resonance.
• Tidal locking
• Tidal resonance
• Titius-Bode law
• Kirkwood gap

A secular resonance is a type of orbital resonance. ... A contour plot of the effective potential of a two-body system (the Sun and Earth here), showing the five Lagrange points. ... Note: This article contains special characters. ... Tidal locking makes one side of an astronomical body always face another, like the Moon facing the Earth. ... In oceanography, tidal resonance is a phenomenon perhaps best exemplified in the Bay of Fundy. ... The Titius-Bode law (or Bodes law) is the observation that orbits of planets in the solar system closely follow a simple geometric rule. ... Kirkwood gaps are gaps that appear in a graph if we classify the asteroids according to their periods, which is proportional to their mean radius from the Sun. ...

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.

Significant figures (also called significant digits and abbreviated sig figs or sig digs, respectively) is a method of expressing errors in measurements. ...

External links

• Locations of Solar System Planetary Mean-Motion Resonances. Web calculator that plots distributions of the semimajor 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).