In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other. 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. ...

History

Ever since the discovery of Newton's laws of motion 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). Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Pierre-Simon Laplace. ... The solar system comprises the Earths Sun and the retinue of celestial objects gravitationally bound to it. ...

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: Look up Eccentricity in Wiktionary, the free dictionary. ... 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 3:2 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.
• The Trojan asteroids may be regarded as being protected by a 1:1 resonance with Jupiter.
• The extrasolar planets Gliese 876b and Gliese 876c are in a 2:1 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. ... Adjective Neptunian Atmospheric characteristics Surface pressure ≫100 MPa Hydrogen - H2 80% ±3. ... Image of the Trojan asteroids in front of and behind Jupiter along its orbital path. ... Infrared image of the star GQ Lupi (A) orbited by a planet (b) at a distance of approximately 20 times the distance between Jupiter and our Sun. ... 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. ...

• There is a series of almost empty lanes in the asteroid belt called Kirkwood gaps corresponding to mean-motion resonances with Jupiter. Almost all asteroids in those regions have been ejected by the repeated perturbations.

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. Image of the main asteroid belt between the orbits of Mars and Jupiter. ... 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. ... Adjective Jovian Atmospheric characteristics Atmospheric pressure 70 kPa Hydrogen ~86% Helium ~14% Methane 0. ... Atmospheric characteristics Atmospheric pressure trace Oxygen 100% Ganymede (gan-ə-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 redirects here. ... 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 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. ... A planet is generally considered to be a relatively large mass of accreted matter in orbit around a star. ... Look up Eccentricity in Wiktionary, the free dictionary. ... Inclination in general is the angle between a reference plane and another plane or axis of direction. ...

• 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°.

Secular resonances can be linear An asteroid is a small, solid object in our Solar System, orbiting the Sun. ... Note: This article contains special characters. ... 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. ... For the Roman god, see Mars (mythology). ... The asteroid belt is a region of the solar system falling roughly between the planets Mars and Jupiter where the greatest concentration of asteroid orbits can be found. ... 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. ...

Mean motion resonances in the Solar System

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

• 2:3 Neptune-Pluto
• 4:2 Mimas-Tethys (Saturn’s moons)
• 2:1 Enceladus-Dione (Saturn’s moons)
• 4:3 Titan-Hyperion (Saturn's moons)
• 1:2:4 Io-Europa-Ganymede (Jupiter’s moons); the only Laplace resonance
• 1:4:6 Charon-P2-P1 (Pluto's moons)

The simple integer ratios between periods are a convenient simplification hiding more complex relations: Adjective Neptunian Atmospheric characteristics Surface pressure ≫100 MPa Hydrogen - H2 80% ±3. ... Adjective Plutonian Atmospheric characteristics Atmospheric pressure 0. ... Atmosphere none Mimas (mye-məs, IPA , Greek Μίμᾱς, rarely Μίμανς) is a moon of Saturn that was discovered in 1789 by William Herschel. ... Atmosphere none Tethys (tee-thəs or teth-əs, IPA , Greek Τηθύς) is a moon of Saturn that was discovered by Giovanni Domenico Cassini in 1684. ... [5] Atmospheric characteristics Pressure trace, significant spatial variability [6] Water vapour 65% [7] Hydrogen 20% [8] Other CO2, CO, N2 [9] Enceladus (en-sel-ə-dəs, IPA , Greek Εγκέλαδος) is the sixth-largest moon of Saturn, discovered in 1789 by William Herschel [10]. Despite its small size, Enceladus has a wide... Atmosphere none Dione (dye-oe-nee, Greek Διώνη) is a moon of Saturn discovered by Giovanni Cassini in 1684. ... Atmospheric characteristics Pressure 146. ... Hyperion (hye-peer-ee-ən, IPA , Greek Ὑπερίων) is a moon of Saturn discovered by William Cranch Bond, George Phillips Bond and William Lassell in 1848. ... Atmospheric characteristics Atmospheric pressure trace Sulfur dioxide 90% Io (eye-oe, IPA: , Greek Ῑώ) is the innermost of the four Galilean moons of Jupiter. ... Atmospheric characteristics Atmospheric pressure 1 µPa Oxygen 100% Europa redirects here. ... Atmospheric characteristics Atmospheric pressure trace Oxygen 100% Ganymede (gan-ə-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. ... Media:Example. ...

• 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 1:2 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 Although the Moon keeps the same side towards Earth, careful observations will reveal you can actually see 59% of the Moons surface. ... Look up Eccentricity in Wiktionary, the free dictionary. ... 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 the 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. The dark moon is Ganymede, the gray moon is Europa, and the yellow moon is Io

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 graphic 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 . ...

'Near' mean motion resonances

Other near resonances exist among the moons including:

Saturn system

• (5:3) Rhea-Dione

Uranus system 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. ...

• (3:1) Umbriel-Miranda
• (5:3) Umbriel-Ariel
• (2:1) Titania-Umbriel
• (3:2) Oberon-Titania

The absence of (precise) resonances in the Uranus system, given their abundance in the Saturn and Jupiter systems is actually a bit of enigma. 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 (mi-ran-da) 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 1851-10-24 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. ...

One can claim 'near' resonances among the Planets e.g.

• (2:1) Neptune-Uranus
• (3:1) Uranus-Saturn
• (5:2) Jupiter-Saturn

(see Titius-Bode law). However, in spite of efforts, no significance has been identified so far for these near commensurabilities. The Titius-Bode law (or Bodes law) is the observation that orbits of planets in the solar system closely follow a simple geometric rule. ...

See also

• Lagrangian points
• Tidal locking
• Tidal resonance
• Titius-Bode law

A contour plot of the effective potential of a two-body system (the Sun and Earth here), showing the 5 Lagrange points. ... 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. ...

External links

Malhotra Orbital Resonances and Chaos in the Solar System, preprint [1]

References

Murray, Dermot Solar System Dynamics, Cambridge University Press, ISBN 0-521-57597-4