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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 4:2:1 resonance of Jupiter's moons Ganymede, Io, and Europa, and the 3:2 resonance between Neptune and Pluto. 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 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: 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°. 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 There are only a few known mean motion resonances in the Solar system involving planets or larger satellites (a much greater number involve asteroids, 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 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 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: 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. The Laplace resonance The most remarkable resonance involving Io-Europa-Ganymede includes the following relation locking the orbital phase of the moons: where 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 Near mean motion resonances Other near resonances exist among the moons including: Saturn system Uranus system The absence of (precise) resonances in the Uranus system, given their abundance in the Saturn and Jupiter systems is actually a bit of an enigma. See also | ||||||||
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