![Galilean moon Laplace resonance animation 2](https://upload.wikimedia.org/wikipedia/commons/e/e5/Galilean_moon_Laplace_resonance_animation_2.gif)
In
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
, orbital resonance occurs when
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
ing bodies exert regular, periodic
gravitational
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
influence on each other, usually because their
orbital period
The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets ...
s are related by a ratio of small
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. Most commonly, this relationship is found between a pair of objects (binary resonance). The physical principle behind orbital resonance is similar in concept to pushing a child on a
swing
Swing or swinging may refer to:
Apparatus
* Swing (seat), a hanging seat that swings back and forth
* Pendulum, an object that swings
* Russian swing, a swing-like circus apparatus
* Sex swing, a type of harness for sexual intercourse
* Swing rid ...
, whereby the orbit and the swing both have a
natural frequency
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force.
The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all par ...
, and the body doing the "pushing" will act in periodic repetition to have a cumulative effect on the motion. Orbital resonances greatly enhance the mutual gravitational influence of the bodies (i.e., their ability to alter or constrain each other's orbits). In most cases, this results in an ''unstable'' interaction, in which the bodies exchange
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be self-correcting and thus stable. Examples are the 1:2:4 resonance of
Jupiter
Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but ...
's moons
Ganymede,
Europa
Europa may refer to:
Places
* Europe
* Europa (Roman province), a province within the Diocese of Thrace
* Europa (Seville Metro), Seville, Spain; a station on the Seville Metro
* Europa City, Paris, France; a planned development
* Europa Cliff ...
and
Io, and the 2:3 resonance between
Pluto
Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of trans-Neptunian object, bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the S ...
and
Neptune
Neptune is the eighth planet from the Sun and the farthest known planet in the Solar System. It is the fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 times ...
. Unstable resonances with
Saturn
Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with an average radius of about nine and a half times that of Earth. It has only one-eighth the average density of Earth; h ...
's inner moons give rise to gaps in the
rings of Saturn
The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirel ...
. The special case of 1:1 resonance between bodies with similar orbital radii causes large
solar system
The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...
bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of
clearing the neighbourhood
"Clearing the neighbourhood" (or dynamical dominance) around a celestial body's orbit describes the body becoming gravitationally dominant such that there are no other bodies of comparable size other than its natural satellites or those otherwise ...
, an effect that is used in the current
definition of a planet
The definition of ''planet'', since the word was coined by the ancient Greeks, has included within its scope a wide range of celestial bodies. Greek astronomy, Greek astronomers employed the term (), 'wandering stars', for star-like objects wh ...
.
A binary resonance ratio in this article should be interpreted as the ''ratio of number of orbits'' completed in the same time interval, rather than as the ''ratio of orbital periods'', which would be the inverse ratio. Thus, the 2:3 ratio above means that Pluto completes two orbits in the time it takes Neptune to complete three. In the case of resonance relationships among three or more bodies, either type of ratio may be used (whereby the smallest whole-integer ratio sequences are not necessarily reversals of each other), and the type of ratio will be specified.
History
Since the discovery of
Newton's law of universal gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
in the 17th century, the
stability of the Solar System
The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable when historically observed, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ...
has preoccupied many mathematicians, starting with
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
. The stable orbits that arise in a
two-body approximation ignore the influence of other bodies. The effect of these added interactions on the stability of the
Solar System
The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...
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.
It was Laplace who found the first answers explaining the linked orbits of the
Galilean moon
The Galilean moons (), or Galilean satellites, are the four largest moons of Jupiter: Io, Europa, Ganymede, and Callisto. They were first seen by Galileo Galilei in December 1609 or January 1610, and recognized by him as satellites of Jupiter ...
s (see below). 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
The ''musica universalis'' (literally universal music), also called music of the spheres or harmony of the spheres, is a philosophical concept that regards proportions in the movements of celestial bodies – the Sun, Moon, and planets – as a f ...
''.
The article on
resonant interaction
In nonlinear systems, a resonant interaction is the interaction of three or more waves, usually but not always of small amplitude. Resonant interactions occur when a simple set of criteria coupling wave-vectors and the dispersion equation are met ...
s describes resonance in the general modern setting. A primary result from the study of
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s 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 analog here would be that a more massive body provides a periodic gravitational kick to a smaller body, as it passes by. The mode-locking regions are named
Arnold tongue
In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamica ...
s.
Types of resonance
![PIA10452 - Saturn A ring spiral density waves](https://upload.wikimedia.org/wikipedia/commons/5/5c/PIA10452_-_Saturn_A_ring_spiral_density_waves.jpg)
In general, an orbital resonance may
*involve one or any combination of the orbit parameters (e.g.
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-center, in geometry
* Eccentricity (graph theory) of a v ...
versus
semimajor axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longe ...
, or eccentricity versus
inclination
Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a Plane of reference, reference plane and the orbital plane or Axis of rotation, axis of direction of the orbiting object ...
).
*act on any time scale from short term, commensurable with the orbit periods, to
secular
Secularity, also the secular or secularness (from Latin ''saeculum'', "worldly" or "of a generation"), is the state of being unrelated or neutral in regards to religion. Anything that does not have an explicit reference to religion, either negativ ...
, measured in 10
4 to 10
6 years.
*lead to either long-term stabilization of the orbits or be the cause of their destabilization.
A ''mean-motion orbital resonance'' occurs when two bodies have periods of
revolution
In political science, a revolution (Latin: ''revolutio'', "a turn around") is a fundamental and relatively sudden change in political power and political organization which occurs when the population revolts against the government, typically due ...
that are a simple integer ratio of each other. Depending on the details, this can either stabilize or destabilize the orbit.
''Stabilization'' may occur when the two bodies move in such a synchronised fashion that they never closely approach. For instance:
*The orbits of
Pluto
Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of trans-Neptunian object, bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the S ...
and the
plutino
In astronomy, the plutinos are a dynamical group of trans-Neptunian objects that orbit in 2:3 mean-motion resonance with Neptune. This means that for every two orbits a plutino makes, Neptune orbits three times. The dwarf planet Pluto is the lar ...
s are stable, despite crossing that of the much larger
Neptune
Neptune is the eighth planet from the Sun and the farthest known planet in the Solar System. It is the fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 times ...
, because they are in a 2:3 resonance with it. The resonance ensures that, when they approach perihelion and Neptune's orbit, Neptune is consistently distant (averaging a quarter of its orbit away). 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 object
In astronomy, a resonant trans-Neptunian object is a trans-Neptunian object (TNO) in mean-motion orbital resonance with Neptune. The orbital periods of the resonant objects are in a simple integer relations with the period of Neptune, e.g. 1:2, 2 ...
s occupying the 1:1 (
Neptune trojan
Neptune trojans are bodies that orbit the Sun near one of the stable Lagrangian points of Neptune, similar to the Trojan (astronomy), trojans of other planets. They therefore have approximately the same orbital period as Neptune and follow rough ...
s),
3:5,
4:7, 1:2 (
twotinos) and
2:5 resonances, among others, with respect to Neptune.
*In the
asteroid belt
The asteroid belt is a torus-shaped region in the Solar System, located roughly between the orbits of the planets Jupiter and Mars. It contains a great many solid, irregularly shaped bodies, of many sizes, but much smaller than planets, called ...
beyond 3.5 AU from the Sun, the 3:2, 4:3 and 1:1 resonances with
Jupiter
Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but ...
are populated by ''clumps'' of asteroids (the
Hilda family
The Hilda asteroids (adj. ''Hildian'') are a dynamical group of more than 5,000 asteroids located beyond the asteroid belt but within Jupiter's orbit, in a 3:2 orbital resonance with Jupiter. The namesake is the asteroid 153 Hilda.
Hildas move i ...
, the few
Thule asteroid
Thule, minor planet designation: 279 Thule, is a large asteroid from the outer asteroid belt. It is classified as a D-type asteroid and is probably composed of organic-rich silicates, carbon and anhydrous silicates. Thule was the first asteroid ...
s, and the numerous
Trojan asteroids
In astronomy, a trojan is a small celestial body (mostly asteroids) that shares the orbit of a larger body, remaining in a stable orbit approximately 60° ahead of or behind the main body near one of its Lagrangian points and . Trojans can shar ...
, respectively).
Orbital resonances can also ''destabilize'' one of the orbits. This process can be exploited to find energy-efficient ways of
deorbit
Atmospheric entry is the movement of an object from outer space into and through the gases of an atmosphere of a planet, dwarf planet, or natural satellite. There are two main types of atmospheric entry: ''uncontrolled entry'', such as the entr ...
ing spacecraft.
For small bodies, destabilization is actually far more likely. For instance:
*In the
asteroid belt
The asteroid belt is a torus-shaped region in the Solar System, located roughly between the orbits of the planets Jupiter and Mars. It contains a great many solid, irregularly shaped bodies, of many sizes, but much smaller than planets, called ...
within 3.5 AU from the Sun, the major mean-motion resonances with
Jupiter
Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but ...
are locations of ''gaps'' in the asteroid distribution, the
Kirkwood gap
A Kirkwood gap is a gap or dip in the distribution of the semi-major axes (or equivalently of the orbital periods) of the orbits of main-belt asteroids. They correspond to the locations of orbital resonances with Jupiter.
For example, there ar ...
s (most notably at the 4:1, 3:1, 5:2, 7:3 and 2:1 resonances).
Asteroid
An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere.
...
s have been ejected from these almost empty lanes by repeated perturbations. However, there are still populations of asteroids temporarily present in or near these resonances. For example, asteroids of the
Alinda family
The Alinda asteroids are a dynamical group of asteroids with a semi-major axis of about 2.5 AU and an orbital eccentricity approximately between 0.4 and 0.65.rings of Saturn
The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirel ...
, the
Cassini Division
The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirel ...
is a gap between the inner
B Ring
The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirel ...
and the outer
A Ring
The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirel ...
that has been cleared by a 2:1 resonance with the moon
Mimas
Mimas may refer to:
*Mimas (Giant), son of Gaia in Greek mythology, one of the Gigantes
* Mimas (''Aeneid''), a son of Amycus and Theono, born the same night as Paris, who escorted Aeneas to Italy
*Karaburun, a town and district in Turkey, formerl ...
. (More specifically, the site of the resonance is the
Huygens Gap, which bounds the outer edge of the
B Ring
The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirel ...
.)
*In the rings of Saturn, the
Encke
Encke may refer to:
*Johann Franz Encke (1791–1865), a nineteenth-century German astronomer
**Encke (crater), a lunar crater
**Encke Division, a dark gap in the rings of Saturn
**Comet Encke, a short-period comet
***Encke (horse) (2009–2014), ...
and
Keeler gaps within the A Ring are cleared by 1:1 resonances with the embedded moonlets
Pan and
Daphnis
In Greek mythology, Daphnis (; grc, Δάφνις, from , ''daphne'', "Bay Laurel") was a Sicilian shepherd who was said to be the inventor of pastoral poetry.
Family
According to tradition, he was the son of Hermes and a nymph, despite which ...
, respectively. The A Ring's outer edge is maintained by a destabilizing 7:6 resonance with the moon
Janus
In ancient Roman religion and myth, Janus ( ; la, Ianvs ) is the god of beginnings, gates, transitions, time, duality, doorways, passages, frames, and endings. He is usually depicted as having two faces. The month of January is named for Janu ...
.
Most bodies that are in resonance orbit in the same direction; however, the
retrograde asteroid
514107 Kaʻepaokaʻawela
514107 Kaʻepaokaʻāwela (), provisionally designated and nicknamed Bee-Zed, is a small asteroid, approximately in diameter, in a resonant, co-orbital motion with Jupiter. It is an unusual minor planet in that its orbit is retrograde, which ...
appears to be in a stable (for a period of at least a million years) 1:−1 resonance with Jupiter.
In addition, a few retrograde
damocloids
Damocloids are a List of minor-planet groups, class of minor planets such as 5335 Damocles and 1996 PW that have Halley-type comet, Halley-type or long-period highly eccentric orbits typical of periodic comets such as Halley's Comet, bu ...
have been found that are temporarily captured in mean-motion resonance with
Jupiter
Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but ...
or
Saturn
Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with an average radius of about nine and a half times that of Earth. It has only one-eighth the average density of Earth; h ...
.
Such orbital interactions are weaker than the corresponding interactions between bodies orbiting in the same direction.
A ''Laplace resonance'' is a three-body resonance with a 1:2:4 orbital period ratio (equivalent to a 4:2:1 ratio of orbits). The term arose because
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
discovered that such a resonance governed the motions of Jupiter's moons
Io,
Europa
Europa may refer to:
Places
* Europe
* Europa (Roman province), a province within the Diocese of Thrace
* Europa (Seville Metro), Seville, Spain; a station on the Seville Metro
* Europa City, Paris, France; a planned development
* Europa Cliff ...
, and
Ganymede. It is now also often applied to other 3-body resonances with the same ratios,
such as that between the
extrasolar planet
An exoplanet or extrasolar planet is a planet outside the Solar System. The first possible evidence of an exoplanet was noted in 1917 but was not recognized as such. The first confirmation of detection occurred in 1992. A different planet, init ...
s
Gliese 876
Gliese 876 is a red dwarf approximately 15 light-years away from Earth in the constellation of Aquarius. It is one of the closest known stars to the Sun confirmed to possess a planetary system with more than two planets, after Gliese ...
c, b, and e.
Three-body resonances involving other simple integer ratios have been termed "Laplace-like"
or "Laplace-type".
A ''
Lindblad resonance
A Lindblad resonance, named for the Swedish galactic astronomer Bertil Lindblad, is an orbital resonance in which an object's epicyclic frequency (the rate at which one periapse follows another) is a simple multiple of some forcing frequency. Res ...
'' drives
spiral density waves both in
galaxies
A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...
(where stars are subject to
forcing by the spiral arms themselves) and in
Saturn's rings
The rings of Saturn are the most extensive ring system of any planet in the Solar System. They consist of countless small particles, ranging in size from micrometers to meters, that orbit around Saturn. The ring particles are made almost entirel ...
(where ring particles are subject to forcing by
Saturn's moons
The moons of Saturn are numerous and diverse, ranging from tiny moonlets only tens of meters across to enormous Titan (moon), Titan, which is larger than the planet Mercury (planet), Mercury. Saturn has 83 natural satellite, moons with confirmed ...
).
A ''
secular resonance A secular resonance is a type of orbital resonance between two bodies with synchronized precessional frequencies. In celestial mechanics, secular refers to the long-term motion of a system, and resonance is periods or frequencies being a simple nu ...
'' occurs when the
precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
of two orbits is synchronised (usually a precession of the
perihelion
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any ellip ...
or
ascending node
An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes.
Planes of reference
Common planes of reference ...
). A small body in secular resonance with a much larger one (e.g. a
planet
A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
) 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
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-center, in geometry
* Eccentricity (graph theory) of a v ...
and
inclination
Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a Plane of reference, reference plane and the orbital plane or Axis of rotation, axis of direction of the orbiting object ...
of the small body.
Several prominent examples of secular resonance involve Saturn. There is a near-resonance between the precession of Saturn's rotational axis and that of Neptune's orbital axis (both of which have periods of about 1.87 million years), which has been identified as the likely source of Saturn's large
axial tilt
In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orbi ...
(26.7°). Initially, Saturn probably had a tilt closer to that of Jupiter (3.1°). The gradual depletion of the Kuiper belt would have decreased the precession rate of Neptune's orbit; eventually, the frequencies matched, and Saturn's axial precession was captured into a spin-orbit resonance, leading to an increase in Saturn's obliquity. (The angular momentum of Neptune's orbit is 10
4 times that of Saturn's rotation rate, and thus dominates the interaction.) However, it seems that the resonance no longer exists. Detailed analysis of data from the
Cassini spacecraft Cassini may refer to:
People
* Cassini (surname)
* Oleg Cassini (1913-2006), American fashion designer
:Cassini family:
* Giovanni Domenico Cassini (1625–1712), Italian mathematician, astronomer, engineer, and astrologer
* Jacques Cassini (1 ...
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. One theory for why the resonance came to an end is that there was another moon around Saturn whose orbit destabilized about 100 million years ago, perturbing Saturn.
The
perihelion secular resonance between
asteroid
An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere.
...
s and
Saturn
Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with an average radius of about nine and a half times that of Earth. It has only one-eighth the average density of Earth; h ...
(''ν
6'' = ''g'' − ''g
6'') helps shape the asteroid belt (the subscript "6" identifies Saturn as the sixth planet from the Sun). 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
The asteroid belt is a torus-shaped region in the Solar System, located roughly between the orbits of the planets Jupiter and Mars. It contains a great many solid, irregularly shaped bodies, of many sizes, but much smaller than planets, called ...
by a close pass to
Mars
Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury (planet), Mercury. In the English language, Mars is named for the Mars (mythology), Roman god of war. Mars is a terr ...
. This resonance forms the inner and "side" boundaries of the
asteroid belt
The asteroid belt is a torus-shaped region in the Solar System, located roughly between the orbits of the planets Jupiter and Mars. It contains a great many solid, irregularly shaped bodies, of many sizes, but much smaller than planets, called ...
around 2
AU, and at inclinations of about 20°.
Numerical simulations have suggested that the eventual formation of a perihelion secular resonance between
Mercury
Mercury commonly refers to:
* Mercury (planet), the nearest planet to the Sun
* Mercury (element), a metallic chemical element with the symbol Hg
* Mercury (mythology), a Roman god
Mercury or The Mercury may also refer to:
Companies
* Merc ...
and Jupiter (''g
1'' = ''g
5'') has the potential to greatly increase Mercury's eccentricity and possibly destabilize the inner Solar System several billion years from now.
The
Titan Ringlet within Saturn's
C Ring represents another type of resonance in which the rate of
apsidal precession
In celestial mechanics, apsidal precession (or apsidal advance) is the precession (gradual rotation) of the line connecting the apsides (line of apsides) of an astronomical body's orbit. The apsides are the orbital points closest (periapsi ...
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
Titan most often refers to:
* Titan (moon), the largest moon of Saturn
* Titans, a race of deities in Greek mythology
Titan or Titans may also refer to:
Arts and entertainment
Fictional entities
Fictional locations
* Titan in fiction, fictiona ...
.
A ''
Kozai resonance
In celestial mechanics, the Kozai mechanism is a dynamical phenomenon affecting the orbit of a binary system perturbed by a distant third body under certain conditions. It is also known as the von Zeipel-Kozai-Lidov, Lidov–Kozai mechanism, Kozai ...
'' 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; as a consequence, such orbits tend to be unstable, since the growing eccentricity would result in small
pericenters, typically leading to a collision or (for large moons) destruction by
tidal forces
The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenom ...
.
In an example of another type of resonance involving orbital eccentricity, the eccentricities of Ganymede and Callisto vary with a common period of 181 years, although with opposite phases.
Mean-motion resonances in the Solar System
![Haumea](https://upload.wikimedia.org/wikipedia/commons/3/3e/Haumea.GIF)
There are only a few known mean-motion resonances (MMR) in the
Solar System
The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar S ...
involving planets,
dwarf planet
A dwarf planet is a small planetary-mass object that is in direct orbit of the Sun, smaller than any of the eight classical planets but still a world in its own right. The prototypical dwarf planet is Pluto. The interest of dwarf planets to p ...
s or larger
satellites
A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisotop ...
(a much greater number involve
asteroid
An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere.
...
s,
planetary ring
A ring system is a disc or ring, orbiting an astronomical object, that is composed of solid material such as dust and moonlets, and is a common component of satellite systems around giant planets. A ring system around a planet is also known as ...
s,
moonlets and smaller
Kuiper belt
The Kuiper belt () is a circumstellar disc in the outer Solar System, extending from the orbit of Neptune at 30 astronomical units (AU) to approximately 50 AU from the Sun. It is similar to the asteroid belt, but is far larger—20 times ...
objects, including many
possible dwarf planets
The number of dwarf planets in the Solar System is unknown. Estimates have run as high as 200 in the Kuiper belt and over 10,000 in the region beyond.
However, consideration of the surprisingly low densities of many large trans-Neptunian objects ...
).
* 2:3
Pluto
Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of trans-Neptunian object, bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the S ...
–
Neptune
Neptune is the eighth planet from the Sun and the farthest known planet in the Solar System. It is the fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 times ...
(also and other
plutino
In astronomy, the plutinos are a dynamical group of trans-Neptunian objects that orbit in 2:3 mean-motion resonance with Neptune. This means that for every two orbits a plutino makes, Neptune orbits three times. The dwarf planet Pluto is the lar ...
s)
* 2:4
Tethys–
Mimas
Mimas may refer to:
*Mimas (Giant), son of Gaia in Greek mythology, one of the Gigantes
* Mimas (''Aeneid''), a son of Amycus and Theono, born the same night as Paris, who escorted Aeneas to Italy
*Karaburun, a town and district in Turkey, formerl ...
(Saturn's moons). Not simplified, because the libration of the nodes must be taken into account.
* 1:2
Dione Dione may refer to:
Astronomy
*106 Dione, a large main belt asteroid
*Dione (moon), a moon of Saturn
*Helene (moon), a moon of Saturn sometimes referred to as "Dione B"
Mythology
*Dione (Titaness), a Titaness in Greek mythology
*Dione (mythology) ...
–
Enceladus
Enceladus is the sixth-largest moon of Saturn (19th largest in the Solar System). It is about in diameter, about a tenth of that of Saturn's largest moon, Titan. Enceladus is mostly covered by fresh, clean ice, making it one of the most refl ...
(Saturn's moons)
* 3:4
Hyperion–
Titan
Titan most often refers to:
* Titan (moon), the largest moon of Saturn
* Titans, a race of deities in Greek mythology
Titan or Titans may also refer to:
Arts and entertainment
Fictional entities
Fictional locations
* Titan in fiction, fictiona ...
(Saturn's moons)
* 1:2:4
Ganymede–
Europa
Europa may refer to:
Places
* Europe
* Europa (Roman province), a province within the Diocese of Thrace
* Europa (Seville Metro), Seville, Spain; a station on the Seville Metro
* Europa City, Paris, France; a planned development
* Europa Cliff ...
–
Io (Jupiter's moons, ratio of ''orbits'').
Additionally,
Haumea
Haumea (minor-planet designation 136108 Haumea) is a dwarf planet located trans-Neptunian object, beyond Neptune's orbit. It was discovered in 2004 by a team headed by Michael E. Brown, Mike Brown of Caltech at the Palomar Observatory in the United ...
is thought to be in a 7:12 resonance with Neptune,
and is thought to be in a 3:10 resonance with Neptune.
The simple integer ratios between periods hide more complex relations:
*the point of
conjunction
Conjunction may refer to:
* Conjunction (grammar), a part of speech
* Logical conjunction, a mathematical operator
** Conjunction introduction, a rule of inference of propositional logic
* Conjunction (astronomy), in which two astronomical bodies ...
can oscillate (
librate) around an equilibrium point defined by the resonance.
*given non-zero
eccentricities, the
nodes
In general, a node is a localized swelling (a "knot") or a point of intersection (a Vertex (graph theory), vertex).
Node may refer to:
In mathematics
*Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two ...
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 motion
In orbital mechanics, mean motion (represented by ''n'') is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the ac ...
s
(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 perfect, but it involves also the precession of
perijove
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any elli ...
(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
In general, a node is a localized swelling (a "knot") or a point of intersection (a Vertex (graph theory), vertex).
Node may refer to:
In mathematics
*Vertex (graph theory), a vertex in a mathematical graph
*Vertex (geometry), a point where two ...
of the two moons.
Laplace resonance
![TheLaplaceResonance2](https://upload.wikimedia.org/wikipedia/commons/0/05/TheLaplaceResonance2.png)
The Laplace resonance involving Io–Europa–Ganymede includes the following relation locking the ''orbital phase'' of the moons:
:
where
are
mean longitude Mean longitude is the ecliptic longitude at which an orbiting body could be found if its orbit were circular and free of perturbations. While nominally a simple longitude, in practice the mean longitude does not correspond to any one physical angle ...
s of the moons (the second equals sign ignores libration).
This relation makes a triple conjunction impossible. (A Laplace resonance in the
Gliese 876
Gliese 876 is a red dwarf approximately 15 light-years away from Earth in the constellation of Aquarius. It is one of the closest known stars to the Sun confirmed to possess a planetary system with more than two planets, after Gliese ...
system, in contrast, is associated with one triple conjunction per orbit of the outermost planet, ignoring libration.) The graph illustrates the positions of the moons after 1, 2 and 3 Io periods.
librates about 180° with an amplitude of 0.03°.
Another "Laplace-like" resonance involves the
moons
A natural satellite is, in the most common usage, an astronomical body that orbits a planet, dwarf planet, or small Solar System body (or sometimes another natural satellite). Natural satellites are often colloquially referred to as ''moons'' ...
Styx
In Greek mythology, Styx (; grc, Στύξ ) is a river that forms the boundary between Earth (Gaia) and the Underworld. The rivers Acheron, Cocytus, Lethe, Phlegethon, and Styx all converge at the centre of the underworld on a great marsh, whic ...
,
Nix
Nix or NIX may refer to:
Places
* Nix, Alabama, an unincorporated community, United States
* Nix, Texas, a ghost town in southwestern Lampasas County, Texas, United States
* Nix (moon), a moon of Pluto
People
* Nix (surname), listing people with ...
and
Hydra
Hydra generally refers to:
* Lernaean Hydra, a many-headed serpent in Greek mythology
* ''Hydra'' (genus), a genus of simple freshwater animals belonging to the phylum Cnidaria
Hydra or The Hydra may also refer to:
Astronomy
* Hydra (constel ...
of Pluto:
:
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
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
); the respective ratio of orbits is 11:9:6. Based on the ratios of
synodic period
The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets, ...
s, there are 5 conjunctions of Styx and Hydra and 3 conjunctions of Nix and Hydra for every 2 conjunctions of Styx and Nix.
As with the Galilean satellite resonance, triple conjunctions are forbidden.
librates about 180° with an amplitude of at least 10°.
Plutino resonances
The dwarf planet
Pluto
Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of trans-Neptunian object, bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the S ...
is following an orbit trapped in a web of resonances with
Neptune
Neptune is the eighth planet from the Sun and the farthest known planet in the Solar System. It is the fourth-largest planet in the Solar System by diameter, the third-most-massive planet, and the densest giant planet. It is 17 times ...
. The resonances include:
*A mean-motion resonance of 2:3
*The resonance of the
perihelion
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any ellip ...
(
libration
In lunar astronomy, libration is the wagging or wavering of the Moon perceived by Earth-bound observers and caused by changes in their perspective. It permits an observer to see slightly different hemispheres of the surface at different tim ...
around 90°), keeping the perihelion above the
ecliptic
The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic again ...
*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 17 AU, while the minimum separation between Pluto and
Uranus
Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus (mythology), Uranus (Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars (mythology), Mars), grandfather ...
is just 11 AU (see
Pluto's orbit for detailed explanation and graphs).
The next largest body in a similar 2:3 resonance with Neptune, called a ''
plutino
In astronomy, the plutinos are a dynamical group of trans-Neptunian objects that orbit in 2:3 mean-motion resonance with Neptune. This means that for every two orbits a plutino makes, Neptune orbits three times. The dwarf planet Pluto is the lar ...
'', is the probable dwarf planet
Orcus
Orcus ( la, Orcus) was a god of the underworld, punisher of broken oaths in Etruscan and Roman mythology. As with Hades, the name of the god was also used for the underworld itself. In the later tradition, he was conflated with Dis Pater.
A ...
. Orcus has an orbit similar in inclination and eccentricity to Pluto's. However, the two are constrained by their mutual resonance with Neptune to always be in opposite phases of their orbits; Orcus is thus sometimes described as the "anti-Pluto".
Naiad:Thalassa 73:69 resonance
Neptune's innermost moon,
Naiad
In Greek mythology, the naiads (; grc-gre, ναϊάδες, naïádes) are a type of female spirit, or nymph, presiding over fountains, wells, springs, streams, brooks and other bodies of fresh water.
They are distinct from river gods, who ...
, is in a 73:69 fourth-order resonance with the next outward moon,
Thalassa
Thalassa (; grc-gre, Θάλασσα, Thálassa, sea; Attic Greek: , ''Thálatta'') was the general word for 'sea' and for its divine female personification in Greek mythology. The word may have been of Pre-Greek origin.
Mythology
According t ...
. 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. The two moons are about 3540 km apart when they pass each other. Although their orbital radii differ by only 1850 km, Naiad swings ~2800 km above or below Thalassa's orbital plane at closest approach. As is common, this resonance stabilizes the orbits by maximizing separation at conjunction, but it is unusual for the role played by orbital inclination in facilitating this avoidance in a case where eccentricities are minimal.
Mean-motion resonances among extrasolar planets
![Resonant planetary system](https://upload.wikimedia.org/wikipedia/commons/8/8e/Resonant_planetary_system.gif)
While most
extrasolar planet
An exoplanet or extrasolar planet is a planet outside the Solar System. The first possible evidence of an exoplanet was noted in 1917 but was not recognized as such. The first confirmation of detection occurred in 1992. A different planet, init ...
ary systems discovered have not been found to have planets in mean-motion resonances, chains of up to five resonant planets
and up to seven at least near resonant planets
have been uncovered. Simulations have shown that during
planetary system formation, the appearance of resonant chains of planetary embryos is favored by the presence of the
primordial gas disc. Once that gas dissipates, 90–95% of those chains must then become unstable to match the low frequency of resonant chains observed.
*As mentioned above,
Gliese 876
Gliese 876 is a red dwarf approximately 15 light-years away from Earth in the constellation of Aquarius. It is one of the closest known stars to the Sun confirmed to possess a planetary system with more than two planets, after Gliese ...
e, b and c are in a Laplace resonance, with a 4:2:1 ratio of periods (124.3, 61.1 and 30.0 days).
In this case,
librates with an amplitude of 40° ± 13° and the resonance follows the time-averaged relation:
:
*
Kepler-223
Kepler-223 (KOI-730, KIC 10227020) is a G5V star with an extrasolar planetary system discovered by the Kepler mission. Studies indicate that the Kepler-223 star system consists of 4 planets orbiting the star.
Planetary system
...
has four planets in a resonance with an 8:6:4:3 orbit ratio, and a 3:4:6:8 ratio of periods (7.3845, 9.8456, 14.7887 and 19.7257 days).
This represents the first confirmed 4-body orbital resonance.
The librations within this system are such that close encounters between two planets occur only when the other planets are in distant parts of their orbits. Simulations indicate that this system of resonances must have formed via
planetary migration
Planetary migration occurs when a planet or other body in orbit around a star interacts with a disk of gas or planetesimals, resulting in the alteration of its orbital parameters, especially its semi-major axis. Planetary migration is the most l ...
.
*
Kepler-80
Kepler-80, also known as KOI-500, is a red dwarf star of the spectral type M0V. This stellar classification places Kepler-80 among the very common, cool, class M stars that are still within their main evolutionary stage, known as the main ...
d, e, b, c and g have periods in a ~ 1.000: 1.512: 2.296: 3.100: 4.767 ratio (3.0722, 4.6449, 7.0525, 9.5236 and 14.6456 days). However, in a frame of reference that rotates with the conjunctions, this reduces to a period ratio of 4:6:9:12:18 (an orbit ratio of 9:6:4:3:2). Conjunctions of d and e, e and b, b and c, and c and g occur at relative intervals of 2:3:6:6 (9.07, 13.61 and 27.21 days) in a pattern that repeats about every 190.5 days (seven full cycles in the rotating frame) in the inertial or nonrotating frame (equivalent to a 62:41:27:20:13 orbit ratio resonance in the nonrotating frame, because the conjunctions circulate in the direction opposite orbital motion). Librations of possible three-body resonances have amplitudes of only about 3 degrees, and modeling indicates the resonant system is stable to perturbations. Triple conjunctions do not occur.
*
TOI-178
TOI-178 is a planetary system in the constellation Sculptor,Requête spécifique à TOI-178sur VizieR. which appears to have at least five, and possibly six, planets in a chain of Laplace resonances, which constitute one of the longest c ...
has 6 confirmed planets, of which the outer 5 planets form a similar resonant chain in a rotating frame of reference, which can be expressed as 2:4:6:9:12 in period ratios, or as 18:9:6:4:3 in orbit ratios. In addition, the innermost planet b with period of 1.91d orbits close to where it would also be part of the same Laplace resonance chain, as a 3:5 resonance with the planet c would be fulfilled at period of ~1.95d, implying that it might have evolved there but pulled out of resonance, possibly by tidal forces.
*
TRAPPIST-1's seven approximately Earth-sized planets are in a chain of near resonances (the longest such chain known), having an orbit ratio of approximately 24, 15, 9, 6, 4, 3 and 2, or nearest-neighbor period ratios (proceeding outward) of about 8/5, 5/3, 3/2, 3/2, 4/3 and 3/2 (1.603, 1.672, 1.506, 1.509, 1.342 and 1.519). They are also configured such that each triple of adjacent planets is in a Laplace resonance (i.e., b, c and d in one such Laplace configuration; c, d and e in another, etc.).
The resonant configuration is expected to be stable on a time scale of billions of years, assuming it arose during planetary migration.
A musical interpretation of the resonance has been provided.
*
Kepler-29
Kepler-29 is a Sun-like star in the northern constellation of Cygnus. It is located at the celestial coordinates: Right Ascension , Declination . With an apparent visual magnitude of 15.456, this star is too faint to be seen with the nak ...
has a pair of planets in a 7:9 resonance (ratio of 1/1.28587).
*
Kepler-36
Kepler-36 is a star in the constellation of Cygnus with two known planets. It has an anomalously large radius, meaning that it is a subgiant.
Planetary system
On June 21, 2012, the discovery of two planets orbiting the star was announced. ...
has a pair of planets close to a 6:7 resonance.
*
Kepler-37
Kepler-37, also known as UGA-1785, is a G-type main-sequence star located in the constellation Lyra 209 light years from Earth. It is host to Extrasolar planet, exoplanets Kepler-37b, Kepler-37c, Kepler-37d and Kepler-37e, all of which o ...
d, c and b are within one percent of a resonance with an 8:15:24 orbit ratio and a 15:8:5 ratio of periods (39.792187, 21.301886 and 13.367308 days).
[
:*And ]
*Of
Kepler-90
Kepler-90, also designated 2MASS J18574403+4918185, is an F-type star located about from Earth in the constellation of Draco (constellation), Draco. It is notable for possessing a planetary system that has the same number of observed pla ...
's eight known planets, the period ratios b:c, c:i and i:d are close to 4:5, 3:5 and 1:4, respectively (4:4.977, 3:4.97 and 1:4.13) and d, e, f, g and h are close to a 2:3:4:7:11 period ratio (2: 3.078: 4.182: 7.051: 11.102; also 7: 11.021).
f, g and h are also close to a 3:5:8 period ratio (3: 5.058: 7.964).
Relevant to systems like this and that of
Kepler-36
Kepler-36 is a star in the constellation of Cygnus with two known planets. It has an anomalously large radius, meaning that it is a subgiant.
Planetary system
On June 21, 2012, the discovery of two planets orbiting the star was announced. ...
, calculations suggest that the presence of an outer gas giant planet facilitates the formation of closely packed resonances among inner super-Earths.
*
HD 41248 has a pair of
super-Earth
A super-Earth is an extrasolar planet with a mass higher than Earth's, but substantially below those of the Solar System's ice giants, Uranus and Neptune, which are 14.5 and 17 times Earth's, respectively.
The term "super-Earth" refers only to ...
s within 0.3% of a 5:7 resonance (ratio of 1/1.39718).
*
K2-138
K2-138, also designated EPIC 245950175 or EE-1, is a large early K-type main sequence star with a system of up to 6 planets discovered by citizen scientists. Four were found in the first two days of the Exoplanet Explorers project on Zoo ...
has 5 confirmed planets in an unbroken near-3:2 resonance chain (with periods of 2.353, 3.560, 5.405, 8.261 and 12.758 days). The system was discovered in the
citizen science
Citizen science (CS) (similar to community science, crowd science, crowd-sourced science, civic science, participatory monitoring, or volunteer monitoring) is scientific research conducted with participation from the public (who are sometimes re ...
project Exoplanet Explorers, using K2 data. K2-138 could host
co-orbital bodies (in a 1:1 mean-motion resonance).
Resonant chain systems can stabilize co-orbital bodies and a dedicated analysis of the K2 light curve and radial-velocity from
HARPS
The High Accuracy Radial Velocity Planet Searcher (HARPS) is a high-precision echelle planet-finding spectrograph installed in 2002 on the ESO's 3.6m telescope at La Silla Observatory in Chile. The first light was achieved in February 2003. H ...
might reveal them.
Follow-up observations with the
Spitzer Space Telescope
The Spitzer Space Telescope, formerly the Space Infrared Telescope Facility (SIRTF), was an infrared space telescope launched in 2003. Operations ended on 30 January 2020. Spitzer was the third space telescope dedicated to infrared astronomy, f ...
suggest a sixth planet continuing the 3:2 resonance chain, while leaving two gaps in the chain (its period is 41.97 days). These gaps could be filled by smaller non-transiting planets. Future observations with
CHEOPS
Khufu or Cheops was an ancient Egyptian monarch who was the second pharaoh of the Fourth Dynasty, in the first half of the Old Kingdom period (26th century BC). Khufu succeeded his father Sneferu as king. He is generally accepted as having com ...
will measure
transit-timing variation
Transit-timing variation is a method for detecting exoplanets by observing variations in the timing of a transit. This provides an extremely sensitive method capable of detecting additional planets in the system with masses potentially as small a ...
s of the system to further analyse the mass of the planets and could potentially find other planetary bodies in the system.
*
K2-32
K2-32 is a G9-type main sequence star slightly smaller and less massive than the sun. Four confirmed transiting exoplanets are known to orbit this star. A study of atmospheric escape from the planet K2-32b caused by high-energy stellar i ...
has four planets in a near 1:2:5:7 resonance (with periods of 4.34, 8.99, 20.66 and 31.71 days). Planet e has a radius almost identical to that of the Earth. The other planets have a size between Neptune and Saturn.
*
V1298 Tauri
V1298 Tauri is a young (23±4 Myr) weakly-lined T Tauri star that is part of the Taurus-Auriga association in the Taurus Molecular Cloud. Alternatively it is part of a proposed moving group, called Group 29 (stellar association), Group 2 ...
has four confirmed planets of which planets c, d and b are near a 1:2:3 resonance (with periods of 8.25, 12.40 and 24.14 days). Planet e only shows a single transit in the K2 light curve and has a period larger than 36 days. Planet e might be in a low-order resonance (of 2:3, 3:5, 1:2, or 1:3) with planet b. The system is very young (23±4
Myr
The abbreviation Myr, "million years", is a unit of a quantity of (i.e. ) years, or 31.556926 teraseconds.
Usage
Myr (million years) is in common use in fields such as Earth science and cosmology. Myr is also used with Mya (million years ago). ...
) and might be a precursor of a compact multiplanet system. The 2:3 resonance suggests that some close-in planets may either form in resonances or evolve into them on timescales of less than 10 Myr. The planets in the system have a size between Neptune and Saturn. Only planet b has a size similar to Jupiter.
*
HD 158259
HD 158259 is a main sequence star located 88 light years away in the constellation Draco, discovered by the SOPHIE échelle spectrograph using the radial velocity method.
Characteristics
HD 158259 is a G0 star with a rotation period of d ...
contains four planets in a 3:2 near resonance chain (with periods of 3.432, 5.198, 7.954 and 12.03 days, or period ratios of 1.51, 1.53 and 1.51, respectively), with a possible fifth planet also near a 3:2 resonance (with a period of 17.4 days). The exoplanets were found with the
SOPHIE échelle spectrograph The SOPHIE (Spectrographe pour l’Observation des Phénomènes des Intérieurs stellaires et des Exoplanètes, literally meaning "spectrograph for the observation of the phenomena of the stellar interiors and of the exoplanets") ''échelle spectrog ...
, using the
radial velocity
The radial velocity or line-of-sight velocity, also known as radial speed or range rate, of a target with respect to an observer is the temporal rate of change, rate of change of the distance or Slant range, range between the two points. It is e ...
method.
*
Kepler-1649
Kepler-1649 is a red dwarf star of spectral type M5V with a radius , a mass , and a metallicity of -0.15 e/H
Planetary system
Two confirmed planets orbit the star: Kepler-1649b and Kepler-1649c. Kepler-1649b is similar to Venus, where ...
contains two Earth-size planets close to a 9:4 resonance (with periods of 19.53527 and 8.689099 days, or a period ratio of 2.24825), including one (
"c") in the habitable zone. An undetected planet with a 13.0-day period would create a 3:2 resonance chain.
*
Kepler-88
Kepler-88 is a Sun-like star in the constellation of Lyra, with three confirmed planets. In April 2012, scientists discovered that a Kepler candidate known as KOI-142.01 (Kepler-88b) exhibited very significant transit-timing variations ca ...
has a pair of inner planets close to a 1:2 resonance (period ratio of 2.0396), with a mass ratio of ~22.5, producing very large
transit timing variation
Transit-timing variation is a method for detecting exoplanets by observing variations in the timing of a transit. This provides an extremely sensitive method capable of detecting additional planets in the system with masses potentially as small as ...
s of ~0.5 days for the innermost planet. There is a yet more massive outer planet in a ~1400 day orbit.
Cases of extrasolar planets close to a 1:2 mean-motion resonance are fairly common. Sixteen percent of systems found by the
transit method
Any planet is an extremely faint light source compared to its parent star. For example, a star like the Sun is about a billion times as bright as the reflected light from any of the planets orbiting it. In addition to the intrinsic difficulty o ...
are reported to have an example of this (with period ratios in the range 1.83–2.18),
as well as one sixth of planetary systems characterized by
Doppler spectroscopy
Doppler spectroscopy (also known as the radial-velocity method, or colloquially, the wobble method) is an indirect method for finding extrasolar planets and brown dwarfs from radial-velocity measurements via observation of Doppler shifts in t ...
(with in this case a narrower period ratio range).
Due to incomplete knowledge of the systems, the actual proportions are likely to be higher.
Overall, about a third of radial velocity characterized systems appear to have a pair of planets close to a
commensurability.
It is much more common for pairs of planets to have orbital period ratios a few percent larger than a mean-motion resonance ratio than a few percent smaller (particularly in the case of first order resonances, in which the integers in the ratio differ by one).
This was predicted to be true in cases where
tidal interactions
The tidal force is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for diverse phenomen ...
with the star are significant.
Coincidental 'near' ratios of mean motion
![Moons of Pluto](https://upload.wikimedia.org/wikipedia/commons/6/69/Moons_of_Pluto.png)
A number of near-
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
-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
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any ellip ...
or other libration to make the resonance perfect (see the detailed discussion in the
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 that 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.
The presence of a near resonance may reflect that a perfect resonance existed in the past, or that the system is evolving towards one in the future.
Some orbital frequency coincidences include:
The least probable orbital correlation in the list 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.
Possible past mean-motion resonances
A past resonance between Jupiter and Saturn may have played a dramatic role in early Solar System history. A 2004
computer model
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be deter ...
by
Alessandro Morbidelli of the
Observatoire de la Côte d'Azur in
Nice
Nice ( , ; Niçard: , classical norm, or , nonstandard, ; it, Nizza ; lij, Nissa; grc, Νίκαια; la, Nicaea) is the prefecture of the Alpes-Maritimes department in France. The Nice agglomeration extends far beyond the administrative c ...
suggested that the formation of a 1:2 resonance between Jupiter and Saturn (due to interactions with
planetesimal
Planetesimals are solid objects thought to exist in protoplanetary disks and debris disks. Per the Chamberlin–Moulton planetesimal hypothesis, they are believed to form out of cosmic dust grains. Believed to have formed in the Solar System a ...
s that caused them to migrate inward and outward, respectively) created a gravitational push that propelled both Uranus and Neptune into higher orbits, and in some scenarios caused them to switch places, which would have doubled Neptune's distance from the Sun. The resultant expulsion of objects from the proto-Kuiper belt as Neptune moved outwards could explain the
Late Heavy Bombardment
The Late Heavy Bombardment (LHB), or lunar cataclysm, is a hypothesized event thought to have occurred approximately 4.1 to 3.8 billion years (Ga) ago, at a time corresponding to the Neohadean and Eoarchean eras on Earth. According to the hypoth ...
600 million years after the Solar System's formation and the origin of Jupiter's
Trojan asteroid
In astronomy, a trojan is a small celestial body (mostly asteroids) that shares the orbit of a larger body, remaining in a stable orbit approximately 60° ahead of or behind the main body near one of its Lagrangian points and . Trojans can sha ...
s. An outward migration of Neptune could also explain the current occupancy of some of its resonances (particularly the 2:5 resonance) within the Kuiper belt.
While Saturn's mid-sized moons Dione and Tethys are not close to an exact resonance now, they may have been in a 2:3 resonance early in the Solar System's history. This would have led to orbital eccentricity and
tidal heating
Tidal heating (also known as tidal working or tidal flexing) occurs through the tidal friction processes: orbital and rotational energy is dissipated as heat in either (or both) the surface ocean or interior of a planet or satellite. When an object ...
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
In geology, a graben () is a depressed block of the crust of a planet or moon, bordered by parallel normal faults.
Etymology
''Graben'' is a loan word from German, meaning 'ditch' or 'trench'. The word was first used in the geologic contex ...
system of
Ithaca Chasma
Ithaca Chasma is a valley (graben) of Saturn's moon Tethys, named after the island of Ithaca, in Greece. It is up to wide, deep and long, running approximately three-quarters of the way around Tethys' circumference, making it one of the long ...
on Tethys.
The satellite system of Uranus is notably different from those of Jupiter and Saturn in that it lacks precise resonances among the larger moons, while the majority of the larger moons of Jupiter (3 of the 4 largest) and of Saturn (6 of the 8 largest) are in mean-motion resonances. 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
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is ...
, 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. A larger relative satellite size increases the strength of their interactions. Both factors lead to more chaotic orbital behavior at or near mean-motion resonances. Escape from a resonance may be associated with capture into a secondary resonance, and/or tidal evolution-driven increases in
orbital eccentricity
In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values betwee ...
or
inclination
Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a Plane of reference, reference plane and the orbital plane or Axis of rotation, axis of direction of the orbiting object ...
.
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.
Evidence for such past resonances includes the relatively high eccentricities of the orbits of Uranus' inner satellites, and the anomalously high orbital inclination of Miranda. 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,
respectively. Miranda probably escaped from its resonance with Umbriel via a secondary resonance, and the mechanism of this escape is believed to explain why its orbital inclination is more than 10 times those of the other
regular Uranian moons (see
Uranus' natural satellites
Uranus, the seventh planet of the Solar System, has 27 known moons, most of which are named after characters that appear in, or are mentioned in, the works of William Shakespeare and Alexander Pope. Uranus's moons are divided into three groups: t ...
).
Similar to the case of Miranda, the present inclinations of Jupiter's moonlets Amalthea and
Thebe
Thebe may refer to:
* Any of several female characters in Greek mythology - see List of mythological figures named Thebe
* Thebe (moon), a moon of Jupiter
* Thebe (currency), 1/100 of a Botswana pula
* Thebe, an Amazons, Amazon
* Thebe, alternate ...
are thought to be indications of past passage through the 3:1 and 4:2 resonances with Io, respectively.
Neptune's regular moons Proteus and Larissa are thought to have passed through a 1:2 resonance a few hundred million years ago; the moons have drifted away from each other since then because Proteus is outside a
synchronous orbit A synchronous orbit is an orbit in which an orbiting body (usually a satellite) has a period equal to the average rotational period of the body being orbited (usually a planet), and in the same direction of rotation as that body.
Simplified meanin ...
and Larissa is within one. Passage through the resonance is thought to have excited both moons' eccentricities to a degree that has not since been entirely damped out.
In the case of
Pluto
Pluto (minor-planet designation: 134340 Pluto) is a dwarf planet in the Kuiper belt, a ring of trans-Neptunian object, bodies beyond the orbit of Neptune. It is the ninth-largest and tenth-most-massive known object to directly orbit the S ...
'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.
The smaller inner moon of the
dwarf planet
A dwarf planet is a small planetary-mass object that is in direct orbit of the Sun, smaller than any of the eight classical planets but still a world in its own right. The prototypical dwarf planet is Pluto. The interest of dwarf planets to p ...
Haumea
Haumea (minor-planet designation 136108 Haumea) is a dwarf planet located trans-Neptunian object, beyond Neptune's orbit. It was discovered in 2004 by a team headed by Michael E. Brown, Mike Brown of Caltech at the Palomar Observatory in the United ...
,
Namaka, is one tenth the mass of the larger outer moon,
Hiʻiaka
In Hawaiian religion, Hiʻiaka is a daughter of Haumea and Kāne.
Attributes and history
Hiiaka is the patron goddess of hula dancers, chant, sorcery, and medicine. Owls are her messengers and are sacred to her. Conceived in Tahiti, Hiiaka was c ...
. Namaka revolves around Haumea in 18 days in an eccentric,
non-Keplerian orbit, and as of 2008 is inclined 13° from Hiʻiaka.
Over the timescale of the system, it should have been tidally damped into a more circular orbit. 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 moons may have been caught in and then escaped from orbital resonance several times. They probably passed through the 3:1 resonance relatively recently, and currently are in or at least close to an 8:3 resonance. Namaka's orbit is strongly
perturbed, with a current precession of about −6.5° per year.
See also
*
1685 Toro
1685 Toro ( ''prov. designation'': ) is an asteroid and near-Earth object of the Apollo group on an eccentric orbit. It was discovered on 17 July 1948, by American astronomer Carl Wirtanen at Lick Observatory on Mount Hamilton, California. The s ...
, an asteroid in 5:8 resonance with the Earth
*
3753 Cruithne
3753 Cruithne is a Q-type, Aten asteroid in orbit around the Sun in 1:1 orbital resonance with Earth, making it a co-orbital object. It is an asteroid that, relative to Earth, orbits the Sun in a bean-shaped orbit that effectively describes a ...
, an asteroid in 1:1 resonance with the Earth
*
Arnold tongue
In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamica ...
*
Commensurability (astronomy)
Commensurability is the property of two orbiting objects, such as planets, satellites, or asteroids, whose orbital periods are in a rational proportion.
Examples include the 2:3 commensurability between the orbital periods of Neptune and Pluto, ...
*
Dermott's law
*
Horseshoe orbit
In celestial mechanics, a horseshoe orbit is a type of co-orbital configuration, co-orbital motion of a small orbiting body relative to a larger orbiting body. The osculating orbit, osculating (instantaneous) orbital period of the smaller body re ...
, followed by an object in another type of 1:1 resonance
*
Kozai resonance
In celestial mechanics, the Kozai mechanism is a dynamical phenomenon affecting the orbit of a binary system perturbed by a distant third body under certain conditions. It is also known as the von Zeipel-Kozai-Lidov, Lidov–Kozai mechanism, Kozai ...
*
Lagrangian point
In celestial mechanics, the Lagrange points (; also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the influence of two massive orbiting bodies. Mathematically, this involves the solution of th ...
s
*
Mercury
Mercury commonly refers to:
* Mercury (planet), the nearest planet to the Sun
* Mercury (element), a metallic chemical element with the symbol Hg
* Mercury (mythology), a Roman god
Mercury or The Mercury may also refer to:
Companies
* Merc ...
, which has a 3:2 spin-orbit resonance
*
Musica universalis
The ''musica universalis'' (literally universal music), also called music of the spheres or harmony of the spheres, is a philosophical concept that regards proportions in the movements of celestial bodies – the Sun, Moon, and planets – as a f ...
("music of the spheres")
*
Resonant interaction
In nonlinear systems, a resonant interaction is the interaction of three or more waves, usually but not always of small amplitude. Resonant interactions occur when a simple set of criteria coupling wave-vectors and the dispersion equation are met ...
*
Resonant trans-Neptunian object
In astronomy, a resonant trans-Neptunian object is a trans-Neptunian object (TNO) in mean-motion orbital resonance with Neptune. The orbital periods of the resonant objects are in a simple integer relations with the period of Neptune, e.g. 1:2, 2 ...
*
Tidal locking
Tidal locking between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit. In the case where a tidally locked b ...
*
Tidal resonance
In oceanography, a tidal resonance occurs when the tide excites one of the resonant modes of the ocean.
The effect is most striking when a continental shelf is about a quarter wavelength wide. Then an incident tidal wave can be reinforced b ...
*
Titius–Bode law
The Titius–Bode law (sometimes termed just Bode's law) is a formulaic prediction of spacing between planets in any given solar system. The formula suggests that, extending outward, each planet should be approximately twice as far from the Sun as ...
*
Transfer operator
Transfer may refer to:
Arts and media
* ''Transfer'' (2010 film), a German science-fiction movie directed by Damir Lukacevic and starring Zana Marjanović
* ''Transfer'' (1966 film), a short film
* ''Transfer'' (journal), in management studies
...
*
Trojan (celestial body)
In astronomy, a trojan is a small celestial body (mostly asteroids) that shares the orbit of a larger body, remaining in a stable orbit approximately 60° ahead of or behind the main body near one of its Lagrangian points and . Trojans can sha ...
, a body in a type of 1:1 resonance
*
Venus
Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never fa ...
, whose Earth conjunction period (584 Earth days) is close to 5 times its
solar day
A synodic day (or synodic rotation period or solar day) is the period for a celestial object to rotate once in relation to the star it is orbiting, and is the basis of solar time.
The synodic day is distinguished from the sidereal day, which is ...
(116.75 days)
Notes
References
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External links
*
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