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. The physics principle behind orbital resonance is similar in concept to pushing a child on a swing, where the orbit and the swing both have a natural frequency, and the other 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 and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter's moons Ganymede, Europa and Io, and the 2:3 resonance between Pluto and Neptune. Unstable resonances with Saturn's inner moons give rise to gaps in the rings of Saturn. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to eject most other bodies sharing their orbits; this is part of the much more extensive process of clearing the neighbourhood, an effect that is used in the current definition of a planet.
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 Pluto completes two orbits in the time it takes Neptune to complete three. In the case of resonance relationships between three or more bodies, either type of ratio may be used (in such cases the smallest whole-integer ratio sequences are not necessarily reversals of each other) and the type of ratio will be specified.
Since the discovery of Newton's law of universal gravitation in the 17th century, the stability of the Solar System has preoccupied many mathematicians, starting with Laplace. 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 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 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 resonances
In general, an orbital resonance may
- involve one or any combination of the orbit parameters (e.g. eccentricity versus semimajor axis, or eccentricity versus orbital 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 stabilization 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 may occur when the two bodies move in such a synchronised fashion that they never closely approach. For instance:
- The orbits of Pluto and the plutinos are stable, despite crossing that of the much larger Neptune, 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 objects occupying the 1:1 (Neptune trojans), 3:5, 4:7, 1:2 (twotinos) and 2:5 resonances, among others, with respect to Neptune.
- In the asteroid belt beyond 3.5 AU from the Sun, the 3:2, 4:3 and 1:1 resonances with Jupiter are populated by clumps of asteroids (the Hilda family, the few Thule asteroids, and the extremely numerous Trojan asteroids, respectively).
Orbital resonances can also destabilize one of the orbits. For small bodies, destabilization is actually far more likely. For instance:
- In the asteroid belt within 3.5 AU from the Sun, the major mean-motion resonances with Jupiter are locations of gaps in the asteroid distribution, the Kirkwood gaps (most notably at the 3:1, 5:2, 7:3 and 2:1 resonances). Asteroids have been ejected from these almost empty lanes by repeated perturbations. However, there are still populations of asteroids temporarily present in or near these resonances. For example, asteroids of the Alinda family are in or close to the 3:1 resonance, with their orbital eccentricity steadily increased by interactions with Jupiter until they eventually have a close encounter with an inner planet that ejects them from the resonance.
- In the rings of Saturn, the Cassini Division is a gap between the inner B Ring and the outer A Ring that has been cleared by a 2:1 resonance with the moon Mimas. (More specifically, the site of the resonance is the Huygens Gap, which bounds the outer edge of the B Ring.)
- In the rings of Saturn, the Encke and Keeler gaps within the A Ring are cleared by 1:1 resonances with the embedded moonlets Pan and Daphnis, respectively. The A Ring's outer edge is maintained by a destabilizing 7:6 resonance with the moon Janus.
Most bodies that are in resonance orbit in the same direction; however, a few retrograde damocloids have been found that are temporarily captured in mean-motion resonance with Jupiter or Saturn. 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 discovered that such a resonance governed the motions of Jupiter's moons Io, Europa, and Ganymede. It is now also often applied to other 3-body resonances with the same ratios, such as that between the extrasolar planets Gliese 876 c, b, and e. Three-body resonances involving other simple integer ratios have been termed "Laplace-like" or "Laplace-type".
A Lindblad resonance drives spiral density waves both in galaxies (where stars are subject to forcing by the spiral arms themselves) and in Saturn's rings (where ring particles are subject to forcing by Saturn's moons).
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.
Several prominent examples of secular resonance involve Saturn. A 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) has been identified as the likely source of Saturn's large axial tilt (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 the spin-orbit resonance, leading to an increase in Saturn's obliquity. (The angular momentum of Neptune's orbit is 104 times that of Saturn's spin, and thus dominates the interaction.)
The perihelion secular resonance between asteroids and Saturn (ν6 = g − g6)[clarification needed] helps shape the asteroid belt. 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 by a close pass to Mars. This resonance forms the inner and "side" boundaries of the asteroid belt around 2 AU, and at inclinations of about 20°.
Numerical simulations have suggested that the eventual formation of a perihelion secular resonance between Mercury and Jupiter (g1=g5)[clarification needed] 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 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.
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; 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.
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
There are only a few known mean-motion resonances in the Solar System involving planets, dwarf planets or larger satellites (a much greater number involve asteroids, planetary rings, moonlets and smaller Kuiper belt objects, including many possible dwarf planets).
- 2:3 Pluto–Neptune (also Orcus and other plutinos)
- 2:4 Tethys–Mimas (Saturn's moons). Not simplified, because the libration of the nodes must be taken into account.
- 1:2 Dione–Enceladus (Saturn’s moons)
- 3:4 Hyperion–Titan (Saturn's moons)
- 1:2:4 Ganymede–Europa–Io (Jupiter's moons, ratio of orbits).
The simple integer ratios between periods are a convenient simplification hiding more complex relations:
- the point of conjunction can oscillate (librate) around an equilibrium point defined by the resonance.
- given non-zero eccentricities, the nodes or periapsides can drift (a resonance related, short period, not secular precession).
As illustration of the latter, consider the well known 2:1 resonance of Io-Europa. If the orbiting periods were in this relation, the mean motions (inverse of periods, often expressed in degrees per day) would satisfy the following
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 remarkable Laplace resonance involving Io–Europa–Ganymede includes the following relation locking the orbital phase of the moons:
where are mean longitudes of the moons (the second equals sign ignores libration). This relation makes a triple conjunction impossible. (A Laplace resonance in the Gliese 876 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°.
This reflects orbital periods for Styx, Nix and Hydra, respectively, that are close to a ratio of 18:22:33 (or, in terms of the near resonances with Charon's period, 3+3/11:4:6; see below); the respective ratio of orbits is 11:9:6. Based on the ratios of synodic periods, 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°.
- A mean-motion resonance of 2:3
- The resonance of the perihelion (libration around 90°), keeping the perihelion above the ecliptic
- The resonance of the longitude of the perihelion in relation to that of Neptune
One consequence of these resonances is that a separation of at least 30 AU is maintained when Pluto crosses Neptune's orbit. The minimum separation between the two bodies overall is 17 AU, while the minimum separation between Pluto and Uranus 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, is the probable dwarf planet Orcus. 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".
Mean-motion resonances among extrasolar planets
While most extrasolar planetary systems discovered have not been found to have planets in mean-motion resonances, some remarkable examples have been uncovered:
- As mentioned above, Gliese 876 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, respectively). In this case, librates with an amplitude of 40° ± 13° and the resonance follows the time-averaged relation:
- KOI-730 d, b, c and e appear to be in a resonance with a 3:4:6:8 orbit ratio, and an 8:6:4:3 ratio of periods (19.72, 14.79, 9.85 and 7.38 days, respectively).
- KOI-500 c, b, e, d and f appear to be in or close to a resonance with a 193:62:41:27:20 ratio of periods (9.522, 7.053, 4.645, 3.072 and 0.9868 days, respectively).
- Both KOI-738 and KOI-787 appear to have pairs of planets in a 7:9 resonance (ratios of 1/1.285871 and 1/1.284008, respectively).
- Kepler-37 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, respectively).
- HD 41248 has a pair of super-Earths within 0.3% of a 5:7 resonance (ratio of 1/1.39718).
Cases of extrasolar planets close to a 1:2 mean-motion resonance are fairly common. Sixteen percent of systems found by the transit method 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 (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 with the star are significant.
Coincidental 'near' ratios of mean motion
A number of near-integer-ratio relationships between the orbital frequencies of the planets or major moons are sometimes pointed out (see list below). However, these have no dynamical significance because there is no appropriate precession of perihelion or other libration to make the resonance perfect (see the detailed discussion in the 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:
|(Ratio) and Bodies||Mismatch after one cycle[a]||Randomization time[b]||Probability[c]|
|(9:23) Venus-Mercury||4.0°||200 y||0.19|
|(8:13) Earth-Venus[d]||1.5°||1000 y||0.065|
|(243:395) Earth-Venus||0.8°||50,000 y||0.68|
|(1:3) Mars-Venus||20.6°||20 y||0.11|
|(1:2) Mars-Earth||42.9°||8 y||0.24|
|(1:12) Jupiter-Earth[e]||49.1°||40 y||0.28|
|(2:5) Saturn–Jupiter[f]||12.8°||800 y||0.13|
|(1:7) Uranus-Jupiter||31.1°||500 y||0.18|
|(7:20) Uranus-Saturn||5.7°||20,000 y||0.20|
|(5:28) Neptune-Saturn||1.9°||80,000 y||0.052|
|(1:2) Neptune-Uranus||14.0°||2000 y||0.078|
|(1:4) Deimos-Phobos||14.9°||0.04 y||0.083|
|(1:1) Pallas - Ceres||0.7°||1000 y||0.0039[g]|
|(7:18) Jupiter - Pallas||0.10°||100,000 y||0.0040[h]|
|87 Sylvia system[i]|
|(17:45) Romulus-Remus||0.7°||40 y||0.067|
|(1:6) Io-Metis||0.6°||2 y||0.0031|
|(3:5) Amalthea-Adrastea||3.9°||0.2 y||0.064|
|(3:7) Callisto-Ganymede||0.7°||30 y||0.012|
|(2:3) Enceladus-Mimas||33.2°||0.04 y||0.33|
|(2:3) Dione-Tethys[j]||36.2°||0.07 y||0.36|
|(3:5) Rhea-Dione||17.1°||0.4 y||0.26|
|(2:7) Titan-Rhea||21.0°||0.7 y||0.22|
|(1:5) Iapetus-Titan||9.2°||4 y||0.051|
|(3:4) Uranus-Chariklo||4.5°||10,000 y||0.073|
|(3:5) Rosalind-Cordelia||0.22°||4 y||0.0037|
|(1:3) Umbriel-Miranda[l]||24.5°||0.08 y||0.14|
|(3:5) Umbriel-Ariel[m]||24.2°||0.3 y||0.35|
|(1:2) Titania-Umbriel||36.3°||0.1 y||0.20|
|(2:3) Oberon-Titania||33.4°||0.4 y||0.34|
|(1:20) Triton-Naiad||13.5°||0.2 y||0.075|
|(1:2) Proteus-Larissa||8.4°||0.07 y||0.047|
|(5:6) Proteus-S/2004 N 1||2.1°||1 y||0.057|
|(1:3) Styx-Charon||58.5°||0.2 y||0.33|
|(1:4) Nix-Charon||39.1°||0.3 y||0.22|
|(1:5) Kerberos-Charon||9.2°||2 y||0.05|
|(1:6) Hydra-Charon||6.6°||3 y||0.037|
|(3:8) Hiʻiaka-Namaka[n]||42.5°||2 y||0.55|
- Mismatch in orbital longitude of the inner body, as compared to its position at the beginning of the cycle (with the cycle defined as n orbits of the outer body – see below). Circular orbits are assumed (i.e., precession is ignored).
- The time needed for the mismatch from the initial relative longitudinal orbital positions of the bodies to grow to 180°, rounded to the nearest first significant digit.
- The probability of obtaining an orbital coincidence of equal or smaller mismatch by chance at least once in n attempts, where n is the integer number of orbits of the outer body per cycle, and the mismatch is assumed to vary between 0° and 180° at random. The value is calculated as 1- (1- mismatch/180°)^n. The smaller the probability, the more remarkable the coincidence. This is a crude calculation that only attempts to give a rough idea of relative probabilities.
- The two near commensurabilities listed for Earth and Venus are reflected in the timing of transits of Venus, which occur in pairs 8 years apart, in a cycle that repeats every 243 years.
- The near 1:12 resonance between Jupiter and Earth causes the Alinda asteroids, which occupy (or are close to) the 3:1 resonance with Jupiter, to be close to a 1:4 resonance with Earth.
- This near resonance has been termed the Great Inequality. It was first described by Laplace in a series of papers published 1784–1789.
- Based on the proper orbital periods, 1684.869 and 1681.601 days, for Pallas and Ceres, respectively.
- Based on the proper orbital period of Pallas, 1684.869 days, and 4332.59 days for Jupiter.
- 87 Sylvia is the first asteroid discovered to have more than one moon.
- This resonance may have been occupied in the past.
- Some definitions of centaurs stipulate that they are nonresonant bodies.
- This resonance may have been occupied in the past.
- This resonance may have been occupied in the past.
- The results for the Haumea system aren't very meaningful because, contrary to the assumptions implicit in the calculations, Namaka has an eccentric, non-Keplerian orbit that precesses rapidly (see below). Hi?iaka and Namaka are much closer to a 3:8 resonance than indicated, and may actually be in it.
The most remarkable (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 by Alessandro Morbidelli of the Observatoire de la Côte d'Azur in Nice suggested that the formation of a 1:2 resonance between Jupiter and Saturn (due to interactions with planetesimals that caused them to migrate inward and outward, respectively) 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 600 million years after the Solar System's formation and the origin of Jupiter's Trojan asteroids. 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 that may have warmed Tethys' interior enough to form a subsurface ocean. Subsequent freezing of the ocean after the moons escaped from the resonance may have generated the extensional stresses that created the enormous graben system of Ithaca Chasma on Tethys.
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 Uranus System, however, due to the planet's lesser degree of oblateness, and the larger relative size of its satellites, escape from a mean-motion resonance is much easier. Lower oblateness of the primary alters its gravitational field in such a way that different possible resonances are spaced more closely together. 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 or inclination.
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).
Similar to the case of Miranda, the present inclinations of Jupiter's moonlets Amalthea and Thebe are thought to be indications of past passage through the 3:1 and 4:2 resonances with Io, respectively.
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 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'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 Haumea, Namaka, is one tenth the mass of the larger outer moon, Hiʻiaka. 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.
- 1685 Toro, an asteroid in 5:8 resonance with the Earth
- 3753 Cruithne, an asteroid in 1:1 resonance with the Earth
- Commensurability (astronomy)
- Dermott's Law
- Horseshoe orbit, followed by an object in another type of 1:1 resonance
- Kozai resonance
- Lagrangian points
- Mercury, which has a 3:2 spin–orbit resonance
- Musica universalis ("music of the spheres")
- Resonant trans-Neptunian object
- Tidal locking
- Tidal resonance
- Titius–Bode law
- Trojan object, a body in a type of 1:1 resonance
- Porco, C.; Nicholson, P. D.; Borderies, N.; Danielson, G. E.; Goldreich, P.; Holdberg, J. B.; Lane, A. L. (1984). "The eccentric Saturnian ringlets at 1.29Rs and 1.45Rs". Icarus 60 (1): 1–16. Bibcode:1984Icar...60....1P. doi:10.1016/0019-1035(84)90134-9.
- Rosen, P. A.; Lissauer, J. J. (1988). "The Titan -1:0 Nodal Bending Wave in Saturn's Ring C". Science 241 (4866): 690–694. Bibcode:1988Sci...241..690R. doi:10.1126/science.241.4866.690. PMID 17839081.
- Chakrabarti, S. K.; Bhattacharyya, A. (2001). "Constraints on the C ring parameters of Saturn at the Titan -1:0 resonance". Monthly Notices of the Royal Astronomical Society 326 (2): L23. Bibcode:2001MNRAS.326L..23C. doi:10.1046/j.1365-8711.2001.04813.x.
- Morais, M. H. M.; Namouni, F. "Asteroids in retrograde resonance with Jupiter and Saturn". Monthly Notices of the Royal Astronomical Society Letters. arXiv:1308.0216. Bibcode:2013MNRAS.436L..30M. doi:10.1093/mnrasl/slt106.
- Barnes, R. (2011). "Laplace Resonance". In Gargaud, M. Encyclopedia of Astrobiology. Springer Science+Business Media. pp. 905–906. doi:10.1007/978-3-642-11274-4_864. ISBN 978-3-642-11271-3.
- Rivera, E. J.; Laughlin, G.; Butler, R. P.; Vogt, S. S.; Haghighipour, N.; Meschiari, S. (2010). "The Lick-Carnegie Exoplanet Survey: A Uranus-mass Fourth Planet for GJ 876 in an Extrasolar Laplace Configuration". The Astrophysical Journal 719 (1): 890–899. arXiv:1006.4244. Bibcode:2010ApJ...719..890R. doi:10.1088/0004-637X/719/1/890.
- Marti, J. G.; Giuppone, C. A.; Beauge, C. (2013). "Dynamical analysis of the Gliese-876 Laplace resonance". Monthly Notices of the Royal Astronomical Society 433 (2): 928–934. arXiv:1305.6768. Bibcode:2013MNRAS.433..928M. doi:10.1093/mnras/stt765.
- Showalter, M. R.; Hamilton, D. P. (2015). "Resonant interactions and chaotic rotation of Pluto's small moons". Nature 522 (7554): 45–49. Bibcode:2015Natur.522...45S. doi:10.1038/nature14469. PMID 26040889.
- Murray, C. D.; Dermott, S. F. (1999). Solar System Dynamics. Cambridge University Press. p. 17. ISBN 978-0-521-57597-3.
- Beatty, J. K. (23 July 2003). "Why Is Saturn Tipsy?". Sky & Telescope. Retrieved 2009-02-25.
- Ward, W. R.; Hamilton, D. P. (2004). "Tilting Saturn. I. Analytic Model". The Astronomical Journal 128 (5): 2501–2509. Bibcode:2004AJ....128.2501W. doi:10.1086/424533.
- Hamilton, D. P.; Ward, W. R. (2004). "Tilting Saturn. II. Numerical Model". The Astronomical Journal 128 (5): 2510–2517. Bibcode:2004AJ....128.2510H. doi:10.1086/424534.
- Laskar, J. (2008). "Chaotic diffusion in the Solar System". Icarus 196 (1): 1–15. arXiv:0802.3371. Bibcode:2008Icar..196....1L. doi:10.1016/j.icarus.2008.02.017.
- Laskar, J.; Gastineau, M. (2009). "Existence of collisional trajectories of Mercury, Mars and Venus with the Earth". Nature 459 (7248): 817–819. Bibcode:2009Natur.459..817L. doi:10.1038/nature08096. PMID 19516336.
- Musotto, S.; Varad, F.; Moore, W.; Schubert, G. (2002). "Numerical Simulations of the Orbits of the Galilean Satellites". Icarus 159 (2): 500–504. Bibcode:2002Icar..159..500M. doi:10.1006/icar.2002.6939.
- Brown, M. E.; Barkume, K. M.; Ragozzine, D.; Schaller, E. L. (2007). "A collisional family of icy objects in the Kuiper belt". Nature 446 (7133): 294–296. Bibcode:2007Natur.446..294B. doi:10.1038/nature05619. PMID 17361177.
- Ragozzine, D.; Brown, M. E. (2007). "Candidate members and age estimate of the family of Kuiper Belt object 2003 EL61". The Astronomical Journal 134 (6): 2160–2167. arXiv:0709.0328. Bibcode:2007AJ....134.2160R. doi:10.1086/522334.
- Buie, M. W. (24 October 2011). "Orbit Fit and Astrometric record for 225088". SwRI (Space Science Department). Retrieved 2014-11-14.
- Sinclair, A. T. (1975). "The Orbital Resonance Amongst the Galilean Satellites of Jupiter". Monthly Notices of the Royal Astronomical Society 171 (1): 59–72. Bibcode:1975MNRAS.171...59S. doi:10.1093/mnras/171.1.59.
- Witze, A. (3 June 2015). "Pluto's moons move in synchrony". Nature News. doi:10.1038/nature.2015.17681.
- Malhotra, R. (1997). "Pluto's Orbit". Retrieved 2007-03-26.
- Brown, M. E. (23 March 2009). "S/2005 (90482) 1 needs your help". Mike Brown's Planets. Retrieved 2009-03-25.
- Laughlin, G. (23 June 2010). "A second Laplace resonance". Systemic: Characterizing Planets. Archived from the original on 2013-12-29. Retrieved 2015-06-30.
- Marcy, Ge. W.; Butler, R. P.; Fischer, D.; Vogt, S. S.; Lissauer, J. J.; Rivera, E. J. (2001). "A Pair of Resonant Planets Orbiting GJ 876". The Astrophysical Journal 556 (1): 296–301. Bibcode:2001ApJ...556..296M. doi:10.1086/321552.
- "KOI-730". Extrasolar Planets Encyclopaedia.
- Beatty, K. (5 March 2011). "Kepler Finds Planets in Tight Dance". Sky and Telescope. Retrieved 2012-10-16.
- Lissauer, J. J.; et al. (2011). "Architecture and dynamics of Kepler's candidate multiple transiting planet systems". The Astrophysical Journal Supplement Series 197 (1): 1–26. arXiv:1102.0543. Bibcode:2011ApJS..197....8L. doi:10.1088/0067-0049/197/1/8.
- "KOI-500". Extrasolar Planets Encyclopaedia.
- Choi, C. Q. (15 October 2012). "Tiniest Alien Solar System Discovered: 5 Packed Planets". Space.com. Retrieved 2012-10-16.
- Barclay, T.; et al. (2013). "A sub-Mercury-sized exoplanet". Nature 494 (7438): 452–454. arXiv:1305.5587. Bibcode:2013Natur.494..452B. doi:10.1038/nature11914. PMID 23426260.
- Jenkins, J. S.; Tuomi, M.; Brasser, R.; Ivanyuk, O.; Murgas, F. (2013). "Two Super-Earths Orbiting the Solar Analog HD 41248 on the Edge of a 7:5 Mean Motion Resonance". The Astrophysical Journal 771 (1): 41. arXiv:1304.7374. Bibcode:2013ApJ...771...41J. doi:10.1088/0004-637X/771/1/41.
- Wright, J. T.; Fakhouri, O.; Marcy, G. W.; Han, E.; Feng, Y.; Johnson, J. A.; Howard, A. W.; Fischer, D. A.; Valenti, J. A.; Anderson, J.; Piskunov, N. (2011). "The Exoplanet Orbit Database". Publications of the Astronomical Society of the Pacific 123 (902): 412–42. arXiv:1012.5676. Bibcode:2011PASP..123..412W. doi:10.1086/659427.
- Terquem, C.; Papaloizou, J. C. B. (2007). "Migration and the Formation of Systems of Hot Super-Earths and Neptunes". The Astrophysical Journal 654 (2): 1110–1120. arXiv:astro-ph/0609779. Bibcode:2007ApJ...654.1110T. doi:10.1086/509497.
- Langford, P. M. (12 March 2012). "Transits of Venus". Astronomical Society of the Channel Island of Guernsey. Retrieved 2016-01-15.
- Bazsó, A.; Eybl, V.; Dvorak, R.; Pilat-Lohinger, E.; Lhotka, C. (2010). "A survey of near-mean-motion resonances between Venus and Earth". Celestial Mechanics and Dynamical Astronomy 107 (1): 63–76. arXiv:0911.2357. Bibcode:2010CeMDA.107...63B. doi:10.1007/s10569-010-9266-6.
- Shortt, D. (22 May 2012). "Some Details About Transits of Venus". The Planetary Society. Retrieved 22 May 2012.
- Goffin, E. (2001). "New determination of the mass of Pallas". Astronomy and Astrophysics 365 (3): 627–630. Bibcode:2001A&A...365..627G. doi:10.1051/0004-6361:20000023.
- Kovacevic, A. B. (2012). "Determination of the mass of Ceres based on the most gravitationally efficient close encounters". Monthly Notices of the Royal Astronomical Society 419 (3): 2725–2736. arXiv:1109.6455. Bibcode:2012MNRAS.419.2725K. doi:10.1111/j.1365-2966.2011.19919.x.
- Taylor, D. B. (1982). "The secular motion of Pallas". Monthly Notices of the Royal Astronomical Society 199 (2): 255–265. Bibcode:1982MNRAS.199..255T. doi:10.1093/mnras/199.2.255.
- Goldreich, P. (1965). "An explanation of the frequent occurrence of commensurable mean motions in the solar system". Monthly Notices of the Royal Astronomical Society 130 (3): 159–181. Bibcode:1965MNRAS.130..159G. doi:10.1093/mnras/130.3.159.
- Chen, E. M. A.; Nimmo, F. (2008). "Thermal and Orbital Evolution of Tethys as Constrained by Surface Observations" (PDF). Lunar and Planetary Science XXXIX. Lunar and Planetary Institute. #1968. Retrieved 2008-03-14.
- Murray, C. D.; Thompson,, R. P. (1990). "Orbits of shepherd satellites deduced from the structure of the rings of Uranus". Nature 348 (6301): 499–502. Bibcode:1990Natur.348..499M. doi:10.1038/348499a0.
- Tittemore, W. C.; Wisdom, J. (1990). "Tidal evolution of the Uranian satellites: III. Evolution through the Miranda-Umbriel 3:1, Miranda-Ariel 5:3, and Ariel-Umbriel 2:1 mean-motion commensurabilities". Icarus 85 (2): 394–443. Bibcode:1990Icar...85..394T. doi:10.1016/0019-1035(90)90125-S.
- Tittemore, W. C.; Wisdom, J. (1988). "Tidal Evolution of the Uranian Satellites I. Passage of Ariel and Umbriel through the 5:3 Mean-Motion Commensurability". Icarus 74 (2): 172–230. Bibcode:1988Icar...74..172T. doi:10.1016/0019-1035(88)90038-3.
- Zhang, K.; Hamilton, D. P. (2007). "Orbital resonances in the inner Neptunian system: I. The 2:1 Proteus–Larissa mean-motion resonance". Icarus 188 (2): 386–399. Bibcode:2007Icar..188..386Z. doi:10.1016/j.icarus.2006.12.002.
- Zhang, K.; Hamilton, D. P. (2008). "Orbital resonances in the inner Neptunian system: II. Resonant history of Proteus, Larissa, Galatea, and Despina". Icarus 193 (1): 267–282. Bibcode:2008Icar..193..267Z. doi:10.1016/j.icarus.2007.08.024.
- Matson, J. (11 July 2012). "New Moon for Pluto: Hubble Telescope Spots a 5th Plutonian Satellite". Scientific American. Retrieved 2012-07-12.
- Ward, W. R.; Canup, R. M. (2006). "Forced Resonant Migration of Pluto's Outer Satellites by Charon". Science 313 (5790): 1107–1109. Bibcode:2006Sci...313.1107W. doi:10.1126/science.1127293. PMID 16825533.
- Ragozzine, D.; Brown, M. E. (2009). "Orbits and Masses of the Satellites of the Dwarf Planet Haumea=2003 EL61". The Astronomical Journal 137 (6): 4766–4776. arXiv:0903.4213. Bibcode:2009AJ....137.4766R. doi:10.1088/0004-6256/137/6/4766.
- Hansen, K. (7 June 2004). "Orbital shuffle for early solar system". Geotimes. Retrieved 2007-08-26.
- Tittemore, W. C. (1990). "Tidal heating of Ariel". Icarus 87 (1): 110–139. Bibcode:1990Icar...87..110T. doi:10.1016/0019-1035(90)90024-4.
- Tittemore, W. C.; Wisdom, J. (1989). "Tidal Evolution of the Uranian Satellites II. An Explanation of the Anomalously High Orbital Inclination of Miranda". Icarus 78 (1): 63–89. Bibcode:1989Icar...78...63T. doi:10.1016/0019-1035(89)90070-5.
- Malhotra, R.; Dermott, S. F (1990). "The Role of Secondary Resonances in the Orbital History of Miranda". Icarus 85 (2): 444–480. Bibcode:1990Icar...85..444M. doi:10.1016/0019-1035(90)90126-T.
- Burns, J. A.; Simonelli, D. P.; Showalter, M. R.; Hamilton, D. P.; Porco, C. C.; Esposito, L. W.; Throop, H. (2004). "Jupiter’s Ring-Moon System" (PDF). In Bagenal, F.; Dowling, T. E.; McKinnon, W. B. Jupiter: The Planet, Satellites and Magnetosphere. Cambridge University Press. ISBN 978-0-521-03545-3.
- C. D. Murray, S. F. Dermott (1999). Solar System Dynamics, Cambridge University Press, ISBN 0-521-57597-4.
- Renu Malhotra Orbital Resonances and Chaos in the Solar System. In Solar System Formation and Evolution, ASP Conference Series, 149 (1998) preprint.
- Renu Malhotra, The Origin of Pluto's Orbit: Implications for the Solar System Beyond Neptune, The Astronomical Journal, 110 (1995), p. 420 Preprint.
- Lemaître, A. (2010). "Resonances: Models and Captures". In Souchay, J.; Dvorak, R. Dynamics of Small Solar System Bodies and Exoplanets. Lecture Notes in Physics 790. Springer. pp. 1–62. doi:10.1007/978-3-642-04458-8. ISBN 978-3-642-04457-1.
- Locations of Solar System Planetary Mean-Motion Resonances. Web calculator that plots distributions of the semimajor axes (or in one case the perihelion distances) of the minor planets in relation to mean-motion resonances of the planets (website maintained by M.A. Murison).