Orbital resonance

For the science fiction novel by John Barnes, see Orbital Resonance (novel).

In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of the bodies, i.e., their ability to alter or constrain each others' 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.

Except as noted in the Laplace resonance figure (below), a 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). The 2:3 ratio above means Pluto completes 2 orbits in the time it takes Neptune to complete 3.

History

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 resonance

In general, an orbital resonance may

• involve one or any combination of the orbit parameters (e.g. eccentricity versus semimajor axis, or eccentricity versus orbit inclination).
• act on any time scale from short term, commensurable with the orbit periods, to secular, measured in 10⁴ to 10⁶ years.
• lead to either long term 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 occurs 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 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 with respect to Neptune.
• In the asteroid belt beyond 3.5 AU from the Sun, the 3:2, 4:3 and 1:1 resonances with Jupiter are populated by clumps of asteroids (the Hilda family, 279 Thule, and the Trojan asteroids, respectively).

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.

A Laplace resonance occurs when three or more orbiting bodies have a simple integer ratio between their orbital periods. For example, Jupiter's moons Ganymede, Europa and Io are in a 1:2:4 orbital resonance. The extrasolar planets Gliese 876e, Gliese 876b and Gliese 876c are also in a 1:2:4 orbital resonance (with periods of 124.3, 61.1 and 30.0 days).^[1][2]

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°).^[3][4][5] 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 that of Saturn's spin, and thus dominates the interaction.)

The ν₆^{[clarification needed]} secular resonance between asteroids and Saturn 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 main asteroid belt around 2 AU, and at inclinations of about 20°.

The Titan Ringlet within Saturn's C Ring exemplifies 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.

Mean motion resonances in the Solar System

The Laplace resonance exhibited by three of the Galilean moons. The ratios in the figure are of orbital periods.

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, smaller Kuiper belt objects, planetary rings and moonlets).

• 2:3 Pluto-Neptune
• 2:4 Tethys-Mimas (Saturn’s moons)
• 1:2 Dione-Enceladus (Saturn’s moons)
• 3:4 Hyperion-Titan (Saturn's moons)
• 1:2:4 Ganymede-Europa-Io (Jupiter’s moons).

The 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 $n\,\!$ (inverse of periods, often expressed in degrees per day) would satisfy the following

$n_{\rm Io} - 2\cdot n_{\rm Eu} = 0$

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), $\dot\omega$. The correct equation (part of the Laplace equations) is:

$n_{\rm Io} - 2\cdot n_{\rm Eu} + \dot\omega_{\rm Io} = 0$

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

$4\cdot n_{\rm Te} - 2\cdot n_{\rm Mi} - \dot\Omega_{\rm Te}- \dot\Omega_{\rm Mi}= 0$

The point of conjunctions librates around the midpoint between the nodes of the two moons.

The Laplace resonance

Illustration of Io-Europa-Ganymede resonance. From the centre outwards: Io (yellow), Europa (gray) and Ganymede (dark)

The most remarkable resonance involving Io-Europa-Ganymede includes the following relation locking the orbital phase of the moons:

$\Phi_L = \lambda_{\rm Io} - 3\cdot\lambda_{\rm Eu} + 2\cdot\lambda_{\rm Ga} = 180^\circ$

where λ are mean longitudes of the moons. This relation makes a triple conjunction impossible. The graph illustrates the positions of the moons after 1, 2 and 3 Io periods. (The Laplace resonance in the Gliese 876 system, in contrast, is associated with one triple conjunction per orbit of the outermost planet.^[2])

Pluto resonances

The dwarf planet Pluto is following an orbit trapped in a web of resonances with Neptune. The resonances include:

• 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^[6] (see Pluto's orbit for detailed explanation and graphs).

The next largest body in a similar 2:3 resonance with Neptune is the candidate 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".^[7]

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[8]	Randomization time[9]	Probability[10]
Planets
(9:23) Venus − Mercury	4.0°	200 y	0.19
(8:13) Earth − Venus	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	49.1°	40 y	0.27
(2:5) Saturn − Jupiter[11]	12.8°	800 y	0.14
(1:7) Uranus − Jupiter	31.1°	500 y	0.17
(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
Mars System
(1:4) Deimos − Phobos	14.9°	0.04 y	0.083
Jupiter System
(3:7) Callisto − Ganymede	0.7°	30 y	0.012
Saturn System
(2:3) Enceladus − Mimas	33.2°	0.04 y	0.33
(2:3) Dione − Tethys	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°	4.0 y	0.051
Uranus System
(1:3) Umbriel − Miranda	24.5°	0.08 y	0.14
(3:5) Umbriel − Ariel	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
Pluto System
(1:4) Nix − Charon	39.1°	0.3 y	0.22
(1:5) P4 − Charon	?	?	?
(1:6) Hydra − Charon	6.6°	3.0 y	0.037

The most remarkable (least probable) orbital correlation in the list is that between Callisto and Ganymede, followed in second place by that between Hydra and Charon.

The two near resonances 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.

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.^[12] 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.^[13]

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.^[14][15] 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,^[16] 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).^[17][18]

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.

Plutoid Haumea's smaller inner moon Namaka is a tenth the mass of Hiʻiaka, orbiting Haumea in 18 days in a highly elliptical, non-Keplerian orbit, and as of 2008 is inclined 13° from the larger moon, which perturbs its orbit.^[19] 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 currently are in or at least close to an 8:3 resonance. This strongly perturbs Namaka's orbit, with a current precession of ~20°.

See also

• 3753 Cruithne, an asteroid in approximate 1:1 orbital resonance with the Earth
• Resonant trans-Neptunian object
• Commensurability (astronomy)
• Kozai resonance
• Dermott's Law
• Lagrangian points
• Mercury, which has a 3:2 spin-orbit resonance
• Tidal locking
• Tidal resonance
• Titius-Bode law
• Trojan object, a body in a type of 1:1 resonance
• Horseshoe orbit, followed by an object in another type of 1:1 resonance
• Musica universalis ("music of the spheres")

References and notes

1. ^ Marcy, Geoffrey W.; Butler, R. Paul; Fischer, Debra; Vogt, Steven S.; Lissauer, Jack J.; Rivera, Eugenio 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. 
2. ^ ^{a b} Rivera, Eugenio J.; Laughlin, Gregory; Butler, R. Paul; Vogt, Steven S.; Haghighipour, Nader; Meschiari, Stefano (June 2010). "The Lick-Carnegie Exoplanet Survey: A Uranus-mass Fourth Planet for GJ 876 in an Extrasolar Laplace Configuration". arXiv:1006.4244v1 [astro-ph.EP]. 
3. ^ Beatty, J. K. (2003-07-23). "Why Is Saturn Tipsy?". SkyAndTelescope.Com. http://www.skyandtelescope.com/news/3306806.html?page=1&c=y. Retrieved 2009-02-25. 
4. ^ Ward, W. R.; Hamilton, D. P. (November 2004). "Tilting Saturn. I. Analytic Model". Astronomical Journal (American Astronomical Society) 128 (5): 2501–2509. Bibcode 2004AJ....128.2501W. doi:10.1086/424533. http://www.iop.org/EJ/abstract/1538-3881/128/5/2501/. Retrieved 2009-02-25. 
5. ^ Hamilton, D. P.; Ward, W. R. (November 2004). "Tilting Saturn. II. Numerical Model". Astronomical Journal 128 (5): 2510–2517. Bibcode 2004AJ....128.2510H. doi:10.1086/424534. http://www.iop.org/EJ/abstract/1538-3881/128/5/2510. Retrieved 2009-02-25. 
6. ^ Renu Malhotra (1997). "Pluto's Orbit". http://www.nineplanets.org/plutodyn.html. Retrieved 2007-03-26. 
7. ^ Michael E. Brown (2009-03-23). "S/2005 (90482) 1 needs your help". Mike Brown's Planets (blog). http://www.mikebrownsplanets.com/2009/03/s1-90482-2005-needs-your-help.html. Retrieved 2009-03-25. 
8. ^ 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).
9. ^ 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.
10. ^ The probability of obtaining an orbital coincidence of equal or smaller mismatch by chance at least once in n attempts, where n is the integral 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.
11. ^ This near resonance has been termed the Great Inequality. It was first described by Laplace in a series of papers published 1784–1789.
12. ^ Hansen, Kathryn (June 7, 2005). "Orbital shuffle for early solar system". Geotimes. http://www.geotimes.org/june05/WebExtra060705.html. Retrieved 2007-08-26. 
13. ^ Chen, E. M. A.; Nimmo, F. (March 2008). "Thermal and Orbital Evolution of Tethys as Constrained by Surface Observations" (PDF). Lunar and Planetary Science XXXIX (2008). http://www.lpi.usra.edu/meetings/lpsc2008/pdf/1968.pdf. Retrieved 2008-03-14. 
14. ^ 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. 
15. ^ 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 ArieI-Umbriel 2:1 Mean-Motion Commensurabilities". Icarus 85 (2): 394–443. Bibcode 1990Icar...85..394T. doi:10.1016/0019-1035(90)90125-S. 
16. ^ 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. 
17. ^ 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. 
18. ^ 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. 
19. ^ D. Ragozzine, M. E. Brown, C. A. Trujillo, E. L. Schaller. "Orbits and Masses of the 2003 EL61 Satellite System". AAS DPS conference 2008. http://www.abstractsonline.com/viewer/viewAbstract.asp?CKey={421E1C09-F75A-4ED0-916C-8C0DDB81754D}&MKey={35A8F7D5-A145-4C52-8514-0B0340308E94}&AKey={AAF9AABA-B0FF-4235-8AEC-74F22FC76386}&SKey={545CAD5F-068B-4FFC-A6E2-1F2A0C6ED978}. Retrieved 2008-10-17. 

• 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 9783642044571. 

External links

• Locations of Solar System Planetary Mean-Motion Resonances. Web calculator that plots distributions of the semimajor axes (or in one case the perihelion distances) of the minor planets in relation to mean motion resonances of the planets (website maintained by M.A. Murison).