Orbital resonance

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.^[25]

Mean-motion resonances in the Solar System

• 2:3 Pluto–Neptune (also Orcus and other plutinos)
• 2:4
  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).

Additionally, Haumea is thought to be in a 7:12 resonance with Neptune,^[26][27] and Gonggong is thought to be in a 3:10 resonance with Neptune.^[28]

The simple integer ratios between periods hide more complex relations:

• the point of
  conjunction can oscillate (librate
  ) around an equilibrium point defined by the resonance.
• given non-zero
  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 ${\displaystyle n\,\!}$ (inverse of periods, often expressed in degrees per day) would satisfy the following

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

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

${\displaystyle {\dot {\omega }}}$

${\displaystyle n_{\rm {Io}}-2\cdot n_{\rm {Eu}}+{\dot {\omega }}_{\rm {Io}}=0}$

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

${\displaystyle 4\cdot n_{\rm {Te}}-2\cdot n_{\rm {Mi}}-{\dot {\Omega }}_{\rm {Te}}-{\dot {\Omega }}_{\rm {Mi}}=0}$

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

Laplace resonance

The Laplace resonance involving Io–Europa–Ganymede includes the following relation locking the orbital phase of the moons:

${\displaystyle \Phi _{L}=\lambda _{\rm {Io}}-3\cdot \lambda _{\rm {Eu}}+2\cdot \lambda _{\rm {Ga}}=180^{\circ }}$

where ${\displaystyle \lambda }$ 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. ${\displaystyle \Phi _{L}}$ librates about 180° with an amplitude of 0.03°.^[29]

Another "Laplace-like" resonance involves the moons Styx, Nix, and Hydra of Pluto:^[16]

${\displaystyle \Phi =3\cdot \lambda _{\rm {S}}-5\cdot \lambda _{\rm {N}}+2\cdot \lambda _{\rm {H}}=180^{\circ }}$

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

As with the Galilean satellite resonance, triple conjunctions are forbidden. ${\displaystyle \Phi }$ librates about 180° with an amplitude of at least 10°.

Sequence of conjunctions of Hydra (blue), Nix (red), and Styx (black) over one third of their resonance cycle. Movements are counterclockwise and orbits completed are tallied at upper right of diagrams (click on image to see the whole cycle).

Plutino 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^[31] (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".^[32]

Naiad:Thalassa 73:69 resonance

Neptune's innermost moon, Naiad, is in a 73:69 fourth-order resonance with the next outward moon, Thalassa. 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.^{[33][34][note 1]}

Mean-motion resonances among extrasolar planets

While most

• 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).^[13][39][40] In this case, ${\displaystyle \Phi _{L}}$ librates with an amplitude of 40° ± 13° and the resonance follows the time-averaged relation:^[13]

${\displaystyle \Phi _{L}=\lambda _{\rm {c}}-3\cdot \lambda _{\rm {d}}+2\cdot \lambda _{\rm {e}}=0^{\circ }}$

• Kepler-223 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).^{[41][42][43][44]} This represents the first confirmed 4-body orbital resonance.^[45] 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.^[44]
• Kepler-80 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.^[46][36]
• TOI-178 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.^[47]
• 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.).^[48][37] The resonant configuration is expected to be stable on a time scale of billions of years, assuming it arose during planetary migration.^[49][50] A musical interpretation of the resonance has been provided.^[50]
• Kepler-29 has a pair of planets in a 7:9 resonance (ratio of 1/1.28587).^[43]
• Kepler-36 has a pair of planets close to a 6:7 resonance.^[51]
• 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).^[52]
• Of Kepler-90'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).^[53][36] f, g and h are also close to a 3:5:8 period ratio (3: 5.058: 7.964).^[54] Relevant to systems like this and that of Kepler-36, calculations suggest that the presence of an outer gas giant planet facilitates the formation of closely packed resonances among inner super-Earths.^[55]
• HD 41248 has a pair of super-Earths within 0.3% of a 5:7 resonance (ratio of 1/1.39718).^[56]
• K2-138 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 project Exoplanet Explorers, using K2 data.^[57] K2-138 could host co-orbital bodies (in a 1:1 mean-motion resonance).^[58] Resonant chain systems can stabilize co-orbital bodies^[59] and a dedicated analysis of the K2 light curve and radial-velocity from HARPS might reveal them.^[58] Follow-up observations with the Spitzer Space Telescope 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.^[60][61] Future observations with CHEOPS will measure transit-timing variations of the system to further analyse the mass of the planets and could potentially find other planetary bodies in the system.^[62]
• K2-32 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.^[63]
• Myr) 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.^[64]
• HD 158259 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, using the radial velocity method.^[65]
• Kepler-1649 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.^[66]
• transit timing variations of ~0.5 days for the innermost planet. There is a yet more massive outer planet in a ~1400 day orbit.^[67]
• HD 110067 has six known planets, in a 54:36:24:16:12:9 resonance ratio.^[68]

Cases of extrasolar planets close to a 1:2 mean-motion resonance are fairly common. Sixteen percent of systems found by the

A number of near-

The least probable orbital correlation in the list – meaning the relationship that seems most likely to have not just be by random chance – 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

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.^[79]

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

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.

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.^[92]

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.^[83][84]

In the case of

The smaller inner moon of the

Notes

References