Resonant trans-Neptunian object

Distribution of trans-Neptunian objects. Objects occupying the stronger resonances are in red.

The diagram illustrates the distribution of the known trans-Neptunian objects. Resonant objects are plotted in red. Orbital resonances with Neptune are marked with vertical bars: 1:1 marks the position of Neptune's orbit and its trojans; 2:3 marks the orbit of Pluto and plutinos; and 1:2, 2:5, etc. mark a number of smaller families. The designation 2:3 or 3:2 both refer to the same resonance for TNOs. There is no ambiguity, because TNOs have, by definition, periods longer than Neptune's. The usage depends on the author and the field of research.

Detailed analytical and numerical studies of Neptune's resonances have shown that the objects must have a relatively precise range of energies.[2][3] If the object's semi-major axis is outside these narrow ranges, the orbit becomes chaotic, with widely changing orbital elements. As TNOs were discovered, more than 10% were found to be in 2:3 resonances, far from a random distribution. It is now believed that the objects have been collected from wider distances by sweeping resonances during the migration of Neptune.[4] Well before the discovery of the first TNO, it was suggested that interaction between giant planets and a massive disk of small particles would, via angular-momentum transfer, make Jupiter migrate inwards and make Saturn, Uranus, and especially Neptune migrate outwards. During this relatively short period of time, Neptune's resonances would be sweeping the space, trapping objects on initially varying heliocentric orbits into resonance.[5]

One of the concerns is that weak resonances may exist and would be difficult to prove due to the current lack of accuracy in the orbits of these distant objects. Many objects have orbital periods of more than 300 years and most have only been observed over a relatively short observation arc of a few years. Due to their great distance and slow movement against background stars, it may be decades before many of these distant orbits are determined well enough to confidently confirm whether a resonance is true or merely coincidental. A true resonance will smoothly oscillate while a coincidental near resonance will circulate. (See Toward a formal definition)

Simulations by Emel'yanenko and Kiseleva in 2007 show that (131696) 2001 XT254 is librating in a 3:7 resonance with Neptune.[15] This libration can be stable for less than 100 million to billions of years.[15]

The orbital period of 2001 XT₂₅₄ around the 3:7 (2.333) resonance of Neptune.

Emel'yanenko and Kiseleva also show that (48639) 1995 TL8 appears to have less than a 1% probability of being in a 3:7 resonance with Neptune, but it does execute circulations near this resonance.[15]

The orbital period of 1995 TL8 missing the 3:7 (2.333) resonance of Neptune.

The classes of TNO have no universally agreed precise definitions, the boundaries are often unclear and the notion of resonance is not defined precisely. The Deep Ecliptic Survey introduced formally defined dynamical classes based on long-term forward integration of orbits under the combined perturbations from all four giant planets. (see also formal definition of classical KBO)

In general, the mean-motion resonance may involve not only orbital periods of the form

${\displaystyle {\rm {p\cdot \lambda -{\rm {q\cdot \lambda _{\rm {N}}}}}}}$

where p and q are small integers, λ and λ_N are respectively the mean longitudes of the object and Neptune, but can also involve the longitude of the perihelion and the longitudes of the nodes (see orbital resonance, for elementary examples)

An object is resonant if for some small integers (p,q,n,m,r,s), the argument (angle) defined below is librating (i.e. is bounded):[16]

${\displaystyle \phi ={\rm {p\cdot \lambda -{\rm {q\cdot \lambda _{\rm {N}}-{\rm {m\cdot \varpi -{\rm {n\cdot \Omega -{\rm {r\cdot \varpi _{\rm {N}}-{\rm {s\cdot \Omega _{\rm {N}}}}}}}}}}}}}}}$

where the ${\displaystyle \varpi }$ are the longitudes of perihelia and the ${\displaystyle \Omega }$ are the longitudes of the ascending nodes, for Neptune (with subscripts "N") and the resonant object (no subscripts).

The term libration denotes here periodic oscillation of the angle around some value and is opposed to circulation where the angle can take all values from 0 to 360°. For example, in the case of Pluto, the resonant angle ${\displaystyle \phi }$ librates around 180° with an amplitude of around 86.6° degrees, i.e. the angle changes periodically from 93.4° to 266.6°.[17]

All new plutinos discovered during the Deep Ecliptic Survey proved to be of the type

${\displaystyle \phi ={\rm {3\cdot \lambda -{\rm {2\cdot \lambda _{\rm {N}}-\varpi }}}}}$

similar to Pluto's mean-motion resonance.

More generally, this 2:3 resonance is an example of the resonances p:(p+1) (for example 1:2, 2:3, 3:4) that have proved to lead to stable orbits.[4] Their resonant angle is

${\displaystyle \phi ={\rm {p\cdot \lambda -{\rm {q\cdot \lambda _{\rm {N}}-({\rm {p-{\rm {q)\cdot \varpi }}}}}}}}}$

In this case, the importance of the resonant angle ${\displaystyle \phi \,}$ can be understood by noting that when the object is at perihelion, i.e. ${\displaystyle \lambda =\varpi }$, then

${\displaystyle \phi =q\cdot (\varpi -\lambda _{\rm {N}})}$

i.e. ${\displaystyle \phi \,}$ gives a measure of the distance of the object's perihelion from Neptune.[4] The object is protected from the perturbation by keeping its perihelion far from Neptune provided ${\displaystyle \phi \,}$ librates around an angle far from 0°.

As the orbital elements are known with a limited precision, the uncertainties may lead to false positives (i.e. classification as resonant of an orbit which is not). A recent approach[18] considers not only the current best-fit orbit but also two additional orbits corresponding to the uncertainties of the observational data. In simple terms, the algorithm determines whether the object would be still classified as resonant if its actual orbit differed from the best fit orbit, as the result of the errors in the observations. The three orbits are numerically integrated over a period of 10 million years. If all three orbits remain resonant (i.e. the argument of the resonance is librating, see formal definition), the classification as a resonant object is considered secure.[18] If only two out of the three orbits are librating the object is classified as probably in resonance. Finally, if only one orbit passes the test, the vicinity of the resonance is noted to encourage further observations to improve the data.[18] The two extreme values of the semi-major axis used in the algorithm are determined to correspond to uncertainties of the data of at most 3 standard deviations. Such range of semi-axis values should, with a number of assumptions, reduce the probability that the actual orbit is beyond this range to less than 0.3%. The method is applicable to objects with observations spanning at least 3 oppositions.[18]

• John K. Davies; Luis H. Barrera, eds. (2004-08-03). The First Decadal Review of the Edgeworth-Kuiper Belt. Springer. ISBN 1-4020-1781-2.
• E. I. Chiang; J. R. Lovering; R. L. Millis; M. W. Buie; L. H. Wasserman & K. J. Meech (June 2003). "Resonant and Secular Families of the Kuiper Belt". Earth, Moon, and Planets. Springer Netherlands. 92 (1–4): 49–62. arXiv:astro-ph/0309250. Bibcode:2003EM&P...92...49C. doi:10.1023/B:MOON.0000031924.20073.d0.
• E. I. Chiang; A. B. Jordan; R. L. Millis; M. W. Buie; L. H. Wasserman; J. L. Elliot; S. D. Kern; D. E. Trilling; K. J. Meech & R. M. Wagner (2003-01-21). "Resonance occupation in the Kuiper Belt: case examples of the 5:2 and trojan resonances". The Astronomical Journal. 126 (1): 430–443. arXiv:astro-ph/0301458. Bibcode:2003AJ....126..430C. doi:10.1086/375207.
• Renu Malhotra. "The Kuiper Belt as a Debris Disk" (PDF). Archived from the original (PDF) on 2005-10-22. Cite journal requires |journal= (help) (as HTML)