Resonant Trans-Neptunian Object - Toward A Formal Definition

Toward A Formal Definition

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

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 a Resonant if for some small integers p,q,n,m,r,s, the argument (angle) defined below is librating (i.e. is bounded)

where the are the longitudes of perihelia and the 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 librates around 180° with an amplitude of around 82° degrees, i.e. the angle changes periodically from 180°-82° to 180°+82°.

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

similar to Pluto's mean motion resonance.

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

In this case, the importance of the resonant angle can be understood by noting that when the object is at perihelion, i.e., then

i.e. gives a measure of the distance of the object's perihelion from Neptune. The object is protected from the perturbation by keeping its perihelion far from Neptune provided librates around an angle far from 0°.