Orbital resonance: Neptune and Pluto

Orbital resonance planets Neptune and PlutoAnother very significant orbital resonance system in our solar system is between Neptune and Pluto.

In fact, there are resonances in our Solar System! The simplest case is the one between Neptune and Pluto. For every two orbits Pluto makes, Neptune makes exactly three. This is why it is impossible for the two bodies every to collide, despite the fact that their orbits cross. This is the only stable resonance involving two planetary (although Pluto is a dwarf planet) bodies, but other objects outside of Neptune’s orbit are in resonances with Neptune, known as Trans-Neptunian objects. Most of these are smaller than Pluto, but one larger one is known: the dwarf planet Eris. However, Eris is not known to be in an exact orbital resonance with Neptune.

Other Trans-Neptunian objects sometimes are in resonances with Neptune, the most common being (object:Neptune) 2:3, corresponding to Pluto and other bodies, 3:5, 4:7, 1:2, and other rarer ones such as 2:5, 3:4, 4:5, 1:4, 1:5, 1:3, 3:7, and 6:11. Some of the latter sometimes correspond to only one known object, and may be coincidental. Some are also unstable, and smaller objects can often be ejected from a resonance by a gravitational pull quite easily. The special case of a 1:1 orbital resonance is addressed in the post, Lagrangian Points.
Orbital Resonance

Pluto lies in the 2:3 mean-motion resonance with Neptune: for every two orbits that Pluto makes around the Sun, Neptune makes three. The two objects then return to their initial positions and the cycle repeats, each cycle lasting about 500 years. This pattern is such that, in each 500-year cycle, the first time Pluto is near perihelion Neptune is over 50° behind Pluto. By Pluto's second perihelion, Neptune will have completed a further one and a half of its own orbits, and so will be a similar distance ahead of Pluto. Pluto and Neptune's minimum separation is over 17 AU. Pluto comes closer to Uranus (11 AU) than it does to Neptune.

The 2:3 resonance between the two bodies is highly stable, and is preserved over millions of years.[68] This prevents their orbits from changing relative to one another; the cycle always repeats in the same way, and so the two bodies can never pass near to each other. Thus, even if Pluto's orbit were not highly inclined the two bodies could never collide.
Relationship with Neptune