### The 3 : 2 Resonance between Neptune and Pluto

For every two orbits of Pluto around the Sun, Neptune completes three orbits. This 3 : 2 resonance has profound consequences for the stability of the orbit of Pluto.

Unstable (left) and stable (right) orbital configurations.

The Harmony of the Spheres

Pythagoras based his musical analysis on two ratios: the octave 2 : 1 and the perfect fifth 3 : 2. He was able to construct an entire diatonic scale (do-re-me- … ) using only these ratios. The idea that musical ratios governe the heavenly bodies was known as The Harmony of the Spheres. Kepler was also convinced that musical ratios were at the heart of the design of the universe. He identified many harmonic relationships. In particular, the variation of the angular speed of Mars generated a musical interval, a perfect fifth: the ratio between the maximum and minimum values of speed was observed to be 3 : 2. Moreover, the relationship between the periods of Mars and Venus was believed to be 3 : 2, again corresponding to a musical fifth.

Kepler’s musical ideas are now regarded as historical curiosities but, in fact, small integer ratios play a central role in solar system dynamics. We look at one example; Neptune and Pluto are in a precise dynamical 3 : 2 resonance. The period of Neptune’s orbit about the Sun is two-thirds that of Pluto. This has critical consequences for the long-term stability of the orbits. The orbit of Pluto is eccentric (e = 0.25) and, for part of its orbit, it is closer to the Sun than Neptune is. Given that the orbit of Pluto is inclined to the eclyptic, there should be no risk of collision; however, a close encounter between the two bodies could greatly perturb the orbit of Pluto. This does not happen. Why?

Stability of the 3 : 2 Resonance

The figure below shows the orbits of Neptune (blue) and Pluto (red) with time-marks: at the initial time (marked 0) the two bodies are on opposite sides of the Sun, with Pluto at perihelion. Successive numbers are half a Neptunean year apart. After one and a half orbits of Neptune, the two bodies are close together, with Pluto again at perihelion, and there is a strong gravitational interaction between them. This configuration, which is unstable, is not found in reality.

At the initial time, Neptune and Pluto are in opposition with Pluto at perihelion. Numbers indicate their positions at intervals of half a Neptunean year, for a period of three years. This configuration is unstable.

The figure below shows the orbits in (approximately) the observed configuration. At time t = 0, they are in conjunction but at the aphelion of Pluto and there is no close encounter. Indeed, all conjunctions are at this point in the orbits. When Pluto is close to the Sun, Neptune is far away from it, so there is no risk of a close encounter. Long-term numerical integrations have shown that the 3 : 2 resonance endures over billions of years. As a result, the distance between Neptune and Pluto never falls below about 20 AU.

At the initial time Neptune and Pluto are in conjunction at the aphelion of Pluto. Numbers indicate their positions at intervals of half a Neptunean year, for a period of three years. This configuration is stable.

Of course, we now know that Pluto is just one of a large collection of trans-Neptunean objects (TNOs) and many of these do not have an orbit-stabilizing resonance configuration like that enjoyed by Pluto. We can see how the 3 : 2 resonance, first found in the context of harmonic musical intervals (perfect fifths), enables the solar system to maintain a regular configuration over billiennia.