# CTK Insights

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17 Nov

### Orbital Resonance and the Existence of Irrational Numbers

I've been reading an outstanding collection An Invitation to Mathematics edited by D. Schleicher and M. Lackmann. There are 14 contributions by leading mathematicians, each introducing a direction of current mathematical research. The remarkable aspect of all the articles is that they all start at a level that could be appreciated by a curious high school student and then gently lead the reader to the frontiers of the Unknown.

On first reading, one article caught my attention, Small Divisors: Number Theory in Dynamical Systems (pp. 43-54) by J.-C. Yoccoz. The article begins thus:

We discuss dynamical systems with two or more particles, such as two planets orbiting around the sun. If the ratio of their rotation periods, say $\alpha$, is rational, then the planets are in resonance, and the mutual interaction will make the dynamics unstable.

I lived many more years to fondly remember my school days than I ever spent on schooling, but that mention of the resonance caused a knee-jerk reaction as if I still was a teenager. On reading that paragraph, I had a flash recollection of an oft-told (but likely untrue) story of Euler embarrassing the encyclopedist Diderot with a frivolous argument that ended with "...hence God exists." The association was rather exhilarating to let the opportunity pass. In my excitement, I rushed to post a message on twitter, "Two planets around a sun with periods T1 & T2. For T1/T2 rational, interaction gives resonance & unstable. Hence irrational numbers exist."

To me, the existence of irrational numbers required no confirmation beyond the argument with sources in the Pythagorean School. As the reaction to my tweet showed, this view is not universally shared. (It was labeled "creationism" and then denied on the basis of the limitations imposed on measurements by the Heisenberg Uncertainty Principle.) In any event, while it may be entertaining (which it was intended to), the argument was rather shaky. It was set straight for me by Colm Mulcahy (@CardColm) and his astronomer friend Carl Murray (Queen Mary College, Univ of London); I am indebted to them both.

The article goes on to model the motion of a single planet with an iterative process (discrete dynamic system) $z_{n+1}=\lambda z_{n}$, where $\lambda=e^{2\pi\alpha}$, $\alpha$ is real, making $\lambda$ a complex number of unit module. As Eugene Wigner (1963 Nobel Prize in Physics) wrote (among much more consequential observation),

Certainly nothing in our experience suggests the introduction of these quantities (complex numbers).

For $\alpha$ rational, the iterations are periodic, while, for irrational $\alpha$, they are chaotic becoming dense on the circle $|z]=|z_{0}|.$

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.

The influence of the second planet is modeled with the introduction of a quadratic term: $z_{n+1}=\lambda z_{n} + z^{2}_{n}$. The behavior of the perturbed system differs much from the unperturbed one. For the rational $\alpha$, the system loses stability: the iterations do not stay close to the origin for any non-zero $z_{0}$. However, for some irrational $\alpha$, viz., those that satisfy certain (Diophantine!) conditions on their proximity to the rational numbers, the system becomes periodic.

All the laws of nature are conditional statements which permit a prediction of some future events on the basis of the knowledge of the present, except that some aspects of the present state of the world, in practice the overwhelming majority of the determinants of the present state of the world, are irrelevant from the point of view of the prediction.

In reality, the planet motions are practically periodic, and the influence the planets exert on each other is often realized as the phenomenon of stable orbital resonance. Bodies involved in an orbital resonance have periods related by rational numbers with small numerators end denominators. Examples are numerous. Jupiter's moons Ganymede, Europa and Io have periods in ratios 4:2:1; the periods of Pluto and Neptune relate as 3:2. In such a configuration the planets periodically disturb each other's trajectory but also undergo mutual trajectory corrections that keep them in stable relation.

On the other hand, the gaps between Saturn's inner moons is believed to be the result of unstable resonance: some moons have been ejected from their trajectories years ago (on the cosmological scale.)

### References

1. K. Dombrowski, Rational Numbers Distribution and Resonance, Progress in Physics, v 1, 2005, pp. 65-67

Oscillating systems, having a peculiarity to change their own parameters because of interactions inside the systems, have a tendency to reach a stable state where the separate oscillators frequencies are interrelated by specific numbers - minima of the rational number density on number line.

2. P. Goldreich, An explanation of the frequent occurrence of commensurable mean motions in the solar system, Monthly Notices of the Royal Astronomical Society, Vol. 130, 1965, pp. 159-181

We now realize that the orbits of a pair of near-commensurate satellites will still evolve as the tides feed angular momentum from the planet's spin into the satellites' orbits. However, we shall see that the satellites will share this angular momentum between them in just the correct proportion to keep their mean motions near-commensurate.

3. A. E. Roy, M. W. Ovenden, On the occurrence of commensurable mean motions in the solar system, Monthly Notices of the Royal Astronomical Society, Vol. 114, 1954, pp. 232-241

The present analysis of the actual mean motions in the solar system indicates that, far from orbits with commensurable mean motions being unstable, there is a distinct preference for them.

Twoo possible interpretations suggest themselves. The observed distribution of orbits may be explained by supposing either that the mechanism of formation of the planets and satellites was such as to favor orbits with commensurable mean motions, or that such orbits are relatively more stable over long periods of time than the neighboring orbits, the planets and satellites thus tending towards commensurable configurations.

5 − = two