Previously, I asked At what point are orbital resonances no longer "ordered" but "chaotic?", and received an answer from @CarlWitthoft:

Perhaps if the calculated fraction had an irrational num or denom, then you could claim "chaos" because the fraction itself can't be evaluated exactly.

I'm a bit confused by this statement because if a resonance was simple but couldn't be represented as an integer fraction, such as 1:$\sqrt2$, would it not be a resonance because the fraction cannot be evaluated. So I am wondering, if the resonance is a ratio with a simple radical (like 2:$\sqrt5$ or 1:$\sqrt3$), would it still be considered a resonance?

  • 4
    $\begingroup$ In my opinion, there is no reason to discuss irrational values in the ratios at all, considering the rational numbers are a dense subset of the real numbers. That is, every real number has a rational number arbitrarily close to it. $\endgroup$
    – Connor Garcia
    Commented Jan 6, 2022 at 17:24
  • $\begingroup$ Just as a side note, non-periodicity is not enough for a system to be chaotic (the system must be ‘sufficiently complex’, normally meaning extremely sensitive initial conditions and perhaps period doubling), but the solar system as it stands is the steady state ending of what once was a chaotic system. $\endgroup$
    – Justin T
    Commented Jan 6, 2022 at 18:14

1 Answer 1


From various sources such as Wikipedia, NASA, and various published papers, an orbital resonance is:

when orbiting bodies exert regular, periodic gravitational influence on each other, usually because their orbital periods are related by a ratio of small integers

So a ratio of $1:\sqrt{2}$ wouldn't count.

What's more it wouldn't even make sense to talk about a ratio of $1:\sqrt{2}$. That requires that the resonance can be defined with infinite precision and that just isn't possible. Orbiting bodies have so many perturbations on them that if you look with enough detail, the resonance is never perfectly $1:\sqrt{2}$ or 1:2 or 2:3 or whatever it may be. You'll never achieve a $1:\sqrt{2}$ resonance.

On a related point, Connor Garcia points out in a comment that the rational numbers are dense in the reals, meaning that you can substitute any rational number with arbitrary precision that is arbitrarily close to $\sqrt{2}$. For example, you might use 1.414213562373095 instead, but then your ratio becomes 1:1.414213562373095 or rather 1000000000000000:1414213562373095 which is a completely meaningless resonance as it will have no significant gravitational effects. Such nonsense resonances result in relative positions of the two bodies which is essentially random and thus not really a resonance.


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