# At what point are orbital resonances no longer "ordered" but "chaotic?"

Orbital resonances are typically in small valued integer ratios, like 2:1, 3:2, or 4:7. However, there are some resonances whose ratios have large reduced values, including the 73:69 Naiad:Thalassa resonance. So I am wondering, at what point is a resonance "no longer" one?

By that I mean when the ratios have such large terms (but nearly exact) that it is no longer considered a resonance (for example 581:137), or will this just be reduced to a simpler but less accurate resonance (from 581:137 to 17:4 with an inaccuracy of 0.01)?

• 73:69 and 581:137 are not "irregular" (whatever that means); they are still ratios of integers. Commented Jan 5, 2022 at 16:55
• It might be a little unsettling that 73:69 is so close to 1:1, and it might feel like there's something that would tend to push them into a lower-order resonance or even to 1:1 with them at 60 or 180 degrees or "tadpoleing", but it seems that it's the inclination of Naiad that keeps this stable and there's now no pressure for them to leave this exact 73:69 resonance. They will continue to repeat this exact same dance every 21.5 days. This is based on a quick reading and so not an answer post.
– uhoh
Commented Jan 5, 2022 at 17:39
• Different but related (and currently unanswered) Just how "locked" are resonant-chains of exoplanets thought to be? (e.g. K2-138 and TOI-178)
– uhoh
Commented Jan 5, 2022 at 18:06
• Asked Is it still called an orbital resonance if the ratio is irrational? (cc @CarlWitthoft) Commented Jan 6, 2022 at 14:05

Consider a child on a stationary swing. The fastest way to get them going is to push once every time they swing (a 1:1 resonance). If you push 581 times for every 137 swings, the pushes will mostly average out and the child won't get very high on the swing.

Similarly, there are infinitely many orbital resonances possible, but when the integers in the ratios get higher, the effects of the resonance on the orbits are smaller. We don't typically describe orbital resonances as "ordered" or "chaotic," but instead describe them as "stable" or "unstable." A good example of this can be found in the asteroid belt. Orbital resonances with Jupiter that are unstable formed gaps in the belt, and orbital resonances that are stable formed concentrations. Credit Britannica

While there are an infinite number of other resonances possible here, practically we only care about the low integer value resonances since they greatly affect the asteroid orbits.

Large integer values for stable orbital resonances (like the 73:69 Naiad:Thalassa resonance) occur increasingly rarely in nature because they have less of a mutual effect on relative orbits than small integer value resonances.

I don't think you are correct. If the resonance can be calculated exactly, then it's a resonance. Chaotic orbits cannot be calculated exactly, as they depend critically on initial conditions.

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.

• 1) the OP clearly is not asking about chaos in dynamical systems sense. 2) the dynamics around a resonance typically are in fact chaotic Commented Jan 5, 2022 at 19:56