<p>According to my understanding, for an orbital resonance to be stable, there must be at least three circumstances present:</p>
<ol>
<li>The forces on the bodies in resonance have to cancel out over time,</li>
<li>The momentary forces shouldn't be that high that the orbits are changed significantly,</li>
<li>after a perturbation, the bodies should pull each other back into resonance.</li>
</ol>
<p>For example, the Trojans are in a 1:1 resonance around the Lagrange points of Jupiter, and the Galilean moons of Jupiter are in a 4:2:1 resonance. A 2:1 resonance wouldn't be stable at it's own because the gravitational forces would always apply at the same point, but the perturbances are cancelled out by the third moon, and the outer and inner moon are in a stable 4:1 resonance.</p>
<p>In general, a resonance can be stable, if the encounters of the objects are equally distributed, so imaginary lines between these points have point symmetry / a regular shape (for example a straight line across the orbits (3:1; 3:2), an equilateral triangle (4:1; 4:3), a square (5:1; 5:2), a pentagon or pentagram, etc.) But, the closer the orbits get to each other, the more unstable they get, because the deflection gets bigger, so those resonances only work if the bodies aren't too massive. 2:1 resonances can be relatively stable too, but without any other resonance preventing it, the eccentricity will rise over time and one of the bodies may be ejected. However, orbital resonances are only stable in certain circumstances, because if the eccentricity and/or the inclination are too high, the forces don't cancel each other out enough. But there still are some configurations in which highly inclined and eccentric orbits are stable.</p>

CC BY-SA 4.0
2021-02-26T13:50:04.207Z
2021-05-08T17:09:59.273Z
2021-05-08T17:09:59.273Z
Jan
<p>I have just started learning about orbital resonance. I understand how bodies in orbital resonance will line up according to the orbital ratio number, and there will be increased gravitational effects upon alignment.</p>
<p>However, I do not understand how resonance can often have a stabilising effect, while other times, it is destabilising? For example, in the Io-Europa-Ganymede system, the 4:2:1 resonance has a stabilising effect. But also, I read that the gravitational perturbances when the moons align cause them to have elliptical orbits, so is it really a stabilising effect? There are other examples where orbital resonance completely prevents stability, like in the Kirkwood Gaps of the asteroid belt.</p>
<p>So my question is, how do orbital resonances form in the first place, and what determines if they are stable/unstable? Thank you.</p>

CC BY-SA 4.0
2021-02-05T17:14:00.190Z
there a study "lyapunov stability" https://www.univie.ac.at/adg/Teaching/ArchitekturvonPlanetensystemen/moser.pdf
Adrian R
CC BY-SA 4.0
2021-02-06T05:06:38.530Z
*Great question!*
uhoh
CC BY-SA 4.0
2021-02-06T21:03:10.473Z
Thanks for the link Adrian, it is a useful read.
Matthew H
CC BY-SA 4.0
2021-02-06T21:07:54.680Z
I'd like to add another point too. The Kirkwood gaps are regions where there is an orbital resonance with Jupiter, which creates regular gravitational perturbances that destabilise any bodies in this region.

This is understandable, but if you consider the Hilda and Trojan asteroid families, these are also in resonance with Jupiter, so would they not be destabilised too? However, it is also important to consider that these families are in the Lagrange points of Jupiter... so maybe that explains it?
Matthew H
CC BY-SA 4.0
2021-02-07T17:24:14.207Z
Aliasing, Gravity being the signal: https://en.wikipedia.org/wiki/Aliasing
Wayfaring Stranger
CC BY-SA 4.0
2021-02-08T20:52:27.613Z
I'd swear I asked this question once before, but I can't find it. I'm not an expert so I am just commenting, but I think there is a destabilizing effect when resonances are too close together.
Jack R. Woods
CC BY-SA 4.0
2021-02-26T16:19:50.607Z
@Matthew Jupiter's Trojan asteroids are nested inside the gravitational valleys at the L4 and L5 Lagrange points (regions).   Jupiter's mass explains why there are so many of them.  That's a specific case of resonance that's well understood and can be highly stable.  Jupiter's Hilda Triangle, while more complicated, also seems synched to Jupiter's large L4 and L5 gravitational valleys.    It may be that most cases of resonance are unstable with L4 and L5 creating exceptions, though Pluto seems stable to Neptune and I don't know if Pluto passes anywhere close to Neptune's L4 and L5 points.
userLTK
<orbit><orbital-resonance>
How can orbital resonance sometimes have a stabilising effect, whilst other times, it has a destabilising effect?
Matthew H
# How can orbital resonance sometimes have a stabilising effect, whilst other times, it has a destabilising effect?

(Asked by: username_0 on 2021-02-04)

I have just started learning about orbital resonance. I understand how bodies in orbital resonance will line up according to the orbital ratio number, and there will be increased gravitational effects upon alignment.

However, I do not understand how resonance can often have a stabilising effect, while other times, it is destabilising? For example, in the Io-Europa-Ganymede system, the 4:2:1 resonance has a stabilising effect. But also, I read that the gravitational perturbances when the moons align cause them to have elliptical orbits, so is it really a stabilising effect? There are other examples where orbital resonance completely prevents stability, like in the Kirkwood Gaps of the asteroid belt.

So my question is, how do orbital resonances form in the first place, and what determines if they are stable/unstable? Thank you.

**Tags:** orbit, orbital-resonance
> **Comments:**
> **username_1**: there a study "lyapunov stability" https://www.univie.ac.at/adg/Teaching/ArchitekturvonPlanetensystemen/moser.pdf  
> **username_2**: *Great question!*  
> **username_0**: Thanks for the link Adrian, it is a useful read.  
> **username_0**: I'd like to add another point too. The Kirkwood gaps are regions where there is an orbital resonance with Jupiter, which creates regular gravitational perturbances that destabilise any bodies in this region.  This is understandable, but if you consider the Hilda and Trojan asteroid families, these are also in resonance with Jupiter, so would they not be destabilised too? However, it is also important to consider that these families are in the Lagrange points of Jupiter... so maybe that explains it?  
> **username_3**: Aliasing, Gravity being the signal: https://en.wikipedia.org/wiki/Aliasing  
> **username_4**: I'd swear I asked this question once before, but I can't find it. I'm not an expert so I am just commenting, but I think there is a destabilizing effect when resonances are too close together.  
> **username_5**: @Matthew Jupiter's Trojan asteroids are nested inside the gravitational valleys at the L4 and L5 Lagrange points (regions).   Jupiter's mass explains why there are so many of them.  That's a specific case of resonance that's well understood and can be highly stable.  Jupiter's Hilda Triangle, while more complicated, also seems synched to Jupiter's large L4 and L5 gravitational valleys.    It may be that most cases of resonance are unstable with L4 and L5 creating exceptions, though Pluto seems stable to Neptune and I don't know if Pluto passes anywhere close to Neptune's L4 and L5 points.  

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## 1 answers

### Answer by username_6 on 2021-02-26. Score: 5 | Accepted answer

According to my understanding, for an orbital resonance to be stable, there must be at least three circumstances present:

1. The forces on the bodies in resonance have to cancel out over time,
2. The momentary forces shouldn't be that high that the orbits are changed significantly,
3. after a perturbation, the bodies should pull each other back into resonance.

For example, the Trojans are in a 1:1 resonance around the Lagrange points of Jupiter, and the Galilean moons of Jupiter are in a 4:2:1 resonance. A 2:1 resonance wouldn't be stable at it's own because the gravitational forces would always apply at the same point, but the perturbances are cancelled out by the third moon, and the outer and inner moon are in a stable 4:1 resonance.

In general, a resonance can be stable, if the encounters of the objects are equally distributed, so imaginary lines between these points have point symmetry / a regular shape (for example a straight line across the orbits (3:1; 3:2), an equilateral triangle (4:1; 4:3), a square (5:1; 5:2), a pentagon or pentagram, etc.) But, the closer the orbits get to each other, the more unstable they get, because the deflection gets bigger, so those resonances only work if the bodies aren't too massive. 2:1 resonances can be relatively stable too, but without any other resonance preventing it, the eccentricity will rise over time and one of the bodies may be ejected. However, orbital resonances are only stable in certain circumstances, because if the eccentricity and/or the inclination are too high, the forces don't cancel each other out enough. But there still are some configurations in which highly inclined and eccentric orbits are stable.


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