<p>The revolution period of a body is the time it takes to return to pericenter. So, for example, Io will return to its perijove after 1.769 days, and Europa to its own perijove after 3.551 days.</p>
<p>But during this time, each moon’s perijove will have moved, which means that the time taken by each moon to return to the same position relative to distant stars will <em>not</em> be 1.769 or 3.551 days.</p>

CC BY-SA 4.0
2023-12-27T05:41:09.840Z
So, it is simply the fact that they do orbit at the same ratio, but that the actual period is different from the equations due to precession of the point where they end? Sort of like just moving the goalpost, right? And that both have their endpoint precess by the same amount, keeping the ratios the same? If so, is there any known way to predict how much precession would occur in a given system, or does it just require observation?
DanceroftheStars
CC BY-SA 4.0
2023-12-27T06:17:39.707Z
You might find the following article interesting: Long-term evolution of the Galilean satellites: the capture of Callisto into resonance https://www.aanda.org/articles/aa/full_html/2020/07/aa37445-20/aa37445-20.html
Pierre Paquette
CC BY-SA 4.0
2023-12-27T07:02:29.903Z
A (very) crude simulation I did shows that if you *don't* take apsidal precession into account, the orbital periods are almost perfectly in 1:2 resonance between Io and Europa. I you *do* take apsidal precession into account, then it's more around 1.75:0.75.
Pierre Paquette
CC BY-SA 4.0
2023-12-27T12:44:30.357Z
@DanceroftheStars Note that the JPL Horizons timespans for the major Jovian satellites is only 6 centuries, and 4 centuries for the minor satellites. https://ssd.jpl.nasa.gov/horizons/time_spans.html
PM 2Ring
CC BY-SA 4.0
2023-12-28T19:10:09.630Z
Question, is there a way to model how much a hypothetical system would precess? Or is it just down to observation?
DanceroftheStars
CC BY-SA 4.0
2023-12-28T23:05:42.087Z
@DanceroftheStars: It is possible to model it. I suggest “Calculation of apsidal precession via perturbation theory” (https://arxiv.org/pdf/1802.07115.pdf), “Nodal Precession in Closely Spaced Planet Pairs” (https://arxiv.org/pdf/2003.07835.pdf), “Nodal and Periastron Precession of Inclined Orbits in the Field of a Rapidly Rotating Neutron Star” (https://articles.adsabs.harvard.edu/pdf/2002AstL...28...83S), and “Classical and relativistic node precessional effects in WASP-33b and perspectives for detecting them” (https://link.springer.com/article/10.1007/s10509-010-0468-x). Enjoy!
Pierre Paquette
CC BY-SA 4.0
2023-12-28T23:07:19.717Z
Oh! and “Introduction to Celestial Mechanics” (https://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/Celestialhtml.html).
Pierre Paquette
CC BY-SA 4.0
2023-12-28T23:30:50.760Z
It doesn't seem like "Calculation of apsidal precession via perturbation theory", has what I am looking for as it seems to just be about planets and not really dealing with resonance (at least it doesn't use the words resonance or resonant in the paper), the second paper, Nodal Precession in Closely Spaced Planet Pairs, seems to actively avoid talking about resonance, per their own claims... Cont...
DanceroftheStars
CC BY-SA 4.0
2023-12-28T23:31:02.203Z
"The goal of this work is to find how the nodal precession rate varies with inclination, eccentricity, and semi-major axis ratio, particularly for the values relevant for packed (though non-resonant) multiplanetary systems like those found by Kepler", does it still wind up being applicable even if it isn't discussing resonant systems?  cont...
DanceroftheStars
CC BY-SA 4.0
2023-12-28T23:31:20.807Z
(Also, that first one seems to have a lot of issues with never mentioning what units are used for any value mentioned, which just seems like poor scientific writing, in my opinion. As I was taught you should make as few assumptions about reader knowledge as possible) Anyway, thanks for the papers, I will look into them more.
DanceroftheStars
CC BY-SA 4.0
2023-12-29T00:36:42.167Z
It’s through the analysis of perturbations that you can deduce whether there are resonances…
Pierre Paquette
CC BY-SA 4.0
2023-12-27T03:04:55.927Z
2023-12-27T20:07:41.637Z
2023-12-27T20:07:41.637Z
Pierre Paquette
<p>I was reading up on orbital resonance and came across something on Wikipedia stating that the 1:2 orbital resonance of <a href="https://en.wikipedia.org/wiki/Io_(moon)" rel="nofollow noreferrer">Io</a>-<a href="https://en.wikipedia.org/wiki/Europa_(moon)" rel="nofollow noreferrer">Europa</a> is &quot;imperfect&quot;. I mean that when you simply take the period of one and multiply it by the ratio, you <em>won't</em> get the correct answer; they won't be equal.</p>
<p>That you have to take into account the precession of the <a href="https://en.wikipedia.org/wiki/Apsis#Etymology" rel="nofollow noreferrer">perijove</a>. Which, when taken into account, makes it a perfect 1:2.</p>
<p>How does this work exactly? I would think that the <a href="https://en.wikipedia.org/wiki/Mean_motion" rel="nofollow noreferrer">mean motion</a> ratio would be the same, regardless of the perijove as I thought that that was the fundamental definition of orbital resonance.</p>
<p>Let's give an example planet, so you can help explain it to me. Let’s say a regular 1 mass Earth, named Kana, with two 0.5 lunar mass moons. Let’s just call them Kana A and B, in a 1:2 resonance. I would think this would mean that they have a period of 27 days and 54 days respectively (which should be 380,944 km for A; and 604,711 km for B, in terms of semi-major axis, according to my previous understanding of orbital resonance), with mean motions in an exact 1:2 ratio. But it appears this is not the case.</p>
<p>Is it due to differing eccentricities? I don't think it would be, as eccentricity doesn't really play a part in orbital period equations. Only relevant body masses and the semi-major axis do.</p>
<p>So how exactly does the precession of the perikana allow for the mean motions to not be perfectly aligned, whilst still having the correct resonance? I don't really understand it at the moment and the page on it explains it very poorly. I want both the physics behind it, and the mathematics behind it, please.</p>

<orbit><orbital-resonance><eccentric-orbit>
Orbital resonance and the precession of the periapsis
DanceroftheStars
# Orbital resonance and the precession of the periapsis

(Asked by: username_0 on 2023-12-27)

I was reading up on orbital resonance and came across something on Wikipedia stating that the 1:2 orbital resonance of [Io](https://en.wikipedia.org/wiki/Io_(moon))-[Europa](https://en.wikipedia.org/wiki/Europa_(moon)) is "imperfect". I mean that when you simply take the period of one and multiply it by the ratio, you *won't* get the correct answer; they won't be equal.

That you have to take into account the precession of the [perijove](https://en.wikipedia.org/wiki/Apsis#Etymology). Which, when taken into account, makes it a perfect 1:2.

How does this work exactly? I would think that the [mean motion](https://en.wikipedia.org/wiki/Mean_motion) ratio would be the same, regardless of the perijove as I thought that that was the fundamental definition of orbital resonance.

Let's give an example planet, so you can help explain it to me. Let’s say a regular 1 mass Earth, named Kana, with two 0.5 lunar mass moons. Let’s just call them Kana A and B, in a 1:2 resonance. I would think this would mean that they have a period of 27 days and 54 days respectively (which should be 380,944 km for A; and 604,711 km for B, in terms of semi-major axis, according to my previous understanding of orbital resonance), with mean motions in an exact 1:2 ratio. But it appears this is not the case.

Is it due to differing eccentricities? I don't think it would be, as eccentricity doesn't really play a part in orbital period equations. Only relevant body masses and the semi-major axis do.

So how exactly does the precession of the perikana allow for the mean motions to not be perfectly aligned, whilst still having the correct resonance? I don't really understand it at the moment and the page on it explains it very poorly. I want both the physics behind it, and the mathematics behind it, please.

**Tags:** orbit, orbital-resonance, eccentric-orbit

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## 1 answers

### Answer by username_1 on 2023-12-27. Score: 8

The revolution period of a body is the time it takes to return to pericenter. So, for example, Io will return to its perijove after 1.769 days, and Europa to its own perijove after 3.551 days.

But during this time, each moon’s perijove will have moved, which means that the time taken by each moon to return to the same position relative to distant stars will *not* be 1.769 or 3.551 days.

> **Comments:**
> **username_0**: So, it is simply the fact that they do orbit at the same ratio, but that the actual period is different from the equations due to precession of the point where they end? Sort of like just moving the goalpost, right? And that both have their endpoint precess by the same amount, keeping the ratios the same? If so, is there any known way to predict how much precession would occur in a given system, or does it just require observation?  
> **username_1**: You might find the following article interesting: Long-term evolution of the Galilean satellites: the capture of Callisto into resonance https://www.aanda.org/articles/aa/full_html/2020/07/aa37445-20/aa37445-20.html  
> **username_1**: A (very) crude simulation I did shows that if you *don't* take apsidal precession into account, the orbital periods are almost perfectly in 1:2 resonance between Io and Europa. I you *do* take apsidal precession into account, then it's more around 1.75:0.75.  
> **username_2**: @DanceroftheStars Note that the JPL Horizons timespans for the major Jovian satellites is only 6 centuries, and 4 centuries for the minor satellites. https://ssd.jpl.nasa.gov/horizons/time_spans.html  
> **username_0**: Question, is there a way to model how much a hypothetical system would precess? Or is it just down to observation?  
> **username_1**: @DanceroftheStars: It is possible to model it. I suggest “Calculation of apsidal precession via perturbation theory” (https://arxiv.org/pdf/1802.07115.pdf), “Nodal Precession in Closely Spaced Planet Pairs” (https://arxiv.org/pdf/2003.07835.pdf), “Nodal and Periastron Precession of Inclined Orbits in the Field of a Rapidly Rotating Neutron Star” (https://articles.adsabs.harvard.edu/pdf/2002AstL...28...83S), and “Classical and relativistic node precessional effects in WASP-33b and perspectives for detecting them” (https://link.springer.com/article/10.1007/s10509-010-0468-x). Enjoy!  
> **username_1**: Oh! and “Introduction to Celestial Mechanics” (https://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/Celestialhtml.html).  
> **username_0**: It doesn't seem like "Calculation of apsidal precession via perturbation theory", has what I am looking for as it seems to just be about planets and not really dealing with resonance (at least it doesn't use the words resonance or resonant in the paper), the second paper, Nodal Precession in Closely Spaced Planet Pairs, seems to actively avoid talking about resonance, per their own claims... Cont...  
> **username_0**: "The goal of this work is to find how the nodal precession rate varies with inclination, eccentricity, and semi-major axis ratio, particularly for the values relevant for packed (though non-resonant) multiplanetary systems like those found by Kepler", does it still wind up being applicable even if it isn't discussing resonant systems?  cont...  
> **username_0**: (Also, that first one seems to have a lot of issues with never mentioning what units are used for any value mentioned, which just seems like poor scientific writing, in my opinion. As I was taught you should make as few assumptions about reader knowledge as possible) Anyway, thanks for the papers, I will look into them more.  
> **username_1**: It’s through the analysis of perturbations that you can deduce whether there are resonances…  

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