I was reading up on orbital resonance and came across something on Wikipedia stating that the 1:2 orbital resonance of Io-Europa 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. Which, when taken into account, makes it a perfect 1:2.

How does this work exactly? I would think that the 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.


1 Answer 1


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.

  • $\begingroup$ 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? $\endgroup$ Dec 27, 2023 at 5:41
  • $\begingroup$ You might find the following article interesting: Long-term evolution of the Galilean satellites: the capture of Callisto into resonance aanda.org/articles/aa/full_html/2020/07/aa37445-20/… $\endgroup$ Dec 27, 2023 at 6:17
  • $\begingroup$ 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. $\endgroup$ Dec 27, 2023 at 7:02
  • $\begingroup$ @DanceroftheStars Note that the JPL Horizons timespans for the major Jovian satellites is only 6 centuries, and 4 centuries for the minor satellites. ssd.jpl.nasa.gov/horizons/time_spans.html $\endgroup$
    – PM 2Ring
    Dec 27, 2023 at 12:44
  • $\begingroup$ Question, is there a way to model how much a hypothetical system would precess? Or is it just down to observation? $\endgroup$ Dec 28, 2023 at 19:10

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