<p>For illustrative purposes, if I assume that the Galilean moons, Io, Europa, Ganymede, and Callisto, have coplanar circular orbits, and their initial configuration are, respectively: 10, 20, 30, and 40 degrees, measured from an arbitrary axis, when this configuration will repeat again? Considering that the four moons have their actual periods.</p>

CC BY-SA 4.0
2021-01-31T02:40:15.500Z
It's a very interesting problem and how to solve it depends on assumptions. You have coplanar and circular which is an excellent start, and though you don't mention it, I assume we also include that three are in "4:2:1 orbital resonance", is that correct? The problem gets complicated because Callisto is not in resonance with the other three, so it's necessary to state *how close* one configuration has to be to another to be considered a repeat. 0.1° or 1° or 10° or...
uhoh
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2021-01-31T02:45:09.313Z
So one set of mathematics is used for the repeat frequency of the resonant triplet and *a different set* is used for the near-repeat frequency of Callisto with those repeats.
uhoh
CC BY-SA 4.0
2021-01-31T18:40:31.427Z
Well, basically, you would take their orbital period and work from there. Io goes around Jupiter every 42 h 28 min, so for it to be back at its 10° of your original position, you need multiples of 42 h 28 min. Then for Europa it’s 3 d 13 h 13.7 min, so that’s a second condition. And so on. Eventually you’ll realize they *never* return to the *exact* same configuration, but as @uhoh said, you can take “close to” as a good-enough answer. Hope it helps!
Pierre Paquette
<orbital-resonance>
How can I calculate how long time a Galilean moon orbital configuration will take to repeat again?
RBN Bat
# How can I calculate how long time a Galilean moon orbital configuration will take to repeat again?

(Asked by: username_0 on 2021-01-31)

For illustrative purposes, if I assume that the Galilean moons, Io, Europa, Ganymede, and Callisto, have coplanar circular orbits, and their initial configuration are, respectively: 10, 20, 30, and 40 degrees, measured from an arbitrary axis, when this configuration will repeat again? Considering that the four moons have their actual periods.

**Tags:** orbital-resonance
> **Comments:**
> **username_1**: It's a very interesting problem and how to solve it depends on assumptions. You have coplanar and circular which is an excellent start, and though you don't mention it, I assume we also include that three are in "4:2:1 orbital resonance", is that correct? The problem gets complicated because Callisto is not in resonance with the other three, so it's necessary to state *how close* one configuration has to be to another to be considered a repeat. 0.1° or 1° or 10° or...  
> **username_1**: So one set of mathematics is used for the repeat frequency of the resonant triplet and *a different set* is used for the near-repeat frequency of Callisto with those repeats.  
> **username_2**: Well, basically, you would take their orbital period and work from there. Io goes around Jupiter every 42 h 28 min, so for it to be back at its 10° of your original position, you need multiples of 42 h 28 min. Then for Europa it’s 3 d 13 h 13.7 min, so that’s a second condition. And so on. Eventually you’ll realize they *never* return to the *exact* same configuration, but as @uhoh said, you can take “close to” as a good-enough answer. Hope it helps!  

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