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For Earth, the lunar month is well known. But Earth has only one moon.

But on Jupiter, how long does it takes for the same moon configuration to appear in the "sky" of the planet?

For the sake of sanity, you can consider only the four Galileian Mooons.

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    $\begingroup$ This is a cool question, and it has an interesting answer if you allow for configurations of a lot less than all 79 moons! $\endgroup$ – uhoh Dec 27 '18 at 14:09
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    $\begingroup$ @uhoh I guess you are right. Let's restrict the scope. $\endgroup$ – Mindwin Dec 28 '18 at 11:07
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The three inner Galilean moons are in orbital resonances. For each orbit of Ganymede, Europa makes two orbits and Io makes four orbits. So after 1 orbit of Ganymede, the two inner moons are in the same configuration. That is 7.154 (Earth) days.

Callisto is not in an orbital resonance, It won't be in the same position, but its orbit (of 16.689 days) is close to a ratio of 7:3 to Ganymede. So after 7 orbits of Ganymede (50.08 days) Callisto will have made just over 3 orbits (50.07 days)

If you want all known moons to be in a similar configuration you will have to wait a while, probably longer than the life of the sun (there are a lot of moons)

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  • $\begingroup$ I've restricted the question to address only the four Galilean moons, I don't think your answer needs change. Just a heads up. XD $\endgroup$ – Mindwin Dec 28 '18 at 11:07
  • $\begingroup$ Seems like you could have a "week", defined by the orbit of Ganymede and a seven week "month" with a very occasional (about 1 in 5000) six week month defined by the near resonance with Callisto. $\endgroup$ – Steve Linton Dec 29 '18 at 14:22
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This is basically a question of asking about the time of simultaneous recurrence of periodic events happening with some given period, and thus is fundamentally mathematical, and it is a type of phenomenon that turns up in many, many, many different places in the study of the universe at all scales.

If you have $N$ cycles of lengths $P_1$, $P_2$, ..., $P_N$, then the shortest time for all of them to complete at once must be the least common multiple of all the times, since that will ensure that at the end, a whole number of every period will have elapsed and thus total recurrence:

$$P_\mathrm{recur} = \mathrm{lcm}(P_1, P_2, \cdots, P_N)$$

However, this lcm is only finite if the periods are all integral or at best rational multiples of some common unit to begin with. If not, and they are related by irrational numbers, then the period will be infinite, i.e. there will be no recurrence, though they may come within arbitrary closeness to their original configuration, but the interval for this will be very sensitive to the amount of proximity to the original configuration desired.

Thus, in fact, actually not only is it good to restrict it to the Galilean moons, and in fact only the inner three, for practicality, it is necessary if you want a finite periodicity because those moons, thanks to their orbital resonances, bear just such a relationship to each other. Even then though, it will have to be approximate, because the resonance is not perfect. Nonetheless, using the 1:2:4 relationship, since each number is a divisor of the next, that means the lcm and thus the period of the system are set by its longest member, the 4x one, which is the period of the futhest moon, Ganymede, so if you want the closest analogue should be Ganymede's orbital period, which is interestingly only a bit longer than our week, at 7.14 days (617 kiloseconds).

So that would be the best "lunar month" that I believe one can do, without too much arbitrary choosing, and you have to leave Callisto out of it. And if one wants, as a close week analogue, and embracing all the weirdness that a month is effectively a week, it could be a useful time unit for the Jupiter system, though I think that it makes more sense to get away with oddly wedding clocks and times to planetary movements as much as possible since they can simply be mapped by computers onto simpler time measures anyways as "events", should they be of concern. Instead I think for astronautical use in both measuring and also organizing we should be moving to just use SI units - seconds, kiloseconds, megaseconds, and gigaseconds - as much as possible use megaseconds for weeks, and for the day analogue, the only non-SI unit still necessary to deal with thanks to humans' biological planetary baggage in the form of the circadian rhythm system and really an organizer and not to be used as a measurer, you can use actually alternatively either 86 or 87 ks (actually 87 ks is almost exactly the natural period of humans' biological clocks), and if you balance both right 23 such days (87 ks each with the last being 86) will equal to exactly two megaseconds, regularizing the system as much as possible. These units will then be understood everywhere.

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