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The cycles/orbital periods of the moons are 12, 20, 20, 24, 32, and 38 days. How often/after how many days would all 6 be full at once?

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    $\begingroup$ I see you didn't take my suggestion to heart to start with something simpler, so that you can learn about orbital mechanics and use that to construct a plausible system, rather than start with something implausible and try to beat it into shape :-/ $\endgroup$ Commented Jun 8 at 20:25
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    $\begingroup$ You have two moons with same period, 20 days. They share same orbit but can not be anywhere close, like closer than 60 degrees. So, never. $\endgroup$ Commented Jun 9 at 17:54

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If they are ever all full at once, and if their orbits are stable, then the time between such grand syzygies is the least common multiple of their periods: 9120 days.

Note that the periods of Io, Europa and Ganymede are in ratio 1:2:4 but they are never all in a line.

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    $\begingroup$ If (as in traditional werewolf lore) there is some slack in the definition of ‘full’, the interval may be much shorter. $\endgroup$ Commented Jun 8 at 20:11
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If your moons have quasi-circular orbits, it implies that the two moons with the same period are on the same orbit by Kepler’s 3rd law, or at least that one is on a Lagrange point of the other.

Only Lagrange points L4 and L5 are stable, but they are 60° ahead and behind respectively. In that case, they will never be full at the same time.

If at least one of these two “moons” has an elliptical orbit, I’m not sure you would still call it a moon, nor if it could be made stable considering that you have 4 other (close) moons plus the star messing with that elliptical orbit (n-body problem). Assuming it is stable, it would depend on whether their initial configuration allows them to be full at the same time – then refer to Anton Sherwood’s answer for that.

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  • $\begingroup$ It is also possible that the orbits of two moons with the same period might be on different orbital planes. This would make the alignment of full moons less likely... but we could also have a precession of the orbital plane and somehow match its period to other occurrences to make it more likely... $\endgroup$ Commented Jun 8 at 12:26
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    $\begingroup$ @MichalPaszkiewicz full moon happens when the planet is almost between the moon and the star, so if the moons have circular orbits, they would be very close to each other at that moment anyway. I doubt this could be stable, so we are back on elliptical orbits. $\endgroup$
    – Didier L
    Commented Jun 8 at 13:05
  • $\begingroup$ Unfortunately there are serious stability issues with Lagrange orbits involving bodies that are of anything like a similar mass. $\endgroup$
    – Ash
    Commented Jun 9 at 8:42
  • $\begingroup$ @Ash OP didn’t mention anything about the size of the moons. I think it’s really the only option to have something stable anyway $\endgroup$
    – Didier L
    Commented Jun 9 at 18:12
  • $\begingroup$ Given the variance in orbital times, they NEED independent orbits. Just like Earth and Mars having independent orbits around the sun. If moons reside in each other's Lagrange points, they'd need to have equal duration orbits. $\endgroup$
    – Gloweye
    Commented Jun 9 at 21:33
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This depends, to some extent, on what you mean by "cycles/orbital periods".

If you're referring to the time between successive "full moons" on each satellite, then Anton Sherwood's answer is correct- if they ever all line up*, it'll happen once every 9120 days.

However, that is not how astronomers use the term "orbital period". An orbital period is the time it takes a body to make one 360° rotation around whatever it's orbiting. This isn't the same thing as time-between-full-moons.

If the times given in the OP are, in fact, orbital periods, and the moons ever line up in the direction of, say, Sagittarius, then they'd line up again in the direction of Sagittarius every 9120 days thereafter. However, that's not the only place or time where they'd all align. The greatest common divisor of your orbital periods is 2, which means halfway through the 9120-day cycle (i.e. 4560 days after the Sagittarius syzygy), they'd all line up in the opposite direction, toward Gemini.

Whether they're full at either or both of those times depends on where your planet is in its orbit around the sun.

If your planet's orbital period (i.e. the length of its year) is a factor of 9120, then the grand syzygies will occur at the same time of year in every year that they occur. If it happens that Sagittarius is in opposition at the time of the Sagittarius syzygy, then all the moons will be full at that time, as well as every 9120 days thereafter.

If the length of your planet's year is an even factor of 9120, then both grand syzygies will occur at the same time of year. If the moons are all full at the Sagittarius syzygy, then they'll all be new during the Gemini syzygy.

If the length of your planet's year is an odd factor of 9120 (i.e. 3, 5, 15, 19, 57, 95, or 285 days), then the moons will be in the same phase on both grand syzygies. If they're full at those times, then you'd have six simultaneous full moons every 4560 days.

If your planet's year-length isn't a factor of 9120, then there wouldn't be a neat 9120-day cycle. The moons would be in a different phase each time they line up. But if the year-length shares any common factors with 9120 (even if the only common factor is 1), then there'd be a longer cycle that does repeat.

*All of this is conditional on the moons actually having stable orbits, which I think is unlikely. Not that that especially matters- if your story requires this improbable arrangement of moons, go for it!

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Their least common denominator is:

$$12=2^2\cdot 3$$ $$20=2^2\cdot 5$$ $$24=2^3\cdot 3$$ $$32=2^5$$ $$38=2\cdot 19$$

So the least common denominator is $$2^5\cdot 3 \cdot 5 \cdot 19 = \underline{\underline{9120}}$$ days.

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Possibly about 9120 days. This is the least common multiple of all the orbital periods, and thus the shortest possible time it would take for all the moons to be back in the same position relative to each other. I’m not sure if there’s a formal astronomical term for this, but I tend to think of it as the ‘system period’ (that is, the period of time it takes the system of bodies to be back in the same relative configuration), and will be referring to it regularly as such for the rest of the answer. Note that irrespective of what phase it is, all six moons will be in the same phase as each other at the start of every system period if they are ever all full at the start of every system period.

Unfortunately, it’s far more complicated than just figuring out the system period.

First, and possibly simplest, is the fact that all the lunar orbital periods share a common divisor of 2. This means that there can be a second time during the system period, exactly half way through, where the moons all align relative to each other in such a way that they can all be full, just 180 degrees around their orbits. In general terms, this will happen a number of times equal to the largest common divisor of all the orbital periods. In this case that’s 2 (they share no other common divisors across all six), but if it were 3 it would happen 3 times, if it were 6 it would happen six times, etc. I will be referring to this possibility later in the answer as ‘double alignment’. Just like with the system period, if the moons are ever all full at either the start of a system period or during double alignment, they will always be in the same phase as each other during each double alignment.

The second issue is that the orbits must all be stable as measured over a time period that is large relative to the system period. For this to be consistent and predictable at all, you need the those orbital periods to be consistent over a long period of time. The system period is a bit less than 25 Earth years, so in this case ‘long period of time’ is likely multiple Earth centuries. This is maybe doable, but the numbers would need to be extremely precise, or some of the orbits would have to be very far apart (or both). Without this criteria, it becomes irrelevant to talk about how frequently they are all full, because it becomes impossible to predict when it will happen without detailed knowledge of how the orbits are changing over time.

This stability requirement brings in an additional complication: You have two moons with the same orbital period. For this to be truly stable, you’re looking at orbital configurations that are not conducive to all the moons being full at the same time, or at least, not all being visible in the sky at the same time while full. Making this work and still allowing for them to all be full at the same time effectively requires that both of the 20 day moons have orbital parameters such that they do not perturb each other’s orbits, which as far as I can tell requires that one of the 20 day moons be in a long elliptical orbit with everything lining up just right. This is at minimum astronomically¹ improbable. This also may make double alignment impossible depending on the parameters of that elliptical orbit.

The third issue has to do with the periods of the individual moons being in phase such that it is actually possible for them to all be full at the same time. As a trivial example, both of the 20 day moons must be full at the same time on each of their orbits, otherwise they will never be full at the same time. This puts a further constraint on the orbits of these moons that moves this from ‘astronomically¹ improbable’ to ‘statistically impossible’. But you can easily hand-wave those parts away (you could even reference this in your story, maybe figuring out how things ended up this way is a major open problem in astronomy).

The fourth issue here is that we don’t know the orbital period of the planet they are orbiting. This is actually required for figuring out the time between full moons, because a full moon only happens if the moon, it’s planet, and the star the planet is orbiting are all lined up relative to each other (referred to as a ‘syzygy’). There are some things we can predict though:

  • If double alignment is not possible:
    • If the planetary orbital period is shorter than the system period and is an integral divisor of the system period, all the moons will be full together once every system period.
    • If the planetary orbital period is longer than the system period and is an integral multiple of the system period, all the moons will be full together once every planetary orbital period.
    • In other cases all moons will be full together after an amount of time equal to the lowest integral multiple of the system period that allows the start of a system period to align with the start of a planetary orbital period. For example, if the planetary orbital period is 1.5 times the system period, all moons will be full together once every 3 system periods, and if the planetary orbital period is 2.5 times the system period, all moons will be full together once every 5 system periods.
  • If double alignment is possible:
    • If the planetary orbital period is shorter than the system period and is an even integral divisor of the system period, then all moons will be full together once every half system period.
    • If the planetary orbital period is shorter than the system period and is an odd integral divisor of the system period, then all moons will be full together only once every system period.
    • If the planetary orbital period is longer than the system period and is an even integral multiple of the system period, then all moons will be full together once every planetary orbital period.
    • If the planetary orbital period is longer than the system period and is an odd integral multiple of the system period, then all moons will be full together once every half planetary orbital period (because double alignment will coincide with the point exactly half way around the planetary orbit).
    • In other cases it becomes a complicated mess to figure things out that requires knowledge of the exact planetary orbital period.

1: Astronomically in this case being in the sense of ‘absurdly large’, not ‘relating to astronomy’.

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