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As I said in the title, I have a world with two moons. One has an orbital period of 17 days Earth days (Moon 1) while the other has an orbital period of 57 Earth days (Moon 2) and the planet they are orbiting orbits it’s star in about 1.4 Earth years. They both have low eccentricity to their orbit (about as much as Earth’s moon). Moon 1 has an Inclination of about 7° while Moon 2 has an inclination of 15°. Moon 1’s axis tilt is 78° while Moon 2’s is 12°. Moon 1’s radius is 523 Km while Moon 2’s is 1,933 Km. Moon 1’s density is 3.2 g/cm^3 and Moon 2’s is 5.3 g/cm^3. I tried finding some type of calculator online or at least some kind of equation I could use, but I have not been able to find either (I probably didn’t think of the right keywords to use to be honest). I was wondering if someone else knew how to calculate this and could provide an answer and/or the way to calculate this. Thank you in advance.

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    $\begingroup$ This is a poorly defined question. The Orbital periods of both moons are a start, but without any information about their orbital inclination, eccentricity, central body tilt, relative sizes and so forth, there is no single answer here. It could happen every full orbit of Moon 2, or be a once in a Millennium event. $\endgroup$
    – ErikHall
    Commented Apr 9 at 12:56
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    $\begingroup$ @ErikHall I added more information as you suggested, I am sorry it was so barebones at first, it probably isn’t a good idea to write questions when I’m planning on sleeping soon. $\endgroup$ Commented Apr 9 at 13:17
  • $\begingroup$ Even ignoring the questions of eccentricity and inclination, one of the parameters that is needed is the orbital period of the planet about the star. $\endgroup$ Commented Apr 9 at 13:18
  • $\begingroup$ Since 17 and 57 are mutually prime, the interval between paired full moons will be on the order of 969 days. $\endgroup$ Commented Apr 9 at 13:21
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    $\begingroup$ If you start from a "paired full moons" day, the next one will be no sooner than the next time both have completed an integer number of orbits. Since 17 and 57 have no common factors (that's what "mutually prime" means), the first multiple of 17 that is also a multiple of 57 happens to be 17×57, which is 969. This assumes that the planetary orbit is either 969 days, or that 969 days is an exact multiple of the planetary orbit. $\endgroup$ Commented Apr 9 at 14:15

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If you have multiple moons and want to keep track of them, I'd suggest using a fantasy calendar. With one, you can get the exact places of the moon, as well as other helpful features.

There is a simple one on the donjon website: https://donjon.bin.sh/fantasy/calendar/

But there's also a more complex calendar with events and weather here: https://app.fantasy-calendar.com/

I hope that helps!

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  1. The duration of a sidereal year of the planet is given as "about" 1.4 Earth years, or about 1.4 × 365.256 ≈ 511.358 days.

  2. Then the duration of a full cycle of phases for Satellite One is

    17 × (511.358 / (511.358 − 17)) ≈ 17.585 days,

    and the duration of a full cycle of phases for satellite Two is

    57 × (511.358 / (511.358 − 57)) ≈ 64.151 days.

    (Because the duration of a full cycle of phases, or a synodic period, is of course longer than the duration of a revolution of the satellite around the planet; that's because during time that the satellite takes to make a full revolution around the planet, the planet itself is moving on its orbit around the star, so that the satellite still has some way to go before getting into the same position with respect to the star.)

  3. Both numbers are only appoximations, and the real average durations of a full cycle of phases have of course many more decimals. The chances of both satellites being absolutely fully full at the exact same time are just about zero.

    But in reality we don't consider our real Moon to be full for just one fleeting moment. We allow some leeway.

    And even with maximum fussiness the Moon is full for a finite time, not only a fleeting instant, because the Sun is not a point source of light, but it has a non-zero angular size. The real life Moon is absolutely completely full for about one full minute every lunar month.

    Now, the question is what counts as a full moon. The real full moon is only an instant, but in practice we accept some extra time around the fleeting instant; for our real Moon we accept about half a day before and after the "true" full Moon if we are fussy, and maybe up to six days around the "true" full Moon if we are practical, when the Moon appears full enough.

    (One of the most important practical uses of knowing the phases of the Moon is do form an idea how well-lit (or badly-lit) will a given night be.)

  4. The discussion therefore splits into fussy and practical.

  5. If we are fussy, the chances of Satellite Two achieving its fullness on the same night as Satellite One is one in 64 every time Satellite One is full, so that a night when both satellites are fully full will happen on the average every 17.585 × 64 ≈ 1,125.414 days.

  6. But if we are practical, we need to make assumptions about what counts as full enough. Let's say that for Satellite One (which has a short phase cycle) we accept about 3 days around the true instant full phase as being full enough, and for Satellite Two (which has a long phase cycle) we accept 10 days around full fullness.

    In this case, the chances that Satellite Two is full enough during the nights that Satellite One is also full enough are 3 × 10/64 ≈ 49%, which means that on the average Satellite One and Satellite Two will both be full enough at least one night every other phase cycle of Satellite One, or in other words, at least one night every 34 nights.

  7. And of course we can play with the numbers. If we allow only 2 days around real complete fullness for Satellite One, and only eight days around real complete fullness for Satellite Two, then we find that Satellite Two is full enough during the nights that Satellite One is also full enough are 2 × 8/64 = 25%, which means that on the average Satellite One and Satellite Two will both be full enough at least one night every fourth phase cycle of Satellite One, or in other words, at least one night every 68 nights.

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