Part One: An introduction.
Some readers might want to skip the long Parts Two and Three and go straight to part Four.
For two astronomical objects which orbit a third astronomical object to have a long synodic period between being lined up they need to haver very similar orbits around the third body.
If you at the synodic periods of other planets in the solar system with respect to Earth, you will see that the greater the difference in orbits, the closer the synodic period of the two planets will be to the orbital period of the inner planet of the two. Because Mars and Venus orbit closer to Earth than other planets, their synodic periods relative to Earth are the longest.
Part Two: Limits to the length of the orbital period of a habitable planet.
There is a limit to the orbital period of a moon of a habitable planet. There is a mass range for the possible mass of a habitable planet and a mass range for the possible mass of a star capable of having a habitable planet. A habitable planet has to orbit its star within the distance range or circumstellar habitable zone where water will be a liquid on its surface. The circumstellar habitable zone of a star can be calculated from the square root of its luminosity compared to that of the Sun and the limits of the Sun's circumstellar habitable zone.
But the limits of the Sun's circumstellar habitable zone aren't well known. The linked table shows that in the last 60 years there have been vastly different estimations of the Sun's circumstellar habitable zone:
https://en.wikipedia.org/wiki/Circumstellar_habitable_zone#Extrasolar_extrapolation
Only one of the estimates is for planets habitable of humans (and beings with the same environmental requirements) in particular. The other estimates are for planets habitable for liquid water using life in general, and the example of Earth shows that some life forms can flourish where unprotected humans would swiftly die. And some of the wider estimates of the habitable zone require specific atmospheric compositions to maintain liquid water temperatures, atmospheric compositions which might be toxic to humans.
So a science fiction writer who a) wants his story to be a plausible as possible
and b) wants the planet to habitable for Earth humans and/or aliens with similar requirements
and c) is certain they only need one such habitable planet in the star system in their story, should probably play it safe and put the habitable planet at the distance from its star where it will receive exactly as much head light from from its star as Earth gets from the Sun.
I call that distance from its star the Earth Equivalent Distance, or EED.
The answer by User17707 to this question: https://astronomy.stackexchange.com/questions/40746/how-would-the-characteristics-of-a-habitable-planet-change-with-stars-of-differe/40751#40751
has a table listing characteristics of stars of different types, including their EEDs, and the orbital periods of planets at their EEDs.
A spectral type F2V star is about the most massive star likely to have a naturally habitable - as opposed to artificially terraformed - planet. According to the table the EED of a F2V star would be at 2.236 Astronomical Units (AU) and the orbital period of a planet at the EED would be 1,018.1 Earth days or about 2.787 Earth years. And of course a habitable planet could be farther from its star and receive somewhat less radiation than Earth gets. So it might be possible to have a habitable planet with a year three or four Earth years long.
Part Three: Limits to the length of the orbital period of a moon of a planet.
In this article:
https://arxiv.org/ftp/arxiv/papers/1209/1209.5323.pdf
The second paragraph on page three says:
The longest possible length of a satellite’s day compatible with Hill stability has been shown to be about P∗p/9, P∗p being the planet’s orbital period about the star (Kipping 2009a).
Citing this as the source:
https://ui.adsabs.harvard.edu/abs/2009MNRAS.392..181K/abstract
This means that the orbital period of a moon in long term stable orbit of a planet has to be less than about one ninth or 0.1111 as long as the orbital period of the planet around its star.
If three to four Earth years are about the maximum orbital period for a naturally human habitable planet, the maximum orbital period of a moon of a human habitable planet would be about 0.333 to 0.444 Earth years, or about 121.7499 to 162.333 Earth days.
And here is another calculation of the maximum possible orbital period of a moon of a habitable planet. A long term stable orbit of a moon has to be within the planet's Hill sphere and closer to the planet than the Hill radius.
https://en.wikipedia.org/wiki/Hill_sphere
The size of the Hill sphere of a planet depends on the mass of the planet, the mass of the star, the semi-major axis of the planet's orbit, and the eccentricity of the planet's orbit.
In the Earth-Sun example, the Earth (5.97×1024 kg) orbits the Sun (1.99×1030 kg) at a distance of 149.6 million km, or one astronomical unit (AU). The Hill sphere for Earth thus extends out to about 1.5 million km (0.01 AU).
And there are limited mass ranges for habitable planets and for the stars which they orbit, and limited ranges for the semi-major axis of the planet's orbits.
This article: https://arxiv.org/ftp/arxiv/papers/1209/1209.5323.pdf discusses the mass range of worlds habitable for liquid water using life in general on pages 3 to 4 and decides that the mass range should be about 0.25 to 2.0 times the mass of Earth.
Stephen H. Dole, in Habitable Planets for Man (1964)
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf
Estimated on pages 53 to 58 That human habitable planets should have masses in the range of about 0.4 to about 2.35 Earth mass. On pages 67 to 72 Dole estimated that a human habitable planet should orbit a star with a mass of about to 0.88 to 1.4 times the mass of the Sun.
And on pages 66 to 67 Dole decided that the eccentricity of human habitable planet's orbits should be less than 0.2.
Here is a link to a Hill sphere calculator.
https://www.vcalc.com/wiki/KurtHeckman/Hill+Sphere+Radius
So I set the orbital eccentricity at Earth's figure of 0.0167 and use a mass range of 0.72 to 1.4 Suns or the star and 0.25 to 2.0 Earths for the planets.
According to the table at:
https://astronomy.stackexchange.com/questions/40746/how-would-the-characteristics-of-a-habitable-planet-change-with-stars-of-differe/40751#40751
A F2V star with a mass of 1.44 Earths would have an EED of about 2.236 AU and planet orbiting at that distance would have an orbital period of 1,018.1 Earth days.
According to the Hill sphere calculator a planet with 0.25 earth mass in that position would have a Hill sphere with a radius of about 1,835,556 kilometers, while a planet with a mass of 2.00 Earths would have a Hill radius of about 3,671,112 kilometers.
According to the table of stellar properties, a K2V star with a mass of 0.78 solar mass would an EED of 0.58 AU and a planet at that distance would have an orbital period of about 182.93 Earth days.
The Hill sphere calculator says that a planet with a mass of 0.25 Earth in that position would have a Hill radius of 574,090 kilometers, while a planet with a mass of 2.0 Earths would have a Hill radius of 1,168,180 kilometers.
So as a general rule human habitable planets should have Hill radii between about 574,090 and 3,671,112 kilometers.
The Hill sphere is only an approximation, and other forces (such as radiation pressure or the Yarkovsky effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius.
https://en.wikipedia.org/wiki/Hill_sphere#True_region_of_stability
So the actual region of stable satellite orbits for a human planet should be between about 191.363 to 287,045 kilometers and 1,223,704 to 1,835,556 kilometers.
So according to this orbital period calculator
https://www.omnicalculator.com/physics/orbital-period
the longest possible stable satellite orbital period for a moon orbiting at a distance of 191,363 to 287,045 kilometers a planet with a mass of 0.25 Earth orbiting a K2V star should be about 18 days and 21 hours to 34 days and 17 hours.
Using a planet with a mass of 2.0 Earth, the longest possible stable satellite orbital period for a moon orbiting at a distance of 191,363 to 287,045 kilometers would be 6 days and 19 hours to 12 days and 11 hours.
if a planet with mass of 2.0 Earth is orbiting a F2V star and has a stable orbital zone out to 1,223,704 to 1,835,556 kilometers, the longest possible orbital period for a moon would be 109 days and 19 hours to 202 days and 1 hours.
And if a planet with mass of 0.25 Earth is orbiting a F2V star and has a stable orbital zone out to 1,223,704 to 1,835,556 kilometers, the longest possible orbital period for a moon would be 305 days and 23 hours to 561 days and 23 hours.
However a planet orbiting a F2V star and having a stable orbital zone out to 1,223,704 to 1,835,556 kilometers would be a planet with a mass of 2 Earths, not one with a mass of 0.25 Earth.
So the maximum possible orbital period for a moon of a habitable planet should be about 109 days and 19 hours to 202 days and 1 hours.
And above I made a calculation that the maximum possible length of the orbital period of a moon of a human habitable planet would be about 0.333 to 0.444 Earth years, or about 121.7499 to 162.333 Earth days.
So it seems fairly safe to say that the maximum possible orbital period of a moon of a human habitable planet should be about 200 Earth days.
What is the minimum possible orbital period of a moon of a habitable planet?
That is determined by the Roach limits of the planets. And for human habitable planets, the Roche limit would usually be so close to the planet that an object orbiting there would have an orbital period less than one Earth day.
Part Four: Almost Co Orbital Moons.
So the orbital period of a moon of a habitable planet should be between about 1 Earth day and about 200 Earth days. A synodic period of two worlds about 250 Earth years or 91,312.5 Earth days long would be about 456.5625 to 91,312.5 times as long as the possible range orbital periods of moons of a human habitable planet.
Can the synodic period of two worlds orbiting a third world be that much longer than their orbital periods?
As I wrote above, the closer the orbits of two objects are, the longer their synodic period between being lined up will be.
And so the example of Janus and Epimetheus, moons of Saturn, is important. They have almost identical orbits around Saturn and are almost exactly co orbital.
Wikipedia says that the orbit of Janus has a semi-major axis of 151,460 (plus or minus 10) kilometers and an orbital period of 0.694660342 Earth days, and the orbit of Epimetheus has a semi-major axis of 151,410 (plus or minus 10) kilometers and an orbital period of 0.694333517 Earth days but that is not always correct. The two moons periodically switch orbits.
Epimetheus's orbit is co-orbital with that of Janus. Janus's mean orbital radius from Saturn is, as of 2006 (as shown by green color in the adjacent picture), only 50 km less than that of Epimetheus, a distance smaller than either moon's mean radius. In accordance with Kepler's laws of planetary motion, the closer orbit is completed more quickly. Because of the small difference it is completed in only about 30 seconds less. Each day, the inner moon is an additional 0.25° farther around Saturn than the outer moon. As the inner moon catches up to the outer moon, their mutual gravitational attraction increases the inner moon's momentum and decreases that of the outer moon. This added momentum means that the inner moon's distance from Saturn and orbital period are increased, and the outer moon's are decreased. The timing and magnitude of the momentum exchange is such that the moons effectively swap orbits, never approaching closer than about 10,000 km. At each encounter Janus's orbital radius changes by ~20 km and Epimetheus's by ~80 km: Janus's orbit is less affected because it is four times more massive than Epimetheus. The exchange takes place close to every four years; the last close approaches occurred in January 2006,[14] 2010, 2014 and 2018. This is the only such orbital configuration of moons known in the Solar System[15] (although, 3753 Cruithne is an asteroid which is co-orbital with Earth).
https://en.wikipedia.org/wiki/Epimetheus_(moon)
If the inner moon advances 0.25 degree beyond the outer moon each Earth day, it should take 1,440 Earth days or 3.9425 Earth years for the inner mmon to pull ahead of and then catch up with the outer moon. Though the article says 4 Earth years, which is 1,461 Earth days.
So if the synodic period of Janus and Epimetheus is 1,440 or 1,461 Earth days, and if the orbital period of the inner moon is exactly 0.694 Earth days, the Synodic period of Janus and Epimetheus as seen from Saturn is about 2,074.9279 or 2,105.1873 times the orbital period of the inner moon.
Thus it is possible for the synodic period of two moons orbiting a planet to over two thousand times the orbital periods of those moons, if those have almost identical orbits. And that is consistent with the factor of about 456.5625 to 91,312.5 times the orbital period necessary to have a synodic period of about 250 Earth years.
Part Five: Synodic Period and Full Moons.
But finding a way for the synodic period of two moons to be as long as about 250 Earth years is not enough. The two moons will not be full every time they are lined up as seen from the planet.
For the moons to be lined up as seen from the planet, there has to be a straight line between the centers of the planet and the centers of the two moons.
For one moon to be full, there has to be a straight line between the centers of the star, the planet, and that moon. And that moon has to be on the opposite side of the planet from the star.
For the two moons to be full at the same time, there has to be straight line between the centers of the star, the planet, and the two moons, and the two moons have to be on the opposite side of the planet from the star.
Since the planet is orbiting the star, the direction from the planet to the star, and thus the direction opposite to the star, changes all the time. And as the two moons orbit around the planet, their phases change all the time.
And it would be a big coincidence if the two moons were always lined up with each other at the same time they were opposite to the star as seen from the planet. So I expect that most times the moons were lined up they wouldn't be full. And using different orbits for the planet around the star and the two moons around the planet, one can come up with many different combinations, some of which can have both the moons being full ever few times they line up, and others can have the moons being full only ever million times they line up, and everything in between.
Part Six: Another Moon or Another Planet?
And it could be possible for three moons to sometimes line up.
Suppose that the inner moon orbits in 2 Earth days, or 180 degrees per day, and the middle moon orbits in 6 Earth days, or 60 degrees per day, and the outer moon orbits in 48 Earth days, or 7.5 degrees per day.
In two days the inner moon will travel back to the starting point, but the middle moon will have traveled 60 degrees from the starting point, and it will take the inner moon 1/3 day to reach there, and the middle moon will have traveled 20 more degrees, and it will take the inner moon 1/9 to reach there, and so on. in 2.4444444 days the inner moon will travel 439.9999 degrees (79.9999 beyond 360) and the middle moon will travel 146.66666 degrees.
In 3 days the inner moon will travel 540 degrees, 180 beyond 360, and the middle moon will travel 180 degrees, And so they will be line up again, though in the opposite direction from the earlier time. In six days they will be lined up again, and in the same direction as the first time.
If the outer moon orbits 7.5 degrees per day over 48 days, the inner moon will gain 172.5 degrees on it each day. after 23 days the inner moon will have gained 23 times 180 degrees on the outer moon, and will catch up with the outer moon in the opposite direction than before, and after 46 days the inner moon will gain 46 times 180 degrees on the outer moon and will catch up with it again in the same direction as before.
If the outer moon orbits 7.5 degrees per day over 48 days, the middle moon will gain 52.5 degrees on it each day. After 8 days days the inner moon will have gained 8 times 52.5 or 420 degrees (60 more than 360) on the outer moon, and will catch up with the outer moon 60 degrees beyond the original position. In 48 days the middle moon will catch up with the outer moon for the sixth time and in the the original direction as seen from the planet.
So the three synodic periods are 3, 23, & 8 days, and 6, 46, and 48 days for lining up up in the same direction. So after 6 X 46 X 48, or 13,248 days, or 36.2710 Earth years, the three moons soon be lined up again on a line between the center of the planet and the distant stars.
The lining up of the three moons in full moon is supposed to happen every 250 planetary years each 392 planetary days long, or 98,000 planetary days. 98,000 planetary days would be 7.3973 times the triple synodic period of 13,248 days calculated above. Thus the direction of the triple synodic point and the direction opposite to star, both moving over time, should change by a number of degrees over the course of each triple synodic period.
You could make the triple full moon happen every 7 or every 8 triple synodic periods, thus making the triple synodic period equal 12,250 or 14,000 planetary days, recalculating the various orbital periods. Or you could keep the triple synodic period at 13,248 planetary days and make the triple full moon happen every 7 or 8 triple synodic periods, or every 92,736 to 105,984 planetary days.
Warning! If you use scientific programs to make calculations, be certain to decide on a ratio between the the lengths of Earth days and planetary days. The scientific programs will not calculate correctly if you use fictional planetary days instead of Earth days, so you will have to convert back and forth between the different types of days.
And instead of using a tripe full moon, you might be able to use a double full moon lined up with a outer planet in the system.
Part Seven: The Sizes of Your Moons.
And all three moons have to be large enough and close enough to the planet that they appear as discs as seen from the surface of the planet, so people can tell their phases.
Theoretically people could tell the difference between phases if the moons appeared as dots of light in the sky, because the apparent brightness of the moons would change with their phases. But people wouldn't know that the moons had crescent phases when they were dim and were full when they were bright. They would simply assume that the moons glowed, and their brightness changed periodically.
So the moons need to appear as discs from the surface of the planet so the phases can be seen.
When the planet Venus is closest to Earth, and looks the largest as seen from Earth, it is between the Sun and Earth, and so we see the dark side, except for a thin illuminated crescent, when we observe Venus through a telescope.
The extreme crescent phase of Venus can be seen without a telescope by those with exceptionally acute eyesight, at the limit of human perception. The angular resolution of the naked eye is about 1 minute of arc. The apparent disk of Venus' extreme crescent measures between 60.2 and 66 seconds of arc,8 depending on the distance from Earth.
https://en.wikipedia.org/wiki/Phases_of_Venus#:~:text=The%20extreme%20crescent%20phase%20of%20Venus%20can%20be%20seen%20without,about%201%20minute%20of%20arc.
I myself have sometimes thought that I might possibly be seeing a crescent Venus.
You would want the moons to have an angular diameter of several arc minutes, in order for people to be able to see the differences between full moons, gibbous moons, half moons, and crescent moons, instead of being barely able to tell the difference between full and crescent moons, and for people with average or poor eyesight and not just people with exceptionally good vision to see the phases of the moons.
So I guess that the minimum angular diameter of the moon which looks the smallest as seen from the planet should be about 5 arc minutes. A full circle contains 360 degrees and thus 21,600 arc minutes. So the radius of circle should equal in length 3,437.7 arc minutes around the circle, and 5 arc minutes should be equal in length to about 1/687.5499 of the radius, or 0.00145 of the radius.
So the physical diameters of the moons should be at least 0.00145 as large as the distances to them, if they are to have angular diameters of 5 arc minutes or more.
Earth's moon has an angular diameter about 30 arc minutes or 0.5 degrees of arc, which varies as its distance in its orbit varies. The Moon's orbit has a semi-major axis of 387,399 kilometers while the Moon has a mean diameter of 3,474.8 kilometers. So the mean diameter of the Moon is about 0.008969 of the semi-major axis of its orbit, to give it an angular diameter as seen from Earth of about 0.5 degree or about 30 arc minutes.
If the diameter and/or the distance of the Moon was changed to give it an angular diameter as seen from Earth of about one degree or 60 arc minutes, the diameter of the Moon would be about 0.01738 of the semi-major axis of its orbit.
If you desire, you can make one or more of your moons have an angular diameter as seen from the planet of one or more degrees of arc. But you will probably run into some problems as you increase the angular diameters of your moons, so don't go too far.
The larger a moon is the more massive it will be both absolutely and relative to the planet and two the other moons. The more massive a moon is relative to the planet and to the other moons the stronger its gravitational force will be on them.
The more massive a moon is relative to the planet, the more its tidal forces will change the rotation rate of the planet and also change the moon's orbit. Depending on the initial conditions of a moon's orbit, tidal interactions can make it spiral inward toward the planet and eventual doom, like Phobos and Triton in our solar system, or outward away from the planet like most major moons. The faster a moon moves inwards or outwards, the more likely it will be to interfere with the orbits of the any other moons.
The more massive a moon is relative to any other moons, the stronger its gravity will change their orbits. Some combinations of moon masses and orbits can be unstable. So the more massive the moons are, the few orbital combinations will be available.
Epimetheus and Janus can orbit so closely together, because their masses are relatively similar, and because the mass of Saturn is so many, many times their masses. A habitable planet will not be so much more massive than any moons large enough to appear as discs as seen from the planet, thus narrowing orbital possibilities.
I note if the moons appear round as seen from the planet they will have to be massive enough to gravitationally rounded. A world has to have enough mass and gravity to compress matter at its core to become gravitationally rounded, a planetary mass object or planemo. There are many quite small planemos in the other solar system, but they are largely made of ice which is soft and compresses easier than rock. Moons orbiting a planet warm enough for liquid water using life can't have a lot of ice and will be almost entirely rock, and so will have to be larger to become planemos.
However, many small solar system objects too small to be planemos are roughly potato shaped. The long axis of such potato shaped moons would probably be pointing at the the planet they orbit, so a more round looking cross section would be seen from the planet, possibly round enough to not mess up the phases of those moons much.
