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This is a followup to my previous question concerning the potential orbits. I've decided on a barycenter with each body in its own orbit--the primary planet having the center orbit, the first moon having the second outermost orbit, and the second having the farthest orbit; none of them form an eccentric orbit. With that established:

Each moon has its own respective properties: one takes four 40-day months (one month per season) to complete a full orbit and is tidally locked, while the other has twelve 40-day months (three months per season) and is not tidally locked. I'm unsure about how exactly the "dark sides" work, but is this setup plausible? If so, can both moons have seasons based on their phases?

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It's a little unclear from the question what everything is taken relative to. I'm assuming the following:

  1. Your planet and its two moons are all orbiting a star.
  2. The moon that's tidally locked is tidally locked relative to the planet, not relative to the sun.
  3. The orbital times quoted are relative to the planet (and the planet's orbital time is not provided).
  4. When using the term "day" to measure the duration of an orbit, you mean 1 Earth-standard day (i.e. 24 hours).
  5. Having seasons implies the presence of an atmosphere on the moons, so each moon must be massive enough to retain one.

Based on that, I would say:

The 'Dark' Side

This is essentially an optical illusion experienced by observers standing on your main planet. Unless the moon is tidally locked to the sun then it doesn't really have a "dark" side in terms of hemispheres that are always in full sunlight or full darkness.

A moon that's tidally locked to its planet will have a normal day/night cycle (probably with some spectacular moments when the planet or the other moon eclipse the sun), the duration of which will depend upon its rotation about its axis. This is where the tidal locking becomes relevant, because it means you only get one day per complete orbit. Or in other words, one day ("day" as in "complete day/night cycle") on the tidally locked moon lasts 160 days ("days" as in "Earth-standard days").

Seasons

Seasons on Earth are caused by the planet being tilted slightly about its axis of rotation. So depending upon where the planet is in its orbit around the sun one hemisphere tends to point more directly towards the sun than the other, thus experiencing longer days and warmer weather.

So your moons can experience seasons the same way (assuming you tilt them appropriately). However the seasons will be tied to the planet's orbit about its sun, not their orbit about the planet (unless their orbits are very wide).

The tidally locked one is problematic, however, given how long each day lasts. On that one the overriding factor won't be which hemisphere is currently pointing towards the sun, but whether it's currently daytime or nighttime. I'd expect brutally hot conditions during the day and frigid at night, unless there's some mechanism that efficiently distributes heat more-or-less evenly across the moon. For instance, maybe persistent storm systems, or some artificial contrivance designed to render the moon habitable despite its long day/night cycle.

But anyways, can the seasons relate to the phase of each moon? I'm going to say tentatively yes, but it requires that the moons' orbital periods must tie into the planet's orbital period about its star. And it may not be workable with two moons that have significantly different orbital periods. A simpler option might be if one moon has seasons that correspond to its phase, while the other does not. Both is tricky if they have vastly different attributes.

Plausibility

I think your orbital durations are quite lengthy for moons, particularly given the barycentric orbit. I don't know at what point the math renders the setup impossible, but I'd think the planet would have to be quite massive in order to retain a moon with an orbital period of 480 days, and looking at our own solar system I'm not finding any example of massive moons that have very long orbital periods (even around very massive planets like Jupiter). Pluto and Charon have a barycentric orbit, and Charon's orbital period is less than 7 days.

And in order to retain an atmosphere (seasons do work better with an atmosphere, I think?) your moons have to be quite massive themselves. Like larger than Mars.

But in terms of what the average person will accept after suspending disbelief, I think you're at least within the realm of plausibility.

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