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I'm currently running a campaign set in a world where the calendar is based on two things.

  1. The moon takes the entire year to cycle through its phases. So instead of full moon to full moon being a month, it takes 400 days.
  2. The year ends on a solar eclipse that happens on the same day every year.

They are weird parameters, I know but was wondering if an planet/moon orbit like this was possible in our universe. I do enjoy mixing science and fantasy and thought it would be neat if it could be possible under natural laws. I don't know where to start to answer that question and suspect that it might not be physically possible. But thought I would ask here anyway. Thanks everyone!

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Regarding orbits, your condition is possible. For this, you want your moon to revolve around the planet with a period of exactly half a year. A depiction follows:

You want your planet have a solar eclipse in a certain time of year. Let's position your solar system so that your planet is on the left of its sun, so the planet, its moon and the sun would be positioned like this:

P---M----...---S

Then, you want the moon to phase in/out for the entire year. Let's check where the moon should be to change phase to 180 degrees in half a year. It will be a full moon, so it should be about behind the planet when viewed from the sun, and the planet itself would complete half a round of its own orbit, now it will be on the right from the sun. Like this:

S---...---P---M

Notice, the moon's position relative to the planet has not changed. Should we use a quarter period (sorry no hand drawings here), the configuration from the same view would be either like this:

^ (to the sun)
|
|
P---M

Or like this:

    ^ (to the sun)
    |
    |
M---P

The first variant means that the moon either does not revolve around the planet, or does 4*X revolutions around it over a year. The non-revolving moon is not physically possible, since it will plain fall to the planet making a big BOOM, the 4*X variant does not satisfy the requirement of changing phases over a year, so that variant is out. The second variant means that the moon did 1/2+X (X is integer) rounds around the planet during the planet's 1/4 years. Solving with both conditions to be satisfies yields X=0, thus the moon's orbital period is (1/4)/(1/2) years, or half a year.

The thing that would be impossible is a solar eclipse, or at least a full solar eclipse, if your moon is just a moon. For example, here on Earth we have a full solar eclipse because our Moon is pretty close to the planet, having its angular diameter about the same as the Sun when viewed from the planet's surface. Should we position the Moon far enough so that its orbital period would be 1/2 of a year (183 days instead of 27.32), it will have its orbital radius of 384.4*(183/27.32)^(2/3) = 1365.9 thousand km, and its angular radius when viewed from Earth would decrease (183/27.32)^(2/3) = 3.553 times, making the Moon's disc cover only 1/(3.553^2) = 0.079 or 7.9% of the Sun's disc. Still plausible to be detected from the planet, though. And for our Earth, this distance is still within its Hill sphere radius which is about 10% larger, so the moon at that distance will still be Earth's satellite, but its orbit would be noticeably affected by the sun, thus a minor correction downwards could be made to the estimated radius in order to retain the moon's period of exactly half a year.

If you would want a moon to be of enough size to cover the entire sun's disc, it will have its radius to be 3.553 times larger, and its mass about 3.553^3 = 44.87 times bigger, this is no longer a moon but another planet, a tad larger than half of Earth. So your planet is no longer a single entity but two planets revolving around their common barycenter at a period of half a year, eclipsing their common sun from each other once a year. Probably that other planet also has an atmosphere, some life on its surface, and there's a possibility for interplanetary travel for either side... Ohhh the possibilities!

So to summarize. Yes it's possible, for that you want to have your planet to be a double, revolving at period of half a year around common barycenter. The masses of counterparts should likely be divided as half-and-half, or a rough estimate of that. There would be a likely problem of tidally locking the planets to each other, or at least some serious tides because the "moon" would be pretty heavy, yet these can be estimated in another question.

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    $\begingroup$ Tidal force goes up with the mass of the tide-inducing body, but it goes down with the cube of the distance. The way you computed it, the sister planet is $3.55^3$ heavier than the Moon, but the distance is also $3.55$ times larger; overall it is a wash -- the tidal force will be about the same. $\endgroup$
    – AlexP
    Sep 29, 2022 at 13:05
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    $\begingroup$ @AlexP Regarding numeric example: Assuming planets of equal mass (=1.0 M_Earth), using formula from here (English counterpart does not have the formula for period vs distance from masses), and taking period of 183 days, I've got 1361000 km distance for such a binary planet, which is within Hill sphere radius for the barycenter and the Sun, thus such an orbit is deemed stable in a system with no other planets, at least. So this can be constructed, barring planets' genesis. $\endgroup$
    – Vesper
    Sep 29, 2022 at 13:18
  • $\begingroup$ So what you're suggesting would be that there are two planets that are orbiting each other while both orbit the sun? I quite like that idea, if that's the case. Is there a planetary system in our universe that is similar to that? $\endgroup$ Sep 30, 2022 at 2:37
  • $\begingroup$ It's true the moon can't be stationary, but it could be retrograde and have a period of 1 year. But getting that to fit inside the hill sphere would be a challenge. $\endgroup$
    – Brianorca
    Sep 30, 2022 at 15:29
  • $\begingroup$ @PeanutNutter not sure if there is one, it's likely early phases of planetary genesis prohibit formation of two bodies of a very similar mass on the same orbit and on roughly the same spot. Also we don't have observation capability enough to find out if a planet around a distant star is a double, we only barely can detect that there is a planet, so natural forming of this star system is questioned. Yet we have our Earth early gotten hit by a something, leaving the Moon in orbit, probably splitting a planet in two by outside force it also possible. $\endgroup$
    – Vesper
    Oct 5, 2022 at 12:04
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Not if you're using real-world physics. A moon's orbit is only stable if it is within the Hill sphere of its parent body; for long-term stability, it needs to be less than about a third of the Hill radius. This works out to a maximum orbital period of about a ninth the parent body's orbital period, or about nine lunar months per year.

The annual solar eclipse isn't a problem: just incline the moon's orbit to the planet's orbit, then give it an orbital period that's an odd-numbered resonance with the parent's orbital period. You can only get a solar eclipse when the new moon happens during a crossing of the planet's orbit, and the odd-numbered resonance (eg. eleven months in a year) means it can only happen once per year. (The Moon being in a nearly 12:1 resonance is why we can get two solar eclipses a year here on Earth.)

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I am afraid it is impossible.

From Kepler's laws we know that the orbital periods depend only from the distances between the bodies.

The moon M taking time T to orbit the planet P with planet P taking T to orbit the star S means that the distance TP and the distance PS are the same.

This would mean that, among the possible configurations, there is one where the moon is overlapping with star and another in which the moon is twice as distant from the star as the planet is.

The first situation is not going to end well for the moon, and I doubt that in the second configuration at that distance the moon would still be in the Hill's sphere of the planet, and therefore it won't be a moon anymore.

The only mass configuration in which the second configuration could be plausible is the star being less than 4 times more massive than the planet, so that at half the distance the planet could have a larger gravitation influence than the star.

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    $\begingroup$ "From Kepler's laws we know that the orbital periods depend only from the distances between the bodies: For the same two bodies. But you don't have the same two bodies; in one case you have the star and the planet, in the other you have the planet and the satellite. But in principle you are right; the moon would have to be very far away to have a revolution period equal to the planet's year, and the planet cannot hold on to it at such a great distance. $\endgroup$
    – AlexP
    Sep 29, 2022 at 7:41
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    $\begingroup$ Re Hill sphere for the moon: I've described that the proper period for requested things to happen is not 1 year but 1/2 years, and at least for Earth a moon with the requested period can exist and be within Earth's Hill sphere. But a year long period would put the moon outside that sphere. $\endgroup$
    – Vesper
    Sep 29, 2022 at 13:25

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