Could there be a way for a solar system to be very precise, so that the lunar calendar and solar calendar align?
I know that this is a bit silly, but I want to make a calendar for my world and I'm really worried about having to do leap years and such to ensure continuing accuracy.
Inaccuracy isn't an option; many worlds could simply have a calendar that gradually gets out of line, but the people in this world are very particular about things being easy to understand yet accurate and informative. This would extend to their calendars, I'm sure.
So, I think that the effects of gravity on other planets can affect rotation - I believe the moon is slowly slowing Earth down, right? So, would it be feasible for the sun and planet to influence each other to make sure that years are a whole number of days and that years are a whole number of months?
(This sounds like a stretch, even to me, but might be something about the frequencies of the minor oscillations in orbit plus some other factor - I have limited knowledge of any physics at all, let alone orbital mechanics, so someone smarter than me might know something.)
Secondly, could it be possible for this to occur for a planet that is a binary planet? My system is quite lopsided, the centre of the two planets' orbit is only JUST outside of the main planet, but it is still enough to be a binary planet.
(If this is impossible, advice on creating a simple yet highly effective calendar for these very fussy people would be nice.)
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I think it's unlikely.
Colloquially a year is 365 days. But if you actually watch Earth complete an orbit in space, it will not be at the same rotation around its axis at the end as when it started. It actually takes about 365.25 rotations to complete an orbit. This leads to the leap year.
The side of the Earth closer to the Sun gets pulled in more than the far side. Because it's constantly rotating, there is a slight, continuous "reshaping" of the Earth by solar gravity. As the insides of Earth move around they experience friction, which bleeds off rotational energy and makes the day get longer and longer over time. So, many years in the future, the length of a day should increase from 1/365.25 years to 1/365 years exactly. An increase of only 0.07%! At that point, you would not need leap years. The problem is that it will not stop there, but keep increasing and then introduce a leap year again. Although, again the process is quite slow, so you can easily have it last entire anthropic time periods (thousands of years).
However, it's also a question of how precise your people are, really. If a year is 365.0001 days, will bother adding a leap year once every millenium? Or will they simply disregard it? As you get more precise with this, the period it lasts for becomes briefer.
In any case, it would be a bit much to expect all the planets in a system to coincide at whole-multiple years. Some of them would surely be fractional even while the others are not. So once your people develop astronomy, they are likely to become disappointed regardless... Unless they place some special importance on their one, precise, home planet.
Synchronizing lunar and solar calendars is very unlikely, I think. You could always claim it is coincidental, but that's a heck of a coincidence. There might be some tidal effects from the Sun as the two planets are orbiting each other, but these would be much weaker than the one synchronizing the day to the solar year, which is already weak unless the planet is close enough to be scorched like Mercury.
There is also no reason why the lunar calendar would sync up with the solar at the same time as the solar year becoming an integral multiple of the day. These two things are not strongly related, and them co-occurring is even more unlikely than one happening on its own.
Honestly, if I saw the kind of system you describe, I would suspect that it came about artificially. It's so unlikely to happen in nature that it suggests there was a deliberate manipulation. Changing the orbits of planets implies a very advanced and powerful civilization - so perhaps the "precursor aliens" trope can help you here.
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Precise alignment for a limited time
Alignment of one ratio
In your world, how long do you require the alignment to last?
As Olin Lathrops's answer explains, stable resonances are very unlikely to arise for large number ratios. A whole number of lunar months in a year, or a whole number of days in a year or in a lunar month, is not going to be stable long term.
However, the drift in the orbital periods of the planet and its moon, and in the period of rotation of the planet, will tend to be slow compared to the lifespan of an individual, which may be sufficient for your purposes depending on how precise you need the alignment to be.
On Earth, the change in the length of a day is so slow that even across millenia, the length of a day varies by only a few seconds. When the number of days in a year eventually reaches a whole number, it will stay a whole number to within a second for centuries. Even if the people in your world can measure time to sub-second accuracy, there may be a period when the length of a year is so close to a whole number of days that the difference is indistinguishable from noise in the measurement.
So a story can be set during a period of history when the number of days in a year just happened to be passing through a whole number. Alternatively the number of lunar months in a year could be temporarily a whole number. Neither of these are unrealistic. During history both these ratios pass through whole numbers several times, very slowly, staying near to a whole number for millenia. On our particular planet, our species is too young to have lived through such a time yet, but the species in your world could have developed writing just as the alignment was being reached.
Alignment of two ratios
What is much less likely to happen is for both ratios to pass near a whole number at the same time. So an Earth-like planet where both the number of days in a year and the number of lunar months in a year are whole numbers may be unrealistic.
If one of the ratios is changing much more slowly than the other, there may be time for the other to catch up and become a whole number while the first has barely moved. This would allow both to be a whole number for a long period of time without being unrealistically improbable.
This is also helped by the fact that the lunar month and the year are both getting shorter when measured in contemporary days (the day length at the time of measurement, rather than our current roughly 24 hour day). In seconds, both the lunar month and the year are getting longer, but days are getting longer at a faster rate, meaning there are fewer days in a lunar month and fewer days in a year than previously. The fact that the ratios are both moving in the same direction means that when they do both align with a whole number, they stay aligned for longer than if they were moving in opposite directions.
Deviation from the average ratio
Bear in mind that even if there is a long period of history when the average lunar month and the average year are both a whole number of days, there will still be variation from the average, particularly in the lunar month. On Earth the length of a lunar month varies by several hours from the average during a year. You could reduce this variation in your story by having the planet and its moon both have orbits that are much closer to circular than for our Earth and Moon.
A 1 day month
In the long term, the length of a day is likely to continue increasing until it matches the length of a lunar month. At this point the Earth will always face the same side to the Moon, in the same way that the Moon already always faces the same side to the Earth.
The drifting of the lunar month length (at least its length in days) will then stop, so that every time the year length passes through a period of being a whole number of days, both ratios will be whole numbers. So if you wait long enough, you can set your story on Earth. There are two problems with this:
- You may not want a 1 day month and a much shorter than 365 day year for your story
- Earth will be uninhabitable by that point, due to the expansion of the Sun.
Alignment when the planet is part of a binary
With a binary planet, the reasoning will be similar if the secondary planet replaces the moon. However, if there is a secondary planet and a separate moon, the dynamics may get more complex. There will now be a year, a lunar month, and what the inhabitants might call a "planetary month" (based on the other planet in the pair). I'd expect it to be even less likely for all 3 of these to have their ratios line up with whole numbers at the same time, and even more variation from the average.
Culture of precision
If the fussiness is part of the story, another approach might be to have ratios that are farther from whole numbers than on Earth, so a calendar based on whole numbers would drift so quickly that the inhabitants would never have tried. Being forced to include fractions in their dating system from early in their history might be part of the reason for them having a culture of precision. Their mathematics may have advanced far more rapidly than ours due to this necessity.
If you decide to go with such an approach, you could try having a much slower rotating planet with much longer days, so the lunar month and year are only a small number of days. Alternatively you could have the moon and sun much closer to the planet, so the lunar month and year are a small number of days without having to make the days longer. You could also use a combination of these two methods to avoid either being too extreme. Either way, the result is that a whole number of days will be such a poor approximation that it might never occur to them to describe a lunar month or a year that way.
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First, the fussiness of the people has absolutely no bearing on how their solar system ended up. It is what it is, whether they like it or not.
As for the physics, you are basically asking for rotations of the planet and its orbit around the star to be integer multiples. Yes, that can happen, but realistically only for small integers. Look up something called "tidal locking" or "tidal resonance". For example, our moon is tidally locked to the earth. In this case its spin and orbit have a ratio of 1:1.
Mercury has a spin to orbit ratio of 3:2. That's only possible because Mercury is so close to the very massive sun.
Tidal locking and resonance rely on the difference in gravity of the body being orbited between the near and far side of the orbiting body. The forces keeping a body in resonance decrease at higher ratios. Therefore, most resonances out there are 1:1, which is full tidal locking. Anything more requires orbiting close to something massive.
If you're looking for something like earth orbiting the sun with a resonance of 365:1, that's not going to happen. First, earth is too far from the sun so that even 1:1 resonance is unlikely. Venus is not tidally locked or in any resonance with the sun, and it's closer to the sun than earth. Second, 365:1 is so extreme that it's not going to happen.
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Something you may not have thought of: the Earth-Luna system has the barycenter inside the Earth's radius. Fairly deep inside, in fact -- about 25% of the way down from the surface.
Your system as described is less binary-like than our own. Either the moon is smaller than Luna, or farther away, or both.
Options:
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Change the fictional setting. Figure out the desired effect, and engineer it to produce that result.
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Months aren't important at all to your civilization. Seasons (assuming significant axial tilt or wild orbital eccentricity) might be the preferred unit, and there's no reason to arbitrarily claim four seasons. Perhaps there are only 2, or they really like some other number -- even seems more likely than odd to me, but perhaps they have some cultural specialty for 5.
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The moon is small and close and therefore fast: in another million years it will be a disaster, but right now it's a speck that is sometimes visible in the daylight, and has visible motion if you have any sort of reference tree or mountain.
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The moon is large and far away -- so far away that it's in the L1 or L2 point, and is either always shading a chunk of the planet (weather effects will be interesting!) or always in the planet's shadow and thus only detectable by occluding other astronomical objects (mythological effects will be interesting!).
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