What should the size of my moons be?

Question

(Not entirely sure if this is the right place to ask this but hopefully I'm right!)

I'm writing a novel and the planet this novel takes place on has two, twin moons of approximately the same size. The idea here is that the seasons are split up by the coming and passings of the moons - essentially, they are determined by where each moon is in the night sky relative to the sun. As a result, the cycle of each moon is approximately the length of a full year - which in itself is ~350 days long.

In addition, the days are pretty much the same length as those on Earth. I didn't want to change too much in case I'd be altering far more than I know. The seasons I'm referring to here just happen to coincide with the positions of the moon - hence why the populace believes that the positions of the two moons are what cause the change in seasons.

In order to have such long moon cycles, and two moons at that, how would I go about doing the calculations for their masses/sizes/mineral makeup? At least how far away should the moons hypothetically be in order to not make the planet a uninhabitable? And how might this affect gravity on the planet - IE is this even possible?

Answer 1

Short answer, what you are asking for is basically impossible.

Longer answer, here is the reason why. But there is something which might possibly give a somewhat similar situation.

The third paragraph in section 2 of the article "Exomoon Habitability Constrained by Illumination and Tidal Heating" says:

The longest possible length of a satellite's day compatible with Hill stability has been shown to be about Pp/9, Pp being the planet's orbital period about the star (Kipping, 2009a).

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3549631/1

That is in the case of a natural satellite tidally locked to its planet, so that the satellite's month and day are the same length. So that means that a natural satellite or moon with a stable orbit has to orbit the planet at least 9 times in each year of the planet.

So a natural satellite that orbits the planet with an orbital period or month equal to the length of the planet's orbital period around its star, or year, will not have a stable orbit and will probably escape from the planet's gravity and orbit the star independently (and possibly collide with and devastate the planet later) very soon by geological standards. By the time that multi celled lifeforms live on the planet, or intelligent life evolves, or the planet develops an oxygen-nitrogen atmosphere breathable for humans, the moon will be long gone.

This is the source the quoted article gives for the sttement:

Kipping D.M. Transit timing effects due to an exomoon. Mon Not R Astron Soc. 2009a;392:181–189.

https://arxiv.org/abs/0810.22432

So if the orbital stability equations are correct, neither of the moons can orbit the planet with a year-long orbit. No matter whether the planet's year is 1 Earth day or 500 Earth years long, the outer moon with the longer month must orbit the planet at least nine times per planetary year, and the inner moon with the shorter month must orbit more times than the outer moon.

But all is not lost. There is something called a synodic period. In the case of two satellites of a planet their synodic period could be the time between successive moments when the satellites are in the same position relative to each other. It could be the time between successive moments when the inner satellite is 90 degrees ahead of the outer satellite, or the time between successive moments when the inner satellite eclipses the outer satellite, for example.

The synodic period may take many orbits of the two satellites until they return to the same (relative) position.

There are four astronomical seasons on Earth, although various regions might have different numbers of climatic seasons for various reasons. But if your fictional planet as an axial tilt like Earth's most regions in the temperate zones should have four seasons, spring, summer, autumn and winter.

So if the two moons have synodic periods of the right length, there could be four synodic periods per year, and the natives of the planet might think that the synodic periods that roughly coincide with the change of seasons somehow cause the change of seasons.

Answer 2

What you are looking for is a synodic (observed relative to an object, like the sun) period for a moon that is one year. The way the motion works out, the moons have an orbital period of half a year.

With a period of 175 days and Earth, that gives your moons a semimajor axis of 1.32 million km, compared to .384 miilion km for our moon. In order for your moons to have the same apparent size as our moon (~30 arcminutes), they need to have a radius of about 6,000 km. This is almost the size of Earth.

This isn't an insurmountable problem. The simpler solution is to crank down the density (the albedo will need to be increased for the moons to be visible anyway, while we're modifying them.) The most reasonable estimate for the least dense planet is 0.7g/cm3, about 1/8 of Earth's density, found for Kepler 253b. This puts 80% of the mass in the planet, which should result in a stable system. We don't know what 253b is made of.

Your moons are either going to be a binary pair, always appearing close to each other in the sky, or they would share horseshoe orbits like Epimethius and Janus. This answer has some more details on that sort of thing.

Answer 3

I suspect the size of the moons, or their mass, do not matter in time it takes to orbit, as long as they are substantially smaller than the main planet. I seem to recall from college physics that mass cancels out in the orbital equations.

Something like measuring how long it takes things to fall in no atmosphere: the bowling ball falls as the same rate as the marble and the ton of bricks falls at the same rate as the ton of feathers.