Can two moons have intersecting orbits yet be guaranteed not to collide?

Working on a D&D campaign. As a physics nerd, I'd like the orbital mechanics of the planet, its sun, and its moons, to follow standard Newtonian/Keplerian mechanics. I'm trying to come up with an interesting set of parameters so that there are neat and tidy alignments at certain times. The system was semi-intelligently designed, and so everything here can be in nice round numbers. That's all mostly just background.

There are two moons, one with a circular orbit and one with a very elliptical orbit. The circular moon has a very short period and the elliptical one a very long period. Here's the thing. I would like the elliptical orbit to have a smaller perigee than the other's altitude, but an apogee several times larger. This means that, if they aren't inclined, their orbits will have to intersect at two points, 90 degrees from the apogee.

I'm not sure if this matters, but I plan for them to have harmonic orbits. Right now the numbers I'm thinking are that the planet has a year of 243 days, the circular moon has a period of 15 days and the elliptical moon has a period of 61 days (seasons, basically). Every 15 orbits/915 days, they align at the apogee and really cool stuff happens.

My question is, with all this, is it possible to say somehow that if these two moons are in the same orbital plane, their orbits intersect at two points, and they both align at the apogee periodically, can it be shown that they either will or will not eventually collide? My rationale for hoping has something to do with the fact that they are harmonic, and at the point 90 degrees around the orbit, where they'd collide, is going to have something to do with pi, so rational and irrational numbers mean they'll never be the same value at the same time. ¯\_(ツ)_/¯

If this isn't the case, either if it can be shown that they definitely will collide, or that it can't be shown one way or another, I can work with that. I know that I can incline one or both orbits as an easy fix, and I know I can ALSO say "yep, magically they never collide" because it's D&D, but it would be super cool if there was a way they could both be in the same plane.

EDIT: This is an aside, in response to Morris' answer, it was getting too long for a comment. Since you mentioned the Dark Crystally-type stuff, there are a few other things going on here, if I may elaborate. :) First, I didn't mention but the planet's year equals its day, as if it were tidally locked. So one half is always baking, the other half is always frozen, and the ring in the middle is roughly habitable. So since the sun never moves and they don't have seasons, they use a lunar calendar. The solar year is 243 days long, the elliptical period is 61 days and the circular period is 15 days. So the elliptical's apogee happens exactly four times a year (1 cycle = 1 "season"), and the circular moon orbits 4 times plus one day for each of the elliptical moon's orbits. So the alignments happen once every 15 of those "seasons", or every 915 days/3.75 years. The alignment happens along the orbital equator at four different points, 90 degrees apart. Each of those four points has an alignment every 60 seasons or 15 years. Very different good/bad things happen depending on which point they overlap. But it works so that every 15 years the planet, sun and moons all align, which is a pretty ominous time.

posted over 4 years ago

Frank Harris‭

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