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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.

Frank Harris
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    I mean, I think we have an example of that in our own solar system, just with planets instead of moons. The orbits of Neptune and Pluto intersect, they just have an orbital resonance that prevents collision, which is true of your proposal as well. The only thing about your proposal that might make things tricky is the alignment at apogee.... not sure if that changes anything. https://en.wikipedia.org/wiki/Pluto#Relationship_with_Neptune – Morris The Cat Dec 04 '19 at 18:49
  • Hey @MorrisTheCat, I forgot all about Neptune and Pluto! Excellent point. I would have thought that Pluto's inclination was the reason for not colliding, but that link does a good job of explaining how their resonance creates a stable equilibrium. However, that might be harder to make happen in the much wackier case of 15:61, and in fact that rationale makes me think they're MAYBE actually more likely to find an equilibrium at 1:4. – Frank Harris Dec 04 '19 at 19:17
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    @MorrisTheCat It's not a resonance. Pluto's orbit does not intersect Neptune's. When Pluto is at the same radius from the sun as part of Neptune's orbit, Pluto is not in the same plane as Neptune. Imagine two "linked" rings tilted to each other but they don't touch. It's like that. Now, precession could move the orbits around so they did intersect. But right now, they don't. – puppetsock Dec 04 '19 at 19:31
  • @MorrisTheCat You should write it up as an answer, since you already did the work to get the link. – puppetsock Dec 04 '19 at 19:32
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    What has magic to do with this? – puppetsock Dec 04 '19 at 19:35
  • Expanding on the Neptune-Pluto thing, since your system was designed, the two moons don't have to be in the same plane at all. Say the circular one is in an equatorial orbit, and the eliptical one in a polar orbit. Then it should be easy to design the orbits so they never intersect. – jamesqf Dec 04 '19 at 19:37
  • @jamesqf I don't think he can get the alignment he wants if they're not in the same orbital plane though. – Morris The Cat Dec 04 '19 at 19:39
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    https://en.wikipedia.org/wiki/Horseshoe_orbit might be of interest – Slarty Dec 04 '19 at 19:47
  • @slarty this was the first thing that came to my mind when I read the question title, but horseshoe orbits (must) have nearly identical periods, rather than the 15:61 in the question. – Zeiss Ikon Dec 04 '19 at 20:04
  • They'll align regardless of inclination, just so long as the alignment happens to happen where their ascending/descending nodes meet. I just think it'll be cooler if they're both equatorial. :) – Frank Harris Dec 04 '19 at 20:33
  • Humm... I wonder if there's a good open-source celestial mechanics program out there? Would be interesting to try some long-term simulations... – jamesqf Dec 05 '19 at 05:12
  • @puppetsock: Pluto's orbit doesn't intersect Neptune's, but the two are in a 2:3 orbital resonance (along with a whole bunch of other objects, known as plutinos). – Vikki Dec 06 '19 at 03:24
  • @jamesqf - no, [KSP] won't help because there's no known solution to the n-body problem, which needs to be solved for any "guarantees". – Mazura Dec 07 '19 at 01:06
  • You could try playing around in Universe Sandbox. It isn't open source (or free), but it is perfect for trying out whether a specific example like this could work or not. – Entropy0 Dec 07 '19 at 11:03
  • @Mazura: While there's no analytic solution to the n-body problem, among many other things, there are perfectly workable numeric solutions, which let NASA &c do things like play billiards around the moons of Saturn. An approximation for a billion years or so would be fine :-) – jamesqf Dec 07 '19 at 17:38
  • Since your planet is tidally locked the sun never sets and the planet doesn't have discrete days and nights. Rather, one side is permanently "day", the other side is permanently "night". After a million years it will still be "day" on the same side of the planet. Effectively, a day has an infinite length. It is not a year long, and the inhabitants probably don't have a concept of "day" and "night". – CJ Dennis Dec 08 '19 at 01:04

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Ok, so you say 'Harmonic Orbits', but actual Space-Talking-Dudes call that 'orbital resonance', and it's the solution to your problem.

We've got an example of something ALMOST exactly like what you're talking about right here in our own solar system with Pluto and Neptune. As puppetsock rightly points out, their orbits don't actually intersect because of Pluto's high inclination, but if they DID intersect, the planets still wouldn't collide because of their 2:3 orbital resonance.

So far so good. At first I was concerned about the alignment possibly creating a problem, but then I realized that ANY two bodies orbiting the same primary are going to align at the convergence of their orbital periods, resonance or not, so now I don't think that's really a problem either.

Now, if you REALLY want to be clever, you'll make the resonant periods of your two satellites harmonic with the planet's orbit around the sun TOO, which means that your alignment will come at the same time of year every time it happens, which is all mystical and stuff and VERY Dark Crystal.

Just don't try to wipe out the Gelflings. It never works.

Morris The Cat
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    The ratio of their orbits is 1.50449. It's not 3:2. This only works for a while. If they were co-planar, eventually they'd have an encounter close enough to drastically alter their orbits. That might take hundreds of orbits, but eventually it would happen. A collision is less likely, but possible. – puppetsock Dec 04 '19 at 20:07
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    @puppetsock that's not what the wikipedia article says, unless I'm missing something. "The 2:3 resonance between the two bodies is highly stable and has been preserved over millions of years.[82] This prevents their orbits from changing relative to one another, and so the two bodies can never pass near each other. Even if Pluto's orbit were not inclined, the two bodies could never collide" – Morris The Cat Dec 04 '19 at 20:15
  • Response to that last bit as an edit to the question. Thanks :) – Frank Harris Dec 04 '19 at 20:26
  • @FrankHarris you didn't ask about this, but having your planet rotate that slowly creates a different problem for you in that it's not rotating anywhere NEAR quickly enough to generate a magnetic field, which means no protection from solar wind, which means your atmosphere gets ripped off like Mars' did. You know, unless magic. – Morris The Cat Dec 04 '19 at 20:29
  • Ahh, yeah, that's a fair point. Although...the planet's core could be rotating independently of the planet's surface, right? One other element is that the planet is quite small and dense, about the radius of Mercury but earthlike gravity. (Fun coincidence, for this to be the case, the average density of the planet is very close to the density of mercury the element).

    So the core might be impossibly dense and rotating impossibly quickly, most likely because of some god-tier magic. It could also be finely controlled to create these orbital perturbations, connecting it all together.

     – Frank Harris Dec 04 '19 at 20:42
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    @FrankHarris I mean, it's D&D, so hand-wave away =) – Morris The Cat Dec 04 '19 at 20:42
  • This of course assumes that they are never, ever perturbed by a 3rd body. – cowlinator Dec 05 '19 at 03:25
  • @puppetsock An orbital resonance indicates that each body has an undue influence on the other(s) due to their repeated gravitational interactions. So, even if that 1.50449 number is correct for now (I don't have any information one way or the other, so I'm just going to assume you're right and move on), it's not going to stay correct. Neptune's gravity will slow Pluto down if it starts to get ahead, and will speed it up if it starts to lag behind. So, yes, it's possible for Neptune and Pluto to orbit in the same plane indefinitely without colliding. – HiddenWindshield Dec 06 '19 at 05:22
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You should have a look at Janus and Epimetheus. They are two moons of Saturn that exchange orbits approximately every four Earth years. This setup is probably not stable for more than a few billion years but it might do for what you want.

Janus and Epimetheus dance explained

Epimetheus orbits closer to Saturn, so has a shorter orbital period and eventually approached Janus from behind.

As they get closer, they tug on each other gravitationally.

The tugging by Epimetheus slows down Janus, which makes it fall toward Saturn in its orbit; Janus speeds up Epimetheus, which makes it rise. Janus has four times the mass of Epimetheus, so it moves inward by less than Epimetheus moves forward.

Closer to Saturn, Janus speeds up in its orbit; farther from Saturn, Epimetheus slows down. Janus will slowly creep ahead of Epimetheus; for years later, they'll do the same dance in reverse.

The Square-Cube Law
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NASA recently discovered a very interesting resonant pair in two moons of Neptune, Naiad and Thalassa. Their orbits (nearly <2000km) intersect and have periods of 7 and 7.5hrs respectively. Even though they are quite close at nearest pass (<4000km) they never actually collide because of this ”unprecedented” 69:73 resonance.

This resonance was mathematically shown to be extremely stable, to where it could persist on the order of billions of years.

Unfortunately for these two, as Triton’s retrograde orbit slowly saps their orbital energy they’ll approach their Roche limit in a few million years and shatter into beautiful rings rivaling Saturn’s.

enter image description here

joshperry
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Oh, maybe that is what the magic is about.

Under normal circumstances, you can't have perfect synchronous orbits. Suppose the orbits are, to pick numbers, 1000 hours and 10,000 hours. That's about 40 days and about 400 days.

So whatever is supposed to happen at the first coincidence might be off by some tiny amount. This tiny discrepancy grows for 400 days until the next encounter. So the second encounter is off by a much larger amount.

Say the first encounter has an error of only 1 centimeter per second. After an hour that's 36 meters. After 10,000 hours that's 360 km. So the second encounter winds up being 360 km off target. Which gives a much larger rate of drift for the next encounter. If the moon is only, say, 1000 km across, then it probably misses entirely on the third encounter.

The most probably occurrence is a near miss that massively alters the orbits.

So you would need some way for the encounters to be tuned. That means you would need to be able to detect the motion of the moons to an accuracy of better than 1 cm/s. Much better. Since 1 cm/s produces 360 km after only one orbit. Probably you need something like no more than a few km per orbit. So call it .01 cm/s, or 3.6 km in 400 days. And you'd need some way to give one or both of the moons just the barest little push, presumably by using one moon against the other. Accurately and in the right direction. And you'd need to do that at the correct time, every time the moons encounter.

That sure looks like magic.

By the way, if I did the math right, to move our moon by .01 cm/s would require the energy equivalent of 100,000 tonnes of TNT. It really starts to look like magic.

puppetsock
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  • Yeah...I had given some thought to perturbations/n-body stuff, but concluded I'd pretty much have to sweep it under the rug. So yes, that's def where the magic comes in. My in-universe rationale is that these orbits were desired and put into being by some very smart and powerful mages or demigods. Long ago, they might have been able to magically redirect an asteroid with some implausible precision, so that it would collide with the system and cause this. Alternately, within the planet, they might have magic things that affect the planet's magnetosphere to produce the desired perturbations. – Frank Harris Dec 04 '19 at 20:20
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    "100,000 tonnes of TNT. It really starts to look like magic." Russia built a bomb that, even at only half power, released 500 times that much energy. – Joseph Sible-Reinstate Monica Dec 05 '19 at 04:10
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    "Under normal circumstances, you can't have perfect synchronous orbits." - um, that's what orbital resonances do. – Martin Bonner supports Monica Dec 05 '19 at 16:34
  • @JosephSible-ReinstateMonica ...and Tsar Bomba doesn't look like magic? – jeffB Dec 05 '19 at 17:53
  • Our moon changes its velocity constantly; it is being accelerated in the prograde direction by the Earth's tidal bulge, which increases its orbital height; the equal and opposite reaction force decreases the rotational energy of the Earth. The result is that the Earth is gradually getting both a longer day and a longer month. There is no need to posit any magic or explosions; the energy content of the rotation of the Earth and the energy content of the tides are both large, and able to act at a distance. – Eric Lippert Dec 05 '19 at 19:13
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    Also, you seem to have some false beliefs about the difficulty and likelihood of "tuning" orbits. Lots of orbits are self-tuning; it is not a coincidence that there are so many resonant orbits in our solar system. Resonances are stable; if orbits are perturbed from a stable resonance, gravity tends to gradually pull the bodies back towards a resonant orbit. – Eric Lippert Dec 05 '19 at 19:16
  • Any sufficiently advanced orbital resonance is indistinguishable from magic. – Amedee Van Gasse Dec 06 '19 at 09:40
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Coincidentally I just watched a Scott Manley video on this topic published in May 2018.

A small asteroid called 2015 BZ509, and a large gas giant named Jupiter have a resonance in their orbits which is self-correcting. Every time they approach, if the smaller body is too fast or too slow (that is, early or late) the influence of the larger body applies a correction.

The video goes into detail of how simulations have been done in reverse to find whether the asteroid was a captured interplanetary body. Upshot is the model is stable over billions of years for multiple possible inputs; that is while it is unusual but not vanishing off toward impossibly-improbable.

An interesting gotcha is that the asteroid is in a retrograde orbit with respect to the planet.

I'm not doing it justice, for great inspiration I suggest you spent 8 minutes watching it.

https://www.youtube.com/watch?v=qMLX2W7OAX0

Criggie
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  • Hopefully this is enough explanation to avoid "link only answer" - if it needs more please comment. Are we permitted to post stills from video ? – Criggie Dec 07 '19 at 11:19