# Binary planets and dual moon alignment occurring once every 200 years

I am trying to figure out a 200 year cycle where Binary planets share 1 moon Planet 1 has its own moon that maintains orbit around it. The cycle would be as illustrated below. Moon 1 would belong to planet 1. Moon 2 would be shared. Moon 1 has a 24 day orbit around Planet 1. What I need to know is what would be the moon orbit cycle in days to complete the dual orbit around both planets so that once every 200 years the moons would align with Moon 2 in the nexus and moon 1 would be on the exact opposite side of planet 1.

1:

• How science-based is this question? Binary planets orbit around a barycenter. In truth, everything orbits around a barycenter, it just doesn't always matter - but it does for binary planets. Based on this Q from Physics.se I'm not convinced that you could ever get the figure-8 orbit you're showing for moon #2. If we throw physics to the wind, this Q becomes one of basic math: velocity of Moon-1, distance-1 and distance-2, work out velocity of Moon-2 to get that alignment every 200 years (whatever a "year" is in your solar system).
– JBH
Commented May 9 at 2:46
• An orbit around two primary bodies that passes between it is not stable. If you want moon 2 to be shared, it should orbit around the outside of the diagram, around both planets at once. Commented May 9 at 2:52
• Math was never my best subject. But I have an explanation for why/how this happens, and funny enough you link has a sub-link to where I got the idea. ( en.wikipedia.org/wiki/Roche_lobe ) It is extremely unlikely but possible and basically all of the conditions would have to be met perfectly. if anything happened to disrupt this it would change everything (which I have plans for that happening as well) Right now I just need to figure out the math to get the orbits figured out. I am basing this on if Earth and Mars in an alternate universe where at one point they were in a binary orbit. Commented May 9 at 3:02
• Candace - I may end up changing it to that but for now I am looking at the possibilities for both. The biggest thing is that this will be science fiction and things don't have to be 100% scientific fact or even theory, but I do want the math to line up. Eventually these 2 planets will end up being modern day comparisons to Earth and Mars. (Size, distance, orbits, etc...) Commented May 9 at 3:07
• At the very least, you would have to give the masses of the bodies involved, and I very much doubt if such an orbital pattern could be contrived, much less occur naturally. More likely, you'd get orbital resonances that result in Moon 2 orbiting mostly on either the Planet 1 or Planet 2 side, depending upon the position of Moon 1. Commented May 9 at 5:34

1. You did not say how many hours are in a day and how many days are in a year.

Note that since the duration of MoonOne's orbit is 24 hours, and after 200 years it must have completed an integer number of orbits, the duration of a year is constrained to be of the form $$k \times 3/25$$ hours, with $$k$$ being an integer.

Justification: 200 years is then $$k \times 3/25 \times 200 = k \times 24$$.

2. It is plain obvious that from a purely arithmetical point of view there are infinitely may solutions.

3. Let's assume that one day is 24 hours, and there are $$N$$ days in a year. You can easily see that any duration of MoonTwo's orbit of the form $$n + 1 / (200 \times N)$$, with $$n$$ an integer fraction of $$N$$, will satisfy the requirement.

Justification: After one year of $$N$$ days, the position of MoonTwo would have advanced, on the average, by $$1/200$$ of its orbit; the $$n$$ vanishes because it is an integer fraction of $$N$$. After 200 years it would have advanced an entire orbit.

• So, yes 24 hours in a day, 364 days a year, Moon 1 will have a 24 day orbit I am trying to figure out the orbit for Moon 2 so that Moon 2 would be at the nexus when Moon 1 is on the exact opposite side of planet 1 every 200 years. I should have put in more information but we are limited with character count and I know what I am thinking but my words didn't relay that... ;) Commented May 9 at 8:30