Stable solutions to simplified 2- and 3-body problems?

[Exy]

TL;DR Summary

Given a stable 2 or 3 body system with each object having different masses and circular orbits around the barycenter, how do I calculate each bodies orbital period?

Hello physics,

While this is about sci-fi worldbuilding, I feel like this belongs on this board more.

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CONTEXT:
I have been building a fictional neighborhood for our star and using the formula

to get the orbital period of orbiting bodies around a stationary mass. Some of the systems have been decided by RNG to be binary and trinary starsystems however, and now I am researching the 2 and 3 body problems.

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The actual problem (more of a thought experiment) goes as follows:

2 bodies with the masses m₁ and m₂ with m₁ ≠ m₂. With both orbiting the barycenter in circular orbits, the above formula does not give usable results.
The question is now either how to get the orbital period of each body, or how to get the relation between the radii of each objects orbit based on their mass (and any other needed parameters)

Then the question turns to stable 3 body systems with each orbiting the joint center in a circle. The goal here again are each bodies orbital period with circular orbits, leading to a stable system. Each body may again have different masses
One of the systems features a large star in the center with 2 smaller ones orbiting it on directly opposite sides. I used the above formula for this with the 3rd stars mass simply being counted as belonging to the stationary center, but is this correct? I would imagine this to be dubious, but I simply do not know.

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What I do unterstand is that these systems are incredibly unstable in reality, but this is just 2 or 3 bodies of different masses orbiting a joint center without being perturbed or losing energy. My current understanding tells me that the 3 body system would need 2 of the bodies to have identical mass, but maybe they don't.

I have looked around the Internet, but have not found anything that I could use. This seems to be a massive rabbithole.
Maybe someone here knows where to go or how to solve this.


EDIT: I have found the answer for the 2 body problem within the wikipedia page for "Barycenter" regarding the distances, but not the orbital period.

 

[anuttarasammyak]

Is the solar system which consists of the sun and planets an answer to your question ?

 

[Exy]

Is the solar system which consists of the sun and planets an answer to your question ?

No I don't think so. I may not have been very good at explaining the problem.
Imagine the 3 body problem with identical masses, all orbiting their barycenter in a circle.
The question is what each bodies distance to the center would be if they were different in mass, but all still orbiting in a circle, now of course at different orbital radii. So a non-hierarchical system.

I have attempted to translate the principles of the binary system over, where a mass 5 times heavier than the other would be five times as close to their shared center, but have had no luck so far.

 

[anuttarasammyak]

Imagine the 3 body problem with identical masses, all orbiting their barycenter in a circle.

(3,3) figure in
https://en.m.wikipedia.org/wiki/File:5_4_800_36_downscaled.gif seems to be your case. I am not sure whether we can extend this pattern to different masses cases or not.

 

[Exy]

(3,3) figure in
https://en.m.wikipedia.org/wiki/File:5_4_800_36_downscaled.gif seems to be your case. I am not sure whether we can extend this pattern to different masses cases or not.

Tl;dr: Here is as far as I have gotten in my failures, but my stomach has now also given up and I cannot work on this until that's fixed. We will see how well I can keep up.

I have been trying to do this since but have not managed to get it to work. I tried to treat it as a 2 body problem with the barycenter of the opposing bodies as the second mass, then placing the masses in such a way as to make their relative distances appropriate at a separation of 120° around the rotational center, but I have not managed to make this work

Now it might be that I just didn't manage to do it, but
http://www.scholarpedia.org/article/Three_body_problem (homographic solutions, figure 3)

Shows an animation of 3 bodies in equilibrium, basically as needed. The interesting thing here is that they seem to be in formation as an equilateral triangle (so with the 120° separation), BUT they are rotating around some other point. So I tried having them orbit their shared barycenter:

But I have also not been able to get this to work. Maybe some other center is needed or I am just not calculating the orbital periods correctly, since they are not coming out as equal.

AND THEN my stomach decided to get an infection and now I can barely write this post. It will take probably at least 1-2 weeks before I can investigate this further. I will try to keep up with this tread, but we will see.

 

[anuttarasammyak]

Shows an animation of 3 bodies in equilibrium, basically as needed. The interesting thing here is that they seem to be in formation as an equilateral triangle (so with the 120° separation), BUT they are rotating around some other point.

Here it seems that the third body is on a Lagrange point.

 

[pbuk]

http://www.scholarpedia.org/article/Three_body_problem (homographic solutions, figure 3)

(animation by R. Moeckel).

Shows an animation of 3 bodies in equilibrium, basically as needed. The interesting thing here is that they seem to be in formation as an equilateral triangle (so with the 120° separation)

Yes, Lagrange showed that this must be the case (I have replaced the image you posted with the animated version).

BUT they are rotating around some other point [than their barycentre]

What makes you think that? Newton's first law requires that they can only orbit the barycentre, and this is how they are depicted (note that $m_{\text{red}} > m_{\text{blue}} \gg m_{\text{black}}$).