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What I'm thinking of is a binary planet system each with rings, 2 moons orbiting the larger one and one moon orbiting the smaller one.
The mass of the larger planet is 5 Earth masses, while the smaller planet's mass is 3 Earth masses. The planets stay about 0.05 AU from each other.
The first planet's moons orbit at 100,000 km and 200,000 km respectively. Whereas the moon of the second planet orbits at 150,000 km.
Would this setup be possible?
This is basically the same as asking if there can be stable planetary orbits in binary star systems, and the answer is (surprisingly, after much arguing to the contrary by astronomers for decades): yes, there can.
Okay, the two situations aren't quite the same, but the basic principles are similar. The main issue is that binary planets are a heck of a lot smaller and closer together than binary stars tend to be, so stable planetary orbits are going to be a lot more constrained. Orbits that enclose the entire binary pair are more available, but orbits around one of the planets are probably going to be low enough for atmospheric friction to be a big factor.
One solution might be to have a pair of tidally locked planetoids that are far enough apart to allow a "polar" orbit over one of the pair to form an elliptical orbit that doesn't approach the saddle point between them. There's a whole bunch of things to do with Roche lobes and 3D gravity maps and all that fun stuff to figure it out, and the odds of something like this forming naturally seem pretty slim, but it could totally work. Probably.
What you'd end up with though is a low orbit, fast-moving satellite that zips across the sky in a matter of minutes. Think ISS rather than Luna. A month would be somewhere between a nice long lunch break and a really short work day.
I don't see why not. Although a cursory check implies you may need a fairly specific setup.
So here we have the solar system. Earth orbits the sun. The moon orbits Earth. Replace the Sun with a gas giant, and move Earth closer in. Now we have a "binary planet system". Can the moon still orbit the Earth?
The link above implies it could be tricky, mainly because we can find no examples of this in our solar system. Jupiter is pretty big, and has a lot of moons, but none of its moons have anything apparent in a stable orbit of the moon (so no moons-of-the-moon). Buuuut that doesn't mean it's impossible (the link seems to imply this as well). What if Jupiter had an earth-sized moon?
You could try playing with the Gravity Simulator though what I have learned in the past 5 minutes is that I have no idea how to set up the scenario I want.
Although.... I'll dip in with an old reliable Worldbuilding rule of thumb, which is that if your question is sufficiently difficult that no one can prove it or disprove it without a PhD in physics then you've passed the test: it is plausible enough to the layman that if you simply set this up in your universe, most people won't question it, and it's already surpassed the realism standards of most Hollywood movies.
Short Answer:
You may need to change some or all of your specifications to have stable orbits for your moons. If you desire that your planets are habitble for humans or for liquid water using life in general, you might have to modify the masses of the planets.
Long Answer:
Part One: Masses and orbits.
The problem of having moons in stable orbits around planets which are part of binary planets is similar to the problem of having subsatellites or moons of moons or moonmoons in orbit around moons of planets.
A subsatellite, also known as a submoon, or moonmoon, is a "moon of a moon" or a hypothetical natural satellite that orbits the moon of a planet.1
It is inferred from the empirical study of natural satellites in the Solar System that subsatellites may be rare, albeit possible, elements of planetary systems. In the Solar System, the giant planets have large collections of natural satellites. The majority of detected exoplanets are giant planets; at least one, Kepler-1625b, may have a very large exomoon, named Kepler-1625b I, which could theoretically host a subsatellite.14 Nonetheless, aside from human-launched satellites in temporary lunar orbit, no notable subsatellite is known in the Solar System or beyond. In most cases, the tidal effects of the planet would make such a system unstable.
Tidal interactions between a planet and its moon will change the length of day of the planet and the orbital distance of the moon.
A moon can orbit a planet in the prograde direction, the same direction as the planet rotates in, or in the retrograde direction, opposite to the direction the planet rotates in.
A moon can orbit a planet closer than the geosynchronous distance, with an orbital period less than the rotation period of the planet, or orbit farther than the geosychronous distance, with an orbital period longer than the rotation period of the planet.
A moon orbiting in the prograde direction above the geosynchronous orbit will be pushed farther and farther away from planet.
A moon orbiting in the retrograde direction will be pulled closer and closer to the planet. The efect on all the retrograde moons in our solar system is too small to be significant, except for the Neptunian moon Triton. Triton is expected to reach Neptune's Roche limit and be destroyed in 3.6 billion years.
Moons orbiting the prograde direction below the geosynchronous orbit will be pulled closer and closer to the planet, eventually being destroyed.
Some stories could use an impending disaster as a moon approaches too close to its planet, while other stories would have no use for such a situation.
If your planets rotate too fast, they will fly apart. If your planets rotate too slowly, the days and nights may get too hot and too cold for any lifeforms you might want to live on them. So planetary rotation periods between a few hours and a few days should be right for habitable planets.
So you can choose rotation periods for your planets and then use orbital period calculations to find the distance of orbits with those periods around planets with masses of 5 and 3 Earth mass, and put your moons inside or outside geosynchronous orbit as you desire.
You want your planets to be about 0.5 AU apart. That is about 7,479,893.535 kilometers.
The Hill sphere of an astronomical object is the volume within which its gravity is strong enough, relative to a more massive nearby object, to have satellites in long term stable orbits.
According to this online Hill sphere calculator https://www.vcalc.com/wiki/KurtHeckman/Hill+Sphere+Radius
the Hill sphere of the planet with 3 Earth mass, relative to the planet with 5 Earth mass, at a distance of 0.05 AU, should have a radius of 4,330,525.59 kilometers.
The Hill sphere is only an approximation, and other forces (such as radiation pressure or the Yarkovsky effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius.
https://en.wikipedia.org/wiki/Hill_sphere
A region of stability out to 1,443,530 to 2,165,297.5 kilometers extends seveal times as far as 100,000, or 150,000, or 200,000 suggested for the orbits of he moons, so the orbits around the planet with 3 Earth mass should be fine.
I tried the calculation with the masses reversed, and it says the Hill radius of the planet with 5 times the mass of Earth should be 6,087,511 kilometers, so the region of stabiity should extend to 2,029,170 to 3,043,755 kilometers, so the moon orbital distances you selected should be fine.
Now assume that the binary planet planet orbits its star at a distance where the planets receive exactly as much radiation from the star as Earth gets from the Sun. I call that the Earth Equivalent Distance or EED. If you assume the star has exactly the same mass and luminosity as the Sun, the EED is 1 AU.
So I tried the Hill sphere calculator for a planet with 3 Earth masses at 1 AU from a star with 1 solar mass. The Hill radius is 2,136,799 kilometers, with a true region of Stability out to 712,266 to 1,068,399 kilometers.
For the planet with 6 Earth masses, the Hill radius is 2,533,431 kilometers, with a true region of stability out to 844,477 to 1,266,715 kilometers.
You want your planets to be about 0.05 AU, or about 7,479,893.535 kilometers apart. That is several times the radii of their regions of stability and their Hill spheres.
So a binary planet with those planetary masses can not have that separation if it orbits 1 AU from a star with 1 solar mass.
If you make the planets orbit with a separation of about a million kilometers they will be within their mutual regions of stabiity. But that will drastically shrink their Hill spheres for their moons.
According to the Hill sphere calculator, at a distance of 1,000,000 kilometers the Hill radius of the 3 Earth mass planet will be 8,353 Kilometers, and that of the 5 Earth mass planet will be 8,353 kilometers.
That will be way too close, within the planetary Roche limits.
So you will have to try different combinations of the many factors, such as the masses of the planets, their orbital separation, the eccentricty of their mutual orbit, the mass of the star, and the distance to the star, to find a combination which might possiblity work.
The answer by user177107 to this question:
Includes a table with characteritics of main sequence stars of difeerent spectral types, including their masses and luminosities, and the orbital characteristics of planetary orbits at their EEDs.
Soyu can take a star of a specific type and see what the Hill radii of planets at planets at their EEDs would be.
If you desire your planets to be habitable for humans, or have oxygen breathing large land animals, or be habitable for liquid water using life in general, there are restrictions on what type of stars you can use.
Pages 67 to 75 of Habitable Planets for Man, Stephen H.Dole, 1964, discuss the spectral types of stars suitable for having planets habitable for humans and beings with similar environmental requirements.
In recent decade s there have been many scientific discussions of planetary habitability. However, on Earth there are many lifeforms which flourish where unprotect humans would swifly die. So those discssions are about the more gneral case of worlds habitable for liquid water using life in genral and not for human in partcilar, and include habitable worlds which would not be habitable for humans.
The Wikipedia article on Planetary Habitability:
https://en.wikipedia.org/wiki/Planetary_habitability
Is a place to start learning about current ideas about which stars are likely to have habitable planets.
A planet doesn't have to orbit its star at the exact EED distance to have a habitable surface temperature. It can orbit a bit closer or a bit farther and still have habitable temperatures. But how large is the"Goldilocks zone" or "circumstelalr habitable zone" of a star?
If you find the inner and outer edges of the Sun's circumstellar habitable zone you can find them for another star by calculating to account for the difference between the star's luminosoty and that of the Sun. So what are the inner and outer limits of the Sun's circumstellar habitable zone?
The table here:
https://en.wikipedia.org/wiki/Circumstellar_habitable_zone#Solar_System_estimates
Shows that there are widely differing estimates of the inner and outer edges, and thus the width, of the Sun's habitable zone. And some of the innermost and outermost edges of the habitable zone are calculated assuming specific compsition of a planet's atmosphere, which may not be breatheable for humans.
So a writer who wants to create a scientifically plausible habitable world will have to investigate the various estimates of the Sun's habitable zone, find the one they consider most convincing, and use that in their calculations for the habitable zones of other stars.
I can imagine that you could try many different configurations of your desired planetary arranement before find one -if any - which has long term stable orbits.
One way to save time would be to use an orbital simulation program and try various configurations in it to see how stable they are. You may find trends indicating the way to go for long term stability.
And possibly you could persude an expert to do some of the calculations for you.
For example, Sean Raymond is an astrophysicist who studies the formation and evolution of planetary systems. He has a blog, PlanetPlanet.net, where he also discusses science fictional solar systems.
And he offers to help design science fictional worlds.
And he actually discusses moons of moons, showing that they might possibly exist under some circumstances.
https://planetplanet.net/2018/10/09/can-moons-have-moons/
https://arxiv.org/pdf/1810.03304.pdf
And the problem of submoons is rather similar to the problem of moons around a planet in a binary paleet system.
Part Two: Planetary mass.
You have chosen to have planetary masses of 3 Earth mass and 5 Earth mass for yur planets. You may have to change those masses up or down to make your binary planet with three moons possible.
I don't know what you want your planets (or your moons) to be like.
But if you want either of your planets to be habitable for human beings, or life forms with similar requirements, or for liqud water using life in general, you will have to choose masses within the posssible mass rnges of habitable worlds, as well as can be calculated with our knowledge.
Dole calculated the mass range for human habitable worlds on pages 53 to 58. becaue he believed that humans would never want to colonize an astronomical object with morethan 1.5 g surface gravity, he calculated probable figures for such a world as 2.35 earth mass, 1.25 Earth radius & diameter, and an escape velocity of 15.3 kilometers per second. But Dole said on pages 53*54 tht various other factors might planet uninhabitble for humans before a mass of 2.35 Earth mass was reached. Of course alien lifeforms could find living on worlds with more than 2.35 the mas sof Earth and over 1.5 the surface gravity of Erth comfortable if they evolve in that surface gravity.
According to this 2017 article "Super-Earths" can't be habitable:
This 2022 article is optimistic about life on Super-Earths.
https://www.space.com/super-earths-more-habitable-finding-more
So you may need to research the habitability of Super-Earths.
This article:
https://arxiv.org/ftp/arxiv/papers/1209/1209.5323.pdf
has a paragraph on ages 3 to 4 discussing the mass range of habitble worlds, and chooses an upper mass a bit less than Dole did.
...An upper mass limit is given by the fact that increasing mass leads to high pressures in the moon’s interior, which will increase the mantle viscosity and depress heat transfer throughout the mantle as well as in the core. Above a critical mass, the dynamo is strongly suppressed and becomes too weak to generate a magnetic field or sustain plate tectonics. This maximum mass can be placed around 2M⊕ (Gaidos et al. 2010; Noack & Breuer 2011; Stamenković et al. 2011).
Thus it is possible that worlds with mass of 3 Earths and 5 Earths might be too massive to behabitable.
